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[
"SUPPORT-VECTOR-MACHINE WITH BAYESIAN OPTIMIZATION FOR LITHOFACIES CLASSIFICATION USING ELASTIC PROPERTIES",
"SUPPORT-VECTOR-MACHINE WITH BAYESIAN OPTIMIZATION FOR LITHOFACIES CLASSIFICATION USING ELASTIC PROPERTIES"
] | [
"Yohei Nishitsuji [email protected]@gmail.com \nDepartment of Geoscience and Engineering\nSummit Exploration and Production Limited London\nDelft University of Technology Delft\nThe Netherlands, UK\n",
"Jalil Nasseri [email protected] \nDepartment of Geoscience and Engineering\nSummit Exploration and Production Limited London\nDelft University of Technology Delft\nThe Netherlands, UK\n"
] | [
"Department of Geoscience and Engineering\nSummit Exploration and Production Limited London\nDelft University of Technology Delft\nThe Netherlands, UK",
"Department of Geoscience and Engineering\nSummit Exploration and Production Limited London\nDelft University of Technology Delft\nThe Netherlands, UK"
] | [] | We investigate an applicability of Bayesian-optimization (BO) to optimize hyperparameters associated with support-vector-machine (SVM) in order to classify facies using elastic properties derived from well data in the East Central Graben, UKCS. The cross-plot products of the field dataset appear to be successfully classified with non-linear boundaries. Although there are a few factors to be predetermined in the BO scheme such as an iteration number to deal with a trade-off between the prediction accuracy and the computational cost, this approach effectively reduces possible human subjectivity connected to the architecture of the SVM. Our proposed workflow might be beneficial in resource-exploration and development in terms of subsurface objective technical evaluations.Quantitative interpretation (QI) is of importance for the purpose of improving insight from subsurface for exploration and development of natural resources. Among many seismic attributes, seismic inversion products are the most commonly used products for QI [e.g., 2, 6, 17]. The elastic properties cross-plots are often used to assess different facies (e.g., sand and shale) and presence of hydrocarbon in reservoirs. When the properties are linearly separable in the original cross-plot space (acoustic impedance, AI and velocity ratio, V P /V S ), a linear projection can be useful because of its simple and fast calculations to distinguish facies. However, the cross-plot data in practice are often exhibited as complex clusters, which essentially requires non-linear separations. To tackle this problem, we come up with support-vector-machines [SVM; e.g., 15, 14], a machine-learning algorithm.The SVM is originally developed to solve a binary (two) classification problem by a linear separation. Having called kernel-function, however, the SVM can handle non-linear problems too. Moreover, not only the binary problems but also multiclass-classification problems can be solved when an appropriate aggregation strategy is implemented. A couple of hyperparameters require to be optimized to perform the SVM effectively and efficiently for any given problems. In terms of finding optimal hyperparameters, the simplest approach can be an exhaustive-grid search [e.g., 16]. This method seeks every single grid within a given searching range, literally one by one. Even though when the number of the SVM's hyperparameters to be optimized is small, this method becomes an inefficient approach when the searching range becomes huge. A time-efficient approach we would like to apply in this study is called Bayesian-optimization [BO; e.g., 7].The BO is a global optimization algorithm that finds hyperparameters of black-box (unknown) functions in as few iterations as possible by searching a good balance between so-called exploration and exploitation [e.g., 4]. There are many industries and applications that come up with the BO in practice such as social network, accommodation arXiv:2204.00081v1 [physics.geo-ph] | 10.48550/arxiv.2204.00081 | [
"https://arxiv.org/pdf/2204.00081v1.pdf"
] | 247,922,653 | 2204.00081 | d1c2c24331a29c7fec7a56dbc4eeef47ee0f254e |
SUPPORT-VECTOR-MACHINE WITH BAYESIAN OPTIMIZATION FOR LITHOFACIES CLASSIFICATION USING ELASTIC PROPERTIES
April 4, 2022
Yohei Nishitsuji [email protected]@gmail.com
Department of Geoscience and Engineering
Summit Exploration and Production Limited London
Delft University of Technology Delft
The Netherlands, UK
Jalil Nasseri [email protected]
Department of Geoscience and Engineering
Summit Exploration and Production Limited London
Delft University of Technology Delft
The Netherlands, UK
SUPPORT-VECTOR-MACHINE WITH BAYESIAN OPTIMIZATION FOR LITHOFACIES CLASSIFICATION USING ELASTIC PROPERTIES
April 4, 2022Deep learning · Bayesian optimization · Hyperparameter · SVM · Oil&Gas · Elastic properties cross-plots · Quantitative interpretation
We investigate an applicability of Bayesian-optimization (BO) to optimize hyperparameters associated with support-vector-machine (SVM) in order to classify facies using elastic properties derived from well data in the East Central Graben, UKCS. The cross-plot products of the field dataset appear to be successfully classified with non-linear boundaries. Although there are a few factors to be predetermined in the BO scheme such as an iteration number to deal with a trade-off between the prediction accuracy and the computational cost, this approach effectively reduces possible human subjectivity connected to the architecture of the SVM. Our proposed workflow might be beneficial in resource-exploration and development in terms of subsurface objective technical evaluations.Quantitative interpretation (QI) is of importance for the purpose of improving insight from subsurface for exploration and development of natural resources. Among many seismic attributes, seismic inversion products are the most commonly used products for QI [e.g., 2, 6, 17]. The elastic properties cross-plots are often used to assess different facies (e.g., sand and shale) and presence of hydrocarbon in reservoirs. When the properties are linearly separable in the original cross-plot space (acoustic impedance, AI and velocity ratio, V P /V S ), a linear projection can be useful because of its simple and fast calculations to distinguish facies. However, the cross-plot data in practice are often exhibited as complex clusters, which essentially requires non-linear separations. To tackle this problem, we come up with support-vector-machines [SVM; e.g., 15, 14], a machine-learning algorithm.The SVM is originally developed to solve a binary (two) classification problem by a linear separation. Having called kernel-function, however, the SVM can handle non-linear problems too. Moreover, not only the binary problems but also multiclass-classification problems can be solved when an appropriate aggregation strategy is implemented. A couple of hyperparameters require to be optimized to perform the SVM effectively and efficiently for any given problems. In terms of finding optimal hyperparameters, the simplest approach can be an exhaustive-grid search [e.g., 16]. This method seeks every single grid within a given searching range, literally one by one. Even though when the number of the SVM's hyperparameters to be optimized is small, this method becomes an inefficient approach when the searching range becomes huge. A time-efficient approach we would like to apply in this study is called Bayesian-optimization [BO; e.g., 7].The BO is a global optimization algorithm that finds hyperparameters of black-box (unknown) functions in as few iterations as possible by searching a good balance between so-called exploration and exploitation [e.g., 4]. There are many industries and applications that come up with the BO in practice such as social network, accommodation arXiv:2204.00081v1 [physics.geo-ph]
fare metasearch and travel route aggregator. The exploration factor seeks where the variance of the current posterior measurement is high for next round of evaluation (iteration), whereas the exploitation factor focuses on where the means of the posterior is low. In other words, the exploration part completely ignores what the posterior has already obtained so far, whilst the exploitation part completely ignores what the posterior has not obtained so far.
In this study, we utilize the BO to optimize the SVM's hyperparameters for the multiclass classification problems associated with elastic impedance cross-plot data obtained from well data in the East Central Graben in UKCS ( Figure 1). We use 10 wells in this study including 22/9-5, 22/14b-5, 22/15-3, 22/15-4, 22/24c-11, 22/30a-16, 23/16d-6, 23/16b-9, 2316f-12, and 23/21-5 (UK National Data Repository: https://ndr.ogauthority.co.uk). The target interval in this study is the Paleocene sediments where the Forties member of Sele Formation is the main reservoir sandstone.
Gas fields Oil fields The chosen wells + Figure 1: The chosen 10 wells (see the introduction for more details) for the study in the East Central Graben in the UK North Sea.
2 The Bayesian-Optimization Applied to The Support-Vector-Machine
Multiclass Classification of The Support-Vector-Machine
In this section, we briefly introduce the multiclass classifications by the SVM. The SVM, which is based on statistical learning theory and structural risk minimization, was initially developed for the purpose of separating binary problems [e.g., 14]. We show a schematic of the binary problem using the SVM in Figure 2. Considering of actual classifications in practice, however, there are many cases that we need to deal with more than three classes. For example, five chosen classes in this study are the geological facies such as shale, tuff, brine-, oil-, gas-sandstone. A plenty number of studies connected to aggregation strategy has been reported to deal with multiclass classifications [e.g., 5]. To date, this is one of ongoing-research topics in machine-learning community.
We assume that we have a training dataset T which has p-th classes for the multiclass classifications by the SVM, such that:
T = {(x 1 , y 1 ) , (x 2 , y 2 ) , · · · (x n , y n )} , x i ∈ R m , y i ∈ {1, 2, · · · , p},(1)
where x i and y i denotes i-th feature vector whose feature dimension is m and their associated classes (labels). For example, p = 2 means a binary-class problem. The SVM minimizes a difference between the expected risk, which we cannot directly obtain due to the lack of probabilistic-distribution information including unknown data which are not observed yet, and empirical risk [e.g., 14]. The minimization allows us to find a maximum margin which divides x i into two classes if p = 2. The decision boundaries drawn by the margins can be either linear or non-linear via hyperplane using so-called kernel function. Among several kernel functions, the gaussian kernel is commonly used because of its better performance than the others [e.g., 10]. Therefore, we adopt the gaussian kernel in this study.
There are two hyperparameters require optimization for a possible gain in performance of the SVM. The first parameter is called sigma, which presents in a gaussian kernel. This parameter determines the complexity (non-linearity) of the margins. The second parameter is penalization, which grants the rate of misclassification. The penalization parameter can be derived from the Lagrangian-dual-problem for the SVM (not shown here for the sake of brevity). Moreover, BO determines the type of aggregation strategy of either one-versus-one (OvO) or one-versus-all (OvA) for solving multi binary class problems.
The Gaussian-Process
We adapt the BO for the optimization framework and determine the aforementioned two hyperparameters and the aggregation strategy for optimizing SVM. With the regime of the BO, a probability distribution to an unknown-target function f , in which hyperparameters h i are present, is assumed to follow the gaussian-process prior:
p f (h 1 ) f (h 2 ) . . . f (h n ) ∼ N µ(h 1 ) µ(h 2 ) . . . µ(h n ) , K(h 1 , h 1 ) K(h 1 , h 2 ) · · · K(h 1 , h n ) K(h 2 , h 1 ) K(h 2 , h 2 ) · · · K(h 2 , h n ) . . . . . . . . . . . . K(h n , h 1 ) K(h n , h 2 ) · · · K(h n , h n ) ,(2)
where N , µ and K indicate normal (gaussian) distribution, the mean function (usually set to be zero) and the covariance function which can be derived from the utilized the gaussian kernel. As one can see from equation 2, the gaussian process, which is completely defined by the mean function as well as the covariance function, is a group of random variables whose distributions are all based on the gaussian distributions. While the covariance is independent of observations, the mean is a linear combination of each observation. For the brevity, we simplify equation 2 using matrix-vector notation:
p(f ) ∼ N (f ;µ, K),(3)
where K is a n × n matrix, whilst f and µ are n × 1 column vectors. p(f ) is the prior information that can be used in the Bayesian regression [e.g., 11], which is given by:
p(f | D) = p(f )p(D | f ) p(D) ,(4)
where D is a set of observation data (also often called evidence) in the matrix-vector notation of D 1:n {h 1:n , f (h 1:n )}.
Given new evidence, a posterior probability, the left-hand side term in equation 4, is updated. Note that we assume noise-free case in equation 4 for simplicity. By solving equation 4, we expect to choose where to observe the function next. The following equation is adapted to make the choice to be fully automated:
h n+1 = argmax h α(h; D),(5)
where α is an acquisition function, which contributes to the objective function argmax h f . Although several policies (tasks and criteria) associated with the acquisition function have been ever proposed in the machine-learning community [e.g., 9,7,12], the following policy explained hereinafter is used in this work.
The Expected-Improvement-Per-Second
The expected improvement, developed by Jones et al. [8] based on the pioneer work by Mockus et al. [7], is commonly used within the number of the acquisition functions. The policy of this function is to evaluate the expected value of possible improvement of the objective function. A searching point where to observe next is based on the largest expected improvement. The acquisition function of the expected improvement can be written as:
α =ˆ∞ −∞ max(0, f − f best , )N (f ; µ, K)df ,(6)
in which the next searching point with the largest expected improvement is automatically selected one after the other. It can be noted that the left-hand side term in the equation 6 increases when the mean function and the covariance (Kernel) function decreases and increases, respectively. Therefore, this policy implicitly has a trade-off between the exploitation factor (driven by low mean) and the exploration factor (driven by high covariance). Tendentially, an explicit trade-off parameter exists in other policies such as the upper-confidence based policy [e.g., 13] which we do not use here.
Snoek et al. [12] improved the performance of the expected improvement by putting the time-weight on its evaluation time, which is called the expected-improvement-per-second (EIPS). The EIPS seeks a point to evaluate the next from where the value of the expected improvement per second is largest. The EIPS finds the optimal parameters faster than using the expected improvement by the Markov-chain-Monte-Carlo [e.g., 12]. In our study, we adopt the EIPS policy for the BO to automatically optimize the hyperparameters of the SVM with respect to solve the facies-classification problems associated with the elastic properties cross-plot data.
Implementation Test
We carry out a synthetic test to check feasibility of our computational implementation on an AI-V P /V S cross-plot prior to the actual classification using the field dataset. By means of the feasibility test we assure that there are no technical problems caused by our implementation per se. The number of target facies defined by the well data for classification is five in this study. We therefore synthesize five classes. We generate 1,000 supervisors (training data) and 50 test data for each class using a gaussian randomizer. After training the SVM with the BO, we check the performance of the test classification. The learning process in general can be optimized within 20 iterations by the BO [3], so we execute 20 iterations. The strategy of the multiclass classification is either the OvO or the OvA depending on the BO. The classification results are shown in Figure 3 using the EIPS acquisition function.
The cost function in Figure 4 shows that the optimized hyperparameters are consequently found in 17th iteration, but major improvement of the cost is obtained in 8th iteration. From Figures 4a-c we find out that the BO searches the hyperparameters from both aspects of the exploitation and the exploration followed by equation 6. This means that several iterations (represented by circles in Figure 4a-c) are focused on neighbouring locations (e.g., where the penalization parameters are higher than about 2 * 10 0 in Figure 4c) by the exploitation factor whilst some iterations are randomly realized by the exploration factor. Based on this experiment, we do not observe major technical issues caused Figure 4, and a confusion matrix of the test accuracy is shown in Figure 5.
Cost ✁ ✂ ✁ ✄ ☎ ✆ ✝ ✞ ✁ ✂ ✁ ✄ ☎ ✆ ✝ ✞ ✁ ✂ ✁ ✄ ☎ ✆ ✝ ✞ (✟) (✠) (✡) (☛) ☞ ✌ ✍ ✎ ✏ ✑ ✌ ✑ ✍ ✑ ✎ ✑ ✏ ✌ ☞ ✑ ☞ ☞ ✒ ☞ ✓ ☞ ✒ ☞ ✎ ☞ ✒ ☞ ✔ ☞ ✒ ☞ ✏ ☞ ✒ ☞ ✕ ☞ ✒ ✑ ☞ ✒ ✑ ✑ ☞ ✒ ✑ ✌ ☞ ✒ ✑ ✖ ☞ ✒ ✑ ✍ ☞ ✒ ✑ ✓ ✗ ✘ ✙ ✚ ✛ ✜ ✚ ✢ ✣ ✤ ✥ ✤ ✣ ✦ ✣ ✧ ★ ✙ ✩ ✪ ✫ ✬ ✭ ✮ ✯ ✰ ✱ ✲ ✳ ✮ ✯ ✴ ✳ ✲ ✰ ✵ ✶ ✷ ✷ ✷
Results and Discussions
The Figure 6 shows the cross-plot of the elastic parameters, AI and V P /V S , for the Palaeocene formations derived from ten chosen wells. There are always overlaps between different facies/classes and we intend to improve and maximize the separation between the classes by implementing the methods described in the sections above. The initial classification results show the optimized parameters are consequently found in 5th iteration out of 20 iterations (Figure 7). The test (prediction) accuracy after 20 iterations result in 66.2 % at this realization ( Figure 8a). Note that the test accuracy might differ slightly after running consequent realizations due to the dynamic nature of the BO. Figure 6: Facies classification result using elastic properties (AI and V P /V S ) from the chosen 10 wells through the SVM after 20th BO iteration of the hyperparameters. The legend and the background colours are same as Figure 3. The details of 20 BO iterations are shown in Figure 7, and a confusion matrix of the test accuracy is shown in Figure 8a.
The Figure 7 indicates the sigma are intensively exploited between 10 0 and 10 -2 in Figure 7b, where the cost is relatively low among all of the 20 iterations. On the contrary, we also see that the BO tries to explore where there are little observations available i.e., the sigma around 10 -4 in Figure 7b. Figure 7: Same as Figure 4, but for the field dataset ( Figure 6).
✁ ✂ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✆ ✠ ✝ ✠ ✞ ✠ ✟ ✆ ☎ ✠ ☎ ☎ ✡ ☛ ☎ ✡ ☛ ☞ ☎ ✡ ✝ ☎ ✡ ✝ ☞ ☎ ✡ ☞ ☎ ✡ ☞ ☞ ☎ ✡ ✞ ☎ ✡ ✞ ☞ ☎ ✡ ✌ ☎ ✡ ✌ ☞ ☎ ✡ ✟ ✍ ✎ ✏ ✑ ✒ ✓ ✑ ✔ ✕ ✖ ✗ ✖ ✕ ✘ ✕ ✙ ✚ ✏ ✛ ✜ ✢ ✣ ✤ ✥ ✦ ✧ ★ ✩ ✪ ✥ ✦ ✫ ✪ ✩ ✧ ✬ ✭ (✮) (✯) (✰) (✱) ✲ ✲ ✲
Condensate
Brine sand Shale Limestone Oil-bearing sand The decision boundaries in the initial classification in general appear to be good visually ( Figure 6), however, there are rooms to improve the separation between the facies. For instances, it looks like that an alternative boundary between the brine sand and the limestone can improve the separation rate such that the limestone class with the white background in Figure 6 could have been more expanded towards left by exchanging with the brine-sand class with the dark grey background. On the contrary, it is too hard to uphold the decision boundaries between classes with significant overlaps such as the oil-and condensate-bearing sands or brine sand and shale visually. However, any further improvement in other scenarios can be examined quantitatively by the confusion matrix, i.e., Figure 8a. A certain number of miss-classified plots is inevitable because of faceis overlaps on AI-V P /V S cross-plot. When we come to think of such over-lapping cases in general, our subjectivity might prone to be dictated unless automatic methods like the BO-based-SVM are adopted.
We investigate further improvement in classification and alternative decision boundaries by increasing the number of iteration; we doubled it to be 40 times. The results are shown in Figures 8b-10.
Although the improvement is consequently found in the 38th iteration, the large improvement of the cost function is rapidly gained within the first several iterations (Figure 10d). The searching range of the hyperparameters in Figure 10 gets broader for both small and large values than those in Figure 7. A possible correlation between the sigma and the penalization parameter in terms of the cost begins to appear in Figure 10a. Figure 10, and a confusion matrix of the test accuracy is shown in Figure 8b. Figure 10: Same as Figure 7 but for 40 iterations (Figure 9).
✁ ✂ ✄ ☎ ✆ ✝ ☎ ✆ ✝ ✞ ☎ ✆ ✟ ☎ ✆ ✟ ✞ ☎ ✆ ✞ ☎ ✆ ✞ ✞ ☎ ✆ ✠ ☎ ✆ ✠ ✞ ☎ ✆ ✡ ☎ ✆ ✡ ✞ ☎ ✆ ☛ ☞ ✌ ✍ ✎ ✏ ✑ ✎ ✒ ✓ ✔ ✕ ✔ ✓ ✖ ✓ ✗ ✘ ✍ ✙ ✚ ✛ ✜ ✚ ✢ ✚ ✢ ✛ ✣ ✚ ✜ ✛ ✤ ✚ ✤ ✛ ✥ ✦ ✧ ★ ✩ ✪ ✫ ✬ ✭ ✮ ✩ ✪ ✯ ✮ ✭ ✫ ✰ ✱ (✲) (✳) (✴) (✵) ✶ ✶ ✶
responsible to the margins' non-linearity whereas the penalization parameter is related to the misclassification. Hence, it can be interpreted that the BO automatically responds to the trade-off nature of these two hyperparameters.
There are certain improvements in the new classification scenario in Figure 9 in comparison with Figure 6 such as visible changes in the boundary between the brine sand and the limestone. Not only the total test accuracy but also the accuracy of the oil-bearing sand in the new classification increases by 1 % (Figures 8a-b). Since we still visually see the potential improvement of the aforementioned boundary between the brine sand and the limestone, we carry out one more additional test with 200 iterations. The results are shown in Figures 8c, 11 and 12.
Like other last two realizations by the 20- (Figure 7d) and the 40-iteration (Figure 10d), the 200-iteration (Figure 12d) also shows a major improvement of the cost function within the first several BO iterations. The possible correlation discussed in Figure 10a becomes more prominent (Figure 12a). In overall, the 200-iteration yields more than 1 % improvement of the test accuracy. For the oil-bearing sand, the accuracy improves to 55.2 %, which is about 15 % higher than when the 40-iteration is tested. Figure 12, and a confusion matrix of the test accuracy is shown in Figure 8c. Figure 12: Same as Figure 7, but for 200 iterations ( Figure 11).
✁ ✂ ✄ ☎ ✆ ✝ ✞ ✝ ✟ ✝ ✠ ✝ ✡ ✆ ✝ ✡ ✞ ✝ ✡ ✟ ✝ ✡ ✠ ✝ ✆ ☎ ✝ ✡ ☎ ✝ ☎ ☛ ☞ ☎ ☛ ☞ ✌ ☎ ☛ ✍ ☎ ☛ ✍ ✌ ☎ ☛ ✌ ☎ ☛ ✌ ✌ ☎ ☛ ✎ ☎ ☛ ✎ ✌ ☎ ☛ ✏ ☎ ☛ ✏ ✌ ☎ ☛ ✑ ✒ ✓ ✔ ✕ ✖ ✗ ✕ ✘ ✙ ✚ ✛ ✚ ✙ ✜ ✙ ✢ ✣ ✔ ✤ ✥ ✦ ✧ ★ ✩ ✪ ✫ ✬ ✭ ✮ ✩ ✪ ✯ ✮ ✭ ✫ ✰ ✱ (✲) (✳) (✴) (✵) ✶ ✶ ✶
As demonstrated hitherto, the hyperparameters of the SVM can be automatically optimized within the framework of the BO. When the cost function appears to be even stable (e.g., Figure 7d), the test accuracy might be improved further by simply increasing the iteration if the current classification is far from the global minima. The iteration number might be determined by a trade-off between the targeted test accuracy and the computational cost. For the field dataset used in the study, there is a fundamental limitation for the classifications relating to the overlapping facies or classes. Some additional elastic properties as input might be considered in future investigations in order to achieve better classification beyond the current limitation, which also would cost more computations. With this regard, so-called multidimensional scaling [e.g., 1] could be useful to be implemented.
The computational cost by the grid search tends to be enormous especially when the searching range is large but utilizing the BO is a versatility choice for geophysical applications as long as we use conventional computers rather than quantum computers which are under development.
Conclusions
We presented an application of Bayesian optimization (BO) with support vector machine (SVM) in order to classify cross-plot products of elastic properties which could be used in seismic QI studies. The applications showed in this study used the data from 10 wells in East Central Graben in the UKCS. We achieved the classification results with visually feasible appearance having non-linear decision boundaries without choosing the SVM hyperparameters subjectively. The cost function related to hyperparameters optimization by the BO exhibited to be stable before finishing the first dozen iterations, but the test accuracy and the visual classification slightly improved by increasing the total iteration number. However, it is difficult to predetermine which iteration number is optimal because there is a trade-off between the prediction accuracy and the computational cost. Nonetheless, the BO application is potentially more efficient in optimizing the hyperparameters of the SVM than a conventional exhaustive-grid search. So, the application of BO with SVM can be useful for well and consequently seismic quantitative interpretations and facies analysis in resource exploration and development subsurface studies.
Figure 3 :
3Classification result of synthetic elastic properties (AI and V P /V S ) using the SVM after 20 BO iterations of the hyperparameters. The filled circles are the training data, the hollow circles are the test data, and the crosses are the ground truth. C, B, O, S and L in the legend box stand for condensate, brine-bearing sand, oil-bearing sand, shale and limestone facies. The five different background colours correspond with the classified facies. The details of 20 BO iterations are shown in
Figure 4 :
4Cross plots of cost values after 20th BO iteration using the synthetic data (Figure 3) for: (a) the penalization parameter and sigma; (b) the cost and the sigma; (c) the penalization parameter and the cost; (d) the number of iterations and the minimum cost. The stars and arrow correspond to where the optimal values are found. by our implementation. InFigure 5, we plot a confusion matrix for the test accuracy of 90.8 %. The numbers in each bin indicate the test accuracy per target class when the sample numbers in the identical bin are used. Note that the accuracy shown here stands for the total accuracy of all kinds of used facies. In this synthetic example, two classes of oil and gas bearing sands are well separated and distinguished from the others as the main target. A prediction accuracy of 90.0 % is achieved with only 20 iterations for the utilized BO.
Figure 5 :
5Confusion matrix of the synthetic data classification inFigure 3. The percentile in each bin is the test accuracy between the target-and resulted classes. The integer shown in each bin is the used sample number.
Figure 8 :
8Confusion matrices of the field data classification with: (a) 20 iterations in Figures 6 and 7; (b) 40 iterations in Figures 9 and 10; (c) 200 iterations in Figures 11 and 12.
Figure 9 :
9The most of pairs with the cost show lower than 0.4 to be likely emerged when both the sigma and the penalization parameter have similar values in logarithmic scale. This is not surprising when we consider of the hyperparameters associated with the SVM. Recall that the sigma is Same asFigure 6but for 40 iterations. The details of 40 BO iterations are shown in
Figure 11 :
11Same as Figure 6 but for 200 iterations. The details of 200 BO iterations are shown in
Schematic image how SVM classifies the binary problem non-linearly. With the kernel trick, the original input dimension is implicitly converted to the feature dimension where the decision boundary can linearly separate classes. The estimated linear boundary in the feature dimension then converts back to the original dimension being as the non-linear boundary.Input dimension
Feature dimension
Output dimension
Kernel trick
by SVM
Decision
Figure 2:
AcknowledgmentsFor SVM applications and Bayesian optimization, LIBSVM and BayesOpt were used in this study, respectively.
Modern Multidimensional Scaling. I Borg, P J F Groenen, https:/link.springer.com/book/10.1007/978-1-4757-2711-1SpringerI. Borg and P. J. F. Groenen. Modern Multidimensional Scaling. Springer, 1997. URL https://link.springer. com/book/10.1007/978-1-4757-2711-1.
AVO inversion of Troll field data. A Buland, M Landrø, M Andersen, T Dahl, 10.1190/1.1444078Geophysics. 61A. Buland, M. Landrø, M. Andersen, and T. Dahl. AVO inversion of Troll field data. Geophysics, 61, 1996. doi: 10.1190/1.1444078.
Robust optimization of SVM hyperparameters in the classification of bioactive compounds. W M Czarnecki, S Podlewska, A J Bojarski, 10.1186/s13321-015-0088-0Journal of Cheminformatics. 7W. M. Czarnecki, S. Podlewska, and A. J. Bojarski. Robust optimization of SVM hyperparameters in the classification of bioactive compounds. Journal of Cheminformatics, 7, 2015. doi: 10.1186/s13321-015-0088-0.
Parallelizing exploration-exploitation tradeoffs in gaussian process bandit optimization. T Desautels, A Krause, J W Burdick, https:/dl.acm.org/doi/10.5555/2627435.2750368Journal of Machine Learning Research. 15T. Desautels, A. Krause, and J. W. Burdick. Parallelizing exploration-exploitation tradeoffs in gaussian process bandit optimization. Journal of Machine Learning Research, 15, 2014. URL https://dl.acm.org/doi/10. 5555/2627435.2750368.
An overview of ensemble methods for binary classifiers in multi-class problems: experimental study on one-vs-one and one-vs-all schemes. M Galar, A Fernández, E Barrenechea, H Bustince, F Herrera, 10.1016/j.patcog.2011.01.017Pattern Recognition. 44M. Galar, A. Fernández, E. Barrenechea, H. Bustince, and F. Herrera. An overview of ensemble methods for binary classifiers in multi-class problems: experimental study on one-vs-one and one-vs-all schemes. Pattern Recognition, 44, 2011. doi: 10.1016/j.patcog.2011.01.017.
Seismic processing, inversion, and avo for gold exploration -case study from western australia. C B Harrison, M Urosevic, 10.1190/geo2011-0506.1Geophysics. 77C. B. Harrison and M. Urosevic. Seismic processing, inversion, and avo for gold exploration -case study from western australia. Geophysics, 77, 2012. doi: 10.1190/geo2011-0506.1.
The Application of Bayesian Methods for Seeking the Extremum. Elsevier. M Jonas, T Vytautas, Z Antanas, M. Jonas, T. Vytautas, and Z. Antanas. The Application of Bayesian Methods for Seeking the Extremum. El- sevier, 1978. URL https://www.researchgate.net/publication/248818761_The_application_of_ Bayesian_methods_for_seeking_the_extremum.
Efficient global optimization of expensive black-box functions. D R Jones, M Schonlau, W J Welch, 10.1023/A:1008306431147Journal of Global Optimization. 13D. R. Jones, M. Schonlau, and W. J. Welch. Efficient global optimization of expensive black-box functions. Journal of Global Optimization, 13, 1998. doi: 10.1023/A:1008306431147.
A new method of locating the maximum point of an arbitrary multipeak curve in the presence of noise. H J Kushner, 10.1115/1.3653121Journal of Basic Engineering. 86H. J. Kushner. A new method of locating the maximum point of an arbitrary multipeak curve in the presence of noise. Journal of Basic Engineering, 86, 1964. doi: 10.1115/1.3653121.
Comparing support vector machines with gaussian kernels to radial basis function classifiers. B Scholkopf, S Kah-Kay, C Burges, F Girosi, P Niyogi, T Poggio, V Vapnik, 10.1109/78.650102IEEE Transactions on Signal Processing. 45B. Scholkopf, S. Kah-Kay, C. Burges, F. Girosi, P. Niyogi, T. Poggio, and V. Vapnik. Comparing support vector machines with gaussian kernels to radial basis function classifiers. IEEE Transactions on Signal Processing, 45, 1997. doi: 10.1109/78.650102.
Taking the human out of the loop: a review of Bayesian optimization. B Shahriari, K Swersky, Z Wang, R P Adams, N De Freitas, 10.1109/JPROC.2015.2494218Proceedings of the IEEE. 106B. Shahriari, K. Swersky, Z. Wang, R. P. Adams, and N. de Freitas. Taking the human out of the loop: a review of Bayesian optimization. Proceedings of the IEEE, 106, 2016. doi: 10.1109/JPROC.2015.2494218.
Practical Bayesian optimization of machine learning algorithms. J Snoek, H Larochelle, R P Adams, https:/dl.acm.org/doi/10.5555/2999325.2999464Advances in Neural Information Processing Systems. J. Snoek, H. Larochelle, and R. P. Adams. Practical Bayesian optimization of machine learning algorithms. In Advances in Neural Information Processing Systems, page 2951-2959. NIPS, 2012. URL https://dl.acm. org/doi/10.5555/2999325.2999464.
Gaussian process optimization in the bandit setting: no regret and experimental design. N Srinivas, A Krause, S M Kakade, M Seeger, https:/dl.acm.org/doi/10.5555/3104322.3104451Proceedings of the 27th International Conference on International Conference on Machine Learning. the 27th International Conference on International Conference on Machine LearningN. Srinivas, A. Krause, S. M. Kakade, and M. Seeger. Gaussian process optimization in the bandit setting: no regret and experimental design. In Proceedings of the 27th International Conference on International Conference on Machine Learning, page 1015-1022. ICML, 2010. URL https://dl.acm.org/doi/10.5555/3104322. 3104451.
The Nature of Statistical Learning Theory. V Vapnik, https:/link.springer.com/book/10.1007/978-1-4757-2440-0SpringerV. Vapnik. The Nature of Statistical Learning Theory. Springer, 1995. URL https://link.springer.com/ book/10.1007/978-1-4757-2440-0.
Pattern recognition using generalized portrait method. Automation and Remote Control. V Vapnik, A Lerner, 24V. Vapnik and A. Lerner. Pattern recognition using generalized portrait method. Automation and Remote Control, 24, 1963. URL https://ci.nii.ac.jp/naid/10020952249/.
Identifying organic-rich Marcellus shale lithofacies by support vector machine classifier in the Appalachian basin. G Wang, T R Carr, Y Ju, C Li, 10.1016/j.cageo.2013.12.002Computers & Geosciences. 64G. Wang, T. R. Carr, Y. Ju, and C. Li. Identifying organic-rich Marcellus shale lithofacies by support vector machine classifier in the Appalachian basin. Computers & Geosciences, 64, 2014. doi: 10.1016/j.cageo.2013.12.002.
Effective pressure prediction from 4D seismic AVO data during CO2-EOR and storage. H Wang, Z Wang, J Ma, L Li, Y Wang, M Tan, Y Zhang, S Cui, Z Qu, 10.1016/j.ijggc.2021.103525International Journal of Greenhouse Gas Control. 1132022H. Wang, Z. Wang, J. Ma, L. Li, Y. Wang, M. Tan, Y. Zhang, S. Cui, and Z. Qu. Effective pressure prediction from 4D seismic AVO data during CO2-EOR and storage. International Journal of Greenhouse Gas Control, 113, 2022. doi: 10.1016/j.ijggc.2021.103525.
| [] |
[
"SCALING OF MAGNETO-QUANTUM-RADIATIVE HYDRODYNAMIC EQUATIONS: FROM LASER-PRODUCED PLASMAS TO ASTROPHYSICS",
"SCALING OF MAGNETO-QUANTUM-RADIATIVE HYDRODYNAMIC EQUATIONS: FROM LASER-PRODUCED PLASMAS TO ASTROPHYSICS"
] | [
"J E Cross \nDepartment of Physics\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUnited Kingdom\n",
"G Gregori \nDepartment of Physics\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUnited Kingdom\n"
] | [
"Department of Physics\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUnited Kingdom",
"Department of Physics\nUniversity of Oxford\nParks RoadOX1 3PUOxfordUnited Kingdom"
] | [] | The relevant equations of magneto-quantum-radiative hydrodynamics are introduced and then written in a dimensionless form in order to extract a set of dimensionless parameters that describe scaledependent ratios of all the characteristic hydrodynamic variables. Under the conditions where such dimensionless number are all large, the equations reduce to the usual ideal magnetohydrodynamics and thus they are scale invariant. We discuss this property with regards to the similarity between astrophysical observations and laboratory experiments. These similarity properties have been successfully exploited in a variety of laboratory experiments where radiative processes can be neglected. On the other hand, when radiation is important, laboratory experiments are much more difficult to scale to the corresponding astrophysical objects. As an example, a recent experiment related to break out shocks in supernova explosions is discussed. | 10.1088/0004-637x/795/1/59 | [
"https://arxiv.org/pdf/1401.7880v2.pdf"
] | 53,511,960 | 1401.7880 | adbc0f0e2ac8acabc90ebf03b965f3a862f31b91 |
SCALING OF MAGNETO-QUANTUM-RADIATIVE HYDRODYNAMIC EQUATIONS: FROM LASER-PRODUCED PLASMAS TO ASTROPHYSICS
30 Jan 2014 Draft version January 31, 2014 Draft version January 31, 2014
J E Cross
Department of Physics
University of Oxford
Parks RoadOX1 3PUOxfordUnited Kingdom
G Gregori
Department of Physics
University of Oxford
Parks RoadOX1 3PUOxfordUnited Kingdom
SCALING OF MAGNETO-QUANTUM-RADIATIVE HYDRODYNAMIC EQUATIONS: FROM LASER-PRODUCED PLASMAS TO ASTROPHYSICS
30 Jan 2014 Draft version January 31, 2014 Draft version January 31, 2014Preprint typeset using L A T E X style emulateapj v. 5/2/11hydrodynamics -MHD -plasmas -radiative transfer -supernovae: individual (SN1993J)
The relevant equations of magneto-quantum-radiative hydrodynamics are introduced and then written in a dimensionless form in order to extract a set of dimensionless parameters that describe scaledependent ratios of all the characteristic hydrodynamic variables. Under the conditions where such dimensionless number are all large, the equations reduce to the usual ideal magnetohydrodynamics and thus they are scale invariant. We discuss this property with regards to the similarity between astrophysical observations and laboratory experiments. These similarity properties have been successfully exploited in a variety of laboratory experiments where radiative processes can be neglected. On the other hand, when radiation is important, laboratory experiments are much more difficult to scale to the corresponding astrophysical objects. As an example, a recent experiment related to break out shocks in supernova explosions is discussed.
INTRODUCTION
Study of astrophysical phenomena using laserproduced plasma is a growing field of research (Remington et al. 1999;Gregori et al. 2012). Modern laser facilities can deliver large amount of energy in very short times, exceeding what is possible from more conventional techniques such as gas guns or pulsed power machines. Pressures near the laser spot (where most of the laser energy is deposited) can reach values in excess of tens of Mbar, thus being comparable to the energy density of bound electrons in atoms. Quantum processes and radiation diffusion can become important in these conditions. The steep density and pressure gradients produce magnetic fields (Haines 1986) which govern the particle transport. These large energies deposited over sub-mm volumes then drive powerful shock waves into the ambient medium (Klein et al. 2003;Foster et al. 2002;Hansen et al. 2005;Robey et al. 2002). The process bears similarities with many astrophysical phenomena where energy is impulsively released in the interstellar medium, such as supernova remnants (Ryutov et al. 1999), Herbig-Haro flows (Hartigan et al. 1987) and accretion shocks (Gregori et al. 2012).
Laboratory experiments offer a viable complementary approach to both astrophysical observations (by providing, for example, the means of directly measuring quantities of interests not accessible by observations) and numerical calculations (thus overcoming limitations in numerical resolutions and potentially addressing non-linear aspects of the dynamical evolutions, and/or validating simulation codes). This is meaningful only if the relevant physics in the laboratory is the same as in the astrophysical object under study. We refer to this as a similarity relation between the two systems. The most obvious situation is the one where the laboratory experiment reaches the exact conditions found in the astrophysical object. This has been exploited, for example, to study the equation [email protected] of state of planetary interiors (Jeanloz et al. 2007) and other compact objects (García Saiz et al. 2008). However it is not always is possible to reach in the laboratory the exact conditions that we are interested in, as the spatial, temporal and energy scales may be outside the range of what is directly reproducible in an experiment. A similarity relation still exists if we can assure that the laboratory and the astrophysical systems evolve in a way that the governing equations are invariant under a scale transformation, i.e. such that the corresponding spatial, density, pressure, time, and so on, values in one system are mapped into the other system by multiplicative constants. This self-similarity can be obtained via fluid equations (Ryutov et al. 1999), or even at the kinetic level (Connor and Taylor 1977;Ryutov et al. 2012) under some conditions. This paper concerns the magnetohydrodynamics (MHD) similarity, and provides a general framework to include effects arising from finite resistivity, thermal conduction, radiation diffusion and quantum non-locality. While fluid similarity has already been discussed in special situations (Ryutov et al. 1999(Ryutov et al. , 2001Falize et al. 2011a,b), this work gives a unified treatment of all of these effects in a simple conceptual form.
GENERAL EQUATIONS
We first start with the full set of equations that describe resistive MHD fluids in presence heat conduction, radiation diffusion and quantum effects (Drake 2006;McClarren et al. 2010;Haas 2011;Zeldovich et al. 2002):
∂ρ ∂t + ∇ · ρu = 0, (1a) ρ ∂u ∂t + u · ∇u = −∇(p+p R )+Φ Bohm +∇·σ ν +F EM ,(1b)∂ ∂t ρǫ + ρu 2 2 + E R + ∇ · ρu ǫ + u 2 2 + pu = −∇ · H − J · E + Φ Bohm · u, (1c) ∂B ∂t = −B(∇ · u) + η∇ 2 B + m e(1 + Z) ∇p × ∇ρ ρ 2 , (1d) H = F R + (p R + E R ) u + Q − σ ν · u.(1e)
These are the continuity (1a), momentum (1b), energy (1c), induction (1d) and energy flux (1e) equations, respectively. Here ρ is the mass density, Z the ionization state, t the time, u the fluid velocity, p the ram pressure, p R the radiation pressure, Φ Bohm the quantum Bohm potential, σ v the stress tensor, F EM electromagnetic volume forces, ǫ the specific internal energy, E R the energy density of the radiation field, H the total energy flux, J the current density, E the electric field, B the magnetic field, η the magnetic diffusivity (η = 1/σ 0 µ 0 where σ 0 is the electric conductivity and µ 0 the vacuum permittivity), m average mass per particle, e elementary charge, Z degree of ionisation, F R the radiative energy flux, and Q the heat flux. Here we write the induction equation to include baroclinic generation of magnetic field via the Biermann battery mechanism (Biermann 1950;Kulsrud and Zweibel 2008). This is the last term on the right hand side of equation (1d). In many laboratory and astrophysical scenarios, the baroclinic term represents the next higher order correction to Ohm's law (Haines 1986). Differently from previous work, the above equations correctly describe quantum hydrodynamics behavior, which becomes important for high density fluids (Schmidt et al. 2012), when the number density reaches values 10 24 cm −3 , as in white dwarfs or neutron star matter. This means that Pauli blocking, tunneling and wave packet spreading begin to exert an effective quantum pressure to the system (Haas 2011). This approach follows from the the fact that deterministic equations can be used to describe both single-particle and many-body distribution functions in the quantum limits if appropriate potential are introduced in the hydrodynamic equations (Bohm 1952;Mostacci et al. 2008).
The source terms in equations (1b), (1c) and (1e) are explicitly given by:
p R = E R 3 = 4σT 4 3c (2a) Φ Bohm = − 2 ρ 2m 2 ∇ ∇ 2 √ ρ √ ρ (2b) σ ν = ρν ∇u + (∇u) T − 2 3 (∇ · u)I + ζ(∇ · u)I (2c) F EM = ρ C E + J × B (2d) F R = − 16σT 3 3χ R ρ ∇T (2e) Q = −κ th ∇T = −χ th ρc p ∇T = − χ th ρk B γ m(γ − 1) ∇T (2f)
where σ is the Stefan-Boltzmann constant, T the temperature, c the speed of light, the reduced Planck's constant, m the particle mass, ν the kinematic viscosity (where ν = µ/ρ, with µ being the (dynamic) viscosity), I the identity tensor and ζ the second coefficient of viscosity, ρ C the charge density, χ R the Rosseland mean opacity, κ th the coefficient of heat conduction, χ th the kinematic coefficient of thermal diffusivity, c p the specific heat capacity at constant pressure, k B Boltzmann's constant, and γ the adiabatic index.
Equation (2a) represents the isotropic thermal radiation pressure within the plasma, and the related energy density of that radiation (Castor 2004), assuming a Planck distribution for the radiation. Equation (2b) accounts for quantum effects. Equation (2c) gives the form of the stress tensor. This forms does not assume that the fluid is incompressible, i.e., ∇ · u does not have to be equal to zero (Drake 2006). Equation (2d) defines the electromagnetic (Lorentz) force on the system. Equation (2e) gives the radiative energy flux, within the local thermodynamic equilibrium (LTE) approximation. In this form, it corresponds to the Rosseland heat flux (Drake 2006;Castor 2004). Finally, equation (2f) describe the thermal heat flux (Landau and Lifshitz 1959).
Quantum potential
Given the presence of the Bohm potential in the above equations and the fact that this term is often omitted, it is important to give a detailed explanation and derivation for its appearance. The form used arises from rewriting the Schrödinger equation in polar form with a wavefunction given by
φ = Re iS/ ,
where R and S are real valued functions. The Schrödinger equation can be thus divided into an imaginary part
∂R ∂t = − 1 2m R∇ 2 S + 2∇R · ∇S ,(3)
and a real part
∂S ∂t = − (∇S) 2 2m + V + Q ,(4)
where V is the external potential and
Q = − 2 2m ∇ 2 R R .
If we now identify, using the correspondence to the classical limit , R 2 = ρ, and u = ∇S/m, then (3) can be re-expressed as a continuity equation, while (4) has the form of an energy equation with the classical potential corrected by the quantum term Q. This leads, for example, to the inclusion of ρQ/m as an energy density correction in the momentum equation.
DIMENSIONLESS ANALYSIS
We now rescale the variables in the hydrodynamic equations by a corresponding characteristic value. This allows us to rewrite the equations in an invariant form, and all the details associated to the physical dimensions of the system are contained in a series of dimensionless numbers, which represents ratios of those characteristic values. Let's assume that the velocity, position, time and density are written as
u → u 0 u * , r → ℓ 0 r * , t → ℓ 0 u 0 t * , ρ → ρ 0 ρ * ,
where u 0 , ℓ 0 , and ρ 0 are the characteristic velocity, length and density of the system, respectively. From now on we will use the convention that starred quantities (i.e., u * ) are dimensionless, while quantities with subscript 0 (i.e., u 0 ) correspond to a characteristic value for that variable. The above assumptions imply
∂ ∂t → u 0 ℓ 0 ∂ ∂t * , ∇ → ∇ * ℓ 0 .
Similarly, we can set
p → p 0 p * , B → B 0 B * , ǫ → ǫ 0 ǫ * .
However, the choice of the values for p 0 , B 0 , and ǫ 0 is not arbitrary. To see this, let's consider the momentum equation (1b), but with the only source terms being the pressure gradient and magnetic field (i.e., in the limit of ideal MHD). Using the relation:
J × B = (B · ∇)B/µ 0 − ∇(B 2 /2µ 0 ) we can rewrite the momentum equation (1b) as ρ 0 ρ * u 0 ℓ 0 ∂u 0 u * ∂t * + u 0 u * · ∇ ℓ 0 * u 0 u * = − ∇ ℓ 0 * p 0 p * + B 2 0 ℓ 0 µ 0 (B * · ∇ * )B * − ∇ * B * 2 2 .
Noticing the common factor of u 2 0 ρ 0 /ℓ 0 on the left and dividing through gives
ρ * ∂u * ∂t * + u * · ∇ * u * = − p 0 ρ 0 u 2 0 ∇ * p * + B 2 0 µ 0 ρ 0 u 2 0 (B * · ∇ * )B * − ∇ * B * 2 2 .(5)
As we require this equation to have the same form as (1b), that is, to be invariant under the scaling transformation, this means that p 0 ≡ ρ 0 u 2 0 and B 0 ≡ u 0 √ µ 0 ρ 0 .
We see that the reference magnetic field has a value such that the fluid velocity and the Alfvén velocity (Alfvén 1942) are the same. We can follow a similar procedure to determine the value for ǫ 0 . Using the energy equation (1c) in the ideal case with no source terms, we get
u 0 ℓ 0 ∂ ∂t * ρ 0 ǫ 0 ρ * ǫ * + ρ 0 u 2 0 ρ * u * 2 2 = − ∇ ℓ 0 * · ρ 0 u 0 ρ * u * ǫ 0 ǫ * + u 2 0 u * 2 2 + ρ 0 u 2 0 p * u 0 u * .
Dividing through by a factor of u 3 0 ρ 0 /ℓ 0 we obtain
∂ ∂t * ǫ 0 u 2 0 ρ * ǫ * + ρ * u * 2 2 = −∇ * · ρ * u * ǫ 0 u 2 0 ǫ * + u * 2 2 + p * u * .
Again, we require this to be invariant under the scaling transformation, which leads to ǫ 0 ≡ u 2 0 . This simple exercise has shown that the equations of ideal MHD are indeed invariant under scaling. This applies for any choice of the scaling transformation. In reality, things are more complex because neither the laboratory system, nor the astrophysical one, can always be assumed to evolve under ideal conditions. To see this, let's consider equation (1d) with the inclusion of the resistive and baroclinic terms. By applying the scaling transformation defined above, we have
u 0 ℓ 0 √ µ 0 ρ 0 u 0 ∂B * ∂t * = − u 0 ℓ 0 √ µ 0 ρ 0 u 0 B * (∇ * · u * )+ η u 0 ℓ 2 0 . √ µ 0 ρ 0 u 0 ∇ * 2 B * + mu 2 0 eℓ 2 0 (1 + Z) ∇ * p * × ∇ * ρ * ρ * 2 .
Dividing through by u 2
0 √ µ 0 ρ 0 /ℓ 0 , ∂B * ∂t * = −B * (∇ * ·u * )+ 1 Re M ∇ * 2 B * + 1 Bi ∇ * p * × ∇ * ρ * ρ * 2 ,(6)1 Bi = m e √ µ 0 ρ 0 ℓ 0 (1 + Z) ,
which we will refer to as the Biermann number. This shows that the equations of resistive MHD are scale invariant only if Re M , and Bi, are the same in both the laboratory and astrophysical systems, or, alternatively, very large in both conditions, such that resistive terms are negligible.
SIMILARITY FOR NON IDEAL EQUATIONS
We must now consider the full system of equations (1c)-(1e). In order to proceed, we need to define additional scaling variables for temperature, current density, electric field and charge:
T → T 0 T * , J → J 0 J * , E → E 0 E * , ρ C → ρ C0 ρ * C .
Momentum equation
We start with the momentum equation (1b) and use the above scaling transformations. Considering each term separately, we have
ρ ∂u ∂t + u · ∇u → ρ 0 u 2 0 ℓ 0 ρ * ∂u * ∂t * + u * · ∇ * u * (7a) − ∇ p + 4σT 4 3c → − ρ 0 u 2 0 ℓ 0 ∇ * p * + 4σT 4 0 3ρ 0 u 2 0 c T * 4 (7b) − 2 ρ 2m 2 ∇ ∇ 2 √ ρ √ ρ → − ρ 0 u 2 0 ℓ 0 ℓ 0 u 0 √ 2m 2 ρ * ∇ * ∇ * 2 √ ρ * √ ρ * (7c) ∇ · ρν ∇u + (∇u) T − 2 3 (∇ · u)I + ζ(∇ · u)I → ρ 0 u 2 0 ℓ 0 ∇ * · µ ρ 0 u 0 ℓ 0 ∇ * u * + (∇ * u * ) T − 2 3 (∇ * · u * )I + ζ ρ 0 u 0 ℓ 0 (∇ * · u * )I (7d) ρ C E + J × B → ρ 0 u 2 0 ℓ 0 ρ C0 ℓ 0 E 0 ρ 0 u 2 0 ρ * C E * + J 0 ℓ 0 u 0 √ µ 0 ρ 0 J * × B * .(7e)
If we divide through by the common term of ρ 0 u 2 0 /ℓ 0 we obtain the momentum equation in dimensionless form
ρ * ∂u * ∂t * + u * · ∇ * u * = −∇ * p * + 1 R T * 4 + 1 H Q ρ * ∇ * ∇ * 2 √ ρ * √ ρ * + ∇ * · 1 Re ∇ * u * + (∇ * u * ) T − 2 3 (∇ * · u * )I + 1 Re ζ (∇ * · u * )I + 1 Ω R ρ * C E * + 1 Ω H J * × B *(8)
The Mihalas number (R) represents the ratio of ram pressure to radiation pressure, and it is related to the more familiar Boltzmann (Bo) number by
1 R = 4σT 4 0 /3c ρ 0 u 2 0 = 4u 0 γ 3c(γ − 1) 1 Bo ,(9)
where Bo = ρ 0 c p T 0 u 0 /σT 4 0 . Here, we have used k B T 0 ∼ mu 2 0 , and c p ∼ γk B /m(γ − 1). The Boltzmann number gives the ratio of the enthalpy flux with the radiation flux. The importance of quantum effects against classical ones within the system is described by the number:
1 H Q = 2 /2mℓ 2 0 mu 2 0 ,(10)
which we will refer to as the Bohm number. We can also recognize the Reynold's number and its obvious extension when considering the second coefficient of viscosity:
1 Re = µ ρ 0 u 0 ℓ 0 ; 1 Re ζ = ζ ρ 0 u 0 ℓ 0(11)
From charge conservation, ρ C0 = J 0 /u 0 , it follows
1 Ω R = ρ C0 ℓ 0 E 0 ρ 0 u 2 0 = J 0 E 0 ℓ 0 ρ 0 u 3 0 ,(12)
which represents the ratio between Ohmic and convective heat transfer. The ratio between convective transport and Hall diffusion is expressed by the coefficient
1 Ω H = J 0 ℓ 0 u 0 √ µ 0 ρ 0 = J 0 B 0 ℓ 0 µ 0 ρ 0 u 2 0 .(13)
4.2. Energy equation Following the same approach as before, but now using the energy equation (1c) and, again, considering each term separately:
∂ ∂t ρǫ + ρu 2 2 + 4σT 4 c → ρ 0 u 3 0 ℓ 0 ∂ ∂t * ρ * ǫ * + ρ * u * 2 2 + 4σT 4 0 ρ 0 u 2 0 c T * 4 , (14a) ∇ · ρu ǫ + u 2 2 + pu → ρ 0 u 3 0 ℓ 0 ∇ * · ρ * u * ǫ * + u 2 2 + p * u * , (14b) ∇ · − 16σT 3 3χ R ρ ∇T → ρ 0 u 3 0 ℓ 0 ∇ * · − 16σT 4 0 3χ R ρ 2 0 ℓ 0 u 3 0 T * 3 ρ * ∇ * T * , (14c) − ∇ · 16σT 4 c · u → − ρ 0 u 3 0 ℓ 0 ∇ * · 16σT 4 0 ρ 0 u 2 0 c T * 4 · u * , (14d) − ∇ · χ th ρk B γ m(γ − 1) ∇T → ρ 0 u 3 0 ℓ 0 ∇ * · − χ th k B T 0 γ ℓ 0 mu 3 0 (γ − 1) ρ * ∇ * T * (14e) ∇ · ρν ∇u + (∇u) T − 2 3 (∇ · u)I + ζ(∇ · u)I · u → ρ 0 u 3 0 ℓ 0 ∇ * · µ ρ 0 u 0 ℓ 0 ∇ * u * + (∇ * u * ) T − 2 3 (∇ * · u * )I + ζ ρ 0 u 0 ℓ 0 (∇ * · u * )I · u * (14f) J · E → ρ 0 u 3 0 ℓ 0 J 0 E 0 ℓ 0 ρ 0 u 3 0 J * · E * (14g) − 2 ρ 2m 2 ∇ ∇ 2 √ ρ √ ρ · u → − ρ 0 u 3 0 ℓ 0 u 0 ℓ 0 √ 2m 2 ρ * ∇ * ∇ * 2 √ ρ * √ ρ * · u * (14h)
A factor of ρ 0 u 3 0 /ℓ 0 has been pulled out from each term. The dimensionless energy equation can thus be written as
∂ ∂t * ρ * ǫ * + ρ * u * 2 2 + 3 R T * 4 + ∇ * · ρ * u * ǫ * + u 2 2 + p * u * = ∇ * · − 1 Π T * 3 ρ * ∇ * T * − 12 R T * 4 · u * + 1 P e γ γ − 1 ρ * ∇ * T * + 1 Re ∇ * u * + (∇ * u * ) T − 2 3 (∇ * · u * )I · u * + 1 Re ζ (∇ * · u * )I · u * − 1 Ω R J * · E * + 1 H Q ρ * ∇ * ∇ * 2 √ ρ * √ ρ * · u * (15)
Analogous to the momentum equation we have new dimensionless numbers. We define a radiation number, Π, which is related to the Bolztmann number by:
1 Π = 16σT 4 0 3χ R ρ 2 0 ℓ 0 u 3 0 = 16 3χ R ℓ 0 ρ 0 1 Bo .(16)
The Péclet number gives the importance of thermal diffusion against convective transport:
1 P e = χ th k B T 0 ℓ 0 mu 3 0 = χ th ℓ 0 u 0 ,(17)
where we have used the k B T ∼ mu 2 0 .
COMPLETE FORM
We are now in a position to write all the complete equations in dimensionless form. Repeating the results obtained above, they are:
∂ρ * ∂t * + ∇ * · ρ * u * = 0 (18) ∂B * ∂t * = −B * (∇ * ·u * )+ 1 Re M ∇ * 2 B * + 1 Bi ∇ * p * × ∇ * ρ * ρ * 2 ,(19)ρ * ∂u * ∂t * + u * · ∇ * u * = −∇ * p * + 1 R T * 4 − 1 H Q ρ * ∇ * ∇ * 2 √ ρ * √ ρ * + ∇ * · 1 Re ∇ * u * + (∇ * u * ) T − 2 3 (∇ * · u * )I + 1 Re ζ (∇ * · u * )I + 1 Ω R ρ * C E * + 1 Ω H J * × B * (20) ∂ ∂t * ρ * ǫ * + ρ * u * 2 2 + 3 R T * 4 + ∇ * · ρ * u * ǫ * + u 2 2 + p * u * = ∇ * · − 1 Π T * 3 ρ * ∇ * T * − 12 R T * 4 · u * + 1 P e ρ * ∇ * T * + 1 Re ∇ * u * + (∇ * u * ) T − 2 3 (∇ * · u * )I · u * + 1 Re ζ (∇ * · u * )I · u * − 1 Ω R J * · E * − 1 H Q ρ * ∇ * ∇ * 2 √ ρ * √ ρ * · u * (21)
The scaling variables and all the dimensionless numbers are given in Table 1. As discussed earlier, similarity between the laboratory and astrophysical object is achieved if the dimensionless numbers are the same or all large in both systems (the ideal MHD case). Under either of these conditions, let's take ℓ
(1)
0 , u(1)0 , ρ(1)0 , J(1)0 , E(1)
0 , and T
(1) 0 the characteristic scaling parameters for the laboratory experiment. The astrophysical system has corresponding values given by
ℓ (2) 0 = g a ℓ (1) 0 , u (2) 0 = g b u (1) 0 , ρ (2) 0 = g c ρ (1) 0 , J (2) 0 = g d J (1) 0 , E (2) 0 = g e E (1) 0 , T (2) 0 = g f T (1) 0 ,
where g a,b,c,d,e,f are scaling constants. From this set of parameters, we can scale all the other characteristic quantities as
t (2) 0 = g a g b t (1) 0 , p(2)0 = g c g 2 b p (1) 0 , B(2)0 = g b √ g c B (1) 0 , ǫ(2)0 = g 2 b ǫ (1) 0 , ρ (2) C0 = g d g b ρ (1) C0 .
All the details concerning the microphysics of the two systems are thus contained only in the dimensionless numbers given in Table 1.
In order to evaluate those numbers, let's assume the plasma is in thermodynamic equilibrium at the temperature T (in eV) and carries a mass density ρ (in g/cm 3 ) from ions of atomic mass A and charge Z. The magnetic field is B (in G). Charge neutrality implies an equal number of negative charges carried by mobile electrons. These assumptions are applicable to both the laboratory and astrophysical plasmas. Following Ryutov et al. (1999); Huba (2002), the kinematic viscosity is, in cases where magnetic field is important or not,
ν (cm 2 /s) = Min 3.3 × 10 −5 A 1/2 T 5/2 Z 4 ρΛ 2.8 × 10 43 ρ 2 Λ A 5/2 Z 2 B 2 T 1/2 3.0 × 10 −9 AT 4 Zρ 2 ,(22)
where Λ is the Coulomb logarithm. The thermal diffusivity is (Ryutov et al. 1999), in the unmagnetised and magnetised case,
χ th (cm 2 /s) = Min 3.3 × 10 −3 AT 5/2 Z(Z+1)ρΛ 8.6 × 10 9 A 1/2 T ZB .(23)
The magnetic diffusivity is given by (Pitaevskii and Lifshitz 1981)
η (cm 2 /s) = 2.4 × 10 5 ZΛ T 3/2 .(24)
The Rosseland opacity, can be written in terms of a cooling function L Λ as
χ R (cm 2 /g) = 1.8 × 10 35 ZρL Λ A 2 T 4 ,(25)
where L Λ ∼ 10 −22 ergs cm 3 /s for typical astrophysical plasmas (Sutherland and Dopita 1993), and for bremsstrahlung-dominated cooling L Λ ∼ 1.7 × 10 −25 Z 2 T 1/2 (Ryutov et al. 1999). In the case of a fully ionized plasma, the opacity is only determined by the free-free absorption, thus (Zeldovich et al. 2002)
χ R (cm 2 /g) = 4.4 × 10 8 Z 3 ρ A 2 T 7/2 .(26)
At higher densities (near and above solid) and when line radiation transport must be included in the calculations, the Rosseland opacity is tabulated as (Tsakiris and Eidmann 1987), where κ 0 , α and β are material dependent constants (see Table 2). The Rosseland opacity is bound to a maximum value given by (Tsakiris and Eidmann 1987) χ R,max (cm 2 /g) = 6.1 × 10 6 Z AT .
χ R (cm 2 /g) = κ 0 ρ α T β(27)
Even in the case that the dimensionless numbers are large in both the laboratory and astrophysical systems, their magnitude can be very different. It is then important to quantify the error in fluid variables in the ideal MHD approximation due to finite values for such dimensionless numbers. We have: , where B id refers to the magnetic field in the ideal MHD approximation, and similarly for the momentum and energy.
∆B B id ∼ 1 Re 2 M + 1 Bi 2 1/2 ,(29)
Characteristic quantity Definition
Length 3.70 × 10 6 0.16 -1.57 Xe 2.00 × 10 8 0.00 -2.00 Ba 5.89 × 10 6 0.14 -1.62 Eu 2.89 × 10 6 0.09 -1.45 W 5.59 × 10 5 0.00 -1.12 Au 6.00 × 10 6 0.30 -1.50 Pb 4.11 × 10 5 0.00 -1.05 U 7.76 × 10 5 0.04 -1.14 Table 2: List of coefficient values for equation (27). Adapted from Tsakiris and Eidmann (1987) and Drake (2006).
ℓ 0 Velocity u 0 Density ρ 0 Current density J 0 Electric field E 0 Temperature T 0 Time t 0 = ℓ 0 /u 0 Pressure p 0 = ρ 0 u 2 0 Magnetic field B 0 = u 0 √ ρ 0 Specific internal energy ǫ 0 = u 2 0 Charge density ρ C0 = J 0 /u 0 Reynolds number Re = ρ 0 u 0 ℓ 0 /µ Reynolds number (bulk) Re ζ = ρ 0 u 0 ℓ 0 /ζ Magnetic Reynolds number Re M = u 0 ℓ 0 /η Biermann number Bi = e(1 + Z) √ µ 0 ρ 0 ℓ 0 /m Mihalas number R = 3cρ 0 u 2 0 /4σT 4 0 Radiation number Π = 3χ R ρ 2 0 ℓ 0 u 3 0 /16σT 4 0 Péclet number P e = ℓ 0 u 0 /χ th Ohmic number Ω R = ρ 0 u 3 0 /J 0 E 0 ℓ 0 Hall number Ω H = ρ 0 u 2 0 /J 0 B 0 ℓ 0 Bohm number H Q = 2m 2 u 2 0 ℓ 2 0 / 2
CONCLUDING REMARKS
The similarity properties have been successfully exploited in a variety of laboratory experiments (Remington et al. 1999), but almost exclusively limited to the condition of radiation free environments. There are astrophysical situations, however, where radiation is important. In order to show the laboratory implications associated to scaling under radiative conditions, we compare the astrophysical case of a shock breakout in a circumstellar medium (Fransson, Lundqvist and Chevalier 1996) to a recent implosion experiment on the National Ignition Facility (NIF) laser (Pak et al. 2013). While, as shown in Table 3, the experiments can indeed reproduce the supernova shock breakout in most aspects, the similarity breaks down when considering the radiation and the Mihalas numbers. This example shows that radiation dominated environments are yet challenging to achieve even on the currently available largest laser facilities. Our work thus provide a useful guide to future experiments towards achieving those conditions. (Pak et al. (2013)) to a supernova breakout shock (Fransson, Lundqvist and Chevalier (1996)).
Table 1 :
1List of scaling variables and dimensionless numbersMaterial
κ 0
α
β
CH
2.00 × 10 6 0.14 -2.00
Al
1.04 × 10 8 0.48 -2.48
Ti
3.07 × 10 7 0.39 -2.21
Fe
6.29 × 10 7 0.31 -2.27
Cu
5.93 × 10 7 0.29 -2.21
Mo
1.99 × 10 6 0.22 -1.49
Sn
Table 3 :
3Example of scaling under radiative conditions from the laboratory
. H Alfvén, Nature. 150H. Alfvén. Nature 150, 405-406 (1942).
. L Biermann, Z. Naturforsch. A. 5L. Biermann. Z. Naturforsch. A 5, 65-71 (1950).
. D Bohm, Phys. Rev. 85D. Bohm. Phys. Rev. 85, 166-179, 1952.
Radiation Hydrodynamics. J I Castor, Cambridge University PressJ.I. Castor. Radiation Hydrodynamics. Cambridge University Press, 2004.
. J W Connor, J B Taylor, Nucl. Fusion bf. 17J.W. Connor and J.B. Taylor. Nucl. Fusion bf 17, 1047-1055, 1977.
P Drake, High-Energy-Density Physics: Fundamentals, Inertial Fusion, and Experimental Astrophysics. SpringerP. Drake. High-Energy-Density Physics: Fundamentals, Inertial Fusion, and Experimental Astrophysics. Springer, 2006.
. E Falize, Astrophys. Space Sci. 336E. Falize et al. Astrophys. Space Sci. 336, 81-85 (2011a).
. E Falize, C Michaut, S Bouquet, ApJ. 730E. Falize, C. Michaut, and S. Bouquet. ApJ 730, 96-104 (2011b).
. M Foster, Phys. Plasmas. 9M. Foster et al. Phys. Plasmas 9, 2251-2263 (2002).
. C Fransson, P Lundqvist, R A Chevalier, ApJ. 461C. Fransson, P. Lundqvist, and R.A. Chevalier ApJ 461, 993-1008 (1996).
. E , García Saiz, Nat. Phys. 4E. García Saiz et al. Nat. Phys. 4, 940-944 (2008).
. G Gregori, Nature. 481G. Gregori et al. Nature 481, 480-3 (2012).
F Haas, Quantum Plasmas: An Hydrodynamic Approach. SpringerF. Haas. Quantum Plasmas: An Hydrodynamic Approach. Springer, 2011.
. M G Haines, Can. J. Phys. 64M.G. Haines. Can. J. Phys. 64, 912-918 (1986).
. F Hansen, Astrophys.Space Sci. 298F. Hansen et al. Astrophys.Space Sci. 298, 61-67 (2005).
. P Hartigan, John Raymond, Lee Hartmann, ApJ. 316P. Hartigan, John Raymond, and Lee Hartmann. ApJ 316, 323-348 (1987).
Revised NRL Plasma Formulary. J D Huba, Office of Naval ResearchJ. D. Huba. Revised NRL Plasma Formulary. Office of Naval Research, 2002.
. R Jeanloz, Proceedings of the National Academy of Sciences of the United States of America. 104R. Jeanloz et al. Proceedings of the National Academy of Sciences of the United States of America 104, 9172-9177 (2007).
. I Klein, ApJ. 583I. Klein et al. ApJ 583, 245-259 (2003).
. M Kulsrud, E G Zweibel, Rep. Prog. Phys. 71M. Kulsrud and E. G. Zweibel. Rep. Prog. Phys. 71 046901-046934 (2008).
Fluid mechanics. L D Landau, E M Lifshitz, Pergamon PressL.D. Landau and E.M. Lifshitz. Fluid mechanics. Pergamon Press, 1959.
. D Landau, E M Lifshitz, L P Pitaevskii, Physical Kinetics. ElsevierD. Landau, E.M. Lifshitz and L.P. Pitaevskii. Physical Kinetics. Elsevier, 1981.
. G Mcclarren, Phys. Plasmas. 17G. McClarren et al. Phys. Plasmas 17 093301-093313 (2010).
. D Mostacci, V Molinari, F Pizzio, Transport Theor. Stat. 37D. Mostacci, V. Molinari, and F. Pizzio. Transport Theor. Stat. 37, 589-600 (2008).
. A Pak, Phys. Plasmas. 20A. Pak et al. Phys. Plasmas 20 056315-056326 (2013).
. B A Remington, Science. 2845419B. A. Remington et al. Science. 284(5419), 1488-1493, 1999.
. H Robey, Phys. Rev. Lett. 89H. Robey et al. Phys. Rev. Lett. 89, 085001-085005 (2002).
. D D Ryutov, R P Drake, J Kane, ApJ. 518D. D. Ryutov, R. P. Drake, and J. Kane. ApJ 518, 821-832 (1999).
. D Ryutov, Phys. Plasmas. 8D. Ryutov et al. Phys. Plasmas 8, 1804-1816 (2001).
. D Ryutov, Plasma Phys. Contr. Fusion. 54D. Ryutov et al. Plasma Phys. Contr. Fusion 54, 105021-105030 (2012).
. D D Ryutov, R P Drake, B A Remington, ApJ. 465D. D. Ryutov, R. P. Drake, and B. A. Remington. ApJ 465, 465-468 (2000).
. R Schmidt, Phys. Rev. E. 85R. Schmidt, et al. Phys. Rev. E 85, 046408-046412 (2012).
. R S Sutherland, M A Dopita, ApJ. 88R. S. Sutherland and M. A. Dopita. ApJ 88, 253-327 (1993).
. G D Tsakiris, K Eidmann, J. Quant. Spectrosc. Radiat. Transfer bf. 8G. D. Tsakiris and K. Eidmann. J. Quant. Spectrosc. Radiat. Transfer bf 8, 353-368 (1987).
Physics of Shock Waves and High-temperature Hydrodynamic Phenomena. Y B , Y P Raizer, DoverY.B. Zel'dovich, and Y.P. Raizer. Physics of Shock Waves and High-temperature Hydrodynamic Phenomena. Dover, 2002.
| [] |
[
"Self-Assembly of Diamondoid Molecules and Derivatives (MD Simulations and DFT Calculations)",
"Self-Assembly of Diamondoid Molecules and Derivatives (MD Simulations and DFT Calculations)"
] | [
"Yong Xue \nDepartment of Physics\nUniversity of Illinois at Chicago\n60607-7052ChicagoILUSA\n",
"G Ali Mansoori \nDepartments of BioEngineering, Chemical Engineering and Physics\nUniversity of Illinois at Chicago\n60607-7052ChicagoILUSA\n"
] | [
"Department of Physics\nUniversity of Illinois at Chicago\n60607-7052ChicagoILUSA",
"Departments of BioEngineering, Chemical Engineering and Physics\nUniversity of Illinois at Chicago\n60607-7052ChicagoILUSA"
] | [
"Int. J. Mol. Sci"
] | We report self-assembly and phase transition behavior of lower diamondoid molecules and their primary derivatives using molecular dynamics (MD) simulation and density functional theory (DFT) calculations. Two lower diamondoids (adamantane and diamantane), three adamantane derivatives (amantadine, memantine and rimantadine) and two artificial molecules (ADM•Na and DIM•Na) are studied separately in 125-molecule simulation systems. We performed DFT calculations to optimize their molecular geometries and obtained atomic electronic charges for the corresponding MD simulation, by which we predicted self-assembly structures and simulation trajectories for the seven different diamondoids and derivatives. Our radial distribution function and structure factor studies showed clear phase transitions and self-assemblies for the seven diamondoids and derivatives. | 10.3390/ijms11010288 | null | 14,495 | 1805.11132 | 995f704d86396c0a4e079181aea226c3d7481cc7 |
Self-Assembly of Diamondoid Molecules and Derivatives (MD Simulations and DFT Calculations)
2010
Yong Xue
Department of Physics
University of Illinois at Chicago
60607-7052ChicagoILUSA
G Ali Mansoori
Departments of BioEngineering, Chemical Engineering and Physics
University of Illinois at Chicago
60607-7052ChicagoILUSA
Self-Assembly of Diamondoid Molecules and Derivatives (MD Simulations and DFT Calculations)
Int. J. Mol. Sci
11201010.3390/ijms11010288Received: 7 December 2009; in revised form: 10 January 2010 / Accepted: 17 January 2010 /International Journal of Molecular Sciences * Author to whom correspondence should be addressed; E-Mail: [email protected] functional theorydiamantanediamondoidsMD simulationmemantinenanotechnologyRDF, rimantadineself-assemblysimulation annealingstructure factor
We report self-assembly and phase transition behavior of lower diamondoid molecules and their primary derivatives using molecular dynamics (MD) simulation and density functional theory (DFT) calculations. Two lower diamondoids (adamantane and diamantane), three adamantane derivatives (amantadine, memantine and rimantadine) and two artificial molecules (ADM•Na and DIM•Na) are studied separately in 125-molecule simulation systems. We performed DFT calculations to optimize their molecular geometries and obtained atomic electronic charges for the corresponding MD simulation, by which we predicted self-assembly structures and simulation trajectories for the seven different diamondoids and derivatives. Our radial distribution function and structure factor studies showed clear phase transitions and self-assemblies for the seven diamondoids and derivatives.
Introduction
Diamondoid molecules (which are cage-hydrocarbons) and their derivatives have been recognized as molecular building blocks in nanotechnology [1][2][3][4][5][6][7][8][9][10]. They have been drawn more and more
OPEN ACCESS
researchers' attentions to their highly symmetrical and strain free structures, controllable nanostructure characteristics, non-toxicity and their applications in producing variety of nanostructure shapes, in molecular manufacturing, in nanotechnology and in MEMS [6,8]. It is important and necessary to study self-assembly of these molecules in order to obtain reference data, such as temperature, pressure, structure factor, bonding properties, etc. for application in nanotechnology e.g., building molecular electronic devices.
Two lower diamondoids (adamantane and diamantane), three adamantane derivatives (amantadine, memantine and rimantadine) and two artificial molecules (substituting one hydrogen ion in adamantane and diamantine with one sodium ion: ADM•Na and DIM•Na) are studied in this report. We classified them into three groups, as shown in Table 1. Table 1. Molecular formulas and 3-D structures of Adamantane, Diamantane, Memantine, Rimantadine, Amantadine, Optimized ADM•Na and Optimized DIM•Na molecules. In these figures black spheres represent "-C", whites represent "-H", Blues represent "-N" and purples represent "-Na".
Group 1 Group 2 Group 3
Adamantane Diamantane Amantadine Rimantadine Memantine Optimized
ADM•Na
Optimized DIM•Na
C 10 H 16 C 14 H 20 C 10 H 17 N C 11 H 20 N C 12 H 21 N C 10 H 15 Na C 14 H 19 Na
Group 1: Adamantane and Diamantane, the lowest two diamondoids. Due to their six or more linking groups, have found major applications as templates and as molecular building blocks in nanotechnology, polymer synthesis, drug delivery, drug targeting, DNA-directed assembly, DNA-amino acid nanostructure formation, and host-guest chemistry [1][2][3][4][5][6][7][8][9][10]. However these diamondoids do not have good electronic properties which are necessary for building molecular electronics, but some of their derivatives do.
Group 2: Amantadine, Memantine and Rimantadine, the three derivatives of adamantane, have medical applications as antiviral agents and due to their amino groups, they could be treated as molecular semiconductors [8,9].
Group 3: ADM•Na, DIM•Na, the two artificial molecules, substituting one hydrogen ion in adamantane and diamantane with a sodium ion could have potential applications in NEMS and MEMS [8][9][10]. This report is aimed at studying the self-assembly and phase transition properties of these seven diamondoids and derivatives for further study of their structures and possibly building molecular electronic devices with them.
We first performed density functional theory (DFT) calculations to optimize initial geometry structures of these seven diamondoids and derivatives and obtained their atomic electronic charges [11][12][13][14][15][16]. Then we performed molecular dynamics (MD) simulation in the study of their self-assembly behaviors. MD simulation methods have been broadly used in studying dynamics of molecules, in spite of its classic approximation [17][18][19][20][21]. As typical plastic crystal, or say molecular crystal, the phase transition behaviors of adamantane molecules have been studied by other researchers using MD simulation method [22,23]. Those studies mainly have focused on the transition of the adamantane from FCC to BCC crystal structure or vice versa, and both of these are in the solid phase. Self-assembly, however, is a transition process from a rather random condition of molecules to an ordered state [1]. In the present case of our interest, this process is similar to condensation transition from gas state to liquid state, and then freezing transition from liquid state to solid state, which is a border range of phase transitions. For this reason we chose the Optimized Potentials for Liquid Simulations All Atom (OPLS-AA) force field in our MD simulations. OPLS-AA force field has shown good results not just in the gas and liquid states but also in crystalline phases in a large range of temperatures [17]; moreover, OPLS-AA is a reliable force field for carbohydrates [18] and as a result, obviously, for hydrocarbons. In the OPLS-AA force field, the non-bonded interactions are represented by the general Lennard-Jones plus Columbic form as shown by Equation 1, where φ ij is the interaction energy of two sites for both intermolecular and intramolecular non-bonded cases: The combining rules used along with this equation are σ ij = (σ ii σj j ) 1/2 and ε ij = (ε ii εj j ) 1/2 . For intermolecular interactions (when i, j corresponds to different molecules) k ij = 1.0. While for intramolecular non-bonded interactions between all pairs of atoms (i < j) separated by three or more bonds, the 1, 4-interactions are scaled down by k ij = 0.5.
Our MD simulations are mainly performed with the use of the Gromacs package [19][20][21] and in order to perform simulations involving small molecules, we added to Gromacs the necessary ITFs "included topology files" for the seven different diamondoids and derivatives we studied. Moreover, Gromacs package is capable enough to accept and apply OPLS-AA force field for MD simulations as well as the fact that it is a fast package with flexible characteristics.
We report a brief summary of our MD simulation strategy: vacuum simulation strategy was used in the first step to determine how many molecules were suitable for our simulation and to obtain equilibrated stable structures. Then we built periodic boundary simulation box; and we used the fast Particle-Mesh Ewald (PME) electrostatics method for the preparation of gas state while we chose the cut-off technique for electrostatics in the self-assembly/freezing procedure.
To simulate the self-assembly (i.e., freezing transition), we applied the simulation annealing technique to gradually decease the system temperature. V-rescale temperature coupling was applied throughout and in all the simulations. This is a temperature-coupling method using velocity rescaling with a stochastic term [18]. The van der Waals force cut-off and neighbor-searching list distance were, both, set to 2.0 nm, which was large enough since above 1.4 nm the van der Waals force was negligible.
Furthermore, in order to obtain the geometries and atomic charges for the MD simulations of all the seven different diamondoids and derivatives we applied the DFT calculation which is a widely used method in quantum chemistry and electronic structure theory.
Density Functional Theory (DFT) Calculations
By applying the DFT calculations through Gaussian 03 package [15], we obtained the optimized initial structures of all the seven diamondoids and derivatives and also their atomic electronic charges for the MD simulations. The B3YLP exchange-correlation functional [12] method was chosen with cc-pVDZ basis [14], for all the molecules except for ADM•Na and DIM•Na for which 6-311 + G(d, p) basis [13] was used since sodium was not included in cc-pVDZ basis; NBO (Natural Bond Orbital) analysis [16] was added to calculate the atomic electronic charges which were used as reference to set the atomic charges of nitrogen and sodium atoms, i.e., −0.76 and 0.65, respectively (in electronic charge units: Coulomb). The atomic charges of carbon and hydrogen atoms were mainly opted from default OPLS-AA force field.
Molecular Dynamic (MD) Simulation Procedures and Results
The first step was to determine the number of molecules in the simulation box which was suitable to resemble an NVT ensemble. We performed short (20- Although these structures are stable, they may not be used for structural analysis, since the simulations are rather too short. We concluded that a 125-molecule system was big enough to represent the self-assembly behavior of larger systems, which also could save us calculation cost. Therefore, we chose 125 molecules in what is reported in the rest of this paper. From this point on, all the MD simulation systems involved 125 molecules for the seven different diamondoids and derivatives. In order to perform self-assembly simulations from gas state to liquid state and eventually solid state, we developed the following procedure for the seven diamondoids and derivatives:
(a) A MD simulation box with 5 × 5 × 5 = 125 molecules was built by the intrinsic tools in the Gromacs software, as shown in Figure 2a.
(b) Then we performed a short MD simulation by letting all the molecules relax in a vacuum at 100 K (as in Figure 1) to obtain the stable self-assembled structure, as shown in Figure 2b.
(c) We boxed those stable structures and set the distances from molecules to the boundaries of the simulation boxes to about +3 nm. In this manner we could build simulation systems with reasonable densities which were in the range of their gas states and also made the systems isolated from the adjacent simulation boxes under the periodic boundary conditions, as shown in Figure 2c.
(d) Longer (more than 1,000 picoseconds) equilibrating simulations were performed in the NVT ensembles, and we applied the PME method with the cut-off at 2.0 nm and high temperatures (in the range of 500-700 K) in order to make sure the entire system went to the gaseous state, i.e., all the molecules separated from each other and distributed randomly in the simulation box as shown in Figure 2d.
By applying the above four-step procedure we could equilibrate the system in a gas state. As a result, we eliminated the problem due to directly equilibrating the initially prepared molecular system, Figure 2a, at high temperatures. The equilibrated and stable-structure gaseous state, Figure 2d, was then ready for cooling down towards the self-assembly.
(e) The next step of the simulation was the cooling down the equilibrated and stable-structure gaseous state, Figure 2d, from the gas to the liquid state as shown in Figure 2e.
(f) Further cooling down of the system resulted in the complete self-assembly of all the molecules (to solid state) as shown in Figure 2f.
The snapshots of the gas-liquid-solid MD simulations for 125 molecules (stages d, e and f of Figure 2) of each of the seven diamondoids and derivatives (Adamantane, Diamantane, Amantadine, Rimantadine, Memantine, ADM•Na, DIM•Na) are reported in Figure 3. From these snapshots we can directly observe clear phase transitions for each kind of molecules, from the gaseous state to the liquid state, and then their aggregation into a highly condensed (self-assembled) state.
In order to find the equilibrium configuration of the collection of the 125 molecules at every given temperature we used the simulation annealing procedure [1,19]. Every change of 1 K occurred within 10 picoseconds (5,000 time-steps and each time-step was 0.002 ps). With these settings we could observe the self-assembly behavior in the cooling step.
It should be mentioned that we applied the cut-off of 2.0 nm for the electrostatics instead of PME, which was used in step d (Figure 2), in order to: (i). make sure all the interactions among molecules are considered, even beyond 1.4 nm, which is the custom cut-off setting, since after 1.4 nm there is no significant van der waals force; (ii). thus save the computation time at the mean time, since PME requires more computation than the cut-off method.
According to Figure 3 adamantane, diamantane, ADM•Na and DIM•Na form ordered (crystalline) condensed states. However, amantadine, rimantadine and memantine, while they self-assemble, they do not seem to form clear ordered (crystalline) condensed states. We further produced the hydrogen bonds locations of the same self-assembled snapshots of amantadine, memantine and rimantadine as shown in Figure 4.
According to Figure 4 we do not observe any ordered format for the location of hydrogen-bonds, which is an indication of the non-crystalline self-assembled states of amantadine, rimantadine and memantine. Obviously the existence of partial hydrogen-bonds between these molecules is the reason for the lack of a clear crystalline state in their self-assemblies. We also produced the radial distribution functions (RDFs) and structure factors (SFs) of the seven different diamondoids and derivatives as presented and discussed below. Study of the RDFs and SFs also reveal further about these phase transition features as discussed below. The RDFs we studied and report here are for the centers of the geometry /mass of molecules.
Adamantane and ADM•Na
The main feature of adamantane is that at low temperatures we can observe ordered crystal structures (the top-right image in Figures 3), which matches the previous experimental and theoretical studies [22]. From the RDF and SF graphs of adamantane at different temperatures ( Figure 5) we can observe gas, liquid and solid characteristics.
In the RDF figures of adamantane, higher and sharper peaks can be observed as the temperature decreases and for the SF graphs, the intensity increases as phase transits from gas to liquid and to solid state. We observed similar features in the RDF and SF figures of of ADM•Na as shown in Figure 6. The phase transition temperatures for ADM•Na are higher than those of adamantane. However, this is attributed to the presence of -Na ion affecting the molecular interactions among adamantanes, i.e., -Na ion in the ADM•Na structure causes stronger bonding than that of the respective -H ion in adamantane. We may also conclude that while the higher phase transition temperatures in ADM•Na, compared to those of adamantane, are due to -Na ion, the geometric structure is determined by the structure of adamantane.
Diamantane and DIM•Na
Diamantane molecules can self-assemble to a certain type of solid state structure which can be observed from simulation snapshots (Figure 3), however, its self-assembled structure is not as neat as the crystal structure of adamantane. In Figure 7 we report the RDF and structure factor of diamantane in various phases.
Figure 7.
Radial distribution functions (left) and structure factors (right) of diamantane at 500 K, 400 K and 300 K for its gas, liquid and solid states, respectively.
From the RDF and structure factor of diamantane, Figure 7, we may also conclude the following: At low temperatures diamantane does not have features of neat crystal structure. DIM•Na shows similar relationship to diamantane as ADM•Na to adamantane, i.e., higher phase transition temperatures due to sodium, while similar self-assembled crystal structures as diamantane molecules (See DIM•Na snapshots in Figure 3 and DIM•Na RDFs in and SFs in Figure 8).
Amantadine, Rimantadine and Memantine
These three adamantane derivatives self-assemble at higher temperatures compared with adamantane, but they do not seem to have well-organized self-assembled structures as shown in their snapshots in Figure 4. The reasoning for the special self-assembled structures of amantadine, rimantadine and memantine is as follows:
(i). Since there are nitrogen ions in the structure of amantadine, rimantadine and memantine, which makes their attractive intermolecular forces much larger than that of adamantane, higher temperatures should be applied to these systems in order to obtain initial gas state structures in the step d of Figure 2.
(ii). During the cooling down process of amantadine, rimantadine and memantine, hydrogen-bonds are formed which, as expected, increases their phase transition temperatures compared to that of adamantane.
As it is observed from their self-assembled structures (snapshots in Figure 4) amantadine, rimantadine and memantine form certain types of self-assembled structure. However, those structures do not seem to be ordered as was in the case of adamantane, i.e., they have no apparent ordered crystalline structure. There may be two main reasons for this: (a). the -NH 2 and -CH 3 groups present in these molecules break the geometrical symmetry of adamantane; (b). we observed that the hydrogen bonds, due to -NH 2 groups are randomly distributed in the bulk structures which makes the entire structures far from an ordered one. In Figures 8-11 we report the RDFs and SFs of amantadine, rimantadine and memantine for their gas, liquid and self-assembled states.
According to Figures 9-11 the self-assembled RDFs and SFs analysis of amantadine, rimantadine and memantine show that for most part there is less obvious ordered self-assembled features, i.e., due to lack of sharp peaks as was the case in adamantane. The liquid-state RDF graphs for amantadine, rimantadine and memantine also, for most part, have less obvious liquid-state features either. We also performed calculations of the hydrogen-bonds saturation as a function of temperature (as shown in Figure 12) as well as hydrogen-bonds distance distribution (as shown in Figure 13-left) and hydrogen-bonds angle distribution (as shown in Figure 13-right) for amantadine, rimantadine and memantine.
According to Figure 12 in the process of phase transition and self-assembly of amantadine, rimantadine and memantine the number of their hydrogen-bonds will reach saturation limit at low temperatures.
From the hydrogen-bond distance-distribution graphs of amantadine, rimantadine and memantine at 60 K ( Figure 13-left), we observe several peaks with their highest peaks at about 0.3 nm. Figure 12. The number of hydrogen bonds for amantadine, memantine and rimantadine vs. temperature. As temperature decreases the numbers of hydrogen-bonds increase, and tend to maximum numbers for the three derivatives. Figure 13. Hydrogen-bond distance distribution (left) and hydrogen-bond angle distribution (right) at 60 K for amantadine, rimantadine and memantine. The most possible hydrogen-bond lengths are around 0.3 nm for the three molecules. The hydrogen-bond angles seem randomly distributed.
From the hydrogen-bond angle distributions graphs of amantadine, rimantadine and memantine at 60 K ( Figure 13-right), we observe that the hydrogen-bond angles are quite randomly distributed for all the three molecules. We already demonstrated the hydrogen-bonds locations and orientations of this group at 50 K ( Figure 4) which also showed no orderly features either. Both Figures 4 and 13 indicate that these structures are not orderly self-assembled structures contrary to the adamantane orderly self-assembled state.
In conclusion these three hydrogen-bonded adamantane derivatives just self-assemble at particular temperatures, but may not form well-organized self-assembled crystal structures. The reason is that the geometry structures of these molecules are not symmetric as was the case for adamantane; and their -NH 2 and -CH 3 segments, make them unable to pack orderly as adamantane does.
In Figures 14 and 15 we report the self-assembled (solid-state) radial distribution functions and structure factors, respectively, of all the seven diamondoids and derivatives with uniform coordinates scales for the purpose of their collective comparison. According to Figures 14 and 15 we can observe that adamantane and ADM•Na have more sharp peaks than the other five, which is the indication of their orderly self-assembled structures. While other five molecules also show self-assembled characteristic peaks they have less number of such sharp peaks than adamantane and ADM•Na. These results match the image observation of simulations (Figure 3), i.e., adamantane and ADM•Na have more orderly structures in their self-assembled states due to their molecular symmetry, which thus proves that self-assembly of those molecules are structure-dependent.
Conclusion
We have performed a detailed molecular dynamics study of the self-assembly process of seven different diamondoids and derivatives due to temperature variations. From the MD simulation study of the seven diamondoids and derivatives, we may conclude the following: (1) The nature of self-assembly in these molecules is a structure-dependent phenomenon. (2) Final self-assembly structures depend on the different bonding types present in the molecular and intramolecular interactions of these various molecules. (3) The artificial molecules (ADM•Na., DIM•Na) still hold neat crystal structures. Although -Na ion increases the phase transition temperature, as does -NH 2 in amantadine, rimantadine and memantine. (4) To a large extent the structural features of diamondoids are retained in ADM•Na and DIM•Na. The reasons for the latter might be that: A. The -Na ion has less topology effect than does the -NH 2 . B. There is no hydrogen-bonding in the structures of ADM•Na and DIM•Na; therefore they can aggregate to ordered self-assembled structures. This feature is very promising, since it allows us to build orderly-shaped self-assemblies suitable for NEMS and MEMS.
40 picoseconds) MD simulations of different numbers of adamantane molecules, (i.e., 8, 27, 64, 125, 216, 343, 512, 729 molecules) in vacuum with the temperature set at 100 K, and we successfully obtained all the stable structures as shown in Figure 1.
Figure 1 .
1Snapshots from MD simulations using various numbers of adamantine molecules in vacuum with the temperature set at 100 K.
Figure 2 .
2Stages of simulation procedure: (a) Initial 5 × 5 × 5 MD simulation box; (b) Molecules relaxed in vacuum at 100 K after a short MD simulation; (c) Boxed simulation system with overall gas density; (d) Gas phase as a result of equilibrating simulation in the NVT ensemble; (e) The liquid state; (f) Final self-assembly of molecules to solid state.
Figure 3 .
3Self-assembly snapshots (left to right) of 125 molecules of the seven diamondoids and derivatives (from top: Adamantane, Diamantane, Amantadine, Rimantadine, Memantine, ADM•Na., DIM•Na) as the temperature [in K shown on every snapshot] is decreased.
Figure 3 .
3Cont.
Figure 4 .
4MD snapshots of (from top to bottom) Amantadine, Rimantadine and Memantine at 50 K and their hydrogen bonds locations and orientations.
Figure 5 .
5Radial distribution functions (left) and structure factors (right) of adamantane at 450 K, 300 K and 150 K for its gas, liquid and solid states, respectively.
Figure 6 .
6Radial distribution functions (left) and structure factors (right) of ADM•Na at 600 K, 450 K and 200 K for its gas, liquid and solid states, respectively.
Figure 8 .
8Radial distribution functions (left) and structure factors (right) of DIM•Na at 600 K, 500 K and 400 K for its gas, liquid and solid states, respectively.
Figure 9 .
9Radial distribution functions (left) and structure factors (right) of amantadine at 450 K, 370 K and 300 K for its gas, liquid and solid states, respectively.
Figure 10 .
10Radial distribution functions (left) and structure factors (right) of rimantadine at 500 K, 430 K and 250 K for its gas, liquid and solid states, respectively.
Figure 11 .
11Radial distribution functions (left) and structure factors (right) of Memantine at 500 K, 430 K and 250 K for its gas, liquid and solid states, respectively.
Figure 14 .
14Radial distribution functions of the seven diamondoids and derivatives (from left: Adamantane, Diamantane, Amantadine, Rimantadine, Memantine, ADM•Na, DIM• Na) in the self-assembled (solid) state.
Figure 15 .
15Structure factors of the seven diamondoids and derivatives (from left: Adamantane, Diamantane, Amantadine, Rimantadine, Memantine, ADM•Na., DIM•Na) in the self-assembled (solid) state.
© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
AcknowledgmentsWe are grateful to the computation facility of the University of Illinois at Chicago. All the MD simulation results are visualized by the VMD package[24].
Principles of Nanotechnology-Molecular-Based Study of Condensed Matter in Small Systems. G A Mansoori, World Scientific Pub. CoHackensack, NJ, USAMansoori, G.A. Principles of Nanotechnology-Molecular-Based Study of Condensed Matter in Small Systems; World Scientific Pub. Co.: Hackensack, NJ, USA, 2005.
Diamondoid molecules. G A Mansoori, Adv. Chem. Phys. 136Mansoori, G.A. Diamondoid molecules. Adv. Chem. Phys. 2007, 136, 207-258.
First-principles simulation of the interaction between adamantane and an atomic-force microscope tip. G P Zhang, T F George, L Assoufid, G A Mansoori, Phys. Rev. B. 0354137Zhang, G.P.; George, T.F.; Assoufid, L.; Mansoori, G.A. First-principles simulation of the interaction between adamantane and an atomic-force microscope tip. Phys. Rev. B 2007, 75, 035413:1-035413:7.
Diamondoids-DNA nanoarchitecture: From nanomodules design to self-assembly. H Ramezani, G A Mansoori, M R Saberi, J. Comput. Theory Nanosci. 4Ramezani, H.; Mansoori, G.A.; Saberi, M.R. Diamondoids-DNA nanoarchitecture: From nanomodules design to self-assembly. J. Comput. Theory Nanosci. 2007, 4, 96-10.
G A Mansoori, T F George, G P Zhang, L Assoufid, Molecular Building Blocks for Nanotechnology: From Diamondoids to Nanoscale Materials and Applications. New York, NY, USASpringer109Mansoori, G.A.; George, T.F.; Zhang, G.P.; Assoufid, L. Molecular Building Blocks for Nanotechnology: From Diamondoids to Nanoscale Materials and Applications (Topics in Applied Physics); Springer: New York, NY, USA, 2007; Volume 109, pp. 44-71.
Molecular Building Blocks for Nanotechnology: Diamondoids as Molecular Building Blocks for Nanotechnology. H Ramezani, G A Mansoori, Topics in Applied Physics. 109SpringerRamezani, H; Mansoori, G.A. Molecular Building Blocks for Nanotechnology: Diamondoids as Molecular Building Blocks for Nanotechnology (Topics in Applied Physics); Springer: New York, NY, USA, 2007; Volume 109, pp. 44-71.
Diamondoids and DNA nanotechnologies. H Ramezani, M R Saberi, G A Mansoori, Int. J. Nanosci. Nanotechnol. 3Ramezani, H.; Saberi, M.R.; Mansoori, G.A. Diamondoids and DNA nanotechnologies. Int. J. Nanosci. Nanotechnol. 2007, 3, 21-35.
Quantum Conductance and Electronic Properties of Lower Diamondoids and their Derivatives. Y Xue, G A Mansoori, Int. J. Nanosci. 7Xue, Y.; Mansoori, G.A. Quantum Conductance and Electronic Properties of Lower Diamondoids and their Derivatives. Int. J. Nanosci. 2008, 7, 63-72.
Opto-electronic properties of adamantane and hydrogen-terminated sila-and germa-adamantane: A comparative study. F Marsusi, K Mirabbaszadeh, G A Mansoori, 41Marsusi, F.; Mirabbaszadeh, K.; Mansoori, G.A. Opto-electronic properties of adamantane and hydrogen-terminated sila-and germa-adamantane: A comparative study. Phys. E 2009, 41, 1151-1156.
Structure and opto-electronic behavior of diamondoids, with applications as mems and at the nanoscale level. G A Mansoori, T F George, G P Zhang, L Assoufid, Progress Nanotechnology Research. New York, NY, USASpringerMansoori, G.A.; George, T.F.; Zhang, G.P.; Assoufid, L. Structure and opto-electronic behavior of diamondoids, with applications as mems and at the nanoscale level. In Progress Nanotechnology Research; Springer: New York, NY, USA, 2009; Chapter 1, pp. 1-19.
Self-consistent equations including exchange and correlation effects. W Kohn, L Sham, Phys. Rev. 140Kohn, W.; Sham, L. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140, A1133-A1138.
Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. C Lee, W Yang, R G Parr, Phys. Rev. B. 37Lee, C.; Yang, W.; Parr, R.G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988, B37, 785-789.
Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. D E Woon, T H Dunning, Jr, J. Chem. Phys. 98Woon, D.E.; Dunning, T.H., Jr. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. J. Chem. Phys. 1993, 98, 1358-1371.
Extension of Gaussian-2 theory to molecules containing third-row atoms Ga-Kr. L A Curtiss, M P Mcgrath, J.-P Blaudeau, N E Davis, R C Binning, Jr, L Radom, J. Chem. Phys. 103Curtiss, L.A.; McGrath, M.P.; Blaudeau, J.-P.; Davis, N.E.; Binning, R.C., Jr.; Radom, L. Extension of Gaussian-2 theory to molecules containing third-row atoms Ga-Kr. J. Chem. Phys. 1995, 103, 6104-6113.
Gaussian 03, Revision D.01. M J Frisch, G W Trucks, H B Schlegel, G E Scuseria, M A Robb, J R Cheeseman, J A Montgomery, Jr, T Vreven, K N Kudin, M Burant, P M W Gill, B Johnson, W Chen, M W Wong, C Gonzalez, J A Pople, Gaussian IncWallingford, CT, USAFrisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Montgomery, J.A., Jr.; Vreven, T.; Kudin, K.N.; Burant, M.; Gill, P.M.W.; Johnson, B.; Chen, W.; Wong, M.W.; Gonzalez, C.; Pople, J.A. Gaussian 03, Revision D.01; Gaussian Inc.: Wallingford, CT, USA, 2004.
Analysis of the geometry of the hydroxyl radical by the ldquo. J E Carpenter, F J Weinhold, Mol. Struct. (Theochem.). 169Carpenter, J.E.; Weinhold, F.J. Analysis of the geometry of the hydroxyl radical by the ldquo. Mol. Struct. (Theochem.) 1988, 169, 41-62.
Development and testing of the opls all-atom force field on conformational energetics and properties of organic liquids. W L Jorgensen, D S Maxwell, J Tirado-Rives, J. Am. Chem. Soc. 118Jorgensen, W.L.; Maxwell, D.S.; Tirado-Rives, J. Development and testing of the opls all-atom force field on conformational energetics and properties of organic liquids. J. Am. Chem. Soc. 1996, 118, 11225-11236.
OPLS all-atom force field for carbohydrates. W Damm, A Frontera, J Tirado-Rives, W L Jorgensen, J. Comput. Chem. 18Damm, W.; Frontera, A.; Tirado-Rives, J.; Jorgensen, W.L. OPLS all-atom force field for carbohydrates. J. Comput. Chem. 1997, 18, 1955-1970.
0: A package for molecular simulation and trajectory analysis. E Lindahl, B Hess, D Van Der Spoel, J. Mol. Mod. 7Lindahl, E.; Hess, B.; van der Spoel, D. GROMACS 3.0: A package for molecular simulation and trajectory analysis. J. Mol. Mod. 2001, 7, 306-317.
GROMACS: Fast, flexible and free. D Van Der Spoel, E Lindahl, B Hess, G Groenhof, A E Mark, H J Berendsen, J. Comp. Chem. 26van der Spoel, D.; Lindahl, E.; Hess, B.; Groenhof, G..; Mark, A.E.; Berendsen, H.J.C. GROMACS: Fast, flexible and free. J. Comp. Chem. 2005, 26, 1701-1718.
Algorithms for highly efficient, load-balanced, and scalable molecular simulation. B Hess, C Kutzner, D Van Der Spoel, E Lindahl, Gromacs, J. Chem. Theory Comput. 4Hess, B.; Kutzner, C.; van der Spoel, D.; Lindahl, E GROMACS 4: Algorithms for highly efficient, load-balanced, and scalable molecular simulation. J. Chem. Theory Comput. 2008, 4, 435-447.
Molecular dynamics simulations of the order-disorder phase transition in adamantane. D W Greig, G S Pawley, Mol. Phys. 89Greig, D.W.; Pawley, G.S. Molecular dynamics simulations of the order-disorder phase transition in adamantane. Mol. Phys. 1996, 89, 677-689.
Structural transition on cooling of plastic adamantane: A molecular-dynamics study. G Ciccotti, M Ferrario, E Memeo, M Meyer, Phys. Rev. Lett. 59Ciccotti, G.; Ferrario, M.; Memeo, E.; Meyer, M. Structural transition on cooling of plastic adamantane: A molecular-dynamics study. Phys. Rev. Lett. 1987, 59, 2574-2577.
VMD-visual molecular dynamics. W Humphrey, A Dalke, K Schulten, J. Mol. Graph. 14Humphrey, W.; Dalke, A.; Schulten, K. VMD-visual molecular dynamics. J. Mol. Graph. 1996, 14, 33-38.
| [] |
[
"Narrow Tetraquarks at Large N",
"Narrow Tetraquarks at Large N"
] | [
"Marc Knecht \nCentre de Physique Théorique CNRS/Aix-Marseille Univ./Univ. du Sud Toulon-Var (UMR 7332\nCNRS-Luminy Case 907\n13288, Cedex 9MarseilleFrance\n",
"Santiago Peris \nDepartment of Physics\nUniversitat Autònoma de Barcelona\n08193BarcelonaSpain\n"
] | [
"Centre de Physique Théorique CNRS/Aix-Marseille Univ./Univ. du Sud Toulon-Var (UMR 7332\nCNRS-Luminy Case 907\n13288, Cedex 9MarseilleFrance",
"Department of Physics\nUniversitat Autònoma de Barcelona\n08193BarcelonaSpain"
] | [] | Following a recent suggestion by Weinberg, we use the large-N expansion in QCD to discuss the decay amplitudes of tetraquarks into ordinary mesons as well as their mixing properties. We find that the flavor structure of the tetraquark is a crucial ingredient to determine both this mixing as well as the decays. Although in some cases tetraquarks should be expected to be as narrow as ordinary mesons, they may get to be even narrower, depending on this flavor structure. | 10.1103/physrevd.88.036016 | [
"https://arxiv.org/pdf/1307.1273v1.pdf"
] | 118,388,961 | 1307.1273 | 376dbfe1913e7dd6edcfa447364f5286e5c12084 |
Narrow Tetraquarks at Large N
4 Jul 2013
Marc Knecht
Centre de Physique Théorique CNRS/Aix-Marseille Univ./Univ. du Sud Toulon-Var (UMR 7332
CNRS-Luminy Case 907
13288, Cedex 9MarseilleFrance
Santiago Peris
Department of Physics
Universitat Autònoma de Barcelona
08193BarcelonaSpain
Narrow Tetraquarks at Large N
4 Jul 2013numbers: 1115Pg1238Lg1325Jx1440Rt
Following a recent suggestion by Weinberg, we use the large-N expansion in QCD to discuss the decay amplitudes of tetraquarks into ordinary mesons as well as their mixing properties. We find that the flavor structure of the tetraquark is a crucial ingredient to determine both this mixing as well as the decays. Although in some cases tetraquarks should be expected to be as narrow as ordinary mesons, they may get to be even narrower, depending on this flavor structure.
PACS numbers: 11.15.Pg, 12.38.Lg, 13.25.Jx, 14.40.Rt
In a recent paper, Weinberg [1] has pointed out that tetraquark mesons (i.e. those formed by two quarks and two antiquarks), if they survive the large-N limit [2], would have decay rates proportional to 1/N and would, therefore, be as narrow as ordinary mesons. In this paper we would like to point out that the flavor structure in a tetraquark meson is crucial to determine the behavior of its decay amplitude at large N and, although Weinberg's result is generically true, there are tetraquark mesons for which there is an extra suppression resulting in a decay width which goes as 1/N 2 . Complications like the effects of mixing between ordinary mesons and tetraquarks need to be discussed in general, although they are not always an essential ingredient to reach this conclusion. N Q Q † Figure 1: Type-A diagram. In this figure, and all the ones that follow, it will be always understood that any number of gluon lines not changing the large-N behavior of the diagram should be added. This diagram is of order N , and this will be explicitly depicted with an "N " in the diagram.
Let us start, following Ref. [1], by considering tetraquark interpolating fields as
Q(x) α;β AB;CD = B α AB (x) B β CD (x) (1) where B α AB (x) = a q a A (x)Γ α q a B (x) ,(2)
and q a A (x) is the quark field with flavor A and color a (in SU(N)) and Γ α are colorless matrices containing spin information. We will always assume that the flavor indices A, B in these quark bilinears are different, so that the vacuum expectation value B α AB (x) 0 identically vanishes [3]. This simplifies the following discussion somewhat. Since the flavor structure in the first bilinear is such that A = B, there are only three nontrivial possibilities for the other bilinear q C q D (note that C = D as well): either C = B, or D = B or, finally, A = B = C = D, i.e. all the flavor indices are different 1 . These three possibilities will determine the possible quark contractions in the relevant Green's function, which in turn will determine its large-N behavior. Before entering the discussion of the different possible decay amplitudes for a tetraquark into ordinary mesons, it is of the utmost importance to identify the tetraquark as a physical state in a Green's function, e.g. as a pole in the correlator 2
√ N √ N N 0 N 0 + + √ N √ N √ N √ NQ(x)Q † (0) 0 .(3)
Therefore, as emphasized above, we now need to look at all the possible quark contractions in this correlator, and this is where the flavor structure comes in. We will call "type-A" diagram a diagram like that in Fig. 1, where the flavor in the tetraquark operator Q(x) is of the form q A q B q B q C which allows an internal quark contraction between the two quarks with flavor index B. This diagram is of order N. 1 The cases A = C and A = D are analogous to the cases B = D and B = C (respectively) and need not be discussed separately. 2 In the following, we will suppress all the indices in Q(x) α;β AB;CD for ease of notation unless required by clarity in the discussion. fig. 2 one finds that the amplitude for the operator Q to create this tetraquark is of order N 1/2 since this amplitude appears squared. This matching has assumed that no cancellations between the different contributions in fig. 2 takes place, as it is customarily done when following large-N reasoning 3 . Similarly, one also infers that the amplitude for Q to create an ordinary meson (second term in fig. 2) is also of order N 1/2 . Using these results, one finally obtains that the mixing between this tetraquark and an ordinary meson (first contribution in fig. 2) is of order N 0 , and it is, therefore, not suppressed in the large-N limit. Because of this, it may be very difficult to disentangle these tetraquarks from ordinary mesons, unless very precise information N Q B on the position of the tetraquark pole is known. Furthermore, since this tetraquark is of the form q A q B q B q C , with the flavor index B contracted, it will fill out the same flavor representation as ordinary q A q C mesons, e.g. a whole nonet in flavour SU (3).
√ N 1/ √ N √ N √ N√ N 1/ √ N N 0 √ N √ N
In order to determine the width of a type-A tetraquark into two ordinary mesons, one may look at the three-point function depicted in fig. 3
4 Q(x) q(y)q(y) q(z)q(z) 0 ,(4)
which is of order N, and look for the tetraquark pole. Its interpretation in terms of physical states is done in fig. 4. One concludes from this figure that the vertex between this tetraquark and two ordinary qq mesons is of order N −1/2 (depicted as an empty circle in fig. 4). Therefore, the decay width of a type-A tetraquark is of order 1/N, i.e. as narrow as an ordinary meson. This is the result obtained in ref. [1]. Fig. 5 helps one understand why a type-A tetraquark behaves like an ordinary meson in its decay: one may reinterpret this process as the mixing of the tetraquark with an ordinary meson (with an unsuppressed amplitude of order N 0 ), plus the subsequent decay of the ordinary meson. The consistency of the conclusions drawn before may be cross-checked by looking 4 The two qq mesons must of course have the right flavor structure to allow the contractions shown in this figure. at the mixed correlator in fig. 6, which is or order N, and its interpretation in terms of physical states, as is done in fig. 7. From the first term in fig. 7, one again obtains that mixing between a tetraquark and an ordinary meson is of order N 0 . Tetraquarks with the flavor structure q A q B q C q B (recall that A = B and C = B), do not allow the contractions of fig. 1. An example of this tetraquark is udsd. These tetraquarks will be called of "type A ′ " because, after a simple rerun of the arguments presented for the previous diagrams in figs. 1-4, the same final conclusion about the behavior of the width as for type-A tetraquarks is reached, namely of order 1/N. An important difference is, however, that these tetraquarks cannot mix with ordinary mesons because any intermediate state always contains 4 quarks, as fig. 8 shows. So, the first two contributions in fig. 2 are absent in this case. We now come to the second type of tetraquarks, which we will call "type B", i.e. with all flavor indices different. This fact prevents any internal quark contraction within the tetraquark operator Q. An example for this type of tetraquark is udcs. The corresponding diagram is depicted in fig. 9. In order to make this contribution connected, a minimum number of two gluons is necessary between the two quark loops, N 0 N 0 Figure 10: Interpretation of the diagram in fig. 9 in terms of intermediate states.
√ N √ N N 0 + + √ N √ N √ N √ N
otherwise the diagram factorizes into two qq meson propagators which can contain no tetraquark pole at leading order (see also below). Of course, in the diagram in fig. 9, any number of gluons should be added as long as they do not alter the large-N counting, which is of order N 0 . This diagram may also be present in the case of the type-A and A ′ tetraquarks, but its contribution remains subleading which is why we have not discussed it until now. When all the four flavor indices in Q are different, the contribution in fig. 9 becomes leading and needs to be discussed separately.
Any intermediate state in fig. 9 always contains four quarks, which means that there is no mixing with ordinary mesons. Since the diagram is of order N 0 , the amplitude for Q to create one of these tetraquarks is also of order N 0 (see fig. 10). Fig. 11 shows the three-point function describing the decay of this tetraquark into ordinary mesons, which is of order N 0 as well. Its interpretation in terms of physical intermediate states in given in fig. 12. One concludes therefore that the width for this type-B tetraquark is of order 1/N 2 and is narrower than the type-A, and A ′ tetraquarks discussed previously. This concludes our discussion of the tetraquarks which may be excited from the vacuum by the action of the four-quark operator Q. We would like to point out, however, about the existence of another logical possibility. This consists of a qq bilinear exciting a tetraquark meson through mixing, like in fig. 13, which shows a diagram of order N 0 . We will call these tetraquarks "type C". Clearly, a vertical cut of the diagram may contain a four quark intermediate state (Fig. 14 shows all the possible intermediate states). Can this state be a tetraquark, i.e. can this singularity become a pole? Although it is true that the color flow in the diagram shows that the intermediate state can be split in the product of two qq color singlets, this does not imply that these two singlets necessarily have to become two separate meson states. We think it is not possible, on the sole basis of large-N counting arguments, to conclusively argue one way or the other without a more detailed dynamical knowledge of the QCD solution in this limit, but we see no reason that would forbid the presence of a tetraquark pole in fig. 13. If this is the case, these tetraquarks will have the flavor structure of the type B q A q B q B q C , i.e. they will also form a nonet, like type-A tetraquarks do. As to the decay properties of a type-C tetraquark, one may look at the three-point function depicted in fig. 15, which is of order N 0 , and its corresponding interpretation in terms of physical states in fig. 16. Since the vertex for this type of tetraquark to decay into ordinary mesons is of order 1/N, its width will be of order 1/N 2 , i.e. as narrow as type-B tetraquarks. Figure 14: Interpretation of the diagram in fig. 13 in terms of physical states.
√ N √ N 1/ √ N 1/ √ N + + N 0 1/ √ N √ N N 0 N 0
In conclusion, we have found that tetraquarks are narrow objects in the large-N limit [1]. They are at least as narrow as ordinary mesons but they may get to be even narrower if the flavor structure is the appropriate one. There have been long discussions in the literature trying to resolve the dichotomy between the description in terms of a four-quark state, or a two-meson molecule [5]. Our conclusion is that, if the large-N expansion is a good guidance, a tetraquark state should be narrow. If the state is broad, it is more likely to be a two-meson bound state (molecule) resulting from an infinite chain of 1/N-suppressed meson-meson interactions. This is plausibly the way states like the f 0 (500) may be formed [6]. Perhaps the most promising tetraquark states from a theory point of view are those we called type B, with all the quark flavors different. For these tetraquarks the complications brought about by mixing will not be an issue since they cannot mix with ordinary mesons and, furthermore, they are expected to be long-lived as their width goes like 1/N 2 . It would be very interesting to see if the lattice could reveal their existence [7]. On the phenomenology side, there is mounting evidence about the existence of mesons with a four-quark content [8]. It remains to be seen whether the large-N expansion can be helpful for explaining this phenomenology, and how it fares as compared to existing alternatives [9].
Figure 3 :
3Three-point function determining the decay of a type-A tetraquark into two ordinary mesons. A circled cross signifies the insertion of the qq operator.
Figure 4 :
4Interpretation in terms of physical states of the contributions in fig. 3. Symbols are like in fig. 2. An example of this type of tetraquark is udds. Cutting vertically through this diagram immediately reveals the presence of a potential four-quark intermediate state. Without knowing the solution to large-N QCD it is not possible to know whether this four-quark intermediate state really exists and forms the necessary pole in this Green's function. But if we assume that the state exists, large-N can be used to predict how narrow it is and how it mixes with ordinary qq mesons. One can see in fig. 2 how to interpret the possible contributions to this Green's function from all the possible intermediate states which are obtained by vertically cutting the diagram. This figure shows the large-N behavior of the different contributions, as determined by
Figure 5 :
5Reinterpretation in terms of mixing between a tetraquark and an ordinary meson of the decay in fig. 3. matching the final value of N obtained in the diagram of fig. 1. For instance, looking at the last term in
Figure 6 :
6Mixed correlator Q(x)q(y)q(y) 0 .
Figure 7 :
7Contributions from the intermediate physical states to the mixed correlator in fig. 6.
Figure 8 :
8Diagram for the two-point correlator of a tetraquark operator of type A ′ .
Figure 9 :
9Diagram for the two-point correlator of a tetraquark operator of type B. A minimum of two gluons are necessary to make this contribution connected.
Figure 11 :
11Diagram for correlator governing the decay of a tetraquark operator of type B.
Figure 12 :
12Interpretation of the diagram infig. 11in terms of physical states.
Figure 13 :
13Diagram showing the presence of a four quark state in the qq two-point correlator.
Figure 15 :
15Diagram depicting the three-point function governing the decay of a tetraquark of type C.
Figure 16 :
16Interpretation of the diagram infig. 15in terms of physical states.
An exception to this rule is the mass of the η ′ meson.[4]
AcknowledgementsWe would like to thank M. Golterman and S. Weinberg for discussions. This work has been supported by FPA2011-25948, SGR2009-894, the Spanish Consolider-Ingenio 2010 Program CPAN (CSD2007-00042).
. S Weinberg, Phys. Rev. Lett. 110261601S. Weinberg, Phys. Rev. Lett. 110, 261601 (2013).
. G Hooft, Nucl. Phys. B. 72461G. 't Hooft, Nucl. Phys. B 72, 461 (1974);
. E Witten, Nucl. Phys. B. 16057E. Witten, Nucl. Phys. B 160, 57 (1979).
. S R Coleman, E Witten, Phys. Rev. Lett. 45100S. R. Coleman and E. Witten, Phys. Rev. Lett. 45, 100 (1980).
. E Witten, Nucl. Phys. B. 156269E. Witten, Nucl. Phys. B 156, 269 (1979).
. R L Jaffe, Nucl. Phys. A. 80425and references thereinR. L. Jaffe, Nucl. Phys. A 804, 25 (2008) and references therein.
. E G See, J R Pelaez, hep-ph/0411107Mod. Phys. Lett. A. 192879and references thereinSee, e.g., J. R. Pelaez, Mod. Phys. Lett. A 19, 2879 (2004) [hep-ph/0411107] and references therein.
. E G See, M Wagner, C Alexandrou, J O Daldrop, M D Brida, M Gravina, L Scorzato, C Urbach, C Wiese, arXiv:1212.1648PoS. ConfinementXhep-lat. and references thereinSee, e.g., M. Wagner, C. Alexandrou, J. O. Daldrop, M. D. Brida, M. Grav- ina, L. Scorzato, C. Urbach and C. Wiese, PoS ConfinementX , 108 (2012) [arXiv:1212.1648 [hep-lat]] and references therein.
. M Ablikim, BESIII CollaborationarXiv:1303.5949hep-exM. Ablikim et al. [ BESIII Collaboration], arXiv:1303.5949 [hep-ex];
. Z Q Liu, Belle CollaborationarXiv:1304.0121Phys. Rev. Lett. 110252002hep-ex. and references thereinZ. Q. Liu et al. [Belle Collaboration], Phys. Rev. Lett. 110, 252002 (2013) [arXiv:1304.0121 [hep-ex]] and references therein.
. L Maiani, F Piccinini, A D Polosa, V Riquer, hep-ph/0412098Phys. Rev. D. 7114028and references thereinL. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D 71, 014028 (2005) [hep-ph/0412098] and references therein.
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"Yuzuru Yoshii \nInstitute of Astronomy\nUniversity of Tokyo\n2-21-1 Osawa181-0015MitakaTokyo\n"
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"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyo",
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"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyo",
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"Department of Astronomy\nUniversity of Massachusetts\n01003AmherstMA",
"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyo",
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"Research Center for the Early Universe (WPI)\nUniversity of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan",
"Institute of Astronomy\nUniversity of Tokyo\n2-21-1 Osawa181-0015MitakaTokyo",
"The Graduate University for Advanced Studies (SOKENDAI)\n2-21-1 Osawa181-0015MitakaTokyo",
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"Institute of Astronomy\nUniversity of Tokyo\n2-21-1 Osawa181-0015MitakaTokyo",
"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyo",
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"Harvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMA",
"Institute of Astronomy\nUniversity of Tokyo\n2-21-1 Osawa181-0015MitakaTokyo"
] | [] | High resolution (∼ 0. ′′ 4) Atacama Large Millimeter/submillimeter Array (ALMA) Cycle 0 observations of HCO + (4-3) and HCN (4-3) toward a mid-stage infrared bright merger VV 114 have revealed compact nuclear (< 200 pc) and extended (∼ 3 − 4 kpc) dense gas distribution across the eastern part of the galaxy pair. We find significant enhancement of HCN (4-3) emission in an unresolved compact and broad (290 km s −1 ) component found in the eastern nucleus of VV114, and we suggest dense gas associated with the surrounding material around an Active Galactic Nucleus (AGN), with a mass upper limit of < ∼ 4 × 10 8 M ⊙ . The extended dense gas is distributed along a filamentary structure with resolved dense gas concentrations (∼ 230 pc; ∼ 10 6 M ⊙ ) separated by a mean projected distance of ∼ 600 pc, many of which are generally consistent with the location of star formation traced in Paα emission. Radiative transfer calculations suggest moderately dense (n H2 = 10 5 -10 6 cm −3 ) gas averaged over the entire emission region. These new ALMA observations demonstrate the strength of the dense gas tracers in identifying both the AGN and star formation activity in a galaxy merger, even in the most dust enshrouded environments in the local universe. | 10.1093/pasj/65.3.l7 | [
"https://arxiv.org/pdf/1305.4535v1.pdf"
] | 119,194,970 | 1305.4535 | 00f2f5eae8d6308137925fab9cde13f8b4c64428 |
Active Galactic Nucleus and Extended Starbursts in a Mid-stage Merger VV114
20 May 2013
Daisuke Iono
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyo
The Graduate University for Advanced Studies (SOKENDAI)
2-21-1 Osawa181-0015MitakaTokyo
Toshiki Saito
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyo
Department of Astronomy
School of Science
The University of Tokyo
7-3-1 Hongo, Bunkyo-ku133-0033Tokyo
Min S Yun
Department of Astronomy
University of Massachusetts
01003AmherstMA
Ryohei Kawabe
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyo
Joint ALMA Observatory
Alonso de Cordova 3107763-0355VitacuraSantiagoChile
Daniel Espada
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyo
Joint ALMA Observatory
Alonso de Cordova 3107763-0355VitacuraSantiagoChile
Yoshiaki Hagiwara
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyo
The Graduate University for Advanced Studies (SOKENDAI)
2-21-1 Osawa181-0015MitakaTokyo
Masatoshi Imanishi
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyo
The Graduate University for Advanced Studies (SOKENDAI)
2-21-1 Osawa181-0015MitakaTokyo
Takuma Izumi
Institute of Astronomy
University of Tokyo
2-21-1 Osawa181-0015MitakaTokyo
Kotaro Kohno
Institute of Astronomy
University of Tokyo
2-21-1 Osawa181-0015MitakaTokyo
Research Center for the Early Universe (WPI)
University of Tokyo
7-3-1 Hongo113-0033BunkyoTokyoJapan
Kentaro Motohara
Institute of Astronomy
University of Tokyo
2-21-1 Osawa181-0015MitakaTokyo
Koichiro Nakanishi
The Graduate University for Advanced Studies (SOKENDAI)
2-21-1 Osawa181-0015MitakaTokyo
Joint ALMA Observatory
Alonso de Cordova 3107763-0355VitacuraSantiagoChile
Hajime Sugai
Kavli Institute for the Physics and Mathematics of the Universe
The Univ. of Tokyo
Ken Tateuchi
Institute of Astronomy
University of Tokyo
2-21-1 Osawa181-0015MitakaTokyo
Yoichi Tamura
Institute of Astronomy
University of Tokyo
2-21-1 Osawa181-0015MitakaTokyo
Junko Ueda
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyo
Department of Astronomy
School of Science
The University of Tokyo
7-3-1 Hongo, Bunkyo-ku133-0033Tokyo
Harvard-Smithsonian Center for Astrophysics
60 Garden Street02138CambridgeMA
Yuzuru Yoshii
Institute of Astronomy
University of Tokyo
2-21-1 Osawa181-0015MitakaTokyo
Active Galactic Nucleus and Extended Starbursts in a Mid-stage Merger VV114
20 May 2013(Received ; accepted )PASJ: Publ. Astron. Soc. Japan , 1-??,telescopes -galaxies:evolution -galaxies:starburst -galaxies:interactions
High resolution (∼ 0. ′′ 4) Atacama Large Millimeter/submillimeter Array (ALMA) Cycle 0 observations of HCO + (4-3) and HCN (4-3) toward a mid-stage infrared bright merger VV 114 have revealed compact nuclear (< 200 pc) and extended (∼ 3 − 4 kpc) dense gas distribution across the eastern part of the galaxy pair. We find significant enhancement of HCN (4-3) emission in an unresolved compact and broad (290 km s −1 ) component found in the eastern nucleus of VV114, and we suggest dense gas associated with the surrounding material around an Active Galactic Nucleus (AGN), with a mass upper limit of < ∼ 4 × 10 8 M ⊙ . The extended dense gas is distributed along a filamentary structure with resolved dense gas concentrations (∼ 230 pc; ∼ 10 6 M ⊙ ) separated by a mean projected distance of ∼ 600 pc, many of which are generally consistent with the location of star formation traced in Paα emission. Radiative transfer calculations suggest moderately dense (n H2 = 10 5 -10 6 cm −3 ) gas averaged over the entire emission region. These new ALMA observations demonstrate the strength of the dense gas tracers in identifying both the AGN and star formation activity in a galaxy merger, even in the most dust enshrouded environments in the local universe.
Introduction
Cosmological simulations have clearly established that galaxy collisions and mergers play major roles in the formation and evolution of galaxies by triggering a rapid mass build-up (e.g., Cole et al. 2000). High-resolution major merger simulations have shown that the star formation physics is more dominated by mass fragmentation and turbulent motion across the merging disks, forming massive clumps of dense gas clouds (M gas = 10 6−8 M ⊙ ) and triggering star formation across the galaxy disks (Teyssier et al. 2010), or in a dense filamentary structure along the merging interface (Saitoh et al. 2009). In some cases, radial streaming can efficiently feed the gas to the central black hole, possibly triggering an 'AGN phase' during the course of the galaxy merger evolution (e.g. Hopkins et al. 2006).
An important observational test is to map the dense gas tracers in merging U/LIRGs (Ultra Luminous Infrared Galaxies) since they show high degree of starburst activity, some of which harbor AGNs in their centers (Sanders & Mirabel 1996). The HCN (4-3) and HCO + (4-3) emis-sion, whose critical densities are n crit ∼ 2 × 10 7 cm −3 and n crit ∼ 4 × 10 6 cm −3 (Meijerink et al. 2007) respectively, are both reliable dense gas tracers and now readily accessible at sub arcsecond angular resolution with the advent of the Atacama Large Millimeter/Submillimeter Array (ALMA). Here we present ALMA Cycle 0 HCN (4-3) and HCO + (4-3) observations of an IR-bright galaxy VV114 with the primary goal to study the distribution of dense gas during the critical stage when the two gas rich galaxies collide and merge.
VV114 is a gas-rich (M H2 = 5.1 × 10 10 M ⊙ ; Yun et al. 1994, Iono et al. 2004and Wilson et al. 2008 interacting pair with high-infrared luminosity (L IR = 4.0 × 10 11 L ⊙ ) located at D = 86 Mpc (1 ′′ = 0.4 kpc). It consists of two optical galaxies (VV114E and VV114W) with a projected separation of 6 kpc ( Figure 1). Evidence for wide-spread star formation activity and shocks across the entire system is found in the UV, optical, and mid-IR (Alonso-Herrero et al. 2002, Goldader et al. 2002, Rich et al. 2011. A dust-obscured AGN in VV114E is also suggested (but not conclusive) from NIR (Le Floc'h et al. 2002;Imanishi et al. 2007) and X-ray observations (Grimes et al. 2006), sug-[Vol. , gesting that both starburst and AGN activities might have been triggered by the ongoing merger.
ALMA Observations
The HCN (4-3) (ν rest = 354.505 GHz) and HCO + (4-3) (ν rest = 356.734 GHz) observations toward VV114 were obtained on June 1 -3, 2012 during the Cycle 0 program of ALMA using the extended configuration. The digital correlator was configured with 0.488 MHz resolution for the spectral window that contains the emission lines. Absolute flux calibration was performed using Uranus, J1924-292 was used for bandpass calibration, and the time dependent gain calibration was performed using J0132-169 (6 degrees away from VV 114). The total on source time was 86 minutes.
We used the delivered calibrated data product and CLEANed the image down to 1.5 sigma using the ALMA data reduction package CASA. Channel maps with 30 km s −1 velocity resolution were made, with a synthesized beam size of 0. ′′ 5 × 0. ′′ 4 (PA = 52 degrees)(equivalent to 200 × 160 pc). The rms noise level was 0.9 mJy beam −1 for the robust=0.5 maps. The continuum was subtracted using all of the line-free channels in the bandpass.
Distribution of HCN (4-3) and HCO + (4-3)
The HCN (4-3) and HCO + (4-3) integrated intensity maps are presented in Figure 1. While the HCN (4-3) emission is only seen near the eastern nucleus of VV114 and resolved into four peaks, the HCO + (4-3) emission is more extended and has at least 10 peaks in the integrated intensity map. The total integrated intensity of HCO + (4-3) and HCN (4-3) are 15.3 ± 0.4 Jy km s −1 and 4.4 ± 0.2 Jy km s −1 , respectively. The higher HCO + (4-3) flux observed by the SMA (≥ 17 ± 2 mJy; Wilson et al. 2008) using a 2. ′′ 8 × 2. ′′ 0 beam is likely attributed to missing flux by the ALMA observation. We show a direct comparison between the J=4-3 (this work) and J=1-0 (taken at the Nobeyama Millimeter Array; Imanishi et al. (2007)) transitions of both species in Figure 2, after convolving the ALMA images with the NMA beam (7. ′′ 5 × 5. ′′ 5). While the J=4-3 transitions of both species are concentrated near the eastern near-infrared nucleus with a slight extension to the west for HCO + (4-3), the distribution of the HCN (1-0) and HCO + (1-0) are different; the HCN (1-0) emission is separated into two clumps in the east-west whereas the HCO + (1-0) emission is extended widely toward the western nucleus.
We label the six HCO + (4-3) peaks in the eastern part of VV114 as E0 -E5 and the four detected in the western part of VV114 as W0 -W3 ( Figure 1, Table 1). The HCO + (4-3) and HCN (4-3) emission peaks are spatially consistent for E0 -E3. The compact component E0 is unresolved with the current resolution, and the size upper limit is < 200 pc. HCN (4-3) emission is not detected in the overlap region (W0 -W2), where the high CO (1-0) velocity dispersion and significant methanol detection both suggest the presence of shocked gas (Saito et al. in preparation).
Dense Gas and AGN /Starburst Activity
Relative Strengths of HCN (4-3) and HCO + (4-3) and a Signature of AGN
We present a comparison between the surface brightness of HCO + (4-3) and HCN (4-3) in Figure 3. Although the statistics are limited, the three molecular clumps (E1, E2 and E3) show an increasing trend between Σ HCN and Σ HCO + . In contrast, the ratio between the beam averaged surface brightness of E0 is a factor of three higher than E1, E2 and E3; E0 is the only component that has HCN (4-3) -HCO + (4-3) integrated flux ratio which is larger than unity (HCN (4-3)/HCO + (4-3) = 1.6). Gaussian fits to the HCO + (4-3) and HCN (4-3) spectra at E0 give peak = 9.0 mJy, σ = 123 km s −1 (for HCN (4-3)) and peak = 6.9 mJy, σ = 93 km s −1 (for HCO + (4-3)). Thus the HCN (4-3) emission is not only brighter at E0, but it is also broader than the gas traced in HCO + (4-3), suggesting that the HCN (4-3) and HCO + (4-3) are tracing physically different gas at < 200 pc scales. Such a high relative intensity of the HCN emission is possibly a signature of a buried AGN, as suggested by previous studies (e.g. Kohno et al. 2001).
It has been known that the brightness of the HCN emission line is enhanced near the AGN compared to star forming regions (Kohno et al. 2001), with higher contrast in high J transitions (Hsieh et al. 2012). Individual galaxies (e.g. NGC 1068, NGC 1097) have been studied extensively in high resolution (Kohno et al. 2003, Hsieh et al. 2012), clearly revealing the over abundance of HCN emission near the Seyfert nucleus, through J = 1 to 4. The exact reasoning for the enhanced intensity ratio is not clearly understood, and it could be due to gas excitation effects (e.g. density and temperature), intensity of the incident radiation field (e.g. PDR vs. XDR), IR pumping (e.g. Garcia-Burillo et al. 2006), or other non-collisional excitation due to star formation or supernovae explosions (see Krips et al. 2008 for a discussion). There is evidence suggesting the dominance of low density (< 10 4.5 cm −3 ) gas in a sample of AGNs (Krips et al. 2008), and hence the difference in critical density is likely not the only reason for the difference in the relative abundance.
Regardless of the exact physical origin of the higher relative intensity of the HCN (4-3) emitting gas, the broad (FWHM = 290 km s −1 ) and compact (< 200 pc) unresolved source E0 is of significant interest, since it coincides with the region where past observations suggest the presence of a buried AGN. We derive the upper limit to the dynamical mass by using M dyn = rσ 2 /G (assuming an inclination of 90 degrees for simplicity), where r is the radius enclosing the emission region, σ is the width of the HCN line, and G is the gravitational constant. The upper limit to the dynamical mass estimated from the linewidth and the beam size is < ∼ 4 × 10 8 M ⊙ . Since the HCN emission is generally believed to be optically thick, we estimate the dense gas mass of the E0 component adopting (Imanishi et al. 2007; in dark contours) is compared with the HCN (4-3) emission convolved to the NMA resolution (in red contours). The contour levels for HCN (1-0) are the same as Imanishi et al. (2007) the conversion factor provided in Gao & Solomon (2004).
Using the integrated intensity of HCN (4-3) (see Section 3) and HCN (1-0) = 7 Jy km s −1 (Imanishi et al. 2007), we derive HCN(4-3)/(1-0) = 0.63. This yields a dense gas mass of ∼ 8.1 × 10 6 M ⊙ , hence > 2% of the total mass is in dense molecular form and significant amount of dense gas is present in a very compact region. Finally, we note that while these are evidences suggesting a compact AGN near the eastern nucleus of VV114, the 350 GHz -8.5 GHz flux ratio suggests the contrary. The ratio is 1.2 ± 0.1 for E0, and 1.1 ± 0.1 for E1 and E2, using the the archival 8.5 GHz radio continuum image obtained from the VLA archive (beam size ∼ 0. ′′ 9) and the 350 GHz image obtained from the ALMA observations. The 350 GHz continuum emission is also unresolved at E0, but E1 and E2 show resolved structure. If the dominant source of radio continuum emission is indeed due to hot plasma surrounding the AGN, then we expect this ratio to be higher near the putative AGN (i.e. E0), which is inconsistent with the current results and argues in favor of a common physical origin (e.g. a massive starburst) in all three regions. Higher resolution radio continuum imaging is necessary to understand the origin of the radio emission in E0.
Extended Dense Gas Filament, Star Formation, and the Global Gas Conditions
The average size of the clumps forming the filamentary structure (i.e. E1-E5, W0-W3) is 230 ± 70 pc, with an average dense gas mass of ∼ 10 6 M ⊙ and a mean projected separation of ∼ 600 pc. We compare the distribution of the HCO + (4-3) and star formation activity traced in Paα line in Figure 4. Spatial correspondence between HCO + (4-3) and the brightest peaks of Paα is generally seen. Such a long filamentary dense gas structure and associated star formation are predicted along the colliding interface of (Tateuchi et al. 2012) is obtained using the NIR camera ANIR (Motohara et al. 2008) mounted on the University of Tokyo Atacama Observatory 1m telescope (miniTAO) (Minezaki et al. 2010). A ∼ 1 kpc long Paα extension to the NE is also seen emanating from the putative AGN, which may be a signature of shock ionization, or star formation activity in the compressed gas along the AGN jet. two colliding galaxies (Saitoh et al. 2009), and the masses are also consistent with the massive star forming clumps predicted in simulations by Teyssier et al. (2010).
Finally, we derive the global physical conditions of gas by comparing the total integrated HCO + (4-3)/(1-0) and HCN(4-3)/(1-0) ratios to the results from radiative transfer modeling (RADEX; Van der Tak et al. 2007). The results are n H2 = 10 5 -10 6 cm −3 and T = 30 -500 K, assuming abundance ratios of [HCO + ]/[H 2 ] = 1.0 × 10 −9 (Irvine et al. 1987) and [HCN]/[HCO + ] = 0.1 to 1 (to be consistent with M82; Krips et al. 2008). Although the range in the derived temperature is too large to be a meaningful constraint, this suggests the presence of moderately dense gas averaged over the entire galaxy pair. We caution here that these are average quantities which are derived without considering the difference in the spatial distribution between the J=4-3 and 1-0 transitions (see Figure 2). Higher angular resolution imaging of the J=1-0 transition is clearly needed in order to determine the spatial distribution of the physical properties.
Summary and Future Prospects
We present 0. ′′ 4 resolution HCN (4-3) and HCO + (4-3) observations toward a mid-stage IR bright merger VV114 obtained during cycle 0 program of ALMA. For the first time, these new high-quality maps allow us to investigate the central regions of this merging LIRG at 200 pc resolution. We find that both the HCN (4-3) and HCO + (4-3) emission in the eastern nucleus of VV 114 are compact (< 200 pc) and broad (290 km s −1 for HCN (4-3)) with high HCN (4-3)/HCO + (4-3) ratio. From the new ALMA observations along with past X-ray and NIR observations, we suggest the presence of an obscured AGN in the eastern nucleus of VV114. We also detect a 3-4 kpc long fil- ament of dense gas, which is likely tracing the active star formation triggered by the ongoing merger. In a forthcoming paper, we will present a comprehensive modeling of VV 114 using our new 12 CO (1-0), 13 CO (1-0) and CO (3-2) ALMA observations, as well as a chemical analysis of the nucleus and the overlap region of VV114 (Saito et al. in prep).
Fig. 1 .
1(left) HST ACS image of VV114 overlaid with the approximate regions of the panels shown on the right, and the approximate field of views of the ALMA 3-point mosaic. (Credit: NASA, ESA, the Hubble Heritage (STScI/AURA)-ESA/Hubble Collaboration, and A. Evans (University of Virginia, Charlottesville/NRAO/Stony Brook University). (right) The distribution of HCO + (4-3) (top) and HCN (4-3) (bottom) in VV 114. The contours are; 0.05, 0.15, 0.25, 0.35, 0.55, 0.85, 1.25, 1.65, 2.05, 2.45 Jy km s −1 .
Fig. 2 .
2(left) The HCN (1-0) emission
Fig. 3 .
3and the HCN (4-3) contours are (2.8 − 4.8) × 10 −2 (in steps of 0.2 × 10 −2 ) Jy/beam km s −1 . The crosses indicate the locations of the near-infrared peaks shown inFigure 2ofImanishi et al. (2007). (right) Similar to lef t but for the HCO + emission. The HCN (4-3) contours are (1.0 − 1.7) × 10 −2 (in steps of 0.1 × 10 −2 ) Jy/beam km s −1 . Relation between the HCN (4-3) and HCO + (4-3) surface brightness for different regions in VV114. The triangles represent upper limits to the HCN (4-3) surface brightness. The solid lines are the surface brightness ratio of 0.5, 1 and 2.
Fig. 4 .
4HCO + (4-3) emission overlaid on a Paα image. The contour levels are the same as Figure 1. The Paα map
Table 1 .
1Properties of the Molecular ClumpsSource S HCO + dv 1 S HCN dv 1 T HCO + 2 T HCN The integrated flux densities. 2 The peak temperature of each molecular clump. The error on each value is 0.05 K.2
The authors thank the anonymous referee for comments that improved the contents of this paper. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2011.0.00467.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.
. A Alonso-Herrero, G H Rieke, M J Rieke, N Z Scoville, AJ. 124166Alonso-Herrero, A., Rieke, G. H., Rieke, M. J., & Scoville, N. Z., 2002, AJ, 124, 166
. S Cole, C G Lacey, C M Baugh, C S Frenk, MNRAS. 319168Cole, S., Lacey, C. G., Baugh, C. M., & Frenk, C. S., 2000, MNRAS, 319, 168
. Y Gao, P M Solomon, ApJ. 606271Gao, Y. & Solomon, P. M., 2004, ApJ, 606, 271
. J D Goldader, M Gerhardt, T M Heckman, M Seiber, D B Sanders, D Calzetti, C C Steidel, ApJ. 568651Goldader, J. D., Gerhardt, M., Heckman, T. M., Seiber, M., Sanders, D. B., Calzetti, D., & Steidel, C. C. 2002, ApJ, 568, 651
. S Garcia-Burillo, ApJ. 64517Garcia-Burillo, S., et al., ApJ, 645, L17
. J P Grimes, T Heckman, C Hoopes, D Strickland, A Aloisi, G Meurer, A Ptak, ApJ. 648310Grimes, J. P., Heckman, T., Hoopes, C., Strickland, D., Aloisi, A., Meurer, G., & Ptak, A. 2006, ApJ, 648, 310
. P F Hopkins, L Hernquist, T J Cox, T Di Matteo, B Robertson, V Springel, ApJS. 1631Hopkins, P. F., Hernquist, L., Cox, T. J., Di Matteo, T., Robertson, B., & Springel, V. 2006, ApJS, 163, 1
. P - Y Hsieh, P T P Ho, K Kohno, C - Y Hwang, S Matsushita, ApJ. 74790Hsieh, P-.Y., Ho, P.T.P., Kohno, K., Hwang, C-.Y., & Matsushita, S. 2012, ApJ, 747, 90
. M Imanishi, K Nakanishi, T Yoichi, N Oi, K Kohno, AJ. 1342366Imanishi, M., Nakanishi, K., Yoichi, T., Oi, N., & Kohno, K. 2007, AJ, 134, 2366
. D Iono, P T P Ho, M S Yun, S Matsushita, A B Peck, K Sakamoto, ApJL. 61663Iono, D., Ho, P. T. P., Yun, M. S., Matsushita, S., Peck, A. B., & Sakamoto, K. 2004, ApJL, 616, 63
Interstellar processes. W M Irvine, P F Goldsmith, A Hjalmarson, A88- 14501 03-90Proceedings of the Symposium. the SymposiumGrand Teton National Park, WY; Dordrecht, DReidel Publishing CoIrvine, W. M., Goldsmith, P. F., & Hjalmarson, A. 1987, Interstellar processes; Proceedings of the Symposium, Grand Teton National Park, WY, July 1-7, 1986 (A88- 14501 03-90). Dordrecht, D. Reidel Publishing Co., 1987, p. 561-609.
. K Kohno, S Matsushita, B Vila-Vilaro, S K Okumura, T Shibatsuka, M Okiura, S Ishizuki, R Kawabe, ASPC. 239672Kohno, K., Matsushita, S., Vila-Vilaro, B., Okumura, S. K. Shibatsuka, T., Okiura, M., Ishizuki, S., & Kawabe, R. 2001, ASPC, 239, 672
. K Kohno, S Ishizuki, S Matsushita, B Vila-Vilaro, R Kawabe, PASJ. 551Kohno, K., Ishizuki, S., Matsushita, S., Vila-Vilaro, B., & Kawabe, R. 2003, PASJ, 55, 1
. M Krips, R Neri, S Garcia-Burillo, S Martin, F Combes, J Gracia-Carpio, A Eckart, ApJ. 67737ApJKrips, M., Neri, R., Garcia-Burillo, S., Martin, S., Combes, F, Gracia-Carpio, J., & Eckart, A., ApJ, 677, 262 2011, ApJ, 736, 37
. M Krips, ApJ. 73637Krips, M., et al., 2011, ApJ, 736, 37
. Le Floc'h, E Charmandaris, V Laurent, O Mirabel, I F Gallais, P Sauvage, M Vigroux, L Cesarsky, C , A&A. 391417Le Floc'h, E., Charmandaris, V., Laurent, O., Mirabel, I. F., Gallais, P., Sauvage, M., Vigroux, L., & Cesarsky, C., 2002, A&A, 391, 417
. T Minezaki, D Kato, S Sako, M Konishi, S Koshida, N Mitani, T Aoki, M Doi, T Handa, Y Ita, K Kawara, K Kohno, T Miyata, K Motohara, T Soyano, T Tanabe, M Tanaka, K Tarusawa, Y Yoshii, L Bronfman, M T Ruiz, M Hamuy, Proceedings of SPIE. 7733Minezaki, T., Kato, D., Sako,S., Konishi, M., Koshida, S., Mitani, N., Aoki, T., Doi, M., Handa, T., Ita, Y., Kawara, K., Kohno, K., Miyata, T., Motohara, K., Soyano, T., Tanabe, T., Tanaka, M., Tarusawa, K., Yoshii, Y., Bronfman, L., Ruiz, M. T., Hamuy, M. 2010, Proceedings of SPIE, 7733, 773356-1
. K Motohara, N Mitani, S Sako, Y K Uchimoto, K Toshikawa, T Yamamuro, T Handa, M Tanaka, T Aoki, M Doi, K Kawara, K Kohno, T Minezaki, T Miyata, T Soyano, T Tanabe, K Tarusawa, Y Yoshii, Proceedings of SPIE. 701470142Motohara, K., Mitani, N., Sako, S., Uchimoto, Y. K., Toshikawa, K., Yamamuro, T., Handa, T., Tanaka, M., Aoki, T., Doi, M., Kawara, K., Kohno, K., Minezaki, T., Miyata, T., Soyano, T., Tanabe, T., Tarusawa, K., Yoshii, Y. 2008, Proceedings of SPIE, 7014, 70142T
. R Meijerink, M Spaans, F P Israel, 461793Meijerink, R., Spaans, M., & Israel, F. P. 2007,å, 461, 793
. J A Rich, L J Kewley, M A Dopita, ApJ. 73487Rich, J. A., Kewley, L. J. & Dopita, M. A., 2011, ApJ, 734, 87
. D B Sanders, I F Mirabel, ARA&A. 34749Sanders, D. B. & Mirabel, I. F., 2012, ARA&A, 34, 749
. T R Saitoh, PASJ. 61481Saitoh, T. R., et al., 2009, PASJ, 61, 481
. K Tateuchi, PKAS. 27297Tateuchi, K., et al., 2012, PKAS, 27, 297
. R Teyssier, D Chapon, F Bournaud, ApJ. 720149Teyssier, R., Chapon, D., & Bournaud, F., 2010, ApJ, 720, 149
. F F S Van Der Tak, J H Black, J H Schoier, D J Jansen, E F Van Dishoeck, A&A. 468627Van der Tak, F. F. S., Black, J. H., Schoier, J. H., Jansen, D. J., van Dishoeck, E. F., 2007, A&A 468, 627
. C D Wilson, G R Petitpas, D Iono, A J Baker, A B Peck, M Krips, B Warren, J Golding, A Atkinson, L Armus, T J Cox, P Ho, M Juvela, S Matsushita, C J Mihos, Y Pihlstrom, M S Yun, ApJS. 178189Wilson, C. D., Petitpas, G. R., Iono, D., Baker, A. J., Peck, A. B., Krips, M., Warren, B., Golding, J., Atkinson, A., Armus, L., Cox, T. J., Ho, P., Juvela, M., Matsushita, S., Mihos, C. J., Pihlstrom, Y., Yun, M. S., 2008, ApJS, 178, 189L
. M S Yun, N Z Scoville, R A Knop, ApJ. 430109Yun, M. S., Scoville, N. Z., & Knop, R. A., 1994, ApJ, 430, 109L
| [] |
[
"Australia's Approach to AI Governance in Security & Defence",
"Australia's Approach to AI Governance in Security & Defence"
] | [
"S Kate Devitt [email protected] \nUniversity of Queensland\n\n\nTrusted Autonomous Systems\n\n",
"Damian Copeland [email protected] \nUniversity of Queensland\n\n",
"Tim Mcfarland ",
"Eve Massingham ",
"Rachel Horne ",
"Tara Roberson "
] | [
"University of Queensland\n",
"Trusted Autonomous Systems\n",
"University of Queensland\n"
] | [] | Australia is a leading AI nation with strong allies and partnerships. Australia has prioritised the development of robotics, AI, and autonomous systems to develop sovereign capability for the military. Australia commits to Article 36 reviews of all new means and methods of warfare to ensure weapons and weapons systems are operated within acceptable systems of control. Additionally, Australia has undergone significant reviews of the risks of AI to human rights and within intelligence organisations and has committed to producing ethics guidelines and frameworks in Security and Defence. Australia is committed to OECD's values-based principles for the responsible stewardship of trustworthy AI as well as adopting a set of National AI ethics principles. While Australia has not adopted an AI governance framework specifically for the Australian Defence Organisation (ADO); Defence Science and Technology Group (DSTG) has published 'A Method for Ethical AI in Defence' (MEAID) technical report which includes a framework and pragmatic tools for managing ethical and legal risks for military applications of AI. Australia can play a leadership role by integrating legal and ethical considerations into its ADO AI capability acquisition process. This requires a policy framework that defines its legal and ethical requirements, is informed by Defence industry stakeholders, and provides a practical methodology to integrate legal and ethical risk mitigation strategies into the acquisition process. 2 | null | [
"https://arxiv.org/pdf/2112.01252v2.pdf"
] | 244,799,416 | 2112.01252 | 2f3fa6cde5ee57e4d9289f88ee396b8e61dc3d26 |
Australia's Approach to AI Governance in Security & Defence
S Kate Devitt [email protected]
University of Queensland
Trusted Autonomous Systems
Damian Copeland [email protected]
University of Queensland
Tim Mcfarland
Eve Massingham
Rachel Horne
Tara Roberson
Australia's Approach to AI Governance in Security & Defence
AustraliaArticle 36systems of controlAI Ethicsmilitary ethics
Australia is a leading AI nation with strong allies and partnerships. Australia has prioritised the development of robotics, AI, and autonomous systems to develop sovereign capability for the military. Australia commits to Article 36 reviews of all new means and methods of warfare to ensure weapons and weapons systems are operated within acceptable systems of control. Additionally, Australia has undergone significant reviews of the risks of AI to human rights and within intelligence organisations and has committed to producing ethics guidelines and frameworks in Security and Defence. Australia is committed to OECD's values-based principles for the responsible stewardship of trustworthy AI as well as adopting a set of National AI ethics principles. While Australia has not adopted an AI governance framework specifically for the Australian Defence Organisation (ADO); Defence Science and Technology Group (DSTG) has published 'A Method for Ethical AI in Defence' (MEAID) technical report which includes a framework and pragmatic tools for managing ethical and legal risks for military applications of AI. Australia can play a leadership role by integrating legal and ethical considerations into its ADO AI capability acquisition process. This requires a policy framework that defines its legal and ethical requirements, is informed by Defence industry stakeholders, and provides a practical methodology to integrate legal and ethical risk mitigation strategies into the acquisition process. 2
Capability) said "The Loyal Wingman project is a pathfinder for the integration of autonomous systems and artificial intelligence to create smart human-machine teams" (de Git 2021). AVM Roberts also confirmed that "[w]e need to ensure that ethical and legal issues are resolved at the same pace that the technology is developed" (Department of Defence 2021b).
Just over six months later, on 25 th September, Australia 1 won the silver medal at the 2021 DARPA Subterranean Challenge, also known as the robot Olympics. In the event, Australia used multiple robotics platforms equipped with AI to autonomously explore, map, and discover models representing lost or injured people, suspicious backpacks, or phones, or navigate tough conditions such as pockets of gas. The outstanding performance confirms Australia's international reputation at the forefront of robotics, autonomous systems and AI research and development (Persley 2021).
In parallel to technology development, Australia is navigating the challenge of developing and promoting AI governance structures inclusive of Australian values, standards, ethical and legal frameworks. National initiatives have been led by: Autonomous Systems in Defence ; Department of Defence 2021b).
Two large surveys of Australian attitudes to AI were conducted in 2020. Selwyn and Gallo Cordoba (2021) found that, based on over 2000 respondents, Australian public attitudes towards AI are informed by an educated awareness of the technologies affordances. Attitudes are generally positive and respondents are proud of their scientific achievements. However, Australians are concerned about the government's trustworthiness using automated decisionmaking algorithms due to incidents such as the Robodebt (Braithwaite 2020) and #CensusFail (Galloway 2017) scandals. A second study of over 2500 respondents found that Australians have low trust in AI systems but generally 'accept' or 'tolerate' AI (Lockey, Gillespie, and Curtis 2020). They found that Australians trust research institutions and Defence organisations the most to use AI and trusted commercial organisations the least. Australians expect AI to be regulated and carefully managed (Lockey, Gillespie, and Curtis 2020).
This chapter will begin with Australia's strategic position, Australia's definition of AI and identifying what ADO wants AI for. It will then move into AI governance initiatives and specific efforts to develop frameworks for ethical AI both in both civilian and military contexts.
The chapter will conclude with likely future directions for Australia in AI governance to reshape military affairs and Australia's role in international governance mechanisms and strategic partnerships.
Australia's strategic position
Australia's current strategic position has been shaped by two major developments, 1) the announcement of AUKUS 15 September 2021 (Morrison, Johnson, and Biden 2021) and 2) public commitment to NATO and AUKUS allies through sanctions, supply of humanitarian support and lethal weapons to the Ukraine to defend against the invasion of Russia (Prime Minister andMinister of Defence 2022, 1 March 2022).
AUKUS confirmed a shift in strategic interests for the United States to Asia Pacific and away from the Middle East with the withdrawal from Afghanistan. "Recognizing our deep defense ties, built over decades, today we also embark on further trilateral collaboration under AUKUS to enhance our joint capabilities and interoperability. These initial efforts will focus on cyber capabilities, artificial intelligence, quantum technologies, and additional undersea capabilities" (The White House 2021b) Commitment to defend Ukraine signals a strengthening global alliance amongst liberal democracies, of which Australia is a part. The Prime Minister calls the conflict ""a very big wake-up call" that reunites liberal democracies to meet a polarising and escalating threat" (Kelly 2022).
The 2020 Strategic Update identified new objectives for Australian defence-see Box 1. (Department of Defence (2020a, 24-25) 1. to shape Australia's strategic environment; 2. to deter actions against Australia's interests; and 3. to respond with credible military force, when required (Department of Defence 2020a 2.12).
Box 1 New Objectives for Australian Defence 2020
These new objectives will guide all aspects of ADO's planning including force structure planning, force generation, international engagement and operations (2.13).
To implement these objectives, ADO will:
• prioritise our immediate region (the north-eastern Indian Ocean, through maritime and mainland South East Asia to Papua New Guinea and the South West Pacific) for the ADF's geographical focus;
• grow the ADF's self-reliance for delivering deterrent effects; expand Defence's capability to respond to grey-zone activities, working closely with other arms of Government;
• enhance the lethality of the ADF for the sorts of high-intensity operations that are the most likely and highest priority in relation to Australia's security; maintain the ADF's ability to deploy forces globally where the Government chooses to do so, including in the context of US-led coalitions; and
• enhance ADO's capacity to support civil authorities in response to natural disasters and crises.
With these in mind, Australia's global position has been elevated with the announcement of a new Australia, UK and United States science and technology, industry, and defence partnership (AUKUS) (Morrison, Johnson, and Biden 2021). This partnership is likely to increase data, information and AI sharing and aligned AI governance structures and interoperability policies (Deloitte Center for Government Insights 2021) to manage joint and cooperative military action, deterrence, cyber-attacks, data theft, disinformation, foreign interference, economic coercion, attacks on critical infrastructure, supply chain disruption and so forth (Hanson and Cave 2021).
In addition to AUKUS, Australia has a number of strategic partnerships including the global 'five-eyes' network of UK, US, Australia, Canada and New Zealand (Office of the Director of National Intelligence); the Quad surrounding China India, Japan, Australia and US (Shih and Gearan 2021) and local partnerships including the Association of Southeast Asian Nations (ASEAN) (Thi Ha 2021) and Pacific family (Blades 2021).
The strategic issues identified in both the Defence White Paper (Department of Defence 2016) and the 2020 Strategic Update (Department of Defence 2020a) puts AI among priority information and communications technology capabilities, e.g. "3.39 Over the next five years, Defence will need to plan for developments including next generation secure wireless networks, artificial intelligence, and augmented analytics" (Department of Defence 2020a).
Australia's definition of artificial intelligence
Australia (Department of Industry Science Energy and Resources 2021) defines AI as: "AI is a collection of interrelated technologies that can be used to solve problems autonomously and perform tasks to achieve defined objectives. In some cases, it can do this without explicit guidance from a human being . AI is more than just the mathematical algorithms that enable a computer to learn from text, images or sounds. It is the ability for a computational system to sense its environment, learn, predict and take independent action to control virtual or physical infrastructure." Australia defines AI by its functions (sensing, learning, predicting, independent action), focus and degree of independence in the achievement of defined objectives with or without explicit guidance from a human being. The Australian definition encompasses the role of AI in digital and physical environments without discussing any particular methodology or technology that might be used. In doing so it aligns itself with the OECD's Council on Artificial Intelligence's definition of an AI system: "An AI system is a machine-based system that can, for a given set of humandefined objectives, make predictions, recommendations, or decisions influencing real or virtual environments. AI systems are designed to operate with varying levels of autonomy" (OECD Council on Artificial Intelligence
2019)
What does Australian Defence want artificial intelligence for?
While the Australian Department of Defence in Australia has not formally adopted a Defence AI Roadmap or Strategy, Australia has prioritised developing sovereign AI capabilities-see Box 2-as well as: robotics, autonomous systems, precision guided munitions, hypersonic weapons, and integrated air and missile defence systems; space; and information warfare and cyber capabilities (Australian Government 2021).
Box 2 Australian Sovereign Industry Capability Priority:
Robotics, Artificial Intelligence and Autonomous Systems (Australian Government (2021) Robotics and autonomous systems are an important element of military capability. They act as a force multiplier and protect military personnel and assets.
The importance of these capabilities will continue to grow over time. Robotics and autonomous systems will become more prevalent commercially and in the battlespace.
Australian industry must have the ability to design and deliver robotic and autonomous systems. This will enhance the ADF's combat and training capability through:
• improving efficiency • reducing the physical and cognitive load to the operator • increasing mass • achieving decision making superiority • decreasing risk to personnel.
These systems will comprise of:
• advanced robots • sensing and artificial intelligence encompassing al gorithms • machine learning and deep learning.
These systems will enhance bulk data analysis. This will facilitate decision making processes and enable autonomous systems.
Australia notes that AI will play a vital role in ADO's future operating environment, delivering on strategic objectives of shape, deter and respond (Department of Defence 2021a). AI will contribute to Australia in maintaining a capable, agile and potent ADO.
Potential military AI applications have been taxonomized into warfighting functions (force application, force protection, force sustainment, situational understanding) and enterprise functions (personnel, enterprise logistics, business process improvement)-see Annex A . Operational contexts help discern the range of purposes of AI within Defence as well as diverse legal, regulatory, and ethical structures required in each domain to govern AI use. For example the use of AI in ensuring abidance with workplace health and safety risk management (Morrison 2021; Centre for Work Health and Safety 2021) is relevant to Defence People Group 2 , whereas the use of AI within new weapons systems and the Article 36 review process is within the portfolio for Defence Legal (Commonwealth of Australia 2018). AI will also be needed to manage Australia's grey zone threat (Townshend, Lonergan, and Warden 2021 Intelligence Research Network (DAIRNET) to develop AI "to process noisy and dynamic data in order to produce outcomes to provide decision superiority" (Defence Science & Technology Group 2021b).
These efforts include some human-centred projects, such as human factors for explainable AI and studies in AI bias in facial recognition (Defence Science Institute 2020b). Defence has committed to develop guidelines on the ethical use of data (Department of Defence 2021a) and
the Australian Government has committed to governance and ethical frameworks for the use of artificial capabilities for intelligence purposes (Attorney-General's Department 2020, Recommendation 154). Ethics guidelines will help Australia respond to public debate on the ethics of facial recognition for military purposes even if biases are reduced (van Noorden 2020).
AI Governance in Australia
AI governance includes social, legal, ethical and technical layer (algorithms and data) that require norms, regulation, legislation, criteria, principles, data governance, algorithm accountability and standards (Gasser and Almeida 2017). Australia's strategic, economic, cultural, diplomatic, and military use of AI will be expected to be governed in accordance with Australian attitudes and values (such as described in Box 3) and international frameworks.
Box 3 Australian Values (Department of Home Affairs 2020)
• Respect for the freedom and dignity of the individual • Freedom of religion (including the freedom not to follow a particular religion), freedom of speech and freedom of association Australia is positioning itself to be consistent with emerging best practice internationally for a for ethical, trustworthy (Ministère des Armées 2019), responsible AI (Fisher 2020) and allied frameworks for ethical AI in Defence (Lopez 2020;Stanley-Lockman 2021).
To this end, Australia is a founding member of The Global Partnership on AI (GPAI) 4 , an international and multi-stakeholder initiative to undertake cutting-edge research and pilot 4 Other GPAI countries include: Canada, France, Germany, India, Italy, Japan, Mexico, New Zealand, Korea, Singapore, Slovenia, the United Kingdom, the United States, and the European Union. In December 2020, Brazil, the Netherlands, Poland, and Spain joined GPAI (Gobal Partnership on AI 2021; Department of Industry Science Energy and Resources 2020a).
• Commitment to the rule of law, which means that all people are subject to the law and should obey it • Parliamentary democracy whereby our laws are determined by parliaments elected by the people, those laws being paramount and overriding any other inconsistent religious or secular 'laws'
• Equality of opportunity for all people, regardless of their gender, sexual orientation, age, disability, race or national or ethnic origin
AI Ethics Principles
In 2019, the Australian Government sought public submissions in response to a CSIRO Data61AI Ethics discussion paper . A voluntary AI Ethics Framework emerged from the Department of Industry Innovation and Science (2019) (DISER) to guide businesses and governments developing and implementing AI in Australia. The framework includes eight AI ethics principles (AU-EP) to help reduce the risk of negative impacts from AI and ensure the use of AI is underpinned by good governance standards-see Box 4. 5 GPAI working groups are focussed on four key themes: responsible AI; data governance; the future of work; and innovation and commercialisation
Box 4 Australia's AI Ethics Principles (AU-EP) (Department of Industry Innovation and
Science (2019) 1. Human, societal, and environmental wellbeing: AI systems should benefit individuals, society, and the environment. 6. Transparency and explainability: There should be transparency and responsible disclosure so people can understand when they are being significantly impacted by AI and can find out when an AI system is engaging with them. 7. Contestability: When an AI system significantly impacts a person, community, group or environment, there should be a timely process to allow people to challenge the use or outcomes of the AI system. 8. Accountability: People responsible for the different phases of the AI system lifecycle should be identifiable and accountable for the outcomes of the AI systems, and human oversight of AI systems should be enabled.
Case studies have been undertaken with industry to evaluate the usefulness and effectiveness of the principles. Many of the findings and due diligence frameworks will be useful in the dialogue between Defence industries, Australian Defence Force and the Department of Defence.
Key findings from Industry (Department of Industry Science Energy and Resources 2020b) include:
• AU-EP are relevant to any organisation involved in AI (private, public, large or small)
• Organisations expect the Australian Government to lead by example and implement AU-EP.
• Implementing AU-EP can ensure that businesses can exemplify best practice and be ready to meet community expectations or any changes in standards or laws
• Ethical issues can be complex, and businesses may need more help from professional or industry bodies, academia or experts and government.
• Businesses need training and education, certification, case study examples, and costeffective methods to help them implement and utilise AU-EP.
The case studies revealed that responsibilities of AI purchasers and AI developers differ. Each group needed internal due diligence, communication and information from external stakeholders, including vendors or customers, to establish accountability and responsibility obligations. Businesses found some principles more challenging to practically implement. The advice given by the Government is that businesses ought to document the process of managing ethical risks (despite ambiguity) and to refer serious issues to relevant leaders.
To support ethical AI businesses are advised by DISER to:
• Set appropriate standards and expectations of responsible behaviour when staff deploy AI. For example, via a responsible AI policy and supporting guidance.
• Include AI applications in risk assessment processes and data governance arrangements.
• Ask AI vendors questions about the AI they have developed.
• Form multi-disciplinary teams to develop and deploy AI systems. They can consider and identify impacts from diverse perspectives.
• Establish processes to ensure there is clear human accountability for AI-enabled decisions and appropriate senior approvals to manage ethical risks. For example, a cross-functional body to approve an AI system's ethical robustness.
• Increase ethical AI awareness raising activities and training for staff.
The Australian Government commits to continuing to work with agencies to encourage greater uptake and consistency with AU-EP (Department of Industry Science Energy and Resources 2020b). In the 2021 AI Action Plan, Australia hopes that widespread adoption of AU-EP among business, government and academia will build trust in AI systems (Department of Industry Science Energy and Resources 2021). AU-EP are included in the Defence Method for Ethical AI in Defence report (Appendix A Comparison of Ethical AI Frameworks Devitt et al. 2021, 48-50). However, some have questioned the value of AU-EP without them being embedded in policy, practise and accountability mechanisms. Contentious uses of AI in government such as facial recognition by police have not gone through ethical review and do not have institutional ethical oversight (ASPI 2022). While federal frameworks may not have seen effective operationalisation, NSW has published their own AI Assurance Framework that mandates AI projects to undergo ethical risk assessment (NSW Government 2022).
Standards
The Future of Humanity Institute at the University of Oxford recommends developing international standards for ethical AI research and development (Cihon 2019;Dafoe 2018). This is consistent with Standards Australia's Artificial Intelligence Standards Roadmap:
Making Australia's Voice Heard (2020)-see Box 5.
Standards Australia seeks to increase cooperation with the United States National Institute for
Standards & Technology (NIST) and other Standards Development Organisations (SDOs).
Australia has a stated aim to participate in ISO/IEC/JTC 1/SC 42, and the National Mirror Committee (IT-043) regarding AI. Standards Australia notes the importance of improving AI data quality as well as ensuring Australia's adherence to both domestic and international best practise in privacy and security by design.
Box 5 Artificial Intelligence Standards Roadmap: Making Australia's Voice Heard
(Standards Australia (2020)
Recommendations:
1. Increase the membership of the Artificial Intelligence Standards Mirror Committee in Australia to include participation from more sectors of the economy and society.
Australian businesses and government agencies develop a proposal for a direct text
adoption of ISO/IEC 27701 (Privacy Information Management), with an annex mapped to local Australian Privacy Law requirements. This will provide Australian businesses and the community with improved privacy risk management frameworks that align with local requirements and potentially those of the GDPR, CBPR and other regional privacy frameworks.
Australian Government stakeholders, with industry input, develop a proposal to
improve data quality in government services, to optimise decision-making, minimise bias and error, and improve citizen interactions. 6. Australian stakeholders channel their concerns about inclusion, through participating in the Standards Australia AI Committee (IT-043), to actively shape the development of an international management system Standard for AI as a pathway to certification. 7. The Australian Government consider supporting the development of a security-bydesign initiative, which leverages existing standards used in the market, and which recognises and supports the work being carried out by Australia's safety by-design initiative.
8. Develop a proposal for a Standards hub setup to improve collaboration between standards-setters, industry certification bodies, and industry participants, to trial new more agile approaches to AI Standards for Australia.
Human Rights
A report by the Australian Human Rights Commissioner (AHRC) (Santow 2021) concerning human rights and AI in Australia makes a suite of recommendations including the establishment of an AI Safety Commissioner (Sadler 2021). Of relevance to this chapter are the recommendations to:
• require human rights impact assessments (HRIA) before any government department or agency uses an AI-informed decision-making system to make administrative
decisions [Recommendation 2]
• the government needs to make AI decision making transparent and explainable to affected individuals and give them recourse to challenge the decision [Recommendations 3,5,6,8].
• AU-EP should be used to encourage corporations and other non-government bodies to undertake a human rights impact assessment before using an AI-informed decisionmaking system.
Human rights impact assessments of AI in Defence will vary depending on the context of deployment, e.g. whether the AI is deployed within the war fighting or rear echelon functions.
It is notable that Defence has established precedence working collaboratively with the Australian Human Rights Commission (AHRC), e.g. on 'Collaboration for Cultural Reform in Defence' examining human rights issues including gender, race and diversity, sexual orientation and gender identity and the impact of alcohol and social media on the cultural reform process (Jenkins 2014). Thus, it is possible that Australia could work again with the AHRC on AI decision making in the Australian Defence Force. As algorithms could unfairly bias recommendations, honours and awards, promotion, duties, or postings against particular groups e.g. women or LGBTIQ+. While not documented in the military yet, Tech giants including Amazon have withdrawn AI powered recruitment when they found that it was biased against women (Parikh 2021). There are also new methods to debias the development, use and iteration of AI tools in human resources (Parikh 2021).
Australian AI Action Plan
In the Australian AI Action Plan (Department of Industry Science Energy and Resources 2021), the Australian government commits to:
• Developing and adopting AI to transform Australian businesses • Creating an environment to grow and attract the world's best AI talent • Using cutting edge AI technologies to solve Australia's national challenges • Making Australia a global leader in responsible and inclusive AI To achieve the final point, Australia commits to AI Ethics Principles (Department of Industry Innovation and Science 2019) and the OECD Principles (2019) on AI to promote AI that is innovative, trustworthy and that respects human rights and democratic values.
The OECD AI principles (2019) are:
1. AI should benefit people and the planet by driving inclusive growth, sustainable development and well-being.
2. AI systems should be designed in a way that respects the rule of law, human rights, democratic values and diversity, and they should include appropriate safeguardsfor example, enabling human intervention where necessaryto ensure a fair and just society.
3. There should be transparency and responsible disclosure around AI systems to ensure that people understand AI-based outcomes and can challenge them.
4. AI systems must function in a robust, secure and safe way throughout their life cycles and potential risks should be continually assessed and managed.
5. Organisations and individuals developing, deploying or operating AI systems should be held accountable for their proper functioning in line with the above principles.
In 2021, the OECD released a report on how Nations are responding to AI Ethics principles. In the report, Australia is noted as:
• deploying a myriad of policy initiatives, including: establishing formal education programmes on STEM and AI-related fields to empower people with the skills for AI and prepare for a fair labour market transition.
• offering fellowships, postgraduate loans, and scholarships to increase domestic AI research capability and expertise and retain AI talent. Australia has dedicated AUD 1.4 million to AI and Machine Learning PhD scholarships • Australia and Singapore, building on their pre-existing trade agreement, also signed the Singapore-Australia Digital Economy Agreement (SADEA) where Parties agreed to advance their co-operation on AI Recently the US and Europe confirmed commitment to OECD principles in a joint statement that:
"The United States and European Union will develop and implement AI systems that are innovative and trustworthy and that respect universal human rights and shared democratic values, explore cooperation on AI technologies designed to enhance privacy protections, and undertake an economic study examining the impact of AI on the future of our workforces" (The White
House 2021a).
Australia is likely to remain aligned with the AI frameworks of allies, particularly the UK (AI Council 2021) and the USA (National Security Commission on Artificial Intelligence 2021).
AI governance in Defence
While Australia has not released an overarching AI governance framework for Defence, this chapter outlines an argument for such a framework that draws from publicly released concepts, strategy, doctrine, guidelines, papers, reports and methods relating to human, AI, and data governance relevant to Defence and Australia's strategic position.
Australia is a founding partner in the US's AI Partnership for Defense (PfD) that includes Canada, Denmark, Estonia, France, Finland, Germany, Israel, Japan, the Republic of Korea, Norway, the Netherlands, Singapore, Sweden, the United Kingdom, and the United States (JAIC Public Affairs 2021, 2020). In doing so, Australia has aligned its AI partnerships with AUKUS, five-eyes (minus New Zealand), the Quad (minus India) and ASEAN via Singapore 6 .
In particular Australia is seeking to increase AI collaboration with the US and UK through AUKUS (Nicholson 2021).
AI in Weapons Systems
Computer software designed to perform computational or control functions has been used in It is assumed that the governance of AI will dovetail with aspects of human governance, particularly where AI augments or replaces human decision-makers, and in some parts similarly to technology governance and in accordance with best practice in data-governance.
Australian Defence has confirmed commitments to non-AI governance of humans and technology as detailed in the section below.
Human Governance in Defence
Expectations of human decision-makers are likely to be applied if not extended whenever AI influences or replaces human decision-making, including moral and legal responsibilities. 2. It also includes responsibility for health, welfare, morale and discipline of assigned personnel.
Within the Definition of Command is authority and responsibility over military decision making including the use of physical or digital resources such as how and when AI is deployed.
When making decisions, the Australian Defence Force Leadership Doctrine (ADF-P-0, Ed.
3)(2021a) states "Ethical leadership is the single most important factor in ensuring the legitimacy of our operations and the support of the Australian people".
Suggesting that Command is expected to deploy digital assets ethically, the Leadership Doctrine argues in no uncertain terms that "your responsibility as a leader is to ensure the pursuit of your goals is ethical and lawful. There are no exceptions" (Australian Defence Force 2021a, 7). The Australian Defence Force -Philosophical -0 Military Ethics Doctrine (Australian Defence Force 2021b) breaks down ethical leadership into a framework including intent, values, evaluate, lawful and reflect (see Figure 1). Education Doctrine (Australian Defence Doctrine Publication 2021) emphasises the importance of "Innovative and inquiring minds" that are "better equipped to adapt to fastchanging technological, tactical and strategic environments". Abilities sought include:
• objectively seek and identify credible information,
• accurately recognise cues and relationships,
• quickly make sense of information and respond appropriately.
Ethical AI could contribute and augment human capabilities for a faster and more agile force.
Australian Defence personnel using AI to augment decision making will be expected to use it ethically and lawfully; to increase the informativeness and evidence-base for decisions; and for decision-makers to agile and accountable both with and without AI.
Ethics in Australian Cybersecurity and Intelligence
Key strategic threats for Australia are cybercrime, ransomware and information warfare. Cyber security incidents are increasing in frequency, scale and sophistication, threatening Australia's economic prosperity and national interests (White 2021). AI is likely to play a role in both decision support and in autonomous defence and offensive campaigns to thwart those who seek to undermine Australia's interests.
Australia has not published an ethics of AI policy for cybersecurity or intelligence. However, ethical behaviours are highlighted in publicly available value statements, such as "we always The Australian Government is also committed to working with businesses on potential legislative changes including the role of privacy, consumer and data protection laws (Cyber Digital and Technology Policy Division 2020).
Defence Data Strategy
In 2021, Defence released a Defence Data Strategy. The strategy promises a Data Security Policy to ensure the adoption of a risk-based approach to data security that allows Defence more latitude to respond to the increase in grey-zone activities, including cyber-attacks, and foreign interference, and a renewed focus on data security and storage processes. Defence Nevertheless, Defence is committed to producing training so that personnel are "equipped to treat data securely and ethically" by 2023 (Department of Defence 2021a, 13).
Framework for Ethical AI in Defence
Australia has not adopted an ethics framework specifically for AI use in Defence. However, a Defence Science and Technology technical report based on outcomes from an evidence-based
Responsibility: Who is responsible for AI?
MEAID notes two key challenges of understanding and responsibility that must be addressed when operating with AI systems. Firstly, in order to effectively and ethically employ a given system (AI or not), the framework argues that a commander must sufficiently understand its behaviour and the potential consequences of its operation (Devitt et al. 2021, 11). Secondly, there can be difficulty in identifying any specific individual responsible for a given decision or action.
Responsibility for critical decisions is spread across multiple decision-makers offering multiple opportunities to exercise authority but also to make mistakes. The allocation of ethical and legal responsibility could be distributed across the nodes/agents in the human-AI network causally relevant for a decision (Floridi 2016). However, legal responsibility ultimately lies with humans. Additionally, AI could help reduce mistakes and augment human makers who bear responsibility (Ekelhof 2018). Decisions made with the assistance of or by AI are captured by accountability frameworks including domestic and international law 9 .
The Department of Defence can examine legal cases of responsibility in the civilian domain to guide some aspects of the relevant frameworks, e.g. the apportioning of responsibility for the test-driver in an Uber automated vehicle accident (Ormsby 2019). Defence could also consider arguments that humans within complex systems without proactive frameworks risk being caught in moral crumple zones (Elish 2019) where the locus of responsibility falls on human operators rather than the broader system of control within which they operate. Defence must keep front of mind that humans, not AI, have legal responsibilities, and that Individuals-not only states-can bear criminal responsibility directly under international law (Cryer, Robinson, and Vasiliev 2019).
Governancehow is AI controlled?
MEAID suggests that AI creators must consider the context in which AI is to be used and how AI will be controlled. The point of interface through which control is achieved will vary, depending on the nature of the system and the operational environment. There must be work conducted to understand how humans can be capable of operating ethically within machinebased systems of control in accordance with Australia's commitment to Article 36 reviews of all new means and methods of warfare (Commonwealth of Australia 2018) 10 . 9 There is sometimes considerable uncertainty about exactly how to apply legal frameworks to decisions made with significant AI involvement. 10
Trusthow can AI be trusted?
Human-AI systems in Defence need to be trusted by users and operators, by commanders and support staff and by the military, government, and civilian population of a nation. MEAID
points out the High-Level Expert Group on Artificial Intelligence of the European Union
"believe it is essential that trust remains the bedrock of societies, communities, economies and sustainable development" (High-Level Expert Group on Artificial Intelligence 2019). They argue that trustworthy AI must be lawful, ethical, and robust.
MEAID suggests that trust is a relation between human-human, human-machine and machinemachine, consisting of two components: competency and integrity. Competence comprises of skills, reliability and experience; Integrity comprises of motives, honesty and character (Devitt 2018). This framework is consistent with the emphasis on character and professional competence in ADF-P-0 ADF Leadership Doctrine (Australian Defence Force 2021a, 4). It is noted that the third value of ADF leadership is understanding, which falls within the responsibility facet discussed above.
Operators will hold multiple levels of trust in the systems they are using depending on what aspect of trust is under scrutiny. In some cases, users may develop a reliance on low integrity technology that they can predict easily, such as using the known flight path of an adversary's drone to develop countermeasures. Users may also depend on technologies because of convenience rather than trust. Finally individual differences exist in the propensity to trust, highlighting that trust is a relational rather than an objective property.
To be trusted, AI systems need to be safe and secure within the Nation's sovereign supply chain. Throughout their lifecycle, AI systems should reliably operate in accordance with their intended purpose (Department of Industry Innovation and Science 2019).
Law: How can AI be used lawfully?
AI developers should be cognisant of the legal obligations within their anticipated use of the technology. Law within a Defence context has specific ethical considerations that must be understood. International humanitarian law (IHL) (lex specialis) and international human rights law (lex generalis) were forged from ethical theories in just war theory jus ad bellum governing the resort to force, jus in bello regulating the conduct of parties engaged in lawful combat (Coates 2016), jus post bellum regarding obligations after combat. The legal frameworks that accompany Defence activities are human-centred, which should mean that AI compliance with them will produce more ethical outcomes (Liivoja and McCormack 2016).
Using AI to augment human decision-making could lead to better humanitarian outcomes.
There are many policies and directives that may apply, some of which have the force of law.
In military context, there will also typically be an extant set of rules called the rules of engagement, which among other things specify the conditions that must be met in order to fire upon a target.
Legal compliance may be able to be 'built into' AI algorithms, but this relies on legal rules being sufficiently unambiguous and well specified that they can be encoded as rules that a computer can interpret and meets stakeholder expectations. In practice, laws are not always that clear, even to humans. Laws can intentionally be created with ambiguity to provide flexibility. In addition, they can have many complicated conditions and have many interconnections to other laws. Further work is needed to clarify how AI can best enable abidance with applicable laws.
Traceability: How are the actions of AI recorded?
MEAID notes that there are legislative requirements for Defence to record its decision-making.
However, the increasing use of AI within human-AI systems means the manner of records must be considered. Records can represent the systems involved, the causal chain of events, and the humans and AIs that were part of decisions.
MEAID suggests that information needs to be accessible and explanatory; the training and expertise of humans must be open to scrutiny; and the background theories and assumptions, training, test and evaluation process of AIs must be retained. Information on AI systems should be available and understandable by auditors. Just as some aspects of human decision-making can be inscrutable, some aspects of the decisions of AIs may remain opaque. Emerging transparency standards may guide best practise for Defence (Winfield et al. 2021).
When decisions lead to expected outcomes or positive outcomes, the factors that lead to those decisions may not come under scrutiny. However, when low likelihood and/or negative outcomes occur, organisations should be able to 'rewind' the decision process to understand what occurred and what lessons might be learned. Noting that decisions made under uncertainty will always have a chance of producing negative outcomes, even if the decision-making process is defensible and operators are acting appropriately.
No matter how an AI is deployed in Defence, its data, training, theoretical underpinning, decision-making models and actions should be recorded and auditable by the appropriate levels of government and, where appropriate, made available to the public.
Method for Ethical AI in Defence
MEAID recommends assessing ethical compliance from design to deployment, requiring repeated testing, prototyping, and reviewing for technological and ethical limitations.
Developers already must produce risk documentation for technical issues. Similar documentation for ethical risks ensures developers identify, acknowledge and attempt to mitigate ethical risks early in the design process and throughout test and evaluation .
MEAID closely aligns to the IEEE Standard Model Process for Addressing Ethical Concerns
During System Design (IEEE 2021)-see Box 7.
Box 7 IEEE 7000-2021 standard (IEEE 2021) provides:
• a system engineering standard approach integrating human and social values into traditional systems engineering and design.
• processes for engineers to translate stakeholder values and ethical considerations into system requirements and design practices.
• a systematic, transparent, and traceable approach to address ethically-oriented regulatory obligations in the design of autonomous intelligent systems.
Australia has developed a practical methodology ) that can support AI project managers and teams to manage ethical risks in military AI projects including three tools:
1. An AI Checklist for the development of ethical AI systems 2. An Ethical AI Risk Matrix to describe identified risks and proposed treatment 3. For larger programs, a data item descriptor (DID) for contractors to develop a formal Legal, Ethical and Assurance Program Plan (LEAPP) to be included in project documentation for AI programs where an ethical risk assessment is above a certain threshold (See APPENDIX G. DATA ITEM DESCRIPTION DID-ENG-SW-LEAPP Devitt et al. 2021)).
AI Checklist
The main components of the checklist are:
A. Describe the military context in which the AI will be employed B. Explain the types of decisions supported by the AI C. Explain how the AI integrates with human operators to ensure effectiveness and ethical decision making in the anticipated context of use and countermeasures to protect against potential misuse D. Explain framework/s to be used E. Employ subject matter experts to guide AI development F. Employ appropriate verification and validation techniques to reduce risk.
Ethical AI Risk Matrix
An Ethical AI Risk Matrix will:
• Define the activity being undertaken
• Indicate the ethical facet and topic the activity is intended to address.
• Estimate the risk to the project objectives if issue is not addressed?
• Define specific actions you will undertake to support the activity • Provide a timeline for the activity • Define action and activity outcomes • Identify the responsible party(ies)
• Provide the status of the activity.
Analysis
MEAID offers practical advice and tools for defence industries and Defence to communicate, document and iterate design specifications for emerging technologies and to identify operational contexts of use considerate of ethical and legal considerations and obligations.
MEAID also offers entry points to explain system function, capability, and limits to both expert and non-expert stakeholders to military technologies.
MEAID aims to practically ensure accountability for a) considering ethical risks, b) assigning person(s) to each risk and c) making humans accountable for decisions on how ethics are derisked. It has been noted on the International Committee of the Red Cross blog (Copeland and Sanders 2021) as establishing an iterative process to engage industry during the design and acquisition phase of new technologies to increase IHL abidance and reduce civilian harms.
Developed by Defence Science and Technology Group, Plan Jericho Air Force (Department of Defence 2020b) and Trusted Autonomous Systems 11 ; the MEAID framework has been adopted by industry 12 , is the ethics framework used in a case study of Allied Impact (Gaetjens, Devitt, and Shanahan 2021) and being trialled by Australia in the TTCP AI Strategic Challenge Australia's Department of Defence may inform Defence industry its legal and ethical requirements to enable AI developers to introduce design measures to mitigate or remove the risks before entering the Defence procurement process, rather than attempting to address legal and ethical risks during the acquisition process. This provides efficacies for both Defence and industry by allowing industry to better focus their development priorities and assists Defence in streamlining its AI capability acquisition process.
MEAID provides Defence with a practical approach that can readily integrate into the existing Product Life Cycle process to inform and enable the transfer of legal and ethical AI technology from Defence industry into Defence. MEAID tools such as the LEAPP provide Defence with visibility of a contractor's plan to mitigate legal and ethical risk and, together with the facets of ethical AI (see Figure 2) and the Article 36 weapon review process, can inform Government decisions to acquire military AI technology that both legal and align with Australia's ethical principles.
Australia can play a leadership role by integrating legal and ethical considerations into its Defence AI capability acquisition process. This requires a policy framework that defines its legal and ethical requirements, is informed by Defence industry stakeholders and provides a practical methodology to integrate legal and ethical risk mitigation strategies into the acquisition process.
Conclusion
This chapter explored Australia's public positioning on AI and AI governance 2018-2021 through published strategies, frameworks, action plans and government reports. While these provide a top-down view, high-level national AI strategies may align with the lived experience of public servants and personnel encountering AI in Defence or Australian Defence Force commanders and operators using AI systems (Kuziemski and Misuraca 2020).
Australia is a leading AI nation with strong allies and partnerships. It has prioritised the development of robotics, AI, and autonomous systems to develop sovereign capability for the military. Australia commits to Article 36 reviews of all new means and method of warfare to ensure weapons and weapons systems are operated within acceptable systems of control.
Additionally, the country has undergone significant reviews of the risks of AI to human rights and within intelligence organisations and has committed to producing ethics guidelines and frameworks in Security and Defence (Department of Defence 2021a; Attorney-General's Department 2020). Australia is committed to OECD's values-based principles for the responsible stewardship of trustworthy AI as well as adopting a set of National AI ethics principles. While Australia has not adopted an AI governance framework specifically for Defence; A Method for Ethical AI in Defence (MEAID) published by Defence Science includes a framework and pragmatic tools for managing ethical and legal risks for military applications of AI.
Key findings of the chapter are that Australia has formed strong international AI governance partnerships likely to reinforce and strengthen strategic partnerships and power relations. Like many nations, Australia's commitment to civilian AI Ethics principles do not provide military guidance or governance. The ADO has the opportunity to adopt a robust AI ethical policy for security and defence that emphasises commitment to existing international legal frameworks and can be applied to AI-driven weapons. A risk-based ethical AI framework suited for military purposes and aligned with best practise, standards and frameworks internationally can ensure defence industries consider ethics-by-design and law-by-design ahead of the acquisitions process. Australia should continue to invest, research and develop AI governance frameworks to meet the technical potential and strategic requirements of military uses of AI.
Introduction
On 27 February 2021, Australia's Loyal Wingman military aircraft hinted at the possibility of fully autonomous flight at Woomera Range Complex in South Australia (Royal Australian Air Force 2021; Insinna 2021). With no human on board, the plane used a pre-programmed route with remote supervision to undertake and complete its mission. The flight's success and the Royal Australian Air Force's announcement to order six aircraft, signalled an intention to incorporate artificial intelligence (AI) to increase military autonomous capability and freedom of manoeuvre. Air Vice-Marshal (AVM) Cath Roberts (Head of Air Force
1 .
1Australia's national research organisation, CSIRO Data61 (Hajkowicz et al. 2019), 2. Government (Department of Industry Innovation and Science 2019; Department of Industry Science Energy and Resources 2021, 2020a, 2020b) in the civilian domain; and 3. Defence Science and Technology Group, Royal Australian Air Force and Trusted
•
A 'fair go' for all that embraces: o mutual respect; o tolerance; o compassion for those in need; o equality of opportunity for all • The English language as the national language, and as an important unifying element of Australian society projects on AI priorities to advance the responsible development and use of AI built around a shared commitment to the OECD Recommendation on Artificial Intelligence 5 . The OECD has demonstrated considerable "ability to influence global AI governance through epistemic authority, convening power, and norm-and agenda-setting" (Schmitt 2021). Since 2018, Australia has used a consultative methodology and public communication of evidence-based ethics frameworks in both civil and military domains. The civil domain work driven by CSIRO's Data61 (Dawson et al. 2019) and the military work driven by Defence Science and Technology Group (DSTG) (Devitt et al. 2021).
2 .
2Human-centred values: AI systems should respect human rights, diversity, and the autonomy of individuals. 3. Fairness: AI systems should be inclusive and accessible and should not involve or result in unfair discrimination against individuals, communities, or groups. 4. Privacy protection and security: AI systems should respect and uphold privacy rights and data protection and ensure the security of data. 5. Reliability and safety: AI systems should reliably operate in accordance with their intended purpose.
2 .
2Explore avenues for enhanced cooperation with the United States National Institute for Standards & Technology (NIST) and other Standards Development Organisations (SDOs) with the aim of improving Australia's knowledge and influence in international AI Standards development. 3. The Australian Government nominate government experts to participate in ISO/IEC/JTC 1/SC 42, and the National Mirror Committee (IT-043). The Australian Government should also fund and support their participation, particularly at international decision-making meetings where key decisions are made, within existing budgetary means.
weapons systems for over 40 years(Department of Defense 1978). Such weapons require thorough test and evaluation to identify and mitigate risks of computer malfunction. This has lead to a recent drive for digital engineering (88th Air Base Wing Public Affairs 2019; National Security Commission on Artificial Intelligence 2021). AI and autonomous weapons system (AWS) do not necessarily coincide, but the application of Australia's international and domestic legal obligations to AI weapon systems will almost certainly affect Australia's ability to develop, acquire and operate autonomous military systems. Australia has stated that it considers a sweeping prohibition of AWS to be premature (Australian Permanent Mission andConsulate-General Geneva 2017; Commonwealth of Australia 2018; Senate Foreign Affairs Defence and Trade Legislation Committee 2019, 65) and emphasises the importance in compliance with the legal obligation to undertake Article 36 reviews to manage the legal risks associated with these systems. the study, development, acquisition or adoption of a new weapon, means or method of warfare, a High Contracting Party is under an obligation to determine whether its employment would, in some or all circumstances, be prohibited by this Protocol or by any other rule of international law applicable to the High Contracting Party." The Article 36 process requires Australia to determine whether it can meet its international legal obligations in operating AWS. Performing a thorough Article 36 review requires consideration of International Humanitarian Law ('IHL') prohibitions and restrictions on weapons, including Customary International Law, and an analysis of the normal or expected use of the AWS against the IHL rules governing the lawful use of weapons (i.e. distinction, proportionality and precautions in attack). This includes ensuring weapon operators understand their functions and limitations as well as the likely consequences of their use. Thus, users of AWS are legally required to be reasonably confident about how they will operate before deploying them (Liivoja et al. 2020). The ADF Concept for Future robotics and autonomous systems (Vine 2020) states: "3.10 Existing international law covers the development, acquisition and deployment of any new and emerging capability, including future autonomous weapons systems." "3.44 Australia has submitted two working papers to the LAWS GGE in an attempt to demonstrate how existing international humanitarian law is sufficient to regulate current and envisaged weapon systems; the first (Commonwealth of Australia 2018) explained the article 36 weapon review process and the second (Australian Government 2019) outlined the 'System of Control' which regulates the use of force by the ADF. Within the domestic legal system, the RAS (particularly drones) is being considered in the development and review of legislation on privacy, intelligence services and community safety." Australia argues that "if states uphold existing international law obligations…there is no need to implement a specific ban on AWS, at this time" (Commonwealth of Australia 2019). However, the 2015 Senate Committee Report on unmanned platforms said 'the committee is not convinced that the use of AWS should be solely governed by the law of armed conflict, international humanitarian law and existing arms control agreements. A distinct arms control regime for AWS may be required in the future" (see para 8.30). The report recommended that: "8.33 … the Australian Government support international efforts to establish a regulatory regime for autonomous weapons systems, including those associated with unmanned platforms." Australia welcomes discussion (e.g. McFarland 2021, 2020) around international legal frameworks on autonomous weapons and how technological advances in weapons systems can comply with international humanitarian law (Senate Foreign Affairs Defence and Trade Legislation Committee 2019).Ethical AI Statements Across the ServicesDifferent defence institutions in Australia have addressed the importance of ethical and legal aspects of AI in their operations. The Royal Australian Navy (2020) stated that "development of trusted autonomous systems is expected to increase accuracy, maintain compliance with Navy's legal and policy obligations as well as regulatory standards, and if utilised during armed conflict, minimise incidental harm to civilians". The Army (2018) said it would "remain cognisant of the ethical, moral and legal issues around the use of RAS technologies as this strategy evolves and is implemented". Finally, the Royal Air Force (RAAF) (2019, 10-11) mentioned that it would explore ways to ensure ethical and moral values and legal accountabilities remain central, including continuously evaluating which decisions can be made by machines and which must be made by humans. The exploration and pursuit of augmented intelligence must be transparent and accountable to the RAAF's legal, ethical, and moral values and obligations. Greater engagement with risk and opportunity must be matched by accountability and transparency.
The authority which a commander in the military Service lawfully exercises over subordinates by virtue of rank or assignment.
Figure 1 :
1Australian Defence Force Ethical Decision-Making Framework (Figure 5.1, Australian Defence Force 2021b). The Lead the Way: Defence Transformation Strategy articulates that Defence wants human decision-makers to be agile, adaptive and ethical with a continuous improvement culture "embedding strong Defence values and behaviours, clear accountabilities and informed and evidence-based decision-making" (Department of Defence 2020c, 21). ADF-P-7 The
act legally and ethically" (Australian Signals Directorate 2019a, 2019b) and communications suggestive that Australia would expect strong governance of AI systems used in these operations including abidance with domestic and international law and the values of government organisations. For example, speaking to the Lowy Institute Director-General of Australian Signals Directorate (ASD), Mike Burgess (2019) highlighted that "rules guide us when people are watching; values guide us when they're not" (Lowy Institute 2019 51:46) and that ASD is "an organisation that is actually incredibly focused on doing the right thing by the public and being lawful that's an excellent part of our culture born out of our values we put a lot of effort focusing on that" (Lowy Institute 2019 52:27). A comprehensive review of the legal framework of the national intelligence community highlights the importance of accountability, transparency and oversight of how the Australian government collects and uses data (Richardson 2020). The government response to the Richardson report (Attorney-General's Department 2020, 40-41) agrees that governance and ethical frameworks should be developed for the use of artificial capabilities for intelligence purposes (recommendation 154), citing values including control, oversight, transparency, and accountability (recommendations 155-156). The Australian government noted the importance of human-in-the-loop decision-making where a person's rights or interests may be affected or where an agency makes an adverse decision in relation to a person (recommendation 155).
identified ethical considerations as a key component of their data strategy-see Box 6. While they commit to being informed by The Australian Code for the Responsible Conduct of Research and the National Statement on Ethical Conduct in Research (Australian Research Council 2020), neither of these codes provides any guidance on the development of AI for Defence or security purposes. Box 6 Ethical data, Defence Data Strategy 2021-2023 (Department of Defence 2021a, 42) Guidelines around the ethical use of data will be developed to ensure we have a shared understanding of our legislative and ethical responsibilities … The Australian Code for the Responsible Conduct of Research and the National Statement on Ethical Conduct in Research will inform these guidelines. The ethical use of data guidelines will form part of the Defence Human and Animal Research Manual and policies.
workshop 7 has recommended a method for ethical AI in Defence (MEAID) (Department of Defence 2021b; Devitt et al. 2021) and an Australia-specific framework to guide ethical risk mitigation. MEAID draws from the workshop for further consideration and does not represent 7 Workshop held in Canberra 30 Jul to 1 Aug 2019 with 104 people from 45 organisations including representatives from Defence, other Australian government agencies, the Trusted Autonomous Systems Defence Cooperative Research Centre (TASDCRC), civil society, universities and Defence industry the views of the Australian Government. Rather than stipulating principles, MEAID identifies five facets of ethical AI and corresponding questions to support science and technical considerations for the potential development of Defence policy, doctrine, research and project management: Responsibilitywho is responsible for AI?; Governancehow is AI controlled?; Trusthow can AI be trusted?; Lawhow can AI be used lawfully? And Traceability -How are the actions of AI recorded?-see Figure 2.
Figure 2 :
2Facets of Ethical AI in DefenceMEAID notes that facets of ethical AI for Defence and the associated questions align with the unique concerns and regulatory regimes to which Defence is subject. For example, in times of conflict, Defence is required to comply with international humanitarian law (IHL, lex specialis) and international human rights law (lex generalis) in armed conflict (jus in bello 8 ). Defence is also required to comply with international legal norms with respect to the use of force when not engaged in armed conflict (jus ad bellum) when applying military force. Australia's inclusion of 'Law' as an ethical facet highlights the values Australia promotes through abidance with international humanitarian law, particularly the concepts of proportionality, distinction and military necessity which have no direct non-military equivalent and as such requires consideration of a specific set of requirements and responsibilities.
. The Protocol Additional to the Geneva Conventions of 12 August 1949, and relating to the Protection of Victims of International Armed Conflicts (Protocol I), 8 June 1977 refers alternately to ''methods or means of warfare'' (Art. 35(1) and (3), Art. 51(5)(a), Art. 55(1)), ''methods and means of warfare'' (titles of Part III and of Section I of Part III), ''means and methods of attack'' (Art. 57(2)(a)(ii)), and ''weapon, means or method of warfare'' (Art. 36) (International Committee of the Red Cross 2006) With regards to the control of lethal autonomous weapons, Australia notes the legal, policy, technical, and professional forms of controls imposed systematically throughout the ': Strategic and Military Controls for the Use of Force Stage Nine: After-Action Evaluation MEAID supports the governance framework of IEEE's Ethically Aligned Design by the IEEE Global Initiative on Ethics of Autonomous and Intelligent Systems (2019). Human-machine collaboration should be optimised to safeguard against poor decision-making including automation bias and/or mistrust of the system (Hoffman et al. 2018; Alexander 2019). AI should provide confidence and uncertainty in the information or choices being offered by an AI (Christensen and Lyons 2017; McLellan 2016).
(
Stanley-Lockman 2021, 43-44).A side-by-side comparison between AU-EP, MEAID and OECD shows significant overlap in responsibility and trust, but also gaps where military uses of AI encounter ethical considerations not applicable to the civilian realm, such as the application of just war principles of distinction and proportionality (Law); and military control of weapons systems (Governance)-see ANNEX B. The AU-EP share many similarities with the OECD. For example, contestability is equivalent to OECD requirement that humans can understand and intervene on AI-based outcomes as well as challenge themWhile not a formally adopted view of the Australian government, MEIAD establishes tools to assess ethical compliance that, "[e]ven as an opinion, the Method is the clearest articulation of ethical AI for defense among the Indo-Pacific allies" (Stanley-Lockman 2021, 21). As Stanley-Lockman (2021) states:"The [MEAID] tools offer a process to validate that contractors have indeed taken the ethical risks they identified into account in their design and testing prior to later acquisition phases….The incorporation of ethics in design through the acquisition lifecycle also intends to build trust in the process and, by extension, the systems by the time they go into service." Like many countries, ADO undertakes a formal capability acquisition process to assist the Government to identify and meet its military capability needs. This process, known as the Product Life Cycle, consist of four phases (strategy and concepts; risk mitigation and requirement setting; acquisition; and in-service and disposal) which are separated by Government decisions gates (Commonwealth of Australia 2020). This process ensures that Government's strategic objectives (see Box 1) drive Defence's acquisition priorities.Australia's Sovereign Industry Capability Priorities (see Box 2) reflects the Government's realisation that future Defence AI capabilities will increasingly rely on research and development in the civil sector. This necessitates closer collaboration between Defence and Defence industry to ensure the timely delivery of cutting-edge technology the reflects Australia's values (see Box 3) and ethical AI principles (see Box 4) and ensures legal and ethical risks associated with military AI technology are identified and mitigated in the earliest stages of development.
) .
)The ADO acknowledges that they need to effectively use their data holdings to harness theopportunities of AI technologies and the Defence Artificial Intelligence Centre (DAIC) has been established to accelerate Defence's AI capability (Department of Defence 2021a, 35; 2020b). The ADO has launched The AI for Decision Making Initiative (Defence Science Institute 2020a, 2021; Defence Science & Technology Group 2021a) and a Defence Artificial
ANNEX A Contexts of AI in Defence Combat/Warfighting Description The conduct of military missions to achieve decisive effects through kinetic and non-kinetic offensive means. AI examples Autonomous weapons (AWs) and autonomous/semi-autonomous combat vehicles and subsystems AI used to support strategic, operational and tactical planning, including optimisation and deployment of major systems AI used in modelling and simulation used for planning and mission rehearsal AI used in support of the targeting cycle including for collateral damage estimation AI used for Information Warfare such as a Generative Adversarial Network (GAN-) generated announcement or strategic communication AI used to identify potential vulnerabilities in an adversary force to attack AI used for discrimination of combatants and non-combatants Description All measures to counter threats and hazards to, and to minimise vulnerabilities of, the joint force in order to preserve freedom of action and operational effectiveness AI examples Autonomous defensive systems (i.e. Close in Weapons Systems) AI used for Cyber Network Defence AI used to develop and employ camouflage and defensive deception systems and techniques Autonomous decoys and physical, electro-optic or radio frequency countermeasures AI to identify potential vulnerabilities in a friendly force that requires protection AI used to simulate potential threats for modelling and simulation or rehearsal activities Autonomous Medical Evacuation/Joint Personnel Recovery systems Description Activities conducted to sustain fielded forces, and to establish and maintain expeditionary bases. Force sustainment includes the provision of personnel, logistic and any other form of support required to maintain and prolong operations until accomplishment of the mission. AI examples Autonomous combat logistics and resupply vehicles Automated combat inventory management Predictive algorithms for the expenditure of resources such as fuel, spares and munitions Medical AI systems used in combat environments and expeditionary bases Predictive algorithms for casualty rates for personnel and equipment Algorithms to optimise supply chains and the recovery, repair and maintenance of equipment Algorithms to support the provision of information on climate, environment and topography AI used for battle damage repair and front-line maintenance Description The accurate interpretation of a situation and the likely actions of groups and individuals within it. Situational Understanding enables timely and accurate decision making. AI examples AI that enables or supports Intelligence, Surveillance and Reconnaissance (ISR) activities including: object recognition and categorisation of still and full motion video removal of unwanted sensor data identification of enemy deception activities anomaly detection and alerts monitoring of social media and other open-source media channels optimisation of collection assets AI that fuses data and disseminates intelligence to strategic, operational and tactical decision makers Decision support tools Battle Management Systems AI that supports Command and Control functions Algorithms used to predict likely actions of groups and individuals AI used to assess individual and collective behaviour and attitudes Enterprise-level and Rear Echelon Functions Description All activities that support the Raising, Training and Sustaining (RTS) of personnel. AI examples AI used for Human Resource Management including: record keeping posting and promotion disciplinary and performance management recruitment and retention modelling of future personnel requirements prediction of HR supply and demand events and anomalies AI used in individual and collective training and education including modelling and simulation AI used for testing and certification of personnel AI used to model the capability and preparedness of permanent and reserve Description Activities that support rear-echelon enterprise-level logistics functions including support of permanent military facilities AI examples Autonomous rear-echelon supply vehicles and warehouses AI used for optimisation of rear-echelon supply chains and inventory management AI used in depot-level and intermediate maintenance, including: Global supply chain analysis, prediction and optimisation Enterprise-level analysis and prediction for resource demand and supply (i.e. national/strategic fuel requirements) ANNEX B Side-by-Side Comparison of AI Ethics Frameworks Throughout their lifecycle, AI systems should respect human rights, diversity, and the autonomy of individuals 1. AI should benefit people and the planet by driving inclusive growth, sustainable development and well-being. 2. AI systems should be designed in a way that respects … human rights, democratic values and diversity, Contestability: When an AI system significantly impacts a person, community, group or environment, there should be a timely process to allow people to challenge the use or output of the AI system 3. There should be transparency and responsible disclosure around AI systems to ensure that people understand AI-based outcomes and can challenge them. respect and uphold privacy rights and data protection, and ensure the security of data 4. AI systems must function in a robust, secure and safe way throughout their life cycles andTag
Force Application (FA)
Tag
Force Protection (FP)
Tag
Force Sustainment (FS)
Tag
Situational Understanding (SU)
Tag
Personnel (PR)
personnel
Tag
Enterprise Logistics (EL)
Digital twinning
Predictive maintenance
Facets of Ethical AI
in Defence
Australian Government's AI
Ethics Principles
OECD
RESPONSIBILITY:
Who is responsible
for AI?
Human, social and
environmental wellbeing:
Throughout their lifecycle, AI
systems should benefit
individuals, society and the
environment
Human-centred values:
GOVERNANCE:
How is AI
controlled?
Accountability: Those
responsible for the different
phases of the AI system
lifecycle should be identifiable
and accountable for the
outcomes of the AI systems,
and human oversight of AI
systems should be enabled
5. Organisations and
individuals developing,
deploying or operating AI
systems should be held
accountable for their proper
functioning in line with the
above principles
Transparency and
explainability: There should be
transparency and responsible
disclosure to ensure people
know when they are being
significantly impacted by an AI
system, and can find out when
an AI system is engaging with
them
TRUST:
How can AI be
trusted?
Reliability and safety:
Throughout their lifecycle, AI
systems should reliably operate
in accordance with their
intended purpose
Fairness: Throughout their
lifecycle, AI systems should be
2. [AI systems] should include
appropriate safeguards -for
example, enabling human
intervention where necessary -
to ensure a fair and just
society.
inclusive and accessible, and
should not involve or result in
unfair discrimination against
individuals, communities or
groups
Privacy protection and
security: Throughout their
lifecycle, AI systems should
potential risks should be
continually assessed and
managed.
LAW:
How can AI be used
lawfully?
No equivalent
2. AI systems should be
designed in a way that respects
the rule of law
TRACEABLILITY:
How are the actions
of AI recorded?
No equivalent (but implied)
No equivalent (but implied)
Team included Commonwealth Scientific and Industrial Research Organisation (CSIRO) Data61 and Emesent; plus International partner Georgia Institute of Technology
See https://www1.defence.gov.au/about/people-group
Note there is little representation from remaining ASEAN nations Brunei, Cambodia, Indonesia, Laos, Malaysia, Myanmar, the Philippines, Thailand and Vietnam or Pacific Nations
'jus in bello' usually refers to specifically to IHL even though human rights law still operates in conflict
See https://tasdcrc.com.au/ 12 See Athena AI at https://athenadefence.ai/software
AI used in the day-to-day operation of permanent military facilitiesTagBusiness Process Improvement (BP)DescriptionActivities that support rear-echelon administrative business processes that are not related to personnel or logistics.AI examples AI used for Information Management and record-keepingInformational assistants such as policy chatbots AI that supports management of policy and procedures AI used to optimise business and administrative processes, including modelling and simulation tools AI used for enterprise business planning at the strategic, operational and tactical level
Digital engineering transformation coming to Air Force weapons enterprise. Air Base Wing Public AffairsAir Base Wing Public Affairs, 23 December, 2019, "Digital engineering transformation coming to Air Force weapons enterprise," https://www.af.mil/News/Article- Display/Article/2046599/digital-engineering-transformation-coming-to-air-force- weapons-enterprise/.
AI Roadmap. A I Council, AI Council. 2021. "AI Roadmap." United Kingdom. https://www.gov.uk/government/publications/ai-roadmap.
Is our reliance on technology creating a new dark age?. Donovan Alexander, Interesting EngineeringAlexander, Donovan. 2019. "Is our reliance on technology creating a new dark age?" Interesting Engineering, 10 May, 2019. https://interestingengineering.com/is-our- reliance-on-technology-creating-a-new-dark-age.
Robotic and Autonomous Systems Strategy. Army, Army. 2018. "Robotic and Autonomous Systems Strategy." https://researchcentre.army.gov.au/library/other/robotic-autonomous-systems- strategy.
. " Aspi, A I Russia-Ukraine War, Darpa An Australian, Policy, Guns and Money. ASPI, "Russia-Ukraine war, policing and AI, and an Australian DARPA," 4 March, 2022, in Policy, Guns and Money, https://www.aspistrategist.org.au/policy-guns-and-money- russia-ukraine-war-policing-and-ai-and-an-australian-darpa/.
Government response to the Comprehensive review of the legal framework of the National Intelligence Community. Accessed 25Attorney-General's Department. 2020. "Government response to the Comprehensive review of the legal framework of the National Intelligence Community." Accessed 25 September 2021. https://www.ag.gov.au/national-security/publications/government- response-comprehensive-review-legal-framework-national-intelligence-community.
ADF-P-0 ADF Leadership, Edition 3. Australian Defence Doctrine Publication. 2021. ADF-P-7 Learning. 1Accessed 15 OctoberAustralian Defence Doctrine Publication. 2021. ADF-P-7 Learning. Australian Defence Force. 2021a. "ADF-P-0 ADF Leadership, Edition 3." The Forge. https://theforge.defence.gov.au/adf-philosophical-doctrine-adf-leadership ---. 2021b. "ADF-P-0 Military Ethics, Edition 1, 2021." Accessed 15 October. https://theforge.defence.gov.au/ethics.
Group of Governmental Experts on Emerging Technologies in the Area of Lethal Autonomous Weapons Systems. Australian Government, Geneva/CCWGGE.12019WP.2Rev.1.pdf. ---. 2021Four new Sovereign Industrial Capability Priorities announcedAustralian Government. 2019. "Australia's System of Control and applications for Autonomous Weapon Systems." Group of Governmental Experts on Emerging Technologies in the Area of Lethal Autonomous Weapons Systems, Geneva, 25-29 March 2019 and 20-21 August 2019. Accessed 10 September. https://docs- library.unoda.org/Convention_on_Certain_Conventional_Weapons_- _Group_of_Governmental_Experts_(2019)/CCWGGE.12019WP.2Rev.1.pdf. ---. 2021. "Four new Sovereign Industrial Capability Priorities announced." 7 September, 2021. https://business.gov.au/cdic/news-for-defence-industry/four-new-sovereign- industrial-capability-priorities-announced.
Australian Permanent Mission and Consulate-General Geneva. LAWS GGE 13-17Australian Statement -General Exchange of ViewsAustralian Permanent Mission and Consulate-General Geneva. 2017. "Australian Statement - General Exchange of Views, LAWS GGE 13-17 November 2017." https://geneva.mission.gov.au/gene/Statement783.html.
Codes and Guidelines. Australian Research Council, Australian Research Council. 2020. "Codes and Guidelines." https://www.arc.gov.au/policies-strategies/policy/codes-and-guidelines.
Aukus pact strikes at heart of Pacific regionalism. Accessed. 25Radio New Zealand PacificAustralian Signals Directorate. 2019a. "ASD Corporate Plan 2019-2020." https://www.asd.gov.au/sites/default/files/2019- 08/ASD_Corporate_Plan_final_12.pdf. ---. 2019b. "Values." Accessed 25 September. https://www.asd.gov.au/about/values. Blades, Johnny. 2021. "Aukus pact strikes at heart of Pacific regionalism." Radio New Zealand Pacific, 2021. https://www.rnz.co.nz/international/pacific- news/451715/aukus-pact-strikes-at-heart-of-pacific-regionalism.
Beyond the bubble that is Robodebt: How governments that lose integrity threaten democracy. Valerie Braithwaite, Australian Journal of Social Issues. 553Braithwaite, Valerie. 2020. "Beyond the bubble that is Robodebt: How governments that lose integrity threaten democracy." Australian Journal of Social Issues 55 (3): 242-259.
What ASD cyber operatives really do to protect Australian interests. M Burgess, Burgess, M. 2019. "What ASD cyber operatives really do to protect Australian interests." 28 March, 2019. https://www.themandarin.com.au/106332-mike-burgess-director- general-asd-speech-to-the-lowy-institute/.
Ethical use of artificial intelligence in the workplace -AI WHS Scorecard. NSW GovernmentCentre for Work Health and SafetyCentre for Work Health and Safety. 2021. "Ethical use of artificial intelligence in the workplace -AI WHS Scorecard." NSW Government. https://www.centreforwhs.nsw.gov.au/knowledge-hub/ethical-use-of-artificial- intelligence-in-the-workplace-final-report.
Trust between Humans and Learning Machines: Developing the Gray Box. James C Christensen, Joseph B Lyons, 10.1115/1.2017-Jun-5Mechanical Engineering. 13906Christensen, James C., and Joseph B. Lyons. 2017. "Trust between Humans and Learning Machines: Developing the Gray Box." Mechanical Engineering 139 (06): S9-S13. https://doi.org/10.1115/1.2017-Jun-5. https://doi.org/10.1115/1.2017-Jun-5.
United Nations Group of Governmental Experts of the High Contracting Parties to the Convention on Prohibitions or Restrictions on the Use of Certain Conventional Weapons Which May Be Deemed to Be Excessively Injurious or to Have Indiscriminate Effects. Peter Cihon, Standards for AI governance: international standards to enable global coordination in AI research & development. GenevaFuture of Humanity Institute. University of Oxford ; pdf Commonwealth of AustraliaTechnical-ReportGroup of Governmental Experts on Emerging Technologies in the Area of Lethal Autonomous Weapons SystemsCihon, Peter. 2019. "Standards for AI governance: international standards to enable global coordination in AI research & development." Future of Humanity Institute. University of Oxford. https://www.fhi.ox.ac.uk/wp-content/uploads/Standards_-FHI-Technical- Report.pdf Commonwealth of Australia. 2018. "The Australian Article 36 Review Process." United Nations Group of Governmental Experts of the High Contracting Parties to the Convention on Prohibitions or Restrictions on the Use of Certain Conventional Weapons Which May Be Deemed to Be Excessively Injurious or to Have Indiscriminate Effects, 30 August 2018. https://docs- library.unoda.org/Convention_on_Certain_Conventional_Weapons_- _Group_of_Governmental_Experts_(2018)/2018_GGE%2BLAWS_August_Working %2Bpaper_Australia.pdf. ---. 2019. "Australia's System of Control and applications for Autonomous Weapon Systems." Group of Governmental Experts on Emerging Technologies in the Area of Lethal Autonomous Weapons Systems, 25-29 March 2019 and 20-21 August 2019, Geneva, 26 March 2019. https://www.unog.ch/80256EDD006B8954/(httpAssets)/16C9F75124654510C12583
Capability Life Cycle Manual (V.2.1). edited by Investment Portfolio Management Branch. C9003A4EBF/$file/CCWGGE.12019WP.2Rev.1.pdf. C9003A4EBF/$file/CCWGGE.12019WP.2Rev.1.pdf. ---. 2020. Capability Life Cycle Manual (V.2.1). edited by Investment Portfolio Management Branch.
Engaging with the industry: integrating IHL into new technologies in urban warfare. D Copeland, L Sanders, ICRC. 8Humanitarian Law and Policy. Copeland, D., and L. Sanders. 2021. "Engaging with the industry: integrating IHL into new technologies in urban warfare." Humanitarian Law and Policy (blog), ICRC. 8
. October, October. https://blogs.icrc.org/law-and-policy/2021/10/07/industry-ihl-new- technologies/.
An introduction to international criminal law and procedure. Robert Cryer, Darryl Robinson, Sergey Vasiliev, Cambridge University PressCryer, Robert, Darryl Robinson, and Sergey Vasiliev. 2019. An introduction to international criminal law and procedure. Cambridge University Press.
Cyber Digital and Technology Policy Division. Department of Home AffairsCyber Digital and Technology Policy Division. 2020. "2020 Cyber Security Strategy." Department of Home Affairs. https://www.homeaffairs.gov.au/about-us/our- portfolios/cyber-security/strategy.
Governance of AI Program, Future of Humanity Institute. Allan Dafoe, 14421443UKUniversity of Oxford: OxfordAI governance: a research agendaDafoe, Allan. 2018. "AI governance: a research agenda." Governance of AI Program, Future of Humanity Institute, University of Oxford: Oxford, UK 1442: 1443.
Artificial Intelligence: Australia's Ethics Framework: A Discussion Paper. Data61 CSIRO. D Dawson, E Schleiger, J Horton, J Mclaughlin, C Robinson, G Quezada, J Scowcroft, S Hajkowicz, Australia Data61 CSIROAustralia (Data61 CSIRO, AustraliaDawson, D., E. Schleiger, J. Horton, J. McLaughlin, C. Robinson, G. Quezada, J. Scowcroft, and S. Hajkowicz. 2019. Artificial Intelligence: Australia's Ethics Framework: A Discussion Paper. Data61 CSIRO, Australia (Data61 CSIRO, Australia: Australia Data61 CSIRO). https://consult.industry.gov.au/strategic-policy/artificial-intelligence- ethics-framework/.
Loyal Wingman uncrewed aircraft completes first flight. Melanie De Git, 2021, Innovation Quarterly. de Git, Melanie 2021. "Loyal Wingman uncrewed aircraft completes first flight." Innovation Quarterly, Boeing, 12 April, 2021. https://www.boeing.com/features/innovation- quarterly/2021/04/loyal-wingman.page.
AI to enable military commanders to make better decisions. Defence Science & Technology GroupDefence Science & Technology Group, 18 November, 2021a, "AI to enable military commanders to make better decisions," https://www.dst.defence.gov.au/news/2021/11/18/ai-enable-military-commanders- make-better-decisions-faster.
Defence Artificial Intelligence Research Network (DAIRNET) Research Call. ---. 2021b. "Defence Artificial Intelligence Research Network (DAIRNET) Research Call." Accessed 1 November. https://www.dst.defence.gov.au/partner-with- us/opportunities/defence-artificial-intelligence-research-network-dairnet-research-call.
The Artificial Intelligence for Decision Making Initiative. Defence Science InstituteArtificial Intelligence for Decision Making' InitiativeDefence Science Institute, 19 May, 2020a, "The Artificial Intelligence for Decision Making Initiative," https://www.defencescienceinstitute.com/news/the-artificial-intelligence- for-decision-making-initiative. ---, 2020b, "'Artificial Intelligence for Decision Making' Initiative," https://www.defencescienceinstitute.com/component/sppagebuilder/?view=page&id= 29&highlight=WyJhcnRpZmljaWFsIiwiJ2FydGlmaWNpYWwiLCJpbnRlbGxpZ2Vu Y2UiLCJhcnRpZmljaWFsIGludGVsbGlnZW5jZSJd.
Applications open for the artificial intelligence for decision making initiative round 2. May, 2021, "Applications open for the artificial intelligence for decision making initiative round 2," https://www.defencescienceinstitute.com/news/initiatives/applications-open-for-the- artificial-intelligence-for-decision-making-initiative-round-2.
Artificial intelligence enhances the impact of air and space power for the Joint Force. 2019. ADDP 00.1Command and Control AL1. edited by Department of Defence. 2020 Defence Strategic Update. Department of Defence Annual Report 2019-2020Deloitte Center for Government Insights. 2021. "The future of warfighting." Deloitte. https://www2.deloitte.com/global/en/pages/public-sector/articles/future-of- warfighting.html. Department of Defence. 2016. "Defence White Paper." https://www1.defence.gov.au/about/publications/2016-defence-white-paper. ---. 2019. ADDP 00.1 Command and Control AL1. edited by Department of Defence. ---. 2020a. "2020 Defence Strategic Update." https://www1.defence.gov.au/about/publications/2020-defence-strategic-update. ---. 2020b. "Artificial intelligence enhances the impact of air and space power for the Joint Force." Department of Defence Annual Report 2019-2020.
Lead the Way: Defence Transformation Strategy. Defence releases report on ethical use of AI. Defence Data Strategy 2021-2023https://www.transparency.gov.au/annual-reports/department-defence/reporting- year/2019-20-31 ---. 2020c. "Lead the Way: Defence Transformation Strategy." https://www1.defence.gov.au/about/publications/lead-way-defence-transformation- strategy. ---. 2021a. "Defence Data Strategy 2021-2023." https://www1.defence.gov.au/about/publications/defence-data-strategy-2021-2023. ---, 16 February, 2021b, "Defence releases report on ethical use of AI," https://news.defence.gov.au/media/media-releases/defence-releases-report-ethical- use-ai.
Managing Weapon System Software. Department Of Defense, Progress and ProblemsUnclassified Digest of a Classified ReportDepartment of Defense. 1978. Managing Weapon System Software: Progress and Problems (Unclassified Digest of a Classified Report).
Department of Home Affairs. Australian ValuesDepartment of Home Affairs. 2020. "Australian Values." https://www.homeaffairs.gov.au/about-us/our-portfolios/social-cohesion/australian- values.
Australia's Artificial Intelligence Ethics Framework. Accessed. 25Department of Industry Innovation and ScienceDepartment of Industry Innovation and Science. 2019. "Australia's Artificial Intelligence Ethics Framework." Accessed 25 September. https://www.industry.gov.au/data-and- publications/australias-artificial-intelligence-ethics-framework/australias-ai-ethics- principles.
The Global Partnership on Artificial Intelligence launches. Department of Industry Science Energy and ResourcesDepartment of Industry Science Energy and Resources, 16 June 2020, 2020a, "The Global Partnership on Artificial Intelligence launches,"
Australia's Artificial Intelligence Action Plan. Accessed. 11Testing the AI Ethics Principleshttps://www.industry.gov.au/news/the-global-partnership-on-artificial-intelligence- launches. ---. 2020b. "Testing the AI Ethics Principles." Accessed 11 December https://www.industry.gov.au/data-and-publications/australias-artificial-intelligence- ethics-framework/testing-the-ai-ethics-principles. ---. 2021. "Australia's Artificial Intelligence Action Plan." https://www.industry.gov.au/data- and-publications/australias-artificial-intelligence-action-plan.
Trustworthiness of autonomous systems. S Devitt, Foundations of Trusted Autonomy. Hussein A. Abbass, Jason Scholz and Darryn J. ReidChamSpringer International PublishingDevitt, S K 2018. "Trustworthiness of autonomous systems." In Foundations of Trusted Autonomy, edited by Hussein A. Abbass, Jason Scholz and Darryn J. Reid, 161-184. Cham: Springer International Publishing.
A Method for Ethical AI in Defence. S K Devitt, M Gan, J Scholz, R S Bolia, Defence Science and Technology Group (Defence Science and TechnologyDevitt, S K, M Gan, J Scholz, and R S Bolia. 2021. A Method for Ethical AI in Defence. Defence Science and Technology Group (Defence Science and Technology).
Necessary conditions for responsible innovation. Erik Fisher, 10.1080/23299460.2020.1774105Journal of Responsible Innovation. 72Fisher, Erik. 2020. "Necessary conditions for responsible innovation." Journal of Responsible Innovation 7 (2): 145-148. https://doi.org/10.1080/23299460.2020.1774105. https://doi.org/10.1080/23299460.2020.1774105.
Ethical AI in Defence Case Study: Allied Impact. D Gaetjens, S K Devitt, C Shanahan, Defence Science & Technology GroupDST Technical ReportGaetjens, D., S.K. Devitt, and C. Shanahan. 2021. Ethical AI in Defence Case Study: Allied Impact. DST Technical Report: Defence Science & Technology Group.
Big Data: A case study of disruption and government power. Kate Galloway, Alternative Law Journal. 422Galloway, Kate. 2017. "Big Data: A case study of disruption and government power." Alternative Law Journal 42 (2): 89-95.
A layered model for AI governance. Urs Gasser, A F Virgilio, Almeida, IEEE Internet Computing. 216Gasser, Urs, and Virgilio AF Almeida. 2017. "A layered model for AI governance." IEEE Internet Computing 21 (6): 58-62.
Gobal Partnership on AI. 2021. //gpai.ai/The Global Partnership on AIGobal Partnership on AI. 2021. "The Global Partnership on AI ". Accessed 25 September https://gpai.ai/.
Artificial Intelligence: Solving problems, growing the economy and improving our quality of life. (CSIRO Data61 and the Department of Industry, Innovation and Science. S A Hajkowicz, Karimi, C Wark, M Chen, N Evans, Rens, Dawson, Charlton, C Brennan, S Moffatt, K J Srikumar, Tong, Australian Government. Hajkowicz, S A, S Karimi, T Wark, C Chen, M Evans, N Rens, D Dawson, A Charlton, T Brennan, C Moffatt, S Srikumar, and K J Tong. 2019. Artificial Intelligence: Solving problems, growing the economy and improving our quality of life. (CSIRO Data61 and the Department of Industry, Innovation and Science, Australian Government). https://data61.csiro.au/en/Our-Research/Our-Work/AI-Roadmap.
Australia well placed to turbocharge its strategic tech capability. Fergus Hanson, Danielle Cave, Australian Strategic Policy Institute. Last Modified. Hanson, Fergus, and Danielle Cave. 2021. "Australia well placed to turbocharge its strategic tech capability." Australian Strategic Policy Institute. Last Modified 20 September. https://www.aspi.org.au/opinion/australia-well-placed-turbocharge-its-strategic-tech- capability.
Ethics Guidelines for Trustworthy AI. High-Level Expert Group on Artificial IntelligenceHigh-Level Expert Group on Artificial Intelligence. 2019. Ethics Guidelines for Trustworthy AI. European Commission. https://ec.europa.eu/digital-single-market/en/news/ethics- guidelines-trustworthy-ai.
Myths of automation and their implications for military procurement. Robert R Hoffman, Nadine Sarter, Matthew Johnson, John K Hawley, 10.1080/00963402.2018.1486615Bulletin of the Atomic Scientists. 744Hoffman, Robert R., Nadine Sarter, Matthew Johnson, and John K. Hawley. 2018. "Myths of automation and their implications for military procurement." Bulletin of the Atomic Scientists 74 (4): 255-261. https://doi.org/10.1080/00963402.2018.1486615.
IEEE 7000™-2021 -IEEE Standard Model Process for Addressing Ethical Concerns During System Design. Ieee, IEEE. 2021. "IEEE 7000™-2021 -IEEE Standard Model Process for Addressing Ethical Concerns During System Design." https://engagestandards.ieee.org/ieee-7000-2021- for-systems-design-ethical-concerns.html.
Ethically Aligned Design: A Vision for Prioritizing Human Well-being with Autonomous and Intelligent Systems (EADe1). IEEE Global Initiative on Ethics of Autonomous and Intelligent Systems. IEEEIEEE Global Initiative on Ethics of Autonomous and Intelligent Systems. 2019. Ethically Aligned Design: A Vision for Prioritizing Human Well-being with Autonomous and Intelligent Systems (EADe1). IEEE. https://standards.ieee.org/content/ieee- standards/en/industry-connections/ec/autonomous-systems.html.
Australia makes another order for Boeing's Loyal Wingman drones after a successful first flight. Valerie Insinna, DefenceNews. Insinna, Valerie. 2021. "Australia makes another order for Boeing's Loyal Wingman drones after a successful first flight." DefenceNews, 3 March, 2021. https://www.defensenews.com/air/2021/03/02/australia-makes-another-order-for- boeing-made-loyal-wingman-drones-after-a-successful-first-flight/.
A Guide to the Legal Review of New Weapons, Means and Methods of Warfare: Measures to Implement Article 36 of Additional Protocol I of 1977. International Review of the Red Cross. 88864International Committee of the Red CrossInternational Committee of the Red Cross. 2006. "A Guide to the Legal Review of New Weapons, Means and Methods of Warfare: Measures to Implement Article 36 of Additional Protocol I of 1977." International Review of the Red Cross 88 (864). https://www.icrc.org/eng/assets/files/other/irrc_864_icrc_geneva.pdf.
JAIC facilitates first-ever International AI Dialogue for Defense. JAIC Public Affairs28JAIC Public Affairs, 16 September 2020, 2020, "JAIC facilitates first-ever International AI Dialogue for Defense," https://www.ai.mil/news_09_16_20-jaic_facilitates_first- ever_international_ai_dialogue_for_defense_.html. ---, 28
DoD Joint AI Center Facilitates Third International AI Dialogue for Defense. May, 2021, "DoD Joint AI Center Facilitates Third International AI Dialogue for Defense," https://www.ai.mil/news_05_28_21- jaic_facilitates_third_international_ai_dialogue_for_defense.html.
Collaboration for cultural reform in Defence. K Jenkins, Jenkins, K. 2014. "Collaboration for cultural reform in Defence." https://defence.humanrights.gov.au/.
New world disorder: Ukraine redefines global landscape. Paul Kelly, Kelly, Paul. 2022. "New world disorder: Ukraine redefines global landscape." 4 March, 2022. https://www.theaustralian.com.au/inquirer/a-new-world-disorder-morrison-calls-on- the-west-to-unite-against-russia-and-china/news- story/433b6ff74637c45454393030f827f1e7.
AI governance in the public sector: Three tales from the frontiers of automated decision-making in democratic settings. Maciej Kuziemski, Gianluca Misuraca, Telecommunications policy. 446101976Kuziemski, Maciej, and Gianluca Misuraca. 2020. "AI governance in the public sector: Three tales from the frontiers of automated decision-making in democratic settings." Telecommunications policy 44 (6): 101976.
Are Autonomous Weapons Systems Prohibited?. Rain Liivoja, Eve Massingham, Tim Mcfarland, Simon Mckenzie, Game Changer. Trusted Autonomous Systems. Liivoja, Rain, Eve Massingham, Tim McFarland, and Simon McKenzie. 2020. "Are Autonomous Weapons Systems Prohibited?". Game Changer. Trusted Autonomous Systems https://tasdcrc.com.au/are-autonomous-weapons-systems-prohibited/.
Trust in artificial intelligence: Australian insights 2020. S Lockey, N Gillespie, C Curtis, The University of Queensland and KPMG AustraliaLockey, S., N. Gillespie, and C. Curtis. 2020. Trust in artificial intelligence: Australian insights 2020. (The University of Queensland and KPMG Australia).
DOD Adopts 5 Principles of Artificial Intelligence Ethics. C T Lopez, DOD NewsLopez, C.T. 2020. "DOD Adopts 5 Principles of Artificial Intelligence Ethics." DOD News, 2020. https://www.defense.gov/Explore/News/Article/Article/2094085/dod-adopts-5- principles-of-artificial-intelligence-ethics/.
Mike Burgess, Director-General on the Australian Signals Directorate (ASD) -Offensive cyber. Lowy Institute, Lowy Institute. 2019. "Mike Burgess, Director-General on the Australian Signals Directorate (ASD) -Offensive cyber" YouTube. https://youtu.be/Th6EKCwhGrs.
Autonomous Weapon Systems and the Law of Armed Conflict: Compatibility with International Humanitarian Law. Cambridge Core. Tim Mcfarland, Law School Policy Review. Autonomous Weapons and The Jus Ad BellumMcFarland, Tim. 2020. Autonomous Weapon Systems and the Law of Armed Conflict: Compatibility with International Humanitarian Law. Cambridge Core. ---. 2021. "Autonomous Weapons and The Jus Ad Bellum." Law School Policy Review. https://lawschoolpolicyreview.com/2021/03/20/autonomous-weapons-and-the-jus-ad- bellum-an-overview/.
Inside the black box: Understanding AI decision-making. Charles Mclellan, ZDNet. 1McLellan, Charles. 2016. "Inside the black box: Understanding AI decision-making." ZDNet, 1
Artificial Intelligence in Support of Defence: Report of the AI Task Force. Ministère des ArméesMinistère des Armées. 2019. Artificial Intelligence in Support of Defence: Report of the AI Task Force.
How will artificial intelligence and machine learning impact OHS?". Australian Institute for Health and Safety. B Morrison, Last Modified. 21Morrison, B. 2021. "How will artificial intelligence and machine learning impact OHS?". Australian Institute for Health and Safety. Last Modified 21 June. https://www.aihs.org.au/news-and-publications/news/how-will-artificial-intelligence- and-machine-learning-impact-ohs.
Remarks by President Biden, Prime Minister Morrison of Australia, and Prime Minister Johnson of the United Kingdom Announcing the Creation of AUKUS. S Morrison, J Johnson, Biden, Morrison, S, B Johnson, and J Biden. 2021. "Remarks by President Biden, Prime Minister Morrison of Australia, and Prime Minister Johnson of the United Kingdom Announcing the Creation of AUKUS." The White House. https://www.whitehouse.gov/briefing-room/speeches-remarks/2021/09/15/remarks- by-president-biden-prime-minister-morrison-of-australia-and-prime-minister-johnson- of-the-united-kingdom-announcing-the-creation-of-aukus/.
United States of America. National Security Commission on Artificial IntelligenceFinal ReportNational Security Commission on Artificial Intelligence. 2021. "Final Report." United States of America. https://www.nscai.gov/wp-content/uploads/2021/03/Full-Report-Digital- 1.pdf.
Morrison says AUKUS will strengthen cooperation on critical technologies. B Nicholson, The Strategist, Australian Strategic Policy InstituteNicholson, B. 2021. "Morrison says AUKUS will strengthen cooperation on critical technologies." The Strategist, Australian Strategic Policy Institute, 17 November, 2021. https://www.aspistrategist.org.au/morrison-says-aukus-will-strengthen- cooperation-on-critical-technologies/.
NSW AI Assurance Framework. Nsw Government, NSW Government. Accessed. 5NSW Government. 2022. "NSW AI Assurance Framework." NSW Government. Accessed 5 March. https://www.digital.nsw.gov.au/policy/artificial-intelligence/nsw-ai-assurance- framework.
The OECD AI Principles. Oecd, OECD. 2019. "The OECD AI Principles." https://www.oecd.org/going-digital/ai/principles/.
Five Eyes Intelligence Oversight and Review Council (FIORC). Accessed. 25OECD Council on Artificial IntelligenceRecommendation of the Council on Artificial IntelligenceOECD Council on Artificial Intelligence. 2019. Recommendation of the Council on Artificial Intelligence. Office of the Director of National Intelligence. "Five Eyes Intelligence Oversight and Review Council (FIORC)." Accessed 25 September. https://www.dni.gov/index.php/ncsc- how-we-work/217-about/organization/icig-pages/2660-icig-fiorc.
Understanding Bias In AI-Enabled Hiring. Nish Parikh, Forbes MagazineParikh, Nish. 2021. "Understanding Bias In AI-Enabled Hiring." Forbes Magazine, 2021. https://www.forbes.com/sites/forbeshumanresourcescouncil/2021/10/14/understandin g-bias-in-ai-enabled-hiring/?sh=33f2d7997b96.
Australia claims historic top two spot in the 'Robot Olympics. Alexandra Persley, Persley, Alexandra, 2021, "Australia claims historic top two spot in the 'Robot Olympics'," https://www.csiro.au/en/news/News-releases/2021/Australia-claims-historic-top-two- spot-in-the-Robot-Olympics.
Australian Support to the Ukraine. Prime Minister, Minister Of Defence, Prime Minister of Australia. Accessed. 2022Prime Minister, and Minister of Defence. 2022, 1 March 2022. "Australian Support to the Ukraine." Prime Minister of Australia. Accessed 5 March 2022. https://www.pm.gov.au/media/australian-support-ukraine.
Report of the comprehensive review of the legal framework of the national intelligence community. D Richardson, Attorney-General's Department. Accessed. 25Richardson, D. 2020. "Report of the comprehensive review of the legal framework of the national intelligence community." Attorney-General's Department. Accessed 25
At the edge: Exploring and exploiting our fifth-generation edges. Loyal Wingman First FlightRoyal Australian Air Force. 2019. "At the edge: Exploring and exploiting our fifth-generation edges." https://www.airforce.gov.au/our-mission/plan-jericho. ---. 2021. "Loyal Wingman First Flight." YouTube, 2 March, 2021. https://youtu.be/BiSHVl7UMRk.
RAS-AI Strategy 2040: Warfare Innovation Navy. Royal Australian NavyRoyal Australian Navy. 2020. "RAS-AI Strategy 2040: Warfare Innovation Navy." https://www.navy.gov.au/media-room/publications/ras-ai-strategy-2040.
HRC calls for an AI Safety Commissioner. Denham Sadler, InnovationAus. Sadler, Denham. 2021. "HRC calls for an AI Safety Commissioner." InnovationAus, 27 May, 2021. https://www.innovationaus.com/hrc-calls-for-an-ai-safety-commissioner/.
Human Rights and Technology Final Report. E Santow, Australian Human Rights CommissionSantow, E. 2021. "Human Rights and Technology Final Report." Australian Human Rights Commission. https://tech.humanrights.gov.au/downloads.
Mapping global AI governance: a nascent regime in a fragmented landscape. Lewin Schmitt, 10.1007/s43681-021-00083-yAI and Ethics. Schmitt, Lewin. 2021. "Mapping global AI governance: a nascent regime in a fragmented landscape." AI and Ethics. https://doi.org/10.1007/s43681-021-00083-y. https://doi.org/10.1007/s43681-021-00083-y.
Australian public understandings of artificial intelligence. Neil Selwyn, Beatriz Gallo Cordoba, https:/link.springer.com/article/10.1007/s00146-021-01268-zAI & SOCIETY. Selwyn, Neil, and Beatriz Gallo Cordoba. 2021. "Australian public understandings of artificial intelligence." AI & SOCIETY: 1-18. https://link.springer.com/article/10.1007/s00146-021-01268-z.
Official Committee Hansard Senate Foreign Affairs, Defence and Trade Legislation Committee Estimates Wednesday. Senate Foreign Affairs Defence and Trade Legislation Committee. query=Id%3A%22co mmittees%2Festimate%2F53068544-efe7-4494-a0f2-2dbca4d2607b%2F0000%22Senate Foreign Affairs Defence and Trade Legislation Committee. 2019. "Official Committee Hansard Senate Foreign Affairs, Defence and Trade Legislation Committee Estimates Wednesday, 23 October 2019." https://parlinfo.aph.gov.au/parlInfo/search/display/display.w3p;query=Id%3A%22co mmittees%2Festimate%2F53068544-efe7-4494-a0f2-2dbca4d2607b%2F0000%22.
As Biden hosts first Quad summit at the White House, China is the background music. Gerry Shih, Anne Gearan, The Washington Post. Shih, Gerry, and Anne Gearan. 2021. "As Biden hosts first Quad summit at the White House, China is the background music." The Washington Post, 24 September, 2021. https://www.washingtonpost.com/world/2021/09/24/quad-us-india-australia-japan- china/.
Artificial Intelligence Standards Roadmap: Making Australia's Voice Heard. Standards Australia, Standards Australia. 2020. Artificial Intelligence Standards Roadmap: Making Australia's Voice Heard. https://www.standards.org.au/news/standards-australia-sets-priorities- for-artificial-intelligence.
Centre for Security and Emerging Technology, Georgetown University's Walsh School of Foreign Service. Z Stanley-Lockman, Responsible and Ethical Military AI Allies and Allied Perspectives: CSET Issue BriefStanley-Lockman, Z. 2021. "Responsible and Ethical Military AI Allies and Allied Perspectives: CSET Issue Brief." Centre for Security and Emerging Technology, Georgetown University's Walsh School of Foreign Service. https://cset.georgetown.edu/wp-content/uploads/CSET-Responsible-and-Ethical- Military-AI.pdf.
establish-common-principles-to-update-therules-for-the-21st-century-economy-at-inaugural-trade-and-technology-councilmeeting/. ---. 2021b. Last Modified. The White House. 2021a. "FACT SHEET: U.S.-EU15Joint Leaders Statement on AUKUSThe White House. 2021a. "FACT SHEET: U.S.-EU Establish Common Principles to Update the Rules for the 21st Century Economy at Inaugural Trade and Technology Council Meeting." https://www.whitehouse.gov/briefing-room/statements- releases/2021/09/29/fact-sheet-u-s-eu-establish-common-principles-to-update-the- rules-for-the-21st-century-economy-at-inaugural-trade-and-technology-council- meeting/. ---. 2021b. "Joint Leaders Statement on AUKUS." Last Modified 15 September 2021. https://www.whitehouse.gov/briefing-room/statements-releases/2021/09/15/joint- leaders-statement-on-aukus/.
The Aukus challenge to Asean. Thi Ha, The Straits TimesThi Ha, Hoang 2021. "The Aukus challenge to Asean." The Straits Times, 2021. https://www.straitstimes.com/opinion/the-aukus-challenge-to-asean.
The U.S.-Australian Alliance Needs a Strategy to Deter China's Gray-Zone Coercion. Ashley Townshend, Thomas Lonergan, Toby Warden, War on the RocksTownshend, Ashley, Thomas Lonergan, and Toby Warden. 2021. "The U.S.-Australian Alliance Needs a Strategy to Deter China's Gray-Zone Coercion." War on the Rocks, 2021. https://warontherocks.com/2021/09/the-u-s-australian-alliance-needs-a- strategy-to-deter-chinas-gray-zone-coercion/.
The ethical questions that haunt facial-recognition research. Richard Van Noorden, Nature. van Noorden, Richard. 2020. "The ethical questions that haunt facial-recognition research." Nature, 18 November, 2020. https://www.nature.com/articles/d41586-020-03187-3.
Concept for robotics and autonomous systems. R Vine, Australian Defence ForceVine, R. 2020. Concept for robotics and autonomous systems. Australian Defence Force. https://www.defence.gov.au/vcdf/forceexploration/adf-concept-future-robotics- autonomous-systems.asp.
Tackling the growing threats to Australia's cyber security. L White, 2021 Mandarin Defence Special Report. White, L. 2021. "Tackling the growing threats to Australia's cyber security." 2021 Mandarin Defence Special Report, 2021. https://www.themandarin.com.au/169281-tackling-the- growing-threats-to-australias-cyber-security/.
IEEE P7001: A Proposed Standard on Transparency. Alan F T Winfield, Serena Booth, Louise A Dennis, Takashi Egawa, Helen Hastie, Naomi Jacobs, Roderick I Muttram, Joanna I Olszewska, Fahimeh Rajabiyazdi, Andreas Theodorou, Mark A Underwood, Robert H Wortham, Eleanor Watson, https:/www.frontiersin.org/article/10.3389/frobt.2021.665729Frontiers in Robotics and AI. 8225Winfield, Alan F. T., Serena Booth, Louise A. Dennis, Takashi Egawa, Helen Hastie, Naomi Jacobs, Roderick I. Muttram, Joanna I. Olszewska, Fahimeh Rajabiyazdi, Andreas Theodorou, Mark A. Underwood, Robert H. Wortham, and Eleanor Watson. 2021. "IEEE P7001: A Proposed Standard on Transparency." Frontiers in Robotics and AI 8 (225). https://doi.org/10.3389/frobt.2021.665729. https://www.frontiersin.org/article/10.3389/frobt.2021.665729.
| [] |
[
"Collective behaviour of swarmalators on a 1D ring",
"Collective behaviour of swarmalators on a 1D ring"
] | [
"Kevin O'keeffe \nSenseable City Lab\nMassachusetts Institute of Technology\n02139CambridgeMA\n",
"Steven Ceron \nSibley School of Mechanical and Aerospace Engineering\nCornell University\n14853IthacaNYUSA\n",
"Kirstin Petersen \nDepartment of Electrical and Computer Engineering\nCornell University\n14853IthacaNYUSA\n"
] | [
"Senseable City Lab\nMassachusetts Institute of Technology\n02139CambridgeMA",
"Sibley School of Mechanical and Aerospace Engineering\nCornell University\n14853IthacaNYUSA",
"Department of Electrical and Computer Engineering\nCornell University\n14853IthacaNYUSA"
] | [] | We study the collective behavior of swarmalators, generalizations of phase oscillators that both sync and swarm, confined to move on a 1D ring. This simple model captures some of the essence of movement in 2D or 3D but has the benefit of being solvable: most of the collective states and their bifurcations can be specified exactly. The model also captures the behavior of real-world swarmalators which swarm in quasi-1D rings such as bordertaxic sperm and vinegar eels.I. INTRODUCTIONSynchronization and swarming are universal phenomena[1][2][3][4][5]that are in a sense spatiotemporal opposites. Synchronizing units self-organize in time, but not in space; laser arrays fire simultaneously[6,7], heart cells trigger all at once [8], but neither system has a notion of spontaneous group movement. Swarming units flip the picture: they self-organize in space, not time. Birds fly in flocks[9], fish swim in schools [10], but neither coordinates the timing of an internal state or rhythm.The units of some systems appear to self-organize in both space and time. In Biology, sperm[11,12], vinegar eels[13,14], and other microswimmers[15][16][17]synchronize the wriggling of their tails which is speculated to hydrodynamically couple to their motion. In Chemistry, magnetic Janus particles[18][19][20], dieletric Quinke rollers[21][22][23], and other active entities [24] lock their rotations enabling a kind of sync-dependent self-assembly. In Engineering, land-based robots and aerial drones can be programmed to swarm based on their synchronizable internal clocks[25][26][27]. Sync and swarming are also suspected to interact in spatial cognition[28], embryology[29][30][31]and the physics of magnetic domain walls[32].The theoretical study of systems which both sync and swarm is just beginning.Takanapioneered the initiative by formulating a model of chemotactic oscillators [33] and found diverse phenomena [34, 35]. Leibchen and Levis generalized the Vicsek model and found new types of long range synchrony [36]. O'Keeffe et al introduced a model of 'swarmalators' [69] whose collective states have been realized in Nature and technology [21, 25, 26] and is being actively extended. The inclusion of noise [37], local coupling [38, 39], periodic forcing [40], mixed sign interactions [41], and finite N effects [42] have been studied. The potential of swarmalators in bio-inspired computing has been explored [43], as has the well-posedness of N → ∞ solutions of the swarmalator model [44, 45].Here we study swarmalators confined to move on a 1D ring. Our aim is two-fold. First, to model swarmalators which swarm purely in 1D. Frogs, nematodes, and other organisms are often bordertaxic, seeking out the ring-like edges of their confining geometry[46][47][48][49][50][51][52]. Janus particles, when acting as microrobots for precision medicine[53][54][55], will need to navigate pseudo 1D groovesFIG. 1: Order parameters of the 2D swaramalator modelwhere φ, θ are the spatial angle and phase of swarmalators. S+ bifurcates from 0 at K1 as the static async state(Fig 2(b)) destabilizes, S− from 0 at K2 as the active phase wave(Fig 2(f)) destabilizes. The nature of these bifurcations, as well as analytic expressions for both K1 and K2, are unknown. and channels[56]. Toy models for these systems will be useful for applied research.Our second aim is theoretical, namely to investigate the original 2D swarmalator model whose physics is not understood[70].Figure 1shows its order parameters S ± := (N ) −1 j e i(φj ±θj ) -where φ, θ denote the spatial angle and phase -dependence on the coupling strength K. At a critical K 1 , S + jumps from zero as the system transitions from a static async state(Fig 2(b)) to an active phase wave state(Fig 2(e)) in which swarmalators run in a space-phase vortex. As expected of order parameters, S + grows as K is increased. But then it begins to decline at a second value K 2 as the swarmalator vortex bifurcates into a broken band of mini-vortices(Fig 2(e)). The cause of this non-monotinicity, as well as analytic values for K 1 , K 2 , are unknown. Like the old puzzles to understand the transitions of the Kuramoto model[2,[57][58][59][60], the bifurcations of the swarmalator model 'cry out for a theoretical explanation' [57].Our hope is that a retreat to a 1D ring will give some first clues on how to provide such an explanation. | 10.1103/physreve.105.014211 | [
"https://arxiv.org/pdf/2108.06901v1.pdf"
] | 237,091,142 | 2108.06901 | 51848676647441616999a51ebc3668576e33cf32 |
Collective behaviour of swarmalators on a 1D ring
Kevin O'keeffe
Senseable City Lab
Massachusetts Institute of Technology
02139CambridgeMA
Steven Ceron
Sibley School of Mechanical and Aerospace Engineering
Cornell University
14853IthacaNYUSA
Kirstin Petersen
Department of Electrical and Computer Engineering
Cornell University
14853IthacaNYUSA
Collective behaviour of swarmalators on a 1D ring
We study the collective behavior of swarmalators, generalizations of phase oscillators that both sync and swarm, confined to move on a 1D ring. This simple model captures some of the essence of movement in 2D or 3D but has the benefit of being solvable: most of the collective states and their bifurcations can be specified exactly. The model also captures the behavior of real-world swarmalators which swarm in quasi-1D rings such as bordertaxic sperm and vinegar eels.I. INTRODUCTIONSynchronization and swarming are universal phenomena[1][2][3][4][5]that are in a sense spatiotemporal opposites. Synchronizing units self-organize in time, but not in space; laser arrays fire simultaneously[6,7], heart cells trigger all at once [8], but neither system has a notion of spontaneous group movement. Swarming units flip the picture: they self-organize in space, not time. Birds fly in flocks[9], fish swim in schools [10], but neither coordinates the timing of an internal state or rhythm.The units of some systems appear to self-organize in both space and time. In Biology, sperm[11,12], vinegar eels[13,14], and other microswimmers[15][16][17]synchronize the wriggling of their tails which is speculated to hydrodynamically couple to their motion. In Chemistry, magnetic Janus particles[18][19][20], dieletric Quinke rollers[21][22][23], and other active entities [24] lock their rotations enabling a kind of sync-dependent self-assembly. In Engineering, land-based robots and aerial drones can be programmed to swarm based on their synchronizable internal clocks[25][26][27]. Sync and swarming are also suspected to interact in spatial cognition[28], embryology[29][30][31]and the physics of magnetic domain walls[32].The theoretical study of systems which both sync and swarm is just beginning.Takanapioneered the initiative by formulating a model of chemotactic oscillators [33] and found diverse phenomena [34, 35]. Leibchen and Levis generalized the Vicsek model and found new types of long range synchrony [36]. O'Keeffe et al introduced a model of 'swarmalators' [69] whose collective states have been realized in Nature and technology [21, 25, 26] and is being actively extended. The inclusion of noise [37], local coupling [38, 39], periodic forcing [40], mixed sign interactions [41], and finite N effects [42] have been studied. The potential of swarmalators in bio-inspired computing has been explored [43], as has the well-posedness of N → ∞ solutions of the swarmalator model [44, 45].Here we study swarmalators confined to move on a 1D ring. Our aim is two-fold. First, to model swarmalators which swarm purely in 1D. Frogs, nematodes, and other organisms are often bordertaxic, seeking out the ring-like edges of their confining geometry[46][47][48][49][50][51][52]. Janus particles, when acting as microrobots for precision medicine[53][54][55], will need to navigate pseudo 1D groovesFIG. 1: Order parameters of the 2D swaramalator modelwhere φ, θ are the spatial angle and phase of swarmalators. S+ bifurcates from 0 at K1 as the static async state(Fig 2(b)) destabilizes, S− from 0 at K2 as the active phase wave(Fig 2(f)) destabilizes. The nature of these bifurcations, as well as analytic expressions for both K1 and K2, are unknown. and channels[56]. Toy models for these systems will be useful for applied research.Our second aim is theoretical, namely to investigate the original 2D swarmalator model whose physics is not understood[70].Figure 1shows its order parameters S ± := (N ) −1 j e i(φj ±θj ) -where φ, θ denote the spatial angle and phase -dependence on the coupling strength K. At a critical K 1 , S + jumps from zero as the system transitions from a static async state(Fig 2(b)) to an active phase wave state(Fig 2(e)) in which swarmalators run in a space-phase vortex. As expected of order parameters, S + grows as K is increased. But then it begins to decline at a second value K 2 as the swarmalator vortex bifurcates into a broken band of mini-vortices(Fig 2(e)). The cause of this non-monotinicity, as well as analytic values for K 1 , K 2 , are unknown. Like the old puzzles to understand the transitions of the Kuramoto model[2,[57][58][59][60], the bifurcations of the swarmalator model 'cry out for a theoretical explanation' [57].Our hope is that a retreat to a 1D ring will give some first clues on how to provide such an explanation.
FIG. 2: Collective states of the 2D swarmalator model introduced in [61] where swarmalators are represented as colored dots where the color refers to the swarmalators phase. As described in Appendix B, the states displayed are from a slightly different intstance of the model to that presented in [61] which produces the same qualitative behavior, but better emphasises the nonmonotonic behavior of the order parameters S± shown in Figure 1. In all panels a Euler method was used with timestep dt = 0.1 for T = 1000 units for N = 1000 swarmalators. (a) Static sync: (J, K, σ) = (1, 1, 10) (b) Static async (J, K, σ) = (1, 1, 10) (c) Static phase wave (J, K, σ) = (1, 1, 10) (d) Splintered phase wave (J, K, σ) = (1, 1, 10) (e) Active phase wave. In the three static states (a)-(c) swarmalators do not move in space or phase. In the splintered phase wave, each colored chunk is a vortex: the swarmalators librate in both space and phase. In the active phase wave, the librations are excited into rotations; the swarmalators split into counter-rotating groups as indicated by the black arrows.
II. MODEL
We study a pair of modified Kuramoto models,
x i = ν i + J N N j sin(x j − x i ) cos(θ j − θ i )(1)θ i = ω i + K N N j sin(θ j − θ i ) cos(x j − x i )(2)
where (x i , θ i ) ∈ (S 1 , S 1 ) are the position and phase of the i-th swarmalator for i = 1, . . . , N and (ν i , ω i ), (J, K) are the associated natural frequencies and couplings. We consider identical swarmalators (ω i , ν i ) = (ω, ν) and by a change of frame set ω = ν = 0 WLOG. Eq. (2) models position-dependent synchronization. The familiar Kuramoto sine term minimizes swarmalators' pairwise phase difference (so they sync) while the new cosine term strengthens the coupling between nearby swarmalators K ij = K cos(x j − x i ) (so the sync is position dependent). Eq. (1) is Eq. (2)'s mirror-image: it models phase-dependent swarming. Now the sine term minimizes swarmalators' pairwise distances (so they swarm / aggregate) and the cosine term strengthens the coupling between similarly phased swarmalators J ij = J cos(θ j − θ i ). You can also think of Eqs. (1) and (2) as modelling synchronization on the unit torus.
Converting the trig functions to complex exponentials and rearranging makes the model even simpleṙ
x i = J 2 S + sin(Φ + − (x i + θ i )) + S − sin(Φ − − (x i − θ i ))(3)θ i = K 2 S + sin(Φ + − (x i + θ i )) − S − sin(Φ − − (x i − θ i ))(4)
where W ± = S ± e iΦ± = 1 N j e i(xj ±θi) .
The terms x i ± θ i occur naturally so we define
ξ i = x i + θ i (6) η i = x i − θ i(7)
And finḋ
ξ i = J + S + sin(Φ + − ξ) + J − S − sin(Φ − − η) (8) η i = J + S + sin(Φ + − ξ) + J + S − sin(Φ − − η)(9)
where J ± = (J ± K)/2 [71]. We see ring swarmalators obey a sum of two Kuramoto models where the traditional order parameter Re iΦ := (N ) −1 j e iθj has been replaced by a pair of new order parameters W ± = S ± e iΦ± = (N ) −1 j e i(xj ±θj ) . The new W ± measure the system's total amount of 'space-phase order'.
What kind of order is this? The limiting cases are trivial: static sync, (x i , θ i ) = (x * , θ * ), which produces maximal order S ± = 1 (which follows from substitution into Eq.(5)), and static async, in which positions x i are fully uncorrelated with phases θ i , which produces minimal order S ± = 0. Between these two extremes, however, lies something more interesting: Perfect correlation between position and phase x i = ±θ i + c for constant c which yields either (S + , S − ) = (0, 1) or (S + , S − ) = (0, 1) [72]. What does x i ± θ i mean physically? Picture swarmalators on a ring as a group of fireflies flying around a circular track, flashing periodically. Let θ i = 0 be the beginning of their phase cycle (when they flash). Then x i ± θ i + c means each firefly flashes at the same point on its circular lap x i = c. Plotting swarmalators as colored dots in space (where color represents phase) illustrates this behavior most evocatively. Then x i = ±θ i + c corresponds to a colored splay state (Fig 2(c), Fig 3(c)). Since one of S ± are maximal in this rainbow-like state, we call S ± the rainbow order parameters.
As we will show, S ± are natural order parameters for the ring model (insofar as they can distinguish between each of its collective states). Figure 2. x corresponds to the angular position on this unit circle (drawn black in (a) and (b) for greater clarity. The point sizes in these panels, and the corresponding panels (f),(g) are also larger, again to make things clearer) while the color corresponds to the phase. Bottom row, swarmalators are drawn as points in the (x, θ) plane. All results were found by integrating Eqs. (1), (2) using an RK4 solver with timestep dt = 0.1 for T = 500 time units for N = 500 swarmalators. Initial positions and phases were drawn from [−π, π] in all panels except panel (a), which were drawn from [0, π] (since this choice of initial conditions realized the static sync state). Motivation for model. Before showing our analysis, we quickly show how the ring model connects to the original 2D model [61]. The 2D model iṡ
x i = v i + 1 N N j=1 I att (x j − x i )F (θ j − θ i ) − I rep (x j − x i ) ,(10)θ i = ω i + K N N j=1 H att (θ j − θ i )G(x j − x i )(11)
We pick out the rotational component of the swarming motion -since that part is analogous to movement on a ring -by converting Eqs. (10), (11) to polar coordinates. For certain choices of I att (x), F (θ) etc this yieldṡ
r i =ν(r i ) + J 2 S + cos Φ + − ξ i +S − cos Φ − − η i(12)ξ i =ω(r i ) +J + (r i )S + sin Ψ + − ξ i +J − (r i )S − sin Ψ − − η i (13) η i =ω(r i ) +J − (r i )S + sin Ψ + − ξ i +J + (r i )S − sin Ψ − − η i(14)
where we have switched to (ξ i , η i ) = (φ i + θ i , φ i − θ i ) coordinates and φ i is the spatial angle of the i-th swarmalator (which is analogous to x i in the ring model).
We put the derivation and definitions of the various new quantities in the Appendix because they are cumbersome and uninformative to display here. Eqs. (13), (14) reveal the ring model hidden in the 2D model's core. The (ξ i ,η i ) equations have the same form as the (ξ i ,η i ) equations in the 1D model (Eqs. (8), (9)); both are a summed pair of Kuramoto models. The only difference is that in the 2D model,ω i (r i ) andJ ± (r) depend on r i . So the ring model is like the 2D model with the radial dynamics turned off. This is why we think studying the ring model will yield hints on how to study the 2D model (one of the paper's aims).
III. RESULTS
A. Numerics
Simulations show the system settles into five collective states depicted in Figure 3. We provide the Mathematica code used for the simulations at [73].
1. Static sync (Fig 3(a),(f)). Swarmalators fully synchronize their positions x i = x * and phases θ i = θ * resulting in maximal space-phase order S ± = 1.
2. Static π state (Fig 3(b),(g)). One group of swarmalators synchronize at (x * , θ * ) and the remaining fraction synchronize π units away (x * + π, θ * + π). S ± = 1 here also. ). Data were collected by integrating the model (1), (2) using an RK4 method with (dt, T ) = (0.1, 500). The first 90% of data were discarded as transients and the mean of the remaining 10% were taken. We chose just N = 10 swarmalators to illustrate the active async state as clearly as possible; the fluctuations in S± that characterize the state decay to 0 for larger N (see main text).
3. Static phase wave (Fig 3(c),(h)). Swarmalators form a static splay state with x i = 2πi/N and θ i = ±x i + c where the offset c is arbitrary and stems from the rotational symmetry in the model [74]. In (ξ i , η i ) coordinates, either ξ i is splayed ξ i = 2πi/N and η i is locked η i = c or vice versa. The order parameters are either (S + , S − ) = (1, 0) (where phase gradient of the rainbow is clockwise) or (S + , S − ) = (1, 0) (where the phase gradient of the rainbow is counter-clockwise).
4. Active async (Fig 3(d),(i)). Swarmalators form a dynamic steady state, moving in clean limit cycles for small N , but in erratic, jiggling patterns for large N . The motion cools and ultimately freezes as N → ∞. There is little space-phase order as indicated by the low, time averaged values of S ± (Fig 1) so we call this state 'active async'.
5.
Static async ( (Fig 3(e),(j))). Swarmalators form a static, asynchronous crystal with S ± = 0. Figure 4 shows the curve S ± (K) can distinguish between all but the static sync and static π states. Now we analyze the stability of the states. The static sync, π, and static async states are analyzed using standard techniques, but are useful as warm ups to the analysis of the much harder static phase wave state (which is the main analytic contribution of the paper). The active async state, being non-stationary, is analyzed mostly numerically.
B. Analysis
Static sync. We calculate the stability of the state by linearizing around the fixed point in (ξ, η) space. We seek the eigenvalues λ of the Jacobian M
M = Z ξ Z η N ξ N η (15) where (Z ξ ) ij = ∂ξ i ∂ξ j (16) (Z η ) ij = ∂ξ i ∂η j (17) (N ξ ) ij = ∂η i ∂ξ j (18) (N η ) ij = ∂η i ∂η j(19)
Evaluating the derivatives in the above using Eqs. (8), (9) results in a clean block structure:
M = J + A(ξ) J − A(η) J − A(ξ) J + A(η)(20)
where A(y) i,i = − 1 n n j=1 cos(y j − y i ) and A(y) i,j = 1 n cos(y j − y i ) for a dummy variable y. Evaluated at the fixed points of the static sync state, ξ i = c 1 and η i = c 2 for constants c 1 , c 2 , this becomes even simpler,
M SS = J + A 0 J − A 0 J − A 0 J + A 0(21)
where
A 0 := − N −1 N 1 N . . . 1 N 1 N − N −1 N . . . 1 N . . . . . . . . . . . . . . . . . . . . . . . . . 1 N 1 N . . . − N −1 N (22)
Notice that Jacobian of the entire system M has dim(N ) = 2N since there are two state variables (x, θ) for each of the N swarmalators, but that dim(A 0 ) = N since it is a subblock of M . Now, the eigenvaluesλ of A 0 are well known: there is 1 with valueλ = 0 stemming from the rotational symmetry of the model and N − 1 λ = −1. We use these to find the desired eigenvalues λ of M SS using the following identity for symmetric block matrices
det E := det C D D C = det(C + D) det(C − D)(23)
This implies the eigenvalues of E are the union of the eigenvalues of C + D and C − D. Applying this identity to M SS (which has the required symmetric structure)
yields λ 0 = 0 (24) λ 1 = −J (25) λ 2 = −K(26)
with multiplicities 2, −1 + N, −1 + N (which sum to the required 2N ). This tells us the static sync is stable for J > 0 and K > 0 consitent with simulations.
Static π-state. The fixed points here are (x i , θ i ) = (c 1 , c 2 ) for i = 1, 2, . . . , N/2 and (x i , θ i ) = (c 1 + π, c 2 + π) for i = N/2, . . . , N . Conveniently, the shift in π for exactly half of swarmalators does not change the form of the Jacobian, M π = M SS , so the stability is the same as before. This means the static sync and π states are bistable for all J > 0, K > 0. Appendix A discusses the basins of attraction for each state.
Static phase wave. We calculate the stability of the static phase wave using the same strategy as before: linearize around the fixed points and exploit the block structure of the Jacobian M . This time, however, the calculations are tougher.
The fixed points of the static phase wave take two forms: Either ξ is splayed ξ i = 2π(i − 1)/N + c 1 and η i is synchronized η i = c 2 (clockwise rainbow) or ξ is synchronized ξ i = c 1 and η splayed η i = 2π(i − 1)/N + c 2 (counter clockwise rainbow). Here i = 1, . . . , N and the constants c 1 , c 2 are offsets. WLOG we analyze the fixed point with ξ splayed and η sync'd. The Jacobian is
M SP W = J + A 1 J − A 0 J − A 1 J + A 0(27)
where A 0 is as before Eq.(22) but A 1 is new:
(A 1 ) ii = − 1 N j =i cos 2π N (j − i) := − 1 N j =i c ij (28) (A 1 ) ij = 1 N cos 2π N (j − i) := 1 N c ij(29)
The following notation will be useful
c ij := cos 2π(i − j)/N (30) s ij := sin 2π(i − j)/N (31) β ij := c ij + Is ij (32) c i := cos 2πi/N (33) s i := sin 2πi/N(34)
where I = √ −1 is the imaginary unit [75], β 0,0 := β = e 2πIN is the primitive root of unity, and β k = β k is the k-th root of unity. The diagonal element A ii may be simplified. Recalling the sum of roots of unities are zero (you can think of β k as a vector pointing to the beginning of the k-th segment of size 1/N of the unit circle; then summing all the vectors around the unit circle results in zero)
N −1 k=0 β k = N −1 k=0 c k + Is k = 0 (35) which implies N −1 k=0 c k = 0 (36) N −1 k=0 s k = 0(37)
(In other words, the discrete sum of cos(2πk/N ) and sin(2πkN ) around the unit circle is zero, which can be seen by symmetry). Applying these identities to Eq. (28) for (A 1 ) ii ,
(A 1 ) ii = 1 N j =i cos 2π N (j − i) = − 1 N N −1 k=1 c k (38) = − 1 N (−c 0 + 1 N N −1 k=0 c k ) = − 1 N (−c 0 + 0) (39) = c 0 N(40)
Notice in the first sum over i the j − th term is excluded which means the second sum over k begins at 1. Since
c 0 = c 0,0 = 1, we get (A 1 ) ij = 1 N c i,j(41)
With A 0 (Eq. (22)) and A 1 (Eq. (41)) in hand, we can begin finding the eigenvalues of M SP W Eq. (27) by using another identity for block matrices. If the consituent matrices A, C of a general block matrix P ,
P = A B C D(42)
commute then
det(P ) = det(AD − BC)(43)
Luckily, the constituent matrices A 0 and A 1 of M SP W do commute. So we apply the above identity to the characteristic equation for λ
det(M SP W − λI) = det J + A 1 − λI J − A 0 J − A 1 J + A 0 − λI = 0 (44) And find det(M SP W − λI) = det (J + A 1 − λI)(J + A 0 − λI) (45) − J 2 − A 1 A 0 := det(G)(46)
where to compactify the RHS we have defined
G := g 0 g 1 . . . g N −1 g N −1 g 0 . . . g N −2 . . . . . . . . . . . . . . . . . . . g 1 g 2 . . . g 0 (47)
where
g 0 = λ 2 + J + N λ(N − 1 − c 0 ) + J 2 − − J 2 + N c 0 (48) g (k>0) = − J + N λ(1 + c k ) + J 2 − − J 2 + N c k(49)
We want the determinent of G which the product of its eigenvaluesλ j (not to be confused M SP W 's eigenvalues λ i which we are trying to find),
det(G) = N −1 j=0λ j = 0(50)
Casting an eye back to Eq. (47), we see G is a circulent matrix so its eigenvaluesλ j are known exactly! -
λ j = N −1 k=0 g k β j * k (51) λ j = λ 2 + λJ + − J + N λ N −1 k=0 (1 + c k )β j * k + J 2 − − J 2 + N N −1 k=0 c k β j * k(52)
where β k is the primitive root of unity as before. Note we mean the product j * k in β j * k = cos 2πj * k/N + I sin 2πj * k/N . Note also that while G has N eigenvalueŝ λ (running from j = 0, . . . N − 1), M SP W still has the required 2N eigenvalues λ since eachλ j is quadratic in λ.
One last push remains: we simplify the summands in Eq. (52) using some basic trig identities, find eachλ j for j = 0, 1, . . . N − 1, and then setλ j = 0 (since det(G) = 0 so each term in the product must be zero) which yields a quadratic equation for our target λ j .
We begin with j = 0 since it is distinguished from the other values j takes. The first summand on the RHS of Eq.(52) becomes (36)). Plugging these in yieldŝ
N −1 k=0 (1 + c k )β 0 = N −1 k=0 (1 + c k )(1) = N + 0 = N . The second summand becomes N −1 k=0 c k β j * k = N −1 k=0 c k (1) = 0 (which follows from from Eq.λ 0 = λ 2 + λJ + − J + N λ(N ) = λ 2(53)
Settingλ 0 = 0 gives our first eigenvalue
λ 0 = 0(54)
with multiplicity 2.
Next we analyzeλ j>0 all at once. First note
N −1 k=0 β j * k = N −1 k=0 cos 2πjk N + I sin 2πjk N(55)
= 0 + 0 × I = 0
for all j > 0 (when j = 0 we get the simple sum N − 1 which is why we considered the case j = 0 separately). This implies the second summand in Eq. (52) simplifies into the third summand
N −1 k=0 (1 + c k )β j * k = N −1 k=0 c k β j * k when j > 0 which in turn becomes N −1 k=0 c k β j * k = N −1 k=0 c k c j * k + Ic k s j * k(57)
Using the standard formulas,
cos(a) cos(b) = 1 2 cos(a + b) + cos(a − b)(58)cos(a) sin(b) = 1 2 sin(a + b) − sin(a − b)(59)
we see
N −1 k=0 c k c j * k = 1 2 N −1 k=0 cos 2π(1 + j)k N + cos 2π(1 − j)k N (60) N −1 k=0 c k s j * k = 1 2 N −1 k=0 sin 2π(1 + j)k N + sin 2π(1 − j)k N(61)
where we have inverted the c j notation for clarity. Both the cosine terms in Eq. (60) and sine terms in Eq. (61) are zero (as per Eq. (56)) except when the arguments are 0. This occurs for j = 1, −1 (where j = −1 is interpreted modulo N and equivalent to j = N − 1. Since cos 0 = 1 and sin = 0, this yields
N −1 k=0 c k c jk = N 2 (δ j,1 + δ j,N −1 ) (62) N −1 k=0 c k s jk = 0(63)
where δ i,j = 1 is the Kronecker delta. Applying the above to Eq. (57) gives
N −1 k=0 c k β j * k = N 2 (δ j,1 + δ j,N −1 )(64)
Plugging this into Eq. (52) forλ ĵ
λ j = λ 2 + λJ + + − J + N λ + J 2 − − J 2 + N (δ j,1 + δ j,N −1 ) = 0(65)
which holds for j = 1, . . . , N −1 since, recall, we have analyzed j = 0 separately and in that case found λ = 0 with multiplicity 2 (Eq. (54)). The desired λ are the roots of the above equations. Both their values and multiplicities depend on N . For N ≥ 4, a general pattern holds, but N = 2, 3 are special cases.
N (Multiplicity, eigenvalue) 2 λ = (2, 0), (1, ± J 2 + − J 2 − ) 3 λ = (2, 0), (2, −J+ ± 9J 2 + − 8J 2 − /4) 4 λ = (3, 0), (1, −J+), (2, −J+ ± 9J 2 + − 8J 2 − /4) N ≥ 4 λ = (0, N − 1), (N − 3, −J+), (2, −J+ ± 9J 2 + − 8J 2 − /4)λ j = λ 2 + λJ + + 2 − J + N λ + J 2 − − J 2 + N = 0 (66) ⇒ λ = ± J + − 2 − J 2 −(67)
Table I row 1 reports these eigenvalues along with the two zero eigenvalues λ = 0 which gives the required 2N = 4 eigenvalues total. N = 3 is special because precisely one of the two kronecker δ i,j functions is on for all j; there is no j for which both δ j,1 , δ j,N −1 are zero simultaneously. For j = 1, δ j,1 = 1 and for j = N − 1, δ j,N −1 = δ j,2 = 1 (recall j from from 1 to N − 1; so when N = 2, j = 1, 2 only). So we get the following for both j = 1, 2
λ j = λ 2 + λJ + + − J + N λ + J 2 − − J 2 + N = 0 (68) ⇒ λ ± = 1 4 −J + ± 9J 2 + − 8J 2 −(69)
and so the pair λ ± has multiplicity two as shown Table I row 2 (i.e. the full set is λ + , λ + , λ − , λ − ). Finally, for N ≥ 4, there are now intermediary values of j = 2, . . . N − 2 for which δ j,1 = δ j,N −1 = 0 simultaneously in which casê to compute the eigenvalues of M SP W for N = 2, . . . 6 (for larger N , Mathematica starts to struggle). Now we use the expressions for λ to predict the bifurcations of the static phase wave. For convenience we write out the expressions below (which are valid for N > 2. For N = 2, λ = J 2 − − J 2 + > 0 for K < 0 meaning the static phase wave is unstable in this case. We will analyze the N = 2 case fully in the next section).
λ j = λ(λ + J + ) = 0 (70) ⇒ λ = 0, −J +(71)λ 0 = 0 (72) λ 1 = −J + (73) λ 2 = 1 4 −J + ± 9J 2 + − 8J 2 −(74)
λ 2 exists for all N ≥ 3 and triggers a Hopf bifurcation at K c = −J which comes from solving Re(λ ± 2 ) = 0 (the determinant 9J 2 + −8J 2 − is negative here so λ ± 2 are complex conjugates). It also triggers a saddle node bifurcation at K = 0 which comes from solving λ + 2 = 0 (λ − 2 is never 0). λ 1 exists for all N ≥ 4 and triggers a saddle node bifurcation also at K c = −J. These results imply the static phase wave is stable when Analysis of active async. The destabilization of the static phase wave via a Hopf bifurcation for N ≥ 2 implies a limit cycle is born at K < K c . Here swarmalators oscillate about their mean position in space and phase (Fig 3(d)) with little overall order S ± ≈ 0 (Fig 4) for most N . In the simple case N = 2 the active state can be studied using the standard transformation to mean and difference coordinates ( y , ∆y) = (y 1 + y 2 , y 2 − y 1 ) where y is a dummy variable. The mean positions and phases are invariant˙ x =˙ θ = 0 because of the pairwise oddness of Eqs.
v = N −1 N i=1 vi. where vi = (v 2 x + v 2 θ ) 1/K cos x sin x dx = J cos θ sin θ dθ (79) sin x K = C sin θ J(80)
For some constant C determined by initial conditions. Figure 6 plots the contours of Eq. (80), which correspond to the limit cycle of the active async state, along with the fixed point for K = −1. These are consistent with the oscillations in x(t) in Figure 5. They also tell us that the active async is stable for all K < 0 when N = 2, which confirms our result from the last section that the static phase wave is unstable when N = 2. For N > 2, analysis of the active async state becomes unwieldy so we numerically examine the state by plotting the mean velocity or activity
v = N −1 N i=1 v i where v i = (v 2
x + v 2 θ ) 1/2 is the velocity of the i-th swarmalator. Since all of the other collective states are stationary, v is a natural order parameter for the active async state. Figure 7, in which J = 1, shows the v persists for large N when K ≈ K c = −1 (recall, we know the state must exist for some K for all finite N via the Hopf bifurcation). For small coupling strengths K ≈ −2, however, the motion dies out for moderately large systems N 5 and the final static async state is born.
Analysis of static async. Here the swarmalators sit at fixed points (x * i , θ * i ) scattered uniformly in space and phase (Fig 3(e)) implying no global order S ± = 0 (Fig 4). This state is hard to analyze for finite N because for most population sizes N ≥ 3, multiple configurations of fixed points (x * i , θ * i ) exist for a given set of parameters (J, K). Simply enumerating this family of fixed points is a hard problem -it has not been done in the regular Kuramoto model -never mind analyzing their stability. In the continuum limit N → ∞, however, the stability may be analyzed since the state has a simple representation: ρ(x, θ, t) = 1/(4π 2 ) where ρ(x, θ, t)dxdθ gives the fraction of swarmalators with positions between x + dx and phases between θ + dθ at time t. The density obeys the continuity equatioṅ
v x = J sin(x − x) cos(θ − θ)ρ(x , θ , t)dx dθ (82) v θ = K sin(θ − θ) cos(x − x)ρ(x , θ , t)dx dθ (83)
The stability is analyzed by plugging a general perturbation
ρ = ρ 0 + η = (4π 2 ) −1 + η(x, θ, t)(84)
into the continuity equation and computing the spectrum. Since such analyses are standard, we put the details in Appendix C. The eventual result is
K < K c = −J(85)
Recall this is only valid as N → ∞. Interestingly, this result also proves the active async state disappears for N → ∞ via a 'squeeze' argument: since the static phase wave also loses stability at K c = −J, there is no room left for the active async state to exist as N → ∞. This completes our analysis. Figure 8 summarizes our findings in a pseudo bifurcation diagram in (K, N ) space for J = 1(pseudo because N is a discrete quantity and so should plot by plotted on a continuous axis, but we wanted to show the active async state dependence on N ). The boundary of the active async state for N ≥ 3 is found numerically by finding the first time v = 0 which indicates the swarmalators are no longer active. Figure 9 reports the bifurcation diagram in (J, K) space when N → ∞.
IV. MATCH TO REAL WORLD SWARMALATORS
Swarmalators are defined as entities with a twoway interaction between swarming and synchronization [61]. Below we list examples which (i) appear to meet this definition [77], (ii) swarm in ring-like geometries, and (iii) display collective behavior similar to the ring model.
Sperm are a classic microswimmer which swarm in solution and sync their tail gaits [11]. Sperm collected from ram semen and confined to 1D rings bifurcate from an isotropic state (analogues to the static async state) to a vortex state in which sperm rotate either clockwise or anticlockwise [50] which implies their positions and orientations are splayed just like the static phase wave (recall the static phase wave is really a state of uniform rotation and only static in the frame co-moving with the natural frequencies ω, ν) [50]. Moreover, the transition has associated transient decay of rotation velocity (see Fig 4a in [50]) consistent with a Hopf bifurcation, just like the ring model (74). Note unlike other studies of syncing sperm [11], here the phase variable is the sperm's orientation, not its tail rhythm.
Vinegar eels are a type of nematode found in beer mats and the slime from tree wounds [13,14]. They can be considered swarmalators because they sync the wriggling of their heads, swarm in solution, and it seems likely based on their behavior said sync and swarming interact [13,14] (neighbouring eels sync more easily than distant eels, so sync interacts with swarming, and sync'd eels presumable affect each local hydrodynamic environment and thereby affect each other's movements, so swarming interacts with sync). When confined to 2D disks, they seek out the 1D ring boundary forming metachronal waves in which the phase of their gait and their spatial positions around the ring are splayed similar to the static phase wave [13,14] (although note the winding number for the metachronal waves is k > 1; a full rotation in physical space x produces k > 1 rotations in phase θ).
C. elegans are another type of microswimmer which also swarm and sync the gait of their tails. When confined to 1D channels they form synchronous clusters analogous to the static sync state [62]. Though not strictly consistent with the ring model, a channel being a 1D line as opposed to a 1D ring, we mention them here because it is natural to expect sync clusters would persist in a 1D ring too.
V. DISCUSSION
Aim one of the paper was to model real-world swarmalators swarming in 1D. This was a success. The model captured the behavior of vinegar eels, sperm, and C. elegans. Even so, the model failed to capture the phenomenology of other 1D swarmalators such as the the two-cluster states of bordertaxic Japanese tree frogs [52] or the cluster dynamics of synthetic microswimmers [63]. Crafting a model that mimics these systems is a challenge for future research.
Aim two was to provide a stepping stone -i.e. the ring model -to an analytic understanding of the 2D swarmalator model's phenomenology, in particular its bifurcations. To this end, we studied the destabilization of the static async state, deriving a 1D version of K 1 . But an analogue of the active phase wave state did not appear in the ring model [78], so we could not study the second bifurcation at K 2 beyond which S + declines (Figure 1). On the upside, a 1D analogue of the static phase wave was observed, and were able to specify its stability (in the 2D model this state was only observed for K = 0 and its stability was not calculated). Moreover, this stability calculation certified the existence of the new active async state (we say certify because we originally thought the state was just a long transient or perhaps a numerical artefact; our results prove it exists for all finite N ) in which the swarmalators execute noisy, Brownian-like motion. What's interesting here is that this motion occurs for identical, noise-free swarmalators. This suggest any jittery behavior observed in real-world swarmalator systems may arise purely from the cross-talk between units' tendency to sync and swarm, and not from thermal agitation or other forms of noise.
Taken together, our analytic findings take us one step closer to understanding the bifurcations of the swarmalator model which for now remain unexplained.
We would love to see future work analyze the ring model using OA theory [64][65][66][67]. A breakthrough from Ott and Antonsen, the theory states that the density of the infinite-N Kuramoto model ρ(θ, t) has an invariant manifold of poisson kernels which allows dynamics for the classic sync order parameter Z := N −1 j e iθj to be derived explicitly. This is a big win. It reduces an N >> 1dimensional nonlinear system to a simple 2D system (one ODE for the complex quantity Z) -a drastic simplification which effectively solves the Kuramoto model. Given the ring model in (ξ, η) coordinates (Eqs.(8),(9)) resembles the Kuramoto model so closely, we suspect it may be solved with OA theory's magic: If regular oscillators are defined on the unit circle θ i ∈ S 1 and have an invariant manifold of poisson kernels, could ring swarmalators, defined on the unit torus (x, θ) ∈ (S 1 , S 1 ), have an invariant manifold of some 'toroidal' poisson kernel? If so, explicit dynamics for the rainbow order parameters W ± may be derivable and in that sense the ring model solved exactly.
[75] We choose the non-standard notation I since we have already used i and j as indices. Are the static sync and π states the only stable collective states when K > 0? Answering this question conclusively is difficult since it would mean integrating the equations of motion for every initial condition in the phase space T N = S N × S N . Figure 10 explores a subset of T N : initial position x i spaced uniformly on (0, aπ) and initial phases θ i spaced uniformly on (0, bπ) where 0 ≤ a, b ≤ 2 and then perturbed slightly with noise of order 10 −3 to both x i , θ i . When (a, b) = (2, 2) these initial conditions correspond to swarmalators spread out evenly over the unit circle in both space and phase. When a, b < 2, the swarmalators are spread out over subsets of the unit circle.
We use the Daido order parameters R 1 e iφ1 := N −1 j e iθj and R 2 e iφ2 = N −1 j e 2iθj to distinguish between the static and π states. The conditions for each state are 1. Condition for static sync: (R 1 , R 2 ) = (1, 1) 2. Condition for π-state: (R 1 , R 2 ) = (0, 1) While R 2 = 1 for both the static sync and π states, and thus can't distinguish between them, we include it to rule out the existence of some other collective state for which R 2 = 0. Figure 10 shows that the static sync is stable when (0 ≤ a ≤ 1) ∩ (0 ≤ b ≤ 1) and the π-state is stable in the remaining space. No other collective states were observed. Here we show how the ring model is contained within the 2D swarmalator model which is given bẏ
x i = v i + 1 N N j=1 I att (x j − x i )F (θ j − θ i ) − I rep (x j − x i ) ,(B1)θ i = ω i + K N N j=1 H att (θ j − θ i )G σ (x j − x i )(B2)
In [61], the choices I att = x/|x|, I rep = x/|x| 2 , F (θ) = 1 + J cos(θ), G(x) = 1/|x|, H att (θ) = sin(θ) were made. However, choosing linear spatial attraction I att (x) = x, inverse square spatial repulsion I rep (x) = x/|x| 2 and truncated parabolic space-phase coupling G(x) = (1 − |x| 2 /σ 2 )H heaviside (σ − |x|)
x i = 1 N N j =i x j − x i 1 + J cos(θ j − θ i ) − x j − x i |x j − x i | 2 (B3) θ i = K N N j =i sin(θ j − θ i ) 1 − |x j − x i | 2 σ 2 H heaviside (σ − |x j − x i |)(B4)
gives the same qualitative behavior but is nicer to work with analytically. First we show it demonstrates the same behavior. Figure 2 shows its collective states are the same as those of the original model, and Figure 1 shows its order parameters S ± (K) have the same shape: monotonic increase from (K 1 , K 2 ) as K → 0 −1 , monontonic decrease from (K 2 , 0) as K → 0 −1 , and then a discontinuous jump to S + = 1 at K = 0 (K < 0 is a singular perturbation). Note the plot of S ± (K) in Figure 6 of [61], the monotonic decrease on (K 2 , 0) of S + is slight and hard to see, so we plot in for finer K and show a zoom in for small K in Figure ??. Now we show the 'linear parabolic' model, so called because I att = x and G(x) is a parabolic, is cleaner analytically. In polar coordinates it takes forṁ
r i = H r (r i , φ i ) − Jr i R 0 cos Ψ 0 − θ i + J 2 S + cos Φ + − (φ i + θ i ) +S − cos Φ − − (φ i − θ i ) φ i = H φ (r i , φ i ) + J 2r i S + sin Ψ + − (φ i + θ i ) +S − sin Ψ − − (φ i − θ i ) θ i = K 1 − r 2 i σ 2 R 0 sin(Φ 0 − θ i ) − K σ 2 R 1 sin(Φ 1 − θ i ) + Kr i σ 2 S + sin Ψ + − (φ i + θ i ) −S − sin Ψ − − (φ i − θ i ) where H r (r i , φ i ) = 1 N j r j cos(φ j − φ i ) − r i (1 − d −2 ij ) (B5) H φ (r i , φ i ) = 1 N j r j r i sin(φ j − φ i )(1 − d −2 ij ),(B6)Z 0 = R 0 e iΨ0 = 1 N j e iθj ,(B7)Z 0 =R 0 e iΨ0 = 1 N j∈Ni e iθj ,(B8)Z 2 = R 2 e iΨ2 = 1 N j r 2 j e iθj ,(B9)Z 2 =R 2 e iΨ2 = 1 N j∈Ni r 2 j e iθj ,(B10)
W ± =S ± e iΨ± = 1 N j r j e i(φj ±θj ) (B11)
W ± =Ŝ ± e iΨ± = 1 N j∈Ni r j e i(φj ±θj )(B12)
where theẐ 0 , . . . order parameters are summed over all the neighbours N i of the i-th swarmalator: those within a distance σ. Notice that rainbow order parametersW here are weighted by the radial distance r j , which is not the case for the ring model Eqs. (10), (11) (that's why we put a tilde over the W). Assuming σ > max(d ij ), we can set Z 0 = Z 0 ,Ẑ 1 = Z 1 ,Ŵ ± = W ± . Then S ± sin(Φ ± − (φ ± θ)) etc of the ring model starting to emerge. If we assume there is no global synchrony Z 0 = Z 2 = 0, which happens generically in the frustrated parameter regime K < 0, J > 0, and Integrating Eq. (C13) with respect to (.)e i(nx±nθ) extracts the evolution equations for each mode. The ones of interest areẆ
1 ± = J + K 4π W 1 ± (C16)
andα n,m =β n,m = 0 for all (n, m) = (1, 1). Setting W 1 ± = w 1 ± e λ±t yields
λ ± = J + K 4π (C17)
which implies the static async state is stable for
K < K c = −J (C18)
consistent with simulations.
FIG. 3 :
3Collective states of ring model. Top row, swarmalators are drawn in as colored points on a unit circle in a dummy (x,ỹ) plane to allow a comparison with the 2D swarmalator model in
(a),(f) Static sync (J, K) = (1, 1) (b),(g) Static π-state (J, K) = (1, 1) (c),(h) Static phase wave (J, K) = (1, −0.5) (d),(i) Active async (J, K) = (1, −1.05). Here the swarmalators jiggle about in (x, θ) as indicated by the double ended arrows with no global space-phase order as indicated by the scatter plot in (i). The amount of motion / jiggling depends on the population size N as discussed in the main text and illustrated in Figure 7. (e), (j) Static async (J, K) = (1, −2).
FIG. 4 :
4Rainbow order parameters S± for the ring swarmalator model with νi = ωi = 0 and J = 1. We have assumed WLOG that S+ > Si (i.e. if S+ < S− in simulations we swapped (S+, S−) → (S−, S+)
I: Spectrum of static phase wave for different population sizes. The cases N = 2, 3 are distinguished. For N ≥ 4 the pattern is fixed. Notice a multiplicity 1 for eigenvalues that are complex conjugates λ± means one instance of the pair; a multiplicity of 2 means two instances of the pair: λ+, λ+, λ−, λ−. N = 2 is special because both of the Kronecker delta functions trigger at the same time: δ j,N −1 = δ j,2−1 = δ j,1 = 1. Eq. (65) then becomeŝ
FIG. 5 :
5Dynamics in active async state. Top: Position x(t) for a typical swarmalator for N = 2, 5, 50 swarmalators showing oscillatory behavior. For small N , the oscillations are smooth and fast. For larger N , they are irregular and slow. The dynamics of the swarmalator's phase θi (unplotted) are similar. Bottom: time series of order parameter S+. The behavior of S− is qualitatively the same. Simulation parameters were (dt, T ) = (0.1, 100) and (J, K) = (1, −1.05).
population sizes N > 2. To recap, at the left boundary K = −J, it destabilizes via a Hopf bifurcation for N = 3 and a simultaneous Hopf and saddle node for N ≥ 4. At the right boundary K = 0, it destabilizes via a saddle node for all N ≥ 3.
FIG. 6 :
6Phase space for N = 2 swarmalators in (∆x, ∆θ) coordinates (Eqs. (76), (77)) for (J, K) = (1, −1). Saddle point are shown as open red circles, nonlinear centers as open green circles. The family of periodic orbits, which correspond to the active async state, are given by Eq. (80) for C = −6, . . . , 6.
Figure 5
5depicts the oscillations in x(t) for a typical swarmalator for different population sizes. For small N , the oscillations are smooth and fast. For larger N , the oscillations are irregular and slow. (The oscillations in phase, unplotted, behave the same way). The bottom panel shows the dynamics of the rainbow order parameters S ± are similar.
(1),(2) which implies x 1 = −x 2 and θ 1 = −θ 2 . The differences, however, evolve according tȯ ∆x = J sin ∆x cos ∆θ (76) ∆θ = K sin ∆θ cos ∆x (77)These ODEs have 8 fixed points. For K > 0, the two fixed points (∆x, ∆θ) = (0, π), (0, π) are stable nodes (which corresponds to static synchrony) while the rest are saddles. For K < 0, four nonlinear centers exist while the other fixed points are again saddles. The family of periodic solution surrounding the centers correspond to the active async state and can be found explicitly by di-FIG. 7: Mean velocity
2 is the velocity of the i-th swarmalator, in the (N, K) plane for J = 1 and νi = ωi = 0. Simulation parameters were (dt, T ) = (0.1, 5000). The first 90% of data were discarded as transients and the mean of the remaining 10% are plotted. viding Eq. (76) by Eq. (77) and integrating, d∆x d∆θ = J sin ∆x cos ∆θ K sin ∆θ cos ∆x (78)
FIG. 8 :
8Pseudo-bifurcation diagram for model (1), (2) in (K, N ) space with J = 1 (pseudo because the population size N , a discrete quantitiy, is plotted on a continous axis; see main text). The N → ∞ limit is represented by plotting ... on the y-axis. The dotted line schematically represents the bifurcation curve of the active async state and thus merges with the line K = J = −1 as N → ∞.
velocity v = (v x , v θ ) is given by the N → ∞ limit of Eqs (1),(2)
FIG. 9 :
9Bifurcation diagram in (J, K) space as N → ∞.
FIG. 10 :
10Left panel: R1(a, b). Right panel: R2(a, b). Data were collected by integrating Eqs.(1), (2) using an RK4 method with (dt, T ) = (0.1, 500) for N = 100 swarmalators. FIG. 11: Left panel: S+ for the unit vector model which has the same qualitative shape as that of the linear parabolic model presented in Figure 1. Results for T = 500, 1000 are shown to establish convergence. Right panel: zoom in of S+(K) showing the non-monotonic dip. Results were generated using an RK4 solver with dt = 0.1. Appendix B: Connection of ring model to 2D swarmalator model
TABLE
Table I
Irow 3 summarizes the λ's and their multiplicities
in this general case which finally completes our calcula-
tion. We have confirmed the expressions for λ are cor-
rect by using Mathematica (notebook provided at [76])
[76] https://github.com/Khev/swarmalators/blob/master/1D/onring/regular/stability-static-phase-wave.nb[77] We say 'appear' to meet these definition because what is means to prove a bidirectional space-phase coupling in an experimental system is somewhat ambiguous. Most experimental studies of swarmalators either infer a spacephase coupling exists based on the observations (like the microswimmers[11,13,62] we describe;[62] do a particularly comprehensive study), but do not specify if the coupling bidirectional. Other studies demonstrate a twoway coupling in the sense that they encoding it in a model that produces behavior similar to observed data (like magnetic domain walls[32] or myxobacteria[68]). In other words, we're trying to be conservative in any claims we make.[78] We note the splintered phase wave and active phase waves are realized in a 1D ring model presented in the Supplementary Information of[61]. But that model does not have the clean format of the current model Eqs.(8),(9) which, being so simple, seem the natural model to study 1D swarmalator phenomena. Moreover, the analogues of the splintered phase wave and active phase wave states there are unsteady, the order parameters varying in time. So in that sense that are not simpler warm-up versions of the states in 2D.Appendix A: Basin of attractions for static sync and π-state
transform to ξ i = φ i + θ i and η i = φ i − θ i coordinates the ring model is revealed (the terms in the square parentheses in the latter two equations.)Appendix C: Stability of static async stateThe density obeys the continuity equatioṅConsider a perturbation around the static async stateNormalization requires ρ(x, θ) = 1 which impliesThe density ansatz (C4) decomposes the velocitywhere v 0 is velocity in the static async state v 0 = 0. The perturbed velocity v 1 is given by Eqs (C9) with ρ replaced by η. Plugging Eqs (C4), (C6) into (C1) yieldṡTo tackle this, first write the v 1 in terms of the orderwhere the perturbed order parameters arePlugging this into the evolution equation for η Eq. (C7) yieldṡOnly a few Fourier modes are distinguished so we expand η(x, θ, t) in terms of complex exponentialsn,m (t)e i(nx+mθ) +β n,m (t)e i(nx−mθ)where c.c denotes the complex conjugate. Unconventionally, we associate the complex conjugateᾱ,β with the first harmonic. This is so that the order parameters are expressed in terms of W 1 + = α 1,1 (t) and W 1 − = β 1,1 (t) (i.e. without the bar overhead). Also, notice the normalization condition (C5) implies α 0,0 (t) = 0.
A T Winfree, The geometry of biological time. Springer Science & Business Media12A. T. Winfree, The geometry of biological time, vol. 12 (Springer Science & Business Media, 2001).
Y Kuramoto, Chemical oscillations, waves, and turbulence. Courier CorporationY. Kuramoto, Chemical oscillations, waves, and turbu- lence (Courier Corporation, 2003).
Synchronization: a universal concept in nonlinear sciences. A Pikovsky, J Kurths, M Rosenblum, J Kurths, Cambridge university press12A. Pikovsky, J. Kurths, M. Rosenblum, and J. Kurths, Synchronization: a universal concept in nonlinear sci- ences, 12 (Cambridge university press, 2003).
. J Toner, Y Tu, Physical review letters. 754326J. Toner and Y. Tu, Physical review letters 75, 4326 (1995).
. T Vicsek, A Czirók, E Ben-Jacob, I Cohen, O Shochet, Physical review letters. 751226T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Physical review letters 75, 1226 (1995).
. Z Jiang, M Mccall, 10155Z. Jiang and M. McCall, JOSA B 10, 155 (1993).
. G Kozyreff, A Vladimirov, P Mandel, Physical Review Letters. 853809G. Kozyreff, A. Vladimirov, and P. Mandel, Physical Re- view Letters 85, 3809 (2000).
. C S Peskin, Mathematical Aspects of Heart Physiology. Courant Institute of Mathematical SciencesC. S. Peskin, Mathematical Aspects of Heart Physiology (Courant Institute of Mathematical Sciences, New York, 1975), pp. 268-278.
. W Bialek, A Cavagna, I Giardina, T Mora, E Silvestri, M Viale, A M Walczak, Proceedings of the National Academy of Sciences. 1094786W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Sil- vestri, M. Viale, and A. M. Walczak, Proceedings of the National Academy of Sciences 109, 4786 (2012).
. Y Katz, K Tunstrøm, C C Ioannou, C Huepe, I D Couzin, Proceedings of the National Academy of Sciences. 10818720Y. Katz, K. Tunstrøm, C. C. Ioannou, C. Huepe, and I. D. Couzin, Proceedings of the National Academy of Sciences 108, 18720 (2011).
. Y Yang, J Elgeti, G Gompper, Physical review E. 7861903Y. Yang, J. Elgeti, and G. Gompper, Physical review E 78, 061903 (2008).
. I H Riedel, K Kruse, J Howard, Science. 309300I. H. Riedel, K. Kruse, and J. Howard, Science 309, 300 (2005).
. A Quillen, A Peshkov, E Wright, S Mcgaffigan, arXiv:2101.06809arXiv preprintA. Quillen, A. Peshkov, E. Wright, and S. McGaffigan, arXiv preprint arXiv:2101.06809 (2021).
. A Quillen, A Peshkov, E Wright, S Mcgaffigan, arXiv:2104.10316arXiv preprintA. Quillen, A. Peshkov, E. Wright, and S. McGaffigan, arXiv preprint arXiv:2104.10316 (2021).
. G I Taylor, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 209447G. I. Taylor, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 209, 447 (1951).
. S L Tamm, T Sonneborn, R V Dippell, The Journal of cell biology. 6498S. L. Tamm, T. Sonneborn, and R. V. Dippell, The Jour- nal of cell biology 64, 98 (1975).
. M Huang, W Hu, S Yang, Q.-X Liu, H Zhang, Proceedings of the National Academy of Sciences. 118M. Huang, W. Hu, S. Yang, Q.-X. Liu, and H. Zhang, Proceedings of the National Academy of Sciences 118 (2021).
. J Yan, M Bloom, S C Bae, E Luijten, S Granick, Nature. 491578J. Yan, M. Bloom, S. C. Bae, E. Luijten, and S. Granick, Nature 491, 578 (2012).
. J Yan, S C Bae, S Granick, Soft Matter. 11147J. Yan, S. C. Bae, and S. Granick, Soft Matter 11, 147 (2015).
. S Hwang, T D Nguyen, S Bhaskar, J Yoon, M Klaiber, K J Lee, S C Glotzer, J Lahann, Advanced Functional Materials. 301907865S. Hwang, T. D. Nguyen, S. Bhaskar, J. Yoon, M. Klaiber, K. J. Lee, S. C. Glotzer, and J. Lahann, Advanced Functional Materials 30, 1907865 (2020).
. B Zhang, A Sokolov, A Snezhko, Nature communications. 111B. Zhang, A. Sokolov, and A. Snezhko, Nature commu- nications 11, 1 (2020).
. A Bricard, J.-B Caussin, D Das, C Savoie, V Chikkadi, K Shitara, O Chepizhko, F Peruani, D Saintillan, D Bartolo, Nature communications. 61A. Bricard, J.-B. Caussin, D. Das, C. Savoie, V. Chikkadi, K. Shitara, O. Chepizhko, F. Peruani, D. Saintillan, and D. Bartolo, Nature communications 6, 1 (2015).
. B Zhang, H Karani, P M Vlahovska, A Snezhko, Soft Matter. B. Zhang, H. Karani, P. M. Vlahovska, and A. Snezhko, Soft Matter (2021).
R K Manna, O E Shklyaev, A C Balazs, Proceedings of the National Academy of Sciences. the National Academy of Sciences118R. K. Manna, O. E. Shklyaev, and A. C. Balazs, Proceed- ings of the National Academy of Sciences 118 (2021).
A Barciś, M Barciś, C Bettstetter, 2019 International Symposium on Multi-Robot and Multi-Agent Systems (MRS). IEEEA. Barciś, M. Barciś, and C. Bettstetter, in 2019 In- ternational Symposium on Multi-Robot and Multi-Agent Systems (MRS) (IEEE, 2019), pp. 98-104.
. A Barciś, C Bettstetter, IEEE Access. 8218752A. Barciś and C. Bettstetter, IEEE Access 8, 218752 (2020).
. J D Monaco, G M Hwang, K M Schultz, K Zhang, Biological cybernetics. 114269J. D. Monaco, G. M. Hwang, K. M. Schultz, and K. Zhang, Biological cybernetics 114, 269 (2020).
J D Monaco, G M Hwang, K M Schultz, K Zhang, Micro-and Nanotechnology Sensors, Systems, and Applications XI (International Society for Optics and Photonics. 10982109822J. D. Monaco, G. M. Hwang, K. M. Schultz, and K. Zhang, in Micro-and Nanotechnology Sensors, Sys- tems, and Applications XI (International Society for Op- tics and Photonics, 2019), vol. 10982, p. 109822D.
. K Uriu, L G Morelli, Development, growth & differentiation. 59351K. Uriu and L. G. Morelli, Development, growth & dif- ferentiation 59, 351 (2017).
. K Uriu, L G Morelli, Biophysical journal. 107514K. Uriu and L. G. Morelli, Biophysical journal 107, 514 (2014).
. C D Tsiairis, A Aulehla, Cell. 164656C. D. Tsiairis and A. Aulehla, Cell 164, 656 (2016).
. A Hrabec, V Křižáková, S Pizzini, J Sampaio, A Thiaville, S Rohart, J Vogel, Physical review letters. 120227204A. Hrabec, V. Křižáková, S. Pizzini, J. Sampaio, A. Thi- aville, S. Rohart, and J. Vogel, Physical review letters 120, 227204 (2018).
. D Tanaka, Physical review letters. 99134103D. Tanaka, Physical review letters 99, 134103 (2007).
. M Iwasa, D Tanaka, Physics Letters A. 3813054M. Iwasa and D. Tanaka, Physics Letters A 381, 3054 (2017).
. M Iwasa, D Tanaka, Physical Review E. 8166214M. Iwasa and D. Tanaka, Physical Review E 81, 066214 (2010).
. D Levis, I Pagonabarraga, B Liebchen, Physical Review Research. 123026D. Levis, I. Pagonabarraga, and B. Liebchen, Physical Review Research 1, 023026 (2019).
. H Hong, Chaos: An Interdisciplinary Journal of Nonlinear Science. 28103112H. Hong, Chaos: An Interdisciplinary Journal of Nonlin- ear Science 28, 103112 (2018).
. H K Lee, K Yeo, H Hong, Chaos: An Interdisciplinary Journal of Nonlinear Science. 3133134H. K. Lee, K. Yeo, and H. Hong, Chaos: An Interdisci- plinary Journal of Nonlinear Science 31, 033134 (2021).
. F Jiménez-Morales, Physical Review E. 10162202F. Jiménez-Morales, Physical Review E 101, 062202 (2020).
. J U Lizarraga, M A De Aguiar, Chaos: An Interdisciplinary Journal of Nonlinear Science. 3053112J. U. Lizarraga and M. A. de Aguiar, Chaos: An In- terdisciplinary Journal of Nonlinear Science 30, 053112 (2020).
. T A Mclennan-Smith, D O Roberts, H S Sidhu, Physical Review E. 10232607T. A. McLennan-Smith, D. O. Roberts, and H. S. Sidhu, Physical Review E 102, 032607 (2020).
. K P O'keeffe, J H Evers, T Kolokolnikov, Physical Review E. 9822203K. P. O'Keeffe, J. H. Evers, and T. Kolokolnikov, Physi- cal Review E 98, 022203 (2018).
K O'keeffe, C Bettstetter, Micro-and Nanotechnology Sensors, Systems, and Applications XI (International Society for Optics and Photonics. 10982109822K. O'Keeffe and C. Bettstetter, in Micro-and Nanotech- nology Sensors, Systems, and Applications XI (Inter- national Society for Optics and Photonics, 2019), vol. 10982, p. 109822E.
S.-Y Ha, J Jung, J Kim, J Park, X Zhang, Kinetic & Related Models. S.-Y. Ha, J. Jung, J. Kim, J. Park, and X. Zhang, Kinetic & Related Models (2021).
. S.-Y Ha, J Jung, J Kim, J Park, X Zhang, Mathematical Models and Methods in Applied Sciences. 292225S.-Y. Ha, J. Jung, J. Kim, J. Park, and X. Zhang, Math- ematical Models and Methods in Applied Sciences 29, 2225 (2019).
. H H Bau, D Raizen, J Yuan, Taylor & Francis41118606H. H. Bau, D. Raizen, and J. Yuan, in Worm (Taylor & Francis, 2015), vol. 4, p. e1118606.
. J Yuan, D M Raizen, H H Bau, Journal of The Royal Society Interface. 1220150227J. Yuan, D. M. Raizen, and H. H. Bau, Journal of The Royal Society Interface 12, 20150227 (2015).
. R Nosrati, P J Graham, Q Liu, D Sinton, Scientific reports. 61R. Nosrati, P. J. Graham, Q. Liu, and D. Sinton, Scien- tific reports 6, 1 (2016).
. S Ketzetzi, M Rinaldin, P Dröge, J De Graaf, D J Kraft, arXiv:2103.07335arXiv preprintS. Ketzetzi, M. Rinaldin, P. Dröge, J. de Graaf, and D. J. Kraft, arXiv preprint arXiv:2103.07335 (2021).
. A Creppy, F Plouraboué, O Praud, X Druart, S Cazin, H Yu, P Degond, Journal of The Royal Society Interface. 1320160575A. Creppy, F. Plouraboué, O. Praud, X. Druart, S. Cazin, H. Yu, and P. Degond, Journal of The Royal Society Interface 13, 20160575 (2016).
. I Aihara, Physical Review E. 8011918I. Aihara, Physical Review E 80, 011918 (2009).
. I Aihara, T Mizumoto, T Otsuka, H Awano, K Nagira, H G Okuno, K Aihara, Scientific reports. 41I. Aihara, T. Mizumoto, T. Otsuka, H. Awano, K. Na- gira, H. G. Okuno, and K. Aihara, Scientific reports 4, 1 (2014).
. T C Le, J Zhai, W.-H Chiu, P A Tran, N Tran, International journal of nanomedicine. 146749T. C. Le, J. Zhai, W.-H. Chiu, P. A. Tran, and N. Tran, International journal of nanomedicine 14, 6749 (2019).
. G Agrawal, R , ACS Applied Nano Materials. 21738G. Agrawal and R. Agrawal, ACS Applied Nano Materi- als 2, 1738 (2019).
. Y Yi, L Sanchez, Y Gao, Y Yu, Analyst. 1413526Y. Yi, L. Sanchez, Y. Gao, and Y. Yu, Analyst 141, 3526 (2016).
. Z Xiao, M Wei, W Wang, ACS applied materials & interfaces. 116667Z. Xiao, M. Wei, and W. Wang, ACS applied materials & interfaces 11, 6667 (2018).
. S H Strogatz, Physica D: Nonlinear Phenomena. 1431S. H. Strogatz, Physica D: Nonlinear Phenomena 143, 1 (2000).
. S H Strogatz, R E Mirollo, Journal of Statistical Physics. 63613S. H. Strogatz and R. E. Mirollo, Journal of Statistical Physics 63, 613 (1991).
. R Mirollo, S H Strogatz, Journal of Nonlinear Science. 17309R. Mirollo and S. H. Strogatz, Journal of Nonlinear Sci- ence 17, 309 (2007).
. J D Crawford, Journal of Statistical Physics. 74J. D. Crawford, Journal of Statistical Physics 74, 1047-1084 (1994).
. K P O'keeffe, H Hong, S H Strogatz, Nature communications. 81K. P. O'Keeffe, H. Hong, and S. H. Strogatz, Nature communications 8, 1 (2017).
. J Yuan, D M Raizen, H H Bau, Proceedings of the National Academy of Sciences. 1116865J. Yuan, D. M. Raizen, and H. H. Bau, Proceedings of the National Academy of Sciences 111, 6865 (2014).
. F Guzmán-Lastra, A Kaiser, H Löwen, Nature communications. 71F. Guzmán-Lastra, A. Kaiser, and H. Löwen, Nature communications 7, 1 (2016).
. E Ott, T M Antonsen, Chaos: An Interdisciplinary Journal of Nonlinear Science. 1837113E. Ott and T. M. Antonsen, Chaos: An Interdisciplinary Journal of Nonlinear Science 18, 037113 (2008).
Chaos: An Interdisciplinary. A Pikovsky, M Rosenblum, Journal of Nonlinear Science. 2597616A. Pikovsky and M. Rosenblum, Chaos: An Interdisci- plinary Journal of Nonlinear Science 25, 097616 (2015).
. J R Engelbrecht, R Mirollo, Physical Review Research. 223057J. R. Engelbrecht and R. Mirollo, Physical Review Re- search 2, 023057 (2020).
. B Chen, J R Engelbrecht, R Mirollo, Journal of Physics A: Mathematical and Theoretical. 50355101B. Chen, J. R. Engelbrecht, and R. Mirollo, Journal of Physics A: Mathematical and Theoretical 50, 355101 (2017).
. O A Igoshin, A Mogilner, R D Welch, D Kaiser, G Oster, Proceedings of the National Academy of Sciences. 9814913O. A. Igoshin, A. Mogilner, R. D. Welch, D. Kaiser, and G. Oster, Proceedings of the National Academy of Sci- ences 98, 14913 (2001).
Short for 'swarming oscillators. Short for 'swarming oscillators'.
Note the Figure displays the order parameters of the instance of the model. see Appendix BNote the Figure displays the order parameters of the in- stance of the model; see Appendix B
Note we could set J = 1 WLOG by rescaling time but we decided against this so as to make the facilate a clean comparison to the 2D model for which both. J, K) appearNote we could set J = 1 WLOG by rescaling time but we decided against this so as to make the facilate a clean comparison to the 2D model for which both (J, K) appear
If the correlation is xi + θi = c, then S+ = 1. If xi− = θi + c, then S− = 1. If the correlation is xi + θi = c, then S+ = 1. If xi− = θi + c, then S− = 1.
Linear transformations xi → xi + C and θi → C do not change the dynamics because only differences xj − xi and θj − θi appear in the model. Linear transformations xi → xi + C and θi → C do not change the dynamics because only differences xj − xi and θj − θi appear in the model.
| [
"https://github.com/Khev/swarmalators/blob/master/1D/onring/regular/stability-static-phase-wave.nb[77]"
] |
[
"Spatial diquark correlations in a hadron Spatial diquark correlations in a hadron",
"Spatial diquark correlations in a hadron Spatial diquark correlations in a hadron"
] | [
"Jeremy Green [email protected] ",
"John Negele ",
"Michael Engelhardt ",
"Patrick Varilly ",
"Jeremy Green ",
"\nCenter for Theoretical Physics\nDepartment of Physics\nMassachusetts Institute of Technology\n02139CambridgeMA\n",
"\nDepartment of Physics\nNew Mexico State University\nLas Cruces88003-0001NM\n",
"\nUniversity of California\n94720Berkeley, BerkeleyCA\n"
] | [
"Center for Theoretical Physics\nDepartment of Physics\nMassachusetts Institute of Technology\n02139CambridgeMA",
"Department of Physics\nNew Mexico State University\nLas Cruces88003-0001NM",
"University of California\n94720Berkeley, BerkeleyCA"
] | [
"The XXVIII International Symposium on Lattice Field Theory"
] | Using lattice QCD, a diquark can be studied in a gauge-invariant manner by binding it to a static quark in a heavy-light-light hadron. We compute the simultaneous two-quark density of a diquark, including corrections for periodic boundary conditions. We define a correlation function to isolate the intrinsic correlations of the diquark and reduce the effects caused by the presence of the static quark. Away from the immediate vicinity of the static quark, the diquark has a consistent shape, with much stronger correlations seen in the good (scalar) diquark than in the bad (vector) diquark. We present results for m π = 293 MeV in N f = 2 + 1 QCD as well as m π = 940 MeV in quenched QCD, and discuss the dependence of the spatial size on the pion mass. | 10.22323/1.105.0140 | [
"https://arxiv.org/pdf/1012.2353v1.pdf"
] | 118,950,640 | 1012.2353 | bc749bb6741d1dd82b0136d1b78ba8bbfab49829 |
Spatial diquark correlations in a hadron Spatial diquark correlations in a hadron
June 14-19,2010
Jeremy Green [email protected]
John Negele
Michael Engelhardt
Patrick Varilly
Jeremy Green
Center for Theoretical Physics
Department of Physics
Massachusetts Institute of Technology
02139CambridgeMA
Department of Physics
New Mexico State University
Las Cruces88003-0001NM
University of California
94720Berkeley, BerkeleyCA
Spatial diquark correlations in a hadron Spatial diquark correlations in a hadron
The XXVIII International Symposium on Lattice Field Theory
Villasimius, Sardinia Italy * SpeakerJune 14-19,2010
Using lattice QCD, a diquark can be studied in a gauge-invariant manner by binding it to a static quark in a heavy-light-light hadron. We compute the simultaneous two-quark density of a diquark, including corrections for periodic boundary conditions. We define a correlation function to isolate the intrinsic correlations of the diquark and reduce the effects caused by the presence of the static quark. Away from the immediate vicinity of the static quark, the diquark has a consistent shape, with much stronger correlations seen in the good (scalar) diquark than in the bad (vector) diquark. We present results for m π = 293 MeV in N f = 2 + 1 QCD as well as m π = 940 MeV in quenched QCD, and discuss the dependence of the spatial size on the pion mass.
Introduction
Diquarks are two-quark systems. Collective behavior of a diquark has been invoked to explain many phenomena of strong interactions [1]. By introducing diquarks as effective degrees of freedom in chiral perturbation theory, they have been used to explain the enhancement of ∆I = 1 2 nonleptonic weak decays [2]. A simple quark-diquark model is quite successful at organizing the spectrum of excited light baryon states [3].
The simplest diquark operators are quark bilinears with spinor part q T CΓq. The favored combinations are color antitriplet, even parity [4]. These are divided into "good" and "bad" diquarks. The good diquarks, q T Cγ 5 q, have spin 0 and are flavor antisymmetric due to fermion statistics. The bad diquarks, q T Cγ i q, have spin 1 and are flavor symmetric.
Both one-gluon exchange in a quark model [5,6] and instanton [7] models give a spin coupling energy proportional to S i · S j , which favors the good diquark over the bad diquark. The strength of this coupling falls off with increasing quark masses. For the instanton model, the effective interaction has a flavor dependence that also favors the good diquark.
Earlier studies in baryons
Since diquarks are not color singlets, studying them within the framework of lattice QCD typically requires that they be combined with a third quark to form a color singlet. Diquark attractions result in spatial correlations between the two quarks in the diquark, which can be probed by computing a wavefunction or two-quark density.
In one study [8], using gauge fixing and r d u R u d r Figure 1: Geometry of the three quarks used in this paper and in [8] (left), and restricted geometry used in [9] (right).
three quarks of equal mass, a wavefunction was computed by displacing quarks at the sink,
ψ(r 1 , r 2 ) ∝ ∑ r s u(r s + r 1 ,t)d(r s + r 2 ,t)s(r s ,t) ×ū(0, 0)d(0, 0)s(0, 0) .
Then, using the more convenient coordinates R = (r 1 + r 2 )/2 and r = (r 1 − r 2 )/2 (2.1) ( Fig. 1, left), the wavefunction of the good and bad diquarks was shown for different fixed R = |R| as a function of r, in both Coulomb gauge and Landau gauge. In all cases, the wave function had a peak near r = 0, but it was found to fall off more rapidly for the good diquark, consistent with the expectation that good diquarks are more tightly bound.
In a second study [9], spatial correlations were investigated by computing the two quark density ρ 2 (r u , r d ) for a (u, d) diquark in the background of a static quark. To isolate correlations caused by the diquark interaction, analysis was restricted to spherical shells |r u | = |r d | = r (Fig. 1, right). For both good and bad diquarks, the density was found to be concentrated near r ud = |r u − r d | = 0, and the effect was much stronger for the good diquark. Fitting ρ 2 for the good diquark to exp(−r ud /r 0 (r)), r 0 reached a plateau for large r, giving a characteristic size r 0 = 1.1 ± 0.2 fm.
Correlation function
In the first study, the wavefunctions of good and bad diquarks were compared for an unrestricted geometry, but they were not compared against an uncorrelated wavefunction. In the second study, the intrinsic clustering caused by the diquark interaction was shown, however this was achieved by using a restricted geometry.
To overcome these limitations, we combined a diquark with a static quark, using the baryon operator B = ε abc u T a CΓd b s c , taking Γ = γ 5 for the good diquark and Γ = γ 1 for the bad diquark, and calculated the single quark density and the simultaneous two-quark density:
ρ 1 (r) = N 1 0|B(0,t f )J u 0 (r,t)B(0,t i )|0 0|B(0,t f )B(0,t i )|0 , ρ 2 (r 1 , r 2 ) = N 2 0|B(0,t f )J u 0 (r 1 ,t)J d 0 (r 2 ,t)B(0,t i )|0 0|B(0,t f )B(0,t i )|0 .
Here, there are insertions of the current J f µ =f γ µ f , and the normalization factors N 1,2 are required since this local current is not conserved on the lattice.
In a system where ρ 1 (r) is uniform, the two-particle correlation can be defined as
C 0 (r 1 , r 2 ) = ρ 2 (r 1 , r 2 ) − ρ 1 (r 1 )ρ 1 (r 2 ).
Deviations from zero are seen as evidence for interactions between particles. This correlation integrates to zero and approaches zero as the relative distance r 12 = |r 1 − r 2 | increases beyond the range of interactions in the system. The situation considered here is not so simple. The single particle density is not uniform: it is concentrated near the static quark. C 0 will still integrate to zero and fall off at large distances, however it is also larger near the static quark and this obscures the diquark correlations.
In order to remove the effect of the static quark, we define the normalized correlation function:
C(r 1 , r 2 ) = ρ 2 (r 1 , r 2 ) − ρ 1 (r 1 )ρ 1 (r 2 ) ρ 1 (r 1 )ρ 1 (r 2 ) . (3.1)
This divides out the tendency to stay near the static quark and retains the property of being zero if the two light quarks are uncorrelated (i.e. if ρ 2 (r 1 , r 2 ) = ρ 1 (r 1 )ρ 1 (r 2 )). The downsides are that C no longer integrates to zero, and it is possible for C(r, r) to increase without bound as |r| → ∞.
Density in a periodic box
We assume the lattice spacing is small enough that in an infinite volume we can treat ρ 2 (r 1 , r 2 ) as a function of R = |R|, r = |r|, and θ , the angle between R and r. Since the calculation is actually carried out on a finite lattice volume, to recover the infinite volume result, we need to deal with the effect of periodic boundary conditions.
The problem of dealing with ρ in periodic boundary conditions has been previously analyzed for the case of a meson [10]. It was found that ρ 1 (r) = ∑ n∈Z 3ρ 1 (r + nL), whereρ 1 differs from the infinite volume result only for r L due to interactions with periodic images.
For this study, we have a baryon, and there is an additional complication: the contribution from "exchange diagrams" in which the two quarks travel in opposite directions across the periodic boundary and can form a color singlet. As the lattice size grows, this becomes dominated by the propagation of the lightest meson and so falls off as exp(−m π L).
Ignoring the exchange diagrams and interactions with periodic images we find
ρ 2 (r 1 , r 2 ) = ∑ n 1 ,n 2 ∈Z 3 ρ 2 (r 1 + n 1 L, r 2 + n 2 L),
where ρ 2 is the infinite volume two quark density. In order to deal with image effects, a phenomenological fit is used. Given a good functional form f 2 (r 1 , r 2 ) for ρ 2 (invariant under simultaneous rotations of r 1 and r 2 as well as exchange of r 1 and r 2 ), the nearest images are added in:
f 2 (r 1 , r 2 ) = ∑ n i 1 ,n j 2 ∈{−1,0,1} f 2 (r 1 + n 1 L, r 2 + n 2 L).
This function and its lattice integral f 1 (r) = ∑ r 2 f 2 (r, r 2 ) can be simultaneously fit to ρ 2 and ρ 1 using a nonlinear weighted least squares method. This allows the images to be subtracted off,
giving ρ 2 ρ 2 − f 2 + f 2 and ρ 1 ρ 1 − f 1 + f 1 , where f 1 (r) = d 3 r 2 f 2 (r, r 2 ).
The so-called ∆ ansatz for the static potential for interacting quarks [11,12] was used as motivation for the functional form of f 2 . We ultimately found that the following eleven parameter functional form gave a reasonably good fit:
f 2 (r 1 , r 2 ) = Ag(r 1 , B, a 1 , 0)g(r 2 , B, a 1 , 0)g(r,C, a 2 , b 2 )e −D r 3/2 1 +r 3/2 2 +Er 3/2 +FR 3/2 +Ge −α √ r 2 1 +r 2 2 , with g(r, A, a, b) = exp(−Ar) r > a c 1 − br − c 2 r 2 r < a ,
where c 1,2 are given by the requirement that g and ∂ g ∂ r are continuous at r = a.
Lattice Calculations
We used a mixed action scheme [13] with domain wall valence quarks on an asqtad sea, with m π = 293(1) MeV and a = 0.1241(25) fm. This ensemble had 453 HYP smeared MILC gauge configurations [14], which have N f = 2 + 1 and volume 20 3 × 64. Propagators were computed every 8 lattice units in the time direction, allowing for 8 measurements per gauge configuration, with source and sink separated by 8 lattice units. Wilson lines were computed using HYP smeared gauge links, and measurements were averaged over seven positions for the static quark: x = 0 and the six nearest neighbors.
For comparison, we also used a heavy quark mass, with m π ≈ 940 MeV. Since the effects of dynamical sea quarks are negligible at that mass, we performed a calculation with κ = 0.153 Wilson fermions on 200 configurations from the OSU_Q60a ensemble [15], which are 16 3 × 32 with quenched β = 6.00 Wilson action. From the static quark potential, this has a/r 0 = 0.186 [16]. Using r 0 = 0.47 fm, the lattice spacing is a = 0.088 fm. We used a source-sink separation of 11 lattice units and averaged measurements over the two central timeslices.
The functions f 1,2 were fit to a restricted set of the lattice measurements ρ 1,2 . Three conditions were imposed to reduce the influence of the points most affected by images: r < 8a for ρ 1 , r 2 1 +r 2 2 < 100a 2 for ρ 2 , and in both cases r image ≥ 11a, where r image is the distance to the nearest periodic image of the static quark. Fits had χ 2 per degree of freedom ranging from 0.25 to 1.85. In the quenched good diquark case, Figure 2 shows the effect of image corrections for ρ 1 , and here this procedure is quite successful, even extrapolating beyond the range included in the fit.
For ρ 2 , the fit isn't as good as for ρ 1 , but it still works well. Figure 3 shows ρ 2 with and without image corrections. The figure on the right looks cleaner for two reasons. First, the fit function is determined using a global fit, which allows for small deviation from the specified R and θ to be compensated for. Second, image corrections have been applied, which are substantial for points distant from the origin. The end result is that the difference between the plotted point and the fit curve is equal to the difference between the raw data point and the fit for that point.
Results and discussion
As a check of how well the correlation function isolates the diquark from the effect of the static quark, we compared different directions of r. Even at R = 0.2 fm, C was independent of the direction of r, indicating that this correlation function works quite well. Finally, we can compare the systems. Fig. 4 shows the profile of the correlation function C at two fixed distances R from the static quark to the center of the diquark, and Fig. 5 shows the full dependence of C on r, at fixed R = 0.4 fm. The good diquark has a large positive correlation at small r that becomes negative at large r. The bad diquark has similar behavior with smaller magnitude. The difference between the good and bad diquarks is larger for the lighter pion mass, as expected from the quark mass dependence of the spin coupling that splits good and bad diquarks. As R increases, both the correlation and the size of the positive region grow, although it is possible that some of this growth of C as R increases may arise from the normalization of the correlation function.
Our main conclusions are seen clearly in Fig. 5. The diquark correlations are highly independent of θ , indicating negligible polarization by the heavy quark, are much stronger in the good rather than the bad channel, and increase strongly with decreasing quark mass. Finally, it is important to note that the diquark radius is approximately 0.3 fm and the hadron half-density radius is also roughly 0.3 fm, so the diquark size is comparable to the hadron size. This is reminiscent of the size of Cooper pairs in nuclei, and argues against hadron models requiring point-like diquarks. Figure 5: Continuous C(r 1 , r 2 ) derived from the fit, as a function of r (in fm) with R = 0.4 fm. The two axes r and r ⊥ indicate directions of r parallel to and orthogonal to R, respectively. The color of the surface is discontinuous at C = 0.
Figure 2 : 940 Figure 3 :
29403ρ 1 (r) without (left) and with (right) image corrections for the good diquark on the quenched m π = ρ 2 (r 1 , r 2 ) without (left) and with (right) image corrections for the good diquark on the quenched m π = 940 MeV ensemble, as a function of r with r ⊥ R and R/a = 0, 4, 6.
Figure 4 :
4C(r 1 , r 2 ), as a function of r (in fm) with R = 0.2 fm (left) and R = 0.4 fm (right), with r ⊥ R for the good and bad diquarks and the two pion masses.
AcknowledgmentsThis work was supported in part by funds provided by the U.S. Department of Energy under Grants No. DE-FG02-94ER40818 and DE-FG02-96ER40965. P.V. acknowledges support by the MIT Undergraduate Research Opportunities Program (UROP). Additional domain wall propagators were computed using the Chroma software suite[17].
. M Anselmino, E Predazzi, S Ekelin, S Fredriksson, D B Lichtenberg, Rev. Mod. Phys. 65M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson and D. B. Lichtenberg, Rev. Mod. Phys. 65 (1993) 1199-1234.
. M Neubert, B Stech, Phys. Rev. 44M. Neubert and B. Stech, Phys. Rev. D44 (1991) 775-793.
A Selem, F Wilczek, hep-ph/0602128New trends in HERA physics. Proceedings, Ringberg Workshop. Tegernsee, GermanyA. Selem and F. Wilczek, in New trends in HERA physics. Proceedings, Ringberg Workshop, Tegernsee, Germany, October 2-7, 2005, pp. 337-356, 2006. hep-ph/0602128.
. R L Jaffe, Phys. Rept. 409R. L. Jaffe, Phys. Rept. 409 (2005) 1-45.
. A De Rujula, H Georgi, S L Glashow, Phys. Rev. 12A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D12 (1975) 147-162.
. T A Degrand, R L Jaffe, K Johnson, J E Kiskis, Phys. Rev. 122060T. A. DeGrand, R. L. Jaffe, K. Johnson and J. E. Kiskis, Phys. Rev. D12 (1975) 2060.
. T Schäfer, E V Shuryak, Rev. Mod. Phys. 70T. Schäfer and E. V. Shuryak, Rev. Mod. Phys. 70 (1998) 323-426.
. R Babich, Phys. Rev. 7674021R. Babich et. al., Phys. Rev. D76 (2007) 074021.
. C Alexandrou, P De Forcrand, B Lucini, Phys. Rev. Lett. 97222002C. Alexandrou, P. de Forcrand and B. Lucini, Phys. Rev. Lett. 97 (2006) 222002.
. M Burkardt, J M Grandy, J W Negele, Ann. Phys. 238M. Burkardt, J. M. Grandy and J. W. Negele, Ann. Phys. 238 (1995) 441-472.
. C Alexandrou, P De Forcrand, O Jahn, Nucl. Phys. Proc. Suppl. 119C. Alexandrou, P. de Forcrand and O. Jahn, Nucl. Phys. Proc. Suppl. 119 (2003) 667-669.
. T T Takahashi, H Suganuma, Y Nemoto, H Matsufuru, Phys. Rev. 65114509T. T. Takahashi, H. Suganuma, Y. Nemoto and H. Matsufuru, Phys. Rev. D65 (2002) 114509.
. J D Bratt, LHPC Collaboration1001.3620Phys. Rev. D. LHPC Collaboration, J. D. Bratt et. al., Phys. Rev. D (to be published), 1001.3620.
. C W Bernard, Phys. Rev. 6454506C. W. Bernard et. al., Phys. Rev. D64 (2001) 054506.
. G Kilcup, D Pekurovsky, L Venkataraman, Nucl. Phys. Proc. Suppl. 53G. Kilcup, D. Pekurovsky and L. Venkataraman, Nucl. Phys. Proc. Suppl. 53 (1997) 345-348.
. S Necco, R Sommer, Nucl. Phys. 622S. Necco and R. Sommer, Nucl. Phys. B622 (2002) 328-346.
. R G Edwards, SciDAC CollaborationB Joó, SciDAC CollaborationNucl. Phys. Proc. Suppl. 140832SciDAC Collaboration, R. G. Edwards and B. Joó, Nucl. Phys. Proc. Suppl. 140 (2005) 832.
| [] |
[
"Strong solutions to stochastic Volterra equations",
"Strong solutions to stochastic Volterra equations"
] | [
"Anna Karczewska [email protected] \nDepartment of Mathematics\nUniversity of Zielona Góra ul\nSzafrana 4a65-246Zielona GóraPoland\n",
"Carlos Lizama [email protected] \nDepartamento de Matemática\nFacultad de Ciencias\nUniversidad de Santiago de Chile\nCasilla 307-Correo 2SantiagoChile\n"
] | [
"Department of Mathematics\nUniversity of Zielona Góra ul\nSzafrana 4a65-246Zielona GóraPoland",
"Departamento de Matemática\nFacultad de Ciencias\nUniversidad de Santiago de Chile\nCasilla 307-Correo 2SantiagoChile"
] | [] | In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main result provide sufficient conditions for strong solutions to stochastic Volterra equations.2000 Mathematics Subject Classification: primary: 60H20; secondary: 60H05, 45D05. | 10.1016/j.jmaa.2008.09.005 | [
"https://arxiv.org/pdf/math/0512532v3.pdf"
] | 16,010,618 | math/0512532 | 10be825ac0e39a9fc312e6c5310d6ad03947d73c |
Strong solutions to stochastic Volterra equations
10 Jan 2007 September 3, 2018
Anna Karczewska [email protected]
Department of Mathematics
University of Zielona Góra ul
Szafrana 4a65-246Zielona GóraPoland
Carlos Lizama [email protected]
Departamento de Matemática
Facultad de Ciencias
Universidad de Santiago de Chile
Casilla 307-Correo 2SantiagoChile
Strong solutions to stochastic Volterra equations
10 Jan 2007 September 3, 2018
In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main result provide sufficient conditions for strong solutions to stochastic Volterra equations.2000 Mathematics Subject Classification: primary: 60H20; secondary: 60H05, 45D05.
Introduction
We deal with the following stochastic Volterra equation in a separable Hilbert space H
X(t) = X 0 + t 0 a(t − τ ) AX(τ )dτ + t 0 Ψ(τ ) dW (τ ) , t ≥ 0 ,(1)
where X 0 ∈ H, a ∈ L 1 loc (R + ) and A is a closed unbounded linear operator in H with a dense domain D(A). The domain D(A) is equipped with the graph norm | · | D(A) of A, i.e. |h| D(A) := (|h| 2 H + |Ah| 2 H ) 1/2 , where | · | H denotes the norm in H. In this work the equation (1) is driven by a cylindrical Wiener process W and Ψ is an appropriate process defined later. Let us emphasize that the results obtained for cylindrical Wiener process W are valid for classical (genuine) Wiener process, too. Equation (1) arises, in the deterministic case, in a variety of applications as model problems. Well-known techniques like localization, perturbation, and coordinate transformation allow to transfer results for such problems to parabolic integro-differential equations on smooth domains, see [16,Chapter I,Section 5]. In these applications, the operator A typically is a differential operator acting in spatial variables, like the Laplacian, the Stokes operator, or the elasticity operator. The function a should be thought of as a kernel like a(t) = e −ηt t β−1 /Γ(β); η ≥ 0, β ∈ (0, 2). Equation (1) is an abstract stochastic version of the mentioned deterministic model problems. The stochastic approach to integral equations has been recently used due to the fact that in applications the level of accuracy for a given deterministic model not always seem to be significantly changed with increasing model complexity. Instead, the stochastic approach provides better results. A typical example is the use of stochastic integral equations in rainfall-runoff models, see [9].
Our main results concerning (1), rely essentially on techniques using a strongly continuous family of operators S(t), t ≥ 0, defined on the Hilbert space H and called the resolvent (precise definition will be given below). Hence, in what follows, we assume that the deterministic version of equation (1) is well-posed, that is, admits a resolvent S(t), t ≥ 0. Our aim is to provide sufficient conditions to obtain a strong solution to the stochastic equation (1). This paper is organized as follows. In section 2 we prove the main deterministic ingredient for our construction; this is an extension of results of [2] allowing here that the operator A in (1) will be the generator of a C 0 -semigroup, not necessarily of contraction type. Section 3 contains the main definitions and concepts used in the paper. In Section 4 we compare mild and weak solutions while in the last section we provide sufficient condition for stochastic convolution to be a strong solution to the equation (1). We note that this is an improvement of the known results about existence of strong solutions for stochastic differential equations.
Convergence of resolvents
In this section we recall some definitions connected with the deterministic version of the equation (1), that is, the equation
u(t) = t 0 a(t − τ ) Au(τ )dτ + f (t), t ≥ 0,(2)
in a Banach space B. In (2), the operator A and the kernel function a are the same as previously considered in the introduction and f is a B-valued function.
Problems of this type have attracted much interest during the last decades, due to their various applications in mathematical physics like viscoelasticity, thermodynamics, or electrodynamics with memory, cf. [16].
By S(t), t ≥ 0, we denote the family of resolvent operators corresponding to the Volterra equation (2), if it exists, and defined as follows.
Definition 1 (see, e.g. [16]) A family (S(t)) t≥0 of bounded linear operators in B is called resolvent for (2) if the following conditions are satisfied:
1. S(t) is strongly continuous on R + and S(0) = I; 2. S(t) commutes with the operator A, that is, S(t)(D(A)) ⊂ D(A) and AS(t)x = S(t)Ax for all x ∈ D(A) and t ≥ 0;
3. the following resolvent equation holds
S(t)x = x + t 0 a(t − τ )AS(τ )xdτ (3)
for all x ∈ D(A), t ≥ 0.
Necessary and sufficient conditions for the existence of the resolvent family have been studied in [16]. Let us emphasize that the resolvent S(t), t ≥ 0, is determined by the operator A and the function a, so we also say that the pair (A, a) admits a resolvent family. Moreover, as a consequence of the strong continuity of S(t) we have sup t≤T ||S(t)|| < +∞ for any T ≥ 0.
We shall use the abbreviation (a ⋆ f )(t) = Kernels with this property have been introduced by Clément and Nohel [2]. We note that the class of completely positive kernels appears naturally in the theory of viscoelasticity. Several properties and examples of such kernels appears in [16,Section 4.2]. Let us note that in contrary to the case of semigroups, not every resolvent needs to be exponentially bounded even if the kernel function a belongs to L 1 (R + ). The resolvent version of the Hille-Yosida theorem (see, e.g., [16,Theorem 1.3]) provides the class of equations that admit exponentially bounded resolvents. An important class of kernels providing such class of resolvents are a(t) = t β−1 /Γ(β), α ∈ (0, 2) or the class of completely monotonic functions. For details, counterexamples and comments we refer to [5].
In this paper the following result concerning convergence of resolvents for the equation (1) in a Banach space B will play the key role. It corresponds to a generalization of the results of Clément and Nohel obtained in [2] for contraction semigroups.
Theorem 1 Let A be the generator of a C 0 -semigroup in B and suppose the kernel function a is completely positive. Then (A, a) admits an exponentially bounded resolvent S(t).
Moreover, there exist bounded operators
A n such that (A n , a) admit resolvent families S n (t) satisfying ||S n (t)|| ≤ Me w 0 t (M ≥ 1, w 0 ≥ 0) for all t ≥ 0 and S n (t)x → S(t)x as n → +∞ (5)
for all x ∈ B, t ≥ 0. Additionally, the convergence is uniform in t on every compact subset of R + .
Proof The first assertion follows directly from [15,Theorem 5] (see also [16,Theorem 4.2]). Since A generates a C 0 -semigroup T (t), t ≥ 0, the resolvent set ρ(A) of A contains the ray [w, ∞) and
||R(λ, A) k || ≤ M (λ − w) k for λ > w, k ∈ N. Define A n := nAR(n, A) = n 2 R(n, A) − nI, n > w(6)
the Yosida approximation of A.
Then
||e tAn || = e −nt ||e n 2 R(n,A)t || ≤ e −nt ∞ k=0 n 2k t k k! ||R(n, A) k || ≤ Me (−n+ n 2 n−w )t = Me nwt n−w .
Hence, for n > 2w we obtain ||e Ant || ≤ Me 2wt .
Taking into account the above estimate and the complete positivity of the kernel function a, we can follow the same steps as in [15,Theorem 5] to obtain that there exist constants M 1 > 0 and w 1 ∈ R (independent of n, due to (7)) such that
||[H n (λ)] (k) || ≤ M 1 (λ − w 1 ) k+1 for λ > w 1 , where H n (λ) := (λ − λâ(λ)A n ) −1 .
Here and in the sequel the hat indicates the Laplace transform. Hence, the generation theorem for resolvent families implies that for each n > 2ω, the pair (A n , a) admits resolvent family S n (t) such that
||S n (t)|| ≤ M 1 e w 1 t .(8)
In particular, the Laplace transformŜ n (λ) exists and satisfieŝ
S n (λ) = H n (λ) = ∞ 0 e −λt S n (t)dt, λ > w 1 .
Now recall from semigroup theory that for all µ sufficiently large we have
R(µ, A n ) = ∞ 0 e −µt e Ant dt
as well as,
R(µ, A) = ∞ 0 e −µt T (t) dt .
Sinceâ(λ) → 0 as λ → ∞, we deduce that for all λ sufficiently large, we have
H n (λ) := 1 λâ(λ) R( 1 a(λ) , A n ) = 1 λâ(λ) ∞ 0 e (−1/â(λ))t e Ant dt ,
and
H(λ) := 1 λâ(λ) R( 1 a(λ) , A) = 1 λâ(λ) ∞ 0 e (−1/â(λ))t T (t)dt .
Hence, from the identity
H n (λ) − H(λ) = 1 λâ(λ) [R( 1 a(λ) , A n ) − R( 1 a(λ) , A)]
and the fact that R(µ, A n ) → R(µ, A) as n → ∞ for all µ sufficiently large (see, e.g. [14,Lemma 7.3], we obtain that
H n (λ) → H(λ) as n → ∞ .(9)
Finally, due to (8) and (9) we can use the Trotter-Kato theorem for resolvent families of operators (cf. [13, Theorem 2.1]) and the conclusion follows.
An analogous result like Theorem 1 holds in other cases.
Theorem 2 Let A be the generator of a strongly continuous cosine family. Suppose any of the following:
(i) a ∈ L 1 loc (R + ) is completely positive;
(ii) the kernel fuction a is a creep function with a 1 log-convex;
(iii) a = c ⋆ c with some completely positive c ∈ L 1 loc (R + ).
Then (A, a) admits an exponentially bounded resolvent S(t). Moreover, there exist bounded
operators A n such that (A n , a) admit resolvent families S n (t) satisfying ||S n (t)|| ≤ Me w 0 t (M ≥ 1, w 0 ≥ 0) for all t ≥ 0, n ∈ N, and S n (t)x → S(t)x as n → +∞
for all x ∈ B, t ≥ 0. Additionally, the convergence is uniform in t on every compact subset of R + .
The proof follows from [16,Theorem 4.3], where the definition of a creep function can be found, or [15,Theorem 6] and proof of Theorem 1. Therefore it is omitted.
Remark 1
Other examples of the convergence (5) for the resolvents are given, e.g., in [2] and [7]. In the first paper, the operator A generates a linear continuous contraction semigroup. In the second of the mentioned papers, A belongs to some subclass of sectorial operators and the kernel a is an absolutely continuous function fulfilling some technical assumptions.
Proposition 1 Let A, A n and S n (t) be given as in Theorem 1. Then S n (t) commutes with the operator A, for every n sufficiently large and t ≥ 0.
Proof For each n sufficiently large the bounded operators A n admit a resolvent family S n (t), so by the complex inversion formula for the Laplace transform we have
S n (t) = 1 2πi Γn e λt H n (λ)dλ
where Γ n is a simple closed rectifiable curve surrounding the spectrum of A n in the positive sense.
On
Solutions and the stochastic convolution
Let H and U be two separable Hilbert spaces and Q ∈ L(U) be a linear bounded symmetric nonnegative operator. A Wiener process W with covariance operator Q is defined on a probability space (Ω, F , (F t ) t≥0 , P ). We assume that the process W is a cylindrical one, that is, we do not assume that TrQ < +∞. In this case, the process W has values in some superspace of U. Let us note that the results obtained in the paper for cylindrical Wiener process are valid in classical Wiener process, too. Namely, when TrQ < +∞, we can take U = H and Ψ = I. This is apparently well-known that the construction of the stochastic integral with respect to cylindrical Wiener process requires some particular terms. We will need the subspace U 0 := Q 1/2 (U) of the space U, which endowed with the inner product u, v U 0 := Q −1/2 u, Q −1/2 v U forms a Hilbert space. Among others, an important role is played by the space of Hilbert-Schmidt operators. The set L 0 2 := L 2 (U 0 , H) of all Hilbert-Schmidt operators from U 0 into H, equipped with the norm |C| L 2 (U 0 ,H) := ( +∞ k=1 |Cf k | 2 H ) 1/2 , where {f k } is an orthonormal basis of U 0 , is a separable Hilbert space.
According to the theory of stochastic integral with respect to cylindrical Wiener process we have to assume that Ψ belongs to the class of measurable L 0 2 -valued processes. Let us introduce the norms
||Ψ|| t := E t 0 |Ψ(τ )| 2 L 0 2 dτ 1 2 = E t 0 Tr(Ψ(τ )Q 1 2 )(Ψ(τ )Q 1 2 ) * dτ 1 2 , t ∈ [0, T ].
By N 2 (0, T ; L 0 2 ) we denote a Hilbert space of all L 0 2 -predictable processes Ψ such that ||Ψ|| T < +∞.
It is possible to consider a more general class of integrands, see, e.g. [12], but in our opinion it is not worthwhile to study the general case. That case produces a new level of difficulty additionally to problems related to long time memory of the system. So, we shall study the equation (1)
and for any t ∈ [0, T ] the equation (1) holds P -a.s.
Let A * denote the adjoint of A with a dense domain D(A * ) ⊂ H and the graph norm | · | D(A * ) .
Definition 5 Let (PA) hold. An H-valued predictable process X(t), t ∈ [0, T ], is said to be a weak solution to (1), if P ( t 0 |a(t−τ )X(τ )| H dτ < +∞) = 1 and if for all ξ ∈ D(A * ) and all t ∈ [0, T ] the following equation holds
X(t), ξ H = X 0 , ξ H + t 0 a(t − τ )X(τ ) dτ, A * ξ H + t 0 Ψ(τ )dW (τ ), ξ H , P −a.s.
Remark 2 This definition has sense for a cylindrical Wiener process because the scalar process t 0 Ψ(τ )dW (τ ), ξ H , for t ∈ [0, T ], is well-defined.
Definition 6
Assume that X 0 is an H-valued F 0 -measurable random variable. An Hvalued predictable process X(t), t ∈ [0, T ], is said to be a mild solution to the stochastic Volterra equation (1), if
E t 0 |S(t − τ )Ψ(τ )| 2 L 0 2 dτ < +∞ for t ≤ T(11)
and, for arbitrary t ∈ [0, T ],
X(t) = S(t)X 0 + t 0 S(t − τ )Ψ(τ ) dW (τ ), P − a.s.(12)
where S(t) is the resolvent for the equation (2), if it exists.
In some cases weak solutions of equation (1) coincides with mild solutions of (1), see e.g. [11]. In consequence, having results for the convolution on the right hand side of (12) we obtain results for weak solutions.
In the paper we will use the following well-known result. In what follows we assume that (2) admits a resolvent family S(t), t ≥ 0. We introduce the stochastic convolution
W Ψ (t) := t 0 S(t − τ )Ψ(τ ) dW (τ ),(13)
where Ψ belongs to the space N 2 (0, T ; L 0 2 ). Note that, because resolvent operators S(t), t ≥ 0, are bounded, then S(t − ·)Ψ(·) ∈ N 2 (0, T ; L 0 2 ), too. Let us formulate some auxiliary results concerning the convolution W Ψ (t).
Proposition 3 Assume that (2) admits resolvent operators S(t), t ≥ 0. Then, for arbitrary process Ψ ∈ N 2 (0, T ; L 0 2 ), the process W Ψ (t), t ≥ 0, given by (13) has a predictable version.
Proposition 4 Assume that Ψ ∈ N 2 (0, T ; L 0 2 ). Then the process W Ψ (t), t ≥ 0, defined by (13) has square integrable trajectories.
For the proofs of Propositions 3 and 4 we refer to [11].
Proposition 5 Let a ∈ BV (R + ) and suppose that (2) admits a resolvent family S ∈ C 1 (0, ∞; L(H)). Let X be a predictable process with integrable trajectories. Assume that X has a version such that P (X(t) ∈ D(A)) = 1 for almost all t ∈ [0, T ] and (11) holds. If for any t ∈ [0, T ] and ξ ∈ D(A * ) (14) then
X(t), ξ H = X 0 , ξ H + t 0 a(t−τ )X(τ ), A * ξ H dτ + t 0 ξ, Ψ(τ )dW (τ ) H , P −a.s.,X(t) = S(t)X 0 + t 0 S(t − τ )Ψ(τ )dW (τ ), t ∈ [0, T ].(15)
Proof For simplicity we omit the index H in the inner product. Since a ∈ BV (R + ), we can see that (14) implies
X(t), ξ(t) = X 0 , ξ(0) + t 0 (ȧ ⋆ X)(τ ) + a(0)X(τ ), A * ξ(τ ) dτ + t 0 Ψ(τ )dW (τ ), ξ(τ ) + t 0 X(τ ),ξ(τ ) dτ, P − a.s.(16)
for any ξ ∈ C 1 ([0, t], D(A * )) and t ∈ [0, T ]. (For details, see [11]). Now, let us take ξ(τ ) := S * (t − τ )ζ with ζ ∈ D(A * ), τ ∈ [0, t]. The equation (16) may be written like
X(t), S * (0)ζ = X 0 , S * (t)ζ + t 0 (ȧ ⋆ X)(τ ) + a(0)X(τ ), A * S * (t − τ )ζ dτ + t 0 Ψ(τ )dW (τ ), S * (t − τ )ζ + t 0 X(τ ), (S * (t − τ )ζ) ′ dτ,
where the derivative ()' in the last term is taken over τ .
Next, using S * (0) = I, we rewrite
X(t), ζ = S(t)X 0 , ζ + t 0 S(t − τ )A τ 0ȧ (τ − σ)X(σ)dσ + a(0)X(τ ) , ζ dτ + t 0 S(t − τ )Ψ(τ )dW (τ ), ζ + t 0 Ṡ (t − τ )X(τ ), ζ dτ.(17)
To prove (15) it is enough to show that the sum of the first integral and the third one in the equation (17) gives zero.
Since S ∈ C 1 (0, ∞; L(H)), we can use properties of resolvent operators and the deriva-tiveṠ(t − τ ) with respect to τ . Then
I := t 0Ṡ (t − τ )X(τ )dτ, ζ = − t 0Ṡ (τ )X(t − τ )dτ, ζ = − t 0 τ 0ȧ (τ − s)AS(s)ds X(t − τ )dτ − t 0 a(0)AS(τ )X(t − τ )dτ , ζ = −([A(ȧ ⋆ S)(τ ) ⋆ X](t) + a(0)A(S ⋆ X)(t)), ζ .
Note that a ∈ BV (R + ) and hence the convolution (a ⋆ S)(τ ) has sense (see [16, Section 1.6]). Remark 3 If (1) is parabolic and the kernel a is 3-monotone, understood in the sense defined by Prüss [16, Section 3], then S ∈ C 1 (0, ∞; L(H)) and a ∈ BV (R + ) respectively.
S(t − τ )A τ 0ȧ (τ − σ)X(σ)dσ , ζ dτ = t 0 AS(t − τ )(ȧ ⋆ X)(τ ), ζ dτ = = A(S ⋆ (ȧ ⋆ X)(τ ))(t), ζ = A((S ⋆ȧ)(τ ) ⋆ X)(t),
Proposition 6
Assume that A is a closed linear unbounded operator with the dense domain D(A), a ∈ L 1 loc (R + ) and S(t), t ≥ 0, are resolvent operators for the equation (2). If Ψ ∈ N 2 (0, T ; L 0 2 ), then the stochastic convolution W Ψ fulfills the equation (14) with X 0 ≡ 0.
Proof Let us notice that the process W Ψ has integrable trajectories. For any ξ ∈ D(A * ) we have from (13)
t 0 a(t − τ )W Ψ (τ ), A * ξ H dτ ≡ t 0 a(t − τ ) τ 0 S(τ − σ)Ψ(σ)dW (σ), A * ξ H dτ.
Hence from Dirichlet's formula and the stochastic Fubini's theorem we get
t 0 a(t − τ )W Ψ (τ ), A * ξ H dτ = t 0 t σ a(t − τ )S(τ − σ)dτ Ψ(σ)dW (σ), A * ξ H = t 0 t−σ 0 a(t − σ − z)S(z)dz Ψ(σ)dW (σ), A * ξ H .
Taking z := τ − σ and from definition of convolution we have
t 0 a(t − τ )W Ψ (τ ), A * ξ H dτ = t 0 A[(a ⋆ S)(t − σ)]Ψ(σ)dW (σ), ξ H
From the resolvent equation (3) and because
A(a ⋆ S)(t − σ)x = (S(t − σ) − I)x, where x ∈ D(A), we obtain t 0 a(t − τ )W Ψ (τ ), A * ξ H dτ = t 0 [S(t − σ) − I]Ψ(σ)dW (σ), ξ H = = t 0 S(t − σ)Ψ(σ)dW (σ), ξ H − t 0 Ψ(σ)dW (σ), ξ H .
Hence, we obtained the following equation
W Ψ (t), ξ H = t 0 a(t − τ )W Ψ (τ ), A * ξ H dτ + t 0 ξ, Ψ(τ )dW (τ ) H for any ξ ∈ D(A * ).
Corollary 1 Assume that A is a linear bounded operator in H, a ∈ L 1 loc (R + ) and S(t), t ≥ 0, are resolvent operators for the equation (2). If Ψ belongs to N 2 (0, T ; L 0 2 ) then
W Ψ (t) = t 0 a(t − τ )AW Ψ (τ )dτ + t 0 Ψ(τ )dW (τ ) .(18)
Remark 4 The formula (18) says that the convolution W Ψ is a strong solution to (1) with X 0 ≡ 0 if the operator A is bounded.
Strong solution
In this section we provide sufficient conditions under which the stochastic convolution W Ψ (t), t ≥ 0, defined by (13) is a strong solution to the equation (1).
Lemma 1 Let A be a closed linear unbounded operator the with dense domain D(A) equipped with the graph norm |·| D(A) . Suppose that assumptions of Theorem 1 or Theorem 2 hold. If Ψ and AΨ belong to N 2 (0, T ; L 0 2 ) and in addition Ψ(·, ·)(U 0 ) ⊂ D(A), P-a.s., then (18) holds.
Proof Because formula (18) holds for any bounded operator, then it holds for the Yosida approximation A n of the operator A, too, that is
W Ψ n (t) = t 0 a(t − τ )A n W Ψ n (τ )dτ + t 0 Ψ(τ )dW (τ ), where W Ψ n (t) := t 0 S n (t − τ )Ψ(τ )dW (τ ) and A n W Ψ n (t) = A n t 0 S n (t − τ )Ψ(τ )dW (τ ).
Recall that by assumption Ψ ∈ N 2 (0, T ; L 0 2 ). Because the operators S n (t) are deterministic and bounded for any t ∈ [0, T ], n ∈ N, then the operators S n (t − ·)Ψ(·) belong to N 2 (0, T ; L 0 2 ), too. In consequence, the difference Φ n (t − ·) := S n (t − ·)Ψ(·) − S(t − ·)Ψ(·)
belongs to N 2 (0, T ; L 0 2 ) for any t ∈ [0, T ] and n ∈ N. This means that
E t 0 |Φ n (t − τ )| 2 L 0 2 dτ < +∞(20)
for any t ∈ [0, T ].
Let us recall that the cylindrical Wiener process W (t), t ≥ 0, can be written in the form
W (t) = +∞ j=1 f j β j (t),(21)
where {f j } is an orthonormal basis of U 0 and β j (t) are independent real Wiener processes.
From (21) we have t 0 Φ n (t − τ ) dW (τ ) = +∞ j=1 t 0 Φ n (t − τ ) f j dβ j (τ ).(22)
Then, from (20)
E t 0 +∞ j=1 |Φ n (t − τ ) f j | 2 H dτ < +∞(23)
for any t ∈ [0, T ]. Next, from (22), properties of stochastic integral and (23) we obtain for any t ∈ [0, T ],
E t 0 Φ n (t − τ ) dW (τ ) 2 H = E +∞ j=1 t 0 Φ n (t − τ ) f j dβ j (τ ) 2 H ≤ E +∞ j=1 t 0 |Φ n (t − τ ) f j | 2 H dτ ≤ E +∞ j=1 T 0 |Φ n (T − τ ) f j | 2 H dτ < +∞.
By Theorem 1 or 2, the convergence (5) of resolvent families is uniform in t on every compact subset of R + , particularly on the interval [0, T ]. Now, we use (5) in the Hilbert space H, so (5) holds for every x ∈ H. Then, for any fixed j,
T 0 |[S n (T − τ ) − S(T − τ )] Ψ(τ ) f j | 2 H dτ(24)
tends to zero for n → +∞. Summing up our considerations, particularly using (23)
E t 0 Φ n (t − τ )dW (τ ) 2 H ≡ sup t∈[0,T ] E t 0 [S n (t − τ ) − S(t − τ )]Ψ(τ )dW (τ ) 2 H ≤ ≤ E +∞ j=1 T 0 |[S n (T − τ ) − S(T − τ )]Ψ(τ ) f j | 2 H dτ → 0
as n → +∞. Then |A n W Ψ n (t) − AW Ψ (t)| 2 H ≤ N 2 n,1 (t) + 2N n,1 (t)N n,2 (t) + N 2 n,2 (t) < 3[N 2 n,1 (t) + N 2 n,2 (t)].
Let us study the term N n,1 (t). Note that the unbounded operator A generates a semigroup. Then we have for the Yosida approximation the following properties:
A n x = J n Ax for any x ∈ D(A), sup n ||J n || < ∞
where A n x = nAR(n, A)x = AJ n x for any x ∈ H, with J n := nR(n, A). Moreover (see [ For the second term of (26), that is N 2 n,2 (t), we can follow the same steps as above for proving (25).
t − s)f (s)ds, t ∈ [0, T ], for the convolution of two functions. Definition 2 We say that function a ∈ L 1 (0, T ) is completely positive on [0, T ] if for any µ ≥ 0, the solutions of the convolution equations s(t) + µ(a ⋆ s)(t) = 1 and r(t) + µ(a ⋆ r)(t) = a(t) (4) satisfy s(t) ≥ 0 and r(t) ≥ 0 on [0, T ].
Definition 3
3Suppose S(t), t ≥ 0, is a resolvent for (2). S(t) is called exponentially bounded if there are constants M ≥ 1 and ω ∈ R such that ||S(t)|| ≤ M e ωt , for all t ≥ 0. (M, ω) is called a type of S(t).
the other hand, H n (λ) := (λ − λâ(λ)A n ) where A n := nA(n − A) −1 , so each A n commutes with A on D(A) and then each H n (λ) commutes with A, on D(A), too. Finally, because A is closed, and all the following integrals are convergent (exist) we have for all n sufficiently large and x ∈ D(A) AS n (t)x = A Γn e λt H n (λ)xdλ = Γn e λt AH n (λ)xdλ = Γn e λt H n (λ)Axdλ = S n (t)Ax .
under the below Probability Assumptions (abbr. (PA)):1. X 0 is an H-valued, F 0 -measurable random variable; 2. Ψ ∈ N 2 (0, T ; L 0 2 ) and the interval [0, T ] is fixed.
Definition 4
4Assume that (PA) hold. An H-valued predictable process X(t), t ∈ [0, T ], is said to be a strong solution to (1), if X has a version such that P (X(t) ∈ D(A)) = 1 for almost all t ∈ [0, T ]; for any t ∈ [0, T ] t 0 |a(t − τ )AX(τ )| H dτ < +∞, P − a.s.
Proposition 2
2(see, e.g. [4, Proposition 4.15]) Assume that A is a closed linear unbounded operator with the dense domain D(A) ⊂ H and Φ(t), t ∈ [0, T ] is an L 2 (U 0 , H)-predictable process. If Φ(t)(U 0 ) ⊂ D(A), P − a.s. for all t ∈ [0, T ] s) dW (s) ∈ D(A) = 1 and A T 0 Φ(s) dW (s) = T 0 AΦ(s) dW (s), P − a.s.
ζ for any ζ ∈ D(A * ), so J = −I, hence J + I = 0. This means that (15) holds for any ζ ∈ D(A * ). Since D(A * ) is dense in H * , then (15) holds.
, Ψ(·, ·)(U 0 ) ⊂ D(A), P − a.s. Because S(t)(D(A)) ⊂ D(A), thenS(t − τ )Ψ(τ )(U 0 ) ⊂ D(A), P − a.s., for any τ ∈ [0, t], t ≥ 0. Hence, by Proposition 2, P (W Ψ (t) ∈ D(A)) = 1. For any n ∈ N, t ≥ 0, we have |A n W Ψ n (t) − AW Ψ (t)| H ≤ N n,1 (t) + N n,2 (t), where N n,1 (t) := |A n W Ψ n (t) − A n W Ψ (t)| H ,N n,2 (t) := |A n W Ψ (t) − AW Ψ (t)| H = |(A n − A)W Ψ (t)| H .
,[[
AS n (t)x = S n (t)Ax for all x ∈ D(A). So, by Propositions 1 and 2 and the closedness of A we can writeA n W Ψ n (t) S n (t − τ ) − S(t − τ )]AΨ(τ )dW (S n (t − τ ) − S(t − τ )]AΨ(τ )dW (τ )| H .From assumptions, AΨ ∈ N 2 (0, T ; L 0 2 ). Then the term [S n (t − τ ) − S(t − τ )]AΨ(τ ) may be treated like the difference Φ n defined by (19).Hence, from(27)and(25), for the first term of the right hand side of (26)
N n,2 (t) = |A n W Ψ (t) − AW Ψ (t)| H ≡ A n t 0 S(t − τ )Ψ(τ )dW (τ ) − A t 0 S(t − τ )Ψ(τ )dW (τ )
Proof In order to prove Theorem 3, we have to show only the condition(10). Let us note that the convolution W Ψ (t) has integrable trajectories. Because the closed unbounded linear operator A becomes bounded on (D(A), |·| D(A) ), see[17,Chapter 5], we obtain that AW Ψ (·) ∈ L 1 ([0, T ]; H), P-a.s. Hence, properties of convolution provide integrability of the function a(T − τ )AW Ψ (τ ) with respect to τ , what finishes the proof.By the convergence (28), for any fixed j, The main result of this section is the following.Theorem 3 Suppose that assumptions of Theorem 1 or Theorem 2 hold. Then the equation (1) has a strong solution. Precisely, the convolution W Ψ defined by(13)is the strong solution to (1) with X 0 ≡ 0.
Analogously, AS(t − ·)Ψ(·) = S(t − ·)AΨ(·) ∈ space. Particularly, sum of two. see e.g. [1, too. Analogously, AS(t − ·)Ψ(·) = S(t − ·)AΨ(·) ∈ space. Particularly, sum of two Hilbert-Schmidt operators is a Hilbert- Schmidt operator, see e.g. [1].
. Therefore, we can deduce that the operator (A n − A) S(t − ·)Ψ(·) ∈ N 2 (0, T ; L 0 2 ), for any t ∈ [0, TTherefore, we can deduce that the operator (A n − A) S(t − ·)Ψ(·) ∈ N 2 (0, T ; L 0 2 ), for any t ∈ [0, T ].
S(t − τ )Ψ(τ ) may be treated like the difference Φ n defined by (19). So, we obtain References. Hence, A Balakrishnan, Applied Functional Analysis. SpringerA n − AHence, the term [A n − A]S(t − τ )Ψ(τ ) may be treated like the difference Φ n defined by (19). So, we obtain References [1] Balakrishnan A., Applied Functional Analysis, Springer, New York, 1981.
Abstract linear and nonlinear Volterra equations preserving positivity. Clément Ph, J A Nohel, SIAM J. Math. Anal. 10Clément Ph., Nohel J.A., Abstract linear and nonlinear Volterra equations preserving pos- itivity, SIAM J. Math. Anal. Vol. 10, (1979), 365-388.
Infinite Dimensional Linear Systems Theory. R F Curtain, A J Pritchard, Lecture Notes in Control and Information Sciences. 8SpringerCurtain R.F., Pritchard A.J., Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences 8, Springer, New York, 1978.
G Da Prato, J Zabczyk, Stochastic equations in infinite dimensions. CambridgeCambridge University PressDa Prato G., Zabczyk J., Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992.
Counterexamples for abstract linear Volterra equations. W Desch, J Prüss, J. Integral Eqns. Appl. 5Desch W., Prüss J., Counterexamples for abstract linear Volterra equations, J. Integral Eqns. Appl. 5 (1993), 29-45.
One-parameter semigroups for linear evolution equations. K J Engel, R Nagel, Graduate texts in Mathematics. 194SpringerEngel K.J., Nagel R., One-parameter semigroups for linear evolution equations, Graduate texts in Mathematics 194, Springer, New York, 2000.
Monotonicity of solutions of Volterra integral equations in Banach space. A Friedman, Trans. Amer. Math. Soc. 198Friedman A., Monotonicity of solutions of Volterra integral equations in Banach space, Trans. Amer. Math. Soc. 198 (1969), 129-148.
Stochastic evolution equations. W Grecksch, C Tudor, Akademie VerlagBerlinGrecksch W., Tudor C., Stochastic evolution equations, Akademie Verlag, Berlin, 1995.
Stochastic Integral Equations in Rainfall-Runoff Modeling. I I Hromadka, T V Whitley, R J , Springer-VerlagHromadka II, T.V., Whitley, R.J., Stochastic Integral Equations in Rainfall-Runoff Model- ing, Springer-Verlag, 1989.
Stability of semilinear stochastic evolution equations. A Ichikawa, J. Math. Anal. Appl. 90Ichikawa A., Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl. 90, (1982), 12-44.
A Karczewska, Properties of convolutions arising in stochastic Volterra equations. Karczewska A., Properties of convolutions arising in stochastic Volterra equations, preprint: http://xxx.lanl.gov/ps/math.PR/0509012
R S Liptser, A N Shiryayev, Statistics of random processes. New YorkSpringerLiptser R.S., Shiryayev A.N., Statistics of random processes, Springer, New York, 1973.
On an extension of the Trotter-Kato theorem for resolvent families of operators. C Lizama, J. Integral Eqns. Appl. 22Lizama C., On an extension of the Trotter-Kato theorem for resolvent families of operators, J. Integral Eqns. Appl. 2 (2) (1990), 269-280.
A Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. New YorkSpringerPazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
Positivity and regularity of hiperbolic Volterra equations in Banach spaces. J Prüss, Math. Ann. 279Prüss J., Positivity and regularity of hiperbolic Volterra equations in Banach spaces, Math. Ann. 279 (1987), 317-344.
Evolutionary integral equations and applications. J Prüss, Birkhäuser, BaselPrüss J., Evolutionary integral equations and applications, Birkhäuser, Basel, 1993.
Linear operators in Hilbert spaces. J Weidmann, SpringerNew YorkWeidmann J., Linear operators in Hilbert spaces, Springer, New York, 1980.
| [] |
[
"Hartree-Fock Gamow basis from realistic nuclear forces",
"Hartree-Fock Gamow basis from realistic nuclear forces"
] | [
"Q Wu \nSchool of Physics, and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n",
"F R Xu [email protected] \nSchool of Physics, and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n"
] | [
"School of Physics, and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina",
"School of Physics, and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina"
] | [] | We present a simplified method to generate the Hartree-Fock Gamow basis from realistic nuclear forces. The Hartree-Fock iteration in the harmonic-oscillator basis is first performed, and then the obtained HF potential is analytically continued to the complex-k plane, finally by solving the Schrödinger equation in the complex-k plane the Gamow basis is obtained. As examples, the method is applied to 4 He and 22 O with the renormalized chiral N 3 LO potential. The basis obtained which includes bound, resonant and scattering states can be further used in many-body calculations to study weakly bound nuclei. * | 10.1360/n972018-00406 | [
"https://arxiv.org/pdf/1806.01014v1.pdf"
] | 103,647,270 | 1806.01014 | 6053f40ce4b243b3c60deb0d0914c835203c8e72 |
Hartree-Fock Gamow basis from realistic nuclear forces
4 Jun 2018
Q Wu
School of Physics, and State Key Laboratory of Nuclear Physics and Technology
Peking University
100871BeijingChina
F R Xu [email protected]
School of Physics, and State Key Laboratory of Nuclear Physics and Technology
Peking University
100871BeijingChina
Hartree-Fock Gamow basis from realistic nuclear forces
4 Jun 2018
We present a simplified method to generate the Hartree-Fock Gamow basis from realistic nuclear forces. The Hartree-Fock iteration in the harmonic-oscillator basis is first performed, and then the obtained HF potential is analytically continued to the complex-k plane, finally by solving the Schrödinger equation in the complex-k plane the Gamow basis is obtained. As examples, the method is applied to 4 He and 22 O with the renormalized chiral N 3 LO potential. The basis obtained which includes bound, resonant and scattering states can be further used in many-body calculations to study weakly bound nuclei. *
I. INTRODUCTION
The description of nuclei far from the valley of stability is a challenge in nuclear physics.
These nuclei exhibit unusual properties such as halo densities and weak bindings. The coupling to the continuum is needed for the theoretical description of these exotic nuclei.
The Gamow shell model [1][2][3][4][5][6][7][8][9][10] which extends the traditional shell model to the complex plane, can efficiently include the continuum coupling. This model unifies nuclear structure and reaction properties, and has proven to be a promising tool for the descriptions of weakly bound and unbound nuclei. The starting point of the Gamow shell model is the Berggren completeness where bound, resonant (or Gamow) and scattering states are treated on an equal footing [11][12][13][14][15]. Employing the Berggren basis, first principle calculations can also be extended to study weak-binding nuclei and nuclear reactions.
A big concern is how to obtain the single-particle Berggren basis which is used to construct the Slater determinants in many-body calculations. The single-particle basis in Gamow shellmodel calculations are normally constructed from a Woods-Saxon potential with parameters fitted to experimental single-particle energies [1,10]. However, in a fully microscopic approach, the single-particle basis constructed from the nucleon-nucleon interaction is preferred. In many ab initio calculations, the Hartree-Fock (HF) iteration is usually performed as a first approximation [16][17][18]. When extending the ab initio calculations to handle resonant states employing the Berggren ensemble, it's very useful to have the single-particle Berggren basis generated from the HF potential.
There have been several studies about how to obtain the Berggren basis. In Refs. [19,20], mature numerical methods have been developed to solve the complex eigenvalue problem of the Schrödinger equation with a local potential. For a nonlocal HF potential, the numerical procedure known as the Hartree-Fock Gamow (GHF) method is adopted in Refs. [1,21,22] to generate a basis that includes resonant and nonresonant states. In all these calculations, the basis is obtained in the coordinate space by taking proper boundary conditions. In Ref. [23], however, the derivation of the Gamow HF basis in momentum space is carried out. In momentum space, there is no need to worry about the boundary conditions.
In this paper, we present an alternative method to generate the Gamow HF basis from realistic nucleon-nucleon interactions. As in Ref. [23], the basis in our method is finally obtained with the contour deformation method in the complex momentum space [9], while we adopt a simplified way to get the HF potential in the complex k-plane. As examples, we apply this method to 4 He and 22 O with the chiral two-body N 3 LO potential [24] softened by V low-k [25].
II. METHOD
The HF approximation is a common many-body method where the wave function of the system is described by a single Slater determinant. This method is also called self-consistent mean field approximation since the solution to the HF equation behaves as if each particle of the system is subjected to the mean field created by all other particles. The mean field which is called HF potential is usually obtained by solving the nonlinear HF equation by iteration.
The HF calculations for nuclei starting from realistic nuclear forces have been successful [17]. The A-body intrinsic nuclear Hamiltonian H is
H =h 2 2mA A i<j (k i − k j ) 2 + A i<j V ij = 1 − 1 A A i=1h 2 k 2 i 2m + A i<j V ij −h 2 mA k i · k j ,(1)
where m is the mass of the nucleon, V ij is the two-body nucleon-nucleon force. With the Hamiltonian (1), the symmetry-preserved HF iteration is first performed in the harmonicoscillator (HO) basis [17]. For open-shell nuclei, the average filling can be adopted as in Ref. [1]. We only deal with the situation where the single-particle states below the Fermi surface are bound, which is true in most cases. When the iteration converges, the HF potential in HO basis denoted as n 1 l|U (l,j,tz) HF |n 2 l is established, where (l, j, t z ) is used to label the channel with orbital angular momentum l, total angular momentum j and isospin t z . |nl is the HO state. The HF potential is generally nonlocal in r-space, thus the use of the shooting method [20] is not straightforward.
It is preferred to solve the one-body Schrödinger equation in momentum space. The HF equation in momentum space can be written formally as
h 2 2µ k 2 ψ nljtz (k) + dk ′ k ′2 k|U (l,j,tz) HF |k ′ ψ nljtz (k ′ ) = E nljtz ψ nljtz (k),(2)
where we have introduced the effective mass µ = (1 − 1/A) −1 m. ψ nljtz is the HF single- [2]. To obtain the resonant states, Eq. (2) needs to be generalized to the complex k-plane. This can be done by replacing the integral path in Eq. (2) from the real axis to a contour in the complex k-plane [9]. Then the Schrödinger equation reads
particle state in (l, j, t z )h 2 2µ k 2 ψ nljtz (k) + L + dk ′ k ′2 k|U (l,j,tz) HF |k ′ ψ nljtz (k ′ ) = E n ψ nljtz (k),(3)
where L + in a contour in the lower half complex k-plane.
The key problem now is how to obtain the HF potential U HF in the complex k-plane.
In Ref. [23], the double Fourier-Bessel transformation is adopted to achieve the analytical continuation from the real k axis to the complex k-plane. Here, we employ a simplified approach where the direct basis transformation is used.
where k|nl is the momentum-space radial wave function of the HO state. k|nl reads
k|nl = (−i) 2n+l e −1/2b 2 k 2 (bk) l 2n!b 3 Γ(n + l + 3/2) L l+1/2 n (b 2 k 2 ),(5)
where L The momentum-space Schrödinger equation (3) is solved by discretizing the integral interval by the Gauss-Legendre quadrature. The discretized equation reads
h 2 2µ k 2 α ψ nljtz (k α ) + β ω β k 2 β k α |U (l,j,tz) HF |k β ψ nljtz (k β ) = E n ψ nljtz (k α ),(6)
where k β are the integration points and ω β are the corresponding quadrature weights. By
introducingψ nljtz (k α ) = ψ nljtz (k α )k α √ ω α , Eq. (6) becomes a matrix eigenvalue problem β h αβψnljtz (k β ) = E nψnljtz (k α ),(7)
where the matrix elements are
h αβ =h 2 2µ k 2 α δ αβ + √ ω α ω β k α k β k α |U (l,j,tz) HF |k β .(8)
The Gamow HF basis which includes bound, resonant and scattering states can be obtained by diagonalizing the complex symmetric matrix (8). The obtained basis can be written in coordinate space as
ψ nljtz (r) = ∞ 0 ψ nljtz (k)j l (kr)k 2 dk = α k α √ ω α j l (k α r)ψ nljtz (k α ).(9)
III. CALCULATIONS AND DISCUSSIONS
To test our method, the single-particle Gamow HF states of the closed-shell 4 He and 22 O are calculated with the chiral N 3 LO inteaction [24]. The interaction is renormalized by
0 → 0.4−0.24i → 0.8 → 4, (b)0 → 0.44−0.33i → 0.8 → 4, and (c) 0 → 0.48−0.4i → 0.8 → 4
(all in fm −1 ). 20 points are taken for the Gauss-Legendre quadrature in each segments of the contour.
In Fig. 3, we present the obtained neutron single-particle energies in d 3/2 partial wave
B. Convergence
We check the numerical convergence of the resonant state with respect to the number of integration points. In Table I
C. Phase shifts from scattering states
If we employ the contour on the real axis, we can only obtain scattering sates with real energies. However, the information of the resonance states can still be extracted from the scattering states. The asymptotic behaviour of the scattering state ψ nl with energy E n is
ψ nl (r) ∼ 2 π (j l (kr) cos δ − y l (kr) sin δ), r → ∞,(10)
whereh 2 k 2 2µ = E n , δ is the phase shift, j l , y l is the first and the second kind spherical bessel functions. By matching the obtained wave function of the scattering states and the asymptotic behaviour Eq. (10), the phase shift can be obtained as follows:
tan δ = kj ′ l (kR)ψ nl (R) − j l (kR)ψ ′ nl (R) ky ′ l (kR)ψ nl (R) − y l (kR)ψ ′ nl (R) ,(11)
where R is the match point which can take an arbitrary large enough value outside the range of the potential. We take the neutron d 3/2 channel of 22 O for example. By taking the contour on the real axis, we obtain many scattering states with discretized energies. In Figure 5 shows the relation of the phase shift against the energy. The rapid change of the phase shift across π/2 indicates the existence of a resonance state there. The energy E r at which gives a phase shift of π/2 is the center energy of the resonance. While the width of the resonance Γ is given by
Γ 2 = − 1 d cot δ/dE E=Er = 1 dδ/dE E=Er .(12)
In Fig. 5, to illustrate the position and the width of the resonance state, we also show the norm square of the scattering amplitude, which is linear related to the scattering cross section. The scattering amplitude reads
f = 1 k cot δ − ik .(13)
The resonance extracted from the phase shift analysis is 0.87−0.04i MeV, which is consistent with the calculation in previous subsection.
IV. SUMMARY
We present a new simplified method to generate the Hartree-Fock Gamow basis from the realistic nuclear force. We first perform the HF iteration in HO basis, and then the HF potential obtained is analytically continued to the complex k-plane. The continuation is accomplished directly by a basis transformation in which the complex Laguerre polynomials are used. By discretizing the integral of the Schrödinger equation in momentum space, the equation becomes a complex symmetric eigenvalue problem. The Hartree-Fock Gamow basis can be obtained by diagonalizing the complex symmetric matrix. The method is tested for
channel. There are several advantages solving the equation in momentum space. First, the boundary conditions are automatically built into the integral equation. Secondly, the momentum representaion of the Gamow states are non-oscillating and rapidly decreasing, as opposed to the coordinate representation. The numerical procedures are often easier to implement. The Gamow (or resonant) states are generalized eigenstates of the Schrödinger equation with complex energy eigenvalues E = E 0 − iΓ/2, where Γ stands for the decay width. These states correspond to the poles of the S matrix in the complex energy plane lying below the positive real axis
|k ′ = k|n 1 l n 1 l|U (l,j,tz) HF |n 2 l n 2 l|k ′ ,
generalized Laguerre polynomials, b is the HO length and is related to the HO frequency ω by b = h/mω. Since the analytical continuation of the generalized Laguerre polynomial is straightforward, the continuation of U HF to the complex k-plane can be achieved by Eq. (4).
VFIG. 1 .FIG. 2 .FIG. 3 .
123low-k method[25] with a cutoff parameter Λ = 2.1 fm −1 . In all the calculations, we take the frequency parameterhω = 22 MeV for the underlying HO basis and truncate the basis with N max = 2n + l = 10.A. Resonant statesFor 4 He, only the lowest neutron and proton s 1/2 single-particle states in our HF calculation are bound. Resonant states may emerge in the p 3/2 partial wave. To obtain the possible resonant states, we employ the contour in the complex k-plane as shown inFig. 1. The resonant state can be identified by changing the contour, since the position of the resonant state is stable with respect to changes of the contour. InFig. 2, we present the calculated neutron single-particle energies in the p 3/2 partial wave for 4 He with different contours. A resonant state with an energy E = 1.412 − 1.046i (in MeV) is clearly found. In the calculations, the adopted three contours in the complex-k plane are: Contour L + in the complex k-plane used in our calculations of resonant states. The contour is made up of three segments defined by three points A, B and C. Obtained neutron single-particle energies (in MeV) of the Gamow HF basis in the p 3/2 partial wave for 4 He with different contours. The arrow indicates the resonant sate, whose position is stable with respect to changes of the contour. The three contours used are detailed in the text. Same as inFig. 2but for 22 O in the d 3/2 partial wave.
for 22 O with three different contours in complex-k plane. The three contours are: (a) 0 → 0.2−0.01i → 0.4 → 4, (b)0 → 0.2−0.02i → 0.4 → 4, and (c) 0 → 0.2−0.03i → 0.4 → 4 (all in fm −1 ), see Fig. 1. The energy of the obtained resonant state indicated by the arrow in Fig. 3 is E = 0.867 − 0.042i (in MeV). In the calculations, we take 20 discretization points in each segments of the contour.
, we display the obtained energies of the d 3/2 resonant state in the 22 O calculations with different numbers of discretization points. The contour L + employed in the calculations is 0 → 0.2 − 0.02i → 0.4 → 4. We can see the discretization of L + with 20 points in each segment yields a precision of the energy calculation better than 0.1 KeV for the resonant state.
Fig. 4 ,
4we show the wave function of the calculated state with an energy E = 50.7 MeV as well as the asymptotic wave function. The match point R = 10 fm is taken and the phase shift that makes the two wave functions match is 1.5 rad. We can see the calculated wave function indeed behaves in coincidence with the asymptotic wave function at large distance. Scattering states with different energies have different phase shifts. By analyzing the energy dependence of the phase shift, we can obtain the position of the resonance. On the other hand, with a complex contour, the resonance state in neutron d 3/2 channel of 22 O has already been found at E = 0.867 − 0.042i in the previous subsection. We can check whether the two calculations are consistent.
FIG. 4 .FIG. 5 .
45The wave function of the calculated 22 O HF single-particle scattering state at E = 50.7MeV as well as the asymptotic wave function in d 3/2 neutron channel. The phase shift as a function of the scattering energy. The inset graph shows the norm square of the scattering amplitude as a function of the energy.
TABLE I .
IThe obtained energies (in MeV) of the d 3/2 resonant state in the 22 O calculations with different numbers of discretization points. The contour used is detailed in the text. N OA , N AB and N BC denote the number of points in the segments OA, AB and BC, respectively, see Fig. 1.N OA
N AB
N BC
Re(E)
Im(E)
5
5
20
0.8708
-0.0446
10
10
20
0.8666
-0.0421
20
20
20
0.8666
-0.0418
25
25
20
0.8666
-0.0418
30
30
30
0.8666
-0.0418
He and 22 O with the renormalized N 3 LO interactions. The basis obtained can be further used for studies of weakly bound nuclei.
. N Michel, W Nazarewicz, M Ploszajczak, Phys. Rev. C. 7064313N. Michel, W. Nazarewicz, and M. Ploszajczak, Phys. Rev. C 70, 064313 (2004).
. N Michel, W Nazarewicz, M Ploszajczak, T Vertse, J. Phys. G. 36113101N. Michel, W. Nazarewicz, M. Ploszajczak, and T. Vertse, J. Phys. G 36, No. 1, 013101 (2009).
. N Michel, W Nazarewicz, M Ploszajczak, K Bennaceur, Phys. Rev. Lett. 8942502N. Michel,W. Nazarewicz, M. Ploszajczak, and K. Bennaceur, Phys. Rev. Lett. 89, 042502 (2002).
. N Michel, W Nazarewicz, M Ploszajczak, J Okolowicz, Phys. Rev. C. 6754311N. Michel, W. Nazarewicz, M. Ploszajczak, and J. Okolowicz, Phys. Rev. C 67, 054311 (2003).
. Z H Sun, Q Wu, Z H Zhao, B S Hu, S J Dai, F R Xu, Phys. Lett. B. 769227Z. H. Sun, Q. Wu, Z. H. Zhao, B. S. Hu, S. J. Dai, and F. R. Xu, Phys. Lett. B 769, 227 (2017).
. R Idbetan, R J Liotta, N Sandulescu, T Vertse, Phys. Rev. Lett. 8942501R. IdBetan,R. J. Liotta,N. Sandulescu, and T.Vertse, Phys. Rev. Lett. 89, 042501 (2002).
. R Idbetan, R J Liotta, N Sandulescu, T Vertse, Phys. Rev. C. 6714322R. IdBetan,R. J. Liotta,N. Sandulescu, and T.Vertse, Phys. Rev. C 67, 014322 (2003).
. R Idbetan, R J Liotta, N Sandulescu, T Vertse, Phys. Lett. 58448R. IdBetan, R. J. Liotta, N. Sandulescu, and T. Vertse, Phys. Lett. B584, 48 (2004).
. G Hagen, J S Vaagen, M Hjorth-Jensen, J. Phys.A:Math. Gen. 378991G. Hagen, J. S.Vaagen, and M. Hjorth-Jensen, J. Phys.A:Math. Gen. 37, 8991 (2004).
. G Hagen, M Hjorth-Jensen, J S Vaagen, Phys. Rev. C. 7144314G. Hagen,M. Hjorth-Jensen, and J. S. Vaagen, Phys. Rev. C 71, 044314 (2005)
. T Berggren, Nucl. Phys. 109265T. Berggren, Nucl. Phys. A109, 265 (1968).
. T Berggren, Nucl. Phys. 169353T. Berggren, Nucl. Phys. A169, 353 (1971).
. T Berggren, Phys. Lett. 73389T. Berggren, Phys. Lett. B73, 389 (1978).
. T Berggren, Phys. Lett. 3731T. Berggren, Phys. Lett. B373, 1 (1996).
. P Lind, Phys. Rev. C. 471903P. Lind, Phys. Rev. C 47, 1903 (1993).
. H Hergert, S K Bogner, S Binder, A Calci, J Langhammer, R Roth, A Schwenk, Phys. Rev. C. 8734307H. Hergert, S.K. Bogner, S. Binder, A. Calci, J. Langhammer, R. Roth, and A. Schwenk, Phys. Rev. C 87, 034307 (2013).
. B S Hu, F R Xu, Z H Sun, J P Vary, T Li, Phys. Rev. 9414303B. S. Hu, F. R. Xu, Z. H. Sun, J. P. Vary, and T. Li, Phys. Rev. C94, 014303 (2016).
. A Tichai, J Langhammer, S Binder, R Roth, Phys. Lett. B. 756283A. Tichai, J. Langhammer, S. Binder and R. Roth, Phys. Lett. B 756, 283 (2016).
. T Vertse, K F Pal, Z Balogh, Comp. Phys. Comm. 27309T. Vertse, K. F. Pal and Z. Balogh, Comp. Phys. Comm. 27, 309 (1982).
. L G Ixaru, M Rizea, T Vertse, Comput. Phys. Commun. 85217L.G. Ixaru, M. Rizea, T. Vertse, Comput. Phys. Commun. 85 (1995) 217.
. D Vautherin, M Veneroni, Phys. Lett. B. 25175D. Vautherin and M. Veneroni, Phys. Lett. B 25, 175 (1967).
. N Michel, Eur. Phys.J.A. 42523N. Michel, Eur. Phys.J.A. 42, 523 (2009).
. G Hagen, N Michel, Phys. Rev. C. 8621602G. Hagen and N. Michel, Phys. Rev. C 86, 021602(R) (2012).
. D R Entem, R Machleidt, Phys. Rev. 6841001D. R. Entem and R. Machleidt, Phys. Rev. C68, 041001 (2003).
. S K Bogner, T T S Kuo, A Schwenk, Phys. Rept. 3861S. K. Bogner, T. T. S. Kuo, and A. Schwenk, Phys. Rept. 386, 1 (2003).
| [] |
[
"The Dynamics of Human Body Weight Change",
"The Dynamics of Human Body Weight Change"
] | [
"C C Chow ",
"K D Hall ",
"\nNational Institute of Diabetes and Digestive and Kidney Diseases\nLaboratory of Biological Modeling\nNational Institutes of Health\nBethesdaMarylandUnited States of America\n",
"\nUniversity of California\nSan DiegoUnited States of America\n"
] | [
"National Institute of Diabetes and Digestive and Kidney Diseases\nLaboratory of Biological Modeling\nNational Institutes of Health\nBethesdaMarylandUnited States of America",
"University of California\nSan DiegoUnited States of America"
] | [
"PLoS Comput Biol"
] | An imbalance between energy intake and energy expenditure will lead to a change in body weight (mass) and body composition (fat and lean masses). A quantitative understanding of the processes involved, which currently remains lacking, will be useful in determining the etiology and treatment of obesity and other conditions resulting from prolonged energy imbalance. Here, we show that a mathematical model of the macronutrient flux balances can capture the long-term dynamics of human weight change; all previous models are special cases of this model. We show that the generic dynamic behavior of body composition for a clamped diet can be divided into two classes. In the first class, the body composition and mass are determined uniquely. In the second class, the body composition can exist at an infinite number of possible states. Surprisingly, perturbations of dietary energy intake or energy expenditure can give identical responses in both model classes, and existing data are insufficient to distinguish between these two possibilities. Nevertheless, this distinction has important implications for the efficacy of clinical interventions that alter body composition and mass.This is an open-access article distributed under the terms of the Creative Commons Public Domain declaration which stipulates that, once placed in the public domain, this work may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. | 10.1371/journal.pcbi.1000045 | null | 3,070,793 | 0802.3234 | 55e359eea8ca336d38a92a7c2d17fbaf03a8fc6c |
The Dynamics of Human Body Weight Change
2008
C C Chow
K D Hall
National Institute of Diabetes and Digestive and Kidney Diseases
Laboratory of Biological Modeling
National Institutes of Health
BethesdaMarylandUnited States of America
University of California
San DiegoUnited States of America
The Dynamics of Human Body Weight Change
PLoS Comput Biol
431000045200810.1371/journal.pcbi.1000045Received November 26, 2007; Accepted February 28, 2008; Published March 28, 2008Editor: Competing Interests: The authors have declared that no competing interests exist. *
An imbalance between energy intake and energy expenditure will lead to a change in body weight (mass) and body composition (fat and lean masses). A quantitative understanding of the processes involved, which currently remains lacking, will be useful in determining the etiology and treatment of obesity and other conditions resulting from prolonged energy imbalance. Here, we show that a mathematical model of the macronutrient flux balances can capture the long-term dynamics of human weight change; all previous models are special cases of this model. We show that the generic dynamic behavior of body composition for a clamped diet can be divided into two classes. In the first class, the body composition and mass are determined uniquely. In the second class, the body composition can exist at an infinite number of possible states. Surprisingly, perturbations of dietary energy intake or energy expenditure can give identical responses in both model classes, and existing data are insufficient to distinguish between these two possibilities. Nevertheless, this distinction has important implications for the efficacy of clinical interventions that alter body composition and mass.This is an open-access article distributed under the terms of the Creative Commons Public Domain declaration which stipulates that, once placed in the public domain, this work may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose.
Introduction
Obesity, anorexia nervosa, cachexia, and starvation are conditions that have a profound medical, social and economic impact on our lives. For example, the incidence of obesity and its co-morbidities has increased at a rapid rate over the past two decades [1,2]. These conditions are characterized by changes in body weight (mass) that arise from an imbalance between the energy derived from food and the energy expended to maintain life and perform work. However, the underlying mechanisms of how changes in energy balance lead to changes in body mass and body composition are not well understood. In particular, it is of interest to understand how body composition is apportioned between fat and lean components when the body mass changes and if this energy partitioning can be altered. Such an understanding would be useful for optimizing weight loss treatments in obese subjects to maximize fat loss or weight gain treatments for anorexia nervosa and cachexia patients to maximize lean tissue gain.
To address these issues and improve our understanding of human body weight regulation, mathematical and computational modeling has been attempted many times over the past several decades [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Here we show how models of body composition and mass change can be understood and analyzed within the realm of dynamical systems theory and can be classified according to their geometric structure in the two dimensional phase plane. We begin by considering a general class of macronutrient flux balance equations and progressively introduce assumptions that constrain the model dynamics. We show that two compartment models of fat and lean masses can be categorized into two generic classes. In the first class, there is a unique body composition and mass (i.e. a stable fixed point) that is specified by the diet and energy expenditure. In the second class, there is a continuous curve of fixed points (i.e. an invariant manifold) with an infinite number of possible body compositions and masses at steady state for the same diet and energy expenditure rate. We show that almost all of the models in the literature are in the second class. Surprisingly, the existing data are insufficient to determine which of the two classes pertains to humans. For models with an invariant manifold, we show that an equivalent one dimensional equation for body composition change can be derived. We give numerical examples and discuss possible experimental approaches that may distinguish between the classes.
Results
General Model of Macronutrient and Energy Flux Balance
The human body obeys the law of energy conservation [20], which can be expressed as
DU~DQ{DW ,ð1Þ
where DU is the change in stored energy in the body, DQ is a change in energy input or intake, and DW is a change in energy output or expenditure. The intake is provided by the energy content of the food consumed. Combustion of dietary macronutrients yields chemical energy and Hess's law states that the energy released is the same regardless of whether the process takes place inside a bomb calorimeter or via the complex process of oxidative phosphorylation in the mitochondria. Thus, the energy released from oxidation of food in the body can be precisely measured in the laboratory. However, there is an important caveat. Not all macronutrients in food are completely absorbed by the body. Furthermore, the dietary protein that is absorbed does not undergo complete combustion in the body, but rather produces urea and ammonia. In accounting for these effects, we refer to the metabolizable energy content of dietary carbohydrate, fat, and protein, which is slightly less than the values obtained by bomb calorimetry. The energy expenditure rate includes the work to maintain basic metabolic function (resting metabolic rate), to digest, absorb and transport the nutrients in food (thermic effect of feeding), to synthesize or break down tissue, and to perform physical activity, together with the heat generated. The energy is stored in the form of fat as well as in lean body tissue such as glycogen and protein. The body need not be in equilibrium for Equation 1 to hold. While we are primarily concerned with adult weight change, Equation 1 is also valid for childhood growth.
In order to express a change of stored energy DU in terms of body mass M we must determine the energy content per unit body mass change, i.e. the energy density r M . We can then set DU = D(r M M). To model the dynamics of body mass change, we divide Equation 1 by some interval of time and take the limit of infinitesimal change to obtain a one dimensional energy flux balance equation:
d dt r M M ð Þ~I{Eð2Þ
where I = dQ/dt is the rate of metabolizable energy intake and E = dW/dt is the rate of energy expenditure. It is important to note that r M is the energy density of body mass change, which need not be a constant but could be a function of body composition and time. Thus, in order to use Equation 2, the dynamics of r M must also be established. When the body changes mass, that change will be composed of water, protein, carbohydrates (in the form of glycogen), fat, bone, and trace amounts of micronutrients, all having their own energy densities. Hence, a means of determining the dynamics of r M is to track the dynamics of the components. The extracellular water and bone mineral mass have no metabolizable energy content and change little when body mass changes in adults under normal conditions [21]. The change in intracellular water can be specified by changes in the tissue protein and glycogen. Thus the main components contributing to the dynamics of r M are the macronutrients -protein, carbohydrates, and fat, where we distinguish body fat (e.g. free fatty acids and triglycerides) from adipose tissue, which includes water and protein in addition to triglycerides. We then represent Equation 2 in terms of macronutrient flux balance equations for body fat F, glycogen G, and protein P:
r F dF dt~I F {f F E ð3Þ r G dG dt~I C {f C Eð4Þr P dP dt~I P { 1{f F {f C ð Þ Eð5Þ
where r F = 39.5 MJ/kg, r G = 17.6 MJ/kg, r P = 19.7 MJ/kg are the energy densities [3], I F ,I C ,I P are the intake rates, and f F , f C , 12f F 2f C are the fractions of the energy expenditure rate obtained from the combustion of fat, carbohydrates (glycogen) and protein respectively. The fractions and energy expenditure rate are functions of body composition and intake rates. They can be estimated from indirect calorimetry, which measures the oxygen consumed and carbon dioxide produced by a subject [22]. The intake rates are determined by the macronutrient composition of the consumed food, and the efficiency of the conversion of the food into a utilizable form. Transfer between compartments such as de novo lipogenesis where carbohydrates are converted to fat or gluconeogenesis where amino acids are converted into carbohydrates can be accounted for in the forms of f F and f C . The sum of Equations 3, 4, and 5 recovers the energy flux balance Equation 2, where the body mass M is the sum of the macronutrients F, G, P, with the associated intracellular water, and the inert mass that does not change such as the extracellular water, bones, and minerals, and r M = (r F F+r G G+r P P)/M. The intake and energy expenditure rates are explicit functions of time with fast fluctuations on a time scale of hours to days [23]. However, we are interested in the long-term dynamics over weeks, months and years. Hence, to simplify the equations, we can use the method of averaging to remove the fast motion and derive a system of equations for the slow time dynamics. We do this explicitly in the Methods section and show that the form of the averaged equations to lowest order are identical to Equations 3-5 except that the three components are to be interpreted as the slowly varying part and the intake and energy expenditure rates are moving time averages over a time scale of a day.
The three-compartment flux balance model was used by Hall [3] to numerically simulate data from the classic Minnesota human starvation experiment [21]. In Hall's model, the forms of the energy expenditure and fractions were chosen for physiological considerations. For clamped food intake, the body composition approached a unique steady state. The model also showed that apart from transient changes lasting only a few days, carbohydrate balance is precisely maintained as a result of the limited storage capacity for glycogen. We will exploit this property to reduce the three dimensional system to an approximately equivalent two dimensional system where dynamical systems techniques can be employed to analyze the dynamics.
Author Summary
Understanding the dynamics of human body weight change has important consequences for conditions such as obesity, starvation, and wasting syndromes. Changes of body weight are known to result from imbalances between the energy derived from food and the energy expended to maintain life and perform physical work. However, quantifying this relationship has proved difficult, in part because the body is composed of multiple components and weight change results from alterations of body composition (i.e., fat versus lean mass). Here, we show that mathematical modeling can provide a general description of how body weight will change over time by tracking the flux balances of the macronutrients fat, protein, and carbohydrates. For a fixed food intake rate and physical activity level, the body weight and composition will approach steady state. However, the steady state can correspond to a unique body weight or a continuum of body weights that are all consistent with the same food intake and energy expenditure rates. Interestingly, existing experimental data on human body weight dynamics cannot distinguish between these two possibilities. We propose experiments that could resolve this issue and use computer simulations to demonstrate how such experiments could be performed.
Reduced Models
Two compartment macronutrient partition model. The three compartment macronutrient flux balance model Equations 3-5 can be reduced to a two dimensional system for fat mass F and lean mass L = M2F, where M is the total body mass. The lean mass includes the protein and glycogen with the associated intracellular water along with the mass that does not change appreciably such as the extracellular water and bone. Hence the rate of change in lean mass is given by
dL dt~1 zh P ð Þ dP dt z 1zh G ð Þ dG dtð6Þ
where h P = 1.6 and h G = 2.7 are reasonable estimates of the hydration coefficients for the intracellular water associated with the protein and glycogen respectively [3,24]. (We note that fat is not associated with any water.) The glycogen storage capacity is extremely small compared to the fat and protein compartments. Thus the slow component of glycogen can be considered to be a constant (see Methods). In other words, on time scales much longer than a day, which are of interest for body weight change, we can consider glycogen to be in quasi-equilibrium so that dG/ dt = 0, as observed in numerical simulations [3]. This implies that f C = I C /E, which can be substituted into Equation 5 to give
r P dP dt~I P zI C { 1{f F ð ÞEð7Þ
Substituting Equation 7 and dG/dt = 0 into Equation 6 leads to the two compartment macronutrient partition model
r F dF dt~I F {fE ð8Þ r L dL dt~I L { 1{f ð ÞEð9Þ
where r L = r P /(1+h P ) = 7.6 MJ/kg, I F and I L = I P +I C are the intake rates into the fat and lean compartments respectively, E = E(I F , I L ,F,L) is the total energy expenditure rate, and f = f(I F ,I L ,F,L) ; f F is the fraction of energy expenditure rate attributed to fat utilization. We note that dG/dt = 0 may be violated if the glycogen content is proportional to the protein content, which is plausible because most of the glycogen mass is stored in muscle tissue and may scale with protein mass. We show that this assumption leads to the same two dimensional system. Substituting
dG dt~k dP dtð10Þ
for a proportionality constant k, into Equation 4 gives f C E = I C 2r C kdP/dt, which inserted into Equation 5 leads to
r P zkr C ð Þ dP dt~I L { 1{f ð ÞEð11Þ
Substituting Equations 10 and 11 into Equation 6 will again result in Equation 9 but with r L = (r P +kr C )/((1+h P +k+kh G ). For k = 0.044%1 as suggested by Snyder et al. [25], r L has approximately the same value as before.
Previous studies have considered two dimensional models of body mass change although they were not derived from the threedimensional macronutrient partition model. Alpert [5][6][7] considered a model with E linearized in F and L and different f depending on context. Forbes [8] and Livingston et al. [9] modeled weight loss as a double exponential decay. Although, they did not consider macronutrient flux balance, the dynamics of their models are equivalent to the two dimensional model with I F and I L zero, and E linear in F and L.
Energy partition model. The two-compartment macronutrient partition model can be further simplified by assuming that trajectories in the L-F phase plane follow prescribed paths satisfying
r F r L dF dL~a F ,L ð Þð12Þ
where a(F,L) is a continuous function [10,11,26] that depends on the mechanisms of body weight change. Forbes first hypothesized this stringent constraint after analyzing body composition data collected across a large number of subjects [26,27]. Forbes postulated that for adults
a~r F r L F 10:4ð13Þ
so that
F~D exp L=10:4 ð Þ ð 14Þ
where D is a free parameter, and the lean and fat masses are in units of kg. Forbes found that his general relationship (14) was similar whether weight loss is induced by diet or exercise [27]. It is possible that resistance exercise or a significant change in the protein content of the diet may result in a different relationship for a [28][29][30]. Infant growth is an example where a is not well described by the Forbes relationship. Jordan and Hall [11] used longitudinal body composition data in growing infants to determine an appropriate form for a during the first two years of life. Equation 12 describes a family of F vs. L curves, parameterized by an integration constant (e.g., D in Equation 14). Depending on the initial condition, the body composition moves along one of these curves when out of energy balance. Dividing Equation 8 by Equation 9 and imposing Equation 12 results in
f F,L ð Þ~I F {aI L zaE 1za ð ÞE~I F E { a 1za I{E Eð15Þ
Hall, Bain, and Chow [10] showed that the two compartment macronutrient partition model with Equation 15 using Forbes's law (Equation 13) matched a wide range of data without any adjustable parameters. Substituting Equation 15 into the macronutrient partition model 8 and 9 leads to the Energy Partition model:
r F dF dt~1 {p ð Þ I{E ð Þð16Þr L dL dt~p I{E ð Þð17Þ
where p = p(F,L) = 1/(1+a) is known as the p-ratio [31]. In the energy partition model, an energy imbalance I-E is divided between the compartments according to a function p(F,L) that defines the fraction assigned to lean body tissue (mostly protein).
Most of the previous models in the literature are different versions of the energy partition model [6,[12][13][14][15][16][17][18], although none of the authors have noted the connection to macronutrient flux balance or analyzed their models using dynamical systems theory. Some of these previous models are expressed as computational algorithms that can be translated to the form of the energy partition model. Despite the ubiquity of the energy partition model, the physiological interpretation of the p-ratio remains obscure and is difficult to measure directly. It can be inferred indirectly from f (which can be measured by indirect calorimetry) by using Equation 15 [10]. Previous uses of the energy partition model often considered p to be a constant [6,[12][13][14][15][16][17][18], which implies that the partitioning of energy is independent of current body composition and macronutrient composition. This is in contradiction to weight loss data that finds that the fraction of body fat lost does depend on body composition with more fat lost if the body fat is initially higher [26,32,33]. However, if a is a weak function of body composition then a constant p-ratio may be a valid approximation for small changes. Flatt [17] considered a model where the p-ratio was constant but included the dynamics for glycogen. His model would be useful when dynamics on short time scales are of interest.
It may sometimes be convenient to express the macronutrient partition model with a unique fixed point as
r F dF dt~1 {p ð Þ I{E ð Þzy ð18Þ r L dL dt~p I{E ð Þ{yð19Þ
for a function y = y(I F ,I L ,F,L), which is zero at the fixed point (F 0 ,L 0 ). We use this form in numerical examples below. The fasting model of Song and Thomas [19]) used this form with I = 0 and y was a function of F representing ketone production. Comparing to Equation 8 and Equation 18 gives
f~I F E { 1{p ð Þ I{E E { y Eð20Þ
One-dimensional models. The dynamics of the energy partition model Equations 16 and 17 move along fixed trajectories in the L-F plane. Thus a further simplification to a one dimensional model is possible by finding a functional relationship between F and L so that one variable can be eliminated in favor of the other. Such a function exists if Equation 12 has a unique solution, which is guaranteed in some interval of L if a(F,L) and ha/hF are continuous functions of F and L on a rectangle containing this interval. These are sufficient but not necessary conditions. Suppose a relationship F = w(L) can be found between F and L. Substituting this relationship into Equation 16 and Equation 17 and adding the two resulting equations yields the one dimensional equation
dL dt~I {E w L ð Þ,L ð Þ r F w 0 L ð Þzr Lð21Þ
We can obtain a dynamical equation for body mass by expressing the body mass as M = L+w(L). If we can invert this relationship uniquely and obtain L as a function of M, then this can be substituted into Equation 21 to obtain a dynamical equation for M.
As an example, assume p to be a constant, which was used in [6,[12][13][14][15][16][17][18]. This implies that the phase orbits are a family of straight lines of the form F = bL+C; w(L) where b = r L (12p)/(r F p) and C is a constant that is specified by the initial body composition. This results in dM dt~1
{p r F z p r L I{E b 1zb M{C ð Þ zC, M{C 1zb !ð22Þ
Linearizing Equation 22 around a mass M 0 gives
r M dM dt~m {e M{M 0 ð Þ ð 23Þ where r M = r F r L /(r L +(r F 2r L )p), m = I2E(F(M 0 ),L(M 0 )), and e~dE=dM M~M0 j
. This is the form used in [18].
0:4 r F D exp L=10:4 ð Þ z10:4r L I{E D exp L=10:4 ð Þ ,L ð Þ ½ ð 24Þ
Similarly, a one dimensional equation for the fat mass has the form
dF dt~F r F F z10:4r L I{E F,10:4 log F =D ð Þ ð Þ ½ ð 25Þ
Since the mass functions M = L+Dexp(L/10.4) or M = F+ 10.4log(F/D) cannot be inverted in closed form, an explicit one dimensional differential equation in terms of the mass cannot be derived. However, the dynamics of the mass is easily obtained using either Equation 24 or Equation 25 together with the relevant mass function. For large changes in body composition, the dynamics could differ significantly from the constant p models 22 or 23. The one dimensional model gives the dynamics of the energy partition model along a fixed trajectory in the F-L plane. The initial body composition specifies the constant C or D in the above equations. A one dimensional model will represent the energy partition model even if the intake rate is time dependent. Only for a perturbation that directly alters body composition will the one dimensional model no longer apply. However, after the perturbation ceases, the one dimensional model with a new constant will apply again.
Existence and Stability of Body Weight Fixed Points
The various flux balance models can be analyzed using the methods of dynamical systems theory, which aims to understand dynamics in terms of the geometric structure of possible trajectories (time courses of the body components). If the models are smooth and continuous then the global dynamics can be inferred from the local dynamics of the model near fixed points (i.e. where the time derivatives of the variables are zero). To simplify the analysis, we consider the intake rates to be clamped to constant values or set to predetermined functions of time. We do not consider the control and variation of food intake rate that may arise due to feedback from the body composition or from exogenous influences. We focus only on what happens to the food once it is ingested, which is a problem independent of the control of intake. We also assume that the averaged energy expenditure rate does not depend on time explicitly. Hence, we do not account for the effects of development, aging or gradual changes in lifestyle, which could lead to an explicit slow time dependence of energy expenditure rate. Thus, our ensuing analysis is mainly applicable to understanding the slow dynamics of body mass and composition for clamped food intake and physical activity over a time course of months to a few years.
Dynamics in two dimensions are particularly simple to analyze and can be easily visualized geometrically [34,35]. The one dimensional models are a subclass of two dimensional dynamics. Three dimensional dynamical systems are generally more difficult to analyze but Hall [3] found in simulations that the glycogen levels varied over a small interval and averaged to an approximate constant for time periods longer than a few days, implying that the slow dynamics could be effectively captured by a two dimensional model. Reduction to fewer dimensions is an oft-used strategy in dynamical systems theory. Hence, we focus our analysis on two dimensional dynamics.
In two dimensions, changes of body composition and mass are represented by trajectories in the L-F phase plane. For I F and I L constant, the flux balance model is a two dimensional autonomous system of ordinary differential equations and trajectories will flow to attractors. The only possible attractors are infinity, stable fixed points or stable limit cycles [34,35]. We note that fixed points within the context of the model correspond to states of flux balance. The two compartment macronutrient partition model is completely general in that all possible autonomous dynamics in the two dimensional phase plane are realizable. Any two or one dimensional autonomous model of body composition change can be expressed in terms of the two dimensional macronutrient partition model.
Physical viability constrains L and F to be positive and finite. For differentiable f and E, the possible trajectories for fixed intake rates are completely specified by the dynamics near fixed points of the system. Geometrically, the fixed points are given by the intersections of the nullclines in the L-F plane, which are given by the solutions of I F 2fE = 0 and I L = (12f )E = 0. Example nullclines and phase plane portraits of the macronutrient model are shown in Figure 1. If the nullclines intersect once then there will be a single fixed point and if it is stable then the steady state body composition and mass are uniquely determined. Multiple intersections can yield multiple stable fixed points implying that body composition is not unique [4]. If the nullclines are collinear then there can be an attracting one dimensional invariant manifold (continuous curve of fixed points) in the L-F plane. In this case, there are an infinite number of possible body compositions for a fixed diet. As we will show, the energy partition model implicitly assumes an invariant manifold. If a single fixed point exists but is unstable then a stable limit cycle may exist around it.
The fixed point conditions of Equations 8 and 9 can be expressed in terms of the solutions of Figure 1C). Each of these infinite possible body compositions will result in a different body mass M = F+L (except for the unlikely case that E is a function of the sum F+L). The body composition is marginally stable along the direction of the invariant manifold. This means that in flux balance, the body composition will remain at rest at any point on the invariant manifold. A transient perturbation along the invariant manifold will simply cause the body composition to move to a new position on the invariant manifold. The one dimensional models have a stable fixed point if the invariant manifold is attracting. We also show in Methods that for multiple stable fixed points or a limit cycle to exist, f must be nonmonotonic in L and be finely tuned. The required fine-tuning makes these latter two possibilities much less plausible than a single fixed point or an invariant manifold.
Data suggest that E is a monotonically increasing function of F and L [36]. The dependence of f on F and L is not well established and the form of f depends on multiple interrelated factors. In general, the sensitivity of various tissues to the changing hormonal milieu will have an overall effect on both the supply of macronutrients as well as the substrate preferences of various metabolically active tissues. On the supply side, we know that free fatty acids derived from adipose tissue lipolysis increase with increasing body fat mass which thereby increase the daily fat oxidation fraction, f, as F increases [37]. Furthermore, reduction of F with weight loss has been demonstrated to decrease f [38]. Similarly, whole-body proteolysis and protein oxidation increases with lean body mass [39,40] implying that f should be a decreasing function of L. In further support of this relationship, body builders with significantly increased L have a decreased daily fat oxidation fraction versus control subjects with similar F [41]. Thus a stable isolated fixed point is consistent with this set of data.
Implications for Body Mass and Composition Change
We have shown that all two dimensional autonomous models of body composition change generically fall into two classes -those with fixed points and those with invariant manifolds. In the case of a stable fixed point, any temporary perturbation of body weight or composition will be corrected over time (i.e., for all things equal, the body will return to its original state). An invariant manifold allows the possibility that a transient perturbation could lead to a permanent change of body composition and mass.
At first glance, these differing properties would appear to point to a simple way of distinguishing between the two classes. However, the traditional means of inducing weight change namely diet or altering energy expenditure through aerobic exercise, turn out to be incapable of revealing the distinction. For an invariant manifold, any change of intake or expenditure rate will only elicit movement along one of the prescribed F vs. L trajectories obeying Equation 12, an example being Forbes's law (14). As shown in Figure 2, a change of intake or energy expenditure rate will change the position of the invariant manifold. The body composition that is initially at one point on the invariant manifold will then flow to a new point on the perturbed invariant manifold along the trajectory prescribed by (12). If the intake rate or energy expenditure is then restored to the original value then the body composition will return along the same trajectory to the original steady state just as it would in a fixed point model (see Figure 2 solid curves). Only a perturbation that moves the body composition off of the fixed trajectory could distinguish between the two classes. In the fixed point case (Figure 2A dashed-dot curve), the body composition would go to the same steady state following the perturbation to Perturbations that move the body composition off the fixed trajectory can be done by altering body composition directly or by altering the fat utilization fraction f. For example, body composition could be altered directly through liposuction and f could be altered by administering compounds such as growth hormone. Resistance exercise may cause an increase in lean muscle tissue at the expense of fat. Exogenous hormones, compounds, or infectious agents that change the propensity for fat versus carbohydrate oxidation (for example, by increasing adipocyte proliferation and acting as a sink for fat that is not available for oxidation [42][43][44]), would also perturb the body composition off of a fixed F vs. L curve by altering f. If the body composition returned to its original state after such a perturbation then there is a unique fixed point. If it does not then there could be an invariant manifold although multiple fixed points are also possible.
We found an example of one clinical study that bears on the question of whether humans have a fixed point or an invariant manifold. Biller et al. investigated changes of body composition pre-and post-growth hormone therapy in forty male subjects with growth hormone deficiency [45]. Despite significant changes of body composition induced by 18 months of growth hormone administration, the subjects returned very closely to their original body composition 18 months following the removal of therapy. However, there was a slight (2%) but significant increase in their lean body mass compared with the original value. Perhaps not enough time had elapsed for the lean mass to return to the original level. Alternatively, the increased lean mass may possibly have been the result of increased bone mineral mass and extracellular fluid expansion, both of which are known effects of growth hormone, but were assumed to be constant in the body composition models. Therefore, this clinical study provides some evidence in support of a fixed point, but it has not been repeated and the result was not conclusive. Using data from the Minnesota experiment [21] and the underlying physiology, Hall [3] proposed a form for f that predicts a fixed point. On the other hand, Hall, Bain, and Chow [10] showed that an invariant manifold model is consistent with existing data of longitudinal weight change but these experiments only altered weight through changes in caloric intake so this cannot rule out the possibility of a fixed point. Thus it appears that existing data is insufficient to decide the issue.
Numerical Simulations
We now consider some numerical examples using the macronutrient partition model in the form given by Equations 18 and 19, with a p-ratio consistent with Forbes's law (13) (i.e. p = 2/(2+F), where F is in units of kg). Consider two cases of the model. If y = 0 then the model has an invariant manifold and body composition moves along a fixed trajectory in the L-F plane. If y is nonzero, then there can be an isolated fixed point. We will show an example where if the intake energy is perturbed, the approach of the body composition to the steady state will be identical for both cases but if body composition is perturbed, the body will arrive at different steady states.
For every model with an invariant manifold, a model with a fixed point can be found such that trajectories in the L-F plane resulting from energy intake perturbations will be identical. All that is required is that y in the fixed point model is chosen such that the solution of y (F,L) = 0 defines the fixed trajectory of the invariant manifold model. Using Forbes's law (14), we choose y = 0.05(F20.4 exp(L/10.4))/F. We then take a plausible energy expenditure rate of E = 0.14L+0.05F+1.55, where energy rate has units of MJ/day and mass has units of kg. This expression is based on combining cross-sectional data [36] for resting energy with a contribution of physical activity of a fairly sedentary person [3]. Previous models propose similar forms for the energy expenditure [5,7,13,18]. Figure 3 shows the time dependence of body mass and the F vs. L trajectories of the two model examples given a reduction in energy intake rate from 12 MJ/day to 10 MJ/day starting at the same initial condition. The time courses are identical for body Figure 2D). Since a y can always be found so that a fixed point model and an invariant manifold model have identical time courses for body composition and mass, a perturbation in energy intake can never discriminate between the two possibilities.
The time constant to reach the new fixed point in the numerical simulations is very long. This slow approach to steady state (on the order of several years for humans) has been pointed out many times previously [3,5,7,13,18]. A long time constant will make experiments to distinguish between a fixed point and an invariant manifold difficult to conduct. Experimentally reproducing this example would be demanding but if the time variation of the intake rates and physical activity levels were small compared to the induced change then the same result should arise qualitatively. Additionally, the time constant depends on the form of the energy expenditure. There is evidence that the dependence of energy expenditure on F and L for an individual is steeper than for the population due to an effect called adaptive thermogenesis [46], thus making the time constant shorter.
Discussion
In this paper we have shown that all possible two dimensional autonomous models for lean and fat mass are variants of the macronutrient partition model. The models can be divided into two general classes -models with isolated fixed points (most likely a single stable fixed point) and models with an invariant manifold. There is the possibility of more exotic behavior such as multi-stability and limit cycles but these require fine-tuning and thus are less plausible. Surprisingly, experimentally determining if the body exhibits a fixed point or an invariant manifold is nontrivial. Only perturbations of the body composition itself apart from dietary or energy expenditure interventions or alterations of the fraction of energy utilized as fat can discriminate between the two possibilities. The distinction between the classes is not merely an academic concern since this has direct clinical implications for potential permanence of transient changes of body composition via such procedures as liposuction or temporary administration of therapeutic compounds.
Our analysis considers the slow dynamics of the body mass and composition where the fast time dependent hourly or daily fluctuations are averaged out for a clamped average food intake rate. We also do not consider a slow explicit time dependence of the energy expenditure. Such time dependence could arise during development, aging or gradual changes in lifestyle where activity levels differ. Thus our analysis is best suited to modeling changes over time scales of months to a few years in adults. We do not consider any feedback of body composition on food intake, which is an extremely important topic but beyond the scope of this paper. Previous efforts to model body weight change have predominantly used energy partition models that implicitly contain an invariant manifold and thus body composition and mass are not fully specified by the diet. If the body does have an invariant manifold then this fact puts a very strong constraint on the fat utilization fraction f. Hall [3] considered the effects of carbohydrate intake on lipolysis and other physiological factors to conjecture a form of f that does not lead to an invariant manifold. However, our analysis and numerical examples show that the body composition could have an invariant manifold but behave indistinguishably from having a fixed point. Also, the decay to the fixed point could take a very long time, possibly as long as a decade giving the appearance of an invariant manifold. Only experiments that perturb the fat or lean compartments independently can tell.
Methods
Method of Averaging
The three compartment macronutrient flux balance Equations 3-5 are a system of nonautonomous differential equations since the energy intake and expenditure are explicitly time dependent. Food is ingested over discrete time intervals and physical activity will vary greatly within a day. However, this fast time dependence can be viewed as oscillations or fluctuations on top of a slowly varying background. It is this slower time dependence that governs long-term body mass and composition changes that we are interested in. For example, if an individual had the exact same schedule with the same energy intake and expenditure each day, then averaged over a day, the body composition would be constant. If the daily averaged intake and expenditure were to gradually change on longer time scales of say weeks or months then there would be a corresponding change in the body composition and mass. Given that we are only interested in these slower changes, we remove the short time scale fluctuations by using the method of averaging to produce an autonomous system of averaged equations valid on longer time scales.
We do so by introducing a second ''fast'' time variable t = t/e, where e is a small parameter that is associated with the slow changes in body composition and let all time dependent quantities be a function of both t and t. For example, if t is measured in units of days and t is measured in units of hours then e,1/24. Inserting into Equations 3-5 and using the chain rule yields
r F LF Lt z 1 e LF Lt I F t,t ð Þ{f F E t,t ð Þ ð28Þ r G LG Lt z 1 e
LG Lt
I C t,t ð Þ{f C E t,t ð Þð29Þr P LP Lt z 1 e LP Lt I P t,t ð Þ{ 1{f F {f C ð Þ E t,t ð Þð30Þ
We then consider the three body compartments to have expansions of the form
F t,t ð Þ~F 0 t ð ÞzeF 1 t,t ð ÞzO e 2 À Á ð31Þ G t,t ð Þ~G 0 t ð ÞzeG 1 t,t ð ÞzO e 2 À Áð32ÞP t,t ð Þ~P 0 t ð ÞzeP 1 t,t ð ÞzO e 2 À Áð33Þ
where AEF 1 ae = AEP 1 ae = AEG 1 ae = 0 for a time average defined by
SX T~1=T ð Þ ð T 0
Xdt and T represents an averaging time scale of a day. The fast time dependence can be either periodic or stochastic. The important thing is that the time average over the fast quantities is of order e or higher. We then expand the energy expenditure rate and expenditure fractions to first order in e:
E F,G,P,t,t ð ÞE 0 t,t ð Þze LE LF F 1 z LE LG G 1 z LE LP P 1 zO e 2 À Áð34Þf i F ,G,P ð Þf i E 0 ,G 0 ,P 0 À Á ze LE LF F 1 z LE LG G 1 z LE LP P 1 zO e 2 À Áð35Þ
where E 0 (t,t);E(F 0 ,G 0 ,P 0 ,t,t)+O(e 2 ) and iM{F,G,P}. We assume that the expenditure fractions depend on time only through the body compartments. Substituting these expansions into Equations 28-30 and taking lowest order in e gives
r F LF 0 Lt z LF 1 Lt I F t,t ð Þ{f 0 F E 0 t,t ð Þð36Þr G LG 0 Lt z LG 1 Lt I C t,t ð Þ{f 0 C E 0 t,t ð Þð37Þr P LP 0 Lt z LP 1 Lt I P t,t ð Þ{ 1{f 0 F {f 0 C À Á E 0 t,t ð Þð38Þ
Taking the moving time average of Equations 36-38 and requiring that AEhF 1 /htae, AEhG 1 /htae, and AEhP 1 /htae are of order e or higher leads to the averaged equations:
r F dF 0 dt~S I F T{f 0 F SE 0 T ð39Þ r G dG 0 dt~S I C T{f 0 C SE 0 Tð40Þr P dP 0 dt~S I P T{(1{f 0 F {f 0 C )SE 0 Tð41Þ
In the main text we only consider the slow time scale dynamics so we drop the superscript and bracket notation for simplicity. Hence, the system (3)(4)(5) can be thought of as representing the lowest order time averaged macronutrient flux balance equations. We note that in addition to the daily fluctuations of meals and physical activity, there can also be fluctuations in food intake from day to day [23]. Our averaging scheme can be used to average over these fluctuations as well by extending the averaging time T.
A difference in the choice of T will only result in a different interpretation of the averaged quantities.
Stability Conditions for Fixed Points
The dynamics near a fixed point (F 0 ,L 0 ) are determined by expanding fE and (12f )E to linear order in dF = F2F 0 and dL = L2L 0 [34,35]. Assuming solutions of the form exp(lt) yields an eigenvalue problem with two eigenvalues given by
: ð43Þ
A fixed point is stable if and only if Tr J,0 and det J.0. In the case of an invariant manifold, detJ = 0, so the eigenvalues are Tr J and 0. The zero eigenvalue reflects the marginal stability along the invariant manifold, which is an attractor if Tr J,0. An attracting invariant manifold implies a stable fixed point in the corresponding one dimensional model. Unstable fixed points are either unstable nodes, saddle points or unstable spirals. In the case of unstable spirals, a possibility is a limit cycle surrounding the spiral arising from a Hopf bifurcation, where Tr J = 0 and det J.0. In this case, body composition and mass would oscillate even if the intake rates were held constant. The frequency and amplitude of the oscillations may be estimated near a supercritical Hopf bifurcation by transforming the equations to normal form. Stability of a fixed point puts constraints on the form of f. Physiological considerations and data imply that hE/hL.hE/ hF.0 [3,36]. Thus we can set hE/hF = dhE/hL where d ,1 (the derivatives are evaluated at the fixed point). Then detJ.0 implies that
Lf =LF wd Lf =LLð44Þ
and Tr J,0 implies
Lf =LF wcLf =LL{K,ð45Þ
where K = [df+c (12f )](hE/hL)/E.0 and c = r F /r L <5.2. Hence hf/hF.0 and hf/hL,0 guarantees stability of a fixed point. In other words, if f increases monotonically with F and decreases monotonically with L then there will be a unique stable fixed point.
For an invariant manifold, f is given by Equation 15, which immediately satisfies detJ = 0; TrJ,0 is guaranteed if E is monotonically increasing in F and L. For a Hopf bifurcation, we require hf/hF = chf/hL2K and Equation 44, implying (c2d)hf/ hL2K.0. Since c.d, f must increase with L for the possibility of a limit cycle. However, to ensure that trajectories remain bounded f must decrease with L for very small and large values of L. Hence, f must be nonmonotonic in L for a limit cycle to exist. This can also be seen from an application of Bendixson's criterion [35], which states that a limit cycle cannot exist in a given region of the L-F plane if
1 r F L LF fE ð Þz 1 r L L LL 1{f ð ÞE ð Þ ð 46Þ
does not change sign in that region. In addition, the other parameters must be fine tuned for a limit cycle (see Figure 1D). Similarly, as seen in Figure 1C), for multi-stability to exist, nonmonotonicity and fine tuning are also required.
If Equations 16 and 17 are constrained to obey the phase plane paths of Forbes's law, then a reduction to a one dimensional equation can also be made. Using Equation 14 (i.e., w(L) = Dexp(L/10.4)) in Equation 21 yields dL dt~1
where I = I F +I L , and we have suppressed the functional dependence on intake rates. These fixed point conditions correspond to a state of flux balance of the lean and fat components. Equation 26indicates a state of energy balance while Equation27 indicates that the fraction of fat utilized must equal the fraction of fat in the diet. Stability of a fixed point is determined by the dynamics of small perturbations of body composition away from the fixed point. If the perturbed body composition returns to the original fixed point then the fixed point is deemed stable. We give the stability conditions in Methods.The functional dependence of E and f on F and L determine the existence and stability of fixed points. As shown in Methods, an isolated stable fixed point is guaranteed if f is a monotonic increasing function of F and a monotonic decreasing function of L. If one of the fixed point conditions automatically satisfies the other, then instead of a fixed point there will be a continuous curve of fixed points or an invariant manifold. For example, if the energy balance condition 26 automatically satisfies the fat fraction condition 27, then there is an invariant manifold defined by I = E(F,L). The energy partition model has this property and thus has an invariant manifold rather than an isolated fixed point. This can be seen by observing that for f given byEquation 15, Equation26 automatically satisfies condition 27. An attracting invariant manifold implies that the body can exist at any of the infinite number of body compositions specified by the curve I = E(F,L) for clamped intake and energy expenditure rates (see
Figure 1 .
1Possible trajectories (solid lines) for different initial conditions and nullclines (dotted lines) in the L-F phase plane for models with a stable fixed point (A), multi-stability with two stable fixed points separated by one unstable saddle point (B), an attracting invariant manifold (C), and a limit cycle attractor (D). doi:10.1371/journal.pcbi.1000045.g001 body composition but for the invariant manifold case (Figure 2B dashed-dot curve), it would go to another steady state.
Figure 2 .
2An example of a situation where the intake or energy expenditure rate is changed from one clamped value to another and then returned. (A) Fixed point case. (B) Invariant manifold case. Dotted lines represent nullclines. In both cases, the body composition follows a fixed trajectory and returns to the original steady state (solid curves). However, if the body composition is perturbed directly (dashed-dot curves) then the body composition will flow to same point in (A) but to a different point in (B). doi:10.1371/journal.pcbi.1000045.g002 composition and mass. The mass first decreases linearly in time but then saturates to a new stable fixed point. The dashed line represents the same intake rate reduction but with 10 kg of fat removed at day 100. For the invariant manifold model, the fat perturbation permanently alters the final body composition and body mass, whereas in the fixed point model it only has a transient effect. In the fixed point model, the body composition can ultimately exist only at one point given by the intersection of the nullclines (i.e., solution of I = E and y = 0). For the invariant manifold, the body composition can exist at any point on the I = E curve (dotted line in
Figure 3 .
3Time dependence of body mass and F vs. L trajectories. In all the Figures, the solid line is for an intake reduction from 12 MJ/day to 10 J/day and the dashed line is for the same reduction but with a removal of 10 kg of fat at day 100. Time dependence of body mass for the fixed point model (A). Trajectories in the F vs. L phase plane for the fixed point model (B). Dotted lines are the nullclines. Time dependence (C) and phase plane (D) of the invariant manifold model for the same conditions. doi:10.1371/journal.pcbi.1000045.g003
l~1 2
l~1TrJ+ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TrJ 2 {4 det Jp
where
TrJ~{
1
r F
L
LF
fE
ð Þz
1
r L
L
LL
1{f
ð
ÞE
ð
Þ
!
F0,L0
ð
Þ
ð42Þ
and
det J~E
r L r F
LE
LL
Lf
LF
{
LE
LF
Lf
LL
!
F0,L0
ð
Þ
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| Volume 4 | Issue 3 | e1000045
| Volume 4 | Issue 3 | e1000045
| Volume 4 | Issue 3 | e1000045
AcknowledgmentsWe thank T. Hwa and two anonymous reviewers for a critical reading of the manuscript.Author ContributionsConceived and designed the experiments: CC KH. Contributed reagents/ materials/analysis tools: CC KH. Wrote the paper: CC KH.
Obesity and the environment: where do we go from here?. J O Hill, H R Wyatt, G W Reed, J C Peters, Science. 299Hill JO, Wyatt HR, Reed GW, Peters JC (2003) Obesity and the environment: where do we go from here? Science 299: 853-855.
Understanding and addressing the epidemic of obesity: an energy balance perspective. J O Hill, Endocr Rev. 27Hill JO (2006) Understanding and addressing the epidemic of obesity: an energy balance perspective. Endocr Rev 27: 750-761.
Computational model of in vivo human energy metabolism during semistarvation and refeeding. K D Hall, Am J Physiol Endocrinol Metab. 291Hall KD (2006) Computational model of in vivo human energy metabolism during semistarvation and refeeding. Am J Physiol Endocrinol Metab 291: E23-E37.
Can a temporary high fat diet cause persistent obesity via acquired leptin resistance? Insights from a mathematical model. K Hall, D Polidori, Diabetes. 51S2607Hall K, Polidori D (2002) Can a temporary high fat diet cause persistent obesity via acquired leptin resistance? Insights from a mathematical model. Diabetes 51(S2): A607.
A two-reservoir energy model of the human body. S S Alpert, Am J Clin Nutr. 32Alpert SS (1979) A two-reservoir energy model of the human body. Am J Clin Nutr 32: 1710-1718.
Growth, thermogenesis, and hyperphagia. S S Alpert, Am J Clin Nutr. 52Alpert SS (1990) Growth, thermogenesis, and hyperphagia. Am J Clin Nutr 52: 784-792.
A limit on the energy transfer rate from the human fat store in hypophagia. S S Alpert, J Theor Biol. 233Alpert SS (2005) A limit on the energy transfer rate from the human fat store in hypophagia. J Theor Biol 233: 1-13.
Weight loss during fasting: implications for the obese. G B Forbes, Am J Clin Nutr. 23Forbes GB (1970) Weight loss during fasting: implications for the obese. Am J Clin Nutr 23: 1212-1219.
Biexponential model for predicting weight loss after gastric surgery for obesity. E H Livingston, J L Sebastian, S Huerta, I Yip, D Heber, J Surg Res. 101Livingston EH, Sebastian JL, Huerta S, Yip I, Heber D (2001) Biexponential model for predicting weight loss after gastric surgery for obesity. J Surg Res 101: 216-224.
How adaptations of substrate utilization regulate body composition. K D Hall, H L Bain, C C Chow, Int J Obes (Lond). 31Hall KD, Bain HL, Chow CC (2007) How adaptations of substrate utilization regulate body composition. Int J Obes (Lond) 31: 1378-1383.
Dynamic coordination of macronutrient balance during infant growth: Insights from a mathematical model. P Jordan, K Hall, Am J Clin Nutr. In pressJordan P, Hall K (2008) Dynamic coordination of macronutrient balance during infant growth: Insights from a mathematical model. Am J Clin Nutr; In press.
The equations governing weight change in human beings. V W Antonetti, Am J Clin Nutr. 26Antonetti VW (1973) The equations governing weight change in human beings. Am J Clin Nutr 26: 64-71.
A model for the prediction of energy balance and body weight. P R Payne, A E Dugdale, Ann Hum Biol. 4Payne PR, Dugdale AE (1977) A model for the prediction of energy balance and body weight. Ann Hum Biol 4: 525-535.
autoregulation of body weight and body composition in man. a systematic approach through models and simulation. L Girardier, Arch Int Physiol Biochim Biophys. 102Girardier L (1994) [autoregulation of body weight and body composition in man. a systematic approach through models and simulation]. Arch Int Physiol Biochim Biophys 102: A23-A35.
Energy intake, physical activity and body weight: a simulation model. K R Westerterp, J H Donkers, E W Fredrix, P Boekhoudt, Br J Nutr. 73Westerterp KR, Donkers JH, Fredrix EW, Boekhoudt P (1995) Energy intake, physical activity and body weight: a simulation model. Br J Nutr 73: 337-347.
Body weight setpoint, metabolic adaption and human starvation. F P Kozusko, Bull Math Biol. 63Kozusko FP (2001) Body weight setpoint, metabolic adaption and human starvation. Bull Math Biol 63: 393-403.
Carbohydrate-fat interactions and obesity examined by a twocompartment computer model. J P Flatt, Obes Res. 12Flatt JP (2004) Carbohydrate-fat interactions and obesity examined by a two- compartment computer model. Obes Res 12: 2013-2022.
Quantitative analysis of the energy requirements for development of obesity. E Christiansen, L Garby, T I Sorensen, J Theor Biol. 234Christiansen E, Garby L, Sorensen TI (2005) Quantitative analysis of the energy requirements for development of obesity. J Theor Biol 234: 99-106.
Dynamics of starvation in humans. B Song, D M Thomas, J Math Biol. 54Song B, Thomas DM (2007) Dynamics of starvation in humans. J Math Biol 54: 27-43.
Experiments on the metabolism of matter and energy in the human body. W O Atwater, F G Benedict, Bulletin. 136US Department of Agriculture, Office of Experiment StationsAtwater WO, Benedict FG (1903) Experiments on the metabolism of matter and energy in the human body. 1900-1902 Bulletin No 136, US Department of Agriculture, Office of Experiment Stations, Washington, DC.
The biology of human starvation. A Keys, University of Minnesota PressMinneapolis, MinnesotaKeys A (1950) The biology of human starvation. Minneapolis, Minnesota: University of Minnesota Press.
A respiration calorimeter with appliances for the direct determination of oxygen. W O Atwater, F G Benedict, 42Washington, D.CCarnegie Institution of WashingtonPublication NoAtwater WO, Benedict FG (1905) A respiration calorimeter with appliances for the direct determination of oxygen. Publication No 42, Carnegie Institution of Washington, Washington, D.C.
Patterns in food intake correlate with body mass index. V Periwal, C C Chow, Am J Physiol Endocrinol Metab. 291Periwal V, Chow CC (2006) Patterns in food intake correlate with body mass index. Am J Physiol Endocrinol Metab 291: E929-E936.
What is the required energy deficit per unit weight loss?. K D Hall, Int J Obes. In pressHall KD (2007) What is the required energy deficit per unit weight loss? Int J Obes (Lond); In press.
. W S Snyder, M J Cook, E S Nasset, L R Karhausen, G P Howells, Report on the Task Group on Reference Man. PergamonSnyder WS, Cook MJ, Nasset ES, Karhausen LR, Howells GP, et al. (1984) Report on the Task Group on Reference Man. Oxford, UK: Pergamon.
Lean body mass-body fat interrelationships in humans. G B Forbes, Nutr Rev. 45Forbes GB (1987) Lean body mass-body fat interrelationships in humans. Nutr Rev 45: 225-231.
Body fat content influences the body composition response to nutrition and exercise. G B Forbes, Ann N Y Acad Sci. 904Forbes GB (2000) Body fat content influences the body composition response to nutrition and exercise. Ann N Y Acad Sci 904: 359-365.
The effects of exercise training on fat-mass loss in obese patients during energy intake restriction. D Hansen, P Dendale, J Berger, J L Van Loon, R Meeusen, Sports Med. 37Hansen D, Dendale P, Berger J, van Loon JL, Meeusen R (2007) The effects of exercise training on fat-mass loss in obese patients during energy intake restriction. Sports Med (Auckland, NZ) 37: 31-46.
Dietary protein and exercise have additive effects on body composition during weight loss in adult women. D Layman, E Evans, J I Baum, J Seyler, D J Erickson, J Nutr. 135Layman D, Evans E, Baum JI, Seyler J, Erickson DJ, et al. (2005) Dietary protein and exercise have additive effects on body composition during weight loss in adult women. J Nutr 135: 1903-1910.
The role of diet and exercise for the maintenance of fat-free mass and resting metabolic rate during weight loss. P Stiegler, A Cunliffe, Sports Med. 36Stiegler P, Cunliffe A (2006) The role of diet and exercise for the maintenance of fat-free mass and resting metabolic rate during weight loss. Sports Med (Auckland, NZ) 36: 239-262.
Pattern of lean and fat deposition in adults. A Dugdale, P Payne, Nature. 266Dugdale A, Payne P (1977) Pattern of lean and fat deposition in adults. Nature 266: 349-351.
Differences in fat, carbohydrate, and protein metabolism between lean and obese subjects undergoing total starvation. M Elia, R Stubbs, C Henry, Obes Res. 7Elia M, Stubbs R, Henry C (1999) Differences in fat, carbohydrate, and protein metabolism between lean and obese subjects undergoing total starvation. Obes Res 7: 597-604.
Body fat and fat-free mass inter-relationships: Forbes's theory revisited. K D Hall, Brit J Nutr. 97Hall KD (2007) Body fat and fat-free mass inter-relationships: Forbes's theory revisited. Brit J Nutr 97: 1059-1063.
Nonlinear dynamics and chaos: With applications to physics, biology, chemistry and engineering. S H Strogatz, Perseus BooksCambridge, MassachusettsStrogatz SH (2000) Nonlinear dynamics and chaos: With applications to physics, biology, chemistry and engineering. Cambridge, Massachusetts: Perseus Books.
Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. J Guckenheimer, Philip Holmes, SpringerNew YorkGuckenheimer J, Holmes, Philip (1997) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer.
Body composition as a determinant of energy expenditure: a synthetic review and a proposed general prediction equation. J J Cunningham, Am J Clin Nutr. 54Cunningham JJ (1991) Body composition as a determinant of energy expenditure: a synthetic review and a proposed general prediction equation. Am J Clin Nutr 54: 963-969.
Obesity as an adaptation to a high-fat diet: evidence from a cross-sectional study. A Astrup, B Buemann, P Western, S Toubro, A Raben, Am J Clin Nutr. 59Astrup A, Buemann B, Western P, Toubro S, Raben A, et al. (1994) Obesity as an adaptation to a high-fat diet: evidence from a cross-sectional study. Am J Clin Nutr 59: 350-355.
24-hour energy expenditure and sympathetic activity in postobese women consuming a highcarbohydrate diet. A Astrup, B Buemann, N J Christensen, J Madsen, Am J Physiol. 262Astrup A, Buemann B, Christensen NJ, Madsen J (1992) 24-hour energy expenditure and sympathetic activity in postobese women consuming a high- carbohydrate diet. Am J Physiol 262: E282-E288.
Relationship of resting metabolic rate to body composition and protein turnover. S Welle, K S Nair, Am J Physiol Endocrinol Metab. 258Welle S, Nair KS (1990) Relationship of resting metabolic rate to body composition and protein turnover. Am J Physiol Endocrinol Metab 258: E990-E998.
Age and aerobic exercise training effects on whole body and muscle protein metabolism. K R Short, J L Vittone, M L Bigelow, D N Proctor, K S Nair, Am J Physiol Endocrinol Metab. 286Short KR, Vittone JL, Bigelow ML, Proctor DN, Nair KS (2004) Age and aerobic exercise training effects on whole body and muscle protein metabolism. Am J Physiol Endocrinol Metab 286: E92-E101.
Twenty-four-hour energy expenditure and substrate utilization in body builders. I Bosselaers, B Buemann, O J Victor, A Astrup, Am J Clin Nutr. 59Bosselaers I, Buemann B, Victor OJ, Astrup A (1994) Twenty-four-hour energy expenditure and substrate utilization in body builders. Am J Clin Nutr 59: 10-12.
Viruses as an etiology of obesity. R Atkinson, Mayo Clin Proc. 82Atkinson R (2007) Viruses as an etiology of obesity. Mayo Clin Proc 82: 1192-1198.
Human adenovirus ad-36 induces adiposity, up-regulates genes of adipocyte differentiation, and increases insulin sensitivity in rats. M Pasarica, M Yu, H Yang, M Mahida, S Mohankumar, Obesity. 14Pasarica M, Yu M, Yang H, Mahida M, MohanKumar S, et al. (2006) Human adenovirus ad-36 induces adiposity, up-regulates genes of adipocyte differenti- ation, and increases insulin sensitivity in rats. Obesity 14: 1905-1913.
Adipogenic human adenovirus-36 reduces leptin expression and secretion and increases glucose uptake by fat cells. S Vangipuram, M Yu, J Tian, K Stanhope, M P , Int J Obes (Lond). 31Vangipuram S, Yu M, Tian J, Stanhope K, M P, et al. (2007) Adipogenic human adenovirus-36 reduces leptin expression and secretion and increases glucose uptake by fat cells. Int J Obes (Lond) 31: 87-96.
Withdrawal of long-term physiological growth hormone (gh) administration: Differential effects on bone density and body composition in men with adultonset gh deficiency. Bmk Biller, G Sesmilo, Hba Baum, D Hayden, D Schoenfeld, J Clin Endotrinol Metab. 85Biller BMK, Sesmilo G, Baum HBA, Hayden D, Schoenfeld D, et al. (2000) Withdrawal of long-term physiological growth hormone (gh) administration: Differential effects on bone density and body composition in men with adult- onset gh deficiency. J Clin Endotrinol Metab 85: 970-976.
Changes in energy expenditure resulting from altered body weight. R L Leibel, M Rosenbaum, J Hirsch, N Engl J Med. 332Leibel RL, Rosenbaum M, Hirsch J (1995) Changes in energy expenditure resulting from altered body weight. N Engl J Med 332: 621-628.
| [] |
[
"U(N) and holomorphic methods for LQG and Spin Foams U(N) and holomorphic methods for LQG and Spin Foams",
"U(N) and holomorphic methods for LQG and Spin Foams U(N) and holomorphic methods for LQG and Spin Foams"
] | [
"Enrique F Borja [email protected] ",
"Jacobo Diaz-Polo [email protected] ",
"Iñaki Garay ",
"\nInstitute for Theoretical Physics III\nDepartamento de Física Teórica and IFIC\nCentro Mixto\nUniversity of Erlangen-Nürnberg\nStaudtstraße 7D-91058Erlangen (Germany\n",
"\nFacultad de Física\nDepartment of Physics and Astronomy\nUniversidad de Valencia-CSIC\nUniversidad de Valencia\nBurjassot-46100ValenciaSpain\n",
"\nInstitute for Theoretical Physics III\nLouisiana State University Baton Rouge\n70803-4001LA\n",
"\n3rd Quantum Gravity and Quantum Geometry School\nUniversity of Erlangen-Nürnberg\nStaudtstraße 7D-91058Erlangen, Zakopane(Germany)., Poland\n"
] | [
"Institute for Theoretical Physics III\nDepartamento de Física Teórica and IFIC\nCentro Mixto\nUniversity of Erlangen-Nürnberg\nStaudtstraße 7D-91058Erlangen (Germany",
"Facultad de Física\nDepartment of Physics and Astronomy\nUniversidad de Valencia-CSIC\nUniversidad de Valencia\nBurjassot-46100ValenciaSpain",
"Institute for Theoretical Physics III\nLouisiana State University Baton Rouge\n70803-4001LA",
"3rd Quantum Gravity and Quantum Geometry School\nUniversity of Erlangen-Nürnberg\nStaudtstraße 7D-91058Erlangen, Zakopane(Germany)., Poland"
] | [] | The U(N) framework and the spinor representation for loop quantum gravity are two new points of view that can help us deal with the most fundamental problems of the theory. Here, we review the detailed construction of the U(N) framework explaining how one can endow the Hilbert space of N-leg intertwiners with a Fock structure. We then give a description of the classical phase space corresponding to this system in terms of the spinors, and we will study its quantization using holomorphic techniques. We take special care in constructing the usual holonomy operators of LQG in terms of spinors, and in the description of the Hilbert space of LQG with the different polarization given by these spinors. | 10.22323/1.140.0024 | [
"https://arxiv.org/pdf/1110.4578v1.pdf"
] | 117,772,834 | 1110.4578 | 303609d501cba3a42f6d24f39a4f80719a34ac41 |
U(N) and holomorphic methods for LQG and Spin Foams U(N) and holomorphic methods for LQG and Spin Foams
20 Oct 2011 February 28 -March 13, 2011
Enrique F Borja [email protected]
Jacobo Diaz-Polo [email protected]
Iñaki Garay
Institute for Theoretical Physics III
Departamento de Física Teórica and IFIC
Centro Mixto
University of Erlangen-Nürnberg
Staudtstraße 7D-91058Erlangen (Germany
Facultad de Física
Department of Physics and Astronomy
Universidad de Valencia-CSIC
Universidad de Valencia
Burjassot-46100ValenciaSpain
Institute for Theoretical Physics III
Louisiana State University Baton Rouge
70803-4001LA
3rd Quantum Gravity and Quantum Geometry School
University of Erlangen-Nürnberg
Staudtstraße 7D-91058Erlangen, Zakopane(Germany)., Poland
U(N) and holomorphic methods for LQG and Spin Foams U(N) and holomorphic methods for LQG and Spin Foams
20 Oct 2011 February 28 -March 13, 2011
The U(N) framework and the spinor representation for loop quantum gravity are two new points of view that can help us deal with the most fundamental problems of the theory. Here, we review the detailed construction of the U(N) framework explaining how one can endow the Hilbert space of N-leg intertwiners with a Fock structure. We then give a description of the classical phase space corresponding to this system in terms of the spinors, and we will study its quantization using holomorphic techniques. We take special care in constructing the usual holonomy operators of LQG in terms of spinors, and in the description of the Hilbert space of LQG with the different polarization given by these spinors.
Introduction
Over the last few years, a new set of tools for approaching some of the fundamental open problems in loop quantum gravity (LQG) has been developed. Since the identification in [1,2] of a characteristic U(N) symmetry in the Hilbert space of SU(2) intertwiners, a new framework has emerged, producing some exciting results, but most importantly, opening a brand new avenue to look at fundamental issues like the dynamics of the theory or the identification of symmetries at a purely quantum level. The U(N) framework can also be derived as the holomorphic quantization of a classical spinor system, providing, among other interesting insights, a novel way to study the semiclassical limit of the theory. In this article we present a short but self-contained pedagogical introduction to the core formulation and techniques that constitute the new U(N) framework for loop quantum gravity, including the related spinor formulation and holomorphic quantization methods.
The organization of the paper is as follows. In section 2 we present the so-called U(N) framework for loop quantum gravity. We introduce a spinorial representation in section 3 and proceed to its quantization using holomorphic methods in section 4, providing a full analogy with the U(N) framework. In section 5 we discuss an action principle for the classical spinor system. Finally, in section 6 we present a short review of the "state of the art" on the holomorphic methods based on the work by Livine and Tambornino [3], giving strong formal consistency to the techniques presented here. We conclude summarizing the main features of the presented framework.
The U(N) framework for intertwiners
Our starting point to introduce the U(N) framework will be to study the structure of the Hilbert space of SU (2) intertwiners, the building blocks of spin network states. This is, given a set of SU (2) representations (spins), the space of invariant tensors
H (J) N ≡ ∑ i j i =J H j 1 ,..,j N .
As shown in [2], intertwiner spaces H (J) N carry irreducible representations of U(N) and the full space H N can be endowed with a Fock space structure with creation and annihilation operators compatible with the U(N) action [4]. This structure is at the foundation of the U(N) techniques, and we will review its basic construction in what follows.
We start by introducing the well-known Schwinger representation of the su(2) algebra. This representation describes the generators of su (2) in terms of a pair of uncoupled harmonic oscillators. In our case, we introduce 2N oscillators -a pair for each leg of the intertwiner-with creation operators a i , b i , i running from 1 to N:
[a i , a † j ] = [b i , b † j ] = δ ij , [a i , b j ] = 0.
The local su(2) generators at each leg of the intertwiner can be expressed then as quadratic operators:
J z i = 1 2 (a † i a i − b † i b i ), J + i = a † i b i , J − i = a i b † i , E i = (a † i a i + b † i b i ). (2.1)
As expected, the J i 's constructed this way satisfy the standard commutation relations of the su(2) algebra while the total energy E i is a Casimir operator:
[J z i , J ± i ] = ±J ± i , [J + i , J − i ] = 2J z i , [E i , J i ] = 0.
The operator E i is the total energy carried by the pair of oscillators a i , b i and its eigenvalue is twice the spin 2j i of the corresponding SU(2) representation. We can express the standard SU(2) Casimir operator in terms of this energy as:
J 2 i = E i 2 E i 2 + 1 = E i 4 (E i + 2) .
As it is well-known, in the context of LQG the spin j i is related to the area associated with the i-th leg of the intertwiner. In this particular framework, the most natural choice for the regularization of the area operator is such that the spectrum is given directly by the Casimir E i /2 (the spin j i ); we will consider that case in this paper.
The key observation now is that one can use these harmonic oscillators to construct operators acting on the Hilbert space of intertwiners, i.e., operators that are invariant under global SU(2) transformations generated by J ≡ ∑ i J i . These constitute the starting point of the U(N) formalism, and they are quadratic invariant operators acting on pairs of (possibly equal) legs i, j [1,2]:
E ij = a † i a j + b † i b j , E † ij = E ji , F ij = (a i b j − a j b i ), F ji = −F ij .
The operators E, F, F † form a closed algebra:
[E ij , E kl ] = δ jk E il − δ il E kj , (2.2) [E ij , F kl ] = δ il F jk − δ ik F jl , [E ij , F † kl ] = δ jk F † il − δ jl F † ik , [F ij , F † kl ] = δ ik E lj − δ il E kj − δ jk E li + δ jl E ki + 2(δ ik δ jl − δ il δ jk ), [F ij , F kl ] = 0, [F † ij , F † kl ] = 0.
We can see that commutators of E ij operators have the structure of a u(N)-algebra (which motivates the name of the U(N) framework). The diagonal operators are precisely the operators giving the energy on each leg, E ii = E i . Then the value of the total energy E ≡ ∑ i E i gives twice the sum of all spins 2 × ∑ i j i , i.e. twice the total area.
The E ij -operators change the energy/area carried by each leg, while still conserving the total energy, while the operators F ij (resp. F † ij ) decrease (resp. increase) the total area E by 2:
[E, E ij ] = 0, [E, F ij ] = −2F ij , [E, F † ij ] = +2F † ij .
This suggests that we can decompose the Hilbert space of N-valent intertwiners into subspaces of constant area:
H N = {j i } Inv ⊗ N i=1 V j i = J∈N ∑ i j i =J Inv ⊗ N i=1 V j i = J H (J) N ,
where as before V j i denote the Hilbert space of the irreducible SU(2)-representation of spin j i , spanned by the states of the oscillators a i , b i with fixed total energy E i = 2j i . In [2], the structure of these subspaces H (J) N of N-valent intertwiners with fixed total area J was studied, identifying the irreducible representations of U(N), generated by the E ij operators, that they naturally carry. Then the operators E ij allow to navigate from state to state within each subspace H Finally, it was also found that the whole set of operators E ij , F ij , F † ij satisfy the following quadratic constraints [5]:
∀i, j, ∑ k E ik E kj = E ij E 2 + N − 2 , (2.3) ∑ k F † ik E jk = F † ij E 2 , ∑ k E jk F † ik = F † ij E 2 + N − 1 , (2.4) ∑ k E kj F ik = F ij E 2 − 1 , ∑ k F ik E kj = F ij E 2 + N − 2 , (2.5) ∑ k F † ik F kj = E ij E 2 + 1 , ∑ k F kj F † ik = (E ij + 2δ ij ) E 2 + N − 1 . (2.6)
As noticed in [5] and further extended in [6], these relations have the structure of constraints on the multiplication of two matrices E ij and F ij . This is one of the main hints that will lead to the derivation of the U(N) framework as a quantization of a classical matrix model, as we are going to see in following sections.
Classical spinor formalism
A very interesting feature of the new U(N)-framework is that it can be re-derived in terms of spinors in a rather straightforward way [3,6]. The operators in the U(N)formalism can be shown to arise as the quantization of classical spinor matrices. This connection can help understand the geometrical meaning of spin network states in LQG, as well as provide hints on the semi-classical limit of the full theory. There is also a connection with the so-called "twisted geometries" [7,8] that express the classical phase space of loop quantum gravity on a given graph as a classical spinor model, unravelling the relation between spin networks and discrete geometry. This could provide new ideas on the study of spin network dynamics through the spinfoam approach.
We are going to present the basic concepts that lead to recast the U(N)-framework in terms of spinors, showing how this is related to standard SU(2) intertwiners in LQG. Let us start by introducing the spinor notation that we will use [3,4,6,8,10]. Let z be a spinor
|z = z 0 z 1 , z| = z 0z1 .
We can associate it to a geometrical 3-vector X(z), defined from the projection of the 2 × 2 matrix |z z| onto Pauli matrices σ a (taken Hermitian and normalized so that (σ a ) 2 = I):
|z z| = 1 2 z|z I + X(z) · σ . (3.1)
It is straightforward to compute the norm and the components of this vector in terms of the spinors:
| X(z)| = z|z = |z 0 | 2 + |z 1 | 2 , X z = |z 0 | 2 − |z 1 | 2 , X x = 2 ℜ (z 0 z 1 ), X y = 2 ℑ (z 0 z 1 ).
With this, the spinor z is entirely determined by the corresponding 3-vector X(z) up to a global phase. We can give the reverse map:
z 0 = e iφ | X| + X z 2 , z 1 = e i(φ−θ) | X| − X z 2 , tan θ = X y X x ,
where e iφ is an arbitrary phase. Let us also introduce now the duality map ς acting on spinors:
ς z 0 z 1 = −z 1 z 0 , ς 2 = −1.
This is an anti-unitary map, ςz|ςw = w|z = z|w , and we will write the related state
as |z] ≡ ς|z , [z|w] = z|w .
This map ς maps the 3-vector X(z) onto its opposite:
|z][z| = 1 2 z|z I − X(z) · σ .
In order now to describe N-valent intertwiners, we consider N spinors z i and their corresponding 3-vectors X(z i ). A standard requirement is to ask the N spinors to satisfy a closure condition, i.e., that the sum of the corresponding 3-vectors vanish, ∑ i X(z i ) = 0.
Recalling the definition of X(z i ), this closure condition can be expressed in terms of 2 × 2 matrices:
∑ i |z i z i | = A(z)I, with A(z) ≡ 1 2 ∑ i z i |z i = 1 2 ∑ i | X(z i )|. (3.2)
This further translates into quadratic constraints on the spinors:
∑ i z 0 iz 1 i = 0, ∑ i z 0 i 2 = ∑ i z 1 i 2 = A(z). (3.3)
In simple terms, this means that the two components of the spinors, z 0 i and z 1 i , form orthogonal N-vectors of equal norm. In order to simplify the notation, let us introduce the matrix elements of the 2 × 2 matrix ∑ i |z i z i | :
C ab = ∑ i z a iz b i .
Then the closure constraints are written very simply:
C 00 − C 11 = 0, C 01 = C 10 = 0.
(Anti-)holomorphic quantization
Let us construct the classical phase space in terms of spinors. We will then proceed to its quantization, following [3,6].
We start by postulating simple Poisson bracket relations for a set of N spinors:
{z a i ,z b j } ≡ i δ ab δ ij , (4.1) with all other brackets vanishing, {z a i , z b j } = {z a i ,z b j } = 0.
They exactly reproduce the Poisson bracket structure of 2N uncoupled harmonic oscillators.
Our expectation now is to have the closure constraints generating global SU(2) transformations on the N spinors. Let us then compute the Poisson brackets between components of the C-constraints :
{C 00 − C 11 , C 01 } = −2iC 01 , {C 00 − C 11 , C 10 } = +2iC 10 , {C 10 , C 01 } = i(C 00 − C 11 ), (4.2) {TrC, C 00 − C 11 } = {TrC, C 01 } = {TrC, C 10 } = 0.
These four components C ab do indeed form a closed u(2) algebra, with the three closure conditions C 00 − C 11 , C 01 and C 10 being generators of a su(2) subalgebra. We will denote as C these three su(2)-generators, with C z ≡ C 00 − C 11 and C + = C 10 and C − = C 01 . One can already guess that these three closure conditions C will be related to the SU(2) generators J at the quantum level, while the operator Tr C will correspond to the total energy/area E.
In correspondence to operators E and F, let us introduce matrices M and Q satisfying M = M † , t Q = −Q and the classical analogs to the quadratic constraints (2.3-2.6). Up to a global phase, these matrices can be written with generality as:
M = λ U∆U −1 , ∆ = 1 1 0 N−2 , Q = λ U∆ ǫ t U, ∆ ǫ = 1 −1 0 N−2 ,
where U is a unitary matrix U † U = I. Then, if we define spinors such that
z i ≡ ū i1 √ λ u i2 √ λ , λ ≡ TrM/2, (4.3)
u ij being the elements of the unitary matrix U, we can write the components of M and Q as
M ij = z i |z j = z j |z i , Q ij = z j |z i ] = [z i |z j = −[z j |z i . (4.4)
One can see that, in this setting, the unitarity condition on the matrices U is equivalent to the closure conditions on the spinors.
If we now compute the Poisson brackets of the M ij and Q ij matrix elements:
{M ij , M kl } = i(δ kj M il − δ il M kj ), {M ij , Q kl } = i(δ jk Q il − δ jl Q ik ), {Q ij , Q kl } = 0, {Q ij , Q kl } = i(δ ik M lj + δ jl M ki − δ jk M li − δ il M kj ),
we observe that they reproduce the expected commutators (2.2) up to the i-factor. We further check that these variables commute with the closure constraints generating the SU(2) transformations:
{ C, M ij } = { C, Q ij } = 0. (4.5)
Finally, one can also compute their commutator with Tr C:
{Tr C, M ij } = 0, {Tr C, Q ij } = { ∑ k M kk , Q ij } = +2i Q ij ,
which confirms that the matrix M is invariant under the full U(2) subgroup and that Tr C acts as a dilatation operator on the Q variables, or reversely that the Q ij acts as creation operators for the total energy/area variable Tr C.
We have already characterized the classical phase space associated to spinors z i and matrices M ij , Q ij . Let us now proceed to its quantization. In order to do so, we will consider Hilbert spaces H (Q) J of homogeneous polynomials in Q ij of degree J: and show that they match the action of the U(N) operators E ij and F † ij described earlier. We choose to quantizez i as multiplication operators while promoting z i to derivative operators:
H (Q) J ≡ {P ∈ P[Q ij ] | P(ρQ ij ) = ρ J P(Q ij ), ∀ρ ∈ C} .z a i ≡z a i × , z a i ≡ ∂ ∂z a i ,(4.7)
which satisfy the commutator [ẑ,ẑ] = 1 as expected for the quantization of the classical bracket {z,z} = i. We can then quantize the matrix elements M ij and Q ij and the closure constraints following this correspondence:
M ij =z 0 i ∂ ∂z 0 j +z 1 i ∂ ∂z 1 j , Q ij =z 0 iz 1 j −z 1 iz 0 j = Q ij , Q ij = ∂ 2 ∂z 0 i ∂z 1 j − ∂ 2 ∂z 1 i ∂z 0 j , C ab = ∑ kz b k ∂ ∂z a k .
It is straightforward to check that the C ab and the M ij respectively form a u(2) and a u(N) Lie algebra, as expected:
[ C ab , C cd ] = δ ad C cb − δ cb C ad , [ M ij , M kl ] = δ kj M il − δ il M kj , [ C ab , M ij ] = 0. (4.8)
which amounts to multiply the Poisson bracket 4.2 and 4.5 by −i. Let us analyze the structure resulting from this quantization. First, by checking the action of the closure constraints on functions of the variables Q ij :
C Q ij = 0, (Tr C) Q ij = 2Q ij , ∀P ∈ H (Q) J = P J [Q ij ], C P(Q ij ) = 0, (Tr C) P(Q ij ) = 2J P(Q ij ),∑ ik M ik M ki = (Tr C) (Tr C) 2 + N − 2 = 2J(J + N − 2).
Therefore, we can conclude that the quantization of our spinors and M-variables exactly matches the u(N)-structure on the intertwiner space (with the exact same ordering):
H (Q) J ∼ H (J) N , M ij = E ij , (Tr C) = E. (4.10)
In the third place, turning to the Q ij -operators, it is straightforward to check that they have the exact same action that the F † ij operators, they satisfy the same Lie algebra commutators (2.2) and the same quadratic constraints (2.4-2.6). Clearly, the simple multiplicative action of an operator Q ij sends a polynomial in P J [Q ij ] to a polynomial in P J+1 [Q ij ]. Reciprocally, the derivative action of Q ij decreases the degree of the polynomials and maps
P J+1 [Q ij ] onto P J [Q ij ].
Finally, let us consider the scalar product on the space of polynomials P[Q ij ]. There seems to be a unique measure (up to a global factor) compatible with the correct Hermiticity relations for M ij and Q ij , Q ij . It is given by:
∀φ, ψ ∈ P[Q ij ], φ|ψ ≡ ∏ i d 4 z i e − ∑ i z i |z i φ(Q ij ) ψ(Q ij ) . (4.11)
Then it is easy to check that we have M † ij = M ji and Q † ij = Q ij as desired. The spaces of homogeneous polynomials P J [Q ij ] are orthogonal with respect to this scalar product. The quickest way to realize that is to consider the operator (Tr C), which is Hermitian with respect to this scalar product and takes different values on spaces P J [Q ij ] for different values of J. Thus these spaces P J [Q ij ] are orthogonal to each other.
This concludes the quantization procedure, showing that the intertwiner space for N legs and fixed total area J = ∑ i j i can be constructed as the space of homogeneous polynomials in Q ij of degree J. We obtain a description of intertwiners as anti-holomorphic wave-functions of spinors z i constrained by the closure conditions 1 .
Classical action and effective dynamics
Once we have constructed the Hilbert space of a single intertwiner, and in order to make contact with the standard spin network formulation of loop quantum gravity, we want to apply this framework to graphs. Therefore, we will have several intertwiners glued together according to the corresponding graph structure. We will then write an action principle in terms of spinors, compatible with the Poisson bracket structure (4.1). In order to do so, we need to take into account the closure constraints, but also the matching conditions coming from the gluing of intertwiners.
The key element to elaborate the connection between the spinor formalism and the standard formulation of LQG is the reconstruction of the SU(2) group element g e (the holonomy) associated to an edge in terms of the spinors [8]. Let us consider an edge e with a spinor at each of its end-vertices z s(e),e and z t(e),e . There exists a unique SU(2) group element mapping one onto the other. More precisely, the requirement that The group elements g e (z s(e),e , z t(e),e ) ∈ SU(2) commute with the matching conditions, ensuring that the energy of the oscillators on the edge e is the same at both vertices s(e) and t(e). However, they are obviously not invariant under SU(2) transformations. As we know from loop quantum gravity, in order to construct SU(2)-invariant observables, we need to consider the trace of holonomies around closed loops, i.e., the oriented product of group elements g e along closed loops L on the graph:
G L ≡ − → ∏ e∈L g e .
For the sake of simplicity, let us assume that all the edges of the loop are oriented the same way, so we can number the edges e 1 , e 2 , ..e n with v 1 = t(e n ) = s(e 1 ), v 2 = t(e 1 ) = s(e 2 ), etc (see figure 1). Then, we can explicitly write the holonomy G L in terms of the spinors:
Tr G L = Trg(e 1 ) . . . g(e n ) = Tr ∏ i |z v i ,e i ] z v i+1 ,e i | − |z v i ,e i [z v i+1 ,e i | ∏ i z v i ,e i |z v i ,e i z v i+1 ,e i |z v i+1 ,e i .
Or, if instead of factorizing this expression by edges, we group the terms by vertices, we obtain:
Tr G L = ∑ r i =0,1 (−1) ∑ i r i ∏ i ς r i−1 z v i ,e i−1 | ς 1−r i z v i ,e i ∏ i z v i ,e i−1 |z v i ,e i−1 z v i ,e i |z v i ,e i ,
where the ς r i records whether we have the term |z v i ,e i ] z v i+1 ,e i | or |z v i ,e i [z v i+1 ,e i | on the edge e i , with r i = 0, 1 (recall that ς is the anti-unitary map sending a spinor |z to each dual |z]).
Depending on the specific values of the r i parameters, the scalar products at the numerators are given by the matrix elements of M i or Q i at the vertex i. Since these matrices are by definition SU(2)-invariant (they commute with the closure conditions), this is a consistency check, a posteriori, that the holonomy Tr G L correctly provides a SU(2)-invariant observable. Taking into account the various possibilities for the signs (−1) r i , we can write the holonomy
Tr G L = ∑ r i =0,1 (−1) ∑ i r i M {r i } L ∏ i z v i ,e i |z v i+1 ,e i ,(5.M {r i } L ≡ ∏ i r i−1 r iQ i i,i−1 + (1 − r i−1 )r i M i i−1,i + r i−1 (1 − r i )M i i,i−1 + (1 − r i−1 )(1 − r i )Q i i,i−1 = ∏ i ς r i−1 z v i ,e i−1 | ς 1−r i z v i ,e i .
Using these ingredients, we are going to write an action principle for this formalism. For this, we have to keep in mind the graph structure, with intertwiners at the vertices, glued together along edges. The phase space therefore consists of spinors z v,e (where e are edges attached to the vertex v, i.e., such that v = s(e) or v = t(e)) which we constrain by the closure conditions C v at each vertex v and the matching conditions on each edge e. The corresponding action thus reads: where the 2 × 2 Lagrange multipliers Λ v , satisfying Tr Λ v = 0, impose the closure constraints and the Lagrange multipliers ρ e ∈ R impose the matching conditions. All the constraints are first class, they generate SU(2) transformations at each vertex and U(1) transformations on each edge e. Analogously, this system can be parameterized in terms of N v × N v unitary matrices U v and the parameters λ v . The matrix elements U v e f refer to pairs of edges e, f attached to the vertex v. As mentioned before, the closure constraints are automatically encoded in the requirement of unitarity for U v . It remains, then, to impose the matching conditions
S Γ 0 [z v,e ] = dt ∑ v ∑ e|v∈∂e (−i z v,e |∂ t z v,e + z v,e |Λ v |z v,e )+ ∑v = λ v U v ∆(U v ) −1 being functions of both λ v and U v )
. So in this case, the action reads:
S Γ 0 [λ v , U v ] = dt ∑ v −i λ v Tr U v ∆∂ t U v † − TrΘ v (U v U v † − I) + ∑ e ρ e (M s(e) ee − M t(e) ee ),
where the ρ e impose the matching conditions while the N v × N v matrices Θ v are the Lagrange multipliers for the unitarity of the matrices U v .
This free action describes the classical kinematics of spin networks on the graph Γ. We can now add interaction terms to this action. Such interaction terms should be built with generalized holonomy observables M {r i } L , thus trivially satisfying the closure and matching conditions. A proposal for a classical action for spin networks with non-trivial dynamics was made in [6]:
S Γ γ {r i } L = S Γ 0 + dt ∑ L,{r i } γ {r i } L M {r i } L ,
where the γ {r i } L are the coupling constants giving the relative weight of each generalized holonomy in the full Hamiltonian. This action was then applied in particular to a specific model based on a 2-vertex graph. We refer the reader to the original paper for more details on the analysis of the dynamics derived from this Hamiltonian.
Spinor representation and Holomorphic methods in LQG
There is an interesting twist in the utility of this spinorial framework for LQG [3]. These methods are extremely powerful in order to understand the geometrical interpretation of the spin network states. The single-edge Hilbert space in LQG is given by H e := L 2 (SU(2), dg), i.e., the space of square integrable functions over the group SU(2). This Hilbert space can be obtained as the quantization of a cotangent bundle over SU (2) acting as the classical phase space (T * SU(2) ≈ SU(2) × su (2)). At this point, instead of the usual coordinatization of this classical phase space given by (g, X) -a group and a Lie algebra element respectively-, a pair of C 2 -spinors (|z , |z ) recovering the classical physics in T * SU(2) can be employed.
As we have already seen before it is possible to obtain the SU(2) group elements associated to such a pair of spinors. We are going to review that discussion simplifying the notation and taking a slightly different choice of mapping between the spinors and the group element and its inverse 2 . We start with spinors |z and |z , living at the source and the target vertices of a given edge, then we can write the corresponding SU(2)-element and its inverse as follows:
g(z,z) = |z [z| − |z] z| z z , g −1 (z,z) = −|z [z| + |z] z| z z .
It is straightforward to show that, under a local SU(2)-transformation h ∈ SU(2) in the defining representation of the group, g(z,z) transforms as the holonomy of a SU(2)connection:
g(z,z) −→ h 1 g(z,z)h −1 2 .
Moreover, each spinor is mapped to R 3 (up to a phase) yielding area vectors corresponding to the different faces of an elementary polyhedron. In this sense, the Hilbert space for a graph with E edges can be regarded as the space of glued polyhedra covering a piecewise flat manifold (this is ensured by the imposition of the proper matching conditions). Following this line of thought, it is easy to show that the su(2)-elements X(z) := X(z) · σ andX(z) := X (z) · σ, associated with the normals to the glued faces of two polyhedra, are related through the expression:
X = g −1 Xg.
On the other hand, we can endow the space C 2 × C 2 with a symplectic structure given by {z i , z j } = iδ j i = {z i ,z j } , thus obtaining a proper phase space. This phase space has to be equipped with a constraint forcing the two spinors to have the same length (matching condition). This constraint is given by z 2 − z 2 and it can be noticed that it generates U(1) transformations. Nevertheless, the group and Lie-algebra elements are invariant under these transformations by construction. The interesting point here is that, using this symmetry, one can arrive to the cotangent bundle of SU(2) by means of a U(1)-gauge reduction:
C 2 × C 2 //U(1)/Z 2 ≃ T * SU(2) \ {|X| = 0} ,
where // denotes a double quotient (see [3] for details). This space has indeed the correct symplectic structure -that arises in terms of g and Xwritten in terms of spinors. In this sense, given an edge of a graph, one can either assign it a pair (g, X) or a pair of spinors (|z , |z ). This way, the degrees of freedom are shifted from the edge to its vertices.
Once we are convinced that this correspondence between spinors and the SU(2)group and algebra elements works, we can also find the expression of the Haar measure dg in terms of spinors. The result is really simple, the Haar measure is just the product of two Gaussian measures
SU(2) dg = C 2 ×C 2 dµ(z)dµ(z).
In order to arrive to this result one has to be careful with the redundancies introduced by the spinors (8 degrees of freedom instead of the 3 in a group element g) which means that one has to make use of a twisted rotation (as defined in [3]) in order to implement a SU(2)-transformation leaving the group element invariant. Taking this into account the former relation can be proven by computing the scalar product between two representation matrices of SU(2) in terms of spinors. We refer the interested reader to the original paper for more details.
The procedure presented here provides an alternative quantization for T * SU(2), where the Hilbert space for an edge e, H spin e is obtained after an appropriate gauge-reduction, implementing the proper matching conditions on that edge, of the Bargmann space of holomorphic square-integrable functions on both spinors. This is, in some sense, an opposite way of thinking with respect to the usual picture, where the group and Lie algebra elements are taken as the fundamental variables. Here, the spinorial variables are considered as fundamental and the group and Lie algebra elements arise as composite objects. We can also build ladder operators in terms of spinors and derive the standard holonomy and flux-operators of the theory.
We are considering two quantizations of the same classical phase space, resulting from the use of two different polarizations, and we arrive at the usual LQG Hilbert space H e with the standard variables or to H spin e in terms of spinors. The relevant question then is, are those Hilbert spaces unitarily equivalent? The answer is in the affirmative. Indeed, one can construct a unitary map between those Hilbert spaces T : H e → H spin e employing a method based on a modification of the Segal-Bargmann transform. This map sends SU(2) representation matrices to holomorphic functions in (|z , |z ) and it can be regarded as the restriction to the holomorphic part of the group element written in terms of spinors (see [3] for details). Furthermore, this map can be straightforwardly generalized to an arbitrary graph and can be shown to be compatible with the inductive limit construction used to define the Hilbert space of the continuum theory. Therefore, there is a strong formal consistency for this reformulation of LQG, which captures the same physics while opening new ways to tackle the open problems in the theory, such as the lack of a well defined semiclassical limit or the geometric interpretation of the spin networks states.
Conclusions
We have reviewed here the U(N) and spinorial framework for LQG. This new point of view has become a very useful tool to deal with certain fundamental problems in LQG. In particular, it was interesting for the study of a simple model (the 2-vertex model) with a pair of nodes and an arbitrary number of links [5,6,9]. In these papers, a plausible dynamics was proposed for this system (at the quantum and classical level) and some striking mathematical analogies with loop quantum cosmology were explored. Also, using this framework, it was possible to study semiclassical states [4] and the simplicity constraints [10].
Let us revisit the main points presented in this review:
• Making use of the Schwinger representation, it is possible to write SU(2) invariant operators acting on the Hilbert space of intertwiners in LQG [1].
• A key point for the U(N) construction is that the LQG Hilbert space of intertwiners with N-legs and fixed total area J carries an irreducible U(N) representation [2].
• One can show that the full space of N-leg intertwiners is endowed with a Fock space structure, with annihilation and creation operators given by the operators F ij and F † ij .
• We have presented a classical framework based on spinors whose quantization (using holomorphic methods) leads back to the U(N) framework for the Hilbert space of intertwiners in LQG [6].
• An action principle for the spinor phase space has been described. Then, using the relation given in [8] between spinors and SU(2) group elements, an expression for the holonomy operators was built. Taking advantage of these SU(2) and U(1) invariant operators, it was possible to give a generic interaction term (whose quantization is direct) for this framework.
• Finally, we gave a glimpse of the formal and rigorous construction of the spinor representation and holomorphic quantization techniques for the whole LQG kinematical Hilbert space, as it was developed by Livine and Tambornino [3]. There, it was shown that although the choice of the spinors as fundamental variables of the theory implies a different choice of polarization (and in general different polarizations lead to inequivalent quantizations), in this case it is possible to construct explicitly a unitary map T between the usual LQG Hilbert space and the one derived in terms of the spinors.
Both the U(N) framework and the spinor representation are promising ways to deal with the most important open problems within LQG, namely, the semiclassical states and the dynamics. They may also become useful computational tools to calculate physical quantities within the theory, such as correlation functions. Besides, this new point of view may be useful in the context of group field theory. The possibility of having Feynman amplitudes expressed as integrals over spinor variables or the study of the renormalization properties of group field theories are interesting problems where the spinor representation can play a crucial role.
To conclude, we would like to remark that this framework certainly constitutes a viewpoint that opens new ways of research, both in loop quantum gravity and in group field theory and spin foams, that are worth exploring.
N
. On the other hand, operators F ij , F † ij allow to go from one subspace H(J) N to the next H (J±1) N , thus endowing the full space of N-valent intertwiners H N with a Fock space structure, with creation operators F † ij and annihilation operators F ij .
N-valent intertwiners with fixed total area J. To this purpose, we will construct the explicit representation of the operators resulting from the quantization of M ij and Q ij on H (Q) J
we see that wave functions P ∈ H(Q) J are SU(2)-invariant (vanish under the closure constraints) and are eigenvectors of the Tr C-operator with eigenvalue 2J. Second, operators M and (Tr C) acting on the Hilbert space H (Q) J (i.e., on SU(2)invariant functions vanishing under the closure constraints) satisfy the same quadratic constraints as the u(N)-generators E ij : (Tr C) = ∑ k M kk , ∑ k M ik M kj = M ij (Tr C) 2 + N − 2 . (4.9) This allows us to get the value of the (quadratic) U(N)-Casimir operator on the space H
(e),e |z t(e),e = |z s(e),e ] z s(e),e |z s(e),e , g e |z t(e),e ] z t(e),e |z t(e),e = − |z s(e),e z s(e),e |z s(e),e , g e ∈ SU(2), i.e., that g e map the (normalized) source spinor to the dual of the target spinor, uniquely fixes the value of g e to: g e ≡ |z s(e),e ] z t(e),e | − |z s(e),e [z t(e),e | z s(e),e |z s(e),e z t(e),e |z t(e),e
Figure 1 :
1The loop L = {e 1 , e 2 , .., e n } on the graph Γ.
e ρ e z s(e),e |z s(e),e − z t(e),e |z t(e),e ,
each edge e (the matrices M
An alternative construction[6], which can be considered as "dual" to the representation defined above, can also be carried out based on coherent states for the oscillators. This approach yields, upon quantization, the framework of the U(N) coherent intertwiner states introduced in[4] and further developed in[10].
In order to remain faithful to the paper by Livine and Tambornino[3], we adopt in this section the notation and the choice of the Poisson bracket structure adopted by them. The only difference with respect the one presented in section 4 is a sign in the Poisson bracket, thus switching from an antiholomorphic to a holomorphic prescription.
AcknowledgementsWe would like to thank Etera R. Livine for endless discussions. This work was in part supported by the Spanish MICINN research grants FIS2008-01980, FIS2009-11893 and ESP2007-66542-C04-01 and by the grant NSF-PHY-0968871. IG is supported by the Department of Education of the Basque Government under the "Formación de Investigadores" program.
Reconstructing quantum geometry from quantum information: Spin networks as harmonic oscillators. Florian Girelli, Etera R Livine, arXiv:gr-qc/0501075Class.Quant.Grav. 22Florian Girelli and Etera R. Livine. Reconstructing quantum geometry from quantum information: Spin networks as harmonic oscillators. Class.Quant.Grav. 22 (2005) 3295-3314 [arXiv:gr-qc/0501075].
The Fine Structure of SU(2) Intertwiners from U(N) Representations. Laurent Freidel, Etera R Livine, arXiv:0911.3553v1J.Math.Phys. 5182502Laurent Freidel and Etera R. Livine. The Fine Structure of SU(2) Intertwiners from U(N) Representations. J.Math.Phys. 51 (2010) 082502 [arXiv:0911.3553v1].
Spinor Representation for Loop Quantum Gravity. R Etera, Johannes Livine, Tambornino, arXiv:1105.3385v1Etera R. Livine and Johannes Tambornino. Spinor Representation for Loop Quantum Gravity. [arXiv:1105.3385v1].
U(N) Coherent States for Loop Quantum Gravity. Laurent Freidel, Etera R Livine, arXiv:1005.2090v1J.Math.Phys. 5252502Laurent Freidel and Etera R. Livine. U(N) Coherent States for Loop Quantum Gravity. J.Math.Phys. 52 (2011) 052502 [arXiv:1005.2090v1].
Dynamics for a 2-vertex Quantum Gravity Model. Enrique F Borja, Jacobo Diaz-Polo, Iñaki Garay, Etera R Livine, arXiv:1006.2451v2Class.Quant.Grav. 27235010Enrique F. Borja, Jacobo Diaz-Polo, Iñaki Garay and Etera R. Livine. Dynamics for a 2-vertex Quantum Gravity Model. Class.Quant.Grav. 27 (2010) 235010 [arXiv:1006.2451v2].
U(N) tools for Loop Quantum Gravity: The Return of the Spinor. Enrique F Borja, Laurent Freidel, Iñaki Garay, Etera R Livine, arXiv:1010.5451v1Class.Quant.Grav. 2855005Enrique F. Borja, Laurent Freidel, Iñaki Garay and Etera R. Livine. U(N) tools for Loop Quantum Gravity: The Return of the Spinor. Class.Quant.Grav. 28 (2011) 055005 [arXiv:1010.5451v1].
Twisted geometries: A geometric parametrisation of SU(2) phase space. Laurent Freidel, Simone Speziale, arXiv:1001.2748v3Phys. Rev. 8284040Laurent Freidel and Simone Speziale. Twisted geometries: A geometric parametrisation of SU(2) phase space. Phys. Rev. D82 (2010) 084040 [arXiv:1001.2748v3].
From twistors to twisted geometries. Laurent Freidel, Simone Speziale, arXiv:1006.0199v1Phys. Rev. 8284041Laurent Freidel and Simone Speziale. From twistors to twisted geometries. Phys. Rev. D82 (2010) 084041 [arXiv:1006.0199v1].
U(N) invariant dynamics for a simplified Loop Quantum Gravity model. Enrique F Borja, Jacobo Diaz-Polo, Iñaki Garay, Etera R Livine, arXiv:1012.3832v1Enrique F. Borja, Jacobo Diaz-Polo, Iñaki Garay and Etera R. Livine. U(N) invariant dynamics for a simplified Loop Quantum Gravity model. [arXiv:1012.3832v1].
Revisiting the Simplicity Constraints and Coherent Intertwiners. Maite Dupuis, Etera R Livine, arXiv:1006.5666v1Class.Quant.Grav. 2885001Maite Dupuis and Etera R. Livine. Revisiting the Simplicity Constraints and Coherent Intertwiners. Class.Quant.Grav. 28 (2011) 085001 [arXiv:1006.5666v1].
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[
"Hubble Space Telescope Imaging of Luminous Extragalactic Infrared Transients and Variables from the SPIRITS Survey *",
"Hubble Space Telescope Imaging of Luminous Extragalactic Infrared Transients and Variables from the SPIRITS Survey *"
] | [
"Howard E Bond \nDepartment of Astronomy & Astrophysics\nPennsylvania State University\n16802University ParkPAUSA\n\nSpace Telescope Science Institute\n3700 San Martin Dr21218BaltimoreMDUSA\n",
"Jacob E Jencson \nSteward Observatory\nUniversity of Arizona\n933 N Cherry Ave85721-0065TucsonAZUSA\n",
"Patricia A Whitelock \nSouth African Astronomical Observatory\nP.O. Box 97935ObservatorySouth Africa\n\nDepartment of Astronomy\nUniversity of Cape Town\nPrivate Bag X3, Rondebosch7701South Africa\n",
"Scott M Adams \nCahill Center for Astronomy & Astrophysics\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n\nPresent address: Orbital Insight\n1201 N Wilson Blvd., Suite 210022209ArlingtonVAUSA\n",
"John Bally \nCenter for Astrophysics & Space Astronomy\nAstrophysical & Planetary Sciences Department\nUniversity of Colorado\nUCB 38980309BoulderCOUSA\n",
"Ann Marie Cody \nSETI Institute\n339 Bernardo Ave., Suite 20094043Mountain ViewCAUSA\n",
"Robert D Gehrz \nSchool of Physics & Astronomy\nMinnesota Institute for Astrophysics\nUniversity of Minnesota\n116 Church St. SE55455MinneapolisMNUSA\n",
"Mansi M Kasliwal \nCahill Center for Astronomy & Astrophysics\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n",
"Frank J Masci \nIPAC\nCalifornia Institute of Technology\n1200 E California Blvd91125PasadenaCAUSA\n"
] | [
"Department of Astronomy & Astrophysics\nPennsylvania State University\n16802University ParkPAUSA",
"Space Telescope Science Institute\n3700 San Martin Dr21218BaltimoreMDUSA",
"Steward Observatory\nUniversity of Arizona\n933 N Cherry Ave85721-0065TucsonAZUSA",
"South African Astronomical Observatory\nP.O. Box 97935ObservatorySouth Africa",
"Department of Astronomy\nUniversity of Cape Town\nPrivate Bag X3, Rondebosch7701South Africa",
"Cahill Center for Astronomy & Astrophysics\nCalifornia Institute of Technology\n91125PasadenaCAUSA",
"Present address: Orbital Insight\n1201 N Wilson Blvd., Suite 210022209ArlingtonVAUSA",
"Center for Astrophysics & Space Astronomy\nAstrophysical & Planetary Sciences Department\nUniversity of Colorado\nUCB 38980309BoulderCOUSA",
"SETI Institute\n339 Bernardo Ave., Suite 20094043Mountain ViewCAUSA",
"School of Physics & Astronomy\nMinnesota Institute for Astrophysics\nUniversity of Minnesota\n116 Church St. SE55455MinneapolisMNUSA",
"Cahill Center for Astronomy & Astrophysics\nCalifornia Institute of Technology\n91125PasadenaCAUSA",
"IPAC\nCalifornia Institute of Technology\n1200 E California Blvd91125PasadenaCAUSA"
] | [] | SPIRITS-the SPitzer InfraRed Intensive Transients Survey-searched for luminous infrared (IR) transients and variables in nearly 200 nearby galaxies from 2014 to 2019, using the warm Spitzer telescope at 3.6 and 4.5 µm. Among the SPIRITS variables are IR-bright objects that are undetected in ground-based optical surveys. We classify them as (1) transients, (2) periodic variables, and (3) irregular variables. The transients include "SPRITE"s (eSPecially Red Intermediate-luminosity Transient Events), having maximum luminosities fainter than supernovae, red IR colors, and a wide range of outburst durations (days to years). Here we report deep optical and near-IR imaging with the Hubble Space Telescope (HST) of 21 SPIRITS variables. They were initially considered SPRITE transients, but many eventually proved instead to be periodic or irregular variables as more data were collected. HST images show most of these cool and dusty variables are associated with star-forming regions in late-type galaxies, implying an origin in massive stars. Two SPRITEs lacked optical progenitors in deep pre-outburst HST images; however, one was detected during eruption at J and H, indicating a dusty object with an effective temperature of ∼1050 K. One faint SPRITE turned out to be a dusty classical nova. About half the HST targets proved to be periodic variables, with pulsation periods of 670-2160 days; they are likely dusty asymptotic-giant-branch (AGB) stars with masses of ∼5-10 M . A few of them were warm enough to be detected in deep HST frames, but most are too cool. Out of six irregular variables, two were red supergiants with optical counterparts in HST images; four were too enshrouded for HST detection. | 10.3847/1538-4357/ac5832 | [
"https://arxiv.org/pdf/2202.11040v1.pdf"
] | 247,025,682 | 2202.11040 | 3db5cfacd30d0eed806653393befe79fcaa14652 |
Hubble Space Telescope Imaging of Luminous Extragalactic Infrared Transients and Variables from the SPIRITS Survey *
FEBRUARY 23, 2022
Howard E Bond
Department of Astronomy & Astrophysics
Pennsylvania State University
16802University ParkPAUSA
Space Telescope Science Institute
3700 San Martin Dr21218BaltimoreMDUSA
Jacob E Jencson
Steward Observatory
University of Arizona
933 N Cherry Ave85721-0065TucsonAZUSA
Patricia A Whitelock
South African Astronomical Observatory
P.O. Box 97935ObservatorySouth Africa
Department of Astronomy
University of Cape Town
Private Bag X3, Rondebosch7701South Africa
Scott M Adams
Cahill Center for Astronomy & Astrophysics
California Institute of Technology
91125PasadenaCAUSA
Present address: Orbital Insight
1201 N Wilson Blvd., Suite 210022209ArlingtonVAUSA
John Bally
Center for Astrophysics & Space Astronomy
Astrophysical & Planetary Sciences Department
University of Colorado
UCB 38980309BoulderCOUSA
Ann Marie Cody
SETI Institute
339 Bernardo Ave., Suite 20094043Mountain ViewCAUSA
Robert D Gehrz
School of Physics & Astronomy
Minnesota Institute for Astrophysics
University of Minnesota
116 Church St. SE55455MinneapolisMNUSA
Mansi M Kasliwal
Cahill Center for Astronomy & Astrophysics
California Institute of Technology
91125PasadenaCAUSA
Frank J Masci
IPAC
California Institute of Technology
1200 E California Blvd91125PasadenaCAUSA
Hubble Space Telescope Imaging of Luminous Extragalactic Infrared Transients and Variables from the SPIRITS Survey *
FEBRUARY 23, 2022DRAFT VERSION Typeset using L A T E X twocolumn style in AASTeX62
SPIRITS-the SPitzer InfraRed Intensive Transients Survey-searched for luminous infrared (IR) transients and variables in nearly 200 nearby galaxies from 2014 to 2019, using the warm Spitzer telescope at 3.6 and 4.5 µm. Among the SPIRITS variables are IR-bright objects that are undetected in ground-based optical surveys. We classify them as (1) transients, (2) periodic variables, and (3) irregular variables. The transients include "SPRITE"s (eSPecially Red Intermediate-luminosity Transient Events), having maximum luminosities fainter than supernovae, red IR colors, and a wide range of outburst durations (days to years). Here we report deep optical and near-IR imaging with the Hubble Space Telescope (HST) of 21 SPIRITS variables. They were initially considered SPRITE transients, but many eventually proved instead to be periodic or irregular variables as more data were collected. HST images show most of these cool and dusty variables are associated with star-forming regions in late-type galaxies, implying an origin in massive stars. Two SPRITEs lacked optical progenitors in deep pre-outburst HST images; however, one was detected during eruption at J and H, indicating a dusty object with an effective temperature of ∼1050 K. One faint SPRITE turned out to be a dusty classical nova. About half the HST targets proved to be periodic variables, with pulsation periods of 670-2160 days; they are likely dusty asymptotic-giant-branch (AGB) stars with masses of ∼5-10 M . A few of them were warm enough to be detected in deep HST frames, but most are too cool. Out of six irregular variables, two were red supergiants with optical counterparts in HST images; four were too enshrouded for HST detection.
cal imaging surveys have been finding OTs in increasingly large numbers, and the discovery rates are now becoming enormous. However, optical surveys are relatively insensitive to objects that are intrinsically cool, dusty, or located in obscured regions. Thus our knowledge of variable and transient phenomena occurring primarily at infrared (IR) wavelengths has, until fairly recently, been limited.
In 2014, our team started a systematic search for luminous IR transients and variables in nearby galaxies, called SPIRITS (SPitzer InfraRed Intensive Transients Survey). Our survey used the Infrared Array Camera (IRAC; Fazio et al. 2004) on the warm Spitzer Space Telescope to search for variable extragalactic objects at wavelengths of 3.6 and 4.5 µm. From 2014-2016, our target list contained ∼190 galaxies, consisting of about 37 galaxies within 5 Mpc, 116 luminous galaxies with distances of about 5 to 15 Mpc, and the 37 most luminous galaxies in the Virgo cluster at 17 Mpc. From 2017 through the end of the survey in 2019 December, we reduced our target list to a subset of the 105 galaxies of the original sample most likely to host new transients, including the most luminous and actively star-forming galaxies, and the 58 galaxies that had previously hosted an IR transient candidate. Observing cadences ranged from a few weeks to several years, augmented with additional data available in the Spitzer archive. 1 The nominal S/N = 5 limiting magnitudes for isolated objects in our exposures are [3.6] = 20.0 and [4.5] = 19.1 (Vega scale). These correspond to absolute magnitudes of −8.5 and −9.4 at 5 Mpc, and −10.9 and −11.8 at 15 Mpc. However, in practice the limiting magnitudes are affected by our ability to remove the background in image subtraction, and the detection limits can be substantially brighter than the nominal values.
Details of the SPIRITS image-processing and variableidentification pipeline are given in Kasliwal et al. (2017, hereafter K17). The pipeline includes subtraction of template reference images, for which we used available archival frames, the Post-Basic Calibrated Data (PBCD)-level mosaics, including Super Mosaics 2 , or images from the Spitzer Survey of Stellar Structure in Galaxies (S4G; PID 61065; PI K. Sheth). Where Super Mosaics or S4G mosaics were not available, we used stacks of archival BCD-level images. For all "saved" sources, those vetted by human scanners and given a SPIRITS designation, we performed aperture photometry on the difference images using a 4 mosaickedpixel (2. 4) aperture and background annulus from 4-12 pixels (2. 4-7. 2). The extracted flux is multiplied by aperture corrections of 1.215 for [3.6] and 1.233 for [4.5], as described in the IRAC Instrument Handbook, 3 and converted to Vega-system magnitudes using the Handbook-defined zeromagnitude fluxes for both IRAC channels. The final photometry that we employ places a grid of apertures near each individual source position to robustly estimate the uncertainties and upper limits, as described in Jencson (2020, hereafter J20). For non-transient, variable sources (where the difference flux measured may be negative; see our definitions below in §2), if there is a plausible, identifiable counterpart source in the reference images, we add the flux of the source from aperture photometry on the reference images to our difference photometry. Otherwise, we offset the difference-flux measurements of a given light curve to bring the minimum, negative value to zero before converting to magnitudes. All photometry presented in this work is provided in an electronic format as "data behind the figures." Full details of the SPIRITS survey and initial discoveries are presented in K17, and updated overviews are given by Jencson et al. (2019b) and J20.
SPIRITS VARIABLES AND "SPRITE"S
Based on what was known about OTs at the beginning of the SPIRITS survey, we anticipated at a minimum that we would discover members of known classes of dusty transients, as well as heavily optically obscured SNe. The category of dusty OTs includes eruptive events with peak luminosities between those of SNe and CNe, which appear to fall into two main groups:
(1) "Intermediate-luminosity red transients" (ILRTs), typified by NGC 300 OT2008-1 (Bond et al. 2009) and SN 2008S in NGC 6946 (Szczygieł et al. 2012). The progenitors of both of these events were detected in archival Spitzer images as luminous mid-IR sources (Prieto 2008;Prieto et al. 2008), which were heavily obscured at optical wavelengths. The outflows from ILRTs form substantial dust, and they are bright IR sources at late times after their optical light has faded (e.g., Kochanek 2011, and references therein). ILRTs are strongly associated with spiral arms, indicating that they arise from fairly massive stars. The origin of ILRT outbursts is debated: proposed mechanisms include electroncapture SNe, low-mass core-collapse SNe, events related to LBV eruptions, and binary interactions (see discussion and references in Adams et al. 2016;Cai et al. 2018Cai et al. , 2019Cai et al. , 2021. A recently discovered likely member of this class, M51 OT2019-1 (AT 2019abn), had a massive, self-obscured progenitor similar to the two class prototypes, and it was shown to be variable in the 12 years of available pre-outburst archival imaging with Spitzer/IRAC (Jencson et al. 2019a). An extended phase of early circumstellar dust destruction observed during the rise of this transient disfavors a terminalexplosion scenario, strengthening the connection between ILRTs and giant eruptions of LBVs (Jencson et al. 2019a;Williams et al. 2020).
(2) "Luminous red novae" (LRNe), a lower-luminosity class of dust-forming transients, generally (but not always) associated with older populations. Examples include M31 RV and V4332 Sgr (Bond 2011(Bond , 2018, and references therein) and V838 Mon (Sparks et al. 2008;Woodward et al. 2021). These events are probably the result of commonenvelope interactions and stellar mergers (e.g., Howitt et al. 2020); this was definitely the case for the LRN V1309 Sco, which was shown to have been a close binary with a rapidly decreasing orbital period before its eruption (Tylenda et al. 2011).
As reported by K17, our initial analysis of the SPIRITS data indeed revealed numerous IR transients and variables. An unexpected finding was a new class of IR transients that lack counterparts in deep optical imaging during outburstunlike the ILRTs and LRNe described above-and have maximum IR luminosities lying between those of CNe and SNe. We refer to these objects as "SPRITE"s (eSPecially Red Intermediate-luminosity Transient Events). These events were defined in K17 as transients with absolute magnitudes at maximum in the range −14 < M [4.5] < −11, IR colors in the range 0.3 < [3.6] − [4.5] < 1.6, and having no optical counterparts detected during the outbursts in deep ground-based images.
A second surprise emerged as the SPIRITS program continued to collect data. Several luminous stars initially considered to be transients based on a small number of observations showing a rising brightness, including several of the candidate SPRITEs, were found to be periodic variables when more Spitzer data became available. In fact, it is striking how many luminous variable stars are conspicuous in latetype galaxies, when Spitzer frames taken over several years are blinked. The contrast with what is seen at optical wavelengths, where bright variables are rare, is remarkable. These objects are most likely pulsating cool and dusty stars, with very long periods, up to several years. Most of them are probably dust-obscured AGB stars, similar to, but more luminous than, those found in the Large Magellanic Cloud and other nearby galaxies, as described by Whitelock et al. (2003), Goldman et al. (2017), Whitelock et al. (2018), and Menzies et al. (2019). They may be analogs of the OH/IR stars found close to the plane in our own Galaxy (e.g., Epchtein & Nguyen-Quang-Rieu 1982;Jones et al. 1982).
These IR-luminous pulsators have been discussed by Karambelkar et al. (2019, hereafter K19), who present a catalog of over 400 periodic or suspected periodic variables found in the SPIRITS survey. Another recent study of pulsating IR variables, based in part on SPIRITS data for galaxies in the Local Group, has been presented by Goldman et al. (2019). However, we cannot rule out that some of these apparently periodic objects may actually be massive binaries with very long orbital periods, rather than pulsators. For example, Williams et al. (2012) have described a binary system containing a carbon-rich Wolf-Rayet (WC) star, which periodically forms dust during periastron passage when its wind collides with the outflow from a companion star. Recently, we reported six extragalactic WC Wolf-Rayet binary candidates displaying such dust-formation episodes as mid-IR SPIRITS variables, including the mid-IR counterpart to the recently discovered colliding-wind WC4+O binary candidate, N604-WRXc, in M33 (Lau et al. 2021).
A few of the SPIRITS transients are so luminous that they are very likely to be obscured SNe (Jencson et al. 2017(Jencson et al. , 2019b. In a few cases these objects did prove to have detectable optical counterparts. An example is SPIRITS 16tn in NGC 3556, at a distance of only 8.8 Mpc, which reached at least M [4.5] = −16.7 (Jencson et al. 2018). It had been missed in optical SN surveys, but following the SPIRITS discovery it was detected in deep Hubble Space Telescope (HST) and ground-based images at I and JHK. Its optical extinction is estimated to be A V 8-9. This event raises the possibility that some of the SPIRITS transients with relatively few observations could have been obscured SNe that happened not to be imaged around their times of maximum light.
Young stellar objects (YSOs) are another potential source of transient phenomena at IR wavelengths. Some low-to moderate-mass YSOs experience sudden ∼5 to 10 magnitude increases at visual wavelengths, followed by a gradual decline lasting years to decades. These "FU Orionis" outbursts are thought to be triggered by enhanced accretion from a circumstellar disk onto the YSO (e.g., Hartmann & Kenyon 1996). The Spitzer YSOVAR project (Rebull et al. 2014) found that most YSOs are variable in the IR, and some experience luminous outbursts. In 2015, two massive YSOs underwent ∼4 × 10 4 L IR flares (Hunter et al. 2017;Caratti o Garatti et al. 2017). More luminous events are thought to be associated with explosions such as the ∼550-yr-old BN/KL outflow in Orion OMC1 (Bally et al. 2020). Bally & Zinnecker (2005) proposed that mergers of the most massive stars can produce IR flares with luminosities comparable to those of SNe. Numerical simulations show that in dense cloud cores, orbital decay can induce massive stars and binaries to migrate rapidly to the core's center to form nonhierarchical systems. Such systems are unstable. N-body interactions tend to rearrange them into hierarchical configurations, compact binaries, and ejected stars (e.g., Reipurth & Mikkola 2015). In sufficiently dense sub-groups, N-body interactions may lead to YSO mergers. In a molecular cloud, dust shifts the UV and visual light from major accretion events and stellar mergers into the IR or sub-mm. The resulting luminous IR flares are expected to be similar to some SPIRITS IR transients.
HST FOLLOW-UP OF SPIRITS VARIABLES
In the present paper we report results of follow-up optical and near-IR imaging observations of SPRITEs and other SPIRITS transients and variables, made with HST and its Wide Field Camera 3 (WFC3). We received allocations of two HST orbits for an initial Director's Discretionary program in Cycle 21 (GO/DD-13935, PI H.E.B.), and eight orbits for a regular Cycle 23 program (GO-14258, PI H.E.B.). Two observing programs from another team (GO-14463 in Cycle 23, GO-14892 in Cycle 24, both with PI B. McCollum) were devoted to imaging of two further SPIRITS transients;
we have obtained these data from the public HST archive 4 and include analyses of them in this paper. In addition to our newly obtained WFC3 images from these programs, we downloaded HST frames of the sites from the archive; these had been secured with the Wide Field Planetary Camera 2 (WFPC2) and the Advanced Camera for Surveys (ACS), as well as WFC3.
The main goals of our HST imaging were: (1) deep searches for optical/near-IR counterparts of SPRITEs while the events were underway; (2) characterization of their stellar environments; and (3) identification of (or limits on) progenitor objects by comparison of our new HST images with pre-outburst archival images.
Our primary considerations for HST target selection were: (1) the outburst event appeared to satisfy the photometric criteria for SPRITEs outlined above, and was expected to be underway during the HST observation 5 ; (2) there was no detected optical counterpart in concomitant optical groundbased imaging with moderate-aperture telescopes (to rule out ordinary CNe, SNe, LBVs, and other known types of OTs); and (3) the site of the SPRITE had been imaged previously by HST at the F814W band (or longward in a few cases). As noted by K17, SPRITEs appear to have a wide duration of outburst timescales, ranging from fast events to long-duration eruptions. We attempted to sample transients spanning this range. Table 1 lists the SPIRITS targets that were observed in the four HST programs. Columns 1 and 2 list the object designations and host galaxies, and column 3 gives the date of the HST observation. Column 4 contains the HST program ID number. Our WFC3 observations were made in the UVISchannel F814W bandpass, and the IR-channel F110W and F160W filters (although in one case we omitted the F110W image). Hereafter in this paper we denote these filters as I, J, and H, respectively, but we emphasize that these should not be confused with the ground-based bandpasses with the same designations.
For most of the observations we used small subarrays, rather than read out the entire detector, in order to obtain more exposure time during the HST visits. In four cases, we adjusted the HST pointing, and used a larger subarray, so as to include one or two additional interesting SPIRITS variables in the WFC3 field of view (FOV), as indicated by the multiple entries in column 1. All exposures were taken at three dither positions, and we downloaded the default pipeline drizzle-combined images from the HST archive. 4 http://archive.stsci.edu/ 5 Our observations were "non-disruptive targets of opportunity," meaning that the HST observations were to be obtained no less than 3 weeks after activation. As discussed below, in a few cases of fast transients, the IR outbursts had already ended by the time of the HST observation.
Columns 5 through 7 in Table 1 list the total exposure times in each filter, which varied slightly due to HST visibility constraints. For nominal exposure times of 1500, 600, and 300 s, the limiting (S/N = 5) HST Vega-scale IJH magnitudes are 26.2, 25.2, and 23.4, respectively, based on the WFC3 exposure-time calculators (ETCs). 6 Our final observation in GO-14258 was actually made on the obscured SN SPIRITS 16tn, rather than a candidate SPRITE. As noted above, this target is discussed in detail by Jencson et al. (2018); it will not be considered further in the present paper.
We also searched the SPIRITS archive for additional variable objects that happened to fall within the FOVs of our HST images, but had not been the primary targets. Five of them proved interesting enough to include in the analyses in this paper. The final column in Table 1 lists these additional "serendipitous" IR variables. Table 2 lists information on the SPIRITS variables targeted in the HST observations discussed in this paper. For each HST pointing in the table, a header gives the host galaxy and its distance modulus, from the literature sources cited in a footnote. Column 1 gives the SPIRITS designations, and columns 2 and 3 the J2000 coordinates. The final column gives our classification of each object's light curve, as described in this section.
IR LIGHT-CURVE CLASSIFICATION
At the time the prime targets in each field were selected for HST imaging, they were all candidate SPRITE transients. However, with the accumulation of substantial additional Spitzer data since the dates of the HST observations, we are able to refine our variability information beyond what was available at the time of selection. In particular (see §2), it became apparent that many of the candidate SPRITEs are actually not transients, but pulsators with very long periodslikely mass-losing intermediate-mass stars approaching the end of their AGB evolution. For the purposes of this paper, we have used the following classification scheme for transients, periodic variables, and irregular variables:
(1) Transients and SPRITEs. These are sources that were undetected, brightened once, and then declined below detectability, and for which there are sufficient data before and/or after the outburst to rule out the categories below. Specifically, we use the criteria outlined in §1.3 of J20 to select bona fide transients. Transients falling within the absolute-magnitude and color ranges given in §2, which were not detected from the ground at optical wavelengths, are called SPRITEs. We subdivide them into "fast" SPRITEs Jencson et al. (2018), and interpreted as a heavily obscured core-collapse SN. Tabulated here for completeness, but not discussed in this paper.
(having outburst durations of less than one typical Spitzer visibility window of 6 months) and "slow" SPRITEs (having outburst durations extending over more than one visibility window).
(2) Periodic and likely periodic variables. As discussed above, K19 identified a set of SPIRITS variables with sufficient data to show that they were varying periodically, most likely due to long-period pulsations. We call the targets in the present study "periodic variables" if we have seen at least one and a half cycles of variation. If there is less time coverage, but the available light-curve shape is similar to those of the known periodic variables, we call the object "periodic?".
(3) Irregular variables. These objects vary irregularly, in a fashion inconsistent with either of the above classes. Specifically, they do not show obvious periodicity, nor an outbursting transient behavior as described above.
ASTROMETRIC REGISTRATION
In order to search for optical counterparts of SPIRITS variables and transients in HST images, it is necessary to carry out a precise astrometric registration of the Spitzer and HST frames. This task is complicated by the fact that a large fraction of stars and background galaxies that are prominent at 3.6 and 4.5 µm are faint or invisible at optical wavelengthsand vice versa. In addition, sources that are isolated at HST resolution (the WFC3 plate scales are 0. 0396 and 0. 128 pixel −1 in the UVIS and IR channels, respectively) are often blended at Spitzer IRAC resolution (0. 6 pixel −1 ).
These considerations make it necessary to blink the IRAC and WFC3 frames visually in order to select a sample of isolated objects common to both images-many of which are either foreground stars or compact, IR-luminous background galaxies.
In many cases, the SPIRITS variables were seen to lie in crowded locations in their host galaxies, including clusters, associations, and H II regions. In these instances, we first subtracted a reference Spitzer image, in which the variable was faint or absent, from a frame showing the variable in a bright phase. We then used these "difference" images to determine the positions of the variables in the same astrometric framework as the reference objects in the direct frames.
We employed standard tasks in IRAF and STSDAS 7 to determine centroid locations for the reference stars and galaxies in the frames. Then we used the geomap task to map the coordinate system of the Spitzer frame to the HST image, followed by the geoxytran task to determine the (x, y) position of the Spitzer variable in the HST frame. The precision of the registrations varied depending on the number and quality of the reference objects, but generally ranged from an 7 The Image Reduction and Analysis Facility (IRAF) was distributed by the National Optical Astronomy Observatory, which was operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. The Space Telescope Science Data Analysis System (STSDAS) was distributed by the Space Telescope Science Institute, which is operated by AURA for NASA. a The distance moduli adopted for the host galaxies are from these sources: rms of about 0.1 to 0.3 Spitzer/IRAC pixels (0. 06 to 0. 18), or about 1.5 to 4.5 WFC3/UVIS pixels in the x and y directions. Depending on the quality of the target's image in the Spitzer frames, the uncertainty of its position could be somewhat larger than for the reference-frame stars.
INFRARED TRANSIENTS AND SPRITES
With the accumulation of Spitzer observations and other information, only three of our targets chosen for HST follow-up have remained classified as transients. We discuss them in this section. Their IR light curves are shown in the three panels of Figure 1. In this figure, and in subsequent lightcurve plots in this paper, the epochs of HST observations are marked with vertical black or gray arrows, marking the dates of our triggered observation or of the archival observations, respectively. Since our targets are faint and extremely red, we generally only consider archival HST images taken in broadband filters at I (F814W) or longer wavelengths.
For the transients and SPRITEs, we also include optical constraints from wide-field, untargeted surveys, namely the intermediate Palomar Transient Factory (iPTF; Cao et al. 2016) and the Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al. 2018;Smith et al. 2020). The iPTF constraints, originally reported in Appendix 2 of J20, consist of forced-photometry measurements on the gand Mould R-band difference images at the locations of our transients using the PTF IPAC/iPTF Discovery Engine (PTFIDE) tool , stacked in 10-day windows to provide deeper limits. ATLAS constraints were obtained from forced photometry 8 on the available ATLAS-c ("cyan") and ATLAS-o ("orange") difference images, again stacked in 10day windows. Table 3 summarizes several properties of the light curves of the three transients, including information on the dates they were detectable, dates of maximum light, their apparent and absolute magnitudes at peak, and the [3.6] − [4.5] color at peak. . 14aje faded quickly, dropping by ∼0.6 mag on 2014 April 24. It was below detection at our next visit on 2014 September 2, and at all of our Spitzer observations since then. As presented in Appendix 2 of J20, limits from iPTF constrain any transient optical emission during the IR outburst to r 21 mag. We classify 14aje as a fast SPRITE, confirming the initial classification by K17.
We triggered our first HST SPIRITS follow-up observations on this transient, obtaining WFC3 frames on 2014 September 22. In addition to our HST data, there was subsequent archival imaging in an unrelated program in 2017 (GO-14678, PI B. Shappee); both dates are marked with arrows in the light-curve figure. An earlier archival observation was obtained in 2003 (GO-9492, PI F. Bresolin), outside the plotted time interval. Our HST observations were discussed briefly by K17, but are updated here.
As the light curve shows, the 14aje event was so fast that the IR outburst unfortunately appears to have ended at an un-certain date before the epoch of our triggered HST observations. We registered Spitzer frames showing 14aje at maximum with the HST ACS images taken in 2003 and 2017, in order to determine its location in the HST frames. (We chose the ACS frames because of their large FOVs, providing more astrometric reference sources than our own WFC3 images obtained with smaller subarrays.) The top panel in Figure 2 shows a color rendition of the SPRITE's environment, taken from the Hubble Legacy Archive 9 (HLA). The site lies in a spiral arm of M101, with several young associations nearby containing blue supergiants and a few red supergiants. The transient is located in a dark dust lane, but does not appear to lie within a rich association. The bottom three panels in Figure 2 zoom in on the site in the WFC3 images we obtained several months after the outburst in, from left to right, I, J, and H. The green circles show the 3σ error locations from the astrometric registration. There are several faint stars within the error circle in the I frame, the brightest of which is more conspicuous in the J and H images. The apparent magnitudes (Vega scale) of this star are I = 25.8, J = 23.3, and H = 22.2, according to photometry from the Hubble Source Catalog (HSC), 10 available from the HLA display of the field. However, none of the stars within the error circle varied significantly in the I-band frames taken in 2003, 2014, and 2017. Thus we conclude that we did not detect the transient with HST a few months after its IR eruption had ended. Moreover, aside from the possibility that the object was able to return to essentially the same quiescent level it had before outburst, we have no compelling identification of an optical progenitor. These observations, along with the rapidity of the IR transient, make this SPRITE event qualitatively different from ILRTs like NGC 300 OT2008-1, SN 2008S, and M51 OT2019-1, which would have been detected easily by HST a few months after their outbursts at I, J, and H.
There is also no pre-eruption IR counterpart detected in the available archival Spitzer/IRAC imaging. We examined the location of the event in the SHA Super Mosaics in all four IRAC channels, which consist of stacks of images taken between 2004-2007, and derived 5σ limiting magnitudes based on the faintest detected sources within a 40 radius in PSF-9 http://hla.stsci.edu/hlaview.html 10 http://archive.stsci.edu/hst/hsc . In this paper we frequently give photometry for sources detected in HST images; in many cases these values are quoted from the HSC (see Whitmore et al. 2016, for an overview). HSC magnitudes are on the AB scale, and are determined using small photometric apertures. Throughout this paper we have corrected the HSC magnitudes to infinite apertures, using the values at https://archive.stsci.edu/hst/ hsc/help/HSC_faq/ci_ap_cor_table_2016.txt. Where appropriate, we then converted the AB magnitudes to the Vega scale, using the zero-points for the WFC3 camera available at http://www.stsci.edu/hst/instrumentation/wfc3/ data-analysis/photometric-calibration , and for the ACS camera from Sirianni et al. (2005).
−14 −13 −12 −11 −10 −9 −8 −7 −6
Absolute Vega magnitude 2 0 1 2 2 0 1 4 2 0 1 6 2 0 1 8 2 0 2 0 Year 15 16 17 18 19 20 21 22 23 Apparent Vega magnitude 14aje −13 −12 −11 −10 −9 −8 −7 −6 −5 Absolute Vega magnitude 2 0 1 1 2 0 1 3 2 0 1 5 2 0 1 7 2 0 1 9 2 0 2 1 Year 15 16 17 18 19 20 21 22 23 Apparent Vega magnitude 17fe −13 −12 −11 −10 −9 −8 −7 −6 −5 Absolute Vega magnitude 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 2 0 1 3 2 0 1 5 2 0 1 7 2 0 1 9 2 0 2 1 Year 15 16 17 18 19 20 21 22 23 Apparent Vega magnitude 14axa Figure 1. Spitzer IRAC light curves of two SPRITEs and a transient. In all of the light-curve figures in this paper, Vega-scale magnitudes are plotted for IRAC measurements, with red squares indicating [4.5] magnitudes, and blue circles [3.6] magnitudes. Error bars are shown when they are larger than the plotting symbols. Open symbols with down arrows indicate upper luminosity limits for non-detections. Open downward triangles represent optical limits (AB magnitudes) in ATLAS-c (turquoise) and -o (light orange) for SPIRITS 17fe and iptf-g (green) and -R (dark orange) for SPIRITS 14aje and SPIRITS 14axa. The detection of an optical counterpart of SPIRITS 14axa reported by Hornoch & Kucakova (2014) at R = 18.9 mag (Vega) is also shown as the orange diamond. The epochs of our triggered HST observations are indicated at the top of each frame by black arrows, and the epochs of available archival HST broad-band images (I band or longward) by gray arrows. Left frame: 14aje in M101, a luminous, very fast, and very red SPRITE, detected in only two observations; middle frame: 17fe in NGC 7793, a slow SPRITE, which brightened suddenly and then declined over at least the subsequent 2.7 years; right frame: 14axa in M81, a likely classical nova caught in a dusty phase. One possibility is that SPIRITS 14aje was a heavily obscured core-collapse SN, similar to those presented in Jencson et al. (2017Jencson et al. ( , 2018Jencson et al. ( , 2019b, but for which the luminous IR peak was missed during the gap in Spitzer/IRAC coverage between 2012 and the start of the SPIRITS survey in 2014. The red [3.6] − [4.5] color would not be unusual for a late-phase core-collapse SN (see, e.g., Tinyanont et al. 2016;Szalai et al. 2019;Jencson et al. 2019b). The deep optical limits from iPTF shown in Figure 1 to R 21 mag would then imply many magnitudes of extinction (A V 9 mag for an SN peaking at M R = −16 mag). Such high obscuration, perhaps by a dense molecular cloud, would explain the lack of a conspicuous progenitor star in the archival HST imaging. Still, given our relatively weak constraints on the timescale and peak brightness of 14aje, we are unable to confirm this scenario, and its definite nature thus remains elusive.
17fe: Archetypal Slow SPRITE
The Spitzer IR light curves of SPIRITS 17fe in the Sculptor-group galaxy NGC 7793 (d 3.6 Mpc) are shown in the middle panel of Figure 1. We caught it rising in brightness on 2017 February 16, and it had brightened another 0.8 mag on 2017 March 16. At that date, marking the maximum brightness seen in our data, it had apparent magnitudes of [3.6] = 16.52 and [4.5] = 15.83, corresponding to absolute magnitudes of −11.3 and −11.9, respectively. Subsequently, Figure 2. HST images of the site of the fast SPRITE SPIRITS 14aje in M101. Top: color rendition of the site, from B, V , and I images in the Hubble Legacy Archive. The frame is 29 high (∼950 pc at the distance of M101). The site of 14aje, marked with a red cross, lies in a dust lane in a spiral arm of M101, with several rich young associations in the vicinity. Bottom row: zooms in on HST frames we obtained several months after the outburst, in I, J, and H. The site of 14aje from astrometric registration of the Spitzer and HST frames is marked with green 3σ error circles. None of the optical stars detected within the error circles in these frames appear to have varied in archival HST images taken at an epoch before, and an epoch after, the outburst (see text). Each frame is 5 high. All HST images presented in this paper have north at the top, east on the left. 17fe slowly faded until going below our detection limit in our final Spitzer observations on 2019 November 1. Thus the outburst duration was at least 988 days. There were no detections of this object prior to the 2017 outburst throughout the available Spitzer imaging at 3.6 and 4.5 µm, as shown by the upper limits in the middle panel of Figure 1. Also shown are the optical limits derived from the ATLAS forcedphotometry light curves in the ATLAS-c and -o bands, which constrain the presence of an optical counterpart to 19-20 AB mag in both bands for nearly the entire duration of the IR transient. We thus classify 17fe as a prototypical slow SPRITE-luminous in the IR, undetected in ground-based optical data.
The site of 17fe serendipitously lies within the HST field that we imaged for SPIRITS 15wt (discussed below in §7.5), for which our triggered observations were obtained on 2016 April 18. This was 304 days before our first Spitzer detection of the 17fe event. We astrometrically registered a Spitzer 4.5 µm difference-image frame, taken at 17fe's max-imum light, with an archival HST ACS I-band frame obtained in 2003 (GO-9774, PI S. Larsen). (The 2003 frame, rather than our 2016 image obtained with a WFC3 subarray, was chosen for the registration because of its larger FOV.) The top picture in Figure 3 shows a color rendition of the site from the HLA. Like the fast SPRITE 14aje, the 17fe event occurred in a spiral arm of its host galaxy, with numerous young blue stars, red supergiants, and dust lanes in its vicinity. The frames in the middle row of Figure 3 zoom in on HST images of the site, with green circles marking the 3σ location from the astrometric registration. From left to right, these frames show I in 2003, and J and H from our own pre-outburst observation in 2016. In addition to these frames, there are archival HST images in the I band obtained in 2001 (two epochs: GO-8599, PI T. Boeker; and GO-9042, PI S. Smartt) and in 2014 (GO-13364, PI D. Calzetti). There were no changes in brightness of any objects inside the error circle in all of these pre-outburst images. Outside the error circles on the southwest side is a bright red star, which is a high-amplitude variable in the HST frames; however, it is too far outside the circle to be related to 17fe.
In addition to the pre-outburst HST images, there are fortuitous archival WFC3/IR frames obtained in the J and H bandpasses on 2018 January 16 (GO-15330, PI D. Calzetti), 306 days after the date of the IR maximum. The IR outburst was still underway at this epoch. Cutouts from these frames are shown in the bottom row of Figure 3. A faint, very red object has appeared near the center of the astrometric error circle at both wavelengths, making it a very likely near-IR counterpart of the SPRITE in outburst. Based on aperture photometry relative to HSC stars in the nearby field, we find Vega-scale magnitudes for this star of J = 24.2 and H = 21.4. There is a hint that this object is present in our 2016 preoutburst J and H frames, but the field is crowded with overlapping faint stars. There is no convincing progenitor in the pre-outburst I frames; there is a partially resolved star inside the error circle in the 2003 I-band image just southwest of the center, with a Vega-scale magnitude of I 27.1. However, this star does not coincide with the object that appeared in the 2018 frames.
In Figure 4, we show the spectral-energy distribution (SED) of 17fe, constructed from the 2018 January 16 WFC3/IR detections and interpolations of the Spitzer/IRAC [3.6] and [4.5] light curves to the same epoch, 333 days after the first detection of the event with Spitzer. The SED is very red, appearing to peak in the IR around 3 µm, at a band luminosity of λL λ ∼ 10 4 L . The near-to mid-IR SED can be approximated by a blackbody spectrum of temperature T BB 1050 K. We also tried fitting the HST and Spitzer points with two separate blackbodies. In this case, the HST data alone indicate a slightly warmer temperature of T BB 1290 K; however, there is not strong evidence for two components. These values are near the temperatures for dust condensation (e.g., Ney & Hatfield 1978;Gehrz et al. 1980a), suggesting the presence of newly formed, warm dust.
A stellar merger is a compelling scenario for the origin of LRNe and at least some slow SPRITEs. For SPRITEs like 17fe, early dust formation is required to obscure or dramatically shorten the associated OT. In models by Pejcha et al. (2016a,b), elaborated by Metzger & Pejcha (2017), the secondary light-curve peak seen in many LRNe can be explained by the shock-interaction of the dynamical merger ejecta with equatorially concentrated material ejected from the binary during the pre-dynamical in-spiral phase. The dense, rapidly cooling regions behind radiative shocks are favorable locations for dust formation. Metzger & Pejcha (2017) find that, for certain binary configurations, namely those involving giant stars and having long phases of pre-merger mass loss, dust may form early enough to completely obscure the associated shock-powered transient at optical wavelengths.
For a similar, but more luminous, slowly evolving IR transient, SPIRITS 19fi, an associated faint, short-duration (≈10 day), red OT was detected in stacked observations from the Zwicky Transient Facility (see J20). For two other slow SPRITES, SPIRITS 17ar and SPIRITS 18nu, their near-IR outburst spectra show strong molecular absorption features akin to those of a late M giant (Jencson et al., in prep.). These features are also seen in late-time spectra of several optically bright LRNe (e.g., Kamiński et al. 2015;Blagorodnova et al. 2017Blagorodnova et al. , 2020. These observations, together with the late-time SED of SPIRITS 17fe suggestive of warm dust, lend credence to a stellar merger accompanied by early dust formation as a viable origin of many slow SPRITEs. (2015 February 7). Our triggered HST observations on 2014 September 26 were obtained between the dates of the Spitzer detection and the subsequent non-detection, so it is unknown whether the IR event was still underway. The absolute magnitude at the single Spitzer detection was M [4.5] −11.5.
We learned later that this event had also been detected at optical wavelengths and reported as a CN, designated PNV J09560160+6903126. 11 The initial discovery was by Hornoch & Kucakova (2014), who reported an unfiltered (approximately R) optical magnitude of 18.9, on 2014 May 21.9. The transient had been fainter than magnitude 21.7 two nights earlier, demonstrating an extremely fast rise time. Five nights later, observed the object with a wide-field camera on the 2.5-m Isaac Newton Telescope. A narrow-band filter confirmed strong emission at Hα, and yielded a Sloan r magnitude of 19.6. All of this information is consistent with the transient being a CN, although to our knowledge there is no direct spectroscopic confirmation of this conclusion. Our detection with Spitzer suggests that the nova was in a dust-forming post-maximum phase at the time of our observation.
The site of 14axa lies between two spiral arms of M81, in a crowded sheet of stars which appears to lack a young population (in contrast to most of the SPIRITS IR transients, which strongly tend to be associated with spiral arms, young associations, and dusty environments). There are three avail- Figure 3. HST images of the site of the slow SPRITE SPIRITS 17fe in NGC 7793. Top: color rendition of the environment, created in the Hubble Legacy Archive from B, V , and I frames. Frame is 30 high (∼525 pc at the distance of NGC 7793). The site of 17fe, marked with a red cross, lies in a spiral arm, with dust lanes and numerous young blue stars and red supergiants in the vicinity. Middle row: zooms in on HST frames taken before the outburst, in I, J, and H. The I frame was taken 13.2 yr before the eruption, and J and H 0.8 yr before. The site of 17fe from astrometric registration of the Spitzer and HST frames is marked with green 3σ error circles. Each frame is 2. 4 high. Bottom row: HST frames taken in J and H 0.8 yr after the maximum of the IR outburst, while the eruption was still underway. A near-IR counterpart is detected at J and is bright at H. Figure 5 shows renditions of the three HST I-band frames, with green circles marking the 3σ error positional locations. Not far from the centers of the circles is a faint star that noticeably brightened in the 2014 image. Approximate I magnitudes (AB scale), determined from aperture photometry relative to nearby stars with HSC magnitudes, are 25.2 in 2002, 25.8 in 2004, and 24.7 in 2014. (In our 2014 J and H frames, the object is badly blended with the neighboring star to the northeast.) The Iband luminosity at the 2014 observation is consistent with a CN about four months past maximum; see, for example, the I-band and near-IR light curves of T Pyx presented by Walter et al. (2012) and . However, the pre-outburst object is unusually luminous compared to the progenitors of typical CNe. Unless it is a chance superposition, the pre-outburst detection suggests that the binary system has a red-giant donor star, similar to Galactic (recurrent) novae such as T Pyx and RS Oph (e.g., see the reviews of Schaefer 2010; Mukai 2015). The object was below detection in archival pre-outburst HST images at the V band, showing that it was indeed red.
1.0 4.0 Wavelength [µm] 10 3 10 4 λL λ [L ] SPIRITS 17fe T BB = 1290 K T BB = 1070 K T BB = 1050 K
As discussed in J20, the majority of SPIRITS transients fainter than M [4.5] −12.5 at peak are similar to 14axa: that is, they were detected only within a single Spitzer visibility window-implying an evolutionary timescale 6 months. At their peak they have red IR colors of [3.6] − [4.5] between 0.5 and 1.2. Two events, SPIRITS 15bb and SPIRITS 15bh, were also preceded by short ( 2 months), faint (M g ≈ −8) optical outbursts recovered in stacked, archival iPTF images. All 15 of these fast and faint IR transients in SPIRITS were found in just seven galaxies within ≈5 Mpc, and nearly half of them were in M81. This high rate from a small number of the nearest galaxies suggests they are common events. CNe are thus an attractive scenario for their origin; the rate in large galaxies like the Milky Way is estimated to be ≈40-80 yr −1 (e.g., Shafter 2017; De et al. 2021). As with 14axa, the optical precursors detected for 15bb and 15bh were in the luminosity range typical of novae, further supporting this hypothesis. Some novae, namely those of the DQ Her class, such as NQ Vul (Ney & Hatfield 1978), LW Ser (Gehrz et al. 1980a), and V705 Cas (Gehrz et al. 1995a;Evans et al. 1997Evans et al. , 2005, form optically thick dust shells, while others that form optically thin shells still produce strong IR emission, peaking on a timescale of ≈50-80 days at M [4.5] ≈ −11 to −12.5 (e.g., V1668 Cyg; Gehrz et al. 1980b). These properties are generally consistent with the fast and faint IR transients discovered by SPIRITS (J20). Further study on the implications of this population for the rate of strongly dust-forming novae and their impact on the chemical enrichment and dust budget of galaxies (e.g., Gehrz 1988Gehrz , 1999 is thus warranted. A less likely alternative is that this was the coronal-emission phase of an ONe nova such as QU Vul (Greenhouse et al. 1988(Greenhouse et al. , 1990. These novae appear to be several absolute magnitudes fainter at 4.5 µm during maximum light than dusty novae (see, e.g., Gehrz et al. 1995b).
LUMINOUS PERIODIC INFRARED VARIABLES
As noted above ( §2), a significant fraction of the suspected transients discovered early in the SPIRITS survey proved eventually to be periodic, or likely periodic, variables when more Spitzer data, over longer time baselines, had accumulated. The periods associated with these sources are nearly all longer than 1000 days. This led to us triggering HST observations of variables that were only recognized as being periodic later on. In typical cases, the objects were in the rising phase of their light curves during the first few SPIRITS observations, leading to our initial classifications of them as candidate SPRITEs.
In view of the importance of the periodicity to understanding these sources, we first summarize our analysis of the Spitzer photometry and the consequent insight into the nature of this sample of periodic sources. We then describe the HST observations of individual objects and the additional understanding they provide.
Characteristics of the periodic and likely periodic variables that we observed with HST are summarized in Table 4. We derived the periods given in column 2 from the Spitzer light curves using the procedure described in K19 (which in turn follows VanderPlas & Ivezić 2015), allowing the [3.6] and [4.5] magnitudes to be analyzed simultaneously. Two of the variables have periods quoted as lower limits, because there is Spitzer photometry covering only about a single cycle. Coverage of the other sources is rather better than for most of the variables discussed by K19, although few have complete coverage for as much as two cycles. Given that Mira light curves are known not to repeat exactly from cycle to cycle, our derived periods can only be good to about 5-10%, and are best judged by examining the individual light curves. The mean apparent magnitudes (denoted [3.6] and [4.5]) are given in columns 3 and 5, the peak-to-peak amplitudes (∆[3.6] and ∆[4.5]) in columns 4 and 6, and the mean colors ([3.6] − [4.5]) in column 7. Columns 8 and 9 give the the mean absolute magnitudes at [3.6] and [4.5], calculated using the distance moduli in A similar class of luminous periodic IR variables discovered in nearby galaxies by SPIRITS was discussed by K19; they suggested that some of these objects are related to the dusty OH/IR stars (intermediate-mass, oxygen-rich Mira variables) found in our Galaxy and the LMC. In Figure 6 we plot the period-luminosity relation for our variables, using filled grey circles for objects with one variation cycle or less, and filled black circles for those with more. Also plotted are the data from K19 for SPIRITS periodic variables in nearby galaxies (filled red circles), and for LMC OH/IR stars from Goldman et al. (2017) (filled cyan squares). All but one of our variables fall on, or close to, the clump of variables that K19 identify with intermediate-mass AGB stars (1000 < P < 2000 days and −11 > M [4.5] > −13), and which itself falls close to the extrapolated period-luminosity relation found for Mira variables in the LMC (Riebel et al. 2015;Whitelock et al. 2017). These sources have large variation amplitudes (e.g., K19, their Figure 4), as do our sources which are in the approximate range 0.8 < ∆[4.5] < 2.2 mag. The range of colors, 0.17 < [3.6] − [4.5] < 1.5, is also comparable to the values discussed by K19 (cf. their Figure 5). Footnotes in column 1 of Table 4 mark the five variables with HST detections or possible detections, as we discuss later in this section.
Most of these variables have extremely red Spitzer colors, indicating very low effective temperatures and circumstellar dust, a consequence of high mass-loss rates. For example, colors of [3.6] − [4.5] 0.6, 1.0, and 1.5 correspond to blackbody temperatures of ∼1000, 800, and 500 K, respectively (cf. Figure 7 in K19).
At such very low effective temperatures, we generally would not expect to detect these objects with our one-orbit HST observations. In a typical case of a variable with an apparent IRAC channel 2 magnitude of [4.5] 16.5, and the nominal exposure times we used for our HST observations ( §3), the WFC3 ETC indicates that the source would not be detected (S/N < 5) in the HST/WFC3 I, J, or H bandpasses if its blackbody temperature were below ∼750 K. Above 750 K it would be detectable only at H. For a temperature above 850 K, it would be detected also at J. For detection at I, a source with [4.5] = 16.5 would have to be hotter than ∼1275 K. However, the use of blackbodies in these estimates is only a rough approximation, since OH/IR stars have strong molecular absorptions, due to H 2 O and CO in particular, which influence the [3.6] − [4.5] and other colors. Moreover, these estimates are optimistic, since they neglect background light and dust extinction.
It is also possible to estimate the anticipated HST magnitudes, using what is known about the LMC OH/IR stars that are illustrated in Figure 6, and assuming that our variables are similar. We use the JHK(L) lightcurves for these OH/IR stars from Whitelock et al. (2003) and I-band light curves from Soszyński et al. (2009); note, however, that the very reddest LMC sources were not detected in the groundbased Ior H-bands. Unfortunately there is only singleepoch Spitzer photometry for these, so we do not have mean denote the peak-to-peak amplitudes of their variations. b Insufficient data at [3.6] to determine mean magnitude, amplitude, and absolute magnitude over the pulsation cycle (see text). The approximate color given in column 7 is based on an estimate of the likely mean [3.6] magnitude. c 15ahg, 15wt, 14bbc, 15mr, and 15mt had detected or suspected optical/near-IR counterparts in HST imaging (see text). 16ea is equivocal (see §7.7). The others had no convincing counterparts at I, J, or H in HST images. large amplitudes of the AGB variables, particularly at the shortest wavelengths, complicates any predictions of HST flux. Furthermore the Goldman et al. (2018) comparison of the SMC with the LMC suggests that the mass-loss rates of these O-rich stars are a function of metallicity; therefore it seems likely that the AGB stars under discussion will have thicker shells than similar stars in the LMC and thus fainter magnitudes at HST wavelengths. Nevertheless, these provide a useful comparison and the following ranges of colors found for the LMC sources are used to estimate the expected HST magnitudes for our sources: 7.6 < [I − 4.5] < 10.7 and 2.6 < H − [4.5] < 4.8. However, additional dust extinction is possible and will make the HST magnitudes even fainter than these values would predict.
We also note that there are significant differences between the bandpasses of the ground-based IJHK filters and the similar filters used by HST. Although these differences are potentially problematic, particularly for cool stars with extended atmospheres where strong molecular absorption dominates the colors, they are not important for the comparisons made here, where the large-amplitude variations dominate.
We now discuss the individual periodic and likely periodic variables that we imaged with HST. Their Spitzer light curves are collected in Figure 7. There are insufficient [3.6] detections to analyze for periodicity, but they can be used for an estimate of the color, which at [3.6] − [4.5] 1.5 makes 15nz the reddest of the periodic variables discussed in this section.
As noted in the introduction to this section, detection of a source as cool as 15nz at HST wavelengths is not expected. We registered a Spitzer frame taken at maximum brightness with HST frames to find the precise location of the variable in the latter. At this site in the disk of M83 there is a dense sheet of faint stars. There are several faint objects within a 3σ error circle, but comparing I-band frames taken at three epochs (2009,2010,2016), we see no significant variation of any of these stars. Only two epochs of HST imaging at J and H are available (2009,2016), but again we see no variation of any detected stars within the error circle. It thus appears that, as was expected, 15nz has no optical or near-IR HST counterpart at I, J, and H. From a comparison with the OH/IR stars in the LMC, we would expect 15nz to have I > 27.4 and H > 21.5, or probably much fainter given how red it is.
15qo and 15aag
SPIRITS 15qo was likewise announced by Jencson et al. (2015) as an IR transient, lying in NGC 1313 (d 4.2 Mpc). A second transient or variable in the same galaxy, SPIRITS 15aag, was announced by Jencson et al. (2016a); it lies only 54 away from 15qo. We adjusted our HST pointing so as to capture both of these objects in the same image frames.
The Spitzer light curves of these two variables are shown in the top-middle and top-right frames of Figure 7. As in the case of 15nz, both objects were caught in rising phases in the first few SPIRITS observations, and were announced as transients. However, as the light curves show, they later proved to be periodic variables. Curiously, their pulsation periods are nearly the same, both being about 1230 days, and they are also nearly in phase. They are, however, definitely distinct objects.
We registered a Spitzer image showing both objects near maximum light with archival HST images as well as the new images obtained in our program. At the site of 15qo there is an extremely rich star field, lying in an actively star-forming region in NGC 1313. Several faint stars fall within a 3σ error circle, including a bluish star near the center. None of these stars appear to be significantly variable in I-band frames taken in 1994, 2003, 2004 (two epochs), and 2016. Our J and H frames show no very red star at the site. We conclude that there is no detectable optical/near-IR counterpart.
The results are similar for 15aag, which is significantly redder than 15qo. It also lies in a rich field, not far from several H II regions. There is again a faint star within the 3σ error circle, which showed no significant variability in Iband frames taken in 2001, 2003, 2004 (two epochs), and 2016. This star is detected in our J and H observations, but the source is not extremely red; unfortunately these are the only frames in J and H in the HST archive. As in the case of 15qo, we find no convincing optical/near-IR counterpart. 7.3. 15ahg, 14al, and 14dd SPIRITS 15ahg is another object announced as an IR transient by Jencson et al. (2016a), lying in the M81-Group spiral galaxy NGC 2403 (d 3.2 Mpc). Two previously identified variables in the SPIRITS database, 14al and 14dd, lie close to the position of 15ahg, and we adjusted the telescope pointing so that we could include all three in the HST frames. The site of these variables lies near a spiral arm on the northwest side of NGC 2403, in an extremely rich star field. Several giant H II regions are nearby.
The Spitzer light curves of 15ahg are shown in the left panel in the second row of Figure 7. Its period is about 1160 days and it is relatively blue, with a mean [3.6] − [4.5] = 0.36 There are two I-band HST observations of this site in the archive, one taken in 2005 (GO-10402, PI R. Chandar), and the other our frame obtained in 2016. We registered these images with a Spitzer IRAC channel 2 frame from the SPIRITS program, taken near maximum brightness of 15ahg. Inside A comparison with the LMC OH/IR stars (see introduction to this section) would predict 18.7 < H < 20.9 and 1.1 < (J − H) < 2.0 for 15ahg. These are consistent with the measured H 18.9 and J − H 1.8 and support our suggestion that this object is a pulsating AGB variable.
The light curves of 14al and 14dd are shown in the second and third panels in the second row of Figure 7. 14al is extremely red, with a mean [3.6] − [4.5] = 1.4, and its ap- proximate period, 2160 days (5.9 years), is the longest of any of the periodic variables. Its peak-to-peak amplitudes, ∆[3.6] 1.0 and ∆[4.5] 0.8 mag, although slightly less than those of the other periodic variables, are still large. 14al lies in a rich star field. There are several faint stars at the site, but none varied significantly in the I frames from 2005 and 2016. None of them are bright at J and H. We conclude there is no convincing optical/near-IR counterpart.
14dd is another likely periodic variable, with a period of 1420 days and a red color of [3.6]−[4.5] 1.0. As in the case of the nearby 14al, there are no variable objects at the site in the two available HST I-band frames, nor any conspicuous objects within the error circle at J and H. Again, we find no credible optical/near-IR counterpart of this very cool object.
15afp
The IR light curves of SPIRITS 15afp are shown in the lefthand panel in the third row of Figure 7. This object rose by over one magnitude from the first SPIRITS observation in 2014 to a peak in late 2015, leading us to announce it as a candidate transient (Jencson et al. 2016a). We triggered our HST imaging in 2016 March. 15afp lies in a spiral arm of the face-on and actively star-forming galaxy NGC 6946 (d 4.5 Mpc). This variable could be periodic, but our observations cover less than one cycle of a period of around 1650 days. We also note our photometry is likely contami-nated by another nearby, but unrelated, variable source. The amplitudes and color are similar to the other periodic variables discussed in this section. However, we cannot rule out that the object is a slow SPRITE transient, based on our relatively limited data.
We registered a Spitzer image taken at the maximum brightness of 15afp with an ACS I-band frame obtained in 2016 October (GO-14786, PI B. Williams). The site lies in a dense sheet of stars, with moderately high extinction. There are several faint stars within a 3σ error circle. None of them appeared to vary between the ACS frame and our own WFC3 I-band frame taken 7 months earlier, nor in comparison with a shallow ACS frame obtained in 2004 (GO-9788, PI L. Ho). There are no conspicuous sources at the site in our J and H frames. On the basis of these fairly limited data, we see no convincing evidence of an optical or near-IR counterpart.
15wt and 14bbc
The Spitzer light curves of SPIRITS 15wt are plotted in the middle panel in the third row of Figure 7. 15wt rose by nearly 1 mag over the first two years of SPIRITS observations of the host galaxy, NGC 7793 in the nearby Sculptor Group (d 3.6 Mpc). This slow eruption of an apparent transient prompted us to trigger our HST imaging, obtained on 2016 April 18. However, our subsequent observations, as well as archival data, clearly reveal that 15wt is actually a periodic variable of the type discussed by K19, with a welldetermined period of 1190 days. It is relatively "blue," with a mean color of [3.6] − [4.5] 0.5.
We registered an archival HST/ACS I-band image of NGC 7793 obtained on 2003 December 10 (GO-9774, PI S. Larsen) with a Spitzer frame taken near maximum brightness of 15wt, in order to locate the site in the HST frames. We then registered the ACS frame with WFC3 I-band frames taken on 2014 January 18 (GO-13364, PI D. Calzetti) and on 2016 April 18 in our own program. Just inside the 3σ registration error circle is a star that was not detected in 2003 and 2014, but had brightened in 2016. This object is very bright in the J and H frames that we obtained at the same time as our I image in 2016, and the date of the HST imaging is close to the time of maximum brightness in the Spitzer frames. Thus we conclude that the optical/near-IR object is the counterpart of the IR variable. In confirmation, there are also archival WFC3/IR frames of the site taken on 2018 January 16 in J and H (GO-15330, PI D. Calzetti), when the IR variable was near minimum light, as shown in Figure 7. The counterpart is significantly fainter in these frames than in 2016. Another IR variable, 14bbc, had been discovered earlier during the SPIRITS program. It lies close to the position of 15wt, and we adjusted the HST pointing for our triggered observation in 2016 so as to include both 15wt and 14bbc in the images. The Spitzer light curves of 14bbc are shown in the right-hand panel in the third row of Figure 7. This is another periodic variable, which has undergone two full cycles during the archival and SPIRITS Spitzer observations (plus an earlier observation in 2004). Its period is 1500 days, and its color is ([3.6] − [4.5] 0.7).
The site of 14bbc lies in an extremely dense star field in a spiral arm of NGC 7793. We located the site in the available HST images through astrometric registration, as just described for 15wt. As shown in the I-band frame in the left panel of Figure 10, there are several faint stars inside the 3σ error circle. None of them varied between archival HST frames obtained in 2003, 2014, and our own frames from 2016, and we believe the variable was not detected in the I band. However, as shown in the figure, a blended source does appear in the J frame, and it is bright at H; the HSC gives a Vega-scale magnitude for this object of H = 19.42. In the archival frames obtained on 2018 January 16 used for 15wt, the candidate has faded significantly at both J and H, consistent with the phasing of 14bbc seen in Figure 7. Based on its extremely red optical color, location near the center of the error circle, and variability at J and H, we conclude that this star is the near-IR counterpart of the Spitzer variable. Its magnitude is consistent with the 20.4 > H > 18.2 anticipated for an AGB variable at the distance of NGC 7793, using the LMC OH/IR star colors (see introduction to this section).
15mr and 15mt
The IR variability of SPIRITS 15mr was reported by Jencson et al. (2015), who suggested it as a transient and possible SPRITE. It belongs to the star-forming barred spiral galaxy NGC 4605 (d 5.5 Mpc). Another SPIRITS variable, 15mt, lies nearby, and for our triggered HST observation on 2016 June 14 we adjusted the telescope pointing to include both objects. The Spitzer light curves of both variables are plotted in the bottom-left and bottom-middle panels in Figure 7. As the figure shows, both objects were rising in brightness up to the date of our 2016 HST imaging, leading at that time to our classification of both as possible transients. However, our subsequent observations show that 15mr is a periodic variable that has gone through two cycles, with a period of 1110 days.
The position of 15mr in the period-luminosity relation ( Figure 6) shows it to be about 2 mag brighter than the other sources with similar periods. However, its color, [3.6] − [4.5] 0.8, and peak-to-peak amplitudes, ∆[3.6] = 1.4 and ∆[4.5] = 1.3, are very similar to those of the other pulsators. As Figure 6 shows, a few of the variables in K19 also lie in this region of the period-luminosity relation. The nature of these luminous variables is unclear, although this is where we expect to find mass-losing red supergiants. The LMC OH/IR supergiant, IRAS 04553−6825, the most luminous source from Goldman et al. (2017), is an example of this population.
15mt is also an apparent periodic variable with a longer period (∼1800 days), but the classification is less certain because it has only gone through a single cycle in the available Spitzer data. It has a very "blue" IR color, [3.6] − [4.5] 0.2, and especially large peak-to-peak amplitudes, ∆[3.6] 1.7 and ∆[4.5] 2.2.
We registered a Spitzer IRAC 4.5 µm frame from 2016 April, showing both 15mr and 15mt near maximum brightness, with an archival HST WFC3 I-band image obtained in 2013 (GO-13364, PI D. Calzetti), and we compared the 2013 image with our triggered frame taken in 2016. 15mr lies in a rich star field with moderate extinction, with several young As discussed in the text, here is no obvious counterpart of 14bbc in the I frame (none of the faint stars inside the circle varied across several HST epochs), but it is detected at J and is bright at H. Height of each frame is 2. 4. associations in the vicinity. As shown in the top left image of Figure 11, inside the 3σ registration error circle are several resolved stars and numerous, partially resolved fainter objects. The site lies near or within a rich young association to the southwest, from which it is separated by a dark dust lane. The brightest star within the error circle is also bright in H, as shown in the top right image in Figure 11, making it a candidate counterpart of 15mr. However, this star is not especially red at optical wavelengths, and is even detected in the u band (F336W filter); the HSC gives magnitudes (Vega scale) of u = 23.39, B = 24.12, V = 23.70, and I = 22.99. Yet the object also has a near-IR excess, as shown by the HSC Vega-scale magnitude of H = 20.36. By comparing the two available I-band images from 2013 and 2016, we see no sig-nificant variation of this star. Without further information, it remains unclear whether the bluish star represents a binary companion of (or a chance alignment with) the IR variable, or whether the situation is more complicated. The LMC OH/IR supergiant mentioned above, IRAS 04553−6825, has I − [4.5] 8.7 and H − [4.5] 3.8, so 15mr would be expected to have I 23.7 and H 18.8. Taking into account the limited information we have on supergiant colors and 15mr's large amplitude, the measured H = 20.36 cannot be used to rule out the possibility that it is an OH/IR supergiant. The shorter-wavelength magnitudes may be too bright for the OH/IR-supergiant interpretation, but, as already noted, they could plausibly originate from a chance alignment in this rich star field, or conceivably a binary companion.
15mt also lies in a rich field, overlain with clumpy dust absorption, as shown in the I-band image in the bottom left frame in Figure 11. There are several resolved stars inside the 3σ error circle, and a brighter star on the northeast side just outside the circle. The H-band image in the lower right panel in the figure shows two partially resolved bright stars, one corresponding to the bright I-band star just outside the circle. This bright star is extremely red; its Vega-scale HSC magnitudes are V = 25.18, I = 22.47, and H = 19.44, making it a plausible optical/near-IR counterpart of 15mt-especially since the variable's [3.6] − [4.5] color is relatively "blue." However, we see no significant variation of this star in the I band between 2013 and 2016. The second H-band star, just inside the error circle, appears to be undetected in the I frame. An LMC OH/IR AGB star (see introduction to this section) would have 27.4 > I > 24.3 and 21.5 > H > 19.3, so it seems possible that the star measured is the Spitzer source, but that it is confused at shorter wavelengths by bluer stars. Alternatively the second H-band star may be the Spitzer source. 15mt is discussed further in the summary below. 7.7. 16ea The Spitzer light curves of 16ea are plotted in the bottomright panel of Figure 7. Unfortunately, the quality of the light curves is relatively poor and difficult to interpret. This is due to the presence of many image and subtraction artifacts in the vicinity of 16ea in our difference images, leading to large uncertainties and numerous nondetections. Still, the handful of detections at [4.5] shows evidence of multiple peaks, consistent with a periodic variable. Hence we classify it as probably periodic, although the properties listed in Table 4 should be viewed with caution. The inferred color (from the limited [3.6] detections) is very red at [3.6] − [4.5] = 1.6 mag. The HST observations in GO-14892 were obtained on 2017 May 17 and 21; unfortunately this imaging was done only in the WFC3 IR channel. The status of the variable at this epoch is uncertain, but it appears to have been near a minimum in the pulsation cycle.
We astrometrically registered a Spitzer 4.5 µm difference image with an archival HST WFC3 I-band frame obtained on 2009 December 23 (GO-11360, PI R. O'Connell). Images in the WFC3 IR channel J and H bandpasses were obtained at the same time in the O'Connell program. Figure 12 shows false-color renditions of the 16ea site from these IJH frames, with the 3σ registration error circle shown in the left panel. Inside the circle is a bright star, which is very red, making it a plausible optical and near-IR counterpart of 16ea. The HSC gives magnitudes (adjusted to Vega scale) for this star of I = 22.30, J = 20.75, and H = 19.88. In comparison with an ACS I-band image of the site obtained on 2004 October 24 (GO-10332, PI H. Ford), we see no significant change in brightness of this candidate. Comparing the 2009 and 2017 frames in J and H, there is again no compelling evidence for variability. Given the fragmentary data from both HST and Spitzer, it is difficult to reach firm conclusions about this object, beyond our identification of a probable optical/near-IR counterpart.
Periodic Variables: Discussion
The analysis of the Spitzer data for the periodic variables suggests that the majority (at least nine) of them are highly evolved AGB stars, similar to those discussed by K19 and to the OH/IR AGB variables in the Galaxy and LMC. Where we have HST detections they support this conclusion. These stars are important in several respects. Their progenitors must have had intermediate masses (approximately 5 to 10 M ), and they represent a brief and poorly understood phase of late stellar evolution dominated by convection and mass loss. They are among the most significant dust producers in their respective galaxies, returning significant quantities of processed material to the interstellar medium. Intermediate-mass AGB variables, such as these, will be among the most distant individual stars observable with the James Webb Space Telescope (JWST), and are likely to be important probes of their respective populations.
It is not surprising that the four variables that we detected at HST wavelengths, and identified as possible AGB stars, are among the bluest objects, lying in the range 0.17 < [3.6] − [4.5] < 0.76 in Table 4. 15qo is the bluest ([3.6] − [4.5] = 0.58) variable that was not detected by HST; however, its host galaxy is 0.4 mag more distant than that of 14bbc, the reddest object that was detected. Thus the detections, or lack thereof, are consistent with our conclusion that these sources are AGB variables.
The fifth variable that was also also detected by HST, but does not appear to be an AGB star, is 15mr. As discussed above, it is more luminous than the AGB variables, and is possibly a red supergiant. However, as discussed by K19 for variables in this part of the period-luminosity relation, it could alternatively be a dust-producing binary system.
One source that lies in a position in the period-luminosity consistent with the AGB variables, but may be something else, is 15mt. Its blue Spitzer colors and very large variation amplitudes are not typical, and alternative explanations should be considered. While we might speculate that its periodicity could be due to orbital motion in a binary, we would expect an IR-luminous binary to be dusty and thus have red colors. We observed only a single cycle of variation, so it remains possible that 15mt is actually a transient. Unfortu- Figure 11. False-color renditions of HST WFC3 images of the site of the periodic IR variables SPIRITS 15mr and 15mt in NGC 4605. Top row: the site of 15mr in Iand H-band frames taken in 2016. The green circle in the I frame marks the 3σ error position of the Spitzer variable. The brightest star inside the circle is bright at H, and may be a counterpart of 15mr. Bottom row: I and H frames from the same two HST images, showing the site of 15mt. A bright star just outside the northeast edge of the error circle is much brighter in H and might be a counterpart of the Spitzer variable. However, there is also a source just inside the circle on the northeast side that is bright at H but undetected at I, making it another candidate counterpart. Height of each frame is 3. 6. A bright, very red star inside the southwestern edge of the circle, which is also detected at J and H, is a likely optical and near-IR counterpart of the Spitzer variable. Height of each frame is 3. 3.
nately we have insufficient information to be confident one way or the other.
14al is a better candidate for an AGB variable, but its period, estimated to be longer than 2160 days, is poorly defined because of insufficient time coverage. There are only a few pulsating stars known with primary periods above 2000 days. If confirmed as a pulsating AGB star, 14al is potentially very interesting. It may be a candidate super-AGB star, i.e., a star with a progenitor mass of about 10 M that could become an electron-capture SN (Siess 2007;Doherty et al. 2015Doherty et al. , 2017. It is well worth a more detailed study with JWST.
LUMINOUS IRREGULAR INFRARED VARIABLES
Several classes of stars may produce irregular, or nonperiodic, variability in the IR. These include LBVs, which may undergo repeated outbursts or eruptive mass-loss events capable of forming copious dust, though the mechanism driving such outbursts is not well understood. In Jencson et al. (2019b), we presented discoveries of two such sources, SPIRITS 17pc and SPIRITS 17qm, which underwent multiple, extremely red outbursts over the course of several years, and for which we identified luminous counterparts in archival HST imaging, likely to be LBVs. Dust-forming, collidingwind WC binaries, like the SPIRITS sources presented in Lau et al. (2021), are in fact periodic, but may be classified as irregular variables under our definition presented in §2 if their orbital periods are longer than the available Spitzer coverage. In this section, we discuss the six SPIRITS variables that we classify as irregular, and for which we have HST imaging.
Their Spitzer light curves are shown in Figure 13. Table 5 gives details of the Spitzer photometry.
14qb
IR variability of SPIRITS 14qb in the actively star-forming galaxy NGC 4631 (d 7.3 Mpc) was discovered at our first two epochs of SPIRITS observations. Comparison of IRAC frames obtained in 2014 March and April, with archival images from 2004, showed that 14qb had risen ∼1.1 and 0.8 mag at 3.6 and 4.5 µm, respectively, as shown in the Spitzer light curve in the top left panel of Figure 13. 2 mag at both wavelengths, and was still fainter 5 months later. At that point, we considered it to be a "slow" SPRITE transient, and triggered our HST observations; these were obtained in 2015 November. Since then, however, 14qb did not continue to fade like a transient; instead, it slowly brightened, up to our most recent, and final, Spitzer observations in 2019 October. Thus it cannot be considered to be a SPRITE, nor is it periodic, and we classify 14qb as irregular.
In addition to our triggered HST WFC3 observations in 2015, there are archival frames obtained with ACS in 2004 (GO-9765, PI R. de Jong). We astrometrically registered the ACS I-band frame with a Spitzer 4.5 µm difference image showing 14qb at maximum brightness. 14qb is located in an extremely crowded star-forming region southwest of the nucleus of NGC 4631, with numerous young stars and dust lanes. Figure 14 shows a color rendition of the location, created in the HLA from ACS frames in V and I. The vicinity of 14qb exhibits fairly high and spatially variable extinction, and the object lies near the northern edge of a rich young association.
There are a few faint stars within a 3σ astrometric error circle in the HST I frame, none of which varied significantly from 2004 to 2015. None of these stars are conspicuous in the 2015 J and H frames. We conclude that there is no credible counterpart at I, J, and H to this bright and very cool IR variable. Based on the limited information available for this source, we cannot conclusively determine its nature. Its high IR luminosity could be consistent with dust formation by an LBV, for example, but the star must be heavily enshrouded to explain the lack of a detectable optical or near-IR counterpart.
14akj and 14atl
As noted in the last column of Table 1, SPIRITS 14akj and 14atl are two IR variables that serendipitously lie near the primary HST target 15nz (a periodic variable discussed above in §7.1). These two objects belong to the actively starforming spiral galaxy M83 (d 4.7 Mpc). Their Spitzer light curves are shown in the top-middle and top-right panels of Figure 13.
We classify both variables as irregular. 14akj slowly and irregularly declined in brightness over the duration of the Spitzer imaging. However, we cannot completely rule out that it might be a periodic variable with a very long period, of order 8 years. It is extremely red, with a color of Figure 15. The top left panel shows a false-color rendition of the I-band image with a 3σ error circle marking the position of 14akj. None of the three bright stars varied significantly between the two available HST I-band frames (the other being a 2016 image from program GO-14059, PI R. Soria). None of these stars are prominent in the one available J frame nor the two archival H frames. The top right panel is a color rendition of the field, obtained from the HLA and created from WFC3 frames obtained in 2016 in u, B, and I. The image shows several rich young stellar associations in the vicinity of 14akj. All of the stars inside the error circle are seen to be quite blue. It could be that the counterpart to 14qkj was not detected in the optical or near-IR HST images, in which case it is likely to be a heavily enshrouded, and fairly massive young object. Alternatively, this source could be consistent with a WC colliding-wind binary system, where one of the bright blue stars detected with HST is, in fact, the counterpart. Optical spectroscopy to search for WC features, as presented for other SPIRITS sources in Lau et al. (2021), could test this scenario.
The site of 14atl was also astrometrically localized in the same HST I frame as described above. The bottom two pictures in Figure 15 depict the I frame and the 3σ location (left panel), and the HLA color rendition (right panel). Inside the error circle is an optically bright star, and the color rendition shows that this object is extremely red. This star is very bright in a single available J frame and two H frames. Between 2009 and 2016 the star brightened in I by about 0.5 mag, and in H by 0.6 mag. These are in accordance with a similar brightening seen in the Spitzer data. There is thus little doubt that this star is the optical and near-IR counterpart of 14atl.
Because the HSC magnitudes are averaged over widely separated epochs, we performed aperture photometry on the available individual WFC3 frames. We used the zero-points and corrections to infinite aperture from the site referenced in footnote 10. Table 6 gives our results. The uncertainties in the magnitudes, including systematics from the camera calibrations, are generally about ±0.02-0.03 mag.
We show the 2009 and 2016 SEDs of 14atl in Figure 16, constructed from our HST aperture photometry of the optical/near-IR counterpart and linear interpolations of the [3.6] and [4.5] magnitudes to the corresponding epochs. The SEDs at both epochs are similar, peaking between ≈1-2 µm at band luminosities of λL λ 10 5 L . In comparison with PHOENIX model photospheres (non-rotating, solar metallicity; Kučinskas et al. 2005Kučinskas et al. , 2006, the SEDs appear consistent with a luminous red supergiant with an effective temperature of T eff ≈ 3500 K and bolometric luminosities of log L bol /L 5.24-5.47. We note an excess of mid-IR flux at [3.6] and [4.5] compared to the stellar models. Interestingly, the mid-IR color is redder when the star is fainter. Together, these facts likely point to emission from warm dust that condenses in a stellar wind. The observed variability is likely associated with semiregular variations arising from pulsational instabilities, which are common in cool supergiants (e.g., Yoon & Cantiello 2010; Yang & Jiang 2011).
15ahp
The IR variable SPIRITS 15ahp is another serendipitous target, lying by chance in the field of our HST observations Figure 14. Color rendition of the site of the irregular variable SPIRITS 14qb in the star-forming galaxy NGC 4631, from HST V and I frames in the Hubble Legacy Archive. Frame is 18 high (∼640 pc at the distance of NGC 4631). The site of 14qb is marked with a red cross. It lies in a region of intense star formation, with dust lanes and young associations in the vicinity. of the periodic variables 15ahg, 14al, and 14dd, which were discussed above ( §7.3). This field is in the M81 Group galaxy NGC 2403, and the site of 15ahp lies in a star-forming spiral arm of the host galaxy. The IR light curves of 15ahp are shown in the bottom-left panel of Figure 13. This object varies in IR brightness by about 0.2 mag peak-to-peak, on a fairly short timescale of a few months. We classify it as an irregular variable. Its IR color also appears to vary. The astrometric registration of Spitzer frames with HST images of this site was described in §7.3. The top two panels in Figure 17 show the 3σ astrometric error circles for the location of 15ahp in two of the three available HST I-band im- The multi-epoch SED, constructed in a similar manner to that of 14atl, is shown in the right panel of Figure 16. As with 14atl, the star appears consistent with a red supergiant, though perhaps a bit cooler and less luminous (T eff ≈ 3300 K; log L bol /L ≈ 5.1), based on our comparisons with PHOENIX models. We again note a mid-IR excess compared to the photospheric models and that the mid-IR color is slightly redder when the star is fainter, likely indicating the presence of circumstellar dust.
14th
SPIRITS 14th is yet another serendipitous target, which happens to lie in the field of 15wt and 14bbc in the nearby galaxy NGC 7793, which we imaged with HST as described above ( §7.5) on 2016 April 18. The bottom-middle panel in Figure 13 shows the Spitzer light curves, which we classify as Figure 15. HST images of the sites of the irregular variables SPIRITS 14akj (top) and 14atl (bottom) in star-forming regions of M83. The left-hand frames depict the I-band HST frames of the sites, with green circles marking the 3σ astrometric locations of the irregular IR variables. The right-hand pictures are color renditions of the sites, from the Hubble Legacy Archive. At the location of 14akj there are no convincing optical counterparts; all of the candidate stars are blue and not conspicuously variable. However, 14atl has a bright, optically variable, and extremely red and luminous counterpart. Height of frames is 12 (∼275 pc at the distance of M83). that of an irregular variable. 14th brightened by about 1 mag from late 2011 to a peak in early 2014, and then declined by nearly the same amount over the next two years. Since then it has remained nearly constant at [4.5] but somewhat "noisy" at [3.6], with our most recent and final observation showing it quite faint at [3.6]. This variable is extremely red, with a typical color of [3.6] − [4.5] 1.5. It is not extremely luminous; the absolute magnitude at our final observation was about M [4.5] −11.5.
We registered a Spitzer frame showing 14th near maximum brightness with HST I-band images, as discussed for 15wt and 14bbc in §7.5. The site is in a very rich star field. There are a few resolved stars within a 3σ registration error circle, on top of an unresolved or partially resolved background. Comparing I frames obtained in 2003, 2014, 2016 (our triggered observation), and 2017 shows no significant variations of any of these stars. Moreover, none of them are conspicuous at J and H, nor varied between JH images obtained in 2016 and 2018. We conclude that 14th is undetected in the optical and near-IR HST images, which is consistent with its extremely red IR color. Again, without an obvious coun-terpart detection in the optical or near-IR, it is difficult to draw strong conclusions about the nature of the source, other than that it is likely a young, fairly massive, and heavily enshrouded object.
16aj
SPIRITS 16aj was announced as a possible transient by Jencson et al. (2016b). It lies in the actively star-forming barred spiral NGC 2903, the most distant of the galaxies discussed in this paper (d 9.2 Mpc). The Spitzer light curves of 16aj are shown in the bottom-right panel of Figure 13. This object rose in brightness by about 1 mag over the first two years of SPIRITS monitoring (early 2014 to early 2016), leading us to consider it a slow SPRITE; at that point we triggered HST imaging of the site, which was obtained on 2016 May 23. However, since early 2016 the object has remained at a nearly constant magnitude, although with possible shortterm variations of a few tenths of a magnitude. Thus we now classify 16aj as an irregular variable, rather than a true transient. It is fairly red ( A PHOENIX model photosphere with Teff = 3300 K, scaled to a luminosity of log(L/L ) = 5.11, is superposed, again providing a reasonable approximation to the data.
We astrometrically registered Spitzer frames, including a difference image showing 16aj, with an HST ACS I-band image taken in 2004 (GO-9788, PI L. Ho) and with our own I frame obtained in 2016. At the site there are a few resolved faint stars, lying on a sheet of unresolved starlight. We see no I-band variations between 2004 and 2016, and the J and H frames we obtained on the same date in 2016 reveal no extremely red stars. The site lies on the edge of a rich young association and H II region, and there are numerous dark dust lanes in the vicinity. We conclude that 16aj lacks a detectable optical or near-IR counterpart. Similarly to 14qb, 14th, we conclude only that the star is likely to be relatively young, massive, and heavily enshrouded. 9. SUMMARY SPIRITS was the first large-scale monitoring survey of nearby galaxies, using the warm Spitzer telescope to search for luminous variable stars and transients at IR wavelengths. In the work described here, we employed new and archival optical and near-IR HST images to study the sites of 21 SPIRITS variables. The selected targets were of special interest because they were undetected or very faint in groundbased optical surveys. Our aims were to search for progenitors, attempt to detect the sources during outburst using deep HST imaging, and characterize their environments.
We classify the SPIRITS variables into three groups based on their photometric behavior: SPRITEs and transients; periodic variables; and irregular variables. Our main results from the HST imaging are as follows.
1. "SPRITEs" are a new class of intermediate-luminosity IR transients that lack counterparts in deep groundbased optical imaging. They are defined as objects having absolute magnitudes at maximum in the range −14 < M [4.5] < −11. We investigated HST images of three SPRITEs, two of them with fast outburst timescales of a few days to a few weeks, and one with a slow timescale of nearly three years. Like most SPRITEs, two of the three occurred in dusty, starforming regions, consistent with an origin from massive stars. No progenitors were found in deep preoutburst archival HST images for the two objects in young regions. We did detect one of them-the slowly evolving 17fe-during outburst at J and H. This allowed us to construct an SED, indicating that 17fe during eruption was a dusty object with a temperature of about 1050 K. Unusually, the third SPRITE candidate, the very fast 14axa, occurred in the old bulge of M81 rather than in a star-forming region. It appears to have been a dusty classical nova. Some, or many, of the fast SPRITEs at the low end of the luminosity range are likely to be classical novae, instead of arising from massive stars. Most SPRITEs, however, are events of an uncertain nature-possibly a mixture of massive stellar mergers, dust-obscured core-collapse supernovae, and eruptions of massive stars related to those of luminous blue variables.
2. Variable stars are much more conspicuous among the brightest members of stellar populations in the IR than they are in the optical, particularly in late-type galaxies. More than half of our SPIRITS targets, although initially considered to be transients, proved to be periodic variables. Their pulsation periods are long, ranging from ∼670 to over 2100 days. These objects are likely to be highly evolved, dusty AGB stars, similar to OH/IR objects known in the Galaxy and Magellanic Clouds. The pulsators found by SPIRITS are strongly associated with star-forming regions in nearby spirals and irregular galaxies, and they likely arise from intermediate-mass stars (∼5-10 M ). Out of the 12 periodic variables for which we have HST images, only five were warm enough to be detected with HST at I, J, and/or H, along with one uncertain case.
3. Six SPIRITS variables did not fit our definitions for transients or (likely) periodic variables, and we classify them as irregular. Two of these sources, 14atl and 15ahp, had relatively blue IR colors (−0.1 [3.6] − [4.5] 0.2 ) and had bright, red counterparts in HST imaging. Their optical-near-IR SEDs are consistent with those of luminous, pulsating red supergiants. The remaining four irregular variables are much redder at IR wavelengths ([3.6] − [4.5] 0.9) and, not surprisingly, we did not identify convincing optical or near-IR counterparts in deep HST images. These objects may be consistent with eruptive, dust-forming massloss events, like those of LBVs, but their massive stellar counterparts are likely heavily enshrouded. For the case of 14akj, however, we noted a possible association with several luminous blue stars. 14akj could therefore instead correspond to a dust-forming colliding-wind Wolf-Rayet (WC type) binary. Optical spectroscopy to search for prominent WC emission features would test this possibility.
4. None of the SPIRITS transients observed by HST appear to be flares associated with YSOs. Although some lie in the dust lanes of the host galaxies, none of the investigated transients are directly associated with star-forming complexes. However, future IR observations with facilities such as JWST and the Nancy Roman Space Telescope may reveal YSO transients in starforming regions.
The SPIRITS project was the first systematic time-domain reconnaissance of stellar variability among IR-luminous stars in nearby galaxies. It revealed several new classes of extremely cool and dusty transients and pulsating or irregular variables. An understanding of the nature of these diverse objects will require more intensive time coverage of their variations and outbursts than was possible with Spitzer. Infrared spectroscopy would be a key new element in such investigations. Considerable progress will be possible with the powerful IR capabilities of JWST and the Roman Space Telescope. Support for HST Program numbers GO-13935, GO-14258, and AR-15005 was provided by NASA through grants from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. Support for this work was also provided by NASA through awards issued by JPL/Caltech. P.A.W. acknowledges a research grant from the South African National Research Foundation and is grateful to John Menzies (SAAO) for the use of his period-finding software.
J.B. is supported by NSF grant AST-1910393. R.D.G. was supported in part by the United States Air Force.
Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. Our work is also based in part on data obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESAC/ESA), and the Canadian Astronomy Data Centre (CADC/NRC/CSA). This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
Facilities: HST (ACS, WFPC2, WFC3), IRSA, Spitzer
M101 :
M101Jang & Lee (2017); NGC 6946: Pejcha & Prieto (2015); NGC 2903: Tully et al. (2016); all others: Tully et al. (2013). b Probably a classical nova; see §6.3.
6. 1 .
114aje: Fast SPRITE The Spitzer light curves of SPIRITS 14aje in M101 (d 6.8 Mpc) are shown in the left-hand panel of Figure 1. This transient was first detected on 2014 March 26, at an apparent magnitude [4.5] = 15.52 ± 0.03. This corresponds to an absolute magnitude of M [4.5] −13.6 at the distance of M101, well above the brightest IR luminosities observed for CNe (see §6.3). Its color was extremely red, [4.5] − [3.6] = 1.73. The transient had been undetected in an archival Spitzer observation at 4.5 µm in 2012 and is not present in available Super Mosaic images (2004-2007 stack; see below for more details)
d
Measured at the time of the light-curve peak, tpeak. photometry catalogs constructed for M101 by K19. Our limiting (and absolute) magnitudes of [3.6] > 19.7 (−9.5), [4.5] > 19.2 (−10.0), [5.8] > 16.9 (−12.3), and [8.0] > 15.9 (−13.3) are sufficient to rule out an obscured progenitor as luminous and massive as those observed for the ILRTs mentioned in the previous paragraph. All of them had M [4.5] < −10 before their outbursts.
) was initially classified by us (K17) as a fast SPRITE. We detected it with Spitzer at only one epoch, 2014 June 13, as shown by its light curve in the right-hand panel ofFigure 1. There were Spitzer non-detections 122 days before this observation (2014 February 11), and 239 days afterward
able I-band HST images of the site, two obtained with ACS in 2002 and 2004 (GO-9353, PI S. Smartt, and GO-10250, PI J. Huchra), and our triggered WFC3 observation in 2014. We performed an astrometric registration of the Spitzer 4.5 µm image showing 14axa with the 2004 ACS image, in order to locate the site in the HST frames.
Figure 4 .
4SED of the slow SPRITE, SPIRITS 17fe, constructed from photometry of the HST WFC3/IR F110W, F128N, and F160W images taken on 2018 January 16.2 (blue diamonds), and from the Spitzer/IRAC [3.6] and [4.5] light curves interpolated to the same epoch (orange squares). Blackbody fits to only the HST points (blue dashed curve), only the Spitzer points (orange dotted curve), and all points together (black solid curve) are shown with their corresponding temperatures, TBB, given in the legend in the upper-left corner.
Figure 5 .
5False-color renditions of I-band HST images of the site of SPIRITS 14axa in M81. Top: ACS image from 2002; middle: ACS image from 2004; bottom: WFC3 image taken by us on 2014 September 26, triggered by the Spitzer discovery 3.5 months earlier. Each frame is 1. 6 high. The green circles show the 3σ positional error circles, based on an astrometric registration of the 2004 image with a Spitzer image showing the transient. A faint star inside the error circle, seen in 2002 and 2004, had brightened in the 2014 image, and is likely to be the optical counterpart of this probable classical nova.
Spitzer magnitudes or colors. The single-epoch colors are in the range 0.3 < [3.6] − [4.5] < 0.8 for the AGB stars, and [3.6] − [4.5] = 0.9 for the single supergiant. The AGB amplitudes are in the ranges 1.5 < ∆H < 1.9 and 2.8 < ∆I < 4.0, while the supergiant has ∆H = 0.4 and ∆I = 1.5. The very
was announced by Jencson et al. (2015) as a possible IR transient in M83 (d 4.7 Mpc), based on our initial few SPIRITS observations. These data showed a slowly rising [4.5] luminosity during 2014 and early 2015. In response to this announcement, a separate team (GO-14463, PI B. McCollum) obtained HST imaging of the site. Since 2015 we have accumulated additional IR observations, and there are also archival Spitzer data obtained in 2010. The IR light curves of 15nz are shown in the top-left panel of Figure 7. The data at [4.5] indicate a period close to 1600 days.
Figure 6 .
6Mean [4.5] absolute magnitude versus pulsation period for the periodic and suspected periodic variables from this paper, shown as black points, or grey points if only one pulsation cycle or less has been covered. Red points mark the periodic variables from K19, and cyan squares the LMC OH/IR stars fromGoldman et al. (2017). The isolated and unusually luminous black point is SPIRITS 15mr (see §7.6).
Figure 7 .
7Spitzer IRAC light curves of periodic and suspected periodic variables from difference imaging. Plotting symbols as in Figure 1. Baseline magnitudes (aperture photometry on reference images) are shown as open symbols, where the dashed horizontal bars indicate the range of epochs that were included in the reference image stacks. the 3σ error circle in the I-band frames is a faint star that brightened significantly from 2005 to 2016, consistent with the phasing of the IR variability shown in Figure 7. In the 2005 image, the variable lies in a blended clump of several faint stars, which is possibly a compact sparse cluster. In 2016 it had risen to an HST I magnitude of about 24.5 (Vega scale, based on aperture photometry relative to several HSC stars in the field). This star is optically very red, and is well detected in our J and H frames. The HSC Vega-scale J and H magnitudes for the object are 20.79 and 18.94, respectively. Thus there is little doubt that this object is the optical/near-IR counterpart of the Spitzer variable. The four panels in Fig-ure 8 show small postage stamps from the two I-band frames (top row), and from our J and H images (bottom row). The astrometric 3σ error circle is shown in green in the top two frames.
Figure 8 .
8False-color renditions of HST images of the site of the periodic variable SPIRITS 15ahg in the nearby galaxy NGC 2403. Top row: I-band frames taken in 2005 (left) and 2016 (right). The green circles mark 3σ error positions of the Spitzer variable. Just north of the center is a star that brightened significantly and is a likely optical counterpart of 15ahg. Bottom row: HST images of the site in 2016, taken in J (left) and H (right). The candidate counterpart is bright at J and very bright at H. Height of each frame is 3. 6.
Figure 9
9presents false-color renditions of the I-band frames from 2014 and 2016 in the top row, revealing the star just inside the error circle that brightened in 2016. The bottom row shows that this star is very bright in J and H, in the WFC3 frames taken during our same HST visit in 2016. The HSC gives the following magnitudes (Vega scale) for this object at the 2016 epoch: I = 24.23, J = 20.25, and H = 18.70. The LMC OH/IR stars (see introduction to this section) would have 26.7 > I > 23.6 and 20.8 > H > 18.6 at this distance, consistent with our suggestion that the Spitzer variable and its HST optical/near-IR counterpart is a pulsating AGB star.
Figure 9 .
9False-color renditions of HST WFC3 images of the site of the periodic IR variable SPIRITS 15wt in the nearby galaxy NGC 7793. Top row: I-band frames taken in 2014 (left) and 2016 (right). The green circles mark 3σ error positions of the Spitzer variable. Inside the circles is a star that brightened significantly around the time of maximum IR luminosity, and is a likely optical counterpart of 15wt. Bottom row: near-IR HST images of the site in 2016, taken in J (left) and H (right). The counterpart is bright at J and very bright at H. Height of each frame is 2. 4.
Figure 10 .
10False-color renditions of HST WFC3 images of the site of the periodic IR variable SPIRITS 14bbc in NGC 7793, obtained on 2016 April 18. From left to right the frames were taken in I, J, and H. The green circle in the I image is the 3σ error position of the Spitzer variable.
SPIRITS 16ea was announced by Jencson et al. (2016b) as a possible IR transient in the nearby starburst irregular galaxy NGC 4214 (d 2.9 Mpc). In response to this publication, a separate team (GO-14892, PI B. McCollum) obtained HST imaging observations of the site of 16ea with WFC3.
Figure 12 .
12False-color renditions of HST WFC3 images of the site of the IR variable SPIRITS 16ea in NGC 4214, obtained in 2009. From left to right the frames were taken in I, J, and H. The green circle in the I image is the 3σ error position of the Spitzer object in the WFC3 frame.
The source was luminous (M [4.5] −14.5 in 2014 March) and extremely red ([3.6] − [4.5] 1.1). At our observation a year later, in 2015 March, 14qb had faded about 0.
[3.6] − [4.5] 1.1; it is also luminous, having fallen from M [4.5] −14.5 to −14.0 from 2008 to 2019. 14atl differs from 14akj in several respects. It slowly rose from 2010 to mid-2018 by about one magnitude, but with two dips in brightness. Our final observations showed that another dip was underway. 14atl is relatively quite "blue," with [3.6] − [4.5] 0.15, and it is less luminous than 14akj, with an absolute magnitude averaging about M [4.5] −12.5 over the final few years of our monitoring. We astrometrically registered a Spitzer IRAC channel 2 image showing both 14akj and 14atl with an HST/WFC3 Iband image obtained in 2009 (GO-11360, PI R. O'Connell), in order to determine their locations in the HST frame. The site of 14akj lies in a very active star-forming region. Inside a 3σ error circle there are three fairly bright stars, and several fainter ones, forming a small cluster, as shown in the top two panels of
Figure 13 .
13Spitzer IRAC light curves of six irregular variables. Plotting symbols as inFigure 1.
ages, an archival one from 2005 (GO-10402, PI R. Chandar) and the other our frame obtained as a result of our triggered 2016 March 7 observation. A third I-band image from 2019 is available (GO-15645, PI D. Sand) but not illustrated. Inside the error circle is a prominent star, which rose in I-band brightness by about 0.25 mag between 2005 and 2016. This is qualitatively consistent with the brightening at [3.6] and [4.5] over the same interval, as seen in Figure 13. 15ahp then faded by 0.17 mag at the 2019 HST observation. The star is very bright at J and H, as shown in the bottom two panels in Figure 17. There is thus little doubt that this object is the optical/near-IR counterpart of SPIRITS 15ahp. The HSC gives magnitudes (AB scale) of this star from the 2005 ACS frames of F658N = 21.74 and I = 20.25, and from the 2016 WFC3 frames of I = 20.07, J = 18.63, and H = 18.17.
[3.6] − [4.5] 0.9) and of relatively high luminosity (M [4.5] −13.9).
Figure 16 .
16Left: SED of the irregular variable SPIRITS 14atl in M83, constructed from HST/WFC3 photometry(Table 6)of the optical and near-IR counterpart obtained in 2009 August (orange circles) and 2016 January (red circles). The Spitzer [3.6] and [4.5] measurements are interpolated to the same two epochs. PHOENIX model stellar photospheres (nonrotating, solar-metallicity) with Teff = 3500 K, scaled to luminosities of log(L/L ) = 5.24 and 5.47) provide reasonable approximations to both data epochs. They are superposed in the corresponding colors. Right: SED of the irregular variable SPIRITS 15ahp in NGC 2403, constructed from HSC magnitudes from 2005 March (orange circles) and 2016 March (red circles), along with [3.6] and [4.5] estimates from interpolations of the Spitzer light curves to the corresponding dates.
Figure 17 .
17False-color renditions of HST images of the site of the irregular variable SPIRITS 15ahp in the nearby galaxy NGC 2403. Top row: I-band frames taken in 2005 (left) and 2016 (right). Green circles mark 3σ error positions of the Spitzer variable. A conspicuous optical counterpart of 15ahp lies within the error circle, and brightened by ∼0.25 mag from 2005 to 2016. Bottom row: HST images of the site in 2016, taken in J (left) and H (right). The counterpart is very bright at J and H. Height of each frame is 4. 8.
Table 1 .
1HST Wide Field Camera 3 Observing Log PI for 13935 and 14258 was H.E.B. PI for programs 14463 and 14892 was B. McCollum. In the 14463 program there were also exposures in F625W, F606W, F125W, and F140W; in 14892 there were also exposures in F105W. b SPIRITS 16tn has been analyzed in detail bySPIRITS
Host
Observation Program
Total Exposure Time [s]
Other SPIRITS
Designations
Galaxy
Date
ID a
F814W F110W F160W Variables in HST Field
14aje
M101
2014-09-22
13935
1575
738
177
. . .
14axa
M81
2014-09-26
13935
1659
671
155
. . .
14qb
NGC 4631 2015-11-02
14258
1575
738
177
. . .
15nz
M83
2016-01-18
14463
584
. . .
796
14akj, 14atl
15qo, 15aag
NGC 1313 2016-02-11
14258
1500
553
317
. . .
15ahg, 14al, 14dd NGC 2403 2016-03-07
14258
1500
553
317
15ahp
15afp
NGC 6946 2016-03-19
14258
1320
484
317
. . .
15wt, 14bbc
NGC 7793 2016-04-18
14258
1290
484
317
14th, 17fe
16aj
NGC 2903 2016-05-23
14258
1425
738
177
. . .
15mr, 15mt
NGC 4605 2016-06-14
14258
1575
. . .
684
. . .
16tn b
NGC 3556 2016-09-25
14258
1425
738
403
. . .
16ea
NGC 4214 2017-05-17
14892
. . .
2385
2385
. . .
a
Table 2 .
2HST Targets and Light-Curve Classifications aSPIRITS
R.A.
Decl.
Light-Curve
Designation
(J2000)
(J2000)
Classification
Field of 14aje in M101, m − M = 29.16 ± 0.02
14aje
14:02:55.51
+54:23:18.5
SPRITE (fast)
Field of 14axa in M81, m − M = 27.79 ± 0.06
14axa
09:56:01.52
+69:03:12.4
Transient b
Field of 14qb in NGC 4631, m − M = 29.33 ± 0.10
14qb
12:41:57.50
+32:32:06.7
Irregular
Field of 15nz in M83, m − M = 28.34 ± 0.07
15nz
13:37:08.37
−29:50:19.7
Periodic
14akj
13:37:03.59
−29:50:57.4
Irregular
14atl
13:37:07.96
−29:50:41.3
Irregular
Field of 15qo in NGC 1313, m − M = 28.14 ± 0.08
15qo
03:18:15.26
−66:30:03.4
Periodic
15aag
03:18:23.63
−66:30:24.2
Periodic
Field of 15ahg in NGC 2403, m − M = 27.51 ± 0.06
15ahg
07:36:37.40
+65:38:02.6
Periodic
14al
07:36:32.38
+65:37:26.1
Periodic?
14dd
07:36:30.04
+65:37:57.8
Periodic
15ahp
07:36:35.68
+65:37:47.2
Irregular
Field of 15afp in NGC 6946, m − M = 28.27 ± 0.07
15afp
20:34:59.65
+60:11:18.1
Periodic?
Field of 15wt in NGC 7793, m − M = 27.77 ± 0.07
15wt
23:57:43.38
−32:35:03.1
Periodic
14th
23:57:46.32
−32:34:41.3
Irregular
14bbc
23:57:46.28
−32:35:20.6
Periodic
17fe
23:57:44.77
−32:34:58.4
SPRITE (slow)
Field of 16aj in NGC 2903, m − M = 29.82 ± 0.45
16aj
09:32:11.64
+21:30:03.0
Irregular
Field of 15mr in NGC 4605, m − M = 28.72 ± 0.10
15mr
12:39:54.87
+61:36:46.3
Periodic
15mt
12:40:06.94
+61:36:22.3
Periodic?
Field of 16ea in NGC 4214, m − M = 27.34 ± 0.08
16ea
12:15:38.61
+36:19:46.9
Periodic?
Table 3 .
3Properties of Transients Time between first and last detections.SPIRITS
t0 a
Max. Age b
∆tLC c
tpeak
[4.5]peak
M[4.5],peak [3.6] − [4.5] d
Designation
[MJD]
[days]
[days]
[MJD]
[mag]
[mag]
[mag]
14aje
56742.84
694.45
28.98
56742.84 15.52 ± 0.03
−13.6
1.7 ± 0.1
14axa
56821.90
122.43
<258.75 56821.90 16.28 ± 0.08
−11.5
0.5 ± 0.1
17fe
57800.71
127.04
957.17 57828.22 15.83 ± 0.04
−10.9
0.69 ± 0.06
a Time of the first Spitzer/IRAC detection.
b Maximum age of transient at the time of discovery, i.e., time between first detection and the previous non-
detection
c
Table 2 .
2The absolute magnitudes cover the range −11.07 > M [4.5] > −12.31, except for one unusually luminous variable, 15mr, at M [4.5] −13.76.
Table 4 .
4Properties of Periodic and Suspected Periodic VariablesSPIRITS
Period [3.6] a ∆[3.6] a [4.5] a ∆[4.5] a [3.6] − [4.5] M[3.6]
M[4.5]
Designation [days] [mag] [mag] [mag] [mag]
[mag]
[mag]
[mag]
15nz b
1614
. . .
. . .
16.66
1.07
1.5
. . .
−12.17
15qo
1232 16.69
1.37
16.11
1.43
0.58
−11.45 −11.68
15aag
1233 18.10
1.63
17.07
1.55
1.04
−10.04 −11.07
15ahg c
1163 16.49
0.93
16.13
1.16
0.36
−11.02 −11.38
14al
∼2160 16.60
0.96
15.20
0.82
1.40
−10.91 −12.31
14dd
1418 16.89
0.95
15.92
1.11
0.97
−10.62 −11.59
15afp
>1650 17.63
0.79
16.87
1.13
0.76
−10.64 −11.40
15wt c
1188 16.55
0.90
16.03
0.94
0.53
−11.22 −11.75
14bbc c
1498 16.29
1.12
15.57
1.13
0.71
−11.48 −12.20
15mr c
1113 15.71
1.37
14.96
1.32
0.76
−13.01 −13.76
15mt c
>1800 16.86
1.67
16.69
2.16
0.17
−11.86 −12.03
16ea b,c
∼670
. . .
. . .
16.51
1.11
1.6
. . .
−10.83
a [3.6] and [4.5] denote the mean apparent magnitudes of the variables, and ∆[3.6] and ∆[4.5]
Table 5 .
5Properties of Irregular Variables Time of the observed light curve peak (Vega magnitudes) at either [3.6] or [4.5]. b Measured at time tpeak.SPIRITS
tpeak a
[3.6] b
∆[3.6]
[4.5] b
∆[4.5] [3.6] − [4.5] b M[3.6] b M[4.5] b
Designation
[MJD]
[mag]
[mag]
[mag]
[mag]
[mag]
[mag] [mag]
14qb
58619.73 15.91 ± 0.07
1.4
14.80 ± 0.02
1.0
1.11 ± 0.07
−13.4 −14.5
14th
56715.49 17.11 ± 0.07
1.2
15.51 ± 0.05
1.0
1.60 ± 0.09
−10.7 −12.3
14akj
54695.30 14.96 ± 0.03
0.7
13.82 ± 0.01
0.5
1.14 ± 0.03
−13.4 −14.5
14atl
58277.58 15.84 ± 0.02
1.3
15.69 ± 0.07
1.0
0.15 ± 0.07
−12.5 −12.7
15ahp
57388.85 15.47 ± 0.01
0.4
15.56 ± 0.03
0.2
−0.09 ± 0.03 −12.0 −12.0
16aj
57425.97 16.84 ± 0.34
0.8
15.94 ± 0.27
1.3
0.90 ± 0.43
−13.0 −13.9
a
Table 6 .
6HST WFC3 Photometry (AB Magnitudes) of Irregular Variable SPIRITS 14atl in M83 a GO-11360, PI R. O'Connell. b GO-14463, PI B. McCollum.c GO-14059, PI R Soria.Filter
2009 Aug 26 a 2016 Jan 18-19 b 2016 Jan 21 c
F555W
23.70
. . .
23.45
F606W
. . .
22.46
. . .
F625W
. . .
22.13
. . .
F814W
20.70
20.20
20.17
F110W
19.42
. . .
. . .
F125W
. . .
18.61
. . .
F140W
. . .
18.48
. . .
F160W
19.01
18.41
. . .
https://irsa.ipac.caltech.edu/Missions/spitzer.html 2 Super Mosaics are available as Spitzer Enhanced Imaging Products through the NASA/IPAC Infrared Science Archive: https://irsa.ipac.caltech. edu/data/SPITZER/Enhanced/SEIP/overview.html 3 https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/ iracinstrumenthandbook/
http://etc.stsci.edu/etc/input/wfc3uvis/imaging; the listed values are optimistic and only approximate because they neglect possible background light from the host galaxy and confusion in crowded star fields.
ATLAS forced-photometry server: https://fallingstar-data.com/ forcedphot/
We are grateful to D. Bishop for maintaining a website devoted to extragalactic novae, at https://www.rochesterastronomy.org/novae.html, which alerted us to this optical detection. The apparent coincidence of 14axa with the nova was also noted byOskinova et al. (2018).
. S M Adams, C S Kochanek, J L Prieto, MNRAS. 4601645Adams, S. M., Kochanek, C. S., Prieto, J. L., et al. 2016, MNRAS, 460, 1645
. J Bally, H Zinnecker, AJ. 1292281Bally, J., & Zinnecker, H. 2005, AJ, 129, 2281
. J Bally, A Ginsburg, J Forbrich, ApJ. 889178Bally, J., Ginsburg, A., Forbrich, J., et al. 2020, ApJ, 889, 178
. N Blagorodnova, R Kotak, J Polshaw, ApJ. 834107Blagorodnova, N., Kotak, R., Polshaw, J., et al. 2017, ApJ, 834, 107
. N Blagorodnova, V Karambelkar, S M Adams, MNRAS. 4965503Blagorodnova, N., Karambelkar, V., Adams, S. M., et al. 2020, MNRAS, 496, 5503
. H E Bond, ApJ. 73717Bond, H. E. 2011, ApJ, 737, 17
. H E Bond, Research Notes of the AAS. 2122Bond, H. E. 2018, Research Notes of the AAS, 2, 122
. H E Bond, L R Bedin, A Z Bonanos, ApJL. 695154Bond, H. E., Bedin, L. R., Bonanos, A. Z., et al. 2009, ApJL, 695, L154
. Y.-Z Cai, A Pastorello, M Fraser, MNRAS. 4803424Cai, Y.-Z., Pastorello, A., Fraser, M., et al. 2018, MNRAS, 480, 3424
. Y.-Z Cai, A Pastorello, M Fraser, A&A. 6326Cai, Y.-Z., Pastorello, A., Fraser, M., et al. 2019, A&A, 632, L6
. Y.-Z Cai, A Pastorello, M Fraser, A&A. 654157Cai, Y.-Z., Pastorello, A., Fraser, M., et al. 2021, A&A, 654, A157
. Y Cao, P E Nugent, M M Kasliwal, PASP. 128114502Cao, Y., Nugent, P. E. & Kasliwal, M. M. 2016, PASP, 128, 114502
. A Caratti O Garatti, B Stecklum, R Garcia Lopez, Nature Physics. 13276Caratti o Garatti, A., Stecklum, B., Garcia Lopez, R., et al. 2017, Nature Physics, 13, 276
. K De, M M Kasliwal, M J Hankins, ApJ. 91219De, K., Kasliwal, M. M., Hankins, M. J., et al. 2021, ApJ, 912, 19
. C L Doherty, P Gil-Pons, L Siess, MNRAS. 4462599Doherty, C. L., Gil-Pons, P., Siess, L., et al. 2015, MNRAS, 446, 2599
. C L Doherty, P Gil-Pons, L Siess, PASA. 3456Doherty, C. L., Gil-Pons, P., Siess, L., et al. 2017, PASA, 34, e056
. N Epchtein, Nguyen-Quang-Rieu, A&A. 107229Epchtein, N., & Nguyen-Quang-Rieu 1982, A&A, 107, 229
. A Evans, T R Geballe, J M C Rawlings, MNRAS. 292192Evans, A., Geballe, T. R., Rawlings, J. M. C., et al. 1997, MNRAS, 292, 192
. A Evans, R D Gehrz, Bulletin of the Astronomical Society of India. 40213Evans, A., & Gehrz, R. D. 2012, Bulletin of the Astronomical Society of India, 40, 213
. A Evans, R D Gehrz, L A Helton, MNRAS. 42469Evans, A., Gehrz, R. D., Helton, L. A., et al. 2012, MNRAS, 424, L69
. A Evans, V H Tyne, O Smith, MNRAS. 3601483Evans, A., Tyne, V. H., Smith, O., et al. 2005, MNRAS, 360, 1483
. G G Fazio, J L Hora, L E Allen, ApJS. 15410Fazio, G. G., Hora, J. L., Allen, L. E., et al. 2004, ApJS, 154, 10
. R D Gehrz, ARA&A. 26377Gehrz, R. D. 1988, ARA&A, 26, 377
. R D Gehrz, PhR. 311405Gehrz, R. D. 1999, PhR, 311, 405
. R D Gehrz, G L Grasdalen, J A Hackwell, ApJ. 237855Gehrz, R. D., Grasdalen, G. L., Hackwell, J. A., et al. 1980a, ApJ, 237, 855
. R D Gehrz, M A Greenhouse, T L Hayward, ApJL. 448119Gehrz, R. D., Greenhouse, M. A., Hayward, T. L., et al. 1995a, ApJL, 448, L119
. R D Gehrz, J A Hackwell, G I Grasdalen, ApJ. 239570Gehrz, R. D., Hackwell, J. A., Grasdalen, G. I., et al. 1980b, ApJ, 239, 570
. R D Gehrz, T J Jones, K Matthews, AJ. 110325Gehrz, R. D., Jones, T. J., Matthews, K., et al. 1995b, AJ, 110, 325
. S R Goldman, M L Boyer, K B W Mcquinn, ApJ. 87749Goldman, S. R., Boyer, M. L., McQuinn, K. B. W., et al. 2019, ApJ, 877, 49
. S R Goldman, J T Van Loon, J F Gómez, MNRAS. 4733835Goldman, S. R., van Loon, J. T., Gómez, J. F., et al. 2018, MNRAS, 473, 3835
. S R Goldman, J T Van Loon, A A Zijlstra, MNRAS. 465403Goldman, S. R., van Loon, J. T., Zijlstra, A. A., et al. 2017, MNRAS, 465, 403
. M A Greenhouse, G L Grasdalen, T L Hayward, AJ. 95172Greenhouse, M. A., Grasdalen, G. L., Hayward, T. L., et al. 1988, AJ, 95, 172
. M A Greenhouse, G L Grasdalen, C E Woodward, ApJ. 352307Greenhouse, M. A., Grasdalen, G. L., Woodward, C. E., et al. 1990, ApJ, 352, 307
. L Hartmann, S J Kenyon, ARA&A. 34207Hartmann, L., & Kenyon, S. J. 1996, ARA&A, 34, 207
K Hornoch, H Kucakova, The Astronomer's Telegram. 6176Hornoch, K., & Kucakova, H. 2014, The Astronomer's Telegram, 6176
K Hornoch, H Stoev, V Tudor, O Vaduvescu, M Frigo, The Astronomer's Telegram. 6188Hornoch, K., Stoev, H., Tudor, V., Vaduvescu, O., & Frigo, M. 2014, The Astronomer's Telegram, 6188
. G Howitt, S Stevenson, A Vigna-Gómez, MNRAS. 4923229Howitt, G., Stevenson, S., Vigna-Gómez, A., et al. 2020, MNRAS, 492, 3229
. T R Hunter, C L Brogan, G Macleod, ApJL. 83729Hunter, T. R., Brogan, C. L., MacLeod, G., et al. 2017, ApJL, 837, L29
. I S Jang, M G Lee, ApJ. 83674Jang, I. S., & Lee, M. G. 2017, ApJ, 836, 74
. J E Jencson, California Institute of Technology (J20PhD. ThesisJencson, J. E. 2020, PhD. Thesis, California Institute of Technology (J20)
. J E Jencson, S M Adams, H E Bond, ApJL. 88020Jencson, J. E., Adams, S. M., Bond, H. E., et al. 2019, ApJL, 880, L20
. J E Jencson, M M Kasliwal, S M Adams, ApJ. 86320Jencson, J. E., Kasliwal, M. M., Adams, S. M., et al. 2018, ApJ, 863, 20
. J E Jencson, M M Kasliwal, S M Adams, ApJ. 88640Jencson, J. E., Kasliwal, M. M., Adams, S. M., et al. 2019, ApJ, 886, 40
. J E Jencson, M M Kasliwal, J Johansson, ApJ. 837167Jencson, J. E., Kasliwal, M. M., Johansson, J., et al. 2017, ApJ, 837, 167
J E Jencson, M M Kasliwal, S Tinyanont, The Astronomer's Telegram. 79291Jencson, J. E., Kasliwal, M. M., Tinyanont, S., et al. 2015, The Astronomer's Telegram 7929, 1
J E Jencson, M M Kasliwal, S Tinyanont, The Astronomer's Telegram. 86881Jencson, J. E., Kasliwal, M. M., Tinyanont, S., et al. 2016a, The Astronomer's Telegram 8688, 1
J E Jencson, M M Kasliwal, S Tinyanont, The Astronomer's Telegram. 89401Jencson, J. E., Kasliwal, M. M., Tinyanont, S., et al. 2016b, The Astronomer's Telegram 8940, 1
. T J Jones, A R Hyland, J L Caswell, ApJ. 253208Jones, T. J., Hyland, A. R., Caswell, J. L., et al. 1982, ApJ, 253, 208
. T Kamiński, E Mason, R Tylenda, A&A. 58034Kamiński, T., Mason, E., Tylenda, R., et al. 2015, A&A, 580, 34
. V R Karambelkar, S M Adams, P A Whitelock, ApJ. 877K19110Karambelkar, V. R., Adams, S. M., Whitelock, P. A., et al. 2019, ApJ, 877, 110 (K19)
. M M Kasliwal, J Bally, F Masci, ApJ. 839K1788Kasliwal, M. M., Bally, J., Masci, F., et al. 2017, ApJ, 839, 88 (K17)
. C S Kochanek, ApJ. 74137Kochanek, C. S. 2011, ApJ, 741, 37
. A Kučinskas, P H Hauschildt, H. -G Ludwig, A&A. 442281Kučinskas, A., Hauschildt, P. H., Ludwig, H. -G., et al. 2005, A&A, 442, 281
. A Kučinskas, P H Hauschildt, I Brott, A&A. 4521021Kučinskas, A., Hauschildt, P. H., Brott, I., et al. 2006, A&A, 452, 1021
. R M Lau, M J Hankins, M M Kasliwal, ApJ. 909113Lau, R. M., Hankins, M. J., Kasliwal, M. M., et al. 2021, ApJ, 909, 113
. F J Masci, R R Laher, U D Rebbapragada, PASP. 12914002Masci, F. J., Laher, R. R., Rebbapragada, U. D., et al. 2017, PASP, 129, 014002
. J W Menzies, P A Whitelock, M W Feast, MNRAS. 4835150Menzies, J. W., Whitelock, P. A., Feast, M. W., et al. 2019, MNRAS, 483, 5150
. B D Metzger, O Pejcha, MNRAS. 4713200Metzger, B. D., & Pejcha, O. 2017, MNRAS, 471, 3200
K Mukai, Acta Polytechnica CTU Proceedings. 2246Mukai, K. 2015, Acta Polytechnica CTU Proceedings, 2, 246
. E P Ney, B F Hatfield, ApJL. 219111Ney, E. P., & Hatfield, B. F. 1978, ApJL, 219, L111
. L M Oskinova, T Bulik, A N Gómez-Morán, A&A. 61310Oskinova, L. M., Bulik, T., & Gómez-Morán, A. N. 2018, A&A, 613, L10
. A Pastorello, E Mason, S Taubenberger, A&A. 63075Pastorello, A., Mason, E., Taubenberger, S., et al. 2019, A&A, 630, A75
. O Pejcha, J L Prieto, ApJ. 799215Pejcha, O., & Prieto, J. L. 2015, ApJ, 799, 215
. O Pejcha, B D Metzger, K Tomida, MNRAS. 4554351Pejcha, O., Metzger, B. D. & Tomida, K. 2016a, MNRAS, 455, 4351
. O Pejcha, B D Metzger, K Tomida, MNRAS. 4612527Pejcha, O., Metzger, B. D. & Tomida, K. 2016b, MNRAS, 461, 2527
. J L Prieto, ATel. 1550Prieto, J. L. 2008, ATel, 1550
. J L Prieto, M D Kistler, T A Thompson, ApJL. 6819Prieto, J. L., Kistler, M. D., Thompson, T. A., et al. 2008, ApJL, 681, L9
. L M Rebull, A M Cody, K R Covey, AJ. 14892Rebull, L. M., Cody, A. M., Covey, K. R., et al. 2014, AJ, 148, 92
. B Reipurth, S Mikkola, AJ. 149145Reipurth, B., & Mikkola, S. 2015, AJ, 149, 145
. D Riebel, M L Boyer, S Srinivasan, ApJ. 8071Riebel, D., Boyer, M. L., Srinivasan, S., et al. 2015, ApJ, 807, 1
. B E Schaefer, ApJS. 187275Schaefer, B. E. 2010, ApJS, 187, 275
. A W Shafter, ApJ. 834196Shafter, A. W. 2017, ApJ, 834, 196
. L Siess, A&A. 476893Siess, L. 2007, A&A, 476, 893
. M Sirianni, M J Jee, N Benítez, PASP. 1171049Sirianni, M., Jee, M. J., Benítez, N., et al. 2005, PASP, 117, 1049
. K W Smith, S J Smartt, D R Young, PASP. 13285002Smith, K. W., Smartt, S. J., Young, D. R., et al. 2020, PASP, 132, 085002
. I Soszyński, A Udalski, M K Szymański, AcA. 59239Soszyński, I., Udalski, A., Szymański, M. K., et al. 2009, AcA, 59, 239
. W B Sparks, H E Bond, M Cracraft, AJ. 135605Sparks, W. B., Bond, H. E., Cracraft, M., et al. 2008, AJ, 135, 605
. T Szalai, S Zsíros, O D Fox, ApJS. 38Szalai, T., Zsíros, S., Fox, O. D., et al. 2019, ApJS, 241, 38
. D M Szczygieł, J L Prieto, C S Kochanek, ApJ. 75077Szczygieł, D. M., Prieto, J. L., Kochanek, C. S., et al. 2012, ApJ, 750, 77
. S Tinyanont, M M Kasliwal, O D Fox, ApJ. 833231Tinyanont, S., Kasliwal, M. M., Fox, O. D., et al. 2016, ApJ, 833, 231
. J L Tonry, L Denneau, A N Heinze, PASP. 13064505Tonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130, 064505
. R B Tully, H M Courtois, A E Dolphin, AJ. 14686Tully, R. B., Courtois, H. M., Dolphin, A. E., et al. 2013, AJ, 146, 86
. R B Tully, H M Courtois, J G Sorce, AJ. 15250Tully, R. B., Courtois, H. M., & Sorce, J. G. 2016, AJ, 152, 50
. R Tylenda, M Hajduk, T Kamiński, A&A. 528114Tylenda, R., Hajduk, M., Kamiński, T., et al. 2011, A&A, 528, A114
. J T Vanderplas, Ž Ivezić, ApJ. 81218VanderPlas, J. T., & Ivezić, Ž. 2015, ApJ, 812, 18
. F M Walter, A Battisti, S E Towers, H E Bond, G S Stringfellow, PASP. 1241057Walter, F. M., Battisti, A., Towers, S. E., Bond, H. E., & Stringfellow, G. S. 2012, PASP, 124, 1057
. P A Whitelock, M W Feast, J T Van Loon, MNRAS. 34286Whitelock, P. A., Feast, M. W., van Loon, J. T., et al. 2003, MNRAS, 342, 86
. P A Whitelock, M Kasliwal, M Boyer, EPJWC. 1521009Whitelock, P. A., Kasliwal, M., & Boyer, M. 2017, EPJWC, 152, 01009
. P A Whitelock, J W Menzies, M W Feast, MNRAS. 473173Whitelock, P. A., Menzies, J. W., Feast, M. W., et al. 2018, MNRAS, 473, 173
. B C Whitmore, S S Allam, T Budavári, AJ. 151134Whitmore, B. C., Allam, S. S., Budavári, T., et al. 2016, AJ, 151, 134
. P M Williams, K A Van Der Hucht, F Van Wyk, MNRAS. 4202526Williams, P. M., van der Hucht, K. A., van Wyk, F., et al. 2012, MNRAS, 420, 2526
. S C Williams, D Jones, P Pessev, A&A. 63720Williams, S. C., Jones, D., Pessev, P., et al. 2020, A&A, 637, A20
. C E Woodward, A Evans, D P K Banerjee, AJ. 162183Woodward, C. E., Evans, A., Banerjee, D. P. K., et al. 2021, AJ, 162, 183
. M Yang, B W Jiang, ApJ. 72753Yang, M., & Jiang, B. W. 2011, ApJ, 727, 53
. S.-C Yoon, M Cantiello, ApJL. 71762Yoon, S.-C., & Cantiello, M. 2010, ApJL, 717, L62
| [] |
[
"Mean Field Linear Quadratic Control: FBSDE and Riccati Equation Approaches",
"Mean Field Linear Quadratic Control: FBSDE and Riccati Equation Approaches"
] | [
"Member, IEEEBingchang Wang ",
"Senior Member, IEEEHuanshui Zhang "
] | [] | [] | This paper studies social optima and Nash games for mean field linear quadratic control systems, where subsystems are coupled via dynamics and individual costs. For the social control problem, we first obtain a set of forward-backward stochastic differential equations (FBSDE) from variational analysis, and construct a feedback-type control by decoupling the FBSDE. By using solutions of two Riccati equations, we design a set of decentralized control laws, which is further proved to be asymptotically social optimal. Two equivalent conditions are given for uniform stabilization of the systems in different cases. For the game problem, we first design a set of decentralized control from variational analysis, and then show that such set of decentralized control constitute an asymptotic Nash equilibrium by exploiting the stabilizing solution of a nonsymmetric Riccati equation.It is verified that the proposed decentralized control laws are equivalent to the feedback strategies of mean field control in previous works. This may illustrate the relationship between open-loop and feedback solutions of mean field control (games). | null | [
"https://arxiv.org/pdf/1904.07522v1.pdf"
] | 119,295,119 | 1904.07522 | a747a85d9b117ad35aa89c2a74b811f8f3d9d236 |
Mean Field Linear Quadratic Control: FBSDE and Riccati Equation Approaches
16 Apr 2019
Member, IEEEBingchang Wang
Senior Member, IEEEHuanshui Zhang
Mean Field Linear Quadratic Control: FBSDE and Riccati Equation Approaches
16 Apr 2019arXiv:1904.07522v1 [math.OC] JOURNAL OF L A T E X CLASS FILES 1
This paper studies social optima and Nash games for mean field linear quadratic control systems, where subsystems are coupled via dynamics and individual costs. For the social control problem, we first obtain a set of forward-backward stochastic differential equations (FBSDE) from variational analysis, and construct a feedback-type control by decoupling the FBSDE. By using solutions of two Riccati equations, we design a set of decentralized control laws, which is further proved to be asymptotically social optimal. Two equivalent conditions are given for uniform stabilization of the systems in different cases. For the game problem, we first design a set of decentralized control from variational analysis, and then show that such set of decentralized control constitute an asymptotic Nash equilibrium by exploiting the stabilizing solution of a nonsymmetric Riccati equation.It is verified that the proposed decentralized control laws are equivalent to the feedback strategies of mean field control in previous works. This may illustrate the relationship between open-loop and feedback solutions of mean field control (games).
I. INTRODUCTION
Mean field games have drawn increasing attention in many fields including system control, applied mathematics and economics [7], [8], [12]. The mean field game involves a very large (e-mail: [email protected]) population of small interacting players with the feature that while the influence of each one is negligible, the impact of the overall population is significant. By combining mean field approximations and individual's best response, the dimensionality difficulty is overcome. Mean field games and control have found wide applications, including smart grids [27], [10], finance, economics [13], [9], [32], and social sciences [5], etc.
By now, mean field games have been intensively studied in the LQ (linear-quadratic) framework [18], [19], [25], [33], [6], [29]. Huang et al. developed the Nash certainty equivalence (NCE) based on the fixed-point method and designed an ǫ-Nash equilibrium for mean field LQ games with discount costs by the NCE approach [18], [19]. The NCE approach was then applied to the cases with long run average costs [25] and with Markov jump parameters [33], respectively.
Bensoussan et al. employed the adjoint equation approach and the fixed-point theorem to obtain a sufficient condition for the unique existence of the equilibrium strategy over a finite horizon [6]. For other aspects of mean field games, readers are referred to [21], [23], [39], [11] for nonlinear mean field games, [37] for oblivious equilibrium in dynamic games, [17], [34], [35] for mean field games with major players, [16], [29] for robust mean field games.
Besides noncooperative games, social optima in mean field models have also attracted much interest. The social optimum control refers to that all the players cooperate to optimize the common social cost-the sum of individual costs, which is usually regarded as a type of team decision problem [30], [14]. Huang et al. considered social optima in mean field LQ control, and provided an asymptotic team-optimal solution [20]. Wang and Zhang [36] investigated a mean field social optimal problem where the Markov jump parameter appears as a common source of randomness. For further literature, see [22] for social optima in mixed games, [3] for team-optimal control with finite population and partial information.
Most previous results on mean field games and control were given by virtue of the fixed-point analysis. However, the fixed-point method is sometimes conservative, particularly for general systems. In this paper, we break away from the fixed-point method and solve the problem by tackling forward-backward stochastic differential equations (FBSDE). In recent years, some substantial progress for the optimal LQ control has been made by solving the FBSDE. See [40], [42], [43], [31] for details. This paper investigates social optima and Nash games for linear quadratic mean field systems, where subsystems (agents) are coupled via dynamics and individual costs. For the finite-horizon social control problem, we first obtain a set of forward-backward stochastic differential equations (FBSDE) by examining the variation of the social cost, and give a centralized feedback-type control laws by decoupling the FBSDE. With mean field approximations, we design a set of decentralized control laws, which is further shown to have asymptotic social optimality. For the infinite-horizon case, we design a set of decentralized control laws by using solutions of two Riccati equations, which is shown to be asymptotically social optimal. Some equivalent conditions are further given for uniform stabilization of the multiagent systems when the state weight Q is semi-positive definite or only symmetric. For the problem of mean field games, we first design a set of decentralized control by variational analysis, whose control gain satisfies a nonsymmetric Riccati equation. With the help of the stabilizing solution of the nonsymmetric Riccati equation, we show that the set of decentralized control laws is an asymptotic Nash equilibrium. It is verified that the proposed decentralized control laws are equivalent representation of the feedback strategies in previous works of mean field control and games. Finally, some numerical examples are given to illustrate the effectiveness of the proposed control laws.
The main contributions of the paper are summarized as follows.
(i) For the social control problem, we first obtain necessary and sufficient existence conditions of finite-horizon centralized optimal control by variational analysis, and then design a feedbacktype decentralized control by tackling FBSDE with mean field approximations.
(ii) In the case Q ≥ 0, the necessary and sufficient conditions are given for uniform stabilization of the systems with the help of the system's observability and detectability.
(iii) In the case that Q is only symmetric, the necessary and sufficient conditions are given for uniform stabilization of the systems using the Hamiltonian matrices.
(iv) For the game problem, we show that the decentralized control laws constitute an ε-Nash equilibrium by exploiting the stabilizing solution of a nonsymmetric Riccati equation.
(v) It is under nonconservative assumptions that we obtain the asymptotically optimal decentralized control, and such control laws are shown to be equivalent to the feedback strategies given by the fixed-point method in previous works [19], [20].
The organization of the paper is as follows. In Section II, the socially optimal control problem is investigated. We first construct asymptotically optimal decentralized control laws by tackling FBSDE for the finite-horizon case, then design asymptotically optimal control for the infinitehorizon case and further give two equivalent conditions of uniform stabilization for different cases. In Section III, we design a decentralized ε-Nash equilibrium for the finite-horizon and infinite-horizon cases, respectively. The proposed decentralized control laws are compared with April 17, 2019 DRAFT JOURNAL OF L A T E X CLASS FILES 4 the feedback strategies of previous works in Section IV. In Section V, some numerical examples are given to show the effectiveness of the proposed control laws. Section VI concludes the paper.
The following notation will be used throughout this paper. · denotes the Euclidean vector norm or matrix spectral norm. For a vector z and a matrix Q, z 2 Q = z T Qz, and Q > 0 (Q ≥ 0) means that Q is positive definite (semi-positive definite). For two vectors x, y, x, y = x T y. C([0, T ], R n ) is the space of all R n -valued continuous functions defined on [0, T ], and
C ρ/2 ([0, ∞), R n ) is a subspace of C([0, ∞), R n ) which is given by {f | ∞ 0 e −ρt f (t) 2 dt < ∞}. L 2 F (0, T ; R k ) is the space of all F -adapted R k -valued processes x(·) such that E T 0 x(t) dt < ∞.
For two sequences {a n , n = 0, 1, · · · } and {b n , n = 0, 1, · · · }, a n = O(b n ) denotes that lim sup n→∞ |a n /b n | ≤ C, and a n = o(b n ) denotes lim sup n→∞ |a n /b n | = 0. For convenience of presentation, we use C, C 1 , C 2 , · · · to denote generic positive constants, which may vary from place to place.
II. MEAN FIELD LQ SOCIAL CONTROL
Consider a large population systems with N agents. Agent i evolves by the following stochastic differential equation:
dx i (t) = [Ax i (t) + Bu i (t) + Gx (N ) (t) + f (t)]dt + σ(t)dW i (t), 1 ≤ i ≤ N,(1)
where x i ∈ R n and u i ∈ R r are the state and input of the ith agent.
x (N ) (t) = 1 N N j=1 x j (t), f, σ ∈ C ρ/2 ([0, ∞), R n ). {W i (t), 1 ≤ i ≤ N} are a sequence of independent 1-dimensional
Brownian motions on a complete filtered probability space (Ω, F , {F t } 0≤t≤T , P). The cost function of agent i is given by
J i (u) = E ∞ 0 e −ρt x i (t) − Γx (N ) (t) − η 2 Q + u i (t) 2 R dt,(2)
where Q, R are symmetric matrices with appropriate dimensions, and R > 0. Denote u = {u 1 , . . . , u i , . . . , u N }. The decentralized control set is given by
U d,i = u i u i (t) is adapted to σ(x i (s), 0 ≤ s ≤ t), E ∞ 0 e −ρt u i (t) 2 dt < ∞ .
For comparison, define the centralized control sets as
U c,i = u i u i (t) is adapted to F t , E ∞ 0 e −ρt u i (t) 2 dt < ∞ , and U c = (u 1 , · · · , u N ) u i is adapted to U c,i , where F t = σ{ N i=1 F i t } and F i t = σ(x i (0), W i (s), 0 ≤ s ≤ t), i = 1, · · · , N.
In this section, we mainly study the following problem.
(PS). Seek a set of decentralized control laws to optimize social cost for the system (1)-(2),
i.e., inf u i ∈U d,i J soc , where J soc = N i=1 J i (u).
Assume A1) x i (0), i = 1, ..., N are mutually independent and have the same mathematical expectation.
x i (0) = x i0 , Ex i (0) =x 0 , i = 1, · · · , N. There exists a constant C 0 (independent of N) such that max 1≤i≤N E x i (0) 2 < C 0 .
A. The finite-horizon problem
For the convenience of design, we first consider the following finite-horizon problem.
(P1) inf u∈L 2 F t (0,T ;R r ) J F soc (u), where J F soc (u) = N i=1 J F i (u) and J F i (u) = E T 0 e −ρt x i (t) − Γx (N ) (t) − η 2 Q + u i (t) 2 R dt.(3)
We first give an equivalent condition for the convexity of Problem (P1).
u i ∈ L 2 Ft (0, T ; R r ), i = 1, · · · , N, N i=1 E T 0 e −ρt y i − Γy (N ) 2 Q + u i 2 R dt ≥ 0,
where y (N ) = N j=1 y j /N and y i satisfies
dy i = [Ay i + Gy (N ) + Bu i ]dt, y i (0) = 0, i = 1, 2, · · · , N.(4)
Proof. Let x i andx i be the state processes of agent i with the control v andv, respectively.
Take any λ 1 ∈ [0, 1] and let λ 2 = 1 − λ 1 . Then
λ 1 J F soc (v) + λ 2 J F soc (v) − J F soc (λ 1 v + λ 2v ) =λ 1 λ 2 E T 0 x i −x i − Γ(x N −x N ) 2 Q + u i −ú i 2 R dt.
Denote u = v −v, and y i = x i −x i . Thus, y i satisfies (4). By the definition of the convexity, the lemma follows.
April 17, 2019 DRAFT By examining the variation of J F soc , we obtain the necessary and sufficient conditions for the existence of centralized optimal control of (P1). Theorem 2.1: Suppose R > 0. Then (P1) has a set of optimal control laws if and only if Problem (P1) is convex in u and the following equation system admits a set of solutions
(x i , p i , β j i , i, j = 1, · · · , N): dx i = Ax i − BR −1 B T p i + Gx (N ) + f dt + σdW i , dp i = − (A − ρI) T p i + G T p (N ) dt − Qx i − Q Γ x (N ) −η dt + N j=1 β j i dW j , x i (0) = x i0 , p i (T ) = 0, i = 1, · · · , N,(5)where Q Γ ∆ = Γ T Q + QΓ − Γ T QΓ,η ∆ = Qη − Γ T Qη, p (N ) = 1 N N i=1
p i , and furthermore the optimal control is given byǔ
i = −R −1 B T p i . Proof. Suppose thatǔ i = −R −1 B T p i , where p i , i = 1, · · · ,dp i = α i dt + β i i dW i + j =i β j i dW j , p i (T ) = 0, i = 1, · · · , N,(6)
where α i , i = 1, · · · , N are to be determined. Denote byx i the state of agent i under the controľ u i . For any u i ∈ L 2 Ft (0, T ; R r ) and θ ∈ R, let u θ i =ǔ i + θu i . Denote by x θ i the solution of the following perturbed state equation
dx θ i = Ax θ i + B(ǔ i + θu i ) + f + G N N i=1 x θ i dt + σdW i , x θ i (0) = x i0 , i = 1, 2, · · · , N.
Let y i = (x θ i −x i )/θ. It can be verified that y i satisfies (4). Then by Itô's formula, for any i = 1, · · · , N,
0 = E[ e −ρT p i (T ), y i (T ) − p i (0), y i (0) ] = E T 0 α i , y i + p i , (A − ρI)y i + Gy (N ) + Bu i dt, April 17, 2019 DRAFT which implies 0 = N i=1 E T 0 e −ρt α i , y i + p i , (A − ρI)y i + Gy (N ) + Bu i dt = N i=1 E T 0 e −ρt α i + (A − ρI) T p i , y i + p i , Bu i dt + E T 0 e −ρt N i=1 p i , G N N i=1 y i dt = N i=1 E T 0 e −ρt α i + (A − ρI) T p i + G T p (N ) , y i + B T p i , u i dt.(7)
From (3), we haveJ
F soc (ǔ + θu) −J F soc (ǔ) = 2θI 1 + θ 2 I 2(8)
whereǔ = (ǔ 1 , · · · ,ǔ N ), and
I 1 ∆ = N i=1 E T 0 e −ρt Q x i − (Γx (N ) + η) , y i − Γy (N ) + Rǔ i , u i dt, I 2 ∆ = N i=1 E T 0 e −ρt y i − Γy (N ) 2 Q + u i 2 R dt. Note that N i=1 E T 0 e −ρt Q x i − (Γx (N ) + η) , Γy (N ) dt =E T 0 e −ρt Γ T Q N i=1 x i − (Γx (N ) + η) , 1 N N j=1 y j dt = N j=1 E T 0 e −ρt Γ T Q N N i=1 x i − (Γx (N ) + η) , y j dt = N j=1 E T 0 e −ρt Γ T Q (I − Γ)x (N ) − η , y j dt.
April 17, 2019 DRAFT From (7), one can obtain that
I 1 =E N i=1 T 0 e −ρt Q x i − (Γx (N ) + η) , y i − Γy (N ) dt + Rǔ i + B T p i , u i dt + N i=1 E T 0 e −ρt α i + (A − ρI) T p i + G T p (N ) , y i = N i=1 E T 0 e −ρt Rǔ i + B T p i , u i dt + N i=1 E T 0 e −ρt Q x i − (Γx (N ) + η) − Γ T Q (I − Γ)x (N ) − η + α i + (A − ρI) T p i + G T p (N ) , y i dt.(9)
From (8),ǔ is a minimizer to Problem (P1) if and only if I 2 ≥ 0 and I 1 = 0. By Proposition
2.1, I 2 ≥ 0 if and only if (P1) is convex. I 1 = 0 is equivalent to α i = − (A − ρI) T p i − Γ T Q (I − Γ)x (N ) − η + Q x i − (Γx (N ) + η)) + G T p (N ) , u i = − R −1 B T p i .
Thus, we have the following optimality system:
dx i = (Ax i − BR −1 B Tp i + Gx (N ) + f )dt + σdW i , dp i = − [(A − ρI) Tp i + G Tp(N ) + Qx i − Q Γx (N ) +η)]t + N j=1 β j i dW j , x i (0) = x i0 ,p i (T ) = 0, i = 1, · · · , N,(10)such thatǔ i = −R −1 B Tp i . This implies that the equation systems (5) admits a solution (x i ,p i ,β j i , i, j = 1, · · · , N). On other hand, if the equation system (5) admits a solution (x i ,p i ,β j i , i, j = 1, · · · , N). Leť u i = −R −1 B Tp i . If (P1) is convex, thenǔ is a minimizer to Problem (P1). It follows from (5) that dx (N ) = (A + G)x (N ) − BR −1 B T p (N ) + f dt + 1 N N i=1 σdW i , dp (N ) = − (A + G − ρI) T p (N ) − (I − Γ) T Q(I − Γ)x (N ) +η dt + 1 N N i=1 N j=1 β j i dW j , x (N ) (0) = 1 N N i=1 x i0 , p (N ) (T ) = 0.(11)
Let p i = P x i + Kx (N ) + s. Then by (5), (11) and Itô's formula,
dp i =P Ax i − BR −1 B T (P x i + Kx (N ) + s) + Gx (N ) + f dt + σdW i + (Ṗ x i +ṡ +Kx (N ) )dt + K (A + G)x (N ) − BR −1 B T ((P + K)x (N ) + s) + f dt + 1 N N i=1 σdW i = − (A − ρI) T (P x i + Kx (N ) + s) + G T ((P + K)x (N ) + s) + Qx i − Q Γ x (N ) −η dt + N j=1 β j i dW j .
This implies that
β i i = 1 N Kσ + P σ, β j i = 1 N Kσ, j = i, ρP =Ṗ + A T P + P A − P BR −1 B T P + Q, P (T ) = 0,(12)ρK =K + (A + G) T K + K(A + G) − P BR −1 B T K − KBR −1 B T P + G T P + P G − KBR −1 B T K − Q Γ , K(T ) = 0,(13)ρs =ṡ + [A + G − BR −1 B T (P + K)] T s + (P + K)f −η, s(T ) = 0.(14)Thenǔ i = −R −1 B T (P x i + Kx (N ) + s).
Theorem 2.2: Assume that A1) holds and Q ≥ 0. Then Problem (P1) has an optimal controľ
u i = −R −1 B T (P x i + Kx (N ) + s),
where P, K and s are determined by (12)- (14).
Proof. Denote Π = P + K. Then from (13) and (14), Π satisfies
ρΠ =Π + (A + G) T Π + Π(A + G) − ΠBR −1 B T Π +Q, Π(T ) = 0,(15)whereQ ∆ = (I − Γ) T Q(I − Γ)
. Note that Q ≥ 0 and R > 0. By [2], [41], (12) and (15) admit unique solutions P ≥ 0 and Π ≥ 0, respectively, which implies that (13) and (14) have unique April 17, 2019 DRAFT solutions K and s, respectively. Then by [26], [42], the FBSDE (5) admits a unique solution.
By Theorem 2.1, Problem (P1) has an optimal control given byǔ
i = −R −1 B T (P x i +Kx (N ) +s),
where P, K and s are determined by (12)- (14).
As an approximation to x (N ) in (11), we obtain
dx dt = (A + G)x − BR −1 B T (Πx + s) + f,x(0) =x 0 .(16)
Then, by Theorem 2.2, the decentralized control law for agent i may be taken aŝ
u i (t) = − R −1 B T (Px i (t) + Kx(t) + s(t)), 0 ≤ t ≤ T, i = 1, · · · , N,(17)
where P, K, and s are determined by (12)- (14), andx andx i satisfy (16) and
dx i = (A − BR −1 B T P )x i − BR −1 B T [Kx + s] + Gx (N ) + f dt + σdW i .(18)
Remark 2.1: In previous works [20], [36], the mean field term x (N ) in cost functions given by (17) has asymptotic social optimality, i.e.,
1 N J F soc (û) − 1 N inf u∈L 2 F t (0,T ;R nr ) J F soc (u) = O( 1 √ N ).
Proof. See Appendix A.
B. The infinite-horizon problem
Based on the analysis in Section II-A, we may design the following decentralized control laws for Problem (PS):
u i (t) = − R −1 B T [Px i (t) + (Π − P )x(t) + s(t)], t ≥ 0, i = 1, · · · , N,(19)
where P and Π are determined by
ρP =A T P + P A − P BR −1 B T P + Q,(20)ρΠ =(A + G) T Π + Π(A + G) − ΠBR −1 B T Π +Q,(21)
and s,x ∈ C ρ/2 ([0, ∞), R n ) are determined by
ρs =ṡ + [A + G − BR −1 B T Π] T s + Πf −η,(22)dx dt = (A + G)x − BR −1 B T (Πx + s) + f,x(0) =x 0 .(23)
Here the existence conditions of P, Π, s andx need to be investigated further.
We introduce some assumptions:
A2) The system (A − ρ 2 I, B) is stabilizable, and (A + G − ρ 2 I, B) is stabilizable. A3) Q ≥ 0, (A − ρ 2 I, √ Q) is observable, and (A + G − ρ 2 I, √ Q(I − Γ)) is observable.
Assumptions A2) and A3) are basic in the study of the LQ optimal control problem. We will
show that under some conditions, A2) is also necessary for uniform stabilization of multiagent systems. In many cases, A3) may be weakened to the following assumption. (20) and (21) admit unique solutions P > 0, Π > 0, respectively,
A3 ′ ) Q ≥ 0, (A − ρ 2 I, √ Q) is detectable, and (A + G − ρ 2 I, √ Q(I − Γ)) is detectable. Lemma 2.1: Under A2)-A3),and (22)-(23) admits a set of unique solutions s,x ∈ C ρ/2 ([0, ∞), R n ).
Proof. From A2)-A3) and [2], (20) and (21) admit unique solutions P > 0, Π > 0 such that
A − BR −1 B T P − ρ 2 I and A + G − BR −1 B T Π − ρ 2 I are Hurwitz, respectively. From an argument in [34, Appendix A], we obtain s ∈ C ρ/2 ([0, ∞), R n ) if and only if s(0) = ∞ 0 e (A+G−BR −1 B T Π−ρI)τ (Πf −η)dτ.
Under this initial condition, we have
s(t) = ∞ t e −(A+G−BR −1 B T Π−ρI)(t−τ ) (Πf −η)dτ. It is straightforward thatx ∈ C ρ/2 ([0, ∞), R n ).
We further introduce the following assumption.
A4)Ā + G − ρ 2 I is Hurwitz, whereĀ ∆ = A − BR −1 B T P . Lemma 2.2: Let A1)-A4) hold. Then for (PS), E ∞ 0 e −ρt x (N ) (t) −x(t) 2 dt = O( 1 N ).(24)
Proof. See Appendix B.
It is shown that the decentralized control laws (17) uniformly stabilize the systems (1) .
Theorem 2.4: Let A1)-A4) hold. Then for any N,
N i=1 E ∞ 0 e −ρt x i (t) 2 + û i (t) 2 dt < ∞.(25)April 17, 2019 DRAFT Proof. See Appendix B.
We now give two equivalent conditions for uniform stabilization of multiagent systems.
Theorem 2.5: Let A3) hold. Then for (PS) the following statements are equivalent:
(i) For any initial condition (x 1 (0), · · · ,x N (0)) satisfying A1), N i=1 E ∞ 0 e −ρt x i (t) 2 + û i (t) 2 dt < ∞.(26)
(ii) (20) and (21) admit unique solutions P > 0, Π > 0, respectively, andĀ + G − ρ 2 I is Hurwitz.
(iii) A2) and A4) hold.
Proof. See the Appendix C.
For the case G = 0, we have a simplified version of Theorem 2.5.
Corollary 2.1: Assume that A3) holds and G = 0. Then for (PS) the following statements are equivalent:
(i) For any (x 1 (0), · · · ,x N (0)) satisfying A1), N i=1 E ∞ 0 e −ρt x i (t) 2 + û i (t) 2 dt < ∞.
(ii) (20) and (21) admit unique solutions P > 0, Π > 0, respectively.
(iii) A2) holds.
When A3) is weakened to A3 ′ ), we have the following equivalent conditions of uniform stabilization of the systems. Theorem 2.6: Let A3 ′ ) hold. Then for (PS) the following statements are equivalent:
(i) For any initial condition (x 1 (0), · · · ,x N (0)) satisfying A1), N i=1 E ∞ 0 e −ρt x i (t) 2 + û i (t) 2 dt < ∞.
(ii) (20) and (21) admit unique solutions P ≥ 0, Π ≥ 0, respectively, andĀ + G − ρ 2 I is Hurwitz.
(iii) A2) and A4) hold.
Proof. See the Appendix C. For the more general case that Q are only symmetric, we have the following equivalent conditions for uniform stabilization of multiagent systems.
Denote
M 1 = A − ρ 2 I BR −1 B T Q −A T + ρ 2 I , M 2 = A + G − ρ 2 I BR −1 B T Q −(A + G) T + ρ 2 I .
Theorem 2.7: Assume that both M 1 and M 2 have no eigenvalues on the imaginary axis. Then for (PS) the following statements are equivalent:
(i) For any (x 1 (0), · · · , x N (0)) satisfying A1), N i=1 E ∞ 0 e −ρt x i (t) 2 + û i (t) 2 dt < ∞.
(ii) (20) and (21) admit ρ-stabilizing solutions 1 , respectively, andĀ + G − ρ 2 I is Hurwitz. (iii) A2) and A4) hold. (20) and (21) admit ρ-stabilizing solutions, respec- Then
tively, andĀ + G − ρ 2 I is Hurwitz. Then N i=1 E ∞ 0 e −ρt x i (t) 2 + û i (t) 2 dt < ∞.M 1 = a − ρ/2 b 2 /r q −a + ρ/2 , M 2 = a + g − ρ/2 b 2 /r q(1 − γ) 2 −(a + g − ρ/2) .
By direct computations, neither M 1 nor M 2 has eigenvalues in imaginary axis if and only if
(a − ρ 2 ) 2 + b 2 r q > 0,(27)(a + g − ρ 2 ) 2 + b 2 r (1 − γ) 2 q > 0.(28)
Note that if q > 0 (or a − ρ/2 < 0, q = 0), i.e., (a − ρ/2, √ q) is observable (detectable), then (27) holds, and if (1 − γ) 2 q > 0 (a + g − ρ/2 < 0, q = 0), i.e., (a + g − ρ/2, √ q(1 − γ)) is observable (detectable), then (28) holds.
For this model, the Riccati equation (20) is written as
b 2 r p 2 − (2a − ρ)p − q = 0.(29)
Let ∆ = 4[(a−ρ/2) 2 +b 2 q/r]. If (27) holds then ∆ > 0, which implies (29) admits two solutions.
If q > 0 then (29) has a unique positive solution such that a − b 2 p/r − ρ/2 = − √ ∆/2 < 0.
If q = 0 and a − ρ/2 < 0 then (29) has a unique non-negative solution p = 0 such that
a − b 2 p/r − ρ/2 = a − ρ/2 < 0.
Assume that (27) and (28) hold. By Theorem 2.7, the system is uniformly stable if and only if (a − ρ/2, b) is stabilizable (i.e., b = 0 or a − ρ/2 < 0), and a − b 2 p/r − ρ/2 + g < 0. Note that a − b 2 p/r − ρ/2 < 0. When g ≤ 0, we have a − b 2 p/r − ρ/2 + g < 0.
Example 2.2:
We further consider the model in Example 2.1 for the case that a + g = ρ/2 and γ = 1 (i.e., (28) does not hold). In this case, the Riccati equation (21)
= 0 in C ρ/2 ([0, ∞), R). Thus,x satisfies dx dt = ρ 2x + f.(30)
Assume that f is a constant. Then (30) does not admit a solution in C ρ/2 ([0, ∞), R) unless
x(0) = −2f /ρ.
We are in a position to state the asymptotic optimality of the decentralized control.
April 17, 2019 DRAFT Theorem 2.8: Let A1)-A4) hold. For Problem (PS), the set of decentralized control laws {û 1 , · · · ,û N } given by (19) has asymptotic social optimality, i.e.,
1 N J soc (û) − 1 N inf u∈Uc J soc (u) = O( 1 √ N ).
Proof. We first prove that for u ∈ U c , J soc (u) < ∞ implies that
E ∞ 0 e −ρt ( x i 2 + u i 2 )dt < ∞,(31)for all i = 1, · · · , N. From J soc (u) < ∞, we have E ∞ 0 e −ρt u i 2 dt < ∞ and E ∞ 0 e −ρt x i − Γx (N ) 2 Q dt < ∞,(32)
which further implies that
E ∞ 0 e −ρt (I − Γ)x (N ) 2 Q ≤ 1 N N i=1 E ∞ 0 e −ρt x i − Γx (N ) 2 Q dt < ∞.(33)
By (1) we have
dx (N ) (t) = (A + G)x (N ) (t) + Bu (N ) (t) + f (t) dt + 1 N N i=1 σ(t)dW i (t),
which leads to for any r ∈ [0, 1],
x (N ) (t) = e (A+G)r x (N ) (t − r) + t t−r e (A+G)(t−τ ) [Bu (N ) (τ ) + f (τ )]dτ + 1 N N i=1 t t−r e (A+G)(t−τ ) σ(τ )dW i (τ ).(34)
By J soc (u) < ∞ and basic SDE estimates, we can find a constant C such that
E ∞ r e −ρt t t−r e (A+G)(t−τ ) Bu (N ) (τ )dτ 2 dt ≤ C.
From (33) and (34) we obtain
E ∞ r e −ρt [x (N ) (t − r)] T e (A+G) T r (I − Γ) T Q(I − Γ) · e (A+G)r x (N ) (t − r)dt ≤ C,
which together with A3) implies that
E ∞ 0 e −ρt x (N ) (t) 2 dt < ∞.(35)
This and (32) lead to
E ∞ 0 e −ρt x i (t) 2 Q dt < ∞.(36)
April 17, 2019 DRAFT By (1), we have
x i (t) = e Ar x i (t − r) + t t−r e A(t−τ ) [Bu i (τ ) + f (τ ) + Gx (N ) (τ )]dτ + t t−r e A(t−τ ) σ(τ )dW i (τ ).(37)
It follows from (35) that
E ∞ r e −ρt t t−r e A(t−τ ) Gx (N ) (τ )dτ 2 dt ≤E ∞ 0 e −ρτ Gx (N ) (τ ) 2 r 0 e (A− ρ 2 I)v 2 dvdτ ≤ C.
From (36) and (37), we obtain that
E ∞ r e −ρt x T i (t − r)e A T r Qe Ar x i (t − r)dt ≤ C.
This together with A3) implies that
E ∞ 0 e −ρt x i (t) 2 dt < ∞,
which gives (31). By Theorem 2.4,
E ∞ 0 e −ρt x i 2 + ũ i 2 dt < ∞.
By a similar argument to the proof of Theorem 2.3 combined with Lemma 2.2, the conclusion follows.
If A3) is replaced by A3 ′ ), the decentralized control (19) still has asymptotic social optimality.
Corollary 2.2:
Assume that A1)-A2), A3 ′ ), A4) hold. The set of decentralized control laws given by (19) is asymptotically socially optimal.
Proof. Without loss of generality, we simply assume A + G = diag{A 1 , A 2 }, where A 1 − (ρ/2)I is Hurwitz, and −(A 2 − (ρ/2)I) is Hurwitz (If necessary, we may apply a nonsingular linear transformation as in the proof of Theorem 2.6). Write
x (N ) = [z T 1 , z T 2 ] andQ 1/2 = [S 1 , S 2 ] such that (I − Γ)x (N ) 2 Q = S 1 z 1 + S 2 z 2 2 ,
and (A 2 − (ρ/2)I, S 2 ) is observable which is due to the detectability of (A + G − (ρ/2)I,Q 1/2 ).
III. MEAN FIELD LQ GAMES
In this section, we investigate the game problem for LQ mean field systems.
(PG).
Seek a set of decentralized control laws to minimize individual cost for each agent in the system (1)-(2).
A. The finite-horizon problem
We first consider the finite-horizon problem. Suppose thatx ∈ C([0, T ], R n ) is given for (1) and (3) byx, we have the following auxiliary optimal control problem.
approximation of x (N ) . Replacing x (N ) in(P2) inf u i ∈L 2 F i t (0,T ;R r )J F i (u i ), where dx i (t) = [Ax i (t) + Bu i (t) + Gx(t) + f (t)]dt + σ(t)dW i (t), 1 ≤ i ≤ N, J F i (u i ) = E T 0 e −ρt x i − Γx − η 2 Q + u i 2 R dt.
By examining the variation ofJ F i , we obtain the unique optimal control of (P2). Theorem 3.1: Assume Q ≥ 0, R > 0. Then the FBSDE
dx i = Ax i − BR −1 B T p i + Gx + f dt + σdW i , dp i = − (A T − ρI)p i + Qx i − QΓx − Qη dt + q i dW i , x i (0) = x i0 , p i (T ) = 0, i = 1, 2, · · · , N(38)
admits a unique solution (x i , p i , q i ), and the optimal controlû i = −R −1 B T p i .
Proof. Since Q ≥ 0 and R > 0, then by [41], (P2) is uniformly convex, and hence admits a unique optimal control. By a similar argument with Theorem 2.1, the conclusion follows.
It follows from (38) that
dx (N ) = Ax (N ) − BR −1 B T p (N ) + Gx + f dt + 1 N N i=1 σdW i , dp (N ) = − (A − ρI) T p (N ) + Qx (N ) − QΓx − Qη dt + 1 N N i=1 q i dW i , x (N ) (0) = 1 N N i=1 x i0 , p (N ) (T ) = 0. April 17, 2019 DRAFT Replacingx (N ) byx, we have dx = (A + G)x − BR −1 B Tp + f dt,x(0) = x 0 , dp = − (A − ρI) Tp + Qx − QΓx − Qη dt,p(T ) = 0.(39)
Letp =Px +ŝ. By Itô's formula, we obtain
P (A + G)x − BR −1 B T (Px +ŝ) + f dt + (Ṗx +ṡ)dt = dp = − (A − ρI) T (Px +ŝ) + Qx − QΓx − Qη dt.
This implies
ρP =Ṗ + A TP +P (A + G) −P BR −1 B TP + Q − QΓ,P (T ) = 0,(40)ρŝ =ṡ + (A − BR −1 B TP ) Tŝ +P f − Qη,ŝ(T ) = 0.(41)Denotep i = p i −p, andx i =x i −x.
Then by (38) and (39) we have
dx i = Ax i − BR −1 B Tp i dt + σdW i ,x i (0) = x i0 −x 0 , dp i = − (A − ρI) Tp i + Qx i dt + q i dW i ,p i (T ) = 0. Letp i = Px i . By Itô's formula, dp i = − (A − ρI) Tp i + Qx i dt + q i dW i =Ṗx i dt + P [(Ax i − BR −1 B T Px i )dt + σdW i ],
which implies that q i = P σ, and ρP =Ṗ + A T P + P A − P BR −1 B T P + Q.
Assume A5) Equation (40) where
A = A + G −BR −1 2 B T QΓ − Q −(A − ρI) T .Letû i = −R −1 B T [Px i + (P − P )x +ŝ],(43)
where P,P andŝ are determined by (42), (40) and (41), respectively, andx andx i satisfy
dx = Ax − BR −1 B T (Px +ŝ) + Gx + f dt,x(0) = x 0 ,(44)dx i = (A − BR −1 B T P )x i − BR −1 B T [(P − P )x +ŝ] + Gx (N ) + f dt + σdW i ,x i (0) = x i0 .(45)
Denote u −i = (u 1 , · · · , u i−1 , u i+1 , · · · , u N ).
Theorem 3.2: Let A1), A5) hold and Q ≥ 0. The set of decentralized strategies {û 1 , · · · ,û N } given by (43) is an ε-Nash equilibrium, i.e.,
inf u i ∈L 2 F t (0,T ;R r ) J F i (u i ,û −i ) ≥ J F i (û i ,û −i ) − ε,(46)where ε = (1/ √ N).
Proof. See the Appendix D.
B. The infinite-horizon problem
For simplicity, we consider the case G = 0.
Based on the analysis in Section III-A, we may design the following decentralized control for (PG):û
i (t) = − R −1 B T [Px i (t) + (P − P )x(t) +ŝ(t)], t ≥ 0, i = 1, · · · , N,(47)
where P andP are determined by
ρP =A T P + P A − P BR −1 B T P + Q,(48)ρP =A TP +P A −P BR −1 B TP + Q(I − Γ),(49)
respectively, andŝ,x ∈ C ρ/2 ([0, ∞), R n ) are determined by
ρŝ =ṡ + [A − BR −1 B TP ] Tŝ +P f − η, (50) dx dt =Ax − BR −1 B T (Px +ŝ) + f,x(0) =x 0 .(51)
April 17, 2019 DRAFT andx i satisfies
dx i = (A − BR −1 B T P )x i − BR −1 B T [(P − P )x +ŝ] +Gx (N ) + f dt + σdW i ,x i (0) = x i0 .(52)
Here the existence conditions of P,P , s andx need to be investigated further.
We introduce the following assumptions.
A6) (A − ρ 2 I, B) is stabilizable, Q ≥ 0 and (A − ρ 2 I, √ Q) is detectable.
A7) (49) admits a stabilizing solution.
Lemma 3.1: Assume that M 3 has n stable eigenvalues (with negative real parts) and n unstable eigenvalues, where
M 3 = A − ρ 2 I BR −1 B T Q(I − Γ) −A T + ρ 2 I . Suppose that M 3 L 1 L 2 = L 1 L 2 H 11 ,(53)
where H 11 is Hurwitz and L 1 is invertible. Then A7) holds.
Proof. LetP = −L 2 L −1 1 . It follows from (53) that M 3 −Ī P = −Ī P L 1 H 11 L −1 1 .(54)
By pre-multiplying by [P I] on both sides, we obtain
[P I]M 3 −Ī P = 0,
which leads to (49). By (54), we have
A − BR −1 B TP − ρ 2 I = L 1 H 11 L −1 1 is Hurwitz. It is straightforward that s,x ∈ C ρ/2 ([0, ∞), R n ).V 21 V −1 11
is the stabilizing solution of (49). V comprises 2n independent vectors, which are called Schur vectors [24].
s,x ∈ C ρ/2 ([0, ∞), R n ), and N i=1 E ∞ 0 e −ρt x i (t) 2 + û i (t) 2 dt < ∞.
Proof. By a similar argument in the proof of Theorem 2.6, the lemma follows. (47) is an ε-Nash equilibrium, i.e.,
inf u i ∈U c,i J i (u i ,û −i ) ≥ J i (û i ,û −i ) − ε, where ε = (1/ √ N).
Proof. See Appendix D.
IV. COMPARISON OF DIFFERENT SOLUTIONS
In this section, we compare the proposed decentralized control laws with the feedback decentralized strategies in previous works [19], [20].
We first introduce a definition from [4].
i = −R −1 B T (P x i +s), i = 1, · · · , N,(55)
where P is the semi-positive definite solution of (48), ands =Kx † + φ. HereK satisfies x =x † and φ = s. From the above discussion, we have the equivalence of the two sets of decentralized control laws.
ρK =KĀ +Ā TK −KBR −1 B TK T − Q Γ , and x † , φ ∈ C ρ/2 ([0, ∞), R n ) are determined by dx † dt =Āx † − BR −1 B T (Kx † + φ),x † (0) =x 0 , dφ dt = − [A − BR −1 B T (P +K) − ρI]φ +η,
Proposition 4.1: The set of decentralized control laws {û 1 , · · · ,û N } in (19) is a representation of {ȗ 1 , · · · ,ȗ N } given by (55).
For Problem (PG), let f = 0, and G = 0. In [19], the decentralized strategies are given by
u * i = −R −1 B T (P x i + s * ), i = 1, · · · , N,(56)
where P is the positive definite solution of (20), s * is determined by the fixed-point equation
ρs * = ds * dt +Ā T s * − QΓ(x * + η), dx * dt =Āx * − BR −1 B T s * ,x * (0) =x 0 .(57)
We now show the equivalence of the decentralized open-loop and feedback solutions to mean field games.
Proposition 4.2:
The set of decentralized control laws {û 1 , · · · ,û N } in (47) is a representation of {u * 1 , · · · , u * N } given by (56). Proof. Let s * = K * x * + ψ. From (57), we have
ds * dt =K * dx * dt + dψ dt =K * Āx * − BR −1 B T (K * x * + ψ) + dψ dt =(ρI −Ā) T (K * x * + ψ) + Q(Γx * + η),
which gives
ρK * = K * Ā +Ā T K * − K * BR −1 B T K * − QΓ, ρψ = dψ dt + [Ā − BR −1 B T K * ] T ψ − Qη.
By comparing this with (48)-(50), one can obtain K =P − P , and ψ =ŝ. Thus, we have u * i ≡û i , i = 1, · · · , N, which implies that {u * 1 , · · · , u * N } is a representation of {û 1 , · · · ,û N } in (47). Finally, we consider the 2-dimensional case of Problem (PS). Take parameters as follows: An interesting generalization is to consider mean field LQ control systems with partial measurements by using variational analysis. Also, the variational analysis may be applied to general nonlinear model to construct decentralized control laws for social control and Nash games.
A = 0.1 0 −1 0.2 , B = 1 0 0 1 , G = −0.5 0 0 −0.3 , B = 1 1 , Q = 1 0 0 1 , Γ = 1 0 1 1 , R = 1 0 0 1 , η = 0 0.5 , f = [1 1] T and σ = [0.5 0.5] T . Denotex i (t) = [x 1 i (t)x 2 i (t)] T .max 0≤t≤T E x (N ) (t) −x(t) 2 = O( 1 N ). (A.1)
Proof. It follows by (18) that
dx (N ) = (Ā + G)x (N ) − BR −1 B T (Kx + s) + f dt + 1 N N i=1 σdW i .d(x (N ) −x) = (Ā + G)(x (N ) −x)dt + 1 N N i=1 σdW i ,
which leads tô
x (N ) (t) −x(t) = e (Ā+G)t [x (N ) (0) −x(0)] + 1 N N i=1 t 0 e (Ā+G)(t−τ ) σdW i (τ ). (A.2)
By A1), one can obtain
E x (N ) (t) −x(t) 2 ≤ 2e (Ā+G)t 2 E x (N ) (0) −x(0) 2 + 1 N t 0 tr σ T e −(Ā T +G T +Ā+G)τ σ dτ ≤ 2 N e (Ā+G)t 2 max 1≤i≤N E x i0 2 + T 0 tr σ T e −(Ā T +G T +Ā+G)τ σ dτ , (A.3)
which completes the proof.
Proof of Theorem 2.3. We first prove that for
u ∈ U c , J F soc (u) < ∞ implies that E T 0 e −ρt ( x i 2 + u i 2 )dt < ∞, for all i = 1, · · · , N. By J F soc (u) < ∞, we have E T 0 e −ρt u i 2 dt < ∞. This leads to E T 0 e −ρt u (N ) 2 dt ≤ 1 N N i=1 E T 0 e −ρt u i 2 dt < ∞, where u (N ) = 1 N N i=1 u i . By (1), dx (N ) (t) = (A + G)x (N ) (t) + Bu (N ) (t) + f (t) dt + 1 N N i=1 σ(t)dW i (t),
which with A1) implies that
max 0≤t≤T E x (N ) (t) 2 ≤ C.
Note that
x i (t) = e At x i0 + t 0 e A(t−τ ) [Gx (N ) (τ ) + Bu i (τ ) + f (τ )]dτ.
We have
E T 0 e −ρt x i 2 dt ≤ C E x i0 2 + max 0≤t≤T E x (N ) (t) 2 + max 0≤t≤T E u i (t) 2 < ∞. (A.4)
By (16) and (18), we obtain that (1) and (18),
E T 0 e −ρt x i 2 + û i 2 + x 2 )dt < ∞. (A.5) Letx i = x i −x i ,ũ i = u i −û i andx (N ) = 1 N N i=1x i . Then bydx i = (Ax i + Gx (N ) + Bũ i )dt,x i (0) = 0. (A.6) From (3), we have J F soc (u) = N i=1 E T 0 e −ρt x i − Γx (N ) − η +x i − Γx (N ) 2 Q + û i +ũ i 2 R dt = N i=1 (J F i (û) +J F i (ũ) + I i ), (A.7) whereJ F i (ũ) ∆ = E T 0 e −ρt x i − Γx (N ) 2 Q + ũ i 2 R dt, I i = 2E T 0 e −ρt x i − Γx (N ) − η T Q x i − Γx (N ) +û T i Rũ i dt. By (A4),J F i (ũ) ≥ 0. We now prove 1 N N i=1 I i = O( 1 √ N ). N i=1 I i = N i=1 2E T 0 e −ρt x T i Q(x i − Γx (N ) − η) − Γ T Q((I − Γ)x (N ) − η) + N i=1û T i Rũ i dt = N i=1 2E T 0 e −ρt x T i Q(x i − Γx − η) − Γ T Q((I − Γ)x − η) + N i=1û T i Rũ i dt + N i=1 2E T 0 e −ρt (x (N ) −x) T Q Γxi dt.
(A.8)
By (12)- (14), (A.6) and Itô's formula,
0 = N i=1 E e −ρTxT i (T )(Px i (T ) + Kx(T ) + s(T )) −x T i (0)(Px i (0) + Kx(0) + s(0)) = E T 0 N i=1 e −ρt −x T i Qx i − Q(Γx + η) − Γ T Q ((I − Γ)x − η) û T i Rũ i ) dt + NE T 0 e −ρt (x (N ) −x) T (G T P + P G)x (N ) dt.
From this and (A.8), we obtain
1 N N i=1 I i = 2E T 0 e −ρt (x (N ) −x) T · (Q Γ + G T P + P G)x (N ) dt.1 N N i=1 I i 2 ≤ CE T 0 e −ρt x (N ) −x 2 dt · E T 0 e −ρt x (N ) 2 dt, which implies | 1 N N i=1 I i | = O(1/ √ N).x (N ) (t) −x(t) = e (Ā+G)t [x (N ) (0) −x(0)] + 1 N N i=1 t 0 e (Ā+G)(t−v) σdW i (v).
Thus,
E ∞ 0 e −ρt x (N ) (t) −x(t) 2 dt ≤ 2E ∞ 0 e (Ā+G− ρ 2 I)t 2 x (N ) (0) −x(0) 2 dt + 2E ∞ 0 e −ρt 1 N t 0 e (Ā+G)(t−v) σdW i (v) 2 dt ≤ 2 ∞ 0 e (Ā+G− ρ 2 I)t 2 E x (N ) (0) −x(0) 2 dt + 2 N E ∞ 0 e −ρt t 0 tr σ T σe (Ā+G+Ā T +G T )(t−v) dvdt ≤ 2 N ∞ 0 e (Ā+G− ρ 2 I)t 2 E max 1≤i≤Nx i (0) 2 dt + C N E ∞ 0 e −ρv σ 2 ∞ v e (Ā+Ḡ− ρ 2 I)(t−v) 2 dtdv ≤ O( 1 N ).
Proof of Theorem 2.4. By A1)-A4), Lemmas 2.1 and 2.2, we obtain thatx ∈ C ρ/2 ([0, ∞), R n ) and
E ∞ 0 e −ρt x (N ) (t) −x(t) 2 dt = O( 1 N ),
which further gives that
E ∞ 0 e −ρt x (N ) (t) 2 dt < ∞. Denote g ∆ = −BR −1 B T ((Π − P )x + s) + Gx (N ) + f . Then E ∞ 0 e −ρt g(t) 2 dt < ∞ and x i (t) = eĀ tx i0 + t 0 eĀ (t−v) g(v)dv + t 0 eĀ (t−v) σdW i .(E ∞ 0 e −ρt x i (t) 2 dt ≤ 3E ∞ 0 e (Ā− ρ 2 I)t 2 x i0 2 dt + 3E ∞ 0 e −ρt t t 0 eĀ (t−v) g(v) 2 dvdt + 3E ∞ 0 e −ρt t 0 tr[eĀ T (t−v) σ T (v)σ(v)eĀ (t−v) ]dvdt ≤ C + 3E ∞ 0 e −ρv g(v) 2 ∞ v t e (Ā− ρ 2 I)(t−v) 2 dtdv + 3CE ∞ 0 e −ρv σ(v) 2 ∞ v e (Ā− ρ 2 I)(t−v) 2 dtdv ≤ C + 3CE ∞ 0 e −ρv g(v) 2 dv + 3CE ∞ 0 e −ρv σ(v) 2 dv ≤ C 1 .
This with (19) completes the proof.
APPENDIX C PROOFS OF THEOREMS 2.5 AND 2.6
Proof. i)⇒ ii). By (18),
dE[x i ] dt =ĀE[x i ] − BR −1 B T ((Π − P )x + s) + GE[x (N ) ] + f, E[x i (0)] =x 0 . (C.1)
It follows from A1) that
E[x i ] = E[x j ] = E[x (N ) ], j = i.
By comparing (23)
x(t) =e (A+G−BR −1 B T Π)t x 0 + t 0 e −(A+G−BR −1 B T Π)τ h(τ )dτ ,
where h = −BR −1 B T s + f . By the arbitrariness ofx 0 with (C.2) we obtain that A + G −
BR −1 B T Π − ρ 2 I is Hurwitz. That is, (A + G − ρ 2 I, B) is stabilizable. By [2], (21) admits a unique solution such that Π > 0. Note that E[x (N ) ] 2 ≤ 1 N N i=1 E[x 2 i ].
Then from (26) we have
E ∞ 0 e −ρt x (N ) (t) 2 dt < ∞. (C.3) This leads to E ∞ 0 e −ρt g(t) 2 dt < ∞, where g= − BR −1 B T ((Π − P )x + s) + Gx (N ) + f . By (B.1), we obtain E x i (t) 2 = E eĀ t x i0 + t 0 e −Āτ g(τ )dτ 2 + E t 0 tr σ T (τ )e (Ā T +Ā)(t−τ ) σ(τ ) dτ.
By (26) and the arbitrariness of x i0 we obtain thatĀ − ρ 2 I is Hurwitz, i.e., (A − ρ 2 I, B) is stabilizable. By [2], (20) admits a unique solution such that P > 0.
From (C.2) and (C.3),
E ∞ 0 e −ρt x (N ) (t) −x(t) 2 dt < ∞. (C.4)
On the other hand, (A.2) gives
E x (N ) (t) −x(t) 2 = E e (Ā+G)t [x (N ) (0) −x 0 ] 2 + 1 N t 0 tr σ T (τ )e (Ā T +G T +Ā+G)(t−τ ) σ(τ ) dτ.
By (C.4) and the arbitrariness of x i0 , i = 1, · · · , N, we obtain thatĀ + G − ρ 2 I is Hurwitz.
(ii)⇒(iii). Define V (t) = e −ρtȳT (t)Πȳ(t), whereȳ satisfies dȳ dt = (A + G)ȳ + Bū,ȳ(0) =ȳ 0 . Denote V by V * whenū =ū * = −R −1 B T Πȳ. By (21) we have dV * dt =ȳ T (t) − ρΠ + (A + G − BR −1 B T Π) T Π + Π(A + G − BR −1 B T Π) ȳ(t) =ȳ T (t) −Q − ΠBR −1 B T Π ȳ(t) ≤ 0.
Note that V * ≥ 0. Then lim t→∞ V * (t) exists, which implies
lim t 0 →∞ [V * (t 0 ) − V * (t 0 + T )] = 0. (C.5)
Rewrite Π(t) in (15) by Π T (t). Then we have Π T +t 0 (t 0 ) = Π T (0). By (15),
T +t 0 t 0 e −ρt (ȳ TQȳ +ū T Rū)dt = e −ρt 0ȳ T (t 0 )Π T +t 0 (t 0 )ȳ(t 0 ) + T 0 e −ρt ū + R −1 B T Π T +t 0 (t 0 )ȳ 2 R dt ≥ e −ρt 0 ȳ(t 0 ) 2 Π T +t 0 (t 0 ) = e −ρt 0 ȳ(t 0 ) 2 Π T (0) .
This with (C.5) implies
lim t 0 →∞ e −ρt 0 ȳ(t 0 ) 2 Π T (0) ≤ lim t 0 →∞ T +t 0 t 0 e −ρt ( ȳ 2Q + ū * 2 R )dt = lim t 0 →∞ [V * (t 0 ) − V * (t 0 + T )] = 0.
April 17, 2019 DRAFT By A3), one can obtain that there exists T > 0 such that Π T (0) > 0 (See e.g. [43], [44] Proof of Theorem 2.6. (iii)⇒(i). From [2], (20) and (21) admit unique solutions P ≥ 0, Π ≥ 0 such that A−BR −1 B T P − ρ 2 I and A−BR −1 B T Π− ρ 2 I are Hurwitz, respectively. Thus, there exists a unique s(0) such that s ∈ C ρ/2 ([0, ∞), R n ). It is straightforward thatx ∈ C ρ/2 ([0, ∞), R n ).
By the argument in the proof of Theorem 2.4, (i) follows.
(i)⇒(ii). The proof of this part is similar to that of (i)⇒(ii) in Theorem 2.5.
(ii)⇒(iii). Since Π ≥ 0, then there exists an orthogonal U such that
U T ΠU = 0 0 0 Π 2 ,
where Π 2 > 0. From (20),
ρU T ΠU =(U TĀ U) T U T ΠU + U T ΠUU TĀ U + U TQ U, (C.6) whereĀ ∆ = A + G − ΠBR −1 B T Π,Q =Q + ΠBR −1 B T Π. Denote U TĀ U = Ā 11Ā12 A 21Ā22 , U TQ U = Q 11Q12 Q 21Q22 .
By pre-and post-multiplying by ξ T and ξ where ξ = [ξ T 1 , 0] T , it follows that 0 = ρξ T U T ΠUξ = ξ T U TQ Uξ.
From the arbitrariness of ξ 1 , we obtainQ 11 = 0. SinceQ is semi-positive definite, thenQ 12 = Q 21 = 0, andQ 22 ≥ 0. By comparing each block matrix of both sides of (C.6), we obtain A 21 = 0. It follows from (C.6) that
ρΠ 2 = Π 2Ā22 +Ā T 22 Π 2 +Q 22 . (C.7)
Let ζ = [ζ T 1 , ζ T 2 ] T = U Tȳ * , whereȳ * satisfiesẏ * =Āȳ * . Then we havė ζ 1 =Ā 11 ζ 1 +Ā 12 ζ 2 ,
ζ 2 =Ā 22 ζ 2 .
April 17, 2019 DRAFT By Lemma 4.1 of [38], the detectability of (A + G,Q 1/2 ) implies the detectability of (Ā,Q 1/2 ).
Take ζ(0) = ξ = [ξ T 1 , 0] T . ThenQ 1/2ȳ =Q 1/2 Uζ = 0, which together with the detectability of (Ā,Q 1/2 ) implies ζ 1 → 0 andĀ 11 is Hurwitz. Denote S(t) = e −ρt ζ T 2 Π 2 ζ 2 . By (C.7),
S(T ) − S(0) = − T 0 ζ 2 (t) TQ 22 ζ 2 (t)dt ≤ 0,
which implies lim t→∞ S(t) exists. By a similar argument with the proof of Theorem 2.5, we obtain lim t 0 →∞ e −ρt 0 ζ 2 (t 0 ) 2 Π 2,T (0) = 0 and Π 2,T (0) > 0, which gives ζ 2 → 0 andĀ 22 is Hurwitz. This with the fact thatĀ 11 is Hurwitz gives that ζ is stable, which leads to (iii). To prove (46), it suffices to only consider u i ∈ L 2
Ft (0, T ; R r ) such that J F i (u i ,û −i ) ≤ J F i (û i ,û −i ) < ∞. By (3), E T 0 e −ρt u i 2 dt < ∞. (D.3)
After the set of strategies (u i ,û −i ) is applied, the corresponding dynamics of N agents can be written as dx i =(Ax i + Bu i + Gx (N ) + f )dt + σdW i , dx j =(Ax j + Bû j + Gx (N ) + f )dt + σdW j , j = 1, · · · , i − 1, i + 1, · · · , N. Note that
x i − (Γx (N ) + η) 2 Q ≥ x i − (Γx + η) 2 Q + 2[x i − (Γx + η)] T Q[(x i −x i ) + Γ(x − x (N ) )],
andJ F i (u i ) < ∞. By Schwarz's inequality, (D.4) and (D.5), we obtain
J F i (u i ,û −i ) ≥J F i (u i ) − E T 0 e −ρt x i − (Γx + η) 2 Q dt 1/2 · E T 0 e −ρt (x i −x i ) + Γ(x − x (N ) 2 Q dt 1/2 ≥J F i (u i ) − O(1/ √ N).
From this and (D.2), the theorem follows.
Proof of Theorem 3.3. Note that {x i (t), i = 1, · · · , N} are mutually independent processes with the expectationx(t). By Lemma 3.2,
E ∞ 0 e −ρt x i (t) −x(t) 2 dt ≤ 1 N E ∞ 0 e −ρt x i (t) 2 dt = O( 1 N ).
We only need to show E
E ∞ 0 e −ρt x i − 1 N x i 2 Q dt ≤ E ∞ 0 e −ρt x i − 1 N x i − 1 N j =ix j 2 Q dt + E ∞ 0 e −ρt 1 N j =ix j 2 Q dt ≤ C 0 + E ∞ 0 e −ρt 1 N − 1 j =i x j 2 Q dt ≤ C,
which with Lemma 3.2 implies
E ∞ 0 e −ρt x i 2 Q dt ≤ C 1 ,
where C 1 is independent of N. The rest of the proof follows by that of Theorem 3.2.
This work was supported by the National Natural Science Foundation of China under Grants 61773241, 61573221 and 61633014. Bingchang Wang is with the School of Control Science and Engineering, Shandong University, Jinan 250061, P. R. China. (e-mail: [email protected]) Huanshui Zhang is with the School of Control Science and Engineering, Shandong University, Jinan 250061, P. R. China.
Furthermore, {x i (0), i = 1, ..., N} and {W i , i = 1, ..., N} are independent of each other.
Proposition 2. 1 :
1Problem (P1) is convex in u if and only if for any
N are a set of solutions to the equation system
(
dynamics) is first substituted by a deterministic functionx. By solving an optimal tracking problem subject to consistency requirements, a fixed-point equation is obtained. The decentralized control is constructed by handling the fixed-point equation. Here, we firstly obtain the centralized open-loop solution by variational analysis. By tackling the coupled FBSDEs combined with mean field approximations, the decentralized control laws are designed. Note that in this case s andx are fully decoupled and no fixed-point equation is needed. Theorem 2.3: Let A1) hold and Q ≥ 0. The set of decentralized control laws {û 1 , · · · ,û N }
Remark 2 . 2 :
22In[43], some similar results were given for the stabilization of mean field systems. However, only the limiting problem was considered in their work and the mean field term in dynamics and costs is Ex(t) instead of x (N ) . Here we study large-population multiagent systems and the number of agents is large but not infinite. The errors of mean field approximationsApril 17, 2019 DRAFT are further analyzed. To obtain asymptotic optimality, an additional assumption A4) is needed later.
Remark 2. 3 :
3M 1 and M 2 are Hamiltonian matrices. The Hamiltonian matrix plays a significant role in studying general algebraic Riccati equations. See more details of the property of Hamiltonian matrices in [1], [28]. To show Theorem 2.7, we need two lemmas. The first lemma is a result from [28, Theorem 6]. Lemma 2.3: Equations (20) and (21) admit ρ-stabilizing solutions if and only if A2) holds and both M 1 and M 2 have no eigenvalues on the imaginary axis. Lemma 2.4: Let A1) hold. Assume that
Proof. 7 .:
7From the definition of ρ-stabilizing solutions, A − BR −1 B T P − ρ 2 I and A + G − BR −1 B T Π − ρ 2 I are Hurwitz. By the argument in the proof of Theorem 2.4, the lemma follows.The Proof of Theorem 2.By Consider a scalar system with A = a, B = b, G = g, Q = q, Γ = γ, R = r > 0.
admits a unique solution Π = 0 .
0(22) becomes ρs =ṡ + ρ 2 s and has a unique solution s
which together with(33) gives E ∞ 0 e −ρt S 2 z 2 2 dt < ∞. This and the observability of(A 2 − (ρ/2)I, S 2 ) leads to E ∞ 0 e −ρt z 2 2 dt < ∞. Thus, E ∞ 0 e −ρt x (N ) 2 dt < ∞.The other parts of the proof are similar to that of Theorem 2.8.April 17, 2019 DRAFT
admits a solution in C([0, T ], R n ). By the local Lipschitz-continuous property of the quadratic function, (40) can admit a unique local solution in a small time duration [T 0 , T ]. It may be referred to [1] for some sufficient conditions of the existence of the solution in [0, T ]. We now provide a necessary and sufficient condition to guarantee the global solvability of (40).
Proposition 3. 1 :
1(40) admits a solution in C([0, T ], R n ) if and only if for any t ∈ [0, T ], det{(0, I)e At (0, I) T } > 0,
Remark 3. 1 :
1The above lemma provides a convenient method to compute the stabilizing solutions of algebraic Riccati equations. Assume there exists an invertible matrix V = , where V 11 is invertible, and H 11 , −H 22 are Hurwitz. Then
Theorem 3. 3 :
3Let A1), A6), A7) hold. For Problem (PG), the set of decentralized strategies {û 1 , · · · ,û N } given by
Definition 4. 1 :
1For a control problem with an admissible control set U, a control law u ∈ U is said to be a representation of another control u * ∈ U if (i) they both generate the same unique state trajectory, and (ii) they both have the same open-loop value on this trajectory. For Problem (PS), let f = 0, and G = 0. In [20, Theorem 4.3], the decentralized control laws are given byȗ
= A−BR −1 B T P . By comparing this with (21)-(23), one can obtain thatK = Π−P ,
consider a scalar system with 50 agents in Problem (PS). Take B = Q = R = 1, G = −0.2, f (t) = 1, σ(t) = 0.1, ρ = 0.6, Γ = −0.2, η = 5 in (1)-(2). The initial states of 50 agents are taken independently from a normal distribution N(5, 0.5). Then, under the control law (19), the state trajectories of agents for the cases with A = 0.2 and A = 1 are shown in Figs. 1 and 2, respectively. After the transient phase, the states of agents behave similarly and achieve agreement roughly.
Fig. 2 :Fig. 3 :
23Curves of 50 agents with A = 1. April 17, 2019 DRAFT Next, we simulate the scalar case of Problem (PG), where the parameters are the same as above, except G = 0. After the control laws (47) are applied, the state trajectories of 50 agents with A = 0.2 and A = 1 are shown in Figs. Curves of 50 agents with A = 0.2.
Fig. 4 :
4Curves of 50 agents with A = 1.
For
the case A = 1 and G = 0, the trajectories ofx andx (N ) in Problems (PS) and (PG) are shown in Fig. 5. It can be seen thatx andx (N ) coincide well, which illustrate the consistency of mean field approximations. Clearly, the state average of agents has significantly lower value in Problem (PS) than in (PG).
Fig. 5 :
5Curves ofx andx (N ) in (PS) and (PG).
Both ofx 1 i (0) andx 2 i (0) are taken independently from a normal distribution N(5, 0.5). Under the control laws (19), the trajectories ofx 1 i andx 2 i , i = 1, · · · , N are shown in Figs. 6 and 7, respectively.
Fig. 6 :
6Curves ofx 1 i , i = 1, · · · , 50.
Fig. 7 :
7Curves ofx 2 i , i = 1, · · · , 50.VI. CONCLUDING REMARKSIn this paper, we have considered uniform stabilization and asymptotic optimality for mean field LQ multiagent systems. For social control and Nash game problems, we design the decentralized open-loop control laws by the variational analysis, respectively, which are further shown to be asymptotically optimal. Two equivalent conditions are further given for uniform stabilization of the systems in different cases. Finally, we show such decentralized control laws are equivalent to the feedback strategies in previous works.
Proof of Theorem 3.2. From(44)and(45), we haved(x (N ) −x) = (Ā + G)(x (N ) −x)dt + σ N N i=1 dW i , whereĀ = A − BR −1 B T P . This implies that sup 0≤t≤T E x (N ) (t) −x(t) i (û i ,û −i ) ≤J F i (û i ) + E T 0 e −ρt x (N ) (t) −x(t)
u i −û i ) − BR −1 B T P (x (N ) x i −x i ) = [A(x i −x i ) + G(x (N ) −x)]dt,
∞ 0 e
0−ρt x i 2 Q dt ≤ C for all u i satisfying J i (u i ,û −i ) ≤ J i (û i ,û −i ) ≤ C 0 .(D.6) From (D.6), we obtain
Note thatĀ − ρ 2 I is Hurwitz. By Schwarz's inequality,B.1)
April 17, 2019
DRAFT
and (C.1), we obtain E[x i ] =x. Note that x 2 = Ex i 2 ≤ E x i 2 . Itfollows from (26) that
∞
0
e −ρt x(t) 2 dt < ∞.
(C.2)
By (23), we havē
). Thus, we have lim t→∞ e −ρt ȳ(t) 2 = 0, which (A + G − ρ 2 I, B) is stabilizable. Similarly, we can show (A − ρ 2 I, B) is stabilizable. (iii)⇒(i). This part has been proved in Theorem 2.4.
For a Riccati equation(20), P is called a ρ-stabilizing solution if P satisfies(20) and all the eigenvalues of A − BR −1 B T P are in left half-plane.April 17, 2019 DRAFT
April 17, 2019DRAFT
Matrix Riccati Equations in Control and Systems Theory. H Abou-Kandil, G Freiling, V Ionescu, G Jank, Birkhiiuser VerlagH. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank, Matrix Riccati Equations in Control and Systems Theory. Birkhiiuser Verlag, 2003.
. B D O Anderson, J B Moore, Optimal Control: Linear Quadratic Methods. Englewood Cliffs. Prentice HallB. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice Hall, 1990.
Team-optimal solution of finite number of mean-field coupled LQG subsystems. J Arabneydi, A Mahajan, Proc. 54th IEEE CDC. 54th IEEE CDCOsaka, JapanJ. Arabneydi and A. Mahajan, "Team-optimal solution of finite number of mean-field coupled LQG subsystems," in Proc. 54th IEEE CDC, Osaka, Japan, 2015, pp. 5308-5313.
T Basar, G J Olsder, Dynamic Noncooperative Game Theory. LondonAcademic PressT. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory. Academic Press, London, 1982.
Opinion dynamics in social networks through mean-field games. D Bauso, H Tembine, T Basar, SIAM J. Control Optim. 546D. Bauso, H. Tembine, and T. Basar, "Opinion dynamics in social networks through mean-field games," SIAM J. Control Optim., vol. 54, no. 6, pp. 3225-3257, 2016.
Linear-quadratic mean field games. A Bensoussan, K C Sung, S C Yam, S P Yung, J. Optimization Theory & Applications. 1692A. Bensoussan, K.C. Sung, S.C. Yam, and S. P. Yung, "Linear-quadratic mean field games," J. Optimization Theory & Applications, vol. 169, no. 2, pp. 496-529, 2016.
A Bensoussan, J Frehse, P Yam, Mean Field Games and Mean Field Type Control Theory. New YorkSpringerA. Bensoussan, J. Frehse, and P. Yam, Mean Field Games and Mean Field Type Control Theory. Springer, New York, 2013.
Mean field games. P E Caines, M Huang, R P Malhame, Handbook of Dynamic Game Theory. BerlinSpringerP. E. Caines, M. Huang, and R. P. Malhame, Mean field games, in Handbook of Dynamic Game Theory, T. Basar and G. Zaccour Eds., Springer, Berlin, 2017.
Bertrand and Cournot mean field games. P Chan, R Sircar, Applied Mathematics & Optimization. 713P. Chan and R. Sircar, "Bertrand and Cournot mean field games," Applied Mathematics & Optimization, vol. 71, no. 3, pp. 533-569, 2015.
State estimation and mean field control with application to demand dispatch. Y Chen, A Busic, S Meyn, Proc. 54th IEEE CDC. 54th IEEE CDCOsakaY. Chen, A. Busic, and S. Meyn, "State estimation and mean field control with application to demand dispatch," in Proc. 54th IEEE CDC, Osaka, 2015, pp. 6548-6555.
Probabilistic analysis of mean-field games. R Carmona, F Delarue, SIAM J. Control Optim. 514R. Carmona and F. Delarue, "Probabilistic analysis of mean-field games," SIAM J. Control Optim., vol. 51, no. 4, pp. 2705-2734, 2013.
Mean field games models-a brief survey. D A Gomes, J Saude, Dyn. Games Appl. 42D. A. Gomes and J. Saude, "Mean field games models-a brief survey," Dyn. Games Appl., vol. 4, no. 2, pp. 110-154, 2014.
Mean field games and applications. O Guéant, J M Lasry, P L Lions, Paris-Princeton Lectures on Mathematical Finance. Heidelberg, GermanySpringer-VerlagO. Guéant, J. M. Lasry, and P. L. Lions, "Mean field games and applications," in Paris-Princeton Lectures on Mathematical Finance, pp. 205-266, Springer-Verlag: Heidelberg, Germany, 2011.
Team decision theory and information structures. Y C Ho, Proc. IEEE. IEEE68Y. C. Ho, "Team decision theory and information structures," in Proc. IEEE, vol. 68, no.6, 1980, pp. 644-654.
Mean field LQG games with model uncertainty. J Huang, M Huang, Proc. 52nd IEEE CDC. 52nd IEEE CDCFlorence, ItalyJ. Huang and M. Huang, "Mean field LQG games with model uncertainty," in Proc. 52nd IEEE CDC, Florence, Italy, 2013, pp. 3103-3108.
Robust mean field linear-quadratic-Gaussian games with model uncertainty. J Huang, M Huang, SIAM J. Control Optim. 555J. Huang and M. Huang, "Robust mean field linear-quadratic-Gaussian games with model uncertainty," SIAM J. Control Optim., vol. 55, no. 5, pp. 2811-2840, 2017.
Large-population LQG games involving a major player: the Nash certainty equivalence principle. M Huang, SIAM J. Control Optim. 485M. Huang, "Large-population LQG games involving a major player: the Nash certainty equivalence principle," SIAM J. Control Optim., vol. 48, no.5, pp. 3318-3353, 2010.
Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and Nash equilibrium solutions. M Huang, P E Caines, R P Malhamé, Proc. 42nd IEEE CDC. 42nd IEEE CDCMaui, HIM. Huang, P. E. Caines, and R. P. Malhamé, "Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and Nash equilibrium solutions," in Proc. 42nd IEEE CDC, Maui, HI, 2003, pp. 98-103.
Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behavior and decentralized ε-Nash equilibria. M Huang, P E Caines, R P Malhamé, IEEE Trans. Autom. Control. 529M. Huang, P. E. Caines, and R. P. Malhamé, "Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behavior and decentralized ε-Nash equilibria," IEEE Trans. Autom. Control, vol. 52, no.9, pp. 1560-1571, 2007.
Social optima in mean field LQG control: Centralized and decentralized strategies. M Huang, P Caines, R Malhame, IEEE Trans. Autom. Control. 577M. Huang, P. Caines, and R. Malhame, "Social optima in mean field LQG control: Centralized and decentralized strategies," IEEE Trans. Autom. Control, vol. 57, no. 7, pp. 1736-1751, 2012.
Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. M Huang, R P Malhamé, P E Caines, Communication in Information and Systems. 6M. Huang, R. P. Malhamé, and P. E. Caines, "Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle," Communication in Information and Systems, vol. 6, pp. 221-251, 2006.
Linear-quadratic mean field teams with a major agent. M Huang, L Nguyen, Proc. 55th IEEE CDC. 55th IEEE CDCLas Vegas, NVM. Huang and L. Nguyen, "Linear-quadratic mean field teams with a major agent," in Proc. 55th IEEE CDC, Las Vegas, NV, 2016, pp. 6958-6963,.
Mean field games. J M Lasry, P L Lions, Japan J. Math. 21J. M. Lasry and P. L. Lions, "Mean field games," Japan J. Math., vol. 2, no. 1, pp. 229-260, 2007.
A Schur method for solving algebraic Riccati equations. A J Laub, IEEE Trans. Autom. Control. 246A. J. Laub, "A Schur method for solving algebraic Riccati equations," IEEE Trans. Autom. Control, vol. 24, no. 6, pp. 913-921, 1979.
Asymptotically optimal decentralized control for large population stochastic multiagent systems. T Li, J.-F Zhang, IEEE Trans. Autom. Control. 537T. Li and J.-F. Zhang, "Asymptotically optimal decentralized control for large population stochastic multiagent systems," IEEE Trans. Autom. Control, vol. 53, no. 7, pp. 1643-1660, 2008.
J Ma, J Yong, Forward-backward Stochastic Differential Equations and their Applications. New YorkSpringer-VerlagJ. Ma and J. Yong, Forward-backward Stochastic Differential Equations and their Applications, Springer-Verlag, New York, 1999.
Decentralized charging control for large populations of plug-in electric vehicles. Z Ma, D Callaway, I Hiskens, IEEE Trans. Control Systems Technology. 211Z. Ma, D. Callaway, and I. Hiskens, "Decentralized charging control for large populations of plug-in electric vehicles," IEEE Trans. Control Systems Technology, vol. 21, no. 1, pp. 67-78, 2013.
The time-invariant linear-quadratic optimal control problem. B P Molinari, Automatica. 134B. P. Molinari, "The time-invariant linear-quadratic optimal control problem," Automatica, vol. 13, no. 4, pp. 347-357, 1977.
Linear quadratic risk-sensitive and robust mean field games. J Moon, T Basar, IEEE Trans. Autom. Control. 623J. Moon and T. Basar, "Linear quadratic risk-sensitive and robust mean field games," IEEE Trans. Autom. Control, vol. 62, no. 3, pp. 1062-1077, 2017.
Team decision problems. R Radner, Annals of Mathematical Statistics. 333R. Radner, "Team decision problems," Annals of Mathematical Statistics, vol. 33, no. 3, pp. 857-881, 1962.
Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems. J Sun, X Li, J Yong, SIAM J. Control Optim. 545J. Sun, X. Li, and J. Yong, "Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems," SIAM J. Control Optim., vol. 54, no. 5, pp. 2274-2308, 2016.
Dynamic production output adjustment with sticky prices: A mean field game approach. B.-C Wang, M Huang, Proc. 45th IEEE CDC. 45th IEEE CDCOsaka, JapanB.-C. Wang and M. Huang, "Dynamic production output adjustment with sticky prices: A mean field game approach," in Proc. 45th IEEE CDC, Osaka, Japan, 2015, pp. 4438-4443.
Mean field games for large-population multiagent systems with Markov jump parameters. B.-C Wang, J.-F Zhang, SIAM J. Control Optim. 504B.-C. Wang and J.-F. Zhang, "Mean field games for large-population multiagent systems with Markov jump parameters," SIAM J. Control Optim., vol. 50, no. 4, pp. 2308-2334, 2012.
Distributed control of multi-agent systems with random parameters and a major agent. B.-C Wang, J.-F Zhang, Automatica. 489B.-C. Wang and J.-F. Zhang, "Distributed control of multi-agent systems with random parameters and a major agent," Automatica, vol. 48, no. 9, pp. 2093-2106, 2012.
Hierarchical mean field games for multiagent systems with tracking-type costs: Distributed ε-Stackelberg equilibria. B.-C Wang, J.-F Zhang, IEEE Trans. Autom. Control. 598B.-C. Wang and J.-F. Zhang, "Hierarchical mean field games for multiagent systems with tracking-type costs: Distributed ε-Stackelberg equilibria," IEEE Trans. Autom. Control, vol. 59, no. 8, pp. 2241-2247, 2014.
Social optima in mean field linear-quadratic-Gaussian models with Markov jump parameters. B.-C Wang, J.-F Zhang, SIAM J. Control Optim. 551B.-C. Wang and J.-F. Zhang, "Social optima in mean field linear-quadratic-Gaussian models with Markov jump parameters," SIAM J. Control Optim., vol. 55, no. 1, pp. 429-456, 2017.
Markov perfect industry dynamics with many firms. G Weintraub, C Benkard, B Van Roy, Econometrica. 766G. Weintraub, C. Benkard, and B. Van Roy, "Markov perfect industry dynamics with many firms," Econometrica, vol. 76, no. 6, pp. 1375-1411, 2008.
On a matrix Riccati equation of stochastic control. W Wonham, SIAM J. Control Optim. 64W. Wonham, "On a matrix Riccati equation of stochastic control," SIAM J. Control Optim., vol. 6, no. 4, pp. 681-697, 1968.
Synchronization of coupled oscillators is a game. H Yin, P G Mehta, S P Meyn, U V Shanbhag, IEEE Trans. Autom. Control. 574H. Yin, P. G. Mehta, S. P. Meyn, and U. V. Shanbhag, "Synchronization of coupled oscillators is a game," IEEE Trans. Autom. Control, vol. 57, no. 4, pp. 920-935, April 2012.
Linear-quadratic optimal control problems for mean-field stochastic differential equations. J Yong, SIAM J. Control Optim. 514J. Yong, "Linear-quadratic optimal control problems for mean-field stochastic differential equations," SIAM J. Control Optim., vol. 51, no. 4, pp. 2809-2838, 2013.
J Yong, X Y Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. New YorkSpringer-VerlagJ. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York, 1999.
Control for Itô stochastic systems with input delay. H Zhang, J Xu, IEEE Trans. Autom. Control. 621H. Zhang and J. Xu, "Control for Itô stochastic systems with input delay," IEEE Trans. Autom. Control, vol. 62, no. 1, pp. 350-365, 2017.
Optimal stabilization control for discrete-time mean-field stochastic systems. H Zhang, Q Qi, M Fu, IEEE Trans. Autom. Control. in pressH. Zhang, Q. Qi, and M. Fu, "Optimal stabilization control for discrete-time mean-field stochastic systems," IEEE Trans. Autom. Control, in press, 2018.
Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion. W Zhang, H Zhang, B S Chen, PLACE PHOTO HERE Bingchang Wang received the M.Sc. degree in Mathematics from Central South University. Changsha, China; Beijing, China; Australia, as a Research Academic53Department of Electrical and Computer Engineering, University of Alberta ; he was with School of Electrical Engineering and Computer Science, University of NewcastleCanadaW. Zhang, H. Zhang, and B. S. Chen, "Generalized Lyapunov equation approach to state-dependent stochastic stabiliza- tion/detectability criterion," IEEE Trans. Autom. Control, vol. 53, no. 7, pp. 1630-1642, 2008. PLACE PHOTO HERE Bingchang Wang received the M.Sc. degree in Mathematics from Central South University, Changsha, China, in 2008, and the Ph.D. degree in System Theory from Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, in 2011. From September 2011 to August 2012, he was with Department of Electrical and Computer Engineering, University of Alberta, Canada, as a Postdoctoral Fellow. From September 2012 to September 2013, he was with School of Electrical Engineering and Computer Science, University of Newcastle, Australia, as a Research Academic.
| [] |
[
"Parton distribution amplitudes: revealing diquarks in the proton and Roper resonance",
"Parton distribution amplitudes: revealing diquarks in the proton and Roper resonance"
] | [
"Cédric Mezrag \nIstituto Nazionale di Fisica Nucleare\nSezione di Roma\nP. le A. Moro 2I-00185RomaItaly\n\nPhysics Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n",
"Jorge Segovia \nInstitut de Física d'Altes Energies (IFAE) and Barcelona Institute of Science and Technology (BIST)\nUniversitat Autònoma de Barcelona\nE-08193Bellaterra, Barcelona)Spain\n",
"Lei Chang \nSchool of Physics\nNankai University\n300071TianjinChina\n",
"Craig D Roberts \nPhysics Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n"
] | [
"Istituto Nazionale di Fisica Nucleare\nSezione di Roma\nP. le A. Moro 2I-00185RomaItaly",
"Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA",
"Institut de Física d'Altes Energies (IFAE) and Barcelona Institute of Science and Technology (BIST)\nUniversitat Autònoma de Barcelona\nE-08193Bellaterra, Barcelona)Spain",
"School of Physics\nNankai University\n300071TianjinChina",
"Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA"
] | [] | We present the first quantum field theory calculation of the pointwise behaviour of the leadingtwist parton distribution amplitudes (PDAs) of the proton and its lightest radial excitation. The proton's PDA is a broad, concave function, whose maximum is shifted relative to the peak in QCD's conformal limit expression for this PDA; an effect which signals the presence of both scalar and pseudovector diquark correlations in the nucleon, with the scalar generating around 60% of the proton's normalisation. The radial-excitation is constituted similarly, and the pointwise form of its PDA, which is negative on a material domain, is the result of marked interferences between the contributions from both types of diquark; particularly, the locus of zeros that highlights its character as a radial excitation. These features originate with the emergent phenomenon of dynamical chiralsymmetry breaking in the Standard Model. | 10.1016/j.physletb.2018.06.062 | [
"https://arxiv.org/pdf/1711.09101v1.pdf"
] | 119,487,646 | 1711.09101 | 48d3fb435263da5a7333e102c7c46c5d88af280b |
Parton distribution amplitudes: revealing diquarks in the proton and Roper resonance
Cédric Mezrag
Istituto Nazionale di Fisica Nucleare
Sezione di Roma
P. le A. Moro 2I-00185RomaItaly
Physics Division
Argonne National Laboratory
60439ArgonneIllinoisUSA
Jorge Segovia
Institut de Física d'Altes Energies (IFAE) and Barcelona Institute of Science and Technology (BIST)
Universitat Autònoma de Barcelona
E-08193Bellaterra, Barcelona)Spain
Lei Chang
School of Physics
Nankai University
300071TianjinChina
Craig D Roberts
Physics Division
Argonne National Laboratory
60439ArgonneIllinoisUSA
Parton distribution amplitudes: revealing diquarks in the proton and Roper resonance
(Dated: 22 November 2017)
We present the first quantum field theory calculation of the pointwise behaviour of the leadingtwist parton distribution amplitudes (PDAs) of the proton and its lightest radial excitation. The proton's PDA is a broad, concave function, whose maximum is shifted relative to the peak in QCD's conformal limit expression for this PDA; an effect which signals the presence of both scalar and pseudovector diquark correlations in the nucleon, with the scalar generating around 60% of the proton's normalisation. The radial-excitation is constituted similarly, and the pointwise form of its PDA, which is negative on a material domain, is the result of marked interferences between the contributions from both types of diquark; particularly, the locus of zeros that highlights its character as a radial excitation. These features originate with the emergent phenomenon of dynamical chiralsymmetry breaking in the Standard Model.
1. Introduction -Wave functions provide insights into composite systems, e.g. they express the presence and extent of correlations between constituents, and their signature in scattering processes; and thereby bridge experiment and theory, delivering understanding from what might otherwise seem arcane observations. This is true within quantum chromodynamics (QCD), the quantum field theory describing strong interactions; but there are difficulties. Everyday hadrons (p = proton, neutron, etc.) are constituted from up (u) and down (d) valencequarks; but the Higgs boson generates current-masses for these fermions which are more than 100-times smaller than the scale associated with the composite systems: m u,d ≈ 2 − 4 MeV cf. m p ≈ 1 GeV. Evidently, the interaction energy greatly exceeds the rest masses of the anticipated constituents, making inapplicable the wave functions typical of Schrödinger quantum mechanics.
The difficulties appear chiefly because particle-number is not conserved by Lorentz boosts; and extreme challenges are faced when constituents are light, e.g. wave functions describing incoming and outgoing scattering states then represent systems with different particle content, so a probability interpretation is lost. Such problems are circumvented by using a light-front formulation because eigenfunctions of the Hamiltonian are then independent of the system's four-momentum [1,2].
The light-front wave function of a hadron with momentum P and spin λ, Ψ(P, λ), is complicated. In terms of perturbation theory's partons, Ψ(P, λ) has a countablyinfinite Fock-space expansion, with the N -parton term depending on 3N momentum variables, constrained such that their sum yields P , with a similar constraint on their spin (and flavour). Were it necessary to use this complete object in analyses of even the simplest processes, then little connection between experiment and theory could be made. Fortunately, collinear factorisation in the treat-ment of hard exclusive processes entails that much can be gained merely by studying hadron leading-twist parton distribution amplitudes (PDAs) [3][4][5]. Such a PDA is obtained from the simplest term in the Fock-space expansion, e.g. meson, quark-antiquark (Ň = 2) or baryon, three-quark (Ň = 3), with the constituents' light-fronttransverse momenta integrated out to a given scale, ζ.
Regarding ground-state S-wave light-meson leadingtwist PDAs, the last decade has seen real progress, not concerning their conformal limit [3][4][5]: ϕ(x; ζ) = 6x(1 − x), m p /ζ 0; but on m p /ζ 1, where they are now known to be broad, concave functions, e.g. ϕ π (x; ζ m p ) ≈ (8/π) x(1 − x) [6][7][8][9][10][11][12][13]. This resolves a long-time conflict, eliminating the notion that such PDAs exhibit a minimum at zero relative momentum [14].
Concerning the proton's leading-twist PDA, however, the situation is as unsatisfactory today as it was for mesons ten years ago. Estimates of low-order Mellin moments exist, obtained using sum rules [14,15] or lattice-QCD (lQCD) [16][17][18], but there are no quantum field theory computations of this PDA's pointwise behaviour; and nothing is known about the PDA of the proton's radial excitation. These issues are addressed herein. 2. Proton PDA: Definition -In the isospin-symmetry limit, the proton possesses one independent leading-twist (twist-three) PDA [19], denoted ϕ([x]; ζ) herein:
0|ε abcũa + (z 1 ) C † / n u b − (z 2 ) / n d c + (z 3 )|P, + =: 1 2 f p n · P / n N + 1 0 [dx] ϕ([x]; ζ)e −in·P i xizi(1)
where n 2 = 0; (a, b, c) are colour indices; ψ ± = H ± ψ := (1/2)(I D ± γ 5 )ψ, / n = γ · n;q indicates matrix transpose; C is the charge conjugation matrix, N = N (P ) is the proton's Euclidean Dirac spinor (Ref. [20], Appendix B);
1 0 [dx]f ([x]) = 1 0 dx 1 dx 2 dx 3 δ(1 − i x i )f ([x]
); and f p measures the proton's "wave function at the origin". arXiv:1711.09101v1 [nucl-th] 24 Nov 2017 ϕ([x]) can be computed once the proton's Poincarécovariant wave function is in hand; and following thirty years of study [21][22][23][24][25], a clear picture has appeared. At an hadronic scale, the proton is a Borromean system, bound by two effects [26]: one originates in non-Abelian facets of QCD, expressed in the effective charge [27] and generating confined, nonpointlike but strongly-correlated colour-antitriplet diquarks in both the isoscalar-scalar and isotriplet-pseudovector channels; and that attraction is magnified by quark exchange associated with diquark breakup and reformation. The presence and character of the diquarks owe to the mechanism that dynamically breaks chiral symmetry in the Standard Model [26].
The proton Faddeev amplitude can be written [20]:
Ψ(P ) = ψ 1 + ψ 2 + ψ 3 ,(2)
where the subscript identifies the bystander quark, i.e. the quark not participating in a diquark, ψ 3 gives ψ 1,2 by cyclic permutation of all quark labels, and
ψ 3 ({p}, {α}, {σ}) = N 0 3 + N 1 3 , (3a) N 0 3 = Γ 0 (k; K) α1α2 σ1σ2 ∆ 0 (K) S( ; P )u(P ) α3 σ3 , (3b) N 1 3 = Γ 1j µ (k; K) α1α2 σ1σ2 ∆ 1 µν (K) A j ν ( ; P )u(P ) α3 σ3 , (3c) ({p}, {α}, {σ})
are the momentum, isospin and spin labels of the dressed-quarks constituting the bound state; The proton's Faddeev wave function, χ, is obtained from Eqs. (2), (3) by attaching the appropriate dressedquark and -diquark propagators. Each quantity involved is known because the nucleon Faddeev equation has been widely studied [20,[28][29][30][31][32][33]. We therefore proceed by using algebraic representations for every element, with each form and their relative strengths, when combined, based on these analyses. The dressed-quark propaga-
P = p 1 + p 2 + p 3 is the total momentum of the baryon; k = p 1 , K = p 1 + p 2 , = −K + (2/3tor S(p) = (−i/ p + M )σ M (p 2 ), σ M (s) = 1/[s + M 2 ], σ M (s) = M 2 /[s + M 2 ]; ∆ 0 (K) = σ M0 (K 2 ), ∆ 1 µν (K) = T µν (K)σ M1 (K 2 ), T µν (K) = [δ µν + K µ K ν /K 2 ]; n 0 Γ 0 (k; K)C † = iγ 5 1 −1 dz ρ(z)σ ΛΓ (k 2 +K ) ,(4a)n 1 Γ 1 µ (k; K)C † = i(γ T µ + r 1 f (k; K)[/ k, γ T µ ]) × 1 −1 dz ρ(z)σ ΛΓ (k 2 +K ) ,(4b)
where
ρ(z) = (3/4)(1 − z 2 ), k +K = k + (z − 1)K/2; γ T µ = T µν (K)γ ν , f (k; K) = k · K/(k 2 K 2 (k − K) 2 ) 1/2 ;
and r 1 = 1/4, n 0,1 are fixed by requiring that the zeroth Mellin moment of the leading-twist PDA of each diquark correlation is [n · K/n · P ]. The final elements are:
n S( ; P ) = i 1 −1 dz ρ(z)σ Λ 0 p (w +P ) , (5a) n A j ν ( ; P ) = r A 1 6 o j γ 5 [γ ν − ir P P ν ] × 1 −1 dz ρ(z)σ Λ 1 p (w +P ) ,(5b)
where
w +P = [− +P + (2/3)P ] 2 ; o + = √ 2, o 0 = −1;
r P = 13/87; r A measures the relative 1 + :0 + diquark strengths in the Faddeev amplitude; and n is that amplitude's canonical normalisation constant, whose value ensures the proton has unit charge [34]. Eqs. (4), (5) define a constrained spectral function model for χ [20,[28][29][30][31][32][33], whose fidelity will subsequently be tested. Crucially, the form is completely general: one can always use perturbation theory integral representations (PTIRs) for the propagators and amplitudes that arise in solving the continuum bound-state problem [35][36][37]; so our subsequent analysis will establish an archetype for the continuum computation of baryon PDAs.
O 21 ϕ = H − C † / n H + , O 3 ϕ = / n H + .(6)
As a concrete illustration, we consider the first diagram on the rhs, whose contribution to the proton's PDA is fully determined by the following Mellin moments:
fp 2 n · P / n N + [dx] x l 1 x m 2 ϕ([x]) =: fp 2 n · P / n N + x l 1 x m 2 = [dx] x l 1 x m 2 d 4 (2π) 4 d 4 k (2π) 4 δ x1 n ( + P/3)δ x2 n (k) × χ 1 ({p}, {α}, {σ}) O 21 ϕ O 3 ϕ ,(7)
where δ x n (p) = δ(n·p−xn·P ); p 1 = +P/3, p 2 = k, p 3 = K − k. Considering only the 0 + diquark component, the second and third contributions in Fig. 1 vanish because this correlation is isoscalar-scalar, and hence the leadingtwist part of the last line in Eq. (7) is γ · L 0 + ,
L 0 + µ = 1 4 tr D S d (p 3 )Γ 0 (k; K)S u (p 2 )H − C † / n H + × S u (p 1 )∆ 0 (K)S ( ; P )γ µ / n H +(8)= 1 4 tr D γ µ / n H + S u (p 1 )∆ 0 (K)S ( ; P ) × 1 2 tr D S d (p 3 )Γ 0 (k; K)S u (p 2 )C † γ 5 / n ,(9)
where we have used properties of tr D , the projection operators H ± , and n µ . Inserting Eq. (9) into Eq. (7), one finds that the scalar diquark contribution to the proton's PDA is obtained from a convolution of the diquark's PDA with that of the bystander-quark in the quark+diquark Faddeev amplitude. Importantly, the result generalises to the isotriplet-pseudovector component of the proton's Faddeev wave function, in which case the proton's PDA receives contributions from all three diagrams in Fig. 1. Continuing our illustrative calculation, one first computes the scalar diquark PDA following the methods described in Refs. [7,13]. Namely, in the k-integration of Eq. (7), use a Feynman parametrisation to rearrange the integrand such that there is a single denominator, a kquadratic form raised to some power; and employ a suitably chosen change of variables in order to evaluate the integral over this relative four-momentum using standard algebraic methods. This yields, withū = 1 − u and
z = −1 + 2[ū − β]/[ū − v], D m 0 (K 2 ) = n 0 (K 2 ) n · K n · P 1+m × du dv dβ β m ρ(z(u, v, β))2M [β(v[β − 2] + β) +ū(v − β 2 )][K 2 + M 2 ]
, (10a)
M 2 = [1 −ū + v]M 2 + [ū − v]Λ 2 Γ β(v[β − 2] + β) +ū(v − β 2 ) [ū − v] .(10b)
In our case, one can straightforwardly obtain the following algebraic result when Λ Γ = M (x 3 = 1 −x 2 ,
x 2 = x 2 /[x 2 + x 3 ], y = M 2 /K 2 ): n 0 (K 2 )ϕ 0 (x 2 ,x 3 ) = 12y(1 − ŷ x 2x3 ln[1 +x 2x3 /y]) ,(11)
where n 0 (K 2 ) ensures 1 = dxϕ 0 (x, 1 − x) at each
K 2 . Notably, when K 2 Λ 2 Γ , ϕ 0 (x 2 ,x 3 ) = 6x 2x3 , viz.
the two-body conformal-limit PDA, which describes a correlation-free system; whereas on K 2 Λ 2 Γ , ϕ 0 (x 2 ,x 3 ) = 1, which is the PDA of a pointlike two-body composite, the most highly-correlated system possible. ; ζ) is obtained by adding the 1 + -diquark contributions. That is readily accomplished by employing the procedure sketched above. The addition is a sum of three integrals, two involving seven Feynman parameters, the third, nine, and each with a denominator that is an -quadratic form.
All integrals required to compute ϕ([x]; ζ) are readily evaluated numerically. We choose the parameters in Eqs. (4), (5) so as to emulate realistic Faddeev wave functions [20,[28][29][30][31][32][33]: M = 2/5, M 0 = 2/3, M 1 = 3/4, Λ Γ = 2/5, Λ 0 p = 1, Λ 1 p = 2/5, in units of m p , with r A = 0.30 ± 0.03 ensuring that the scalar diquark contribution to the proton's baryon number is 65 ± 5%. The distribution thus obtained is that associated with the hadronic scale ζ = ζ H = 0.51 GeV [38]. We evolve ϕ([x]; ζ H ) to ζ = ζ 2 = 2 GeV by adapting the algorithm in Refs. [7,8] to the case of baryons, i.e. generalising the functional representation in Ref. [39] and using the leading-order evolution equation in Ref. [5]. The result is depicted in Fig. 2 and efficiently interpolated using (w 00 = 1)
ϕ([x]) = n ϕ x α− 1 (x 2 x 3 ) β− 2 j=0 j i=0 w ij P 2[i+β];α− j−i (2x 1 − 1) ×(x 2 + x 3 ) i C β i ([x 3 − x 2 ]/[x 2 + x 3 ]) ,(13)where n ϕ ensures [dx]ϕ([x]) = 1; (α, β) − = (α, β) − 1/2;
P is a Jacobi function, C a Gegenbauer polynomial; and the interpolation parameters are listed in Table IA. Table IB lists the four lowest-order moments of our proton PDA. They reveal valuable insights, e.g. when the proton is drawn as solely a quark+scalar-diquark correlation, x 2 u = x 3 d , because these are the two participants of the scalar quark+quark correlation; and the system is very skewed, with the PDA's peak being shifted markedly in favour of x 1 u > x 2 u . This outcome conflicts with lQCD results [17,18]. On the other hand, realistic Faddeev equation calculations indicate that pseudovector diquark correlations are an essential part of the proton's wave function. Naturally, when these {uu} and {ud} correlations are included, momentum is shared more evenly, shifting from the bystander u(x 1 ) quark into u(x 2 ), d(x 3 ). Adding these correlations with the known weighting, the PDA's peak moves back toward the centre and our computed values of the first moments align with those obtained using lQCD. This confluence delivers a significantly more complete understanding of the lQCD simulations, which are now seen to validate a picture of the proton as a bound-state with both strong scalar and pseudovector diquark correlations, in which the scalar diquarks are responsible for ≈ 60% of the Faddeev amplitude's canonical normalisation. Importantly, as found with ground-state S-wave mesons [7][8][9][10][11][12], the leading-twist PDA of the ground-state nucleon is both broader than ϕ cl N ([x]) and decreases monotonically away from its maximum in all directions. Our framework is readily extended to describe the quark core of the proton's first radial excitation: m R = (3/2)m p [29,33]. The scalar functions in this system's Faddeev amplitude possess a zero at quark-diquark relative momentum √ 2 ≈ 0.4 GeV≈ 1/[0.5 fm]. This feature can be implemented via Eq. (5):
ρ(z) = (1 − z 2 ) → ρ R (z) = s qq R (1 − z 2 )(z + z qq R ), where (s S R , z S R ) = (−1, 1/50) = (s A,γ R , z A,γ R )
, (s A,P R , z A,P R ) = (1, 3/10) and Λ 0 p → Λ 0 R = 4/5 were all fitted to reproduce known solutions for the first radial excitation. We therewith obtain the PDA in the rightmost panel of Fig. 2, which is efficiently interpolated using Eq. (13) with the parameters in Table IA; and whose first four moments are listed in Table IB. This prediction reveals some curious features, e.g.: the excitation's PDA is not positive definite and there is a prominent locus of zeros in the lower-right corner of the barycentric plot, both of which echo features of the wave function for the first radial excitation of a quantum mechanical system and have also been seen in the leading-twist PDAs of radially excited mesons [40,41]; and the impact of pseudovector correlations within this excitation is opposite to that in the ground-state, viz. they shift momentum into u(x 1 ) from u(x 2 ), d(x 3 ). 4. Epilogue -We used simple perturbation theory integral representations (PTIRs) for all elements in the Faddeev wave functions, therewith defining models constrained by the best available solutions of the continuum three-valence-body bound-state equations. Crucially, the technique we introduced is completely general: it can readily be used with any realistic Poincarécovariant bound-state wave function, once it is expressed via PTIRs. Hence, the veracity of our PDA predictions can straightforwardly be tested in future studies. In the interim, the PDAs we have determined will, e.g. enable the first realistic assessments to be made of the scale at which exclusive experiments involving baryons may properly be compared with predictions based on perturbative-QCD hard scattering formulae and thereby assist contemporary and planned facilities to refine and reach their full potential [42][43][44]. The value of such estimates has recently been demonstrated in studies of mesons [10,45,46].
)P ; and the j sum runs over the (1, 1) = + and (1, 0) = 0 isospin projections. The matrix-valued functions Γ in Eqs. (3) are diquark correlation amplitudes; ∆ 0 , ∆ 1 µν are associated dressed-propagators; and S, A j µ are matrix-valued quarkdiquark amplitudes, describing the relative-momentum correlation between the diquark and bystander quark.
3 .
3Proton PDA: Calculation -Whenever the proton's Faddeev amplitude is as specified by Eqs. (2), (3), then Eq. (1) can be written as depicted in Fig. 1, where
FIG. 1 .
1Studies of the continuum three-body bound-state problem reveal that diquark correlations are an integral part of the proton's Poincaré-covariant wave function, Eqs. (2), (3), in which case Eq. (1) is the sum of three terms, with the spinor projection operators given in Eq. (6). The large (dark blue) ovals represent the (S , A) elements in ψ1,2,3, the (green) circles are the diquark correlation amplitudes, and the single and double lines are dressed-quark and -diquark propagators, respectively.
Using
Eqs.(10), suppressing n in Eq. (5), one can rewrite Eq. (7) in the form (p 1 = + P/3, K = −p 1 + P ):fp 2 [n · P ] 2 / n N + x l 1 x m 2 = d 4 (2π) 4 n · p 1 n · P l × / n H + S u (p 1 )∆ 0 (K)S ( ; P )D m 0 (K 2 ) ,(12)at which point the analysis leading to Eqs. (10) can be adapted to solve this final "two-body" (quark+diquark) convolution problem for the 0 + -diquark component of ϕ([x]; ζ). The result is an equation that expresses this contribution to ϕ([x]; ζ) as an integral over five Feynman parameters in which the denominator is a single -quadratic form. The complete result for ϕ([x]
FIG. 2 .
2Barycentric plots: left panel -conformal limit PDA, ϕ cl N ([x]) = 120x1x2x3; middle panel -computed proton PDA evolved to ζ = 2 GeV, which peaks at ([x]) = (0.55, 0.23, 0.22); and right panel -Roper resonance PDA at ζ = 2 GeV. The white circle in each panel serves only to mark the centre of mass for the conformal PDA, whose peak lies at ([x]
TABLE I .
IA -Eq. (13) interpolation parameters for the proton and Roper PDAs in Fig. 2. B -Computed values of the first four moments of the PDAs. Our error on fN reflects a scalar diquark content of 65 ± 5%; and values in rows marked with " ⊃ av" were obtained assuming the baryon is constituted solely from a scalar diquark. (All results listed at ζ = 2 GeV.) 14.4 1.42 0.78 −0.93 0.22 −0.21 −0.057 −1.24A nφ
α
β
w01
w11
w02
w12
w22
p 65.8 1.47 1.28
0.096 0.094
0.15 −0.053
0.11
R B
10 3 fN /GeV 2
x1 u
x2 u
x3 d
conformal PDA
0.333
0.333
0.333
lQCD [17]
2.84(33) 0.372(7) 0.314(3) 0.314(7)
lQCD [18]
3.60(6)
0.358(6) 0.319(4) 0.323(6)
herein proton
3.78(14) 0.379(4) 0.302(1) 0.319(3)
herein proton ⊃ av 2.97
0.412
0.295
0.293
herein Roper
5.17(32) 0.245(13) 0.363(6) 0.392(6)
herein Roper ⊃ av
2.63
0.010
0.490
0.500
Office of the Director's Postdoctoral Fellowship Program; European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie Grant Agreement No. 665919; Spanish MINECO's Juan de la Cierva-Incorporación programme. Grant Agreement No. IJCI. by: Argonne National Laboratoryby: Argonne National Laboratory, Office of the Di- rector's Postdoctoral Fellowship Program; European Union's Horizon 2020 research and innovation pro- gramme under the Marie Sk lodowska-Curie Grant Agree- ment No. 665919; Spanish MINECO's Juan de la Cierva- Incorporación programme, Grant Agreement No. IJCI- 2016-30028;
Industria y Competitividad under Contract Nos. FPA2014-55613-P and SEV-2016-0588; the Chinese Government's Thousand Talents Plan for Young Professionals; the Chinese Ministry of Education, under the International Distinguished Professor programme; and U.S. Department of Energy. Spanish Ministerio De Economía, Office of Nuclear Physics. Office of Scienceunder contract no. DE-AC02-06CH11357Spanish Ministerio de Economía, Industria y Competitividad under Contract Nos. FPA2014-55613- P and SEV-2016-0588; the Chinese Government's Thou- sand Talents Plan for Young Professionals; the Chinese Ministry of Education, under the International Distin- guished Professor programme; and U.S. Department of Energy, Office of Science, Office of Nuclear Physics, un- der contract no. DE-AC02-06CH11357;
. B D Keister, W N Polyzou, Adv. Nucl. Phys. 20225B. D. Keister and W. N. Polyzou, Adv. Nucl. Phys. 20, 225 (1991).
. S J Brodsky, H.-C Pauli, S S Pinsky, Phys. Rept. 301299S. J. Brodsky, H.-C. Pauli and S. S. Pinsky, Phys. Rept. 301, 299 (1998).
. G P Lepage, S J Brodsky, Phys. Lett. B. 87359G. P. Lepage and S. J. Brodsky, Phys. Lett. B 87, 359 (1979).
. A V Efremov, A V Radyushkin, Phys. Lett. B. 94245A. V. Efremov and A. V. Radyushkin, Phys. Lett. B 94, 245 (1980).
. G P Lepage, S J Brodsky, Phys. Rev. D. 222157G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
. S J Brodsky, G F De Teramond, Phys. Rev. Lett. 96201601S. J. Brodsky and G. F. de Teramond, Phys. Rev. Lett. 96, 201601 (2006).
. L Chang, Phys. Rev. Lett. 110132001L. Chang et al., Phys. Rev. Lett. 110, 132001 (2013).
. C Shi, Phys. Rev. D. 9214035C. Shi et al., Phys. Rev. D 92, 014035 (2015).
. V M Braun, Phys. Rev. D. 9214504V. M. Braun et al., Phys. Rev. D 92, 014504 (2015).
. T Horn, C D Roberts, J. Phys. G. 4373001T. Horn and C. D. Roberts, J. Phys. G. 43, 073001 (2016).
. F Gao, L Chang, Y.-X Liu, Phys. Lett. B. 770551F. Gao, L. Chang and Y.-x. Liu, Phys. Lett. B 770, 551 (2017).
. J.-H Zhang, J.-W Chen, X Ji, L Jin, H.-W Lin, Phys. Rev. D. 9594514J.-H. Zhang, J.-W. Chen, X. Ji, L. Jin and H.-W. Lin, Phys. Rev. D 95, 094514 (2017).
. C Mezrag, H Moutarde, J Rodriguez-Quintero, Few Body Syst. 57729C. Mezrag, H. Moutarde and J. Rodriguez-Quintero, Few Body Syst. 57, 729 (2016).
. V L Chernyak, A R Zhitnitsky, Phys. Rept. 112173V. L. Chernyak and A. R. Zhitnitsky, Phys. Rept. 112, 173 (1984).
. N G Stefanis, M Bergmann, Phys. Rev. D. 473685N. G. Stefanis and M. Bergmann, Phys. Rev. D 47, R3685 (1993).
. V M Braun, Phys. Rev. D. 7934504V. M. Braun et al., Phys. Rev. D 79, 034504 (2009).
. V M Braun, Phys. Rev. D. 8994511V. M. Braun et al., Phys. Rev. D 89, 094511 (2014).
. G S Bali, JHEP. 0270G. S. Bali et al., JHEP 02, 070 (2016).
. V Braun, R J Fries, N Mahnke, E Stein, Nucl. Phys. B. 589433Nucl. Phys. BV. Braun, R. J. Fries, N. Mahnke and E. Stein, Nucl. Phys. B 589, 381 (2000), [Erratum: Nucl. Phys. B 607, 433 (2001)].
. J Segovia, I C Cloët, C D Roberts, S M Schmidt, Few Body Syst. 551185J. Segovia, I. C. Cloët, C. D. Roberts and S. M. Schmidt, Few Body Syst. 55, 1185 (2014).
. R T Cahill, C D Roberts, J Praschifka, Austral. J. Phys. 42129R. T. Cahill, C. D. Roberts and J. Praschifka, Austral. J. Phys. 42, 129 (1989).
. C J Burden, R T Cahill, J Praschifka, Austral. J. Phys. 42147C. J. Burden, R. T. Cahill and J. Praschifka, Austral. J. Phys. 42, 147 (1989).
. R T Cahill, Austral. J. Phys. 42171R. T. Cahill, Austral. J. Phys. 42, 171 (1989).
. H Reinhardt, Phys. Lett. B. 244316H. Reinhardt, Phys. Lett. B 244, 316 (1990).
. G V Efimov, M A Ivanov, V E Lyubovitskij, Z. Phys. C. 47583G. V. Efimov, M. A. Ivanov and V. E. Lyubovitskij, Z. Phys. C 47, 583 (1990).
. J Segovia, C D Roberts, S M Schmidt, Phys. Lett. B. 750100J. Segovia, C. D. Roberts and S. M. Schmidt, Phys. Lett. B 750, 100 (2015).
. D Binosi, C Mezrag, J Papavassiliou, C D Roberts, J Rodriguez-Quintero, Phys. Rev. D. 9654026D. Binosi, C. Mezrag, J. Papavassiliou, C. D. Roberts and J. Rodriguez-Quintero, Phys. Rev. D 96, 054026 (2017).
. S.-S Xu, Phys. Rev. D. 92114034S.-S. Xu et al., Phys. Rev. D 92, 114034 (2015).
. J Segovia, Phys. Rev. Lett. 115171801J. Segovia et al., Phys. Rev. Lett. 115, 171801 (2015).
. J Segovia, C D Roberts, Phys. Rev. C. 9442201J. Segovia and C. D. Roberts, Phys. Rev. C 94, 042201(R) (2016).
. G Eichmann, C S Fischer, H Sanchis-Alepuz, Phys. Rev. D. 9494033G. Eichmann, C. S. Fischer and H. Sanchis-Alepuz, Phys. Rev. D 94, 094033 (2016).
. Y Lu, Phys. Rev. C. 9615208Y. Lu et al., Phys. Rev. C 96, 015208 (2017).
C Chen, arXiv:1711.03142Structure of the nucleon's lowlying excitations. nucl-thC. Chen et al., (2017), Structure of the nucleon's low- lying excitations, arXiv:1711.03142 [nucl-th].
. M Oettel, M Pichowsky, L , Eur. Phys. J. A. 8251M. Oettel, M. Pichowsky and L. von Smekal, Eur. Phys. J. A 8, 251 (2000).
. N Nakanishi, Phys. Rev. 1301230N. Nakanishi, Phys. Rev. 130, 1230 (1963).
. N Nakanishi, Prog. Theor. Phys. Suppl. 431N. Nakanishi, Prog. Theor. Phys. Suppl. 43, 1 (1969).
N Nakanishi, Graph Theory and Feynman Integrals. New YorkGordon and BreachN. Nakanishi, Graph Theory and Feynman Integrals (Gordon and Breach, New York, 1971).
. C Chen, L Chang, C D Roberts, S Wan, H.-S Zong, Phys. Rev. D. 9374021C. Chen, L. Chang, C. D. Roberts, S. Wan and H.-S. Zong, Phys. Rev. D 93, 074021 (2016).
. V M Braun, S E Derkachov, G P Korchemsky, A N Manashov, Nucl. Phys. B. 553355V. M. Braun, S. E. Derkachov, G. P. Korchemsky and A. N. Manashov, Nucl. Phys. B 553, 355 (1999).
. B L Li, Phys. Rev. D. 93114033B. L. Li et al., Phys. Rev. D 93, 114033 (2016).
. B.-L Li, L Chang, M Ding, C D Roberts, H.-S Zong, Phys. Rev. D. 9494014B.-L. Li, L. Chang, M. Ding, C. D. Roberts and H.-S. Zong, Phys. Rev. D 94, 094014 (2016).
. J Dudek, Eur. Phys. J. A. 48187J. Dudek et al., Eur. Phys. J. A 48, 187 (2012).
. V D Burkert, Few Body Syst. 57873V. D. Burkert, Few Body Syst. 57, 873 (2016).
. A Accardi, Eur. Phys. J. A. 52268A. Accardi et al., Eur. Phys. J. A 52, 268 (2016).
. L Chang, I C Cloët, C D Roberts, S M Schmidt, P C Tandy, Phys. Rev. Lett. 111141802L. Chang, I. C. Cloët, C. D. Roberts, S. M. Schmidt and P. C. Tandy, Phys. Rev. Lett. 111, 141802 (2013).
. F Gao, L Chang, Y.-X Liu, C D Roberts, P C Tandy, Phys. Rev. D. 9634024F. Gao, L. Chang, Y.-X. Liu, C. D. Roberts and P. C. Tandy, Phys. Rev. D 96, 034024 (2017).
| [] |
[
"RealAnt: An Open-Source Low-Cost Quadruped for Education and Research in Real-World Reinforcement Learning",
"RealAnt: An Open-Source Low-Cost Quadruped for Education and Research in Real-World Reinforcement Learning"
] | [
"Rinu Boney ",
"Jussi Sainio ",
"Mikko Kaivola ",
"Arno Solin ",
"Juho Kannala "
] | [] | [] | Current robot platforms available for research are either very expensive or unable to handle the abuse of exploratory controls in reinforcement learning. We develop RealAnt, a minimal low-cost physical version of the popular 'Ant' benchmark used in reinforcement learning. RealAnt costs only ∼350 A C ($410) in materials and can be assembled in less than an hour. We validate the platform with reinforcement learning experiments and provide baseline results on a set of benchmark tasks. We demonstrate that the RealAnt robot can learn to walk from scratch from less than 10 minutes of experience. We also provide simulator versions of the robot (with the same dimensions, state-action spaces, and delayed noisy observations) in the MuJoCo and PyBullet simulators. We open-source hardware designs, supporting software, and baseline results for educational use and reproducible research. | null | [
"https://arxiv.org/pdf/2011.03085v2.pdf"
] | 249,395,648 | 2011.03085 | a49d8fa196605760cbc07e4972d666a78242f2a1 |
RealAnt: An Open-Source Low-Cost Quadruped for Education and Research in Real-World Reinforcement Learning
Rinu Boney
Jussi Sainio
Mikko Kaivola
Arno Solin
Juho Kannala
RealAnt: An Open-Source Low-Cost Quadruped for Education and Research in Real-World Reinforcement Learning
Current robot platforms available for research are either very expensive or unable to handle the abuse of exploratory controls in reinforcement learning. We develop RealAnt, a minimal low-cost physical version of the popular 'Ant' benchmark used in reinforcement learning. RealAnt costs only ∼350 A C ($410) in materials and can be assembled in less than an hour. We validate the platform with reinforcement learning experiments and provide baseline results on a set of benchmark tasks. We demonstrate that the RealAnt robot can learn to walk from scratch from less than 10 minutes of experience. We also provide simulator versions of the robot (with the same dimensions, state-action spaces, and delayed noisy observations) in the MuJoCo and PyBullet simulators. We open-source hardware designs, supporting software, and baseline results for educational use and reproducible research.
I. INTRODUCTION
The field of reinforcement learning (RL) has advanced significantly in recent years, with numerous success stories in solving challenging control problems. This is largely due to the availability of simulators that allow for rapid testing of algorithmic performance, which are inexpensive, fast, and can be run in parallel. However, simulators often make unrealistic assumptions about the world. For example, the popular simulator benchmarks for RL [1]- [3] present no communications delays or noise, have simple dynamics, allow for environment resets, and have no concerns about the safety or durability of the robot hardware [4]. We have to bridge this gap by grounding the development of reinforcement learning algorithms on real-world problems such as robot learning.
Most research on robotics is conducted on industrial robots that are very expensive, costing thousands of dollars. This is not very affordable to all researchers, let alone educational use. Traditional control algorithms require precise hardware that is easy to model. This places a lot of limitations on robot design. Reinforcement learning algorithms are able to learn controllers without modeling the dynamics and can also handle noisy observations and controls. However, aggressive exploratory actions taken by RL algorithms can easily damage the components of a robot. For example, plastic gears in RC servos or naively designed 3D printed parts can easily break during random exploration and learning. Also, many educational robot platforms do not offer a pose estimation or tracking solution, which means one cannot utilize reinforcement learning or any closed-loop algorithms for control. In this paper, we present a minimal low-cost quadruped robot platform that can support and sustain reinforcement learning.
We develop and validate RealAnt, a physical version of the popular Ant benchmark available in OpenAI Gym [1], DeepMind Control Suite [2], and PyBullet [3] simulators. The Ant benchmark, introduced in [5], involves learning a controller for an 8 DoF quadruped robot to walk forward as fast as possible. The Ant benchmark is very widely used in the RL community, allowing for a shallow learning curve in using the RealAnt robot introduced in this paper.
RealAnt is a quadruped robot and learning to walk requires delicate balancing and coordination of the leg joints. Learning controllers using RL for such a quadruped robot is challenging. While state-of-the-art RL algorithms are able to learn to walk in a reasonable amount of time [6], there is still room for improvements in sample-efficiency and safe exploration, which are active areas of research in RL. The Ant benchmark strikes a good balance between simplicity and complexity in terms of the difficulty in building the robot and controlling it. The simple design of the 8 DoF quadruped allows us to build it in a cost-efficient and scalable manner.
The contributions of this paper are as follows. (i) We develop a low-cost minimal quadruped robot called RealAnt, a physical version of the popular Ant benchmark used in reinforcement learning research. (ii) We develop the supporting software stack to perform RL on the physical platform. We also provide simulated versions of the RealAnt robot (with same dimensions and state-action spaces, and delayed noisy observations) in PyBullet and MuJoCo simulators for rapid testing. (iii) We validate that the robot is suitable for
II. RELATED WORK
Scalable low-cost robot platforms can enable a plethora of real-world applications like last-mile delivery and automate highly repetitive manually laborious tasks like object stacking. The robotics community is actively working towards more affordable robots, and in this section, we review recent works on low-cost platforms for robotics research.
Recent works have proposed designs for affordable quadruped robots. The MIT mini-cheetah [7], costing around $10k, is a small quadruped robot that can perform a wide range of locomotion behaviors. Solo [8], costing around 4k A C, is an open-source, lightweight, and torque-controlled quadruped robot based on low complexity actuator modules using brushless motors. Stanford Doggo [9], costing less than 3k A C, is an open-source quadruped robot based on a quasidirect-drive mechanism. While these robots are designed for motion planning controllers, we propose RealAnt, costing less than 500 A C, with a focus on research in real-world reinforcement learning. Unlike other available quadruped robots, RealAnt is designed as a direct analogy of the popular Ant benchmark.
Similar to RealAnt, ROBEL [10] is a recently introduced open-source platform for benchmarking real-world reinforcement learning. The ROBEL platforms consist of two robots: D'Claw and D'Kitty, for manipulation and locomotion tasks respectively. D'Kitty is a a 12 DoF quadruped robot, costing around $4.2k, with Dynamixel XM430-W210-R smart actuators. RealAnt is significantly cheaper as it is an 8 DoF robot with cheaper Dynamixel AX-12A servos. While D'Kitty relies on a HTC Vive Tracker setup (costing more than 500 A C) for pose estimation, we use simple fiducial marker tracking (only requiring a web camera) for the same. Being so cheap, it is possible to build more than ten RealAnt robots at the cost of a D'Kitty, enabling scalable and broader real-world experiments. While only simulated or sim2real experiments have been demonstrated using D'Kitty [10], we demonstrate data-efficient RL from scratch directly on the RealAnt robot.
While there exist several cheap quadruped robots, mostly proposed for educational purposes, they were not designed to sustain the abuse of reinforcement learning. Reinforcement learning involves aggressive exploratory actions that can easily damage the servos or the 3D printed body of the robot. We validate that the proposed RealAnt robot can sustain such aggressive actions.
Wheeled robots tend to be more affordable. Examples of such research platforms include Donkey car, DeepRacer [11], JetBot, DuckieBot [12] and their costs fall in the range of $250 to $500. RealAnt enables research and education on contact-rich legged locomotion in a similar affordable cost range.
There has also been progress in reducing the cost of robot platforms for manipulation tasks. Recent works [13], [14] have proposed such platforms in the cost range of $2k to $5k. REPLAB [15], costing around $2k, is an easily reproducible benchmark for vision-based manipulation tasks.
Compared to existing solutions, RealAnt a direct analogy of the popular Ant benchmark, and is hence well suited for bridging the gap to real-world applications of RL.
III. REALANT
We design RealAnt, a minimal and low-cost physical version of the Ant benchmark for research in real-world reinforcement learning. Similar to the Ant benchmark, RealAnt is an 8 DoF quadruped robot (see Fig. 1 for a photo). RealAnt is based on easily available electronic components and a 3D printed body. List of all components used in RealAnt and their costs are in Table I. The RealAnt can be assembled from these components in less than an hour, by using a Phillips screwdriver, side cutters, and a soldering iron.
A. Mechanical Design
The minimally designed body of the robot consists of 1) four 3D printed legs, 2) eight Dynamixel AX-12A servos (and eight FP04-F2 frames sold with them), and 3) a 3D printed top and bottom torso. Each leg of the robot constitutes of two Dynamixel servos joints affixed to each other using Robotis FP04-F2 frames. Four of the leg assemblies are joined together using a 3D printed torso top and bottom plates. 3D printers are easily accessible and allow for rapid prototyping and cost-effective manufacturing. The schematic details are illustrated in Fig. 2.
The parts were printed in PLA (Prusament filament) using a consumer 3D printer (Creality Ender 3 v2). A complete set of parts requires 13.5 h to print, for two torso plates and four legs. To lower the printing time and produce rigid enough parts, they were printed using 0.2 mm layer height, 20% gyroid [16] infill, and with open top and bottom layers. The printed parts weigh 106 g in total, costing 2.65 A C in filament costs, assuming a filament price of 25 A C/kg. The total robot weight is around 710 g. The parts are designed to be longlasting, but if they break, they can be easily reprinted locally and since PLA is biodegradable, they can be recycled.
Economical servo motors are challenging to use in an RL setting. The random actuation and hard hits to the floor can wear and break down the small gears in servo gearboxes. Also in long continuous operation under load, some servos tend to overheat and break. To overcome these issues, we designed the legs short and the platform lightweight enough to reduce sharp jerks, and we opted to do 10-second episodes. Between episodes, the robot position is reset manually if necessary. Also, in software, we limited the maximum torque of the servo motors to half.
Initially, we used high-torque RC hobby servos (such as Turnigy TS-910) for trials, but Robotis Dynamixels were eventually selected for the design due to their longevity in testing, owing mostly to adjustable torque limits, in-built temperature sensors, and overall build quality.
B. Electrical Design
See Fig. 3 for an illustration of the electrical connectivity of the RealAnt robot. For a simple and reliable experimental setup, we use an external lab power supply and directly control the robot using a wired USB connection from a computer. Both the USB and power wires are connected to the OpenCM9.04 microcontroller board. The leg servos are daisy-chained and each leg is connected to one of the four 3-pin servo ports on the board. Alternatively, a LiPo battery, a buck DC-DC voltage converter, a low battery voltage buzzer and a Bluetooth serial converter can be used for completely wireless operation without using wired power and data cables.
C. Pose Estimation
The state of the RealAnt robot includes the 6 DoF pose of the robot and this information is also necessary to derive reward functions for reinforcement learning. For example, the reward used in the simulated Ant benchmark is the forward velocity of the robot. We rely on augmented reality (AR) tag tracking using ArUco tags and the OpenCV version of the popular ArUco library [17]- [19] for pose estimation by detection of square fiducial markers. The tags are printed on A4 office paper and glued to cardboard for rigidity. We attach the tracking tag to the top of the robot body. We use a Logitech Brio 4K web camera and place a frame reference tag within the camera view. The camera model is calibrated by taking pictures of a chessboard pattern.
Using a consumer web camera for the position estimation can be challenging, due to camera latency and frame timing jitter. The latency can be as long as several hundred milliseconds, and frame timing jitter considerable. Using the Logitech Brio 4K web camera with 1280×720 resolution at 60 fps, we measured latency of around 110 ms. This latency requires a robust RL approach and is accordingly added and tested in the simulation model.
Tag-based pose estimation is noisy and since velocity estimation is important in our tasks as a reward signal (see Sec. IV), we used Holoborodko's smooth noise-robust differentiator [20] to improve the velocity estimates. Furthermore, as the web camera is positioned on top of the tags, the z (depth) axis measurement is very noisy. We additionally smoothed this with a lowpass filter. We also add and test this estimation noise in our simulation model.
D. Software Design
We provide supporting software for the RealAnt platform so that it can be easily used and our experiments easily reproduced. We decouple the software into four processes that communicate using ZeroMQ (much like a lightweight ROS environment): a training client, a rollout server, a pose estimation process, and a control process. See Fig. 4 for an illustration of how these processes communicate. The training client controls the whole learning process. It sends the latest policy weights to the rollout server to initiate a training episode. The rollout server loads the policy weights, collects the latest observations from the control process and the pose estimation process, and sends the action computed using the policy network back to the control process. The control process continuously collects servo measurements from the microcontroller and publishes them to the rollout server. It also subscribes to actions from the rollout server and applies them to the robot through the microcontroller. The pose estimation process continuously collects images from web camera, computes pose estimates and publishes them to the rollout server. After completing an episode, the rollout server sends back the collected episode data to the train client. The newly collected data is added to a replay buffer and the agent is updated a few times by sampling from this replay buffer. Decoupling of these processes allow them to run seamlessly in different machines. For example, the data collection (with rollout server and control process) can be performed on a low-end computer and training can be performed on a high-end computer.
The rollouts were performed on a ThinkPad L390 Yoga laptop, which was attached to the RealAnt board and to the webcam. Training was done on a Linux desktop machine equipped with an Nvidia GTX1080 GPU. One episode cycle of REDQ took approximately 110 s wall clock time, including rollout and training.
Based on our hardware design, we provide simulated versions of the robot on PyBullet and MuJoCo simulators, for rapid testing and development. The simulated robot has the same physical dimensions, state-action spaces, and roughly the same dynamics. We find that MuJoCo simulation is more stable and easy to modify while the PyBullet simulator is free and easily accessible. We run our experiments on both simulators for ease of reproducibility.
IV. BENCHMARK TASKS
In this section, we define the RL specifics of the robot including the observation and action spaces. We then introduce three benchmark tasks used to evaluate RL algorithms on simulators and the physical platform.
We use an episodic formulation of reinforcement learning. Each training session is split into episodes lasting 200 steps. We manually reset the robot to the starting position of the task before each episode. We use a control frequency of 20Hz so each episode corresponds to 10 seconds of experience. The three benchmark tasks introduced below share the same observation and action spaces. Only the rewards R t are different. a) Stand / Sleep: This is a very simple task that involves attaining a goal torso height (z g ). That is,
R t = − z t − z g 2 ,
where z t is the z position of the robot torso at timestep t. In the simulator, the robot starts each episode standing upright and the goal is to sleep (that is, z g = 0) and in the physical experiments, the robot starts each episode lying down and the goal is to stand upright (that is, z g = 0.12). b) Turn: This task involves the robot turning 180 • to face the other direction. Each episode starts with the robot lying down. The robot has to balance itself and coordinate all joints to turn the whole body around. The initial yaw γ 0 of the robot is 0 and the goal in this task is to rotate to a yaw of 180 • or 3.14 radians. The rewards are computed based on this angular distance:
R t = − γ t − 3.14 2 .
This challenging task can also be used to ensure that pose estimation and tracking is accurate. c) Walk: This task is the same as for the original Ant benchmark used in simulated RL experiments: learning to walk forward as fast as possible. Each episode starts with the robot lying down. This challenging task involves the robot coordinating all its joints to walk forward as fast as possible. The reward in this task is the forward velocity of the robot:
R t =ẋ t ,
whereẋ t is the velocity of the robot along the x axis.
V. EXPERIMENTS
We begin each training session with ten episodes of data collected using a random policy. We alternate between data collection and training for every episode. We use the stateof-the-art REDQ [21] algorithm and the popular TD3 [22] and soft actor-critic (SAC) [6], [23] algorithms in this paper. REDQ allows for data-efficient RL due to high number of gradient updates of a randomized ensemble of 10 Qnetworks. We perform 2000 learning updates of REDQ or 200 updates of SAC/TD3 at the end of each episode. We use the Adam optimizer with learning rate 0.0003 to update all parameters.
We use fully-connected networks with dense connections for our policy and value networks. As reported in [24], we find that dense connections (concatenation of network inputs 6. Left: Results on the PyBullet simulator: Similar to MuJoCo, we observe that TD3 performs better than SAC and dense connections improve the sample-efficiency of TD3. Middle: Results of our ablation study on the effect of observation latency. Observation stacking is able to effectively handle latencies of up to 100 ms without any impact on performance. Larger latencies degrade learning efficiency. Right: Results of our ablation study on the effect of tracking noise. We add zero-mean Gaussian noise to body xyz and rpy observations with standard deviations of σxyz and σrpy, respectively. Observation stacking is able to deal with noisy observations but noisy rewards (computed from noisy xyz estimates) clearly degrades learning efficiency.
to the input of each layer) enable stable training of deeper networks, which allows for improved sample-efficiency. We use dense connections, three hidden layers, and 256 hidden units for our policy and value networks.
Reinforcement learning on the physical robot has to inevitably deal with latencies and noise in the observations. This makes the environment non-Markovian. To deal with this, we construct the robot state (to be used by the RL algorithm) by stacking the past four observations. Note that noise in the observations also leads to noisy rewards, which makes learning even more challenging. We introduce Gaussian noise with standard deviation 0.01 and an observation latency of 2 steps (100ms) into the simulator environments to better match real-world conditions.
A. Results on Simulator
We first test the state-of-the-art REDQ, SAC and TD3 algorithms on the simulated versions of our RealAnt robot. The results of our experiments on the MuJoCo simulator are shown in Fig. 5. We observe that REDQ significantly outperforms TD3 and SAC on all tasks, to learn them in less than 15 minutes of experience. TD3 is also able to learn all tasks but requires more data. SAC is able to perform slightly better than TD3 in the turn task but performs poorly in sleep and walk tasks. We also perform ablation studies on the walk task to study the importance of dense connections used by all algorithms and the high number of learning updates used by REDQ. The use of dense connections lead to significantly better learning performance. TD3 with 10x learning updates (matching the REDQ algorithm) learns faster in the first 150 episodes but then its performance deteriorates.
We also test the RL algorithms on the PyBullet simulator. We find that RL algorithms are able to exploit the instabilities of the PyBullet simulator, so we only test on the walk task. The results of our experiments are shown in Fig. 6. Similar to our observations in experiments on the MuJoCo simulator, we find that REDQ outperforms TD3 and both algorithms significantly outperform SAC.
B. Ablation Studies on Simulator
Real-world experiments inevitably consist of delays and noise. Inexpensive USB-connected web cameras often have Fig. 7. RealAnt robot can learn all tasks (using the REDQ algorithm) within 10 minutes of data. We focus on data efficiency and terminate the experiment once the policy performs consistently well for 10 consecutive episodes. In the walk task, the robot has learned to successfully walk outside the tracking area. We plot the mean and std of three runs on each task. relatively high communication bus and processing latency of several hundred milliseconds. Furthermore, the pose estimation and tracking is usually noisy. To ensure our method works on the real robot, we study the effect of such delays and noise on the learning efficiency of TD3 (since it is similar to REDQ, runs significantly faster in wall-clock time, and can robustly learn all tasks). We use the MuJoCo simulator for these ablation studies.
1) Effect of tracking latency:
We test the effect of tracking latency by delaying the observation of body position and orientation values in the simulator. To deal with this latency, we simply stack observations from past steps (larger than latency). The results of our experiments are shown in Fig. 6. We find that observation stacking is able to effectively deal with even significant delays of 10 steps or 500 ms.
2) Effect of tracking noise: We test the effect of tracking noise by adding different levels of noise to observations of body position and orientation values. The results of our experiments are shown in Fig. 6. We find that observation stacking used to deal with latency is also effective in dealing with observation noise. The learning algorithm is sensitive to measurements such as body position which is used to derive rewards. However, it is very robust to the other measurements such as body orientation.
C. Results on Physical Robot
In this section, we validate the RealAnt robot by evaluating the best performing REDQ algorithm on the three proposed benchmark tasks of stand, turn, and walk. We use the same network architecture (with dense connections) and hyperparameters as in our simulator experiments.
We focus on learning a reasonable policy in a data-efficient manner and terminate the training once the policy performs consistently well (based on visual observation) for more than 10 subsequent episodes. Between episodes, the robot position is optionally reset manually so as to maintain accurate pose estimation throughout the training episode. The results of our experiments are shown in Fig. 7. We are able to successfully learn all three tasks from just 60 episodes or 10 min of realworld experience, which demonstrates that training multiple different tasks with the RealAnt physical robot is feasible within a single work-day. An example of a learned walking gait and a turn is shown in Fig. 8. In the walk task, while it is possible to learn better gaits with longer training, the policy learns to successfully and consistently walk outside the field of view of our AR tag tracking setup. Implementing other challenging tasks that can be performed within the field of view or use of better tracking systems with a larger tracking area is a line of future work.
To test the robustness of our learned walk policy, we test it's performance of different surfaces. The training was performed using REDQ algorithm on vinyl flooring and tested on a hardwood floor, a polyester rug, a flat cardboard, and an IKEA KOLON plastic floor protector. We measure the walking speed of the robot on these different surfaces and report them in Table II. The results are averaged across three pre-trained policies evaluated for 5 episodes each. The robot trained on a vinyl flooring walks at a similar speed on a flat cardboard and a plastic floor protector. It walks even faster on the polyester rug as it provides more friction. It walks slower on a hardwood floor as it is slippery. While the walking speed varies according to the friction of the surfaces, the robot is able to robustly walk forward in all of them.
VI. DISCUSSION
A. Pose Estimation
Getting the AR tag-based pose estimation to work well with an aggressively moving robot is a challenge due to motion blur. This can make the AR library incapable of detecting the AR tag and consequently providing a pose estimation. The exposure time on web camera needs to be short enough to overcome the motion blur. Therefore, the tracking performance was tested before attempting robot learning, by adding enough lighting and by tweaking camera exposure settings. Some LED floodlights can also produce flickering light (similar to fluorescent lights), which becomes evident only when using short exposure times. This can affect the AR tag detection performance, but slight flicker was not an issue for our experiments. The AR tag detection is also sensitive to camera angle and height adjustment. Some trial and error is required to find a sweet spot where it works well. This ambiguity possibly could be removed with 3D-printed structure under the AR tags (e.g. a shallow pyramid).
Learning to control the robot based on feedback from tagbased pose estimation can be challenging. Especially, for the walking task, the reward only consists of the forward velocity. Estimating the forward velocity from noisy position measurements by naive numerical differentiation leads to even more noisy velocity estimates, which greatly affected the learning performance in our experiments. Therefore, we use Holoborodko's smooth noise-robust differentiator [20] to obtain better velocity estimates from real-world sensor data, which smoothed the estimates enough to learn efficiently.
B. Robot Design
During tests of several iterations of the robot design, several hardware design considerations became evident. During random exploration and training, the robot and its joints get a harsh beating. Hard impacts to the floor can break 3Dprinted rigid structures and strip the servo motor gearbox gears. Also, continuous random actions can destroy servo motor electronics by overheating.
For joint actuation, we initially used RC servos (such as Turnigy TS-910). The problem with general-purpose RC servos is that they don't have easily controllable torque limits and no overheat protection. Thus, we opted for Dynamixel smart actuators that have torque (current) limits so that hard impacts and jerks are reduced. The Dynamixels have also an embedded temperature sensor that is used for motor and electronics overheat protection. We use Dynamixel AX-12A servos in our robot but the newer and similarly priced Dynamixel XL430-W250-T is a good alternative, even though the AX-12A is cheaper in bulk pricing.
We designed the the 3D-printed parts to be lightweight and slightly elastic. Especially, the legs and body servo mounts have a little bit of flex in them, to reduce impacts. To obtain lightness and elasticity, we utilized a gyroid infill pattern which likely spreads stresses more evenly across the parts. We also printed the parts without top and bottom layers, which also helped to make it more lightweight and more elastic. We experimented with PETG and PLA filaments, and chose PLA because it has better availability, it is easier to print and it is also biodegradable.
C. Use in Education
A quadruped robot learning to walk by itself could inspire curiosity in students to learn about programming, robotics, and artificial intelligence. Building and programming the RealAnt robot involves simple yet interesting steps that could serve as an introduction to robotics and programming. For example, a curriculum designed around RealAnt could involve introductions to: programming a microcontroller, Python programming language, controlling servo motors, 3D modelling, 3D printing, augmented reality, and machine learning. Such a curriculum could equip students with the knowledge to design and build similar robots by themselves. Since RealAnt was designed to sustain aggressive explorations in RL, students can explore servo controls without fear of breaking parts. Any broken part can be easily and cheaply 3D printed and replaced.
VII. CONCLUSIONS AND FUTURE WORK
We introduce a very low-cost and minimal robot platform called RealAnt, for real-world educational and research use in reinforcement learning of legged locomotion. RealAnt is based on the Ant benchmark that is very familiar to researchers in RL, allowing for straightforward testing of existing models and algorithms. We provide the supporting software to perform RL research on RealAnt. We validate the robot with RL experiments on three benchmark tasks and demonstrate successful learning using REDQ and TD3 algorithms. Model-based RL algorithms have demonstrated significantly better sample-efficiency on the simulated Ant benchmark [25], [26]. However, such algorithms also require significantly more computational resources, and testing them on RealAnt is a line of future work.
Fig. 1 .
1The low-cost RealAnt robot imitating the 'Ant' RL benchmark.
Fig. 2 .
2Schematic details of the RealAnt robot (all units in millimeters).
Fig. 3 .
3Electrical connectivity of the RealAnt robot.
Fig. 4 .
4Software design of the RealAnt robot.
The 8 -
8dimensional action space of the RealAnt robot defines the set-point for the angular position of the robot joints. The observation space of RealAnt is computed from 6D pose estimates and joint positions. The 3D positions (x, y, and z) and 3D angles (roll α, pitch β, and yaw γ) of the torso are obtained from pose estimation using AR tag tracking. The torso velocities are computed using differences of consecutive pose estimates. The angular positions and velocities of the joints are obtained from the joint servos.
Fig. 5 .
5Results on the MuJoCo simulator: SAC and TD3 learning algorithms, with and without dense connections. We plot the mean and std of the cumulative rewards of 10 runs for each setting. TD3 performs better than SAC and dense connections lead to better sample-efficiency in the walk task.
Fig. 8 .
8Example of learned walking (top row) and turning (bottom row) gaits after 10 minutes of training data on the RealAnt robot.
TABLE I BILL
IOF MATERIALS GROUPED INTO CORE ROBOT PARTS, TRACKINGEQUIPMENT, AND A WIRELESS EXTENSION PACKAGE. Hardware design, supporting software, RL baselines, and a video of learned gaits are available at: https: //github.com/AaltoVision/realant-rl/.UNIT
TOTAL
COMPONENT
QTY PRICE (A C) PRICE (A C)
Robot
Dynamixel AX-12A servos
8
40
320
OpenCM9.04-A microcontroller
1
10
10
OpenCM9.04 accessory set
1
6
6
3D printed parts
-
-
3
Screws and cables
-
-
15
Total
354
Track
Web cam (e.g., Logitech Brio 4K)
1
250
250
Printed tags on office paper
1
1
1
Wireless
Bluetooth-serial converter HC-06
1
9
9
LiPo battery 4S 3.3Ah
1
30
30
LM2596 12V buck converter
1
2
2
Low voltage buzzer
1
3
3
real-world RL research, propose three benchmark tasks, and
report baseline results on these tasks using REDQ, TD3 and
SAC algorithms. We evaluate the state-of-the-art REDQ rein-
forcement learning algorithm for the first time on a physical
platform.
TABLE II WALKING
IISPEED OF REALANT ON DIFFERENT SURFACES USING REDQ POLICIES PRETRAINED ON A VINYL FLOORINGVinyl
Hardwood Polyester Rug Cardboard
Plastic
5.96 cm/s
4.27 cm/s
6.51 cm/s
6.02 cm/s
5.83 cm/s
We found that off-the-shelf tracking cameras such as Intel RealSense T265 are highly sensitive to the high-frequency vibrations in legged robots like RealAnt, leading to significant drift in tracking. Visual-inertial odometry algorithms that are robust to such vibrations can be used for pose estimation and tracking without any external AR tag tracking or motion capture systems. While reinforcement learning could damage the components of a robot, the learning does not take this into account. Feedback from energy usage, foot contact sensors, servo temperature, etc. can be used for safety-aware learning. Sparse negative feedback can also be provided when RL causes significant damage to the robot such as breaking of 3D printed parts or servo gears. Safe exploration in RL is an active research area.ACKNOWLEDGMENTThe authors thank Risto Bruun (Ote Robotics) and Filip Granö for their contributions to the hardware and software design along the development of the RealAnt, Sagar Vaze (University of Oxford) for his suggestions with the manuscript and also Curious AI founders Harri Valpola, Antti Rasmus, Mathias Berglund, and Timo Haanpää for support and inspiring early model-based experiments with the simulated Ant.
OpenAI Gym. G Brockman, V Cheung, L Pettersson, J Schneider, J Schulman, J Tang, W Zaremba, arXiv:1606.01540arXiv preprintG. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schul- man, J. Tang, and W. Zaremba, "OpenAI Gym," arXiv preprint arXiv:1606.01540, 2016.
Deepmind control suite. Y Tassa, Y Doron, A Muldal, T Erez, Y Li, D D , . L Casas, D Budden, A Abdolmaleki, J Merel, A Lefrancq, arXiv:1801.00690arXiv preprintY. Tassa, Y. Doron, A. Muldal, T. Erez, Y. Li, D. d. L. Casas, D. Budden, A. Abdolmaleki, J. Merel, A. Lefrancq, et al., "Deepmind control suite," arXiv preprint arXiv:1801.00690, 2018.
PyBullet, a Python module for physics simulation for games, robotics and machine learning. E Coumans, Y Bai, E. Coumans and Y. Bai, "PyBullet, a Python module for physics simulation for games, robotics and machine learning," http://pybullet. org, 2016-2019.
Challenges of real-world reinforcement learning. G Dulac-Arnold, D Mankowitz, T Hester, arXiv:1904.12901arXiv preprintG. Dulac-Arnold, D. Mankowitz, and T. Hester, "Challenges of real-world reinforcement learning," arXiv preprint arXiv:1904.12901, 2019.
Highdimensional continuous control using generalized advantage estimation. J Schulman, P Moritz, S Levine, M Jordan, P Abbeel, Proceedings of the International Conference on Learning Representations (ICLR). the International Conference on Learning Representations (ICLR)J. Schulman, P. Moritz, S. Levine, M. Jordan, and P. Abbeel, "High- dimensional continuous control using generalized advantage estima- tion," in Proceedings of the International Conference on Learning Representations (ICLR), 2016.
Soft actor-critic algorithms and applications. T Haarnoja, A Zhou, K Hartikainen, G Tucker, S Ha, J Tan, V Kumar, H Zhu, A Gupta, P Abbeel, arXiv:1812.05905arXiv preprintT. Haarnoja, A. Zhou, K. Hartikainen, G. Tucker, S. Ha, J. Tan, V. Kumar, H. Zhu, A. Gupta, P. Abbeel, et al., "Soft actor-critic algorithms and applications," arXiv preprint arXiv:1812.05905, 2018.
Mini cheetah: A platform for pushing the limits of dynamic quadruped control. B Katz, J Di Carlo, S Kim, 2019 International Conference on Robotics and Automation (ICRA). IEEEB. Katz, J. Di Carlo, and S. Kim, "Mini cheetah: A platform for push- ing the limits of dynamic quadruped control," in 2019 International Conference on Robotics and Automation (ICRA). IEEE, 2019, pp. 6295-6301.
An open torque-controlled modular robot architecture for legged locomotion research. F Grimminger, A Meduri, M Khadiv, J Viereck, M Wüthrich, M Naveau, V Berenz, S Heim, F Widmaier, T Flayols, IEEE Robotics and Automation Letters. 52F. Grimminger, A. Meduri, M. Khadiv, J. Viereck, M. Wüthrich, M. Naveau, V. Berenz, S. Heim, F. Widmaier, T. Flayols, et al., "An open torque-controlled modular robot architecture for legged locomotion research," IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 3650-3657, 2020.
Stanford doggo: An open-source, quasi-direct-drive quadruped. N Kau, A Schultz, N Ferrante, P Slade, 2019 International Conference on Robotics and Automation (ICRA). IEEEN. Kau, A. Schultz, N. Ferrante, and P. Slade, "Stanford doggo: An open-source, quasi-direct-drive quadruped," in 2019 International Conference on Robotics and Automation (ICRA). IEEE, 2019, pp. 6309-6315.
ROBEL: Robotics benchmarks for learning with low-cost robots. M Ahn, H Zhu, K Hartikainen, H Ponte, A Gupta, S Levine, V Kumar, Conference on Robot Learning. PMLR, 2020. M. Ahn, H. Zhu, K. Hartikainen, H. Ponte, A. Gupta, S. Levine, and V. Kumar, "ROBEL: Robotics benchmarks for learning with low-cost robots," in Conference on Robot Learning. PMLR, 2020, pp. 1300- 1313.
DeepRacer: Educational autonomous racing platform for experimentation with sim2real reinforcement learning. B Balaji, S Mallya, S Genc, S Gupta, L Dirac, V Khare, G Roy, T Sun, Y Tao, B Townsend, arXiv:1911.01562arXiv preprintB. Balaji, S. Mallya, S. Genc, S. Gupta, L. Dirac, V. Khare, G. Roy, T. Sun, Y. Tao, B. Townsend, et al., "DeepRacer: Educational au- tonomous racing platform for experimentation with sim2real reinforce- ment learning," arXiv preprint arXiv:1911.01562, 2019.
Duckietown: an open, inexpensive and flexible platform for autonomy education and research. L Paull, J Tani, H Ahn, J Alonso-Mora, L Carlone, M Cap, Y F Chen, C Choi, J Dusek, Y Fang, 2017 IEEE International Conference on Robotics and Automation (ICRA). IEEEL. Paull, J. Tani, H. Ahn, J. Alonso-Mora, L. Carlone, M. Cap, Y. F. Chen, C. Choi, J. Dusek, Y. Fang, et al., "Duckietown: an open, inexpensive and flexible platform for autonomy education and research," in 2017 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2017, pp. 1497-1504.
Robot learning in homes: Improving generalization and reducing dataset bias. A Gupta, A Murali, D P Gandhi, L Pinto, Advances in Neural Information Processing Systems. A. Gupta, A. Murali, D. P. Gandhi, and L. Pinto, "Robot learning in homes: Improving generalization and reducing dataset bias," in Advances in Neural Information Processing Systems, 2018, pp. 9094- 9104.
Quasi-direct drive for low-cost compliant robotic manipulation. D V Gealy, S Mckinley, B Yi, P Wu, P R Downey, G Balke, A Zhao, M Guo, R Thomasson, A Sinclair, 2019 International Conference on Robotics and Automation (ICRA). IEEED. V. Gealy, S. McKinley, B. Yi, P. Wu, P. R. Downey, G. Balke, A. Zhao, M. Guo, R. Thomasson, A. Sinclair, et al., "Quasi-direct drive for low-cost compliant robotic manipulation," in 2019 International Conference on Robotics and Automation (ICRA). IEEE, 2019, pp. 437-443.
REPLAB: A reproducible low-cost arm benchmark platform for robotic learning. B Yang, J Zhang, D Jayaraman, S Levine, ICRA. B. Yang, J. Zhang, D. Jayaraman, and S. Levine, "REPLAB: A reproducible low-cost arm benchmark platform for robotic learning," ICRA, 2019.
Infinite periodic minimal surfaces without selfintersections. National Aeronautics and Space Administration. A H Schoen, A. H. Schoen, Infinite periodic minimal surfaces without self- intersections. National Aeronautics and Space Administration, 1970.
Generation of fiducial marker dictionaries using mixed integer linear programming. S Garrido-Jurado, R Munoz-Salinas, F J Madrid-Cuevas, R Medina-Carnicer, Pattern Recognition. 51S. Garrido-Jurado, R. Munoz-Salinas, F. J. Madrid-Cuevas, and R. Medina-Carnicer, "Generation of fiducial marker dictionaries using mixed integer linear programming," Pattern Recognition, vol. 51, pp. 481-491, 2016.
Speeded up detection of squared fiducial markers. F J Romero-Ramirez, R Muñoz-Salinas, R Medina-Carnicer, Image and vision Computing. 76F. J. Romero-Ramirez, R. Muñoz-Salinas, and R. Medina-Carnicer, "Speeded up detection of squared fiducial markers," Image and vision Computing, vol. 76, pp. 38-47, 2018.
The OpenCV Library. G Bradski, Dr. Dobb's Journal of Software Tools. G. Bradski, "The OpenCV Library," Dr. Dobb's Journal of Software Tools, 2000.
Smooth noise robust differentiators. P Holoborodko, P. Holoborodko, "Smooth noise robust differentiators," http://www.holoborodko.com/pavel/numerical-methods/numerical- derivative/smooth-low-noise-differentiators/, 2008.
Randomized ensembled double q-learning: Learning fast without a model. X Chen, C Wang, Z Zhou, K W Ross, International Conference on Learning Representations. X. Chen, C. Wang, Z. Zhou, and K. W. Ross, "Randomized ensembled double q-learning: Learning fast without a model," in International Conference on Learning Representations, 2021.
Addressing function approximation error in actor-critic methods. S Fujimoto, H Hoof, D Meger, International Conference on Machine Learning. S. Fujimoto, H. Hoof, and D. Meger, "Addressing function approxi- mation error in actor-critic methods," in International Conference on Machine Learning, 2018, pp. 1587-1596.
Soft actor-critic: Offpolicy maximum entropy deep reinforcement learning with a stochastic actor. T Haarnoja, A Zhou, P Abbeel, S Levine, International Conference on Machine Learning. T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine, "Soft actor-critic: Off- policy maximum entropy deep reinforcement learning with a stochastic actor," in International Conference on Machine Learning, 2018, pp. 1861-1870.
D2RL: Deep dense architectures in reinforcement learning. S Sinha, H Bharadhwaj, A Srinivas, A Garg, arXiv:2010.09163arXiv preprintS. Sinha, H. Bharadhwaj, A. Srinivas, and A. Garg, "D2RL: Deep dense architectures in reinforcement learning," arXiv preprint arXiv:2010.09163, 2020.
When to trust your model: model-based policy optimization. M Janner, J Fu, M Zhang, S Levine, Advances in Neural Information Processing Systems. M. Janner, J. Fu, M. Zhang, and S. Levine, "When to trust your model: model-based policy optimization," in Advances in Neural Information Processing Systems, 2019, pp. 12 519-12 530.
Regularizing model-based planning with energy-based models. R Boney, J Kannala, A Ilin, Conference on Robot Learning. PMLR, 2020. R. Boney, J. Kannala, and A. Ilin, "Regularizing model-based plan- ning with energy-based models," in Conference on Robot Learning. PMLR, 2020, pp. 182-191.
| [] |
[
"Semiparametric bivariate extreme-value copulas",
"Semiparametric bivariate extreme-value copulas"
] | [
"Javier Fernández Serrano \nDepartamento de Matemáticas\nUniversidad Autónoma de Madrid\nMadridSpain\n"
] | [
"Departamento de Matemáticas\nUniversidad Autónoma de Madrid\nMadridSpain"
] | [] | Extreme-value copulas arise as the limiting dependence structure of component-wise maxima. Defined in terms of a functional parameter, they are one of the most widespread copula families due to their flexibility and ability to capture asymmetry. Despite this, meeting the complex analytical properties of this parameter in an unconstrained setting remains a challenge, restricting most uses to models with very few parameters or nonparametric models. Focusing on the bivariate case, we propose a novel semiparametric approach. Our procedure relies on a series of transformations, including Williamson's transform and starting from a zero-integral spline. Without further constraints, wholly compliant solutions can be efficiently obtained through maximum likelihood estimation, leveraging gradient optimization. We successfully conducted several experiments on simulated and real-world data. Our method outperforms another well-known nonparametric technique over small and medium-sized samples in various settings. Its expressiveness is illustrated with precious data gathered by the gravitational wave detection LIGO and Virgo collaborations. 2020 MSC: Primary 62H05, 62H12, Secondary 62-08 where w = (1 − t)α + tβ. Nonparametric methods are currently the only alternative.Related work. Vettori, Huser, and Genton provide a comprehensive review of EVCs[55]. In general, nonparametric models | null | [
"https://arxiv.org/pdf/2109.11307v3.pdf"
] | 237,605,199 | 2109.11307 | 8565c0a6489571875c743948f016d13c098f7af8 |
Semiparametric bivariate extreme-value copulas
Javier Fernández Serrano
Departamento de Matemáticas
Universidad Autónoma de Madrid
MadridSpain
Semiparametric bivariate extreme-value copulas
bivariate copulacompositional splineextreme-value copulasemiparametric modelWilliamson's transform 2020 MSC: Primary 62H0562H12Secondary 62-08
Extreme-value copulas arise as the limiting dependence structure of component-wise maxima. Defined in terms of a functional parameter, they are one of the most widespread copula families due to their flexibility and ability to capture asymmetry. Despite this, meeting the complex analytical properties of this parameter in an unconstrained setting remains a challenge, restricting most uses to models with very few parameters or nonparametric models. Focusing on the bivariate case, we propose a novel semiparametric approach. Our procedure relies on a series of transformations, including Williamson's transform and starting from a zero-integral spline. Without further constraints, wholly compliant solutions can be efficiently obtained through maximum likelihood estimation, leveraging gradient optimization. We successfully conducted several experiments on simulated and real-world data. Our method outperforms another well-known nonparametric technique over small and medium-sized samples in various settings. Its expressiveness is illustrated with precious data gathered by the gravitational wave detection LIGO and Virgo collaborations. 2020 MSC: Primary 62H05, 62H12, Secondary 62-08 where w = (1 − t)α + tβ. Nonparametric methods are currently the only alternative.Related work. Vettori, Huser, and Genton provide a comprehensive review of EVCs[55]. In general, nonparametric models
Introduction
A copula C is an extreme-value copula (EVC) if it is the weak limit of copulas emerging from component-wise maxima [28]. In the bivariate case, EVCs can be expressed as C(u, v) = exp log(uv)A log(u) log(uv) , for u, v ∈ (0, 1) 2 ,
where A : [0, 1] −→ R, known as the Pickands function (PF), satisfies the following two constraints:
1. max{t, 1 − t} ≤ A(t) ≤ 1, for all t ∈ [0, 1].
2.
A is convex.
The segments making the lower bound for A are called the support lines of the PF. Fig. 1a shows the PF geometry. The PF constraints are not inherently satisfied by conventional approximation methods. Thus, most EVC modelling depends on well-known one-parameter symmetrical families, like This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Declarations of interest: none.
Abbreviations: BM binary merger; CDF cumulative distribution function; CLR centred log-ratio; COBS constrained B-splines; EVC extreme-value copula; GC Gini coefficient; GW gravitational wave; iff if and only if; MLE maximum likelihood estimation; pdf probability density function; PF Pickands function; PLL penalized log-likelihood; RMISE root mean integrated squared error; rv random variable; SBEVC semiparametric bivariate extreme-value copula; SS simulation study; TVD total variation distance; WT Williamson transform; ZBS zero-integral B-spline.
Email address: [email protected] (Javier Fernández Serrano) On the left, PF geometry. The admissible region for its graph appears in grey. The support lines show in red. An example of PF, namely A(t) = t 2 −t +1, is drawn in blue. On the right, the geometry of a 2-monotone function derived from a PF through an affine transformation mapping A, B and C to A , B and C . The transformed version of the PF on the left, W(x) = x − 2 √ x + 1, is drawn in blue. The graph of the function W ranges from A to B and never crosses the dashed line segment between these two points.
in Table 1. Khoudraji's device allows obtaining asymmetrical EVCs from the latter, somewhat extending the applicability of parametric models [38,19]. If A is a PF, given α, β ∈ (0, 1], the following will also be a PF:
A α,β (t) = (1 − t)(1 − α) + t(1 − β) + w A tβ w ,(1)
Family [37]. The above families are symmetrical. Khoudraji's procedure provides a means for introducing asymmetry. demonstrate greater flexibility than parametric ones. Parametric models using Khoudraji's device perform well against nonparametric ones in dimensions higher than two and with mild asymmetry.
A θ (t) θ range Gumbel t θ + (1 − t) θ 1/θ [1, ∞) Galambos 1 − t −θ + (1 − t) −θ −1/θ (0, ∞)
One of the first estimators for the PF was proposed by Pickands in bivariate survival analysis [46].
However, Pickands's method produces almost surely [35] nonconvex PFs over [0,1]. Pickands himself proposed in [46] to use the greatest convex minorant of the original estimator, which remains one of the most practical and efficient approaches.
Perhaps the most widespread nonparametric method is due to Capéraà, Fougères, and Genest [9], from which it borrows its name CFG. They observe that, given a random sample {(U i , V i )} n i=1 from an EVC with PF A, the transformation Z i = log U i / log(U i V i ) is distributed according to the cumulative distribution function (CDF)
H(z) = z + z(1 − z) A (z) A(z) .(2)
One can empirically estimate H with someH and solve (2) for an estimatorÃ
(t) = exp t 0H (z) − z z(1 − z) dz .(3)
The estimatorà is not convex in general either. Jiménez, Villa-Diharce, and Flores propose two modified versions of the CFG that satisfy the convexity constraint [35]. Most estimation methods until the early 2010s are variants of either Pickands', CFG or both [55]. More recent advances have focused on polynomials and splines. For instance, Guillotte and Perron study the conditions under which a polynomial, expressed in Bernstein form, is a PF [30]. Marcon et al. use Bernstein-Bézier polynomials to enforce some PF constraints [42]. Cormier, Genest, and Nešlehová use constrained quadratic smoothing B-splines to develop a compliant PF in a nonparametric fashion using the R cobs package [13].
Previously, Einmahl and Segers had introduced a compliant nonparametric estimator requiring constrained optimization and targeting an equivalent definition of PFs [18]. The PF can be expressed [30] as
A(t) = 1 0 max{t(1 − z), z(1 − t)} dH(z) ,
where H is the so-called spectral measure on [0, 1]: a finite measure satisfying
where 1 B denotes the indicator function on B, η = A almost everywhere on (0, 1), H 0 = 1 + A (0 + ) and H 1 = 1 − A (1 − ).
The concept of Williamson's transform has recently irrupted in copula theory [5]. McNeil and Nešlehová employ it in their study of d-monotone Archimedean generators [44,45]. Charpentier et al. also use it to model multivariate Archimax copulas [11]. Even though they do not consider it in their work, Fontanari, Cirillo, and Oosterlee introduce a subclass of Archimedean copulas called Lorenz copulas, where Williamson's transform could play a crucial role in estimation, as we later specify.
Goals. Some accepted methods fail to meet all the constraints required by the PF, even in the bivariate case [55]. Semiparametric approaches, like the one introduced by Hernández-Lobato and Suárez for Archimedean copulas [32], have not been explored in the context of EVCs.
The research community is currently focusing on n-variate extensions [29]. However, a more flexible and sound construction is missing in the bivariate context. The work by Kamnitui et al. suggests that the bivariate EVC family is not as narrow, especially under asymmetry [37]. Our method will thus exclusively focus on the bivariate setting.
The semiparametric procedure we introduce here offers the following advantages over state-of-the-art methods:
Built-in PF constraint compliance, mapping any θ ∈ R d , for d large, to some PF A θ ranging in a broad spectrum of dependence strengths and asymmetries.
Optimization of θ via maximum likelihood estimation (MLE), taking the most advantage of each observation, even in small samples. Ability to penalize model complexity during the optimization process, especially for large d, reducing overfitting and opening opportunities for Bayesian analysis.
The approach by Cormier, Genest, and Nešlehová [13], also focusing on the bivariate case, allows for a large d, but does not retain control over the domain of θ. Given a sample
{(U i , V i )} n i=1 from an EVC with PF A, the random variables (Z i , T i ), where Z i = log U i / log(U i V i ), T i = logĈ(U i , V i )/ log(U i V i )
, andĈ is the empirical estimate of C, lie close to A's graph. Then one can perform a constrained B-splines regression on those points. However, the estimation procedure chooses the coordinates to satisfy the PF constraints since not all parameters would be valid. Hence, their method lacks a proper structure, falling into the nonparametric category. Difficulties are bound to appear with small samples after relying on the empirical copulaĈ and a regression approach.
Outline. We introduce in Section 2 our semiparametric method. We formally construct and estimate a large subclass of EVCs and explore their properties. Appendix A includes all the proofs. We then test our method on a simulation study (SS) and a real-world case study in Section 3. Section 4 provides further comments on our method's performance and general possibilities. Finally, we offer some concluding remarks in Section 5.
Method
In the following sections, we will cover (i) the construction of a new semiparametric EVC, (ii) some of its properties, (iii) estimation algorithms, (iv) simulation, and (v) a possible solution to one of its limitations.
Construction
A copula arising from our construction will be called a semiparametric bivariate extreme-value copula (SBEVC). We will also refer to our method as SBEVC. The construction of SBEVCs encompasses several steps. The following sections will go through them from our PF goal to a coordinates vector. In each stage, the complexity of the parameter decreases, from an infinite-dimensional functional parameter with stringent constraints to an arbitrary θ ∈ R d . That is not the natural order in the estimation phase, but it constitutes the safest path to weigh the sacrifices we make along the way. Notwithstanding, we briefly summarize the journey in its final form:
1. Given θ ∈ R d , we build a zero-integral B-spline (ZBS) p θ defined in [0, 1].
2. We apply to p θ the inverse centred log-ratio (CLR) transformation to obtain a probability density function (pdf) f θ supported on [0, 1].
3. We integrate f θ using the Williamson transform (WT) to obtain a 2-monotone function W θ supported on [0, 1].
4. We affinely transform W θ to arrive at the PF A θ , as depicted in Fig. 1.
Affine transformation
The support lines of the PF resemble a pair of coordinate axes if rotated and scaled. Let M be the unique 2-dimensional affine transformation mapping (0, 1), (0, 0) and (1, 0) to (0, 1), (1/2, 1/2) and (1, 1), respectively. M and M −1 take the form
M(x, w) = 1 2 1 + x − w 1 + x + w ,(5)M −1 (t, a) = t + a − 1 a − t .(6)
Under certain conditions, the inverse mapping M −1 transforms the graph of a PF, {(t, A(t)) | t ∈ [0, 1]}, into the graph of a 2-monotone function W defined on [0, 1] and satisfying W(0) = 1 and W(1) = 0. Here, 2-monotone stands for non-increasing and convex [44,11]. Fig. 1 shows the transition from one to the other. Note that for a twice differentiable function W with the above boundary constraints, 2-monotonicity is equivalent to W (x) ≤ 0 and W (x) ≥ 0, for all x ∈ (0, 1). Proposition 1. Let all the upcoming functions be twice differentiable on (0, 1). By means of (5) and its inverse (6), there is a one-to-one correspondence between PFs A satisfying A(t) > 1 − t, for all t ∈ (0, 1/2], and 2-monotone functions W defined on [0, 1] and satisfying W(0) = 1, W(1) = 0. Namely, A can be obtained from W as
t(x) = 1 2 (1 + x − W(x)) A(t(x)) = 1 2 (1 + x + W(x))(7)
and conversely, W from A, as
x(t) = t + A(t) − 1 W(x(t)) = A(t) − t ,(8)
where both t(x) and x(t) are automorphisms of [0, 1]. Moreover,
A (t(x)) = 1 + W (x) 1 − W (x) , (9) A (t(x)) = 4 W (x) (1 − W (x)) 3 ,(10)
and
W (x(t)) = A (t) − 1 A (t) + 1 , (11) W (x(t)) = 2 A (t) (1 + A (t)) 3 .(12)
Remark 1. We do not need the smoothness assumption in Proposition 1 in either direction. We can argue that affine transformations map convex epigraphs into convex epigraphs 2 . Hence, the convexity of A is equivalent to that of W. Nonetheless, differentiability is a convenient requirement for our construction.
Example 1. Elaborating on Fig. 1, plugging A(t) = t 2 − t + 1 into (8) yields the W(x) = x − 2 √ x + 1 we see in Fig. 1b. Example 2. The family of functions W θ (x) = (1−x) θ , where θ ∈ [1, ∞)
, meet the conditions in Proposition 1 and thus produce EVCs.
A W function like the one defined in Proposition 1 induces a spectral measure (4) by means of
η(z) = 4 W (t −1 (z)) 1 − W (t −1 (z)) 3(13)
and
H 0 = 2 1 − W (0 + ) ,(14)H 1 = −2W (1 − ) 1 − W (1 − ) .(15)
Such a W fails to attain the comonotonic copula, which has PF
A(t) = max{1 − t, t}. However, it can still model independence if W(x) = 1 − x.
Williamson's transform
Transitioning from A to W is cheap. However, W still poses stringent constraints on derivatives and boundary conditions. We can solve them by taking W as the WT of a rv supported on [0, 1] that places no mass at zero. Definition 1 (Williamson's transform). Let F be the CDF of a non-negative rv satisfying F(0) = 0. We define the WT of F as
W{F}(x) = ∞ x 1 − x r dF(r) .
A fundamental result in [44] states that Ψ = W{F} iff Ψ is 2-monotone and satisfies the boundary conditions Ψ(0) = 1 and Ψ(∞) = lim x→∞ Ψ(x) = 0. Moreover, such an F is unique and can be retrieved from Ψ as F(x) = 1 − Ψ(x) + x Ψ (x + ). It can be easily checked that the support of F is [0, x * ], where x * = inf{x ∈ R∪{∞} | Ψ(x) = 0}. In our case, the support is bounded, since W(1) = 0. Therefore, we get the following corollary.
Corollary 1. A function W : [0, 1] → R is 2-monotone 3 with W(0) = 1 and W(1) = 0 iff it can be expressed as W(x) = 1 x 1 − x r dF(r) ,
for some unique CDF F supported on [0, 1] and such that F(0) = 0.
We can further simplify the construction of W by imposing F to be absolutely continuous with pdf f :
W(x) = 1 x 1 − x r f (r) dr .(16)
The form (16) adds smoothness to W. Differentiating (16) we get
W (x) = − 1 x f (r) r dr ,(17)W (x) = f (x) x .(18)
All in all, the W function satisfies the equation
W(x) =F(x) + x W (x) ,(19)whereF(x) = 1 − F(x) = 1
x f is the survival function of F. Equation (19) is useful for computational purposes. From (17), it directly follows W (1 − ) = 0, thus H 1 in (15) is equal to zero. This feature prevents SBEVC from reaching the independence copula, for which W(x) = 1 − x. The value of W (0 + ) (and subsequently of H 0 ) is, however, dependant on the behaviour of f near zero.
Example 3. Expanding on Example 1, by using (18), we find that
F(x) = √ x. Hence, F is the CDF of U 2 , where U ∼ Unif[0, 1].
Example 4. Elaborating on Example 2, if θ > 1, by (18), we get f (x) = θ(θ − 1)x(1 − x) θ−2 , which is the pdf of the Beta(α = 2, β = θ − 1) distribution. 3 Non-negative, non-increasing and convex.
Example 5. Example 3 is a special case of WTs of positive 4 powers U θ of the uniform distribution on [0, 1]. The general formulas for their densities and CDFs are f U θ (x) = x 1/θ−1 /θ and F U θ (x) = x 1/θ , respectively, whereas their WTs are given by
W U θ (x) = 1 + 1 θ − 1 x − θ θ − 1 x 1 θ , if θ 1 1 − x + x log x , if θ = 1 .(20)
Bayes space
For modelling f , we will resort to the Bayes space, i.e., the Hilbert space (B 2 , ⊕, ) of probability density functions of square-integrable logarithm [17,40]. The space B 2 can be injected into L 2 ([0, 1]) employing the CLR transformation
clr[ f ](x) = log f (x) − 1 0 log f (y) dy .(21)
However, not every element in L 2 ([0, 1]) is attainable, since (21) introduces the constraint
clr −1 [p](x) = exp p(x) 1 0 exp p(y) dy .(22)
What is more, (21) is an isometry between B 2 and L 2 0 ([0, 1]). Example 6. The densities of positive powers U θ of the uniform distribution in Example 5 have CLR transforms clr[U θ ](x) = (1 − θ)(1 + log x)/θ. It immediately follows that all U θ are linearly dependent.
Utilizing the isometry (22), we can search for a suitable function in L 2 0 ([0, 1]) and then transform it back to a pdf. However, this space is infinite-dimensional. In practice, we shall work on a finite subspace. In general, we will build a pdf f θ as a linear combination
f θ (x) = n i=1 (θ i clr −1 [ϕ i ])(x) = clr −1 n i=1 θ i ϕ i (x) ,(23)
where we can assume the (ϕ i ) n i=1 are orthonormal, i.e., ϕ i , ϕ j L 2 ([0,1]) = δ i j , the Kronecker delta, and satisfy the zerointegral constraint.
The null element f 0 ∼ Unif[0, 1] produces the WT (20) for θ = 1. After rotation (7), the resulting PF has an explicit form that involves the Lambert W function [12]. Should the parameters in (23) be normally distributed with zero mean vector, we expect the PF to lie close to the graph in Fig. 2a. In this sense, SBEVC presents a very slight bias towards asymmetry. This bias can be corrected by considering an affine subspace instead of a pure vector space, using a convenient ω ∈ L 2 ([0, 1]) as a centre: 4 For θ ≤ 0, the resulting rv is not bounded. (26). All θ i were sampled from a normal distribution with zero mean and σ = 0.1. We randomly drew a total of 1,000 PFs. The solid line represents the mean function, while the grey envelope represents the confidence interval between quantiles 1% and 99%. The mean line is close to the A(t) = t 2 − t + 1 in Fig. 1a.
f θ = clr −1 [ω] ⊕ n i=1 (θ i clr −1 [ϕ i ]) .(24)
Compositional splines
Machalová et al. in [40] formalize the construction of a compliant ZBS as a linear combination of the usual B-splines. We shall approximate the CLR space with the ZBS subspace. We refer the reader to [40] for further details on how to compute ZBSs and [6] for more profound knowledge on B-splines, in general.
Splines are bounded functions. Committing to them, we would definitely have W (0 + ) = −∞ and thus H 0 = 0 in (14). Therefore, the resulting spectral measure would be absolutely continuous with respect to the Lebesgue measure on [0, 1] with Radon-Nikodym derivative equal to (13).
Given 0 = κ 0 < · · · < κ n+1 = 1, where n ≥ 0, and assuming 2d additional coincidental 5 knots κ −d = · · · = κ −1 = 0 and κ n+2 = · · · = κ n+d+1 = 1 at the endpoints, the space of splines p ∈ Z d κ of degree less than or equal to d and n+2 different knots κ = (κ i ) n+1 i=0 has dimension n + d. The case n = 0 corresponds to zero-integral polynomials over [0, 1]. Altogether, any ZBS can be expressed as
p θ (x) = n+d i=1 θ i Z i (x) , for θ = (θ 1 , . . . , θ n+d ) ∈ R n+d ,(25)
where 1 0 Z i = 0 and we can further assume an orthonormal basis [40], i.e., Z i , Z j L 2 = δ i j .
Furthermore, we can place a convenient centre for our ZBS space to correct the asymmetry bias. We propose to take ω in (24) to be the orthogonal projection z of −(1 + log x)/2, the case θ = 2 in Example 6, onto the space (25): Fig. 3a shows that a spline can effectively approximate the logarithmic centre, despite the divergence near zero. Fig. 3b depicts the underlying orthonormal ZBS basis {Z i }. Fig. 2b shows the effectiveness of the bias correction. x
z(x) = n+d i=1 clr[U 2 ], Z i L 2 Z i (x) .(26){(κ, z(κ)) | κ ∈ κ} z(x) − 1 2 (1 + log x)
Properties
The following sections provide some insights on the relation between the core pdf and the resulting EVC.
Convergence
We will present some results on how convergence on the Bayes space relates to convergence for the resulting EVCs through SBEVC. We shall use the supremum norm f ∞ = sup x∈X | f (x)| of a bounded function f : X → R to measure distances between some objects. X will typically be a compact subset of R n , namely [0, 1], for functions W and A, and [0, 1] 2 , for copulas C. The supremum norm defines a distance
d ∞ ( f, g) = f −g ∞ . A sequence of functions { f n } ∞ n=1
converging on the latter distance to some f is said to converge uniformly. Sometimes, however, uniform convergence is a too strong property. For instance, the function f may not be bounded on the whole X. Another convergence exists in those cases, only requiring the sequence converging uniformly to f on every compact subset K ⊂ X. Then, the sequence of { f n } ∞ n=1 is said to be compactly convergent to f . The following related concept applies to probability measures. Definition 2. Let P and Q be probability measures on the space [0, 1] equipped with the Borel σ-algebra B. The total variation distance (TVD) between P and Q is defined as
d TV (P, Q) = sup B∈B |P(B) − Q(B)| .
TVD satisfies all three axioms of a proper metric. By Scheffé's theorem [54], it can also be expressed in terms of the pdfs f and g of P and Q, respectively, as
d TV (P, Q) ≡ d TV ( f, g) = 1 2 1 0 | f (x) − g(x)| dx .(27)
Our first result links convergence in TVD of a sequence of pdfs with convergence of the corresponding sequence of WTs and their derivatives.
Proposition 2. Let { f n } ∞ n=1 be a sequence of pdfs supported on [0, 1] such that lim n→∞ d T V ( f, f n ) = 0 for some pdf f also on [0, 1].
Let W n and W be the corresponding Williamson transforms of f n and f , respectively. Then, the sequence {W n } ∞ n=1 uniformly converges to W on [0, 1]. Moreover, {W n } ∞ n=1 compactly converges to W on (0, 1]. The next step links the uniform convergence of WTs with that of PFs. Uniform convergence of pointwise-convergent sequences of PFs can be established by other means [20]. Nonetheless, the following result also states the uniform convergence of the first derivatives of PFs under the same hypotheses, which cannot be taken for granted. Uniform convergence of function derivatives is mainly unconnected to uniform convergence of the functions themselves. The reason why it works, in this case, comes down to Williamson's transform, whose first derivative is also in a convenient integral form that allows applying TVD convergence of the internal pdfs to both the function and its derivative simultaneously.
Proposition 3. Let {W n } ∞
Some topological arguments allow establishing uniform convergence of copulas from pointwise convergence alone [48]. Notwithstanding, there exists a connection between the supremum norms of copulas and those of their respective PFs [37]. Namely,
C 1 −C 2 ∞ ≤ 2γ/(1+2γ) 1+1/(2γ) , where γ = A 1 −A 2 ∞ .
We turn next to some assumptions making convergence in L 2 0 ([0, 1]) sufficient for pdfs to converge in TVD. Proposition 4. Let {p n } ∞ n=1 ⊂ L 2 0 ([0, 1]) continuous and uniformly bounded, i.e., p n ∞ ≤ K for some K > 0 and for all n. Suppose lim
n→∞ p − p n 2 = 0, for some p ∈ L 2 0 ([0, 1]). Let f n = clr −1 [p n ] and f = clr −1 [p]. Then, lim n→∞ d T V ( f, f n ) = 0.
SBEVC, acting on convergent sequences in L 2 0 ([0, 1]), produces EVCs that not only uniformly converge but also whose partial derivatives do. Since copula partial derivatives correspond to conditional CDFs, they are paramount in copula sampling algorithms [7]. Corollary 2 guarantees that the resulting samples tend to fit the copula uniformly over the support.
Corollary 2. Let {z n } ∞ n=1 ⊂ L 2 0 ([0, 1]
) be a sequence of uniformly bounded smooth cubic ZBSs that converge in the · 2 norm. The corresponding EVCs from SBEVC {C n } ∞ n=1 uniformly converge to some EVC C, satisfying ∂ i C n · ∞ − −−− → n→∞ ∂ i C compactly over (0, 1] 2 , for i = 1, 2.
Dependence
In the case of an EVC, Kendall's tau and Spearman's rho take the form of integrals involving the PF [37]. The substitution of the PF by the equivalent WT form and a change of variables afterwards do not provide any meaningful insight into the latter's role. However, the apparent relation between WTs and Lorenz curves reveals a new path for measuring association.
The WT W of a PF A satisfies the definition of a Lorenz [21] curve L after the change of variable L(x) = W(1 − x), for x ∈ [0, 1]. Lorenz curves appear in econometrics for assessing wealth inequality through the Gini coefficient (GC) G = 1 − 2 1 0 L. The G index has a geometrical interpretation as the area between L and x → x divided by the area under x → x, which is equal to 1/2. The same interpretation applies to W and, since
1 0 W = 1 0 L, we have G = 1 − 2 1 0 W. A value of G = 0 means
wealth is uniformly distributed (the p% wealthiest proportion of the population accumulate p% of the total wealth, for all p ∈ [0, 1]), whereas G 1 means that nearly all wealth belongs to a tiny fraction of the population. In summary, G = 0 represents perfect equality, while G = 1 represents perfect inequality.
In our context, a PF A is the affine transformation of some W. Since affine transformations change areas by applying a constant factor, the latter cancels out in ratio measures, leaving them invariant. Therefore, the GC is the area between A and the upper bound line {y = 1} divided by the area between the support lines and the upper bound line. This argument leads to , producing the independence copula. This way, the bivariate positive association can be interpreted in econometric terms: comonotonicity is equivalent to perfect inequality, whereas independence corresponds to perfect equality.
G = 4 1 −
The GC is an uncommon association measure in the context of EVCs, despite its simplicity. We have only found a brief mention of it in [31], where the measure was not scaled to lie on [0, 1]. Moreover, PF symmetry was further assumed. The index can be reformulated for any positively quadrant dependent copula C, i.e., C(u, v) ≥ uv, for all u, v ∈ [0, 1] 2 , as 6
G = 4 1 − [0,1] 2 log C(u, v) log uv dudv .
The GC, as defined above, satisfies the axioms of a dependence measure [7]. The GC takes values on [0, 1], unlike Kendall's tau and Spearman's rho, which belong to a more general family of concordance measures, ranging in [−1, 1] and allowing for negative association.
Remark 3. Interestingly, after integrating by parts twice, the area of W can be expressed in terms of the inner pdf f , yielding
G = 1 − E[X]
, where X ∼ f . This means that every GC in (0, 1) is attainable through SBEVC.
Estimation
The estimation process builds upon the various constructions explored above. Given an orthonormal ZBS basis, we aim to find the parameter vector θ that best fits a dataset. Knowing the one-to-one relation [9] between the random vector (U, V) following an EVC and the rv Z = log U/ log(UV), we reduce our problem to fitting the latter, which has a more straightforward pdf, derived from (2):
h(z) = 1 + (1 − 2z) A (z) A(z) + z(1 − z) A (z) A(z) − A (z) A(z) 2 . (28)
Given a random sample D = {z i } m i=1 from Z and a model h θ derived from p θ up to A θ , the frequentist approach to the estimation addresses the maximization of the penalized log-likelihood
(PLL) of h θ (θ|D) = m i=1 log h θ (z i ) − λ 1 0 (p θ (x)) 2 dx ,(29)
for some regularization hyper-parameter λ ≥ 0. The square norm term involving p θ is the linearized curvature of the spline: a simplified non-intrinsic form of the curvature that can be expressed as a covariant tensor θ Ωθ, where Ω = (Ω i j ) = ( 1 0 Z i Z j ). Splines may exhibit complex shapes prone to overfitting, as shown in Fig. (3b). Penalizing the curvature is the proposed method in [40] in the context of compositional data regression. Hernández-Lobato and Suárez applied this approach in semiparametric copula models before [32]. Taking λ = 0 removes regularization, retrieving the usual loglikelihood.
Estimating the parameters of such a model poses some challenges. Evaluating the resulting PF from a parameter vector and a single argument implies several non-trivial operations, most notably the integral and affine transformations (16) and (7). These steps can be applied with near-perfect accuracy, with proper algorithms and without time constraints. However, in an iterative optimization, time is scarce. Therefore, we propose critical approximations at each step that trade some accuracy off for processing speed without compromising the overall stability. The effectiveness of our proposal will be thoroughly tested in a SS in Section 3.
First, note that the evaluation of A in (7) at a specific value t 0 requires solving for x in t(x) = t 0 . The latter will generally be a nonlinear equation that can only be solved through numerical methods at a relatively high computational cost. Hence, in most cases, evaluating (28) in (29) at each point z i becomes rapidly unaffordable as m increases. Moreover, any root finding procedure would prevent us from applying gradient optimization, stopping backpropagation. We propose h θ be approximated by a piecewise linear interpolatorh with sufficiently numerous and carefully selected knots.
Since (29) is based on an empirical univariate sample D, a good knot selection utilizes uniform quantiles of D. This way, the knots will be more spaced on low probability regions and accumulate on high probability ones. This criterion, which was employed in a similar setting in [32], reduces the variance of the parameter vector θ. Once fixed the quantiles
{q i } k i=1 , we need to estimate some {x i } k i=1 such that t i ≡ t θ (x i ) ≈ q i and then take h i = h θ (t i )
as the linear interpolator value at knot t i . Note that the t i 's are approximations for the q i 's. To estimate the required x i 's, we may apply (8) over the q i 's grid using an empirical nonparametric estimate of the PF, like (3). We can state the procedure as follows.
Algorithm 1 (Selection of an interpolation grid for h θ ). Let D = {z i } m i=1 be a random sample following the H distribution. To build an interpolation grid {x i } k+1 i=0 in the W space such that {t(x i )} k+1
i=0 are roughly distributed according to D, follow these steps:
1. Pick k uniform quantiles 0 = q 0 < · · · < q k+1 = 1 of D. 2. Build the empirical CDFH of D.
Build an empirical estimateà usingH and (3). 4. Ensure boundary constraints takinĝ
A(t) = min{1, max{t, 1 − t,Ã(t)}} . 5. Set x 0 = 0 and x k+1 = 1. Then, for every i ∈ {1, . . . , k}, set x i = q i +Â(q i ) − 1. 6. Sort ascendingly the resulting {x i } k+1
i=0 and remove duplicates if needed.
We can reuse the grid obtained in the last algorithm throughout the estimation process, at every gradient descent step and with different values for the parameter vector θ. With this grid and a parameter vector θ, we can now build a light version of h θ to evaluate the PLL. Early experiments suggest that selecting spline knots for p θ according to Algorithm 1 is key to constructing an unbiased estimator A θ .
Along with the interpolation grid, we need to estimate the values of W θ and its derivatives from p θ .
Algorithm 2 (Approximation of the WT and its derivatives). Let p θ be the ZBS corresponding to the parameter vector θ. Let
{r i } n+1
i=0 be an strictly increasing real sequence such that r 0 = 0 and r n+1 = 1. Let 0 such that < r 1 . To build an approximation to the corresponding WT W θ and its first and second derivatives, follow these steps:
1. For i ∈ {0, . . . , n + 1}, set p i = exp p θ (r i ). 2. Compute I ≈ 1 0 exp p θ using the composite trapezoidal rule over {(r i , p i )} n+1 i=0 . 3. For i ∈ {0, . . . , n + 1}, set f i = p i /I. 4. Set s 0 = . Then, for i ∈ {1, . . . , n + 1}, set s i = r i . 5. For i ∈ {0, . . . , n + 1}, setW i = f i /s i . 6. For i ∈ {0, . . . , n}, set ∆ i = s i+1 − s i . 7. For i ∈ {0, . . . , n}, set P i = ∆ i ( f i + f i+1 )/2 and Q i = ∆ i (W i +W i+1 )/2. 8. SetW n+1 = 0.
Then, for i from n down to 0, computeW i using the recurrence relation
W i =W i+1 − Q i .(30)
9. For i ∈ {0, . . . , n + 1}, set δ i = s iW i .
10. SetW n+1 = 0. Then, for i from n down to 0, computeW i using the recurrence relation
W i =W i+1 + δ i − δ i+1 + P i .(31)
11. For i ∈ {0, . . . , n + 1},
set W i =W i /W 0 , W i =W i /W 0 , W i =W i /W 0 . 12. Build a piecewise linear interpolator W 0 from {(r i , W i )} n+1 i=0 for W θ . 13. Build a piecewise linear interpolator W 1 from {(r i , W i )} n+1 i=0 for W θ . 14. Build a piecewise linear interpolator W 2 from {(r i , W i )} n+1 i=0 for W θ .
Now, we are ready to build a light version of h θ .
Algorithm 3 (Approximation of h θ from WT estimates). Let {x i } k+1 i=0 be the interpolation grid from Algorithm 1. Let W 0 , W 1 and W 2 be the piecewise linear approximations to W θ , W θ and W θ from Algorithm 2, respectively. To build an approximation for h θ , follow these steps:
1. For i ∈ {1, . . . , k}, set W i = W 0 (x i ), W i = W 1 (x i ), W i = W 2 (x i ). 2. Set t 0 = 0 and t k+1 = 1. Then, for i ∈ {1, . . . , k}, set t i = (1 + x i − W i )/2. 3. For i ∈ {1, . . . , k}, set A i = (1 + x i + W i )/2. 4. For i ∈ {1, . . . , k}, set M i = 1 − W i . 5. For i ∈ {1, . . . , k}, set A i = (1 + W i )/M i . 6. For i ∈ {1, . . . , k}, set A i = 4W i /M 3 i . 7. For i ∈ {1, . . . , k}, set D i = A i /A i . 8. Set h 0 = h k+1 = 0. Then, for i ∈ {1, . . . , k}, set h i = 1 + (1 − 2t i )D i + t i (1 − t i ) A i A i − D 2 i .(32)
9. Build a piecewise linear interpolatorh from
{(t i , h i )} k+1 i=0 . 10. Compute I ≈ 1 0h using the composite trapezoidal rule over {(t i , h i )} k+1
i=0 . 11. Useĥ =h/I as an approximation for h θ over [0, 1]. Algorithm 3 deals with problems like the approximation of h θ and the rotation of W θ . On the other hand, Algorithm 2 formalizes an efficient computation scheme for W θ . Both together allow computing (θ|D). Fig. 4 shows the full computation graph. Gradients flow from the top PLL down the parameter vector using backpropagation. We recommend using the autograd package [41], capable of performing automatic differentiation on native Python operations. Some representative routines in our implementation are numpy's trapz, for calculating integrals using the trapezoidal rule, and cumsum, for computing recurrences (30) and (31).
Implementation tips. Algorithm 2 and Algorithm 3 use discretization to approximate functions and integrals. The finergrained the discretization steps, the lower the error and the higher the computation time. A trade-off between those dimensions is needed. On the other hand, knowing W θ (0 + ) = −∞ and W θ (0 + ) = ∞, we recommend choosing the grid in Algorithm 2 so that points accumulate near zero, making the linear interpolation more effective. Chebyshev nodes are a standard option.
To facilitate SBEVC's estimation process, we propose to change the copula variable ordering whenever a steep slope is likely to appear for W θ near zero, which coincides with the minimum of A θ being placed at t < 0.5. We can heuristically assess this situation by calculating the mode of the pdf h, as suggested in [19]. If the mode appears at t < 0.5, the PF's minimum will likely be placed at t < 0.5. We support the hypothesis of [19] based on our own experience. Therefore, whenever the mode peaks at t < 0.5, we recommend changing the variable ordering before estimating and then flipping the resulting PF A asÃ(t) = A(1 − t).
Simulation
Once the parameters θ have been estimated, we propose to build W θ and A θ subsequently. From that point on, querying the model (simulating, estimating probabilities, among others) will be equivalent to evaluating the PF A θ , as with any other EVC.
The algorithms in Section 2.3 stand valid, with some minor and convenient changes. Since we only need to build the functions once, and not once per iteration, we may employ more expensive and accurate approximations. In particular, W θ can be evaluated without approximations. More sophisticated procedures should replace trapezoidal rules and linear interpolations. On the other hand, Algorithm 1 is no longer required. Instead, we may employ a root-finding algorithm to invert the automorphism t.
The interpolation points of A θ could be input to the shapepreserving interpolation procedure by Schumaker, which would guarantee that the resulting spline is convex over the whole domain [49]. However, in general, the second derivative of such a spline would not be continuous, which would hinder the simulation process. In practice, we recommend smoothness and accuracy over shape preservation, provided a sufficiently fine interpolation grid is used.
Finally, to draw samples from C θ , we recommend the general algorithm in [7], which only requires inverting one of its partial derivatives [16,19]. The root-finding algorithm in [3] proves to be highly effective.
Refinement
One of the limitations of SBEVC is the fact that an estimated PF A always satisfies A (0 + ) = −1 and A (1 − ) = 1. These constraints are a consequence of our construction, which imposes W (0 + ) = −∞ and W (1 − ) = 0 on the WT. In practice, however, these boundary constraints do not hinder the expressiveness of the resulting model. Remember that, for instance, upper tail dependence does not relate to either boundary derivative of the PF, but the mid-point value A(1/2). This fact contrasts with the nature of another semiparametric procedure like [32], where a slope value entirely determined the tail index.
In SBEVC, misspecified slopes for the PF have a much lower impact on the concordance (Blomqvist's beta) and upper tail dependence. Nonetheless, since it might produce a slight bias, we propose a refinement step that could complement SBEVC.
Khoudraji's method is best known for inducing asymmetry in symmetrical EVCs [38]. However, there is no reason why it could not apply to asymmetrical ones [47]. Consider a PF A obtained through SBEVC. Differentiating (1), we arrive at
A α,β (0 + ) = βA (0 + ) = −β A α,β (1 − ) = αA (1 − ) = α ,(33)
where, remember, α, β ∈ (0, 1], retrieving A for α = β = 1. Even though A is, in general, asymmetrical, we see from (33) that Khoudraji's method serves our purpose of freely parameterizing the boundary slopes. 7 We believe that adding two more parameters through Khoudraji's method may improve the fitness of the resulting model in some particular cases, especially for weak correlations. However, the inclusion of the new parameters in the gradient-based optimization seems unworkable, as it would invalidate the interpolation grid in Algorithm 3. A derivative-free optimization involving both the spline parameter vector θ and the asymmetry parameters α and β could be run, starting from α = β = 1 and some initial guess θ = θ 0 obtained through a gradient-based method.
Results
We will test SBEVC on simulated and actual data. In both scenarios, we will compare SBEVC with the methodology by Cormier, Genest, and Nešlehová [13]. We shall refer to their method as constrained B-splines (COBS). We have chosen COBS for its flexibility and simplicity, sharing three relevant traits with SBEVC: complying with PF constraints, using splines and exclusively addressing bivariate EVCs.
Preliminaries
Before diving into the specific settings of each experiment, let us clarify some shared configuration aspects.
Optimization
We performed all SBEVC estimates using standard Python scientific packages like numpy and scipy, and automatic differentiation, thanks to autograd [41]. Namely, we employed scipy's implementation of the L-BFGS-B algorithm by Byrd et al. [8]. We assessed convergence by setting the ftol=1e-6 configuration parameter in the minimize routine, which targets the relative change in the loss function between iterations. Even though this value is very conservative, the procedure converges well, with reasonable execution times, as we will see.
All estimation runs started at the null spline, with all coordinates equal to zero, regardless of using a fixed affine centre. Despite the caveats by Hernández-Lobato and Suárez [32], as demonstrated in [50], current optimization methods can deal with complex problems even if the initial parameter values are far from the optimal solution. Notwithstanding, we agree with Hernández-Lobato and Suárez that good initial guesses would speed up the process.
COBS is very easy to implement on top of the cobs R package [13]. The R execution environment can be accessed from Python thanks to the rpy2 Python package with little coding overhead. Specifically, we employed the main cobs routine, selecting a smoothing splines regression of degree two by entering lambda=-1. We raised the maximum number of iterations until convergence to 1,000 using the maxiter parameter. We kept the maximum number of spline knots to the default value of 20. Both the knots selection and the smoothing penalty was internally chosen by cobs. The convexity requirement was introduced by setting constraint="convex". The boundary constraints were enforced over a fine 1,001-point equally-spaced grid over [0, 1] using the pointwise argument. Finally, we interpolated cobs' result over the former grid using cubic splines to allow for continuous second derivatives.
Resources
Both the SS and actual data application would not have been possible without the vast repertoire of software artefacts and services currently available.
First, SBEVC, fully implemented in Python, was containerized using Docker [15], which, apart from being ideal for achieving reproducible research, also helped to move our execution environment to the cloud with Kubernetes [52]. Docker was also helpful for preparing a maintainable execution environment with Python and R, as required by cobs.
While developing and testing, we employed a local minikube cluster [53] on an Intel ® Core © i7-4700MQ CPU laptop with eight 2.40 GHz cores and 15.6 GiB of memory and operating system Ubuntu 20.04.3 LTS. We entrusted the bulky SS final executions to a cloud provider. The Kubernetes service comprised 50 dynamically allocated nodes running on possibly different 8 Intel ® Xeon © architectures with Ubuntu 18.04. Overall, each node counted on two virtual CPUs and seven GiB of memory at any given time. Due to Kubernetes' requirements, only one CPU was available for Spark per node.
We sped up the experiments parallelizing specific tasks with Spark [51]. To prepare the Spark setting with Kubernetes, two artefacts were of great help: the Docker image for Apache Spark [14] and the Spark Operator [27].
Finally, Argo CD [4] turned out to be helpful to manage all our Kubernetes experiments from the same friendly user interface, connecting the Kubernetes cluster to the Git repository.
Supplementary materials. Access to source code and other deliverables will be provided upon acceptance for publication.
Simulation study
We conducted a SS to test the effectiveness of SBEVC on a broad spectrum of cases with high confidence. The SS consists of three experiments. The first one addresses the bias and variance tradeoffs by repeating the estimation process for many random samples drawn from a fixed copula in Table 1 for several parameter configurations. The second experiment covers an even more extensive array of EVCs while focusing on validation through TVD. Then, the third one compares SBEVC with COBS in terms of the root mean integrated squared error (RMISE) in several scenarios with varying dependence strengths, asymmetries and sample sizes.
SBEVC entails numerous non-trivial analytic and geometric transformations. Even though we could argue that none of them exceeds a reasonable level of complexity, clever algorithms and powerful computational resources are still needed for it to work in practice. In particular, optimization algorithms are vital to finding solutions that maximize (29) under memory and time constraints. The joint behaviour of all these pieces is difficult to assess from a purely theoretical perspective without simulations.
Common settings. Throughout the SS, copula models build upon a cubic orthonormal ZBS basis. The grid size was 200 both in Algorithm 2 and Algorithm 3 (n + 2 = 200 and k + 2 = 200, respectively). Also, we took = 10 −9 in Algorithm 2. These settings express an adequate balance between approximation accuracy and reasonable execution times.
Bias and variance
The first part of the SS consisted of 30 individual experiments, focusing on a particular instance of a copula family. For each copula instance, we performed an estimation run with SBEVC on each of 100 different random samples from the copula for 3,000 runs. Then, for each 100-sample experiment, we collected the pointwise means and pointwise 98% confidence intervals of the estimated PFs and compared both functional statistics with the original PF. We handled each random sample as an independent Spark task to speed up computations.
We employed the two families in Table 1: the Gumbel and the Galambos families. These are probably two of the most well-known EVCs. Genest and Nešlehová even studied and found a relation between the two in [23], knowing the similarity of their PFs. In each copula family, we tested up to five different values of the unique parameter θ, giving rise to different correlation levels. Finally, apart from the pure form of each Gumbel or Galambos copula, we introduced asymmetry through Khoudraji's device, taking either α or β equal to 0.5 and leaving the other as 1. This configuration was precisely the one that demonstrated higher asymmetry in [24]. As a side note, we remind that the asymmetrical extensions of the Gumbel and Galambos families are known as the Tawn and Joe families, respectively.
Each of the random copula samples consisted of 1,000 observations. Our models were fit using 13 parameters in all cases: 10 more than the ground truth copula families. We believe that the specific number of parameters has less impact in a semiparametric context, where one typically employs a large number and then reduces overfitting by penalizing curvature. In the end, the target of this semiparametric method is a function that lives in an infinite-dimensional space. In practice, both the Gumbel and the Galambos families need fewer parameters than 13, but in this case, we have preferred to stick to a large number to showcase the method's performance in a general setting. Finally, for both the Gumbel and Galambos copulas, we used a curvature penalty factor λ = 10 −5 . Fig. 5 shows the results for the Gumbel copulas, while Fig. 6 presents those of the Galambos. The results are qualitatively very similar. SBEVC displays low biases and variances in all cases. If any, the highest biases appear under asymmetry and low correlations. This behaviour matches the known limitation of SBEVC as regards the boundary slopes, which have fixed values. Variance is also higher for small correlations, in agreement with [37].
In all simulations, we employed the trick mentioned in Section 2.3 for selecting the a priori more convenient variable ordering to avoid numerical instabilities. The procedure worked well, as demonstrated by the nearly identical results obtained for either α = 0.5 or β = 0.5.
Execution details. We approximately recorded the execution times for the 30 experiments with the assistance of the Spark Web UI. Roughly 50% finished in three minutes, 75% in four and 90% in seven. Consistently below eight minutes, the most time-consuming experiments correspond to the highest correlated Gumbel and Galambos copulas. That is presumably because the optimal solution was furthest from the starting null vector. Since there were 100 tasks on each job and the cluster only had 50 nodes, we could expect the average execution time to be half of the previous values.
Total variation
In the second part of the SS, we generated n = 200 random SBEVCs. We chose the affine spline model with centre (26) as the first building block, assuming uniformly distributed knots and d = 13 parameters. Let us call θ 0 ∈ R d the coordinates of the centre of the affine model. We then ran an MCMC simulation assuming the model coordinates θ in (24) were distributed according to the following pdf: where Ω is the curvature matrix of the underlying spline, as described in Section 2.3, and λ and R are tuning parameters. The previous model is the truncated version of an improper prior based on curvature penalization, with factor λ. The support of the distribution is the hyperball of radius R.
p(θ) ∝ e −λθ Ωθ , if θ 2 ≤ R 0 , if θ 2 > R , forθ = θ + θ 0 ,
We tuned the parameters with values λ = 10 −4 and R = 5 so that the resulting splines covered a wide range of correlations (in the sense of the GC) and were, at the same time, smooth. Finally, to prevent any asymmetry, we replaced the even elements in the sequence with their corresponding mirrored ver-sionsÃ(t) = A(1−t). Fig. 7 shows a subsample of the generated random PFs. They cover the area between the support lines and the upper bound line in a reasonably balanced way. We also employed the heuristic to determine the most suitable variable ordering in this part of the SS. Hence, we expected SBEVC to perform well regardless of the orientation of the PF. For each element in the sequence {θ i } n i=1 , we built the EVC C θ i and performed several estimation runs on random samples of different sizes {S j } m j=1 . All fitted models had the same number of parameters as the ground truth splines (d = 13) and employed the same affine translation. The penalty factor in the loss function (29) was also set to λ = 10 −4 . The only aspect in which estimated models differed from ground truth is spline knot placement, which was uniform for the latter, but empirically assessed for the former. Then, for each sample size S j , we estimated a copula C i j using SBEVC and assessed divergence from ground truth through TVD
d TV (C θ i , C i j ) = 1 2 [0,1] 2 |c θ i (u, v) − c i j (u, v)| dudv ,(34)
where c θ i and c i j are the pdfs of C θ i and C i j , respectively. The TVD defined in (34) is the bivariate counterpart of Definition 2 and thus provides an upper bound on the difference between the measured values of each copula on any measurable set B ⊂ [0, 1] 2 . Therefore, (34) is a very conservative evaluation measure. Table 2 presents the main summary statistics from the experiment. Each Spark task targeted a different random EVC. Then, each task comprised four estimation runs. The table shows promising results, considering the complexity of the ground truth models and the finiteness of samples. Mean values are typically below 0.05, whereas the 75% and 90% quantiles do not surpass the 0.10 threshold. decreases as the sample size increases. Fig. 8 reveals some outliers, which become rarer with larger sample sizes. Besides, Fig. 8 suggests SBEVC can handle even the highest correlated samples well. The outliers are due to the sensitivity of the TVD metric to deviations in highly correlated samples. As Fig. 9 shows, the TVD metric positively correlates with the GC even for moderate values of the latter. Execution details. The total execution time for the 200 random EVCs was roughly one hour. Considering the cluster only had 50 nodes, the average execution time for a task in this experiment was approximately 15 minutes.
RMISE
The third part of the SS consisted of 20 individual experiments with different settings. We generated 100 random sam- ples in each experiment and fitted them using SBEVC and COBS. Then, we collected the squared L 2 ([0, 1]) distances between the ground-truth PF and the estimated PF through SBEVC and COBS. For each estimation method, these measures were averaged to approximate the RMISE like in [55]. The lower the RMISE, the better the technique. We also assessed the statistical significance of the results through a Wilcoxon signed-rank test applied on the unaggregated squared L 2 ([0, 1]) distances, assuming the null hypothesis that COBS produces better results than SBEVC. As in the previous parts of the SS, we used for SBEVC d = 13 ZBS elements and the curvature penalty factor was λ = 10 −4 . The RMISE results are presented in Table 3. Each row corresponds to a different experiment. The left-most column shows the experiment settings. The underlying parametric copula belongs to an asymmetrical Gumbel family (see Table 1) using Khoudraji's device (1). Then, θ is the parameter of the Gumbel copula and β is one of the asymmetry parameters, fixing α = 1 as a constant throughout all configurations. The combinations of θ, and β are precisely those that appear in the mid and right columns in Fig. 5. We draw samples from each copula configuration with a medium (n = 1, 000) and a small (n = 250) sample size. As we can see, SBEVC significantly outperforms COBS in all circumstances, roughly halving the RMISE.
Case study
The following sections will solve a statistical modelling and simulation problem on LIGO and Virgo's precious gravitational wave (GW) detection data. The aim of this case study is twofold. On the one hand, we will examine the steps in the construction and estimation of SBEVCs with an authentic hands-on experience. On the other hand, we aim to compare SBEVC with COBS on non-synthetic samples. After some sensible transformations, LIGO and Virgo's data have an EVC dependence structure. The underlying copula presents a very different look than what we have seen in Fig. 5 and Fig. 6.
History
In 2015, the LIGO 9 Scientific Collaboration and the Virgo Collaboration announced the first direct detection of a GW, produced by the merger of a binary black hole [1]. The existence of GWs, ripples in the fabric of space-time, was predicted by Einstein's theory of general relativity in 1916 as a mathematical construct that many thought to have no physical meaning [10]. It took nearly a century from its prediction and 60 years of search to experimentally ascertain the discovery, opening a new era for astronomy.
Only the most extreme events in the Universe, in terms of energy, can generate GWs strong enough to be detected by current experimental procedures due to the small value of the gravitational constant [10], which expresses the rigidity of spacetime. A significant amount of human and material resources are needed to detect GWs. Specifically, sufficiently sensitive interferometers need to have arms several kilometres long. Additionally, in order to discriminate between true detections and spurious local signals (like electromagnetic radiation or earthquakes), several detectors, far apart from each other, are needed.
LIGO, settled in the United States, with two laboratories, was the first detector of an advanced global network that aims to increase discoveries' accuracy and exhaustiveness [1], soon to be joined by others, most notably Virgo, in Italy. Despite LIGO and Virgo joining efforts, it was LIGO that reported the first detection since the Virgo facilities were not operating at that time for upgrading reasons. Since the first detection in 2015, the collaboration of LIGO and Virgo has confirmed 50 events. They all correspond to massive body mergers, mainly black holes and neutron stars.
Data
We have chosen the GW detection dataset gathered by the LIGO and Virgo collaborations during their first three observation runs to test the applicability of SBEVC. It consists of 50 rows and two columns. Each row represents a merger event, while each column features one of the masses involved in the event, measured in solar mass units (M ). During the first and second observation runs, 11 events were detected, while the third run provided 39. The first event was GW150914, in September 2015, and the last one, GW190930 133541, in September 2019. LIGO and Virgo report the larger of the two masses, the primary mass, as the first tuple component, followed by the secondary mass.
We believe that very few datasets better represent bivariate data, considering the very nature of binary mergers. Bivariate models are usually building blocks for higher-dimensional ones, but in this case, all the attention is focused on two mass values of high scientific relevance. Another aspect that adds to this significance is the scarcity of data, for only 50 events have been recorded during five years. This scarcity contrasts with the increasingly large amounts of information coming from IoT, social networks, finance, among others, in the current era of Big Data.
Model
As mentioned above, the dataset consists of 50 bivariate ob-
servations D = {(M (i) 1 , M (i) 2 )} 50 i=1 , where M (i) 1 ≥ M (i) 2 .
The last censoring constraint makes the dataset not directly tractable by usual copulas, supported on the whole [0, 1] 2 , unless conveniently preprocessed.
LIGO and Virgo perform a statistical analysis of the joint mass distribution [2]. They consider two separate univariate models. The first one models the primary mass M 1 unchanged, whereas the second one models the mass ratio Q = M 2 /M 1 conditioning on M 1 . Since M 1 ≥ M 2 , by definition, the resulting model captures by construction the censoring constraint. The final joint model is formed by the vector (M 1 , QM 1 ).
Instead of considering an auxiliary ratio variable, we directly model a bivariate mass vector. We turned the censoring problem into an exchangeable one, where both masses played the same role. The original dataset D does not allow such a treatment, so we hypothesized a new sample space where primary masses are detected with 50% probability at the first vector component and 50% at the second one. This scenario corresponds to detections reporting masses without considering their relative order. Therefore, we built a new sample D =
{( M (i) 1 , M (i) 2 )} 100 i=1 , where M (i) j = M (i) j or M (i) j = M (i−50)
1+ j mod 2 , respectively, if i ≤ 50 or i > 50. We then targeted a random vector ( M 1 , M 2 ). To retrieve the original primary-secondary mass model, we just had to take M 1 = max{ M 1 , M 2 } and M 2 = min{ M 1 , M 2 }.
Using the previous up-sampled and symmetrical dataset, we fitted (i) a single univariate mass model f for both margins and (ii) a copula model C of the dependency between mass ranks.
Univariate margin model. We decided to employ a semiparametric model for the univariate margin mass model. We successfully tried the same technique we used for modelling the density f in (16): Bayes space pdfs built from ZBSs.
The result of our experiment is shown in Fig. 10. We selected 17 parameters, with knots distributed according to the original sample between 1 M and 100 M , and a curvature penalty factor of 10. The first mode, near 1 M , mostly corresponds to neutron stars; black hole masses typically range beyond 5 M .
Bivariate copula model. LettingF be the empirical CDF of the
univariate sample { M (i) 1 } 100 i=1 (equivalently, from { M (i) 2 } 100 i=1 ), we decided to fit a copula pseudo-sample D cop = {( U (i) 1 , U (i) 2 )} 100 i=1 = {(F( M (i) 1 ),F( M (i) 2 ))} 100 i=1
independent of the fitted margin model from the previous section.
The applicability of EVCs was readily made clear after inspecting D cop , where the mirrored data points resembled some characteristic patterns we saw during a random EVC generation runà la Fig. 7. Namely, they outlined two curved paths that met at both the lower and upper tail corners. Data inspection also revealed the absence of upper tail dependence, while lower tail dependence was present. This behaviour did not match the features of EVCs: in practice, they never have lower tail dependence, but they do exhibit dependence in the upper tail. Interestingly, we can resort to survival copulas whenever a switch between lower and upper tails is needed [19]. If a random vector with uniform margins (U, V) is distributed according to a copula C, then (1 − U, 1 − V) follows the survival copula [7,25]
Č(u, v) = u + v − 1 + C(1 − u, 1 − v).
The bivariate copula sample D cop was accordingly transformed
into D surv = {(1 − U (i) 1 , 1 − U (i) 2 ))} 100 i=1 .
OnceČ fits D surv , the original copula can be retrieved by taking C equal to the survival copula ofČ.
An extreme-value dependence test [26], implemented in the function evTestK of the R package copula [33,34,36,43], confirmed our intuition about the applicability of EVCs, yielding a p-value higher than 0.35.
The SBEVC model builds upon a cubic (orthonormal) ZBS basis with 13 elements, a curvature penalty factor of 10 −5 and interpolation grid sizes of k+2 = 80, in Algorithm 3, and n+2 = 200, in Algorithm 2. The value of the latter setting is lower than the one employed in the SS based on the reduced sample size. On the other hand, we used the same COBS configuration as in Section 3.2. Fig. 11 shows the final state of SBEVC's internal functions defined in the WT domain. The resulting Bayes density has two main modes, yielding a WT with a linear region. On the other hand, Fig. 12 shows the estimated PF and its correspondent h density (28). Despite the sample D surv being exchangeable, the h estimate fails to be perfectly symmetrical, with the left peak a bit higher than the one on the right. This behaviour was not wholly unexpected, given that SBEVC does not address symmetry specifically. Taking that into account, Fig. 12 shows that symmetry is reasonably well captured. Notwithstanding, before reversing the survival model, we decided to apply a symmetrization procedure on the resulting PF A, consideringÃ(t) = [A(t) + A(1 − t)]/2. Fig. 13a shows the Bayesian posterior distribution of SBEVC PFs, using the previous PLL result as an initial guess for the MCMC sampling. We drew a million random observations from the posterior distribution. The job was divided into 100 Spark tasks corresponding to MCMC runs with 100 independent walkers [22], each one generating 200 observations, with a burn-in period of 100. The confidence interval turned out to be wider than expected but preserving the overall shape. In turn, Fig. 13b shows the COBS model, which happens to have a very different shape from Fig. 13a, lacking a flat central region. Fig. 13b demonstrates that COBS captures symmetry well. Interestingly, the COBS model falls outside the confidence interval in Fig. 13a, indicating that SBEVC and COBS have very different approaches to data fitting. Fig. 14 shows the corresponding sample-density plots for SBEVC and COBS after reversal of the survival transformation. The pdfs capture the presence of lower tail dependence and the absence of upper tail dependence in both cases. The correlation is also very similar. However, there is a remarkable density gap in Fig. 14a in the region surrounding the diagonal {u = v} that is not present in Fig. 14b. This is how the presence or absence in Fig. 13 of a flat region translates to pdfs. Consequently, SBEVC and COBS disagree when evaluating the chances of BMs involving similar masses. However, the fitted observations from D cop seem to better support SBEVC's hypothesis than COBS'. Table 4 encompasses log-likelihood values of SBEVC and COBS on D surv . The SBEVC model considered is the original asymmetrical one in Fig. 12. Given the apparent similarity between Fig. 13b and the instances in Fig. 5, we included a Gumbel copula fitted via MLE in the comparison. The results confirm the superiority of SBEVC to COBS and the parametric model by a large margin. The latter is the least fit of the three, just below COBS. Joint model. Once fitted both the univariate margin mass model (ZBSs) and the copula models (SBEVC and COBS), the final joint model immediately followed. Fig. 15 plots the original LIGO-Virgo dataset against a random sample generated from each SBEVC and COBS model. The first and second components are the maximum and the minimum, respectively. There are ten times more random samples than original data points, for a total of 500. Fig. 15a and Fig. 15b show very similar simulations. Both capture three main clusters, concentrated in the regions [0, 20] 2 , [20,40] × [0, 20] and [20,40] 2 . It is also worth mentioning that there seems to be a barrier at {M 2 = 40}; it seems unlikely that giant masses merge. As pointed out by Fig. 14, pictures Fig. 15a and Fig. 15b slightly differ in BMs with similar masses, being the diagonal just a bit denser in Fig. 15b
Discussion
SBEVC is fundamentally different from existing EVC estimation approaches. It provides a flexible semiparametric structure, admitting many unconstrained parameters without breaking PF assumptions. Even in the bivariate case, those two feats are difficult to achieve simultaneously [55]. Moreover, retaining complete control of the parameter space opens up exciting possibilities for statistical modelling and data analysis. Indeed, Fig. 7 and Fig. 13a represent breakthroughs in EVC theory. Fig. 7 advances the exploration of the PF space with additional smoothing and expressiveness, extending the seminal work by Kamnitui et al. [37]. Fig. 13a shows a Bayesian posterior sample analysis of PFs, contributing to solving inferential problems. Nonparametric approaches, lacking a proper structure, depend on specific samples to build models and can only answer a limited array of inferential questions.
The results obtained in Section 3 demonstrate the fitting power of SBEVC on a broad spectrum of EVC configurations coming from parametric models or even random SBEVCs like Fig. 7. Specifically, SBEVC outperforms a similarlyintended nonparametric approach like COBSs [13] on small and medium-sized samples. This superiority does not lie in the number of parameters, similar in both, but in the more efficient fitting strategy by SBEVC, especially when data is scarce. Comparing the top picture in Fig. 12 with Fig. 13b, we see that SBEVC fits a univariate pdf via MLE relying on exact observations, whereas COBS attempts a constrained regression on points derived from an empirical copula. The latter approach will generally be more sensitive to deviations from the EVC hypothesis, as implied by the fact that some of the fitted points in Fig. 13b lie outside the admissible region in Fig. 1a. Nevertheless, it is remarkable that SBEVC managed to beat COBS in the RMISE metric, not directly targeted by SBEVC, which shows the far-reaching capabilities of MLE.
SBEVCs represent a vast class of EVCs. Two notable copulas fall outside our construction, namely the independence and comonotonic copulas, which correspond to boundary cases of the PF geometry. These limiting cases are usually handled by other means separately and can be approximated in practice through SBEVCs, as demonstrated in Fig. 7. Additionally, Khoudraji's device could be applied to refine SBEVCs in very low correlation settings. Nonetheless, the construction of SBEVCs holds the key for fitting even more expressive models by replacing ZBSs with neural network architectures [39], perhaps at the expense of losing identifiability and a higher risk of overfitting.
Despite all the previous theoretical and practical arguments favouring SBEVC, the reader may wonder if it is worth the extra execution time and software complexity. After all, COBS can be easily implemented, is already available in R, and provides almost instantaneous results. What is more, some may even question the practical relevance of complying with PF constraints. Unfortunately, there is no definitive answer to those questions: it depends on the user's goals. The seeming complexity of SBEVC is comparable to that of Hernández-Lobato and Suárez's proposal. Theoretical guarantees on the PF are nice to have, ensuring the EVC dependence structure and proper random behaviour in simulation. In most situations, for exploratory data analysis, a compliant nonparametric method like COBS may be the right choice. SBEVC may not be a good option if there are tight time constraints. However, if a more powerful fit is required, data deviates from EVC assumptions, there is not enough data to confidently apply COBS, or one would wish to explore inferential aspects, then SBEVC might be the better, if not the only one.
Conclusions
We have introduced a novel semiparametric approach for estimating bivariate EVCs. To our knowledge, it is the first time such an attempt has been made. SBEVC allows many parameters while complying with PF constraints. The construction harbours an intriguing potential for Bayesian inference and deep learning. SBEVCs represent a vast class of EVCs, encompassing a broad spectrum of dependence strengths and asymmetries. Several SBEVCs' convergence and association properties have been explored. We have also presented all the algorithms required for effectively and efficiently running the estimation process. The SS shows promising results for SBEVC in a wide range of sampling configurations. Specifically, SBEVC produces significantly lower RMISE values than COBS. Finally, the case study demonstrates that SBEVC fits small samples more flexibly than conventional methods.
1 https://orcid.org/0000-0001-5270-9941
Fig. 1 :
1Fig. 1: On the left, PF geometry. The admissible region for its graph appears in grey. The support lines show in red. An example of PF, namely A(t) = t 2 −t +1, is drawn in blue. On the right, the geometry of a 2-monotone function derived from a PF through an affine transformation mapping A, B and C to A , B and C . The transformed version of the PF on the left, W(x) = x − 2 √ x + 1, is drawn
1 0
1z dH(z) = 1. Under absolute continuity of A [30], H admits a decomposition H(B) = H 0 1 B (0) + B η(z) dz + H 1 1 B (1) ,
f ] = 0. If we define the subspace L 2 0 ([0, 1]) of the functions with zero integral, then (21) is a bijection from B 2 to L 2 0 ([0, 1]) with inverse
Fig. 2 :
2On the left, PF A(t) = 1 − t + exp{W −1 (−2t/e 2 ) + 2} arising from the Williamson family (20) for θ = 1. Here, W −1 denotes the k = −1 branch of the complex Lambert W function. On the right, random PFs built as perturbations around centre
Fig. 3 :
3On the left, the projection z(x) of −(1 + log x)/2 onto an orthonormal ZBS 13-dimensional basis with knots κ. The logarithmic function diverges to infinity at zero. On the right, we have the underlying orthonormal cubic ZBS basis with 13 elements.
1 0
1A . The value G = 1 happens when A is identically equal to the lower support lines (1 0 A = 3/4), producing the comonotonic copula. On the other hand, G = 0 occurs when A is identical to the upper bound line ( 1 0 A = 1)
Fig. 4 :
4Computation graph for the estimation process, from the bottom parameter vector θ up to the PLL (θ|D).
) θ = 4, α = 1, β = 0.5
Fig. 5 :
5SS for the Gumbel family. The blue line designates the ground-truth PF, whereas the black corresponds to the estimations' pointwise mean. The shaded areas represent 98% pointwise confidence intervals for the estimates.
Fig. 6 :
6SS for the Galambos family. The blue line designates the ground-truth PF, whereas the black corresponds to the estimations' pointwise mean. The shaded areas represent 98% pointwise confidence intervals for the estimates.
Fig. 7 :
7A subsample of size 100 from the whole population of random PFs used in the second part of the SS.
Fig. 8 :
8Box plots of the TVD distributions for each sample size.
Fig. 9 :
9Regression tree (depth 2) of TVD on the GC for sample size 2,000. On average, SBEVC performs well even on the seven instances with GC above 98%.
Fig. 10 :
10Univariate margin mass model. The pdf is displayed on the left, whereas the quantile function is on the right. In both cases, the fitted model shows in red and the empirical estimate is in blue. The vertical cuts on the left correspond to the underlying spline knots.
Fig. 11 :
11Internal function constructs z (zero-integral spline), f (Bayes density) and W (WT). The plot displays the underlying spline knots of z.
Fig. 12 :Fig. 13 :
1213PF A and target density h. The knots represent the function values at the t i 's grid defined in Algorithm On the left, pointwise confidence interval (98%) and mean of the posterior PFs sample from the MCMC simulation. On the right, the COBS PF model fitting the empirical graph (Z i , T i ).
Fig. 14 :Fig. 15 :
1415Final copula pdfs for SBEVC and COBS after reversal of the survival transformation. The data points shown belong to the D cop dataset. Original masses (red) against random samples from SBEVC (blue, on the left) and COBS (blue, on the right).
Table 1 :
1Main one-parameter EVC families
Table 2 shows that TVD
2mean
10%
25%
50%
75%
90%
sample size
250
.06642 .0226
.03237 .04578 .06279 .08914
500
.04591 .01562 .02492 .03797 .05527 .07525
1000
.03701 .01333 .0213
.03093 .04301 .05994
2000
.0322
.01157 .01681 .02518 .03421 .05841
Table 2 :
2Main summary statistics from the second part of the SS, mean and quantiles, for each sample size. TVD decreases as sample size increases.
Table 3 :
3A comparison between SBEVC and COBS based on RMISE. All
settings refer to the parametric Gumbel family in Table 1. RMISE is statistically
significantly lower for SBEVC.
.
SBEVC COBS Gumbel(θ = 1.87)55.47
33.39
32.48
Table 4 :
4Log-likelihood of different models on the dataset D surv .
A function is convex if and only if (iff) its epigraph is a convex set.
Coincidental knots at the interval endpoints convey maximum smoothness at each interior knot[6]. For splines of degree less than or equal to d, we have (d − 1)-continuous differentiability everywhere in [0, 1].
The CDF (2) and its stochastic interpretation provide a shortcut to check this.
By convexity, the only EVC with either boundary slope equal to zero is the independence copula, with A(t) = 1 for all t ∈ [0, 1]. Therefore, except for this limiting case, both slopes are allowed to vary freely.
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+ x − W θ (x) x(1 − W θ (x)) 3 .Repeatedly applying L'Hôpital's rule, we can check that the denominator tends to zero and, eventually, the whole limit also tends to zero. Finally, the step involving the integral ensures 1 0h = 1, making ah a true pdf, which was not automatically granted by the linear interpolation strategy.
Appendix A. ProofsProof of Proposition 1. Let W be as defined above. We will see that A as defined in(7)is a PF with the additional constraint above.First, note that t(0) = 0, t(1) = 1. By continuity of W, thisProof of Proposition 2. It suffices to check that, for all x ∈ [0, 1],where (·) + denotes the non-negative part of the argument, and then apply Scheffé's theorem(27). Similarly, considering the compact subset [x 0 , 1], for someProof of Proposition 3. It follows from the equivalence between uniform convergence and function graph convergence[56]for functions with compact domain and range. Since the W n 's uniformly converge to a continuous function W, the sequence of the graphs of the W n 's has its limit in the graph of W. Then, note that the graphs of A n and A are affine transformations (7) of the graphs of W n and W, respectively. This ensures, by continuity, that the graphs of the A n 's tend to that of A. Finally, graph convergence for the A n 's implies uniform convergence to A itself. The result for the first derivatives follows similarly. Instead of an affine map, the functions mapping the graph of W to that of A and vice versa are, respectively,Both are the inverse of one another because of(9)and(11). Both functions are continuous. Hence, they preserve compactness and graph convergence.To see that {A n } ∞ n=1 compactly converges to A , consider any compact set K = [t 0 , 1], for t 0 > 0. Then, consider the sequence of restricted function graphs {G[A n | K ]} ∞ n=1 and apply X to every element to obtain another sequence {G[W n | X(K) ]} ∞ n=1 . Now, X(K) is a compact set, so W n | X(K) uniformly converges to W | X(K) and, because of[56], the graph sequence approaches G[W | X(K) ]. The argument finishes by noting that, since T is continuous and since T(G[W n | X(K) ]) = G[A n | K ] and T(G[W | X(K) ]) = G[A | K ], the graphs of the A n | K 's tend to that of A | K .Proof of Proposition 4. First, note that continuity ensures that the limit p is also bounded by the same K. Denoting I q = 1 0 e q , some easy calculations show that | f (x) − f n (x)| ≤ e p(x) |I p − I p n | + I p |e p(x) − e p n (x) | I p I p n .Now, the integrals are bounded, namely e −K ≤ I q ≤ e K . On the other hand, |e p(x) − e p n (x) | ≤ e K |p(x) − p n (x)|, using the mean value theorem and the fact that both functions are bounded by K. All in all,Integrating both sides of the last inequality and using Jensen's inequality, we finally get d TV ( f, f n ) ≤ e 4K p − p n 2 .Proof of Corollary 2. It follows from all the previous convergence results and the form of the partial derivative of an EVC[19,16], where all the terms uniformly converge on compact sets.18Proof of Algorithm 2. Let us set aside the normalization byW 0 for a moment. Clearly,W i is a straight approximation for W θ (r i ). It only remains to check that(31)and(30)provide good approximations for W θ (r i ) and W θ (r i ), respectively. It suffices to see thatand, taking into account(19),where we have used that P i and Q i are the trapezoidal rule approximations for r i+1 r i f θ and r i+1 r i f θ (r)/r dr, respectively. Also, implicit in the previous argument was the approximation s 0 = ≈ 0, used to avoid infinite values. Finally, the last normalization step aims to stabilize the estimation process against numerical errors, enforcing the constraint W θ (0) = 1.Proof of Algorithm 3. The rationale of the algorithm is relatively straightforward. Equation (32) mimics(28), where A θ and its derivatives are evaluated over t i indirectly through equations (7),(9)and(10), requiring only the x i 's and approximations of W θ and its derivatives at those points. On the other hand, the reader can easily check that h θ (0) = h θ (1) = 0, considering all the constraints imposed by SBEVC: A θ (0 + ) = −1, A θ (1 − ) = 1 and f θ (x) > 0 for all x ∈ [0, 1], among others. The case at the 0 endpoint is not trivial, but nearly so. After simplification, we arrive at h θ (0) = 2 f θ (0) lim x→0 +
Let us suppose that t(x 1 ) = t(x 2 ) for some x 1 , x 2 ∈ [0, 1], x 1 < x 2 . Then, W(x 2 ) − W(x 1 ) = x 2 − x 1 > 0, which leads to a contradiction with W being non-increasing. Then, it suffices to see that it is one-to-one. Therefore, t(x) is an automorphism of [0, 1], so A inThen, for t(x) to be an automor- phism of [0, 1], it suffices to see that it is one-to-one. Let us suppose that t(x 1 ) = t(x 2 ) for some x 1 , x 2 ∈ [0, 1], x 1 < x 2 . Then, W(x 2 ) − W(x 1 ) = x 2 − x 1 > 0, which leads to a contra- diction with W being non-increasing. Therefore, t(x) is an au- tomorphism of [0, 1], so A in (7
it is easy to check that t ± (x) = (1 ± x ∓ W(x)) /2 and, since both x and W(x) are non-negative (otherwise W would not be non-increasing. − , with Ran(W) = [0, 1]), we may conclude A(t(x)) ≥ max{t + (x), t − (x)}. Furthermore, A(t(x− t(x), it is easy to check that t ± (x) = (1 ± x ∓ W(x)) /2 and, since both x and W(x) are non-negative (otherwise W would not be non-increasing, with Ran(W) = [0, 1]), we may conclude A(t(x)) ≥ max{t + (x), t − (x)}. Furthermore, A(t(x)
Since W (x) ≤ 0 and W (x) ≥ 0. it follows that A (t) ≥ 0, for all t ∈ (0, 1), and hence A is convex. This finishes the proof that (7) defines a PF such that A(t) > 1 − t, for all t ∈ (0, 1/2Since W (x) ≤ 0 and W (x) ≥ 0, it follows that A (t) ≥ 0, for all t ∈ (0, 1), and hence A is convex. This finishes the proof that (7) defines a PF such that A(t) > 1 − t, for all t ∈ (0, 1/2].
We will similarly show that W as defined in (8) is 2-monotone and satisfies W(0) = 1 and W(1) = 0. First, note that x(0) = 0 and x(1) = 0. By continuity of A, this implies that Ran(x) = [0, 1]. Then, for x(t) to be an automorphism, it suffices to see that x(t) is one-to-one. Conversely, let A be a PF with the latter additional constraint. Let us suppose that x(t 1 ) = x(t 2 ) for some t 1 , t 2 ∈ [0, 1], t 1 < t 2 . This implies that [A(t 2 ) − A(t 1 )]/(t 2 − t 1 ) = −1 and, since A is convex, we must conclude that A(t) , on the other hand, A(t) ≥ max{t, 1 − t}. Therefore, (t 1 , t 2 ] ⊂ (0, 1/2], which leads to a contradiction with A(tConversely, let A be a PF with the latter additional con- straint. We will similarly show that W as defined in (8) is 2- monotone and satisfies W(0) = 1 and W(1) = 0. First, note that x(0) = 0 and x(1) = 0. By continuity of A, this implies that Ran(x) = [0, 1]. Then, for x(t) to be an automorphism, it suffices to see that x(t) is one-to-one. Let us suppose that x(t 1 ) = x(t 2 ) for some t 1 , t 2 ∈ [0, 1], t 1 < t 2 . This implies that [A(t 2 ) − A(t 1 )]/(t 2 − t 1 ) = −1 and, since A is convex, we must conclude that A(t) , on the other hand, A(t) ≥ max{t, 1 − t}. Therefore, (t 1 , t 2 ] ⊂ (0, 1/2], which leads to a contradiction with A(t)
x(t) must be one-to-one and. Hence. all in all, an automorphism of [0, 1Hence, x(t) must be one-to-one and, all in all, an automorphism of [0, 1].
This, in turn, means that W in (8) is well-defined as a function of a single variable. 0, 1This, in turn, means that W in (8) is well-defined as a function of a single variable in [0, 1].
A(1) = 1. Since A(t) > 1 − t for t ∈ (0, 1/2] and A being convex, we have A (t) > −1 and the denominator in both (11) and (12) is well-defined. Moreover, A (t) ≤ 1, otherwise we would have A(1 − ) < 1 − for a sufficiently small . Therefore, W (x) ≤ x for all x ∈ (0, 1). Next, it is easy to check both W(0) = 1 and W(1) = 0, bearing in mind that A(0) =. On the other hand, the convexity of W follows directly from A (t) ≥ 0. Finally, the derivatives. on the other, directly follow by differentiating (7) andNext, it is easy to check both W(0) = 1 and W(1) = 0, bearing in mind that A(0) = A(1) = 1. Since A(t) > 1 − t for t ∈ (0, 1/2] and A being convex, we have A (t) > −1 and the denominator in both (11) and (12) is well-defined. Moreover, A (t) ≤ 1, otherwise we would have A(1 − ) < 1 − for a sufficiently small . Therefore, W (x) ≤ x for all x ∈ (0, 1). On the other hand, the convexity of W follows directly from A (t) ≥ 0. Finally, the derivatives (9) and (10), on the one hand, and (11) and (12), on the other, directly follow by differentiating (7) and (8).
Observation of Gravitational Waves from a Binary Black Hole Merger. B Abbott, 10.1103/physrevlett.116.061102Physical Review Letters. 1166B. Abbott et al. "Observation of Gravitational Waves from a Binary Black Hole Merger". In: Physical Review Letters 116.6 (Feb. 2016). doi: 10.1103/physrevlett.116.061102.
Population Properties of Compact Objects from the Second LIGO-Virgo Gravitational-Wave Transient Catalog. R Abbott, 10.3847/2041-8213/abe949doi: 10.3847/ 2041-8213/abe949The Astrophysical Journal Letters. 91317R. Abbott et al. "Population Properties of Compact Objects from the Second LIGO-Virgo Gravitational-Wave Transient Catalog". In: The Astrophysical Journal Letters 913.1 (May 2021), p. L7. doi: 10.3847/ 2041-8213/abe949.
Algorithm 748: enclosing zeros of continuous functions. G E Alefeld, F A Potra, Y Shi, 10.1145/210089.210111ACM Transactions on Mathematical Software. 213G. E. Alefeld, F. A. Potra, and Y. Shi. "Algorithm 748: enclosing zeros of continuous functions". In: ACM Transactions on Mathematical Soft- ware 21.3 (Sept. 1995), pp. 327-344. doi: 10.1145/210089.210111.
On Some Applications of Williamson's Transform in Copula Theory. T Bacigál, 10.1007/978-3-319-59306-7_3doi: 10 . 1007/978-3-319-59306-7_3Advances in Intelligent Systems and Computing. Springer International PublishingT. Bacigál. "On Some Applications of Williamson's Transform in Copula Theory". In: Advances in Intelligent Systems and Computing. Springer International Publishing, May 2017, pp. 21-30. doi: 10 . 1007/978-3-319-59306-7_3.
Spline Basics". In: Handbook of Computer Aided Geometric Design. C De Boor, 10.1016/b978-044451104-1/50007-1doi: 10 . 1016 / b978 - 044451104-1/50007-1ElsevierC. de Boor. "Spline Basics". In: Handbook of Computer Aided Geo- metric Design. Elsevier, 2002, pp. 141-163. doi: 10 . 1016 / b978 - 044451104-1/50007-1.
Copulas for Finance -A Reading Guide and Some Applications. E Bouyé, 10.2139/ssrn.1032533doi: 10 . 2139 / ssrn.1032533SSRN Electronic Journal. E. Bouyé et al. "Copulas for Finance -A Reading Guide and Some Applications". In: SSRN Electronic Journal (2000). doi: 10 . 2139 / ssrn.1032533.
A Limited Memory Algorithm for Bound Constrained Optimization. R H Byrd, 10.1137/0916069SIAM Journal on Scientific Computing. 165R. H. Byrd et al. "A Limited Memory Algorithm for Bound Con- strained Optimization". In: SIAM Journal on Scientific Computing 16.5 (Sept. 1995), pp. 1190-1208. doi: 10.1137/0916069.
A nonparametric estimation procedure for bivariate extreme value copulas. P Capéraà, A L Fougères, C Genest, 10.1093/biomet/84.3.567Biometrika. 843P. Capéraà, A. L. Fougères, and C. Genest. "A nonparametric esti- mation procedure for bivariate extreme value copulas". In: Biometrika 84.3 (Sept. 1997), pp. 567-577. doi: 10.1093/biomet/84.3.567.
A Brief History of Gravitational Waves. J Cervantes-Cota, S Galindo-Uribarri, G Smoot, 10.3390/universe2030022doi: 10. 3390/universe203002222In: Universe 2.3 (Sept. 2016J. Cervantes-Cota, S. Galindo-Uribarri, and G. Smoot. "A Brief History of Gravitational Waves". In: Universe 2.3 (Sept. 2016), p. 22. doi: 10. 3390/universe2030022.
Multivariate Archimax copulas. A Charpentier, 10.1016/j.jmva.2013.12.013doi: 10.1016/j. jmva.2013.12.013Journal of Multivariate Analysis. 126A. Charpentier et al. "Multivariate Archimax copulas". In: Journal of Multivariate Analysis 126 (Apr. 2014), pp. 118-136. doi: 10.1016/j. jmva.2013.12.013.
On the LambertW function. R M Corless, 10.1007/bf02124750doi: 10 . 1007/bf02124750Advances in Computational Mathematics 5.1 (Dec. 1996). R. M. Corless et al. "On the LambertW function". In: Advances in Computational Mathematics 5.1 (Dec. 1996), pp. 329-359. doi: 10 . 1007/bf02124750.
Using B-splines for nonparametric inference on bivariate extreme-value copulas. E Cormier, C Genest, J Nešlehová, 10.1007/s10687-014-0199-4Extremes 17. 4E. Cormier, C. Genest, and J. Nešlehová. "Using B-splines for non- parametric inference on bivariate extreme-value copulas". In: Extremes 17.4 (Aug. 2014), pp. 633-659. doi: 10.1007/s10687-014-0199-4.
On Densities of Extreme Value Copulas. G Doyon, MA thesis. Universität ZürichG. Doyon. "On Densities of Extreme Value Copulas". MA thesis. Uni- versität Zürich, 2013.
Hilbert Space of Probability Density Functions Based on Aitchison Geometry. J J Egozcue, J L Díaz-Barrero, V Pawlowsky-Glahn, 10.1007/s10114-005-0678-2Acta Mathematica Sinica, English Series. 22J. J. Egozcue, J. L. Díaz-Barrero, and V. Pawlowsky-Glahn. "Hilbert Space of Probability Density Functions Based on Aitchison Geome- try". In: Acta Mathematica Sinica, English Series 22.4 (Jan. 2006), pp. 1175-1182. doi: 10.1007/s10114-005-0678-2.
Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. J H J Einmahl, J Segers, 10.1214/08-aos677The Annals of Statistics. 37J. H. J. Einmahl and J. Segers. "Maximum empirical likelihood estima- tion of the spectral measure of an extreme-value distribution". In: The Annals of Statistics 37.5B (Oct. 2009). doi: 10.1214/08-aos677.
Properties of extreme-value copulas. P Eschenburg, MA thesisP. Eschenburg. "Properties of extreme-value copulas". MA thesis. 2013.
Projection estimators of Pickands dependence functions. A Fils-Villetard, A Guillou, J Segers, 10.1002/cjs.5550360303Canadian Journal of Statistics. 363A. Fils-Villetard, A. Guillou, and J. Segers. "Projection estimators of Pickands dependence functions". In: Canadian Journal of Statistics 36.3 (Sept. 2008), pp. 369-382. doi: 10.1002/cjs.5550360303.
Lorenz-generated bivariate Archimedean copulas. A Fontanari, P Cirillo, C W Oosterlee, 10.1515/demo-2020-0011Dependence Modeling. 8A. Fontanari, P. Cirillo, and C. W. Oosterlee. "Lorenz-generated bivari- ate Archimedean copulas". In: Dependence Modeling 8.1 (Jan. 2020), pp. 186-209. doi: 10.1515/demo-2020-0011.
emcee: The MCMC Hammer. D Foreman-Mackey, 10.1086/670067Publications of the Astronomical Society of the Pacific. 125D. Foreman-Mackey et al. "emcee: The MCMC Hammer". In: Publi- cations of the Astronomical Society of the Pacific 125.925 (Mar. 2013), pp. 306-312. doi: 10.1086/670067.
When Gumbel met Galambos. C Genest, J Nešlehová, 10.1007/978-3-319-64221-5_6Copulas and Dependence Models with Applications. Springer International PublishingC. Genest and J. Nešlehová. "When Gumbel met Galambos". In: Cop- ulas and Dependence Models with Applications. Springer International Publishing, 2017, pp. 83-93. doi: 10.1007/978-3-319-64221-5_6.
Tests of symmetry for bivariate copulas. C Genest, J Nešlehová, J.-F Quessy, 10.1007/s10463-011-0337-6Annals of the Institute of Statistical Mathematics. 64C. Genest, J. Nešlehová, and J.-F. Quessy. "Tests of symmetry for bi- variate copulas". In: Annals of the Institute of Statistical Mathematics 64.4 (Sept. 2011), pp. 811-834. doi: 10.1007/s10463-011-0337-6.
Multivariate Survival Modelling: A Unified Approach with Copulas. P Georges, 10.2139/ssrn.1032559doi: 10. 2139/ssrn.1032559In: SSRN Electronic Journal. P. Georges et al. "Multivariate Survival Modelling: A Unified Ap- proach with Copulas". In: SSRN Electronic Journal (2001). doi: 10. 2139/ssrn.1032559.
On the Ghoudi, Khoudraji, and Rivest test for extreme-value dependence. Christian Noomen Ben Ghorbal, Johanna Genest, Nešlehová, 10.1002/cjs.10034Canadian Journal of Statistics. 37Noomen Ben Ghorbal, Christian Genest, and Johanna Nešlehová. "On the Ghoudi, Khoudraji, and Rivest test for extreme-value dependence". In: Canadian Journal of Statistics 37.4 (Dec. 2009), pp. 534-552. doi: 10.1002/cjs.10034.
Extreme-Value Copulas. G Gudendorf, J Segers, 10.1007/978-3-642-12465-5_6Copula Theory and Its Applications. Berlin HeidelbergSpringerG. Gudendorf and J. Segers. "Extreme-Value Copulas". In: Cop- ula Theory and Its Applications. Springer Berlin Heidelberg, 2010, pp. 127-145. doi: 10.1007/978-3-642-12465-5_6.
Nonparametric estimation of multivariate extreme-value copulas. G Gudendorf, J Segers, 10.1016/j.jspi.2012.05.007Journal of Statistical Planning and Inference 142. 12G. Gudendorf and J. Segers. "Nonparametric estimation of multivari- ate extreme-value copulas". In: Journal of Statistical Planning and In- ference 142.12 (Dec. 2012), pp. 3073-3085. doi: 10.1016/j.jspi. 2012.05.007.
Polynomial Pickands functions. S Guillotte, F Perron, 10.3150/14-bej656Bernoulli 22.1 (Feb. 2016). S. Guillotte and F. Perron. "Polynomial Pickands functions". In: Bernoulli 22.1 (Feb. 2016), pp. 213-241. doi: 10.3150/14-bej656.
A new extreme value copula and new families of univariate distributions based on Freund's exponential model. S Guzmics, G C Pflug, 10.1515/demo-2020-0018Dependence Modeling. 8S. Guzmics and G. C. Pflug. "A new extreme value copula and new families of univariate distributions based on Freund's exponential model". In: Dependence Modeling 8.1 (Jan. 2020), pp. 330-360. doi: 10.1515/demo-2020-0018.
Semiparametric bivariate Archimedean copulas. J M Hernández-Lobato, A Suárez, 10.1016/j.csda.2011.01.018Computational Statistics & Data Analysis. 556J. M. Hernández-Lobato and A. Suárez. "Semiparametric bivariate Archimedean copulas". In: Computational Statistics & Data Analysis 55.6 (June 2011), pp. 2038-2058. doi: 10.1016/j.csda.2011.01. 018.
Multivariate Dependence with Copulas. M Hofert, R package version 1.0-1. 2020M. Hofert et al. copula: Multivariate Dependence with Copulas. R package version 1.0-1. 2020.
Modeling Multivariate Distributions with Continuous Margins Using the copula R Package. Ivan Kojadinovic, Jun Yan, Journal of Statistical Software. 349Ivan Kojadinovic and Jun Yan. "Modeling Multivariate Distributions with Continuous Margins Using the copula R Package". In: Journal of Statistical Software 34.9 (2010), pp. 1-20.
Nonparametric Estimation of the Dependence Function in Bivariate Extreme Value Distributions. J R Jiménez, E Villa-Diharce, M Flores, 10.1006/jmva.2000.1931Journal of Multivariate Analysis. 762J. R. Jiménez, E. Villa-Diharce, and M. Flores. "Nonparametric Esti- mation of the Dependence Function in Bivariate Extreme Value Dis- tributions". In: Journal of Multivariate Analysis 76.2 (Feb. 2001), pp. 159-191. doi: 10.1006/jmva.2000.1931.
Enjoy the Joy of Copulas: With a Package copula. Jun Yan, Journal of Statistical Software. 21Jun Yan. "Enjoy the Joy of Copulas: With a Package copula". In: Jour- nal of Statistical Software 21.4 (2007), pp. 1-21.
On the size of the class of bivariate extreme-value copulas with a fixed value of Spearman's rho or Kendall's tau. N Kamnitui, 10.1016/j.jmaa.2018.11.057Journal of Mathematical Analysis and Applications. 4721N. Kamnitui et al. "On the size of the class of bivariate extreme-value copulas with a fixed value of Spearman's rho or Kendall's tau". In: Journal of Mathematical Analysis and Applications 472.1 (Apr. 2019), pp. 920-936. doi: 10.1016/j.jmaa.2018.11.057.
Contributionsà l'étude des copules età la modélisation des valeurs extrêmes bivariées. A Khoudraji, Québec, CanadaUniversité LavalPhD thesisA. Khoudraji. "Contributionsà l'étude des copules età la modélisation des valeurs extrêmes bivariées". PhD thesis. Université Laval, Québec, Canada, 1995.
Deep Archimedean Copulas. C K Ling, F Fang, J Z Kolter, arXiv:2012.0313734th Conference on Neural Information Processing Systems. Vancouver, Canadacs.LGC. K. Ling, F. Fang, and J. Z. Kolter. "Deep Archimedean Copu- las". In: 34th Conference on Neural Information Processing Systems (NeurIPS 2020). Vancouver, Canada, Dec. 5, 2020. arXiv: 2012 . 03137 [cs.LG].
Compositional splines for representation of density functions. J Machalová, 10.1007/s00180-020-01042-7Computational Statistics. 362J. Machalová et al. "Compositional splines for representation of density functions". In: Computational Statistics 36.2 (Oct. 2020), pp. 1031- 1064. doi: 10.1007/s00180-020-01042-7.
Autograd: Effortless gradients in numpy. D Maclaurin, D Duvenaud, R P Adams, ICML 2015 AutoML Workshop. 2385D. Maclaurin, D. Duvenaud, and R. P. Adams. "Autograd: Effortless gradients in numpy". In: ICML 2015 AutoML Workshop. Vol. 238. 2015, p. 5.
Multivariate nonparametric estimation of the Pickands dependence function using Bernstein polynomials. G Marcon, 10.1016/j.jspi.2016.10.004Journal of Statistical Planning and Inference 183. G. Marcon et al. "Multivariate nonparametric estimation of the Pickands dependence function using Bernstein polynomials". In: Jour- nal of Statistical Planning and Inference 183 (Apr. 2017), pp. 1-17. doi: 10.1016/j.jspi.2016.10.004.
Nested Archimedean Copulas Meet R: The nacopula Package. Marius Hofert, Martin Mächler, Journal of Statistical Software. 39Marius Hofert and Martin Mächler. "Nested Archimedean Copulas Meet R: The nacopula Package". In: Journal of Statistical Software 39.9 (2011), pp. 1-20.
Multivariate Archimedean copulas, d-monotone functions and 1-norm symmetric distributions. A J Mcneil, J Nešlehová, 10.1214/07-aos556doi: 10.1214/ 07-aos556The Annals of Statistics. 37A. J. McNeil and J. Nešlehová. "Multivariate Archimedean copulas, d-monotone functions and 1-norm symmetric distributions". In: The Annals of Statistics 37.5B (Oct. 2009), pp. 3059-3097. doi: 10.1214/ 07-aos556.
From Archimedean to Liouville copulas. A J Mcneil, J Nešlehová, 10.1016/j.jmva.2010.03.015Journal of Multivariate Analysis. 101A. J. McNeil and J. Nešlehová. "From Archimedean to Liouville copu- las". In: Journal of Multivariate Analysis 101.8 (Sept. 2010), pp. 1772- 1790. doi: 10.1016/j.jmva.2010.03.015.
Multivariate extreme value distribution. J Pickands, Proceedings 43th, Session of International Statistical Institution. 43th, Session of International Statistical Institution1981J. Pickands. "Multivariate extreme value distribution". In: Proceedings 43th, Session of International Statistical Institution, 1981 (1981).
Minimum-distance statistics for the selection of an asymmetric copula in Khoudraji's class of models. J.-F Quessy, O Kortbi, 10.5705/ss.202014.0082Statistica Sinica. J.-F. Quessy and O. Kortbi. "Minimum-distance statistics for the selec- tion of an asymmetric copula in Khoudraji's class of models". In: Sta- tistica Sinica (2016). doi: 10.5705/ss.202014.0082.
Copulas and Stochastic Processes. Volker Schmitz, PhD thesis. Aachen: Rheinisch-Westfälischen Technischen Hochschule Aachen. isbnVolker Schmitz. "Copulas and Stochastic Processes". PhD thesis. Aachen: Rheinisch-Westfälischen Technischen Hochschule Aachen, Jan. 21, 2003. isbn: 3-8322-1278-7.
On Shape Preserving Quadratic Spline Interpolation. L I Schumaker, 10.1137/0720057SIAM Journal on Numerical Analysis. 20L. I. Schumaker. "On Shape Preserving Quadratic Spline Interpola- tion". In: SIAM Journal on Numerical Analysis 20.4 (Aug. 1983), pp. 854-864. doi: 10.1137/0720057.
Modelling Multivariate Dependencies with Semiparametric Archimedean Copulas. Javier Fernández Serrano, 172MA thesis. Madrid: Universidad Autónoma de MadridJavier Fernández Serrano. "Modelling Multivariate Dependencies with Semiparametric Archimedean Copulas". MA thesis. Madrid: Universi- dad Autónoma de Madrid, July 8, 2016, p. 172.
Introduction to Nonparametric Estimation. A B Tsybakov, 10.1007/b13794SpringerNew YorkA. B. Tsybakov. Introduction to Nonparametric Estimation. Springer New York, 2009. doi: 10.1007/b13794.
A comparison of dependence function estimators in multivariate extremes. S Vettori, R Huser, M G Genton, 10.1007/s11222-017-9745-7Statistics and Computing. 28S. Vettori, R. Huser, and M. G. Genton. "A comparison of dependence function estimators in multivariate extremes". In: Statistics and Com- puting 28.3 (May 2017), pp. 525-538. doi: 10.1007/s11222-017- 9745-7.
Uniform Convergence and Graph Convergence. W C Waterhouse, 10.2307/2319894The American Mathematical Monthly. 83641W. C. Waterhouse. "Uniform Convergence and Graph Convergence". In: The American Mathematical Monthly 83.8 (Oct. 1976), p. 641. doi: 10.2307/2319894.
Argo CD. Declarative continuous deployment for Kubernetes. Argo Project. Argo CD. Declarative continuous deployment for Kuber- netes.
Data Mechanics. Docker image for Apache Spark. Data Mechanics. Docker image for Apache Spark.
Container technology. Docker, Inc. DockerDocker, Inc. Docker. Container technology.
Spark Operator. Kubernetes operator for managing the lifecycle of Apache Spark applications on Kubernetes. Google Cloud Platform, Google Cloud Platform. Spark Operator. Kubernetes operator for man- aging the lifecycle of Apache Spark applications on Kubernetes.
The Apache Software Foundation. Spark. A unified analytics engine for large-scale data processing. Version 3.1.1.The Apache Software Foundation. Spark. A unified analytics engine for large-scale data processing. Version 3.1.1. Mar. 2, 2021.
The Cloud Native Computing Foundation. Kubernetes. Production-Grade Container Scheduling and Management. The Cloud Native Computing Foundation. Kubernetes. Production- Grade Container Scheduling and Management.
The Minikube Community. minikube. Run Kubernetes locally. 20The Minikube Community. minikube. Run Kubernetes locally. 20
| [] |
[
"Discretization of Time Series Data",
"Discretization of Time Series Data"
] | [
"Elena S Dimitrova \nBioinformatics Facility\nVirginia Bioinformatics Institute at Virginia Tech\nWashington St. (0477)24061BlacksburgVAUSA\n",
"John J Mcgee \nBioinformatics Facility\nVirginia Bioinformatics Institute at Virginia Tech\nWashington St. (0477)24061BlacksburgVAUSA\n",
"Reinhard C Laubenbacher \nBioinformatics Facility\nVirginia Bioinformatics Institute at Virginia Tech\nWashington St. (0477)24061BlacksburgVAUSA\n"
] | [
"Bioinformatics Facility\nVirginia Bioinformatics Institute at Virginia Tech\nWashington St. (0477)24061BlacksburgVAUSA",
"Bioinformatics Facility\nVirginia Bioinformatics Institute at Virginia Tech\nWashington St. (0477)24061BlacksburgVAUSA",
"Bioinformatics Facility\nVirginia Bioinformatics Institute at Virginia Tech\nWashington St. (0477)24061BlacksburgVAUSA"
] | [] | Data discretization, also known as binning, is a frequently used technique in computer science, statistics, and their applications to biological data analysis. We present a new method for the discretization of real-valued data into a finite number of discrete values. Novel aspects of the method are the incorporation of an information-theoretic criterion and a criterion to determine the optimal number of values. While the method can be used for data clustering, the motivation for its development is the need for a discretization algorithm for several multivariate time series of heterogeneous data, such as transcript, protein, and metabolite concentration measurements. As several modeling methods for biochemical networks employ discrete variable states, the method needs to preserve correlations between variables as well as the dynamic features of the time series. A C++ implementation of the algorithm is available from the authors at | 10.1089/cmb.2008.0023 | [
"https://arxiv.org/pdf/q-bio/0505028v2.pdf"
] | 24,515,133 | q-bio/0505028 | 0b46767f3ba17d2f60132daf52327f254a838f50 |
Discretization of Time Series Data
Elena S Dimitrova
Bioinformatics Facility
Virginia Bioinformatics Institute at Virginia Tech
Washington St. (0477)24061BlacksburgVAUSA
John J Mcgee
Bioinformatics Facility
Virginia Bioinformatics Institute at Virginia Tech
Washington St. (0477)24061BlacksburgVAUSA
Reinhard C Laubenbacher
Bioinformatics Facility
Virginia Bioinformatics Institute at Virginia Tech
Washington St. (0477)24061BlacksburgVAUSA
Discretization of Time Series Data
8/29/2005 1
Data discretization, also known as binning, is a frequently used technique in computer science, statistics, and their applications to biological data analysis. We present a new method for the discretization of real-valued data into a finite number of discrete values. Novel aspects of the method are the incorporation of an information-theoretic criterion and a criterion to determine the optimal number of values. While the method can be used for data clustering, the motivation for its development is the need for a discretization algorithm for several multivariate time series of heterogeneous data, such as transcript, protein, and metabolite concentration measurements. As several modeling methods for biochemical networks employ discrete variable states, the method needs to preserve correlations between variables as well as the dynamic features of the time series. A C++ implementation of the algorithm is available from the authors at
INTRODUCTION
Discretization of real data into a typically small number of finite values is often required by machine learning algorithms (Dougherty et al., 1995), Bayesian network applications (Friedman and Goldszmidt, 1996), and any modeling algorithm using discrete-state models. Binary discretizations are the simplest way of discretizing data, used, for instance, for the construction of Boolean network models for gene regulatory networks (Kauffman, 1969;Albert and Othmer, 2003). The expression data are discretized into only two qualitative states as either present or absent. An obvious drawback of binary discretization is that labeling the realvalued data according to a present/absent scheme generally causes the loss of a large amount of information. Discrete models and modeling techniques allowing multiple states have been developed and studied in, e.g., (Laubenbacher and Stigler, 2004;Thieffry and Thomas, 1998). But experimental data are typically continuous, or, at least, represented by computer floating point numbers. For the case of small samples of biological data, many statistical methods for discretization are not applicable due to the insufficient amount of the data. Some other existing discreti-zation techniques assume that the number of discrete classes to be obtained is given, e.g., (Friedman et al., 2000). While this number is extremely important, it is not clear how to properly select it in many cases. In this paper we introduce a new method for the discretization of experimental data into a finite number of states. While of interest for other purposes, this method is designed specifically for the discretization of multivariate time series, such as those used for the construction of discrete models of biochemical networks built from time series of experimental data. We employ a graph-theoretic clustering method to perform the discretization and an information-theoretic technique to minimize loss of information content. One of the most useful features of our method is the determination of an optimal number of discrete states that is most appropriate for the data. Our C++ program takes as input one or more vectors of real data and discretizes their entries into a number of states that best fits the data. Our main objective was to construct a method that preserves correlations between variables as well as information about network dynamics inherent in the time series. We have validated the method in two ways: by using published DNA microarray data to test for preservation of correlations and by comparing the dynamics of a discrete and continuous model constructed using the modeling method in (Laubenbacher and Stigler, 2004).
DISCRETIZATION PROBLEM
In order to place our method in a general context we first give a definition of discretization (Hartemink, 2001):
A discretization of a real-valued vector v = (v 1 ,…,v N ) is an integer-valued vector d = (d 1 ,…,d N ) with the following properties:
(1) Each element of d is in the set {0, 1,…, D -1} for some (usually small) positive integer D, called the degree of the discretization.
(2) For all 1
≤ i, j ≤ N, we have j d i d ≤ if j v i v ≤ .
Spanning discretizations of degree D are a special case of the discretizations we consider in this paper. They are defined in (Hartemink, 2001) as discretizations that satisfy the additional property that the smallest element of d is equal to 0 and that the largest element of d is equal to D -1. Both methods, however, assume extra knowledge about the data source, which may not always be available. For example, the sample size may be insufficient to estimate distributions. For time series of transcript data the number of time points is typically much smaller than the number of genes considered, so that statistical approaches to discretization become problematic. Also, in these cases it is rarely known what the appropriate discretization thresholds for each gene might be. Another common discretization technique is based on clustering (Jain and Dubes, 1988). One of the most common clustering algorithms is the k-means clustering developed by MacQueen (1967). The goal of the k-means algorithm is to minimize dissimilarity in the elements within each cluster while maximizing this value between elements in different clusters. The algorithm takes as input a set of points S to be clustered and a fixed integer k. It partitions S into k subsets by choosing a set of k cluster centroids. The choice of centroids determines the structure of the partition since each point in S is assigned to the nearest centroid. Then for each cluster the centroids are re-computed based on which elements are contained in the cluster. These steps are repeated until convergence is achieved. Many applications of the kmeans clustering such as the MultiExperiment Viewer (Saeed et al., 2003) start by taking any random partition into k clusters and computing their centroids. As a consequence, a different clustering of S may be obtained every time the algorithm is run. Another inconvenience is that the number k of clusters to be formed has to be specified in advance. Another method is single-link clustering (SLC) with the Euclidean distance function on vectors of real data to produce a spanning discretization. SLC is a divisive (top-down) hierarchical clustering that defines the distance between two clusters as the minimal distance of any two objects belonging to different clusters (Jain and Dubes, 1988). In the context of discretization, these objects will be the real-valued entries of the vector to be discretized, and the distance function that measures the distance between two vector entries v and w will be the one-dimensional Euclidean distance |v -w|. Top-down clustering algorithms start from the entire data set and iteratively split it until either the degree of similarity reaches a certain threshold or every group consists of one object only. For the purpose of data analysis, it is impractical to let the clustering algorithm produce clusters containing only one real value. The iteration at which the algorithm is terminated is crucial since it determines the degree of the discretization, and one of the most important features of our discretization method is a definition of the termination criteria. SLC with the Euclidean distance function satisfies one of our major requirements: very little starting information is needed -only distances between points. It may result, however, in a discretization where most of the points are clustered into a single partition if they happen to be relatively close to one another. This negatively affects the information content of the discrete vector (to be discussed later in the paper). Another problem with SLC is that its direct implementation takes D, the desired number of discrete states, as an input. However, we would like to choose D as small as possible, without losing correlation and dynamic information, so that an essentially arbitrary choice is unsatisfactory. These two issues were addressed by modifying the SCL algorithm: our method begins by discretizing a vector in the same way as SLC but instead of providing D as part of the input, the algorithm contains termination criteria which determine the appropriate number D. After that each discrete state is checked for information content and if it is determined that this content can be considerably increased by further discretization, then the state is separated into two states in a way that may not be consistent with SLC. The details of these procedures are given next.
METHOD
The method assumes that the data to be discretized consist of one or several vectors of real-valued entries. It is appropriate for applications when there is no knowledge about distribution, range, or discretization thresholds of the data and arranges the data points into clusters only according to their relative distance with respect to each other and the resulting information content. The algorithm employs graph theory as a tool to produce a clustering of the data and provides a termination criterion.
Discretization of one vector
Even if more than one vector is to be discretized, the algorithm discretizes each vector independently and for some applications this may be sufficient. The example of such a vector to keep in mind is a time series of expression values for a single gene. If the vector contains m distinct entries, a complete weighted graph on m vertices is constructed, where a vertex represents an entry and an edge weight is the Euclidean distance between its endpoints. The discretization process starts by deleting the edge(s) of highest weight until the graph gets disconnected. If there is more than one edge labeled with the current highest weight, then all of the edges with this weight are deleted. The order in which the edges are removed leads to components, in which the distance between any two vertices is smaller than the distance between any two components, a requirement of SLC. We define the distance between two components G and H to be
{ } H h G g h g ∈ ∈ − , | | min
. The output of the algorithm is a discretization of the vector, in which each cluster corresponds to a discrete state and the vector entries that belong to one component are discretized into the same state.
Example
Suppose that vector v = (1, 2, 7, 9, 10, 11) is to be discretized. The corresponding SLC dendrogram that would be obtained by SLC algorithms such as the Johnson's algorithm (Johnson, 1967) is given in Figure 1. We start with constructing the complete weighted graph based on v which corresponds to iteration 0 of the dendrogram ( Figure 2).
Fig. 2.
The complete weighted graph constructed from vector entries 1, 2, 7, 9, 10, 11. Only the edge weights of the outer edges are given.
Eight edges with weights 10, 9,9,8,8,7,6, and 5, respectively, have to be deleted to disconnect the graph into two components: one containing vertices 1 and 2 and another having vertices 7, 9, 10, and 11; this is the first iteration. Having disconnected the graph, the next task is to determine if the obtained degree of discretization is sufficient; if not, the components need to be further disconnected in a similar manner to obtain a finer discretization. A component is further disconnected if one of the following four conditions is satisfied ("disconnect further" criteria):
(1) The average edge weight of the component is greater than half the average edge weight of the complete graph.
(2) The distance between its smallest and largest vertices is greater than or equal to half this distance in the complete graph. For the complete graph, the distance is the graph's highest weight.
(3) The minimum vertex degree of the component is less than the number of its vertices minus 1. The con-trary implies that the component is a complete graph by itself, i.e. the distance between its minimum and maximum vertices is smaller than the distance between the component and any other component. (4) Finally, if the above conditions fail, a fourth one is applied: disconnect the component if it leads to a substantial increase in the information content carried by the discretized vector.
The result of applying only the first three criteria is analogous to SLC clustering with the important property that the algorithm chooses the appropriate level to terminate. Applying the fourth condition, the information measure criterion may, however, result in a clustering which is inconsistent with any iteration of the SLC dendrogram. This criterion is discussed next.
Information measure criterion
Discretizing the entries of a real-valued vector into a finite number of states certainly reduces the information carried by the discrete vector in the sense defined by Shannon (1948). In his paper, Shannon developed a measure of how much information is produced by a discrete source. The measure is known as entropy or Shannon's entropy. Suppose there is a set of n possible events whose probabilities of occurrence are known to be p 1 , p 2 ,…, p n . Shannon proposed a measure of how much choice is involved in the selection of the event or how certain one can be of the outcome, which is given by . log
1 2 ∑ = − = n i i i p p H
The base 2 of the logarithm is chosen so that the resulting units may be called bits. In our context the Shannon's entropy of a vector discretized into n states is given by
∑ − = = 1 0 2 log n i i i w n n w H ,
where w i is the number of entries discretized into state i (assuming a spanning discretization). An increase in the number of states implies an increase in entropy, with an upper bound of log 2 n. However, we want the number of states to be small. That is why it is important to notice that H increases by a different amount depending on which state is split and the size of the resulting new states. For example, if a state containing the most entries is split into two new states of equal size, H will increase more than if a state of fewer entries is split or if we split the larger state into two states of different sizes. To see that splitting a given state into two states of equal size results in maximum entropy increase, consider a vector whose entries have been divided into n states, one of which, labeled with 0, contains 0 w entries. As a function of 0 w , the entropy is given by Suppose that we split state 0 into two states containing m and 0 w m − entries, respectively, where 0 < m < 0 w . This will change only the first term of the right-hand side of the above entropy expression and leave the summation part the same. It is easy to verify that
∑ = + =0 0 2 0 ( ) log w n h w n w = achieves its maximum value over 0 < m < 0 w at 0 2 w m = .
Therefore, splitting a state into two states of equal size maximizes the entropy increase.
As explained in the previous section, the information measure criterion is applied to a component only after the component has failed the other three conditions. Once this happens, we consider splitting it further only if doing so would provide a very significant increase of the entropy, i.e. if the component corresponds to a "large" collection of entries (recurring entries are included since all entries have to be considered when computing the information content of a vector). In our implementation a component gets disconnected further only if it contains at least half the vector entries. Unlike with the other criteria, if a component is to be discretized under the information condition, the corresponding sorted entries are split into two parts: not between the two most distant entries but into two equal parts (or with a difference of one entry in case of an odd number of entries). This is to guarantee a maximum increase of the information measure. In Example 3.2, the two components that were obtained by removing the edges of heaviest weight both fail the "disconnect further" Conditions 1 -3. If the discretization process stopped at this iteration, then the vector d = (0, 0, 1, 1, 1, 1) has Shannon's entropy 0.78631. Having most of the entries of v discretized into the same state, 1, reduces the information content of d. Suppose discretization of v continues according to SLC, i.e., without enforcing the fourth condition of "disconnect further". The next step is to remove the edges of highest weight until a component gets disconnected. This yields the removal of the four edges of weights 4, 3, 2, and 2, respectively, to obtain discretization d = (0, 0, 1, 2, 2, 2). The Shannon's entropy of the new discretization of v is 1.43534. Still half of the entries of v remain at the same discrete level, now 2, which does not allow for a maximal increase in the information content of d. If instead discretization proceeded by applying the information criterion to the bigger component, the resulting discretization becomes d = (0, 0, 1, 1, 2, 2) with Shannon's entropy 1.58631, as opposed to the previous entropy of 1.43534. As illustrated by Example 3.2, the proposed discretization algorithm produces a discretization which is consistent with the definition given above, keeps the number of discrete states small, and maximizes information content over traditional SLC.
Algorithm summary
Input: set S r = {v i | i = 1,…,m} where each v i = (v i1 ,…,v iN )
is a real-valued vector of length N to be discretized.
Output: set S d = {d i | i = 1,…,m} where each d i = (d i1 ,…,d iN )
is the discretization of v i for all i = 1,…,m.
(1) For each i = 1,…,m, construct a complete weighted graph G i where each vertex represents a distinct v ij and the weight of each edge is the Euclidean distance between the incident vertices.
(2) Remove the edge(s) of highest weight.
(
3) If G i is disconnected into components C i1 Gi ,…,C iMi
Gi , go to 4. Else, go to 2.
(4) For each C ik Gi , k = 1,…,M i , apply "disconnect further" criteria 1-3. If any of the three criteria holds, set G i = C ik Gi and go to 2. Else, go to 5.
(5) Apply "disconnect further" 4. If criterion 4 is satisfied, go to 6. Else, go to 7.
(6) Sort the vertex values of C ik Gi and split them into two sets: if |V(C ik Gi )| is even, split the first |V(C ik Gi )|/2 sorted vertex values of C ik Gi into one set and the rest -into another. If |V(C ik Gi )| is odd, split the first |V(C ik Gi )|/2 +1 sorted vertex values of C ik Gi into one set and the rest -into another.
(7) Sort the components C ik Gi , k = 1,…,M i , by the smallest vertex value in each C ik Gi and enumerate them 0, …, D i -1, where D i is the number of components into which G i got disconnected. For each j = 1,…,N, d ij is equal to the label of the component in which v ij is a vertex.
Algorithm complexity
Given M variables, with N time points each, we compute N(N-1)/2 distances to construct the distance matrix so the complexity of this step is O(N 2 ). The distance matrix is used to create the edge and vertex sets of the complete distance graph, containing N(N-1)/2 edges. This can also be accomplished in O(N 2 ) time. These edges are then sorted into decreasing order, so that the largest edges are removed first. A standard sorting algorithm, such as merge sort, has complexity O(N logN) (Knuth, 1998). As each edge is removed, the check for graph disconnection involves testing for the existence of a path between the two vertices of the edge. This test for graph disconnection can be accomplished with a breadth-first search, which has order O(E+V) (Pemmaraju, 2003), with E the number of edges and V the number of vertices in the component. In our case this translates to complexity O(N 2 ). Edge removal is typically performed for a large percentage of the N(N-1)/2 edges, so this step has overall complexity O(N 4 ). The edge removal step dominates the complexity so that the overall complexity is O (M N 4 ) to discretize all M variables. While this is the theoretical worst-case performance, because of the heuristics we have added the typical performance is significantly better.
Requirements on the number of states
While for some applications any number of discretization states is acceptable, there are some cases when there are limitations on this number. For example, if the purpose of discretizing the data is to build a model of polynomials over a finite field as in (Laubenbacher and Stigler, 2004), then the number of states must be a power of a prime since every finite field has cardinality p n , where p is prime and n is a positive integer. Our method deals with this problem in the following way. Suppose that a vector has been discretized into m states in the way described above. The next step is to find the smallest integer k = p n such that m ≤ k. This value for k gives the number of states that needs to be obtained. Since the discretization algorithm yielded m clusters, the remaining k -m can be constructed by sorting the entries in each cluster and splitting the one that contains the two most distant entries with respect to Euclidean distance. The splitting should take place between these entries. This is repeated until k clusters are obtained. This approach has a potential problem. For instance, if a vector got discretized into 14 states and the total number of distinct entries of the vector is 15, then k = 16 cannot be reached. In this case the two closest states could be merged together to obtain 13 states. In general it may not be desirable to reduce the number of states because this results in loss of information. We would rather increase the number of states unless it is impossible as in the above example.
Discretization of several vectors
Some applications may require that all vectors in a data set be discretized into the same number of states. For example the approach adopted by Laubenbacher and Stigler (2004) imposes such a requirement on the discretization. The way we deal with this is by first discretizing all vectors separately. Suppose that for N vectors, the discretization method discretized each into m 1 , m 2 , …, m N states, respectively. Let m = max{ m i | i = 1,…, N }. Now find the least possible n k p = such that m k ≤ . Finally, discretize all variables into k states in the same way that was described for the discretization of a single vector into the required number of states.
PRESERVATION OF CORRELATIONS
While any discretization inevitably results in information loss, a good multivariate discretization should preserve some important features of the data such as correlation between the variables. To demonstrate that our method has this property, we use the temporal map of fluctuations in mRNA expression of a set of genes related to rat central nervous system development presented in (Wen et al., 1998). Wen et al. (1998) focus on the cervical spinal cord and the genes included in their study are from families that are believed to be important for spinal cord development. They used an RT-PCR protocol to measure the expression of 112 genes in central nervous system development. The data consist of nine expression measurements for each gene: cervical spinal cord tissue was dissected from animals in embryonic days 11, 13, 15, 18, and 21 and in postnatal days 0, 7, 14, and 90. We calculated the Spearman rank correlation coefficient between each gene's analog time series and discretized time series according to (Walpole et al., 1998). Then we identified each coefficient value as representing a "significant" or "not significant" correlation based on a critical value of 0.683, which corresponds to a confidence level of 0.025 for a time series of nine points. This means that the probability that the correlation coefficient for a pair of uncorrelated time series of length 9 will be greater than or equal to 0.683 is 0.025 (Walpole et al., 1998). The result is that 71 out of 112 genes were found to be significantly correlated to their discrete version, which is 63.39% of the total number of genes. By augmenting the level of confidence to 0.05, this percentage increases to 76.79%. We also calculated the Spearman rank correlation coefficient for each pair of genes before and after discretizing the data, considering only the genes that discretized into exactly three states -56 of them. For the original data given in (Wen et al., 1998), out of the 1540 gene pairs, 234 pairs were identified as significantly correlated with a confidence level of 0.025. Based on the discretized data, 132 pairs were correctly identified as significantly correlated and 1181 pairs were correctly recognized as not significantly correlated. That is, 1313, or more than 85%, of the correlations were correctly classified after discretization. These results imply that our algorithm can be successfully applied to cases when preserving the relationships between the variables in a system is important.
APPLICATION TO THE REVERSE ENGINEERING OF AN ARTIFICIAL GENE NETWORK
As mentioned in the introduction, our main motivation was the need to discretize time series in order to construct dynamic models using a finite state set. To demonstrate the effectiveness of our method, we used it to generate appropriate data for the reverse-engineering method developed in (Laubenbacher and Stigler, 2004). Since data from real gene networks are limited, we chose to test the method on an artificial gene network.We used the A-Biochem software system developed by P. Mendes and his collaborators (Mendes et al., 2003). A-Biochem automatically generates artificial gene networks with particular topological and kinetic properties. These networks are embodied in kinetic models, which are used by the biochemical-network simulator Gepasi (Mendes, 1993(Mendes, , 1997 to produce simulated gene expression data. We generated an artificial gene network with five genes and ten total input connections using the Albert-Barabási algorithm (Albert and Barabási, 2000). Gepasi uses a continuous representation of biochemical reactions, based on ordinary differential equations (ODE).
With the parameters we specified, Gepasi generated an ODE system that represents the network. For example, the synthesis rate of gene G 1 is given by
) ( 1 ) ( 3 01 . 0 ) ( 1 01 . 0 ) ( 1 01 . 0 1 t G t G t G t G dt dG − + + = .
Analyzing the dynamics of the ODE system, one finds that it has two stable steady states (of which only the first is biochemically meaningful): S 1 = (1.99006, 1.99006, 0.000024814, 0.997525, 1.99994) and S 2 = (-0.00493694, -0.00493694, -0.0604538, -0.198201, 0.0547545). As Laubenbacher and Stigler (2004) demonstrated, the performance of their algorithm dramatically improves if knockout time series for genes are incorporated. For this reason we supplied seven time series of 11 points each: two wildtype time series and five knockout time series, one for each gene. The first wild-type time series is generated by solving the ODE system numerically for t = 0, 2, 6,…, 20 with initial conditions G i (0) = 1 for all i = 1,…, 5. Figure 3 shows a plot of the numerical solution of the ODE system with these initial conditions. One can simulate a gene knockout in Gepasi by setting the corresponding variable and initial condition to zero. The second time series is generated like the first one but this time with t = 0, 1,…, 10 and initial conditions (G 1 (0), G 2 (0), G 3 (0), G 4 (0), G 5 (0)) = (1, -1, -0.6, -1, 0.5).
(We emphasize that we are including the steady state S 2 in order to show that the discretized data preserve information about the dynamics of the ODE system.) Fig.3. Plot of the numerical solution of the ODE system with initial condition (G 1 (0), G 2 (0), G 3 (0), G 4 (0), G 5 (0)) = (1, 1, 1, 1, 1).
In this case, the time points from each of the seven time series constitute the input vectors. The discretization algorithm chose a state set X of cardinality 5 and, based on the discrete data, the reverse engineering method generated five polynomials describing the discrete model. For example, the polynomial that describes the dynamics of G 1 (which is now denoted by x1) is f1 = -x1*x5^2-2*x2*x5^2+2*x4*x5^2+2*x5^3 +x1*x2+x2^2-2*x1*x3+x3^2-2*x1*x4+2*x2*x4-x3*x4+x4^2-x2*x5+x4*x5-x1-2*x2-x3-2*x5-1.
That is, the discrete model is given by the time-discrete dynamical system f = (f 1 , … , f 5 ) : X 5 → X 5 . Now we compare the dynamics of the two models. First, the discretization maps steady state S 1 to the fixed point FP 1 = (4, 4, 1, 4, 2) of f and steady state S 2 to the fixed point FP 2 = (0, 1, 1, 1, 0). The time series produced by solving the ODE system and converging to S 1 is given in the top part of Figure 4. The corresponding discrete points from the time series in the bottom part of Figure 4 form a trajectory that ends at FP 1 ( Figure 5). The discrete model trajectory can be superimposed over the discretization of the continuous one, illustrating the matching dynamics of the two models. The same can be observed for the second steady-state S 2 that is mapped to fixed point FP 2 . the ODE system for t = 0,…,10 with initial conditions (G 1 (0),G 2 (0),G 3 (0),G 4 (0),G 5 (0)) = (1, 1, 1, 1, 1); bottom: corresponding discrete point time series.
DISCUSSION
The discretization method presented here has two novel features. Firstly, it uses Shannon's information criterion to determine clusters and, secondly, it determines the optimal number of clusters for a given data set. In addition to its use as a novel clustering method, it is particularly suitable for the discretization of multivariate time series, since it preserves a large degree of variable correlation and information about dynamic features. It thus provides a valuable tool for any application that requires discretization of continuous data when the number of discrete classes that best fits the data is unknown. So far our discretization method has only been tested on noiseless data. An important advantage of using discrete states is that a significant portion of the noise is absorbed in the process. The next step is to quantify this portion precisely. Based on preliminary experiments we expect that for data that discretize into a relatively small number of states and that contain a degree of noise common to many biological data, the majority of the noise is absorbed into the discrete states.
Fig. 5.
Trajectories formed by the discretized wild-type time series. (Vastani et al., 2004)
Fig. 1 .
1Dendrogram representing the SLC algorithm applied to the data of Example 3.2. The column on the right gives the corresponding Shannon's entropy increasing at each consecutive level.
Fig. 4 .
4Top: wild-type time series generated by solving numerically
ACKNOWLEDGEMENTSThis work has been partially supported by NIH Grant Nr. 1RO1 GM068947-01. The authors thank A. Jarrah, P. Mendes, and B. Stigler for contributing insights and help with editing, and D. Camacho and W. Sha for helpful discussions.
Topology of evolving networks: local events and universality. R Albert, A Barabási, Phys. Rev. Lett. 85Albert, R. and Barabási, A. (2000) Topology of evolving networks: local events and universality. Phys. Rev. Lett. 85, 5234-5237.
The topology of the regulatory interactions predict the expression pattern of the segment polarity genes in Drosophila melanogaster. R Albert, H Othmer, J. Theor. Biol. 223Albert, R. and Othmer, H. (2003) The topology of the regulatory interactions predict the expression pattern of the segment polar- ity genes in Drosophila melanogaster. J. Theor. Biol. 223, 1-18.
A genome-wide transcriptional analysis of the mitotic cell cycle. R Cho, M Campbell, E Winzeler, L Steinmetz, A Conway, L Wodicka, T Wolfsberg, A Gabrielan, D Landsman, D Lockhart, Davis , R , Moll. Cell. 2Cho, R., Campbell, M., Winzeler, E., Steinmetz, L., Conway, A., Wodicka, L., Wolfsberg, T., Gabrielan, A., Landsman, D., Lockhart, D., and Davis, R. (1998) A genome-wide transcrip- tional analysis of the mitotic cell cycle. Moll. Cell 2, 65-73.
Exploring the metabolic and genetic control of gene expression on a genomic scale. J Derisi, V Iyer, P Brown, Science. 282DeRisi, J., Iyer, V., and Brown, P. (1997) Exploring the metabolic and genetic control of gene expression on a genomic scale. Sci- ence 282, 699-705.
Supervised and unsupervised discrimination of continuous Features. J Dougherty, R Kohavi, M Sahami, Machine learning: Proceedings of the 12 th International Conference. Prieditis, A. and Russell, S.San Francisco, CAMorgan KauffmanDougherty, J., Kohavi, R., Sahami, M. (1995) Supervised and un- supervised discrimination of continuous Features. In Prieditis, A. and Russell, S. (eds.), Machine learning: Proceedings of the 12 th International Conference, Morgan Kauffman, San Fran- cisco, CA.
Discretization of continuous attributes while learning Bayesian networks. N Friedman, M Goldszmidt, Proc. Of the 13 th International Conference on Machine Learning. Saitta, L.Of the 13 th International Conference on Machine LearningSan Francisco, CAMorgan KauffmanFriedman, N. and Goldszmidt, M. (1996) Discretization of con- tinuous attributes while learning Bayesian networks. In Saitta, L. (ed.), Proc. Of the 13 th International Conference on Machine Learning, 157-165, Morgan Kauffman, San Francisco, CA.
Using Bayesian networks to analyze expression data. N Friedman, M Linial, I Nachman, D Pe'er, J. Comput. Biol. 7Friedman, N., Linial, M., Nachman, I., and Pe'er, D. (2000) Using Bayesian networks to analyze expression data. J. Comput. Biol. 7, 601-620.
Principled computational methods for the validation and discovery of genetic regulatory networks. Massachusetts Institute of Technology. A Hartemink, Ph. D. dissertationHartemink, A. (2001) Principled computational methods for the validation and discovery of genetic regulatory networks. Mas- sachusetts Institute of Technology, Ph. D. dissertation.
Algorithms for clustering data. A Jain, R Dubes, Prentice HallJain, A. and Dubes, R., 1988. Algorithms for clustering data. Pren- tice Hall. 58-89.
Hierarchical clustering schemes. S C Johnson, Psychometrika. 32Johnson, S. C. (1967) Hierarchical clustering schemes. Psycho- metrika 32, 241-254.
Metabolic stability and epigenesist in randomly constructed genetic nets. S A Kauffman, J. Theor. Biol. 22Kauffman, S. A. (1969) Metabolic stability and epigenesist in ran- domly constructed genetic nets. J. Theor. Biol. 22, 437-467.
Sorting and Searching, 2 nd edition. D E Knuth, Addison-Wesley3Reading, MassachusettsThe Art of Computer ProgrammingKnuth, D.E., 1998. The Art of Computer Programming, Volume 3: Sorting and Searching, 2 nd edition. Addison-Wesley, Reading, Massachusetts.
A computational algebra approach to the reverse engineering of gene regulatory networks. R Laubenbacher, B Stigler, J. Theor. Biol. 229Laubenbacher, R. and Stigler, B. (2004) A computational algebra approach to the reverse engineering of gene regulatory net- works. J. Theor. Biol. 229, 523-537.
Some methods for classification and analysis of multivariate observations. J Macqueen, Proceedings of the 5 th Berkeley Symposium of Mathematical Statistics and Probability. the 5 th Berkeley Symposium of Mathematical Statistics and ProbabilityBerkeley, CAUniversity of California Press1MacQueen, J. (1967) Some methods for classification and analysis of multivariate observations. Proceedings of the 5 th Berkeley Symposium of Mathematical Statistics and Probability 1, 281- 297. University of California Press, Berkeley, CA.
GEPASI: a software package for modeling the dynamics, steady states and control of biochemical and other systems. P Mendes, Comput. Appl. Biosci. 9Mendes, P. (1993) GEPASI: a software package for modeling the dynamics, steady states and control of biochemical and other systems. Comput. Appl. Biosci. 9, 563-571.
Biochemistry by numbers: simulation of biochemical pathways with Gepasi 3. P Mendes, Trends Biochem. Sci. 22Mendes, P. (1997) Biochemistry by numbers: simulation of bio- chemical pathways with Gepasi 3. Trends Biochem. Sci. 22, 361-363.
Artificial gene networks for objective comparison of analysis algorithms. P Mendes, W Sha, Ye , K , Bioinformatics. 19Mendes, P., Sha, W., and Ye, K. (2003) Artificial gene networks for objective comparison of analysis algorithms. Bioinformatics 19, ii122-ii129.
Inferring subnetworks from perturbed expression profiles. D Pe'er, A Regev, G Elidan, N Friedman, Bioinformatics. 17Pe'er, D., Regev, A., Elidan, G., Friedman, N. (2001) Inferring subnetworks from perturbed expression profiles. Bioinformatics 17, S215-224.
S Pemmaraju, S Skiena, Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Cambridge University PressPemmaraju, S., Skiena, S. (2003) Computational Discrete Mathe- matics: Combinatorics and Graph Theory with Mathematica. Cambridge University Press.
a free, open-source system for microarray data management and analysis. A Saeed, V Sharov, J White, J Li, W Liang, N Bhagabati, J Braisted, M Klapa, T Currier, M Thiagarajan, A Sturn, M Snuffin, A Rezantsev, D Popov, A Ryltsov, E Kostukovich, I Borisovsky, Z Liu, A Vinsavich, V Trush, J Quackenbush, BioTechniques. 42Saeed, A., Sharov, V., White, J., Li, J., Liang, W., Bhagabati, N., Braisted, J., Klapa, M., Currier, T., Thiagarajan, M., Sturn, A., Snuffin M., Rezantsev, A., Popov, D., Ryltsov, A., Kos- tukovich, E., Borisovsky, I., Liu, Z., Vinsavich, A., Trush, V., Quackenbush, J. (2003) TM4: a free, open-source system for microarray data management and analysis. BioTechniques 34(2), 374-378.
A Mathematical Theory of Communication. C Shannon, The Bell Systems Technical Journal. 27Shannon, C. (1948) A Mathematical Theory of Communication. The Bell Systems Technical Journal 27, 379-423, 623-656.
Qualitative analysis of gene networks. D Thieffry, R Thomas, Proc. Pacific Symp. on Biocomputing. Pacific Symp. on BiocomputingSingaporeWorld ScientificThieffry, D. and Thomas, R. (1998) Qualitative analysis of gene networks. Proc. Pacific Symp. on Biocomputing, 77-88, World Scientific, Singapore.
. H Vastani, A Jarrah, R Laubenbacher, Vastani, H., Jarrah, A., Laubenbacher, R. (2004) Available at http://dvd.vbi.vt.edu.
Probability and statistics for engineers and scientists. R Walpole, R Myers, S Myers, Prentice Hall6th editionWalpole, R., Myers, R., Myers, S. (1998) Probability and statistics for engineers and scientists, 6 th edition. Prentice Hall. 630-632.
Large-scale temporal gene expression mapping of central nervous system development. X Wen, S Fuhrman, G Michaelis, D Carr, S Smith, J Barker, R Somogyi, Proc. Natl. Acad. Sci. USA. 95Wen, X., Fuhrman, S., Michaelis, G., Carr, D., Smith, S., Barker, J., Somogyi, R. (1998) Large-scale temporal gene expression mapping of central nervous system development. Proc. Natl. Acad. Sci. USA. 95, 334-339.
| [] |
[
"Spectral Reconstruction with Deep Neural Networks",
"Spectral Reconstruction with Deep Neural Networks"
] | [
"Lukas Kades \nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany\n",
"Jan M Pawlowski \nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany\n\nExtreMe Matter Institute EMMI\nGSI\nPlanckstr. 1D-64291DarmstadtGermany\n",
"Alexander Rothkopf \nFaculty of Science and Technology\nUniversity of Stavanger\nNO-4036StavangerNorway\n",
"Manuel Scherzer \nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany\n",
"Julian M Urban \nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany\n",
"Sebastian J Wetzel \nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany\n",
"Nicolas Wink \nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany\n",
"Felix Ziegler \nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany\n"
] | [
"Institut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany",
"Institut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany",
"ExtreMe Matter Institute EMMI\nGSI\nPlanckstr. 1D-64291DarmstadtGermany",
"Faculty of Science and Technology\nUniversity of Stavanger\nNO-4036StavangerNorway",
"Institut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany",
"Institut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany",
"Institut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany",
"Institut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany",
"Institut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany"
] | [] | We explore artificial neural networks as a tool for the reconstruction of spectral functions from imaginary time Green's functions, a classic ill-conditioned inverse problem. Our ansatz is based on a supervised learning framework in which prior knowledge is encoded in the training data and the inverse transformation manifold is explicitly parametrised through a neural network. We systematically investigate this novel reconstruction approach, providing a detailed analysis of its performance on physically motivated mock data, and compare it to established methods of Bayesian inference. The reconstruction accuracy is found to be at least comparable, and potentially superior in particular at larger noise levels. We argue that the use of labelled training data in a supervised setting and the freedom in defining an optimisation objective are inherent advantages of the present approach and may lead to significant improvements over state-of-the-art methods in the future. Potential directions for further research are discussed in detail. | 10.1103/physrevd.102.096001 | [
"https://arxiv.org/pdf/1905.04305v1.pdf"
] | 152,282,829 | 1905.04305 | 305957d4da59f7d0d531b0b0688b000cb9441237 |
Spectral Reconstruction with Deep Neural Networks
Lukas Kades
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 16D-69120HeidelbergGermany
Jan M Pawlowski
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 16D-69120HeidelbergGermany
ExtreMe Matter Institute EMMI
GSI
Planckstr. 1D-64291DarmstadtGermany
Alexander Rothkopf
Faculty of Science and Technology
University of Stavanger
NO-4036StavangerNorway
Manuel Scherzer
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 16D-69120HeidelbergGermany
Julian M Urban
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 16D-69120HeidelbergGermany
Sebastian J Wetzel
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 16D-69120HeidelbergGermany
Nicolas Wink
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 16D-69120HeidelbergGermany
Felix Ziegler
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 16D-69120HeidelbergGermany
Spectral Reconstruction with Deep Neural Networks
We explore artificial neural networks as a tool for the reconstruction of spectral functions from imaginary time Green's functions, a classic ill-conditioned inverse problem. Our ansatz is based on a supervised learning framework in which prior knowledge is encoded in the training data and the inverse transformation manifold is explicitly parametrised through a neural network. We systematically investigate this novel reconstruction approach, providing a detailed analysis of its performance on physically motivated mock data, and compare it to established methods of Bayesian inference. The reconstruction accuracy is found to be at least comparable, and potentially superior in particular at larger noise levels. We argue that the use of labelled training data in a supervised setting and the freedom in defining an optimisation objective are inherent advantages of the present approach and may lead to significant improvements over state-of-the-art methods in the future. Potential directions for further research are discussed in detail.
I. INTRODUCTION
Machine Learning has been applied to a variety of problems in the natural sciences. For example, it is regularly deployed in the interpretation of data from highenergy physics detectors [1,2]. Algorithms based on learning have shown to be highly versatile, with their use extending far beyond the original design purpose. In particular, deep neural networks have demonstrated unprecedented levels of prediction and generalisation performance, for reviews see e.g. [3,4]. Machine Learning architectures are also increasingly deployed for a variety of problems in the theoretical physics community, ranging from the identification of phases and order parameters to the acceleration of lattice simulations [5][6][7][8][9][10][11][12][13][14][15].
Ill-conditioned inverse problems lie at the heart of some of the most challenging tasks in modern theoretical physics. One pertinent example is the computation of real-time properties of strongly correlated quantum systems. Take e.g. the phenomenon of energy and charge transport, which so far has defied a quantitative understanding from first principles. This universal phenomenon is relevant to systems at vastly different energy scales, ranging from ultracold quantum gases created with optical traps to the quark-gluon plasma born out of relativistic heavy-ion collisions.
While static properties of strongly correlated quantum systems are by now well understood and routinely computed from first principles, a similar understanding of real-time properties is still subject to ongoing research. The thermodynamics of strongly coupled systems, such as the quark gluon plasma, has been explored using the combined strength of different non-perturbative approaches, such as functional renormalisation group methods and lattice field theory calculations. There are two limitations affecting most of these approaches: Firstly, in order to carry out quantitative computations, time has to be analytically continued into the complex plane, to socalled Euclidean time. Secondly, explicit computations are either fully numerical or at least involve intermediate numerical steps.
This leaves us with the need to eventually undo the analytic continuation of Euclidean correlation functions, which are known only approximately. The most relevant examples are two-point functions, the so-called Euclidean propagators. The spectral representation of quantum field theory relates the propagators, be they in Minkowski or Euclidean time, to a single function encoding their physics, the so-called spectral function. The number of different structures contributing to a spectral function are in general quite limited and consist of poles and cuts. If we can extract from the Euclidean two-point correlator its spectral function, we may immediately compute the corresponding real-time propagator.
If we know the Euclidean propagator analytically, this information allows us in principle to recover the corresponding Minkowski time information. In practice, however, the limitation of having to approximate correlator data (e.g. through simulations) turns the computation of spectral functions into an ill-conditioned problem. The most common approach to give meaning to such inverse problems is Bayesian inference. It incorporates additional prior domain knowledge we possess on the shape of the spectral function to regularise the inversion task. The positivity of hadronic spectral functions is one prominent example. The Bayesian approach has seen continuous improvement over the past two decades in the context of spectral function reconstructions. While originally it was restricted to maximum a posteriori estimates for the most probable spectral function given Euclidean data and prior information [16][17][18], in its most modern form it amounts to exploring the full posterior distribution [19]. An important aspect of the work is to develop appropriate mock data tests to benchmark the reconstruction performance before applying it to actual data. Generally, the success of a reconstruction method stands or falls with its performance on physical data. While this seems evident, it was in fact a hard lesson learned in the history of Bayesian arXiv:1905.04305v1 [physics.comp-ph] 10 May 2019 reconstruction methods, a lesson which we want to heed.
Inverse problems of this type have also drawn quite some attention in the machine learning (ML) community [20][21][22][23]. In the present work we build upon both the recent progress in the field of ML, particularly deep learning, as well as results and structural information gathered in the past decades from Bayesian reconstruction methods. We set out to isolate a property of neural networks that holds the potential to improve upon the standard Bayesian methods, while retaining their advantages, utilising the already gathered insight in their study.
Consider a feed-forward deep neural network that takes Euclidean propagator data as input and outputs a prediction of the associated spectral function. Although the reasoning behind this ansatz is rather different, one can draw parallels to more traditional methods. In the Bayesian approach, prior information is explicitly encoded in a prior probability functional and the optimisation objective is the precise recovery of the given propagator data from the predicted spectral function. In contrast, the neural network based reconstruction is conditioned through supervised learning with appropriate training data. This corresponds to implicitly imposing a prior distribution on the set of possible predictions, which, as in the Bayesian case, regularises the reconstruction problem. Optimisation objectives are now expressed in terms of loss functions, allowing for greater flexibility. In fact, we can explicitly provide pairs of correlator and spectral function data during the training. Hence, not only can we aim for the recovery of the input data from the predictions as in the Bayesian approach, but we are now also able to formulate a loss directly on the spectral functions themselves. This constitutes a much stronger restriction on potential solutions for individual propagators, which could provide a significant advantage over other methods. The possibility to access all information of a given sample with respect to its different representations also allows the exploration of a much broader set of loss functions, which could benefit not only the neural network based reconstruction, but also lead to a better understanding and circumvention of obstacles related to the inverse problem itself. Such an obstacle is given, for example, by the varying severity of the problem within the space of considered spectral functions. By employing adaptive losses, inhomogeneities of this type could be neutralised.
Similar approaches concerning spectral functions that consist of normalised sums of Gaussian peaks have already been discussed in [24,25]. In this work, we investigate the performance of such an approach using mock data of physical resonances motivated by quantum field theory, and compare it to state-of-the-art Bayesian methods. The data are given in the form of linear combinations of unnormalised Breit-Wigner peaks, whose distinctive tail structures introduce additional difficulties. Using only a rather naive implementation, the performance of our ansatz is demonstrated to be at least comparable and potentially superior, particularly for large noise levels. We then discuss potential improvements of the architecture, which in the future could establish neural networks to a state-of-the-art approach for accurate reconstructions with a reliable estimation of errors.
The paper is organised as follows. The spectral reconstruction problem is defined in Section II A. State-of-theart Bayesian reconstruction methods are summarised in Section II B. In Section II C we discuss the application of neural networks and potential advantages. Section III contains details on the design of the networks and defines the optimisation procedure. Numerical results are presented and compared to Bayesian methods in Section IV. We summarise our findings and discuss future work in Section V.
II. SPECTRAL RECONSTRUCTION AND POTENTIAL ADVANTAGES
A. Defining the problem Typically, correlation functions in equilibrium quantum field theories are computed in imaginary time after a Wick rotation t → it ≡ τ , which facilitates both analytical and numerical computations. In strongly correlated systems, a numerical treatment is in most cases inevitable. Such a setup leaves us with the task to reconstruct relevant information, such as the spectrum of the theory, or genuine real-time quantities such as transport coefficients, from the Euclidean data.
The information we want to access is encoded in the associated spectral function ρ. For this purpose it is most convenient to work in momentum space both for ρ and the corresponding propagator G. The relation between the Euclidean propagator and the spectral function is given by the well known Källen-Lehmann spectral representation,
G(p) = ∞ 0 dω π ω ρ(ω) ω 2 + p 2 ≡ ∞ 0 dω K(p, ω)ρ(ω) ,(1)
which defines the corresponding Källen-Lehmann kernel. The propagator is usually only available in the form of numerical data, with finite statistical and systematic uncertainties, on a discrete set of N p points, which we abbreviate as G i = G(p i ). The most commonly used approach is to work directly with a discretised version of (1). We utilise the same abbreviation for the spectral function, i.e. ρ i = ρ(ω i ), discretised on N ω points. This lets us state the discrete form of (1) as
G i = Nω j=1 K ij ρ j ,(2)
where K ij = K(p i , ω j )∆ω j . This amounts to a classic illconditioned inverse problem, similar in nature to those encountered in many other fields, such as medical imaging or the calibration of option pricing methods. Typical Examples of mock spectral functions reconstructed via our neural network approach for the cases of one, two and three Breit-Wigner peaks. The chosen functions mirror the desired locality of suggested reconstructions around the original function (red line). Additive, Gaussian noise of width 10 −3 is added to the discretised analytic form of the associated propagator of the same original spectral function multiple times. The shaded area depicts for each frequency ω the distribution of resulting outcomes, while the dashed green line corresponds to the mean. The results are obtained from the FC parameter network optimised with the parameter loss. The network is trained on the largest defined parameter space which corresponds to the volume Vol O. The uncertainty for reconstructions decreases for smaller volumes as illustrated in Figure 5. A detailed discussion on the properties and problems of a neural network based reconstruction is given in Section IV A.
errors on the input data G(p i ) are on the order of 10 −2 to 10 −5 when the propagator at zero momentum is of the order of unity.
To appreciate the problems arising in such a reconstruction more clearly, let us assume we have a suggestion for the spectral function ρ sug and its corresponding propagator G sug . The difference to the measured data is encoded in
G(p)−G sug (p) = ∞ 0 dω π ω ω 2 + p 2 ρ(ω) − ρ sug (ω) ,(3)
with a suitable norm . . Evidently, even if this expression vanishes point-wise, i.e. G(p i ) − G sug (p i ) = 0 for all p i , the spectral function is not uniquely fixed. Experience has shown that with typical numerical errors on the input data, qualitatively very different spectral functions can lead to in this sense equivalent propagators. This situation can often be improved on by taking more prior knowledge into account, c.f. the discussion in [26]. This includes properties such as:
1. Normalisation and positivity of spectral functions of asymptotic states. For gauge theories, this may reduce to just the normalisation to zero, expressed in terms of the Oehme-Zimmermann superconvergence [27,28].
2. Asymptotic behaviour of the spectral function at low and high frequencies.
3. The absence of clearly unphysical features, such as drastic oscillations in the spectral function and the propagator.
Additionally, the parametrisation of the spectral function in terms of frequency bins is just one particular basis. In order to make reconstructions more feasible, other choices, and in particular physically motivated ones, may be beneficial, c.f. again the discussion in [26]. In this work, we consider a basis formulated in terms of physical resonances, i.e. Breit-Wigner peaks.
B. Existing methods
The inverse problem as defined in (1) has an exact solution in the case of exactly known, discrete correlator data [29]. However, as soon as noisy inputs are considered, this approach turns out to be impractical [30]. Therefore, the most common strategy to treat this problem is via Bayesian inference. This approach is based on Bayes' theorem, which states that the posterior probability is essentially given by two terms, the likelihood function and prior probability:
P (ρ|D, I) ∝ P (D|ρ, I) P (ρ|I) .(4)
It explicitly includes additionally available prior information on the spectral function in order to regularise the inversion task. The likelihood P (D|ρ) encodes the probability for the input data D to have arisen from the test spectral function ρ, while P (ρ) quantifies how well this test function agrees with the available prior information.
The two probabilities fully specify the posterior distribution in principle, however they may be known only implicitly. In order to gain access to the full distribution, one may sample from the posterior, e.g. through a Markov Chain Monte Carlo process in the parameter space of the spectral function. However, in practice one is often content with the maximum a posteriori (MAP) solution. Given a uniform prior, the Maximum Likelihood Estimate (MLE) corresponds to an estimate of the MAP.
C. Advantages of neural networks
In order to make genuine progress, we set out in this study to explore methods in which our prior knowledge of the analytic structure can be encoded in different ways. To this end, our focus lies on the use of Machine Learning in the form of artificial neural networks. These feature a high flexibility in the encoding of information by learning abstract internal representation. They possess the advantageous properties that prior information can be explicitly provided through the training data, and that the solution space can be regularised by choosing appropriate loss functions.
Minimising (3), while respecting the constraints discussed in Section II A, can be formulated as minimising a loss function
L G (ρ sug ) = G[ρ sug ] − G[ρ] .(5)
This corresponds to indirectly working on a norm or loss function for ρ,
L ρ (ρ sug ) = ρ sug − ρ ,(6)
Of course, the optimisation problem as given by (6) is intractable, since it requires the knowledge of the true spectral function ρ. Minimising L ρ (ρ sug ) for a given set of {ρ sug } also minimises L G , since the Källén-Lehmann representation (1) is a linear map. In turn, however, minimising L G does not uniquely determine the spectral function, as has already been mentioned. Accordingly, the key to optimise the spectral reconstruction is the ideal use of all known constraints on ρ, in order to better condition the problem. The latter aspect has fueled many developments in the area of spectral reconstructions in the past decades. Given the complexity of the problem, as well as the interrelation of the constraints, this calls, in our opinion, for an application of supervised machine learning algorithms for an optimal utilisation of constraints. To demonstrate our reasoning, we generate a training set of known pairs of spectral functions and propagators and train a neural network to reconstruct ρ from G by minimising a suitable norm, utilising both L G and L ρ during the training. When the network has converged, it can be applied to measured propagator data G for which the corresponding ρ is unknown.
Estimators learning from labelled data provide a potentially significant advantage due to the employed supervision, because the loss function is minimised a priori for a whole range of possible input/output pairs. Accordingly, a neural network aims to learn the entire set of inverse transformations for a given set of spectral functions. After this mapping has been approximated to a sufficient degree, the network can be used to make predictions. This is in contrast to standard Bayesian methods, where the posterior distribution is explored on a case by case basis. Both approaches may also be combined, e.g. by employing a neural network to suggest a solution ρ sug , which is then further optimised using a traditional algorithm.
The given setup forces the network to regularise the illconditioned problem by reproducing the correct training spectrum in accord with our criteria for a successful reconstruction. It is the inclusion of the training data and the free choice of loss functions that allows the network to fully absorb all relevant prior information. This ability is an outstanding property of supervised learning methods, which could yield potentially significant improvements over existing frameworks. Examples for such constraints are the analytic structure of the propagator, asymptotic behaviors and normalisation constraints.
The parametrisation of an infinite set or manifold of inverse transformations by the neural network also enables the discovery of new loss functions which may be more appropriate for a reliable reconstruction. This includes, for example, the exploration of correlation matrices with adapted elements, in order to define a suitable norm for the given and suggested propagators. Existing, iterative methods may also benefit from the application of such adaptive loss functions. These may include parameters, point-like representations and arbitrary other characteristics of a given training sample.
Formulated in a Bayesian language, we set out to explicitly train the neural network to predict MAP estimates for each possible input propagator, given the training data as prior information. By salting the input data with noise, the network learns to return a denoised representation of the associated spectral functions.
III. A NEURAL NETWORK BASED RECONSTRUCTION
Neural networks provide high flexibility with regard to their structure and the information they can process. They are capable of learning complex internal representations which allow them to extract the relevant features from a given input. A variety of network architectures and loss functions can be implemented in a straightforward manner using modern Machine Learning frameworks. Prior information can be explicitly provided through a systematic selection of the training data. The data itself provides, in addition to the loss function, a regularisation of possible suggestions. Accordingly, the proposed solutions have the advantage to be similar to the ones in the training data.
The section starts with notes on the design of the neural networks we employ and ends with a detailed introduction of the training procedure and the utilised loss functions.
A. Design of the neural networks
We construct two different types of deep feed-forward neural networks. The input layer is fed with the noisy
G(p 1 ) G(p 2 ) G(p 3 ) G(p 4 ) A M Γ input layer hidden layers output layer (a)
input layer hidden layers output layer propagator data G(p). The output for the first type is an estimate of the parameters of the associated ρ in the chosen basis, which we denote as parameter net (PaNet).
G(p 1 ) G(p 2 ) G(p 3 ) G(p 4 ) ρ(ω 1 ) ρ(ω 2 ) ρ(ω 3 ) ρ(ω 4 ) (b)
For the second type, the network is trained directly on the discretised representation of the spectral function. This network will be referred to as point net (PoNet).
A consideration of a variable number of Breit-Wigners is feasible per construction by the point-like representation of the spectral function within the output layer. This kind of network will in the following be abbreviated by PoVarNet. See Figure 2 for a schematic illustration of the different network types. Note that in all cases a basis for the spectral function is provided either explicitly through the structure of the network or implicitly through the choice of the training data. If not stated otherwise, the numerical results presented in the following always correspond to results from the PaNet. We compare various types of hidden layers and the impact of depth and width within the neural networks.
In general, choosing the numbers of layers and neurons is a trade-off between the expressive power of the network, the available memory and the issue of overfitting. The latter strongly depends on the number of available training samples w.r.t. the expressivity. For fully parametrised spectral functions, new samples can be generated very efficiently for each training epoch, which implies an, in principle, infinite supply of data. Therefore, in this case, the risk of overfitting is practically nonexistent. The specific dimensions and hyperparameters used for this work are provided in Appendix C. Numerical results can be found in Section IV.
B. Training strategy
The neural network is trained with appropriately labelled input data in a supervised manner. This approach allows to implicitly impose a prior distribution in the Bayesian sense. The challenge lies in constructing a training dataset that is exhaustive enough to contain the relevant structures that may constitute the actual spectral functions in practical applications.
From our past experience with hadronic spectral functions in lattice QCD and the functional renormalisation group, the most relevant structures are peaks of the Breit-Wigner type, as well as thresholds. The former present a challenge from the point of view of inverse problems, as they contain significant tail contributions, contrary e.g. to Gaussian peaks, which approach zero exponentially fast. Thresholds on the other hand set in at finite frequencies, often involving a non-analytic kink behavior. In this work, we only consider Breit-Wigner type structures as a first step for the application of neural networks to this family of problems.
Mock spectral functions are constructed using a superposed collection of Breit-Wigner peaks based on a parametrisation obtained directly from one-loop perturbative quantum field theory. Each individual Breit-Wigner is given by
ρ (BW ) (ω) = 4AΓω (M 2 + Γ 2 − ω 2 ) 2 + 4Γ 2 ω 2 .(7)
Here, M denotes the mass of the corresponding state, Γ its width and A amounts to a positive normalisation constant. Spectral functions for the training and test set are constructed from a combination of at most N BW = 3 different Breit-Wigner peaks. Depending on which type of network is considered, the Euclidean propagator is obtained either by inserting the discretised spectral function into (2), or by a computation of the propagator's analytic representation from the given parameters. The propagators are salted both for the training and test set with additive, Gaussian noise
G noisy i = G i + .(8)
This is a generic choice which allows to quantify the performance of our method at different noise levels.
The advantage of neural networks to have direct access to different representations of a spectral function implies a free choice of objective functions in the solution space. We consider three simple loss functions and combinations thereof. The (pure) propagator loss L G (ρ sug ) defined in (5) represents the most straightforward approach. This objective function is accessible also in already existing frameworks such as the BR or GrHMC methods and is implemented in this work to facilitate a quantitative comparison. In contrast, the loss functions that follow are only accessible in the neural network based reconstruction framework. This unique property is owed to the possibility that a neural network can be trained in advance on a dataset of known input and output pairs. As pointed out in Section II C, a loss function can e.g. be defined directly on a discretised representation of the spectral function ρ. This approach is implemented through L ρ (ρ sug ), see (6). The optimisation of the parameters
θ = {A i , M i , Γ i | 0 ≤ i < N BW } of
our chosen basis is an even more direct approach. In principle, the space of all possible choices of parameters is R 3·N BW + , assuming they are all positive definite. Of course, only finite subvolumes of this space ever need to be considered as target spaces for reconstruction methods. Therefore, we will often refer to a finite target volume simply as the parameter space for a specific setting. The respective loss function defined in this space is given by:
L θ (θ sug ) = θ sug − θ .(9)
All losses are evaluated using the 2-norm. In the case of the parameter net, we have ρ sug ≡ ρ(θ sug ). Apart from the three given loss functions, we also investigate a combination of the propagator and the spectral function loss,
L G,ρ (ρ sug , α) = L ρ (ρ sug ) + αL G (ρ sug ) .(10)
The type of loss function that is employed as well as the selection of the training data have major impact on the resulting performance of the neural network. Given this observation, it seems likely that a further optimisation regarding the choice of the loss function can significantly enhance the prediction quality. However, for the time being, we content ourselves with the types given above and postpone the exploration of more suitable training objectives to future work. In the following section, we continue with a thorough assessment of the performance of the neural network based approach and compare numerical results with the existing methods introduced in Section II B.
IV. NUMERICAL RESULTS
In this section we present numerical results for the neural network based reconstruction and validate the discussed potential advantages by comparing to existing
Reliable reconstructions
FIG. 3. This figure serves to illustrate the impact which the details of the training procedure and of the inverse problem itself have on the quality of the reconstructions. The term reliable reconstruction refers to a homogeneous distribution of losses within the parameter space. This involves a reliable error estimation on the given reconstruction and ensures locality of proposed solutions. In essence, we want to emphasise the importance of realising that aiming for reliable reconstructions is a complicated, multifactorial problem whose facets need to be sufficiently disentangled in order to understand all contributions to systematic errors.
methods. Details on the training procedure as well as the training and test datasets can be found in Appendix C, together with an introduction to the used performance measures. We start now with a brief summary of the main findings for our approach. Furthermore, a detailed numerical analysis and discussion of different network setups w.r.t performance differences are provided. Subsequently, additional post-processing methods for an improvement of the neural network predictions are covered. The section ends with a discussion of results from the PoNet. Readers who are interested in a comparison of the neural network based reconstruction to existing methods may proceed directly with Section IV B.
A. Reconstruction with neural networks
Our findings concerning the optimal setup of a feedforward network can be summarised as follows. As pointed out in Section II C, the network aims to learn an approximate parametrisation of a manifold of (matrix) inverses of the discretised Källén-Lehmann transformations. The inverse problem grows more severe if the propagator values are afflicted with noise. In Bayesian terms, this is caused by a wider posterior distribution for larger noise. The network needs to have sufficient expressivity, i.e. an adequate number of hyperparameters, to be able to learn a large enough set of inverse transformations. We assume that for larger noise widths a smaller number of hyperparameters is necessary to learn satisfactory The results on the left hand side imply that for larger errors, the choice of a specific network architecture has negligible impact on the quality of the reconstructions. All performance measures can be lowered for the given noise widths by applying a post-processing procedure on the suggested parameters of the network. In particular, the propagator loss can be minimised. The comparison on the right hand side shows that the choice of the loss function has major impact on the resulting performance of the network. The results underpin the importance of an appropriate loss function and support our argument of potential advantages of neural networks compared to existing approaches in Section II C. Contour plots in parameter space are illustrated for the respective measures in Figure 12 and Figure 13.
transformations, since the available information content about the actual inverse transformation decreases for a respective exact reconstruction. A varying severity of the inverse problem within the parameter space leads to an optimisation of the spectral reconstruction in regions where the problem is less severe. This effect occurs naturally, since there the network can minimise the loss more easily than in regions where the problem is more severe. Besides the severity of the inverse problem, the form of the loss function has a large impact on global optima within the landscape of the solution space. Based on these observations, an appropriate training of the network is non-trivial and demands a careful numerical analysis of the inverse problem itself, and of different setups of the optimisation procedure. A sensible definition of the loss function or a non-uniform selection of the training data are possible approaches to address the disparity in the severity of the inverse problem. A more straightforward approach is to iteratively reduce the covered parameter ranges within the learning process, based on previous suggested outcomes. This amounts to successively increasing the prediction accuracy by restricting the network to smaller and smaller subvolumes of the original solution space. However, one should be aware that this approach is only sensible if the reconstructions for different noise samples on the original propagator data are sufficiently close to each other in the solution space. A successive optimisation of the prediction accuracy in such a way can also be applied to existing methods. All approaches ultimately aim at a more homogeneous reconstruction loss within the solution space. This allows for a reliable control of systematic errors, as well as an accurate estimation of statistical errors. The desired outcome for a generic set of Breit-Wigner parameters is illustrated and discussed in Figure 1. The essence of our discussion here is summarised pictorially in Figure 3.
The impact of the net architecture and the loss function on the overall performance within the parameter space is illustrated in Figure 4. Associated contour plots can be found in Figure 12 and Figure 13. These plots demonstrate that the minima in the loss landscape highly depend on the employed loss function. In turn, this leads to different performance measures. This observation confirms our previous discussion and the necessity of an appropriate definition of the loss function. It also reinforces our arguments regarding potential advantages of neural Figure 1 for different volumes of the parameter space, again using a noise width of 10 −3 . The plots demonstrate how the quality of the reconstruction improves if the parameter space which the network has to learn is decreased. The volumes of the corresponding parameter spaces are listed in Table I. The results are computed from the Conv PaNet. The systematic deviation of the distribution of reconstructions for large volumes shows that the network has not captured the manifold of inverse transformations completely for the entire parameter space. This is in concordance with the results discussed in Figure 12 and Figure 14. Table I. The results demonstrate the potential advantage of an iterative restriction of the parameter ranges of possible solutions. The contour plots in Figure 14 depict changes of the performance measures within the parameter space. More strongly peaked prior distributions lead to better reconstructions. The comparison with results of the GrHMC approach illustrates the improvement of the performance of neural networks for larger errors and smaller volumes. These observations confirm the discussions of Figure 5 and Figure 8. Adding a post-processing step leads in particular for the propagator loss and for smaller noise widths to an improvement of the reconstruction, as has also been discussed in Figure 4. networks in comparison to other approaches for spectral reconstruction. The comparison of different feed-forward network architectures shows that the specific details of the network structure are rather irrelevant, provided that the expressivity is sufficient.
Differences in the performance of the networks that are trained with the same loss function become less visible for larger noise. This is illustrated by a comparison of contour plots with different noise widths, see e.g. Figure 12. The severity of the inverse problem grows with the noise and the information content about the actual matrix transformation decreases. These properties lead to the observation of a generally worse performance for larger noise widths, as can be inferred from Figure 5, Figure 8 and Figure 9, for example. They also imply that for specific noise widths, the neural network possesses enough hyperparameters to learn a sufficient parametrisation of the inverse transformation manifold. Furthermore, the local optima into which the network converges are mainly determined by differences in the local severity of the inverse problem. Hence, the issue remains that generic loss functions are inappropriate to address the varying local severity of the inverse problem. This issue implies the existence of systematic errors for particular regions within the parameter space, as can be seen e.g. in the left plot of Figure 5.
The results shown in Figure 5, Figure 6 and Figure 14 confirm our discussion regarding the expressive power of the network w.r.t. the complexity of the solution space and the decreasing information content for larger errors. The parameter space is gradually reduced, effectively increasing the expressivity of the network relative to the severity of the problem and improving the behavior of the loss function for a given fixed parameter space. The respective volumes are listed in Table I. Shrinking the parameter space leads to a more homogeneous loss landscape due to the increased locality, thereby mitigating the issue of inappropriate loss functions. The necessary number of hyperparameters decreases for larger noise widths and smaller parameter ranges in the training and test dataset. The arguments above imply a better performance of the network for smaller parameter spaces. A reduction of the parameter space effectively corresponds effectively to a sharpening of the prior information, which also has positive effects on the spread of the posterior distribution. More detailed discussions on the impact of different elements of the training procedure can be found in the captions of the respective figures.
Since increasing the expressivity of the network is limited by the computational demand required for the training, one can also apply post-processing methods to improve the suggested outcome w.r.t. the initially given, noisy propagator. These methods are motivated by the in some cases large observed root-mean-square deviation of the reconstructed suggested propagator to the input, see for example Figure 4. The application of standard optimisation methods on the suggested results of the network represents one possible approach to address this problem.
Here, the network's prediction is interpreted as a guess of the MAP estimate, which is presumed to be close to the true solution. For the PaNet, we minimise the propagator loss a posteriori with respect to the following loss function:
min θsug L PP [θ sug ] = min θsug G noisy − G [ρ (θ sug )] .(11)
This ensures that suggestions for the reconstructed spectral functions are in concordance with the given input propagator. Results obtained with an additional postprocessing are marked by the attachment PP in this work. The numerical results in Figure 9 and Figure 5 show that the finite size of the neural network can be partially compensated for small errors. The resulting low propagator losses are noteworthy, and are close to state-of-the-art spectral reconstruction approaches. One reason for this similarity is the shared underlying objective function. However, the situation is different for larger noise widths. For our choice of hyperparameters, the algorithm quickly converges into a local minimum. For large noise widths, the optimisation procedure may even lead to worse results than the initially suggested reconstruction. This is due to the already mentioned systematic deviations which are caused by the inappropriate choice of the loss function for large parameter spaces. This kind of post-processing should therefore be applied with caution, since it may cancel out the potential advantages of neural networks w.r.t. the freedom in the definition of the loss function.
The following alternative post-processing approach preserves the potential advantages of neural networks while nevertheless minimising the propagator loss. The idea is to include the network into the optimisation process through the following objective:
min Ginput L input [G input ] = min Ginput G noisy − G [ρ (θ sug )] ,(12)
where G input corresponds to the input propagator of the neural network and θ sug to the associated outcome. This facilitates a compensation of badly distributed noise samples and allows a more accurate error estimation. The approach is only sensible if no systematic errors exist for reconstructions within the parameter space, and if the network's suggestions are already somewhat reliable. We postpone a numerical analysis of this optimisation method together with the exploration of more appropriate loss functions and improved training strategies to future work, due to a currently lacking setup to train such a network.
In Figure 7 and Figure 15, results of the PoNet and the PaNet are compared. We observe that spectral reconstructions based on the PoNet structure suffer from similar problems as the PaNet, cf. again Figure 3. The point-like representation of the spectral function introduces a large number of degrees of freedom for the solution space. The training procedure implicitly regularises It can be seen that the reconstructed spectral function of the neural network exhibits in particular for larger errors a lower deviation to the original spectral function than the GrHMC method. This mirrors the in general observable better performance of the neural network for larger errors, as can be seen in Figure 6 and in Figure 9. The green and the red curve correspond to reconstructions of the Conv PaNet and the GrHMC method for the same given noisy propagator. The prior is in both cases given by the parameter range of volume Vol B. The uncertainty of the reconstructions for the neural network is depicted by the grey shaded areas as described in Figure 1.
For small errors, this area is covered by the corresponding reconstructed spectral functions. this problem, however, a visual analysis of individual reconstructions shows that in some cases the network struggles with common issues known from existing methods, such as partly non-Breit-Wigner like structures and wiggles. An application of the proposed post-processing methods serves as a possible approach to circumvent such problems. An inclusion of further regulator terms into the loss function, concerning e.g. the smoothness of the reconstruction, is also possible.
B. Benchmarking and discussion
In this section, we want to emphasise differences of our proposed neural network approach to existing methods. Our arguments are supported by an in-depth numerical comparison.
Within all approaches the aim is to map out, or at least to find the maximum of, the posterior distribution P (ρ|G) for a given noisy propagator G. The BR and GrHMC methods represent iterative approaches to accomplish this goal. The algorithms are designed to find the maximum for each propagator on a case-by-case basis. The GrHMC method additionally provides the possibility to implement constraints on the functional basis of the spectral function in a straightforward manner. In contrast, a neural network aims to learn the full manifold of inverse Källen-Lehman transformations for any noisy propagator (at least within the chosen parameter space). In this sense, it needs to propose for each given propagator an estimate of the maximum of P (ρ|D). A complex parametrisation, as given by the network, an exhaustive training dataset and the optimisation procedure itself are essential features of this approach for tackling this tough challenge. The computational effort to find a solution in an iterative approach is therefore shifted to LG with respect to the parameters. A reconstruction resulting in an averaged peak with the other parameter set effectively removed, as outlined in [26], results in the spiking parameter losses for the GrHMC reconstructions with large errors.
the training process as well as the memory demand of the network. Accordingly, the neural network based reconstruction can be performed much faster after training has been completed, which is in particular advantageous when large sets of input propagators are considered.
The numerical results in Figure 6, Figure 8, Figure 9, Figure 10 and Figure 11 demonstrate that the formal arguments of Section II C apply, particularly for comparably large noise widths as well as smaller parameter ranges. For both cases, the network successfully approximates the required inverse transformation manifold. Smaller noise widths and a larger set of possible spectral functions can be addressed by increasing the number of hyperparameters and through the exploration of more appropriate loss functions, as was already discussed previously.
V. CONCLUSION
In this study we have explored artificial neural networks as a tool to deal with the ill-conditioned inverse problem of reconstructing spectral functions from noisy Euclidean propagator data. We systematically investigated the performance of this approach on physically motivated mock data and compared our results with existing methods. Our findings demonstrate the importance of understanding the implications of the inverse problem itself on the optimisation procedure as well as on the resulting predictions.
The crucial advantage of the presented ansatz is the superior flexibility in the choice of the objective function. As a result, it can outperform state-of-the-art methods if the network is trained appropriately and exhibits sufficient expressivity to approximate the inverse transformation manifold. The numerical results demonstrate that defining an appropriate loss function grows increasingly important for an increased variability of considered spectral functions and of the severity of the inverse problem.
In future work, we aim to further exploit the advantage of neural networks that local variations in the severity of the inverse problem can be systematically compensated. The goal is to eliminate systematic errors in the predictions in order to facilitate a reliable reconstruction with an accurate error estimation. This can be realised by finding more appropriate loss functions with the help of implicit and explicit approaches [31,32]. A utilisation of these loss functions in existing methods is also possible if they are directly accessible. Varying the prior distribution will also be investigated, by sampling non-uniformly over the parameter space during the creation of the training data. Furthermore, we aim at a better understanding of the posterior distribution through the application of invertible neural networks [23]. This novel architecture provides a reliable estimation of errors by mapping out the entire posterior distribution by construction.
In conclusion, we believe that the suggested improvements will boost the performance of the proposed method to an as of yet unprecedented level and that neural networks will eventually replace existing state-of-the-art methods for spectral reconstruction.
ACKNOWLEDGMENTS
We thank Ion-Olimpiu Stamatescu and the ITP machine learning group for discussions and work on related topics. M. Scherzer acknowledges financial support from DFG under STA 283/16-2. F.P.G. Ziegler is supported by the FAIR OCD project. The work is supported by EMMI, the BMBF grant 05P18VHFCA, and is part of and supported by the DFG Collaborative Research Centre "SFB 1225 (ISOQUANT)" as well as by Deutsche Forschungsgemeinschaft (DFG) under Germany's Excellence Strategy EXC-2181/1 -390900948 (the Heidelberg Excellence Cluster STRUCTURES). In contrast to the previous plots, the neural network and the GrHMC method now use different priors for each case in order to allow for a reasonable comparison with the BR method, see Table II. We observe that all approaches qualitatively capture the features in the spectral function. Due to the comparably large error on the input data, all methods are expected to face difficulties in finding an accurate solution. The reconstructions of the neural network approach and the GrHMC method are comparable, whereas the BR method struggles in particular with thin peaks and the three Breit-Wigner case. The results demonstrate that, generally, using suitable basis functions and incorporating prior information lead to a superior reconstruction performance.
Appendix A: BR method
Different Bayesian methods propose different prior probabilities, i.e. they encode different types of prior information. The well known Maximum Entropy Method e.g. features the Shannon-Jaynes entropy
S SJ = dω ρ(ω) − m(ω) − ρ(ω)log ρ(ω) m(ω) ,(A1)
while the more recent BR method uses a variant of the gamma distribution
S BR = dω 1 − ρ(ω) m(ω) + log ρ(ω) m(ω) .(A2)
Both methods e.g. encode the fact that physical spectral functions are necessarily positive definite but are otherwise based on different assumptions. As Bayesian methods they have in common that the prior information has to be encoded in the functional form of the regulator and the supplied default model m(ω). Note that discretising ρ by choosing a particular functional basis also introduces a selection of possible outcomes. The dependence of the most probable spectral function, given input data and prior information, on the choice of S, m(ω) and the discretised basis comprises the systematic uncertainty of the method.
One major limitation to Bayesian approaches is the need to formulate our prior knowledge in the form of an optimisation functional. The reason is that while many of the correlation functions relevant in theoretical physics have very well defined analytic properties it has not been possible to formulate these as a closed regulator functional S. Take the retarded propagator for example (for a more comprehensive discussion see [26]). Its analytic structure in the imaginary frequency plane splits into two parts, an analytic half-plane, where the Euclidean input data is located, and a meromorphic half-plane which contains all structures contributing to the real-time dynamics. Encoding this information in an appropriate regulator functional has not yet been achieved.
Instead the MEM and the BR method rather use concepts unspecific to the analytic structure, such as smoothness, to derive their regulators. Among others this e.g. manifests itself in the presence of artificial ringing, which is related to unphysical poles contributing to the real-time propagator, which however should be suppressed by a regulator functional aware of the physical analytic properties.
Appendix B: GrHMC method
The main idea of the setup is already stated in the main text in Section II and was first introduced in [26]. Nevertheless, for completeness we outline the entire reconstruction process here. The approach is based on formulating the basis expansion in terms of the retarded propagator. The resulting set of basis coefficients are then determined via Bayesian inference. This leaves us with two objects to specify in the reconstruction process, the choice of a basis/ansatz for the retarded propagator and suitable priors for the inference.
Once a basis has been chosen it is straightforward to write down the corresponding regression model. As in the reconstruction with neural nets we use a fixed number of Breit-Wigner structures, c.f. (7), corresponding to simple poles in the analytically continued retarded propagator. The logarithm of all parameters is used in the model in order to enforce positivity of all parameters.
The uniqueness of the parameters is ensured by using an ordered representation of the logarithmic mass parameters.
The other crucial point is the choice of priors, which are of great importance to tame the ill-conditioning practically and should therefore be chosen as restrictive as possible. For comparability to the neural net reconstruction, the priors are matched to the training volume in parameter space. However, it is more convenient to work with a continuous distribution. Hence the priors of the logarithmic parameters are chosen as normal distributions where we have fixed the parameters by the condition that the mean of the distribution is the mean of the training volume and the probability at the boundaries of the trainings volume is equal. Details on the training volume in parameters space can be found in Appendix C.
All calculations for the GrHMC method are carried out using the python interface [33] of Stan [34].
Appendix C: Mock data, training set and training procedure
We consider three different levels of difficulty for the reconstruction of spectral functions to analyse and compare the performance of the approaches in this work. These levels differ by the number of Breit-Wigners that need to be extracted based on the given information of the propagator. We distinguish between training and test sets with one, two and three Breit-Wigners. A variable number of Breit-Wigners within a test set entails the task to determine the correct number of present structures. This can be done a priori or a posteriori based either on the propagator or on the quality of the reconstruction. We postpone this problem to future work.
The training set is constructed by sampling parameters uniformly within a given range for each parameter. The ranges for the parameters of a Breit-Wigner function of ( (7)) are as follows: M ∈ [0.5, 3.0], Γ ∈ [0.1, 0.4] and A ∈ [0.1, 1.0]. In addition, we investigate the impact of the size of the parameter space on the performance of the network for the case of two Breit-Wigner functions. This is done by decreasing the ranges of the parameters Γ and A gradually. We proceed differently for the two masses to guarantee a certain finite distance between the two Breit-Wigner peaks. Instead of decreasing the mass range, the minimum and maximum distance of the peaks is restricted. Details on the different parameter spaces can be found in Table I. The propagator function is parametrised by N p = 100 data points that are evaluated on a uniform grid within the interval ω ∈ [0, 10]. For a training of the point net, the spectral function is discretised by N ω = 500 data points on the same interval. Details about the training procedure can be found at the end of the section. The parameter ranges deviate for the comparison of the neural network approach with existing methods. The corresponding ranges are listed in Table II. I. Parameter ranges of the different volumes in parameter space. Two Breit-Wigner functions are sampled uniformly based on these parameters for the training and test sets. The difference ∆M = M2 − M1 is limited to restrict the minimum possible distance between two peaks. The volumes V θ in Figure 6 are computed based on these parameter ranges. Figure 11.
BR Comparison
The different approaches are compared by a test set for each number of Breit-Wigners consisting of 1000 random samples within the parameter space. Another test set is constructed for two Breit-Wigners with a fixed scaling A 1 = A 2 = 0.5, a fixed mass M 1 = 1 and equally chosen widths Γ := Γ 1 = Γ 2 . The mass M 2 and the width Γ are varied according to a regular grid in parameter space. This test set allows the analysis of contour plots of different loss measures. It provides more insights into the minima of the loss functions of the trained networks and into the severity of the inverse problem. The contour plots are averaged over 10 samples for the noise width of 10 −3 (except for Figure 10).
We investigate three different performance measures and different setups of the neural network for a comparison to existing methods. The root-mean-squaredeviation of the predicted parameters in parameter space, of the reconstructed spectral function and of the reconstructed propagator are considered. For the latter case, the error is computed based on the original propagator without noise. The measures are denoted as parameter loss, spectral function loss and propagator loss in this work. The spectral function loss and the propagator loss are computed based on the discretised representations on the uniform grid. Representative error bars for all methods are depicted in Figure 9.
The training procedure for the neural networks in this work is as follows. A neural network is trained separately for each training set, i.e., for each error and for each range of parameters. The learning rates are between 10 −5 and 10 −7 . The batch size is between 128 and 500 and the number of generated training samples per epoch is around 6 × 10 5 . Depending on the kind of network, the nets are trained for 80 to 160 epochs. The used loss functions are described at the end of Section III B. The implemented net architectures are provided in Table III. The upper three rows correspond to reconstructions of propagators with a noise width of 10 −5 , the lower ones with 10 −3 . The plots illustrate the loss measures in a hyperplane within the parameter space whose properties are described in Appendix C. The networks are trained with the parameter loss on the training set of volume Vol O. The contour plots show that the local minima are slightly different for small noise widths, whereas the global structures remain similar for all network architectures. These differences are caused by a slightly differing utilization of the limited number of hyperparameters. The differences between the network architectures become less visible for larger errors due to the growing severity of the inverse problem and a decreasing knowledge about the correct inverse transformations. Interestingly, the loss landscape of the convolutional neural network, which intrinsically operates on local structures, and of the fully connected networks are almost equal. The non-locality of the inverse integral transformation represents a possible reason for why the specific choice of the network structure is largely irrelevant. We conclude that the actual architecture is rather negligible in comparison to other attributes of the learning process, such as the selection of training data and the choice of the loss function. Figure 12, but with a comparison o different loss functions. The considered loss functions are introduced in Section III B. The results are based on the Conv PaNet that is trained on volume Vol O. The optima in the loss function differ and, consequently, lead to different mean squared errors for the different measures. It is interesting that the network with the pure propagator loss function leads to a rather homogeneous propagator loss distribution. In contrast, the networks with the pure parameter and the pure spectral function loss do not result in homogeneous distributions for their corresponding loss function. The large set of nearly equal propagators for different parameters explains this observation. It confirms also once more the necessity of approaches that can be trained using loss functions with access to more information than just the reconstructed propagator. 14. Analysis of prior information (parameter space of the training data) and of local differences in the severity of the inverse problem -The evolution of the landscape of different loss measures is shown for networks that are trained on different parameter spaces. All contour plots are based on the same section of the parameter space, namely the range that is spanned by volume D. The upper three and lower rows correspond again to reconstructions of propagators with noise widths 10 −5 and 10 −3 . The gradual reduction of the parameter space allows the analysis of different levels of complexity of the problem. A general improvement of performance can be observed besides a shift of the global optima. The more homogeneous loss landscape demonstrates that the problem of a different severity of the inverse problem is still present, but damped. Table. The size of the output layer is determined by the use of a parameter net or a point net and the considered number of Breit-Wigners. An attached PP indicates that a post-processing procedure is applied on the suggested parameters, for more details, see Section IV A.
FIG. 1. Examples of mock spectral functions reconstructed via our neural network approach for the cases of one, two and three Breit-Wigner peaks. The chosen functions mirror the desired locality of suggested reconstructions around the original function (red line). Additive, Gaussian noise of width 10 −3 is added to the discretised analytic form of the associated propagator of the same original spectral function multiple times. The shaded area depicts for each frequency ω the distribution of resulting outcomes, while the dashed green line corresponds to the mean. The results are obtained from the FC parameter network optimised with the parameter loss. The network is trained on the largest defined parameter space which corresponds to the volume Vol O. The uncertainty for reconstructions decreases for smaller volumes as illustrated in Figure 5. A detailed discussion on the properties and problems of a neural network based reconstruction is given in Section IV A.
FIG. 2 .
2Sketches of the (a) PaNet, shown here for the case of predicting the parameters A, M, Γ of a single Breit-Wigner peak, as well as the (b) PoNet (and by extension also the Po-VarNet). The specific network dimensions in this figure serve a purely illustrative purpose, explicit details on the employed architectures are given in Appendix C.
of different net architecures. All networks are trained based on the parameter loss. The associated architures can be found in Table III. of different loss functions. Details on the loss functions are described at the end of Section III B. All results are based on networks with the archticture Conv.FIG. 4. The performance of different net architectures and loss functions is compared for additive Gaussian noise with widths of 10 −3 and 10 −5 on the given input propagator. Shown here are the respective losses for the predicted parameters, for the discretised reconstructed spectral function and for the reconstructed propagator to the true, noise-free propagator. For both figures, the performance measures and the training of the neural networks are based on the training and test set of the largest volume in parameter space, Vol O. The definitions of the performance measures are given at the end of Appendix C.
FIG. 5 .
5The uncertainties of reconstructions of spectral functions on the same original propagator are illustrated in the same manner as described in
FIG. 6 .
6The plots in this figure quantify the impact of the parameter space volume used for the training on the performance of the network. The performance measures are computed based on the test set of the smallest volume, Vol D. The parameter ranges in the training set are gradually reduced to analyse different levels of complexity of the problem. A network is trained separately for each volume, which are listed in
FIG. 7 .
7Comparison of reconstruction errors of the PaNet and PoNet. The performance measures are computed based on the test set of the largest parameter space volume Vol O for one, two and three Breit-Wigners. The overall smaller losses for the point nets are due to the large number of degrees of freedom for the point-like representation of the spectral function. The partly competitive performance of the PoVarNet compared to the results of the PoNet encourage the further investigation of networks that are trained using a more exhaustive set of basis functions to describe physical structures in the spectral functions.
FIG. 8 .
8The quality of the reconstruction of two Breit-Wigner peaks is compared for different strength of additive noise on the same propagator. The labels indicate the noise width on the original propagator.
FIG. 9 .
9The performance of the reconstruction of spectral functions is benchmarked for the neural network approach with respect to results of the GrHMC method. The neural network approach is in particular for large noise widths competitive. The worse performance for smaller noise widths is a result of an inappropriate training procedure and a too low expressive power of the neural network. The problems are caused by a varying severity of the inverse problem and by a too large parameter space that needs to be covered by the neural network, as discussed in Section IV A. The error bars of the results for the FC network are representative for typical errors within all methods and plots of this kind.
FIG. 10 .
10Comparison of performance measures for the reconstruction of two Breit-Wigners with neural networks and with the GrHMC method for input propagators with noise width 10 −3 within the parameter space volume Vol O. The similar loss landscape emphasises the high impact of variations of the severity of the inverse problem within the parameter space on the quality of reconstructions. Contrary to expectations, the parameter network mimics, despite an optimisation based on the parameter loss L θ , the reconstruction of the GrHMC method which relies on an optimisation of the propagator loss
FIG. 11 .
11Reconstructions of one, two and three Breit-Wigners are compared for our proposed neural network approach, the GrHMC method and the BR method. The reconstructions of the first two methods are based on a single sample with noise width 10 −3 , while the results of the BR method are obtained from multiple samples with larger errors, but an average noise width of 10 −3 as well.
FIG. 12 .
12Comparison of network architectures -Contour plots of loss measures are shown for different net architectures.
FIG. 13 .
13Comparison of loss functions -Contour plots of loss measures are illustrated in the same manner as in
FIG. 14. Analysis of prior information (parameter space of the training data) and of local differences in the severity of the inverse problem -The evolution of the landscape of different loss measures is shown for networks that are trained on different parameter spaces. All contour plots are based on the same section of the parameter space, namely the range that is spanned by volume D. The upper three and lower rows correspond again to reconstructions of propagators with noise widths 10 −5 and 10 −3 . The gradual reduction of the parameter space allows the analysis of different levels of complexity of the problem. A general improvement of performance can be observed besides a shift of the global optima. The more homogeneous loss landscape demonstrates that the problem of a different severity of the inverse problem is still present, but damped.
FIG. 15 .
15Comparison of the parameter net and the point net -Root-mean-squared-deviations are compared between the parameter net and the point net, trained on two Breit-Wigner like structures (PoNet) and trained on a variable number of Breit-Wigners (PoNetVar), with respect to different loss functions. The two upper rows correspond to results from input propagators with a noise width of 10 −5 and the two lower ones with a noise width of 10 −3 . Problems concerning a varying severity of the inverse problem and concerning an information loss caused by the additive noise remain independent of the chosen basis for the representation of the spectral function. Name CenterModule Number of parameters FC FC(6700) ⇒ ReLU ⇒ FC(12168) ⇒ ReLU ⇒ FC(1024) 95 × 10 6 Deep FC FC(512) ⇒ ReLU ⇒ FC(1024) ⇒ ReLU ⇒ (FC(4056) ⇒ ReLU) 3 ⇒ (FC(2056) ⇒ ReLU) 2 50 × 10 6 Narrow Deep FC FC(512) ⇒ ReLU ⇒ (FC(1024) ⇒ ReLU) 3 ⇒ (FC(2056) ⇒ ReLU) 5 ⇒ (FC(1024) ⇒ ReLU) 3 ⇒ FC(512) ⇒ ReLU ⇒ FC(256) 96 × 10 6 Straight FC (FC(4112) ⇒ BatchNorm1D ⇒ ReLU ⇒ Dropout(0.2)) 7 102 × 10 6 Conv Conv(64, 10) ⇒ ReLU ⇒ Conv(256, 10) ⇒ ReLU ⇒ (FC(4096) ⇒ ReLU) 2 ⇒ FC(1024) 41 × 10 6 TABLE III. Details on the implemented network architectures. The general setup is: Input(100) ⇒ ReLU ⇒ CenterModule ⇒ ReLU ⇒ FC(3/6/9/500) ⇒ Output, whereas the CenterModule is given along with the associated name in the
475, 0.525] [0.5, 3.0] [0.1125, 0.1375] [0.875, 1.125]Vol
A
M
Γ
∆M
O
[0.1, 1.0] [0.5, 3.0]
[0.1, 0.4]
[0.0, 2.5]
A
[0.3, 0.7] [0.5, 3.0]
[0.1, 0.3]
[0.25, 1.75]
B
[0.4, 0.6] [0.5, 3.0]
[0.1, 0.2]
[0.5, 1.5]
C [0.45, 0.55] [0.5, 3.0]
[0.1, 0.15]
[0.75, 1.25]
D [0.
TABLE
TABLE II. Parameter ranges for the training of the neural networks for the comparison inA
M
Γ
1BW
[0.1, 1.0] [0.5, 3.0] [0.1, 0.4]
2BWa
[0.2, 1.8] [0.8, 3.8] [0.2, 1.0]
2BWb
[0.3, 1.2] [0.8, 3.8] [0.002, 0.02]
3BW
[0.2, 1.8] [1.0, 6.0] [0.2, 1.0]
. D Guest, K Cranmer, D Whiteson, 10.1146/annurev-nucl-101917-021019arXiv:1806.11484Ann. Rev. Nucl. Part. Sci. 68161hepexD. Guest, K. Cranmer, and D. Whiteson, Ann. Rev. Nucl. Part. Sci. 68, 161 (2018), arXiv:1806.11484 [hep- ex].
. A Radovic, M Williams, D Rousseau, M Kagan, D Bonacorsi, A Himmel, A Aurisano, K Terao, T Wongjirad, 10.1038/s41586-018-0361-2Nature. 56041A. Radovic, M. Williams, D. Rousseau, M. Kagan, D. Bonacorsi, A. Himmel, A. Aurisano, K. Terao, and T. Wongjirad, Nature 560, 41 (2018).
. Y Lecun, Y Bengio, G Hinton, 10.1038/nature14539Nature. 521436Y. LeCun, Y. Bengio, and G. Hinton, Nature 521, 436 (2015).
. J Schmidhuber, 10.1016/j.neunet.2014.09.003arXiv:1404.7828J. Schmidhuber, (2014), 10.1016/j.neunet.2014.09.003, arXiv:1404.7828.
. J Carrasquilla, R G Melko, 10.1038/nphys4035Nature Physics. 13431J. Carrasquilla and R. G. Melko, Nature Physics 13, 431 (2017).
. P E Shanahan, D Trewartha, W Detmold, 10.1103/PhysRevD.97.094506Phys. Rev. D. 9794506P. E. Shanahan, D. Trewartha, and W. Detmold, Phys. Rev. D 97, 094506 (2018).
. G Carleo, M Troyer, 10.1126/science.aag2302Science. 355602G. Carleo and M. Troyer, Science 355, 602 (2017).
. S J Wetzel, 10.1103/PhysRevE.96.022140Phys. Rev. E. 9622140S. J. Wetzel, Phys. Rev. E 96, 022140 (2017).
. S J Wetzel, M Scherzer, 10.1103/PhysRevB.96.184410Phys. Rev. B. 96184410S. J. Wetzel and M. Scherzer, Phys. Rev. B 96, 184410 (2017).
. L Wang, 10.1103/PhysRevB.94.195105Phys. Rev. B. 94195105L. Wang, Phys. Rev. B 94, 195105 (2016).
. W Hu, R R P Singh, R T Scalettar, 10.1103/PhysRevE.95.062122Phys. Rev. E. 9562122W. Hu, R. R. P. Singh, and R. T. Scalettar, Phys. Rev. E 95, 062122 (2017).
. L Huang, L Wang, 10.1103/PhysRevB.95.035105Phys. Rev. B. 9535105L. Huang and L. Wang, Phys. Rev. B 95, 035105 (2017).
. J Liu, Y Qi, Z Y Meng, L Fu, 10.1103/PhysRevB.95.041101Phys. Rev. B. 9541101J. Liu, Y. Qi, Z. Y. Meng, and L. Fu, Phys. Rev. B 95, 041101 (2017).
. J Karpie, K Orginos, A Rothkopf, S Zafeiropoulos, 10.1007/JHEP04(2019)057arXiv:1901.05408JHEP. 0457hep-latJ. Karpie, K. Orginos, A. Rothkopf, and S. Zafeiropou- los, JHEP 04, 057 (2019), arXiv:1901.05408 [hep-lat].
. J M Urban, J M Pawlowski, arXiv:1811.03533J. M. Urban and J. M. Pawlowski, arXiv:1811.03533.
. M Jarrell, J E Gubernatis, 10.1016/0370-1573(95)00074-7Phys. Rept. 269133M. Jarrell and J. E. Gubernatis, Phys. Rept. 269, 133 (1996).
. M Asakawa, T Hatsuda, Y Nakahara, 10.1016/S0146-6410(01)00150-8arXiv:hep-lat/0011040Prog. Part. Nucl. Phys. 46459heplatM. Asakawa, T. Hatsuda, and Y. Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001), arXiv:hep-lat/0011040 [hep- lat].
. Y Burnier, A Rothkopf, 10.1103/PhysRevLett.111.182003arXiv:1307.6106Phys. Rev. Lett. 111182003hep-latY. Burnier and A. Rothkopf, Phys. Rev. Lett. 111, 182003 (2013), arXiv:1307.6106 [hep-lat].
A Rothkopf, 13th Conference on Quark Confinement and the Hadron Spectrum (Confinement XIII) Maynooth. A. Rothkopf, in 13th Conference on Quark Confinement and the Hadron Spectrum (Confinement XIII) Maynooth,
. Ireland, arXiv:1903.02293hep-phIreland, July 31-August 6, 2018 (2019) arXiv:1903.02293 [hep-ph].
. V Shah, C Hegde, arXiv:1802.08406V. Shah and C. Hegde, arXiv:1802.08406.
. H Li, J Schwab, S Antholzer, M Haltmeier, arXiv:1803.00092H. Li, J. Schwab, S. Antholzer, and M. Haltmeier, arXiv:1803.00092.
. R Anirudh, J J Thiagarajan, B Kailkhura, T Bremer, arXiv:1805.07281R. Anirudh, J. J. Thiagarajan, B. Kailkhura, and T. Bre- mer, arXiv:1805.07281.
. L Ardizzone, J Kruse, S Wirkert, D Rahner, E W Pellegrini, R S Klessen, L Maier-Hein, C Rother, U Köthe, arXiv:1808.04730cs.LGL. Ardizzone, J. Kruse, S. Wirkert, D. Rahner, E. W. Pellegrini, R. S. Klessen, L. Maier-Hein, C. Rother, and U. Köthe, arXiv:1808.04730 [cs.LG].
. R Fournier, L Wang, O V Yazyev, Q Wu, arXiv:1810.00913physics.comp-phR. Fournier, L. Wang, O. V. Yazyev, and Q. Wu, arXiv:1810.00913 [physics.comp-ph].
. H Yoon, J.-H Sim, M J Han, 10.1103/PhysRevB.98.245101arXiv:1806.03841Physical Review B. 98245101cond-mat.str-elH. Yoon, J.-H. Sim, and M. J. Han, Physical Review B 98, 245101 (2018), arXiv:1806.03841 [cond-mat.str-el].
. A K Cyrol, J M Pawlowski, A Rothkopf, N Wink, 10.21468/SciPostPhys.5.6.065arXiv:1804.00945SciPost Phys. 565hep-phA. K. Cyrol, J. M. Pawlowski, A. Rothkopf, and N. Wink, SciPost Phys. 5, 065 (2018), arXiv:1804.00945 [hep-ph].
. R Oehme, W Zimmermann, 10.1103/PhysRevD.21.1661Phys. Rev. 211661R. Oehme and W. Zimmermann, Phys. Rev. D21, 1661 (1980).
. R Oehme, 10.1016/0370-2693(90)90499-VPhys. Lett. 252641R. Oehme, Phys. Lett. B252, 641 (1990).
. G Cuniberti, E De Micheli, G A Viano, 10.1007/s002200000324arXiv:cond-mat/0109175Commun. Math. Phys. 21659cond-mat.str-elG. Cuniberti, E. De Micheli, and G. A. Viano, Commun. Math. Phys. 216, 59 (2001), arXiv:cond-mat/0109175 [cond-mat.str-el].
. Y Burnier, M Laine, L Mether, 10.1140/epjc/s10052-011-1619-0arXiv:1101.5534Eur. Phys. J. 711619hep-latY. Burnier, M. Laine, and L. Mether, Eur. Phys. J. C71, 1619 (2011), arXiv:1101.5534 [hep-lat].
. C Nogueira, K Santos, B Wadhawan, Zhou, 1707.02198 [cs.LGC. Nogueira dos Santos, K. Wadhawan, and B. Zhou, 1707.02198 [cs.LG].
. L Wu, F Tian, Y Xia, Y Fan, T Qin, J Lai, T.-Y Liu, arXiv:1810.12081cs.LGL. Wu, F. Tian, Y. Xia, Y. Fan, T. Qin, J. Lai, and T.-Y. Liu, arXiv:1810.12081 [cs.LG].
Pystan: the python interface to stan. Stan Development Team, version 2.17.1.0Stan Development Team, "Pystan: the python interface to stan, version 2.17.1.0," http://mc-stan.org (2018).
. B Carpenter, A Gelman, M D Hoffman, D Lee, B Goodrich, M Betancourt, M Brubaker, J Guo, P Li, A Riddell, Journal of statistical software. 76B. Carpenter, A. Gelman, M. D. Hoffman, D. Lee, B. Goodrich, M. Betancourt, M. Brubaker, J. Guo, P. Li, and A. Riddell, Journal of statistical software 76 (2017).
| [] |
[
"Automatic evaluation of human oocyte developmental potential from microscopy images",
"Automatic evaluation of human oocyte developmental potential from microscopy images"
] | [
"Denis Baručić \nDepartment of Cybernetics\nFaculty of Electrical Engineering\nCzech Technical University\nPragueCzech Republic\n",
"Jan Kybic \nDepartment of Cybernetics\nFaculty of Electrical Engineering\nCzech Technical University\nPragueCzech Republic\n",
"Olga Teplá \nDepartment of Obstetrics and Gynecology\nThe First Faculty of Medicine and General Teaching Hospital\nCzech Republic\n",
"Zinovij Topurko \nDepartment of Obstetrics and Gynecology\nThe First Faculty of Medicine and General Teaching Hospital\nCzech Republic\n",
"Irena Kratochvílová \nInstitute of Physics\nCzech Academy of Sciences\nCzech Republic\n"
] | [
"Department of Cybernetics\nFaculty of Electrical Engineering\nCzech Technical University\nPragueCzech Republic",
"Department of Cybernetics\nFaculty of Electrical Engineering\nCzech Technical University\nPragueCzech Republic",
"Department of Obstetrics and Gynecology\nThe First Faculty of Medicine and General Teaching Hospital\nCzech Republic",
"Department of Obstetrics and Gynecology\nThe First Faculty of Medicine and General Teaching Hospital\nCzech Republic",
"Institute of Physics\nCzech Academy of Sciences\nCzech Republic"
] | [] | Infertility is becoming an issue for an increasing number of couples. The most common solution, in vitro fertilization, requires embryologists to carefully examine light microscopy images of human oocytes to determine their developmental potential. We propose an automatic system to improve the speed, repeatability, and accuracy of this process. We first localize individual oocytes and identify their principal components using CNN (U-Net) segmentation. Next, we calculate several descriptors based on geometry and texture. The final step is an SVM classifier. Both the segmentation and classification training is based on expert annotations. The presented approach leads to a classification accuracy of 70%. | 10.1117/12.2604010 | [
"https://arxiv.org/pdf/2103.00302v2.pdf"
] | 232,076,210 | 2103.00302 | 2106e1f07133ff3d76acc47089493236d5a5aa28 |
Automatic evaluation of human oocyte developmental potential from microscopy images
Denis Baručić
Department of Cybernetics
Faculty of Electrical Engineering
Czech Technical University
PragueCzech Republic
Jan Kybic
Department of Cybernetics
Faculty of Electrical Engineering
Czech Technical University
PragueCzech Republic
Olga Teplá
Department of Obstetrics and Gynecology
The First Faculty of Medicine and General Teaching Hospital
Czech Republic
Zinovij Topurko
Department of Obstetrics and Gynecology
The First Faculty of Medicine and General Teaching Hospital
Czech Republic
Irena Kratochvílová
Institute of Physics
Czech Academy of Sciences
Czech Republic
Automatic evaluation of human oocyte developmental potential from microscopy images
human oocytesfertilizationmicroscopyclassificationsegmentation
Infertility is becoming an issue for an increasing number of couples. The most common solution, in vitro fertilization, requires embryologists to carefully examine light microscopy images of human oocytes to determine their developmental potential. We propose an automatic system to improve the speed, repeatability, and accuracy of this process. We first localize individual oocytes and identify their principal components using CNN (U-Net) segmentation. Next, we calculate several descriptors based on geometry and texture. The final step is an SVM classifier. Both the segmentation and classification training is based on expert annotations. The presented approach leads to a classification accuracy of 70%.
INTRODUCTION
Infertility has been an issue for several years and is expected to grow further. Nowadays, the most common solution for an infertile couple is in vitro fertilization (IVF). One of the crucial steps is choosing the best oocytes to be fertilized, since, for practical, ethical, and legal reasons, it is not feasible to fertilize more than a few of them -and even fewer can be implanted.
The situation is easier when a patient's own oocytes are used and are, therefore, more readily available. In this case, the embryologists attempt to fertilize almost all collected oocytes that are not apparently damaged. The knowledge gained from these attempts can be used later to determine the a priori developmental potential of newly collected oocytes, reducing both the cost and the failure rate of the IVF. This is especially important when the oocytes come from a donor. In this case, each oocyte is very valuable, and it is important to determine its developmental potential reliably. Currently, this is performed by an expert who carefully examines the oocytes under a microscope. The selection process requires extensive experience, is time-consuming, and is done outside the optimal environment, so it is desirable to shorten it as much as possible.
In this work, we aim to replace this subjective process with an automatic evaluation of the developmental ability of oocytes from digitized light microscopy images to improve its speed, repeatability, and accuracy. We present a proof-of-concept solution on a small dataset, showing the viability of this approach.
Previous work
To the best of our knowledge, the task of fully automatic oocyte developmental potential assessment has not been addressed before. The closest work to ours is Manna et al., 1 who feed LBP texture features to an ensemble of shallow neural networks and try to predict whether an oocyte or embryo (i.e., fertilized oocyte) will lead to birth. However, their approach requires manual segmentation, and obtaining the ground truth data takes much more time. Since multiple embryos are usually implanted, the outcome is only certain in the relatively few cases (namely 12 1 ) when all or none of the implanted embryos lead to birth. Furthermore, many embryos are not implanted but frozen, and the pregnancy outcome might not be known for years or not at all. For this reason, here we have chosen a different classification target and decided to predict an embryo's ability to start the correct development, which can be determined relatively quickly and for all embryos. Recently, deep learning has been used for oocyte segmentation. 2, 3 Other works analyzed microscopic images of embryos. 4,5 An extensive database of 50 000 embryo images allowed to train a model based on deep learning to classify the embryos into three quality classes. 4 Raudonis et al. 5 described a method for analyzing time-lapse sequences of embryo images.
Viswanath et al. 6 classified swine cumulus oocyte complexes (i.e., oocytes with the cumulus cells, unlike in our data). The authors examined the number of cumulus cell layers and the homogeneity of the cytoplasm. A semi-automatic (snake) method was used for segmentation and random forests for the classification of oocytes. Pure texture analysis turned out to be useful for cytoplasm clustering 7 or cytoplasm segmentation 8 .
Proposed approach
This paper describes a fully automatic approach to classify oocytes in light microscopy images (see Fig. 1) into two categories, viable and nonviable, where viable oocytes have a good potential of becoming well-developed embryos. We learn from subjective expert annotations of individual oocytes. Our approach consists of five consecutive stages -localization, extraction of individual oocytes, segmentation, feature extraction, and classification.
DATA
Our anonymized dataset consists of 34 grayscale images of groups of oocytes after cumulus cell denudation. Each image contains 1 ∼ 7 oocytes. The images of 1392 × 1040 pixels were acquired using Nikon Diaphot 300 inverted microscope, Eppendorf (Hamburg, Germany) micromanipulation system equipped with a thermoplate (Tokai Hit, Japan). The acquisition of the images was approved by the Ethics Committee of the General University Hospital, Prague, Czech Republic, on October 12, 2018, reference number 79/17. The ground truth (GT) segmentations (see Fig. 1) were created using the GIMP graphical editor. We considered four classes: background, cytoplasm, zona pellucida, polar body, and cumulus cells. The individual oocytes were classified as viable or nonviable by the embryology expert (OT). Skipping incompletely visible oocytes yielded 50 viable and 53 nonviable oocytes. Furthermore, from observations of the development, we learned the true number of viable oocytes y j in each image j. This information is used to check the expert annotations in Sec. 4.5.
METHOD
Our method consists of five stages, explained in the following subsections. Fig. 2 gives a high-level overview.
Oocyte localization
We first perform a binary segmentation of cytoplasm versus the remaining classes because the cytoplasm is clearly distinguishable in the images. We use a U-Net 9 CNN with MobileNetV2 10 as the encoder, the standard mirroring decoder, and a pixel-wise softmax final layer. MobileNetV2 is a fast architecture with a relatively low number of parameters. The extra speed is desirable when the tool is deployed in a production environment. The network was trained for 500 epochs until convergence. We use the Dice loss function in combination with the ADAM optimizer (learning rate 10 −4 ). To prevent overfitting, multiple data augmentation methods (shifting and rotation, contrast and brightness adjustments, and blurring) are applied to the training images during training.
Connected foreground components smaller than 10 4 pixels are suppressed. The threshold was picked so that it does not rule out any viable oocytes, the area of which is always more than 4 · 10 4 pixels. Finally, regions of interest (ROI) of size 416 × 416 are extracted from around the centers of gravity of the remaining foreground components (see Fig. 3).
Oocyte segmentation
Once the ROIs are extracted, they are segmented into the five classes described in Sec. 2 using another CNN. Since the polar body or cumulus cells classes are challenging to segment, we use a U-Net with the powerful ResNet50 architecture 11 as the encoder, trained for 600 epochs, which was sufficient for convergence. The rest of the procedure is identical to Sec. 3.1. Example segmentations are shown in Fig. 4.
Feature extraction
Given the small number of available training images, we could not use deep learning for feature extraction. Instead, using the segmentation from the previous section (ROI) we compute for each oocyte the 24 features described below.
First, to handle the case where the ROI contains parts of several oocytes, we keep only the largest cytoplasm and zona pellucida components. We also suppress polar body components smaller than 500 pixels (for a bad oocyte, it is possible to have multiple polar body components, so we cannot just keep the largest).
Ellipses are fitted to the boundary of the cytoplasm class and to the outer boundary of the zona pellucida class by least squares fitting of the boundary pixels (see Fig. 5). We calculate the following features based on the cytoplasm ellipse:
• mean axis µ c = ac+bc 2 ,
• eccentricity e c = 1 −
a 2 c b 2 c , a c ≥ b c ,
• compactness γ c = acbcπ Sc , where a c , b c are the estimated ellipse semi-axes, and S c is the area of the cytoplasm component. The features µ z , e z , γ z are calculated similarly for the zona pellucida class. We also define the misalignment m = c c − c z , the Euclidean distance between the ellipse centers, and the ratio of the cytoplasm and zona pellucida areas, 7 r = Sc Sz . Regarding polar bodies, two features are used: the number of connected components, n pb , and the total area, S pb . The presence of cumulus cells is not related to the oocyte quality but may influence the other features, hence we also calculate the total area of cumulus cells S cc , which completes the 11 geometrical features.
The remaining 13 features describe the texture of the cytoplasm. 1 The first 10 texture features are calculated efficiently from a three-level undecimated Haar wavelet transform 12 and correspond to the energies in the low pass channel and 9 high frequency channels. The remaining three features are the mean, variance, and entropy of the pixel intensities in the cytoplasm.
Oocyte classification
For each oocyte, the extracted feature vector is normalized and fed into a binary classifier to produce a binary label, viable or nonviable. Several classifiers were tried with similar results. For the sake of space, only kernel SVM results are reported. The RBF SVM kernel with γ = 10 −2 and the cost parameter C = 1.0 were selected using stratified cross-validation and grid search on a training dataset of 83 randomly selected oocytes (ROIs). The remaining 20 ROIs were left for testing (Sec. 4.3). This yielded the mean validation accuracy A val ≈ 74%.
RESULTS
First, we evaluate the three stages of the pipeline separately. After that, we examine the significance of the designed features. Finally, we analyze the expert annotations.
Oocyte localization evaluation
Cross-validation was performed to evaluate the oocyte localization described in Sec. 3.1 using the 34 training images. For our data, the method worked perfectly. The number of detected oocytes was always equal to the ground truth and the localization error between the centers of gravity of the cytoplasm class, and the ground truth cytoplasm segmentation was inferior to 10 pixels (corresponding to approximately 5% of the oocyte diameter) in 98% of cases.
Oocyte segmentation evaluation
The five-class oocyte segmentation (Sec. 3.2) was evaluated using a 10-fold cross-validation on the 103 ROIs, each approximately centered on one oocyte.
The Intersection over Union (IoU) metric computed over the folds for the cytoplasm, zona pellucida, polar body, and cumulus cells classes was 95.48%, 89.72%, 42.85%, and 60.29%, respectively. While cytoplasm and zona pellucida are segmented very well, the polar body class suffers from false detections and is more difficult to segment due to its small size. Although the cumulus cells segmentation performance is also far from perfect, it is mostly confused with background, which is not critical for our task (see Fig. 4).
Oocyte classification evaluation
The performance of the SVM from Sec. 3.4 to classify the oocytes as viable or not was evaluated using the 20 testing ROIs omitted during the training.
The classifier obtained the testing accuracy A test = 70%, sensitivity Se test = 70%, specificity Sp test = 70%, precision P test = 70%, and the area under the ROC curve was AU C test = 0.69 (see Fig. 6).
Feature significance
To evaluate the features' significance, we ran a leave-one-out cross-validation on the whole dataset for different subsets of the features. Table 1 contains the results for four selected subsets. The most significant feature is the number of polar bodies n pb . A viable oocyte contains a single polar body (n pb = 1). However, in some images, the polar body may be hidden. Both the textural and geometrical features improve the classification accuracy, with a small additional improvement if using both. Table 1. Accuracy achieved using four feature subsets. Each subset contains the number of polar bodies n pb . The 13 texture features are denoted as "texture", and "geometry" denotes the remaining 10 features. The table shows the average accuracy computed using the leave-one-out procedure.
Expert annotation evaluation
The oocyte annotations created by the embryologist are used to train the classifier. Since these labels are subjective, we evaluate their expected accuracy using the reliably known number of viable oocytes y j as a reference.
Given image j, we denote n j the number of oocytes andŷ j the number of oocytes labeled as viable by the expert. The mean absolute error of the expert MAE = 1 N N j=1 |y j −ŷ j |, where N is the number of images, was 0.44 with approximately 3 oocytes per image on the average. Fig. 7 displays a histogram of the differences in the number of viable oocytes per image for the annotations given by the expert and for those predicted by the leave-one-out cross-validation of our automatic method. The histogram reveals only a slight difference between the two annotations. The Kolmogorov-Smirnov statistic computed for the two histograms is D = 0.11, which suggests no significant difference.
DISCUSSION AND CONCLUSIONS
In this paper, a proof-of-concept solution was proposed to automatically detect oocytes with good developmental potential and are therefore viable for fertilization. When interpreting the results, it is important to realize that the expert assessment of the oocyte quality from a single image is difficult and rather subjective. Our performance (AUC=0.69) is nevertheless comparable to the performance achieved by Manna et al. 1 (AUC=0.68), who worked with the more reliable information of whether an oocyte leads to birth. The errors in the number of viable oocytes per image are also comparable to that of the expert. We expect our accuracy to improve further when more data is available. This should enable us to employ deep learning techniques instead of hand-crafted features, boosting the performance. We are also working on learning directly from the number of viable oocytes.
Figure 1 .
1Example of an input image and the corresponding expert segmentation. The green and red frames denote viable and nonviable oocytes, respectively (left image). Classes background, cytoplasm, zona pellucida, polar body, and cumulus cells are denoted in black, blue, green, red, and yellow, respectively (right image).
Figure 2 .
2The classification pipeline. (1) Binary segmentation. (2) ROI extraction. (3) Five class segmentation. (4) Feature extraction. (5) Classification.
Figure 3 .
3Example of an input image (left) and two ROIs extracted from the image.
Figure 4 .
4Four extracted ROIs in the top row and the corresponding predicted segmentations (black, green, blue, red and yellow denote background, zona pellucida, cytoplasm, polar body and cumules cells, respectively) and expert segmentations (outlined).
Figure 5 .
5Fitted ellipses for cytoplasm (blue) and zona pellucida (green).
Figure 6 .
6ROC curve obtained for the SVM classifier on the testing data. The dot represents our operating point.
Figure 7 .
7Histogram of annotation errors committed by a human expert and an SVM.
ACKNOWLEDGMENTSThe authors acknowledge the support of the OP VVV funded project "CZ.02.1.01/0.0/0.0/16 019/0000765 Research Center for Informatics" and the Grant Agency of the Czech Technical University in Prague, grant No. SGS20/170/OHK3/3T/13.
Artificial intelligence techniques for embryo and oocyte classification. C Manna, Reproductive biomedicine online. 261Manna, C. et al., "Artificial intelligence techniques for embryo and oocyte classification," Reproductive biomedicine online 26(1), 42-49 (2013).
A robust deep learning-based multiclass segmentation method for analyzing human metaphase II oocyte images. S Firuzinia, S M Afzali, F Ghasemian, S A Mirroshandel, Computer Methods and Programs in Biomedicine. 201105946Firuzinia, S., Afzali, S. M., Ghasemian, F., and Mirroshandel, S. A., "A robust deep learning-based mul- ticlass segmentation method for analyzing human metaphase II oocyte images," Computer Methods and Programs in Biomedicine 201, 105946 (2021).
Semantic segmentation of human oocyte images using deep neural networks. A Targosz, P Przysta Lka, R Wiaderkiewicz, G Mrugacz, BioMedical Engineering OnLine. 201Targosz, A., Przysta lka, P., Wiaderkiewicz, R., and Mrugacz, G., "Semantic segmentation of human oocyte images using deep neural networks," BioMedical Engineering OnLine 20(1), 1-26 (2021).
Deep learning enables robust assessment and selection of human blastocysts after in vitro fertilization. P Khosravi, NPJ Digital Medicine. 21Khosravi, P. et al., "Deep learning enables robust assessment and selection of human blastocysts after in vitro fertilization," NPJ Digital Medicine 2(1), 1-9 (2019).
Towards the automation of early-stage human embryo development detection. V Raudonis, Biomedical Engineering OnLine. 181Raudonis, V. et al., "Towards the automation of early-stage human embryo development detection," Biomed- ical Engineering OnLine 18(1), 1-20 (2019).
Grading of mammalian cumulus oocyte complexes using machine learning for in vitro embryo culture. P S Viswanath, IEEE-EMBS International Conference on Biomedical and Health Informatics (BHI). IEEEViswanath, P. S. et al., "Grading of mammalian cumulus oocyte complexes using machine learning for in vitro embryo culture," in [2016 IEEE-EMBS International Conference on Biomedical and Health Informatics (BHI)], 172-175, IEEE (2016).
A texture-based image processing approach for the description of human oocyte cytoplasm. T M A Basile, IEEE Transactions on Instrumentation and Measurement. 5910Basile, T. M. A. et al., "A texture-based image processing approach for the description of human oocyte cytoplasm," IEEE Transactions on Instrumentation and Measurement 59(10), 2591-2601 (2010).
Cytoplasm image segmentation by spatial fuzzy clustering. L Caponetti, SpringerInternational Workshop on Fuzzy Logic and ApplicationsCaponetti, L. et al., "Cytoplasm image segmentation by spatial fuzzy clustering," in [International Workshop on Fuzzy Logic and Applications ], 253-260, Springer (2011).
U-Net: Convolutional networks for biomedical image segmentation. O Ronneberger, International Conference on Medical Image Computing and Computer-Assisted Intervention. SpringerRonneberger, O. et al., "U-Net: Convolutional networks for biomedical image segmentation," in [Interna- tional Conference on Medical Image Computing and Computer-Assisted Intervention ], 234-241, Springer (2015).
M Sandler, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionMobileNetV2: Inverted residuals and linear bottlenecksSandler, M. et al., "MobileNetV2: Inverted residuals and linear bottlenecks," in [Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition ], 4510-4520 (2018).
Deep residual learning for image recognition. K He, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionHe, K. et al., "Deep residual learning for image recognition," in [Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition], 770-778 (2016).
Texture classification and segmentation using wavelet frames. M Unser, IEEE Transactions on Image Processing. 411Unser, M., "Texture classification and segmentation using wavelet frames," IEEE Transactions on Image Processing 4(11), 1549-1560 (1995).
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"MNRAS"
] | Many focal-reducer spectrographs, currently available at state-of-the art telescopes facilities, would benefit from a simple refurbishing that could increase both the resolution and spectral range in order to cope with the progressively challenging scientific requirements but, in order to make this update appealing, it should minimize the changes in the existing structure of the instrument. In the past, many authors proposed solutions based on stacking subsequently layers of dispersive elements and record multiple spectra in one shot (multiplexing). Although this idea is promising, it brings several drawbacks and complexities that prevent the straightforward integration of a such device in a spectrograph. Fortunately nowadays, the situation has changed dramatically thanks to the successful experience achieved through photopolymeric holographic films, used to fabricate common Volume Phase Holographic Gratings (VPHGs). Thanks to the various advantages made available by these materials in this context, we propose an innovative solution to design a stacked multiplexed VPHGs that is able to secure efficiently different spectra in a single shot. This allows to increase resolution and spectral range enabling astronomers to greatly economize their awarded time at the telescope. In this paper, we demonstrate the applicability of our solution, both in terms of expected performance and feasibility, supposing the upgrade of the Gran Telescopio CANARIAS (GTC) Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy (OSIRIS). | null | [
"https://arxiv.org/pdf/1704.08150v1.pdf"
] | 119,359,319 | 1704.08150 | 3b5104927c49fec8e78451e0ff018b106fc1fa20 |
Spectral multiplexing using stacked VPHGs -Part I
2017
A Zanutta
INAF -Osservatorio Astronomico di Brera
via E. Bianchi 4623807MerateLCItaly
& M Landoni
INAF -Osservatorio Astronomico di Brera
via E. Bianchi 4623807MerateLCItaly
†
INAF -Osservatorio Astronomico di Brera
via E. Bianchi 4623807MerateLCItaly
M Riva
INAF -Osservatorio Astronomico di Brera
via E. Bianchi 4623807MerateLCItaly
A Bianco
INAF -Osservatorio Astronomico di Brera
via E. Bianchi 4623807MerateLCItaly
Spectral multiplexing using stacked VPHGs -Part I
MNRAS
0002017Accepted XXX. Received YYY; in original form ZZZPreprint 27 April 2017 Compiled using MNRAS L A T E X style file v3.0instrumentation: spectrographs -techniques: spectroscopic -telescopes -methods: observational
Many focal-reducer spectrographs, currently available at state-of-the art telescopes facilities, would benefit from a simple refurbishing that could increase both the resolution and spectral range in order to cope with the progressively challenging scientific requirements but, in order to make this update appealing, it should minimize the changes in the existing structure of the instrument. In the past, many authors proposed solutions based on stacking subsequently layers of dispersive elements and record multiple spectra in one shot (multiplexing). Although this idea is promising, it brings several drawbacks and complexities that prevent the straightforward integration of a such device in a spectrograph. Fortunately nowadays, the situation has changed dramatically thanks to the successful experience achieved through photopolymeric holographic films, used to fabricate common Volume Phase Holographic Gratings (VPHGs). Thanks to the various advantages made available by these materials in this context, we propose an innovative solution to design a stacked multiplexed VPHGs that is able to secure efficiently different spectra in a single shot. This allows to increase resolution and spectral range enabling astronomers to greatly economize their awarded time at the telescope. In this paper, we demonstrate the applicability of our solution, both in terms of expected performance and feasibility, supposing the upgrade of the Gran Telescopio CANARIAS (GTC) Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy (OSIRIS).
INTRODUCTION
The current state-of-the-art spectroscopic facilities could be divided in two main groups depending on the resolution. The first one is characterized by a low resolution (R < 2000), particularly suitable for multi-object spectroscopy or Integral Field Unit (IFU). Among them we can find examples like ESO VIMOS (Le Fèvre et al. 2003), FORS1-2 (Appenzeller et al. 1998), MOSFIRE (McLean et al. 2012), the FOSC at ESO-NTT (Snodgrass et al. 2008;Buzzoni et al. 1984) or OSIRIS at GTC (Cepa 2010). The second one is featured by a high resolution (R >> 4000), which is guaranteed through diffractive elements like echelle or echellette grating. Among this group, successfully examples are HIRES at Keck (Vogt 2002), ESO UVES (Dekker et al. 2000), CRIRES (Käufl 2008) or the most recent ESO X-SHOOTER (Vernet et al. 2011).
The resolution plays a key role in the era of 10 m class telescopes since the only way to increase the sensitivity of a spectrum, in terms of detectable features and accuracy, is Contact e-mail: [email protected] † Contact e-mail: [email protected] to increase it as much as possible while maintaining a good signal-to-noise ratio (SNR) over a wide spectral range. Current focal reducer spectrographs, like the GTC-OSIRIS, already provide diffraction gratings that allow to secure spectra with R ≥ 2000 but, unfortunately, their spectral range is very narrow. Therefore, to obtain a spectrum from 4000 to 10000Å, it would require to observe the same source multiple times, thus wasting an enormous amount of awarded time. On the other hand, deciding to operate in alternative facilities based on echelle gratings (e.g. ESO-XSHOOTER or Keck-ESI), a scientist could secure spectra with wide wavelength coverage at high resolution (R > 4000). Nevertheless, spectrographs like GTC-OSIRIS or EFOSC are much simpler, widely diffused in a large number of optical telescopes and with a flexible design that includes imaging capabilities. Therefore, an improvement of these facilities that allow to fill the gap between the two layouts is really attractive.
In most astrophysical topics, the increase of the resolving power of secured spectra permits to achieve many scientific advantages. For example, the use of ESO-XSHOOTER spectrograph produced a vast number of spectra of QSOs at R > 5000, allowing to better understand the physical state and chemical composition of the IGM (López et al. 2016;D'Odorico et al. 2013D'Odorico et al. , 2016. Another interesting topic tackled with the same instrument is the determination of the redshift of BL Lac objects. These sources are active nuclei of massive elliptical galaxies whose emission is dominated by a strong non-thermal continuum Shaw et al. 2013;Massaro et al. 2016;Sandrinelli et al. 2013) that prevents the determination of their distance that is, however, mandatory for constraining models of their emission or to better understand absorption of hard γ-rays by Extragalactic Background Light (EBL). In fact for example, Pita et al. (2014); Landoni et al. (2014) demonstrated that this problem could be mitigated by securing spectra with high SNR and increased resolution to detect fainter spectral features, necessary to estimate the redshift (and thus the distance) of the source.
As already pointed out, it is possible to collect spectra of astrophysical sources at high resolution (R > 4000) with focal reducer instruments but the price to pay is a limited spectral range. A clever solution could be the substitution of the single high R element, whose range is limited, with a new dispersive device capable to increase spectral coverage, simultaneously recording multiple high R spectra in different ranges (multiplexing). Being able to deliver such dispersive element will achieve a great improvement of the expected throughput of already commissioned spectrographs. In this paper, we focus on the GTC-OSIRIS as a candidate and demonstrator for housing our innovative device. In particular, we report on two different cases. The first one allows to secure two spectra in a single shot increasing the resolution by a factor of ∼ 2 while the second challenges instruments like X-SHOOTER combining three high resolution spectral regions simultaneously. This work is organized as follows: in Section 2 we report on the theoretical background and the design principle of these multiplexed devices. In Section 3 we discuss the expected performances on the sky through simulations, while in Section 4 we give our conclusions.
GRATING MULTIPLEXING -THEORY AND APPLICABILITY
As highlighted in the introduction, being able to simultaneously record multiple spectra of wavelength ranges, or alternatively, to have a high resolution element that covers a very wide spectral range, brings a huge advantage to the astronomical community. In particular, depending on the optical layout, a spectrograph would benefit of the possibility to increase the resolution or the spectral range, maintaining the same exposure times. Otherwise, a typical focal reducer imager, like FORS or OSIRIS, would benefit from the combination of very low dispersion gratings to acquire multiple snapshots of the same field in different bands simultaneously, as depicted in the cartoon of Figure 1.
In the present study we have tried to answer to these needs, testing the feasibility of a new type of dispersive element, that can result in a huge technological boost for those instruments that are becoming obsolete and for the new ones that are yet to be built.
The general idea is to place multiple gratings (multiplexed), stacked subsequently, in a way that they will produce simultaneously spectra of different wavelength regions. Figure 1. Scheme of a possible application of a multiplexed device in GRISM 2 mode. Multiple and high dispersive VPHG layers compose the multiplexed element which produces on the CCD, spectra of the slit in different spatial locations (one for each grating layer). In the inset are reported the possible uncombined efficiencies, peaked in different spectral ranges. Figure 2. Scheme of the layers composing a multiplexed grating. VPHG1 and VPHG2 may have different line density and orientation (clock) in order to separate the two spectra along the y direction. Filter and glass are not mandatory elements and can be replaced with other layers such as prisms.
The basic concept of a transmissive element is sketched in Figure 2. Each spectrum in the instrument's detector is designed to cover a specific wavelength range, according to the scientific case that has to be studied. Consequently, the design phase is indeed a crucial part in the definition of the characteristics of the multiplexed dispersive element. Moreover, strategies to separate the spectra avoiding their overlapping should be considered.
In this particular configuration, since the grating layers are superimposed, the key idea is to apply a small rotation along optical axis (ε in Figure 2) between the layers, in order to separate (along the y direction) the different spectra appearing on the detector.
Being able to secure multiple spectra with one exposure, the analyzed spectral range is extended (maintaining the same resolution R) or the resolution of the system in the same spectral range is significantly increased. This system is therefore suitable for upgrading an already built instrument, giving a great enhancement by a simple replacement of the dispersive element, which preserves all the existing abilities (e.g. the imaging in FOSC).
The type of dispersive element that we have considered in this study is the transmissive Volume Phase Holographic Grating, VPHG (Barden et al. 1998). They consist in a periodic modulation of the refractive index (∆n) in a thin layer of a photosensitive material. These elements represent today the most used dispersive elements in astronomy and yet the element whose performances are most difficultly surpassed in both low and medium resolution spectrographs (Spanò et al. 2006;Baldry et al. 2004;Pazder & Clemens 2008).
Since many different VPHGs are usually integrated inside astronomical spectrographs and each of them is a custom designed grating, each astronomical observation can take advantage of specific dispersive elements with features tailored for achieving the best performances. Accordingly, the design and manufacturing of highly efficient and reliable VPHGs require photosensitive materials where it is possible to control both the refractive index modulation and the film thickness d, in order to tune the device's efficiency.
Regarding the holographic materials, up to now Dichromated Gelatins (DCG) is considered the reference material thanks to the very large modulation of the refractive index that can be stored (Liang-Wen et al. 1998;Bianco et al. 2012), which turns into relative large bandwidth in high dispersion gratings. Unfortunately this material requires a complex chemical developing process making it difficult for large scale and large size production. Moreover, the material is sensitive to humidity, therefore, it is necessary to cover the grating with a second substrate, burdening the control of the wavefront error.
The availability of holographic materials with similar performances, but with self-developing properties is desirable, because they will not require any chemical post-process and moreover, the ∆n formation could be monitored and set during the writing step.
Photopolymers are a promising class of holographic materials and today, they are probably the best alternative to DCG, thanks to the improved features in terms of refractive index modulation, thickness control and dimension stability (Lawrence et al. 2001;Bruder et al. 2011;Ortuño et al. 2013;Fernández et al. 2015). A lot of studies have been carried out to understand deeply the behavior of this class of materials. Moreover, the formation of the refractive index modulation has been recently studied (Gleeson & Sheridan 2009;Gleeson et al. 2011;Li et al. 2014a,b), through the development of models that predict the trends as function of the material properties and writing conditions (Kowalski & McLeod 2016).
We already demonstrated in other papers the use of photopolymers for making astronomical VPHGs with performances comparable to those provided by VPHGs 2 A GRISM is a combination of a prism and grating arranged so that light at a chosen central wavelength passes straight through. The advantage of this arrangement is that the same camera can be used both for imaging (without the grism) and spectroscopy (with the grism) without having to be moved. Grisms are inserted into a camera beam that is already collimated. They then create a dispersed spectrum centered on the object's location in the camera's field of view.
based on DCGs and good aging performances (Zanutta et al. 2014a), but with a much simpler production process (Zanutta et al. 2016b). Therefore, we think that the big advantages of this novel holographic material could be the key point to realize the multiplexed dispersive element.
The newly (Bruder & Fäcke 2010;Berneth et al. 2011Berneth et al. , 2013 developed photopolymer film technology (Bayfol HX film) evolved from efforts in holographic data storage (HDS) (Dhar et al. 2008) where any forms of post processing is unacceptable. These new instant developing recording media open up new opportunities to create diffractive optics and have proven to be able to record predictable and reproducible optical properties (Bruder et al. 2009). Depending on the application requirements, the photopolymer layer can be designed towards e.g. (high or low) index modulation, transparency, wavelength sensitivity (monochromatic or RGB) and thickness to match the grating's wavelength and/or angular selectivity.
Since the material consists in the holographic layer coupled with a polymeric substrate with a total thickness of ca. 60 -150 µm, it can be laminated or deposited one on top of the other after having been recorded, making straightforward the stacking realization.
Clearly, another possibility is to holographically record multiple gratings inside the same layer but, as described later, in order to optimize the efficiency curves, usually very different thicknesses are required for each grating, therefore this strategy will not let us have the advantage to tune the response curves in the design process.
Working principle
As stated at the beginning of this section, the design concept consists in placing a set of transmission VPHGs stacked subsequently (multiplexed) (see Figure 2). As highlighted in the figures, this device will form one single optical element whose dimensions are comparable to standard VPHGs already available in the target instrumentation.
Some attempts have been made to explore this idea (Muslimov et al. 2016;Battey et al. 1996) but, although steps have been made in the right direction, the proposed solutions are limited by the necessity of a newly designed spectrograph, and do not take into consideration the crucial efficiency optimization that, without proper design, will make the device ineffective.
Hence, to preserve integration simplicity, one has to mix materials, design strategies and required performances, in order to produce multiplexed dispersive elements that could be easily integrated in an available instrument. This gives to astronomers the possibility to enhance the resolution (and spectral coverage) by simply replacing the disperser already installed in the optical path.
Regarding the material, thanks to the crucial capability to finely tune the refractive index modulation ∆n (Zanutta et al. 2016a) and the slenderness of the film containing the grating, Bayfol HX photopolymers by Covestro gave us the possibility to design the multiplexing element to:
(i) realize a compact and thin device that can be integrated as replacement in many already existing instruments (Landoni et al. 2016a;Zanutta et al. 2014b);
(ii) tune the single stacked efficiency in such a precise way that they will not interfere with each other and obviate to all the problems related to the realization of these devices; (iii) match the design requirements and obtain high efficiency;
(iv) stack multiple layers of gratings in one single device for the simultaneous acquisition of multiple spectra with a broad wavelength band.
In the multiplexed device, each layer will generate a portion of the spectrum that all together will compose the total dispersed range required. Such pieces, on the detector, will be disposed one on the top of each other, resulting in a total spectral range that is far wider than the one obtainable using a single grating with a comparable dispersion.
Issues in details: the geometrical effect
Although the stack of subsequent diffraction elements brings many advantages, some constraints and critical points arise and should be discussed.
The first one is purely a geometrical effect and is related to the propagation of the incoming beam throughout multiple dispersing elements that must not interfere with each other. For this reason, it has to be taken into account that the incoming beam is diffracted multiple times (since it encounters two or more dispersive elements) according to the grating equation
mλΛ = sin α + sin β(1)
with m the number of the diffracted order, Λ the line density of the VPHG and α, β the incidence and diffracted angles respectively.
Fixing λ, different combinations of diffraction orders can occur, resulting in light diffracted in different directions. In particular let us consider for simplicity an example of two stacked multiplexed elements as shown in Figure 3. Each grating is optimized for dispersing efficiently a specific wavelength range (labelled B for blue and R for red). Let's suppose that the two possess the same line density. If a red monochromatic wavelength λ R (case i) enters the multiplexed device, it will be firstly diffracted by the R grating in all the possible orders, that will eventually enter the second grating. Each of these beams are in turn recursively diffracted by the B grating, but only few of them possess the correct direction for further propagation (total internal reflection TIR can occur) to the detector.
Otherwise, when a blue monochromatic beam λ B is considered (case ii, for equal incidence angle) each diffracted order will possess a smaller diffraction angle β (with respect of the previous case) due to the shorter wavelength, therefore it is possible that some overlapping between blue and red orders would occur since more diffraction orders have a direction that can potentially enter the detector.
Issues in details: the efficiency depletion due to subsequent gratings
In the design of VPHGs for an astronomical spectrograph, after having satisfied the dispersion and resolution requirements, which fix parameters like the line density (Λ) of the Figure 3. Monochromatic beam propagation in a twomultiplexed device (i -red wavelength case, ii -blue wavelength case). In the beam notation J.K, the number J is the diffraction order of the first grating and K the one of the second grating. R identifies the grating layer designed for dispersing efficiently the red wavelengths, while B the blue ones. Bold lines are the orders that we want to exploit in the detector gratings and the incidence and diffraction angles (α and β), the optimization of the diffraction efficiency (both peak efficiency and bandwidth) is necessary.
λ R R B 0.0 0.1 1.0 i) ii) λ B R B 0.0 0.1 0.2 1.-1 1.0 α 0 β R 0.-1 1.1
To perform this task, the main parameters to be considered are:
(i) the refractive index modulation ∆n; (ii) the active film thickness d ; (iii) the slanting angle φ (i.e. the angle between the normal of the grating surface and the normal of the refractive index modulation plane).
Considering a sinusoidal refractive index modulation, and working in the Bragg regime (the light is sent only in one diffraction order other than the zero), the well-known Kogelnik model can be used to compute the grating's efficiency (Kogelnik 1969). For small angles, large diffraction efficiency is achieved when the product ∆n · d is equal to half of the wavelength and this is the starting point in the optimization process. As already stressed, during the VPHG design, not only the peak efficiency is important, but also the efficiency at the edges of the spectral range. According to the Kogelnik model, the spectral bandwidth (∆λ) of the diffraction efficiency curve is proportional to (Barden et al. 2000):
∆λ λ ∝ cot α Λd(2)
In this equation, α is the incidence angle, Λ is the line density of the grating and it is evident that the bandwidth is inversely proportional to Λ and the thickness of the grating d. Hence, the optimization of the diffraction efficiency curves, acting on the ∆n and d, provides large differences in the grating response.
If a grating works in the Bragg regime, the largest peak efficiency and bandwidth is obtained for very thin films and large ∆n. Undoubtedly, the ∆n upper value is determined by the performances of the holographic material. If the VPHG works in the Raman-Nath regime (Moharam & Young 1978), it diffracts the light with a non-negligible efficiency in more than one diffraction order and this is the case of low dispersion gratings and should be considered to avoid the further explained second order contamination (see Section 2.4).
For such gratings, the light diffracted in high orders is proportional to ∆n 2 , ergo it is better to increase the film thickness and reduce the ∆n in order to achieve a large peak efficiency.
The availability of an holographic material that can exploit a precise ability to tune the ∆n (Zanutta et al. 2014b(Zanutta et al. , 2016b, is therefore crucial for the design of multiplexed elements, in order to be able to adjust the efficiency response of each dispersive layer.
In the multipexing context, it is important to evaluate how a grating with a certain efficiency affects the response of the following one. In order to give a feeling of the complexity of the problem, let us reconsider the two-multiplexing device in Figure 3. The total multiplexed efficiency on the detector will not merely be the sum of the single layer efficiencies η B,1st (λ, α 0 ), η R,1st (λ, α 0 ), with λ the wavelength and α 0 the initial incidence angle.
The notation " 1st " indicates the diffraction order at which the efficiency η refers to, meaning that the system is aligned to work out the 1 st order.
The spectrum generated by the R grating, before reaching the detector, has to pass through the B grating, and this will eventually diminish its intensity. To complicate that, we add the fact that each wavelength of the R spectrum possess different diffraction angles β R (which became the incidence angles for the B grating) and therefore this varies the response from the second grating, according to the grating equation (eq. 1). The resulting R efficiency η * R,1st (λ, α 0 ) will be then:
η * R,1st (λ, α 0 ) = η B,0th (λ, β R ) · η R,1st (λ, α 0 )(3)
Moreover, the light that enters the second layer has already been processed by the previous gratings, therefore, its final efficiency will be the product of the leftover intensity, times the efficiency of the B grating:
η * B,1st (λ, α 0 ) = η R,0th (λ, α 0 ) · η B,1st (λ, α 0 )(4)
Practically, the goal is to obtain gratings with negligible overlapping efficiencies.
Issues in details: spectral range and Second Order Contamination
A critical point in spectroscopy is the contamination of recorded spectra, usually obtained through the first diffraction order, by light coming from other diffraction orders, usually the second. Since signals of the different orders are overlapped, there is no possibility to remove the unwanted light a posteriori with data reduction. Therefore, dispersive elements with spectral range greater than [λ to 2λ] will inevitably suffer of this problem. This issue is well known in astronomy and it is usually avoided by placing order-sorting filters coupled with the dispersing element or in a filter wheel in the optical path of the instrument. The filters serve to block the light at lower wavelength that can overlap to the acquired spectrum. Another approach is to reduce as much as possible the efficiency of the unwanted orders.
Although in VPHGs, it is possible to mitigate these effect by varying parameters, such as thickness and ∆n, in Wavelength (Angstrom) Ratio between theoretical spectrum and actual one Figure 4. Ratio between contaminated and non-contaminated spectrum (magenta line) for a source with a power law spectral energy distribution at SNR of ∼ 100. Black lines correspond to the detection limits dictated by SNR. The 2nd order contamination appears as a recognizable peak surpassing the black lines which corresponds to the noise level.
order to finely tune the efficiency curve, we decided to limit the wavelength range of the multiplexed device, adopting a spectral band where no contamination occurs. However, we will show another strategy that deals with the second orders in the forthcoming "Part II" of this work.
In Figure 4 we report the effects of second order contamination with a multiplexed device designed to work in an extended wavelength range [4200 -10000 ]. We have simulated two different spectra: the first one contaminated with photons coming from the second order, while the other one considering only the contributions from the first order. We also assumed a SNR of about 100. The ratio between the two signals (magenta line) is a rough estimation of how much unwanted light appears as a bump in the collected data. In fact, according to this figure, the ratio between the two is within the noise level (solid black lines) up to λ 9000, indicating the two spectra are undistinguishable. Surpassing this wavelength the ratio is higher than the noise level, meaning that photons from the second order are superimposing in the spectrum.
DESIGN CONCEPTS AND EXPECTED PERFORMANCES
Theoretical framework
In order to understand the spectral behavior of a multiplexing dispersive element, we choose to study the feasibility of this system considering an astronomical instrument that could take advantage of the multiplexing technology.
The resolving power of a spectrograph, R (or simply resolution) is:
R = mΛλW χD T(5)
where W is the length of the illuminated area on the grating by the collimated beam, χ is the angular slit width (projected on the sky) and D T is the diameter of the telescope.
For a correct interpretation of the results, it has to be pondered that, the optical layout of the spectrograph (such as the ratio of telescope diameter and the collimated beam in the spectrograph) can be used as a rule of thumb to quantify the advantage of this approach. In this paper we present the case of the focal reducer OSIRIS, installed at the 10 m telescope Gran Telescopio Canarias, as candidate facility for the on-sky commissioning of the multiplexed device. We have chosen to exploit two different case studies, changing the number of elements in the multiplexed device. A two-stacked multiplexed device with an approximate resolution of R ∼ 2000, and a three-stacked multiplexed device with a resolution of about R ∼ 5000. The first one was intended to compete with observations carried out at the same facility for the determination of redshift lower limit of the TeV γ-ray BL Lacertae (BL Lac) object S4 0954+258 (see Landoni et al. (2015a) and the next sections) while the second one is intended to be compared with medium-high state-of-the-art resolution spectrographs such as ESO-XSHOOTER (Vernet et al. 2011;López et al. 2016). In this section, we demonstrate the design and applicability of the two cases. For each of them, we present the analysis to determine the efficiency behavior, taking into account the dispersion and spectral range that each VPHG should show in relation to the instrument specifications. This activity is carried out both through optical ray-tracing and Rigorous Coupled Wave Analysis RCWA simulations (Moharam & Gaylord 1981). The outputs of this calculation are the most suitable efficiency curves for each stack that will guarantee the higher overall diffraction efficiency and are computed varying the key parameters described in Section 2.3.
After the grating design, the subsequent step is to assess, through simulations, the expected on-sky performances of each device. Thus, we build up syntethic simulated spectra (starting from powerlaw model, as in the case of BL Lac, or template spectrum as in the case of QSO) of the targets according to the expected signal-to-noise ratio (SNR) in each pixel defined as:
S N = N * N * + N sky + n pix · RON 2(6)
where N * is the number of expected counts from the target source evaluated as:
N * = f (λ) · n i=1 ·η i (λ) · A · t ex p(7)
where f (λ) is the input spectrum in ph sec −1 cm −2Å −1 , A is the collecting area of the telescope, η i are the efficiencies of atmosphere transmission, telescope, spectrograph, multiplexed device and CCD (we consider a slit efficiency of about ∼ 0.80 arcsec) and t ex p is the total integration time. The quantity N sky is evaluated in the same way considering a flat spectrum normalized in V o R band to a flux that corresponds to ∼ 21 mag arcsec −2 , which is a typical value for the La Palma sky brightness. The read-out-noise (RON) of the detector is assumed equal to 7 e-/pix. For each simulation, we consider a seeing of ∼ 0.80 with a slit width of ∼ 0.60 to be comparable with the current available instrumentation specifications and performances (Cepa 2010). The plate scale of the system is assumed to be ∼ 0.30 arcsec/pix while the efficiencies of the telescope and optics of the spectrograph are derived from literature (Cepa 2010).
Two-multiplexed grating case
This first case that we took into consideration is a two stacked multiplexed device, for OSIRIS and in GRISM mode, that can cover in one single exposure a spectral range from 4000 to 8000 with a resolution of approximately 2000.
In order to achieve that, the dispersive element splits the wavelength range in two parts, that are imaged on the detector one on top of the other. The inter-spectraseparation depends on the tilt of the two gratings in the diffraction element. In this particular scenario, the minimum distance between the two is approximately 2' (projected angle on the sky), but merely because we have chose arbitrarily a tilt value of 2.5°(see Figure 5). The two dispersive elements share the same prisms and thus the same incidence angle.
In Table 1 the specifications of the gratings that have been designed are reported, while in Figure 6 we presented the calculated efficiency curves of the layers that compose the device. With respect of this last figure, a long-pass filter at 4000 is installed in the device in order to avoid contamination from the second order. Moreover the VPHG, which disperses the light in the range 5500 -8000 , has been designed to suppress as much as possible the contribution from the second order, which remains outside the spectral range.
As highlighted in the previous sections, an important effect that has to be taken into account is that the diffracted intensity will be dimmed as light gradually passes through the VPHG layers but, in this configuration, thanks to the precise design process, this effect is minimized. Indeed, for each grating layer, a specific value of ∆n, d and slanting angle φ was chosen in order to ensure the compatibility between the efficiency curves.
In the hypothesis that the sequence is first the RED grating and second the BLUE grating, the wavelengths that are diffracted by the RED grating (dotted green in Figure 6), are then transmitted through the BLUE layer with the resulting efficiency plotted in solid green. On the other hand, the wavelengths that have to be diffracted by the BLUE grating, must firstly pass though the RED layer, with a resulting efficiency that is plotted in solid blue.
After accounting for all of these effects, the obtained efficiency curve for the multiplexed dispersive element is re- Figure 6. Diffraction efficiencies of the gratings composing the multiplexed element. The dotted lines refer to the single layer efficiencies (1st and 2nd diffraction orders), while the solid lines referto the corrected efficiencies (labelled "eaten") at the exit of the multiplexed element, due to the reciprocal interference of the dispersive layers. Vertical lines identify the wavelength boundaries of the spectra in the CCD for each VPHG. "Blue 2" is a VPHG with ∆n = 0.038 and d = 6 µm while "Red 2" with ∆n = 0.024, d = 12 µm and φ = 0.5 o .
ported in Figure 7. The bump in the central region is due to light diffracted by both gratings and that falls on the detector in different places. Finally we remind that the efficiencies presented in the simulations do not take into consideration the material absorptions or the reflection losses that could arise to the presence of interfaces inside the device. Nevertheless we expect that these effects could be negligible at this level and are of the order of few percent points.
The case of S4 0954+65
S4 0954+65 is a bright BL Lac object identified for the first time by Walsh et al. (1984) which exhibits all the properties of its class. In particular, the source presents a strong variability in optical, with R apparent magnitudes usually ranging between 15 and 17 (Raiteri et al. 1999), linear polarisation (Morozova et al. 2014) and a radio map that shows a complex jet-like structure. This BL Lac has recently caught attention since it was detected with the Cherenkov telescope MAGIC with a 5-σ significance (Mirzoyan 2015). The determination of redshift of BL Lac objects (in particular for TeV sources) is mandatory to assess their cosmological role and evolution, which appears to be controversial due to redshift incompleteness (Ajello et al. 2014) and to properly understand their radiation mechanism and energetics (see e.g In the era of 10 m class telescopes (like the GTC), the research in this field has reached the so called "photon starvation regime" since the only way to significantly increase the SNR is the adoption of extremely large aperture telescope (like ELT) ).
On the other end, one can greatly increase the resolution of the secured spectra, maintaining a high SNR, decreasing the minimum Equivalent Width (EW min ), allowing to measure fainter spectral features (see e.g. Sbarufatti et al. (2006), Shaw et al. (2013)).
In particular, S4 0954+65 has been observed by Landoni et al. (2015b) after its outburst on the night of February 28th, 2015. The object was observed with two grisms (R1000B and R1000R) in order to ensure a spectral coverage from 4200 to ∼ 10000Å adopting a slit of 1.00 with a resolution of R ∼ 800. For each grism, the total integration time was 450s that corresponds to about 0.5 hrs of telescope allocation time (including overheads). The collected data allowed to disprove previous redshift claims of z = 0.367 (Stickel et al. 1993;Lawrence et al. 1996) and to infer a lower limit to the distance of z ≥ 0.45 thanks to EW min ∼ 0.15Å and SNR > 100.
In order to further increase the lower limit to the redshift or, even better, detect faint spectral features arising from the host galaxies that harbors this BL-Lac, the only straightforward solution with this state-of-the-art instrumentation is to drastically increase the resolution of the collected spectrum. Considering the case of GTC and OSIRIS, the only available opportunity is to observe the target with the GRISMs R2500. Unfortunately these gratings possess a very narrow spectral range so in order to ensure the wavelength coverage similar to the required (4000 -8000Å), one must collect four different spectra. This turns out in a telescope allocation time of about 2 hrs (including overheads).
By the adoption of the two VPHG multiplexed device, the observer is able to collect simultaneously two spectra with a whole spectral range from 4000 to 8000Å with a resolution of approximately 2000. The simulated spectra obtained with this device is reported in Figure 8 along with the comparison of the R1000B+R observed one.
We also report the distribution of minimum detectable Equivalent Width, estimated following the recipes detailed in Sbarufatti et al. (2005) (histogram in the bottom right corner of Figure 8).
The detectable EW min on the spectrum simulated by assuming the new dispersing element is 0.03, which is a factor of 5 lower than the compared one. This turns in a lower limit to the redshift of z 0.55 putting the source at a plausible redshift region where the absorption from the EBL becomes severe and making this TeV object an excellent probe for the study of the EBL through absorption.
In this figure we also report the expected SNR obtained with our device (solid magenta in the top left box), and the one simulated assuming the currently available R2500 devices at the GTC.
Three-multiplexed grating case
For the science cases that require a wide spectral range with a moderate resolution, nowadays the only possibility to fulfill the requirements is to adopt an echelle grating based instrument, which is capable to secure wide wavelength ranges in a reasonable number of shots . Otherwise according to OSIRIS GRISMs specifications, in the GTC manual, up to six different setups (and exposures) are required to obtain the same result just in terms of spectral range, since the maximum resolution is approximately R max = 2500.
In this section we present a possible application of multiplexed VPHGs, aiming to refurbish the dispersive elements of OSIRIS at GTC, in order to reach the closest possible performance with respect to UV and VIS arm of X-SHOOTER.
In order to cover a wide spectral range with a resolution of approximately 4500, we have designed two multiplexed dispersive elements, each one composed by three stacked layers, therefore they will produce on the detector three spectra for each single exposure. With these two devices together, in just two exposures, we can cover a range from 3500 to 10000 .
While the number of dispersive layers could be theoretically further increased, due to complexities in calculations, possible transparency issues and manufacturing alignment, in this work we decided to set the limit to three elements.
BLUE device, from 350 to 600 nm
The first (of two) multiplexed device will be responsible for the dispersion of the light in the range 3500 -6000 , there- fore hereafter it will be identified as the BLUE device. It is composed by three dispersing layers, each of them generating the peaks in the summed efficiency displayed in Figure 9 (solid blue curve). For this case, we did not report the plot with the contributions that generate the overall efficiency, since the general procedure is the same that in the two-multiplexed case. As highlighted in the previous case, the vertical solid lines identify the size of the detector with respect to each spectra: since the total range will appear divided in three parts, the upper is displayed with solid blue boundaries, the central with green and the lower with red. As some small portions of the range will overlap, bumps in efficiency in the regions between the peaks appear.
In Table 2, we report the specifications of the three gratings that have been designed for this BLUE element along with the calculated resolution and dispersion that is achievable integrating this device in the OSIRIS spectrograph.
RED device, from 600 to 1000 nm
This second multiplexed device will be responsible to disperse the light in the spectral range from 6000 to 10000 , therefore hereafter it will be identified as the RED device. Figure 10 (solid blue curve) reports the overall efficiency curve that can be produced by the three dispersing layers composing this device.
In Table 3, we report the specifications of the gratings that have been designed for this RED element along with the calculated resolution and dispersion that is achievable integrating this device in the OSIRIS spectrograph.
Application to Extragalactic Astrophysics. The characterisation of Intergalactic medium
The study of the Intergalactic and Circumgalactic medium (IGM and CGM) is a powerful tool to investigate the properties of the cool (and clumpy) gaseous halos between the observer and the source, that lies at a certain z. The only way to investigate the IGM or the CGM is through absorption lines imprinted in the spectra of distant QSO, as demonstrated in the last few years by e. (2015)). This research field is actively growing and, recently, has begun to probe not only the physical state and the chemical composition of the IGM but also the three-dimensional distribution of the gas allowing scientists to build up an actual tomography of the cool Universe between background quasars and the Earth. For example, in this context one of the most recent and successfully survey is the CLAMATO survey (Lee et al. 2014). In this projects, authors aims to to collect spectra for 500 background Lyman-Break galaxies (LBGs) in ∼ 1 sq degree area to reconstruct a 3D map with an equivalent volume of (100 h −1 Mpc) 3 .
The key step in these spectroscopic studies is the availability of moderate-high resolving power (R ∼ 5000) and wide spectral coverage, in order to probe as much as possibile absorption lines, perform diagnostic ratio to probe the interplay between galaxies and the intergalactic medium (IGM). However, such observations are typically time consuming and require good SNR at moderate R, a particularly high challenge for distant QSOs which tend to be faint. For all of these reason, the availability of new instrumentation available to collect spectra in a wide spectral range (3500-1 µm) at R > 4000 would be really advantageous allowing to further increase the availability of telescopes able to tackle these kind of surveys, especially for facilities with moderate telescope aperture. In order to demonstrate the applicability of our new device, we simulated the expected performance by assuming to observe for t ex p = 200s for each grating a QSO (template taken from López et al. (2016)) at redshift z = 3.78 with m R ∼ 17. The overall obtained spectrum is reported in Figure 11. In particular the solid blue line corresponds to emission spectrum of the quasar secured with the BLUE multiplexed device. The absorption lines, imprinted by Lyman-α intervening systems and used to probe the IGM, are clearly detected and resolved in most of the cases. The solid red line, instead, report the spectrum recorded with the RED multiplexed device where emission line from C IV and C III] are visibile. Results reported in Figure 11 are obtained with a total integration time of about 400s while, by comparison, to obtain the same results at half resolution with grisms available at GTC-OSIRIS would require more than 1000s, since it should be observed four times with 4 different gratings.
As highlighted in the previous paragraphs, the X-SHOOTER spectrograph is able to obtain similar results with a broader band in a single snapshot. Although this outcome is obviously outside the capabilities of our proposed solution, the multiplexing VPHG allows to cover in just two snapshot a comparable quality (in terms of R, SNR and spectral range) in the UV and visible band. Therefore, the integration of such element in a facility like OSIRIS would allow to scientifically compete with key-science projects that require spectroscopic capabilities otherwise available only with major instrument commissioning.
CONCLUSIONS
We have demonstrated the theoretical feasibility and the advantages of an innovative dispersive element, able to greatly increase the performances of the existing spectrograph at the state of the art 10 m telescope GTC. Thanks to the advantages derived by the adoption of the photopolymeric material considered in the simulations, we achieved to increase by at least a factor of two in terms of resolution (and thus in the spent observing time), without changes in the optical layout of the spectroscopic instrument. We have also shown that in the case of the three-multiplexed VPHG, it is possible to reach with GTC OSIRIS, approximately the performances of the UV and VIS arm of X-SHOOTER (when operating in medium resolution) in just two exposures of the same target. Even though in this work we have selected GTC OSIRIS for the simulations, the philosophy behind this multiplexing design could be applied to almost every focal reducer spectrograph, donating the discussed advantages to all the instruments, allowing them to handle scientific cases that would be otherwise out of reach for these facilities. In the forthcoming second part of this work, we will realize and integrate the multiplexed device in a spectrograph for science verification, focusing on the observational cases highlighted in the simulations in this paper. Figure 11. Simulation of a quasar spectrum at z ∼ 3.75, observed with our double Three-Multiplexed system. The solid blue line corresponds to the expected emission spectrum of the quasar recorded by the BLUE multiplexed device, it is worth noted that absorption lines from intervening systems are clearly visible and resolved in most cases. The solid red line corresponds to the part of the spectrum recorded with the RED multiplexed device Walsh D., Beckers J., Carswell R., Weymann R., 1984, Monthly Notices of the Royal Astronomical Society, 211, 105 Zanutta A., Bianco A., Insausti M., Garzón F., 2014a, in SPIE Astronomical Telescopes+ Instrumentation. pp 91515X-91515X Zanutta A., Landoni M., Bianco A., Tomasella L., Benetti S., Giro E., 2014b, Publications of the Astronomical Society of the Pacific, 126, 264 Zanutta A., Orselli E., Fäcke T., Bianco A., 2016a, in SPIE Astronomical Telescopes+ Instrumentation. pp 99123B-99123B Zanutta A., Orselli E., Fäcke T., Bianco A., 2016b, Optical Materials Express, 6, 252 This paper has been typeset from a T E X/L A T E X file prepared by the author.
Figure 5 .
5Simulated spectra onto the OSIRIS detector with the two layers multiplexed grating.
Figure 7 .
7Blue line: overall efficiency of the two-multiplexing dispersive element. Green lines: single grating efficiencies of the spectra that are reaching the detector's focal plane. Vertical lines identify detector's boundaries.are detected at TeV regime, the knowledge of their distance is unavoidable since they could be exploited as a probe of the Extragalactic Background Light (EBL, see e.gDomínguez et al. (2011), Franceschini et al. (2008) allowing to understand how extremely high energy photons propagate from the source to the Earth and interact with the EBL through γ-γ absorption. Unfortunately, the determination of the redshift of BL Lacs has proven to be difficult (see e.g.Shaw et al. (2013),Landoni et al. (2013),Massaro et al. (2016)) since their very faint spectral features are strongly diluted by their non-thermal emission (see the review ofFalomo et al. (2014)).
Figure 8 .
8Two-Multiplexed grating case: Simulation of the S4 0954+65 spectrum (magenta) and comparison with the real observed spectrum (solid blue) secured with R1000B+R GRISMS. Spectral regions where telluric absorptions are severe are shaded and not included in the analysis. Bottom right box reports the comparison of the histograms of EW mi n between the observed spectrum in feb. 2015 (shaded blue) and the one obtained with the new dispersing device. Top left box reports the SNR of the spectrum of S4 0954+65 obtained with the new device (magenta) and the one estimated with GRISMs R2500 (cyan).
Figure 9 .
9Multiplexing dispersive element designed for the BLUE band. Blue line: overall efficiency, green lines: single layer efficiencies of the VPHGs. In the same way explained for the two-multipexed case, vertical lines identify detector's boundaries. "Blue 3.1" possess ∆n = 0.055, d = 4 µm, "Blue 3.2" ∆n = 0.037, d = 6 µm and φ = -0.2 o , "Blue 3.3" ∆n = 0.035, d = 7.5 µm.
Figure 10 .
10Multiplexing dispersive element designed for the RED band. Blue line: overall efficiency, green lines: single layer efficiencies of the VPHGs. In the same way explained for the two-multipexed case, vertical lines identify detector's boundaries. "Red 3.1" possess ∆n = 0.055, d = 6 µm, "Red 3.2" ∆n = 0.044, d = 10 µm and φ = -0.5 o , "Red 3.3" ∆n = 0.038, d = 12 µm.
g Prochaska et al. (2014); Landoni et al. (2016b); López et al. (2016) since its surface brightness is extremely faint to be probed directly, and only few examples are know to succeed in the detection of emission of Ly-α lines in the CGM (e.g Arrigoni Battaia et al.
Table 1 .
1Parameters of the stacked grating composing the twomultiplexed device for OSIRIS, with prisms' apex angle of 36.0°.grating l/mm
λ range
λ c e nt r .
R 0.6
dispersion
[nm]
[nm]
@λ c e nt r al
[Å/ px]
blue 2
1500
400-558
475
2232
0.52
red 2
1000
550-800
675
2086
0.78
0.35
wavelength [μm]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
diffraction efficiency
1st BLUE
1st BLUE eaten
1st RED
1st RED eaten
2nd RED
2nd RED eaten
0.45
0.55
0.65
0.75
0.85
Table 2 .
2Parameters of the BLUE stacked grating composing the three multiplexed device for OSIRIS, with prisms' apex angle of 52.3°.grating
l/mm
λ c e nt r .
λ range
R 0.6
dispersion
[nm]
[nm]
@λ c e nt r al
[Å/ px]
BLUE 3.1
2850
385
354-425
4339
0.24
BLUE 3.2
2400
460
411-498
4430
0.29
BLUE 3.3
1980
550
493-600
4346
0.35
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
diffraction efficiency
wavelength [μm]
0.35
0.4
0.45
0.5
0.55
0.6
Table 3 .
3Parameters of the RED stacked grating composing the three multiplexed device for OSIRIS, with prisms' apex angle of 55.1°.grating
l/mm λ c e nt r .
λ range
R 0.6
dispersion
[nm]
[ o ]
@λ c e nt r al
[Å/ px]
RED 3.1
1750
655
605-721
4825
0.39
RED 3.2
1480
775
707-846
4851
0.46
RED 3.3
1240
920
843-1000
4814
0.55
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
diffraction efficiency
wavelength [μm]
0.65 0.7
0.75 0.8
0.85 0.9
0.95
MNRAS 000, 1-12(2017)
ACKNOWLEDGEMENTSThis work was partly supported by the European Community (FP7) through the OPTICON project (Optical Infrared Co-ordination Network for astronomy) and by the INAF through the TECNO-INAF 2014 "Innovative tools for high resolution and infrared spectroscopy based on non-standard volume phase holographic gratings".
M Ajello, D Gasparrini, R W Romani, M S Shaw, American Astronomical Society Meeting Abstracts #224. 9Ajello M., Gasparrini D., Romani R. W., Shaw M. S., 2014, in American Astronomical Society Meeting Abstracts #224. p. 410.09
The Messenger. I Appenzeller, 941Appenzeller I., et al., 1998, The Messenger, 94, 1
. Arrigoni Battaia, F Yang, Y Hennawi, J F Prochaska, J X Matsuda, Y Yamada, T Hayashino, T , 10.1088/0004-637X/804/1/26ApJ. 80426Arrigoni Battaia F., Yang Y., Hennawi J. F., Prochaska J. X., Matsuda Y., Yamada T., Hayashino T., 2015, ApJ, 804, 26
I K Baldry, J Bland-Hawthorn, J Robertson, Publications of the Astronomical Society of the Pacific. 116403Baldry I. K., Bland-Hawthorn J., Robertson J., 2004, Publica- tions of the Astronomical Society of the Pacific, 116, 403
S C Barden, J A Arns, W S Colburn, Astronomical Telescopes & Instrumentation. Barden S. C., Arns J. A., Colburn W. S., 1998, in Astronomical Telescopes & Instrumentation. pp 866-876
. S C Barden, J A Arns, W S Colburn, J B Williams, Publications of the Astronomical Society of the Pacific. 112809Barden S. C., Arns J. A., Colburn W. S., Williams J. B., 2000, Publications of the Astronomical Society of the Pacific, 112, 809
Spectrograph with multiplexing of different wavelength regions onto a single opto-electric detector array. D E Battey, H Owen, J M Tedesco, Battey D. E., Owen H., Tedesco J. M., 1996, Spectrograph with multiplexing of different wavelength regions onto a single opto-electric detector array
H Berneth, F.-K Bruder, T Fäcke, R Hagen, D Hönel, D Jurbergs, T Rölle, M.-S Weiser, Proc. SPIE. SPIE79570Berneth H., Bruder F.-K., Fäcke T., Hagen R., Hönel D., Jurbergs D., Rölle T., Weiser M.-S., 2011, in Proc. SPIE. p. 79570H
H Berneth, F.-K Bruder, T Fäcke, R Hagen, D Hönel, T Rölle, G Walze, M.-S Weiser, SPIE Optics+ Optoelectronics. Berneth H., Bruder F.-K., Fäcke T., Hagen R., Hönel D., Rölle T., Walze G., Weiser M.-S., 2013, in SPIE Optics+ Optoelec- tronics. pp 877603-877603
A Bianco, G Pariani, A Zanutta, C Bertarelli, SPIE Astronomical Telescopes+ Instrumentation. Bianco A., Pariani G., Zanutta A., Bertarelli C., 2012, in SPIE Astronomical Telescopes+ Instrumentation. pp 84502W- 84502W
. F Bruder, T Fäcke, International Journal of Materials Research. 101199Bruder F., Fäcke T., 2010, International Journal of Materials Re- search, 101, 199
. F.-K Bruder, Journal of Photopolymer Science and Technology. 22257Bruder F.-K., et al., 2009, Journal of Photopolymer Science and Technology, 22, 257
. F.-K Bruder, R Hagen, T Roelle, M.-S Weiser, T Fäcke, Angewandte Chemie International Edition. 504552Bruder F.-K., Hagen R., Roelle T., Weiser M.-S., Fäcke T., 2011, Angewandte Chemie International Edition, 50, 4552
The Messenger. B Buzzoni, 389Buzzoni B., et al., 1984, The Messenger, 38, 9
. J Cepa, 10.1007/978-3-642-11250-8_2Astrophysics and Space Science Proceedings. 1415Cepa J., 2010, Astrophysics and Space Science Proceedings, 14, 15
. V D'odorico, 10.1093/mnras/stt1365MNRAS. 4351198D'Odorico V., et al., 2013, MNRAS, 435, 1198
. V D'odorico, 10.1093/mnras/stw2161MNRAS. 4632690D'Odorico V., et al., 2016, MNRAS, 463, 2690
H Dekker, S D'odorico, A Kaufer, B Delabre, H Kotzlowski, 10.1117/12.395512Proc. SPIEVol. 4008, Optical and IR Telescope Instrumentation and Detectors. Iye M., Moorwood A. F.SPIEVol. 4008, Optical and IR Telescope Instrumentation and DetectorsDekker H., D'Odorico S., Kaufer A., Delabre B., Kotzlowski H., 2000, in Iye M., Moorwood A. F., eds, Proc. SPIEVol. 4008, Optical and IR Telescope Instrumentation and Detectors. pp 534-545, doi:10.1117/12.395512
. L Dhar, K Curtis, T Fäcke, Nature photonics. 2403Dhar L., Curtis K., Fäcke T., 2008, Nature photonics, 2, 403
. A Domínguez, 10.1111/j.1365-2966.2010.17631.xMNRAS. 4102556Domínguez A., et al., 2011, MNRAS, 410, 2556
. R Falomo, E Pian, A Treves, 10.1007/s00159-014-0073-zA&ARv. 2273Falomo R., Pian E., Treves A., 2014, A&ARv, 22, 73
. R Fernández, S Gallego, J Francés, I Pascual, A Beléndez, Optical Materials. 4418Fernández R., Gallego S., Francés J., Pascual I., Beléndez A., 2015, Optical Materials, 44, 18
. A Franceschini, G Rodighiero, M Vaccari, 10.1051/0004-6361:200809691A&A. 487837Franceschini A., Rodighiero G., Vaccari M., 2008, A&A, 487, 837
Journal of optics A: pure and applied optics. M R Gleeson, J T Sheridan, 1124008Gleeson M. R., Sheridan J. T., 2009, Journal of optics A: pure and applied optics, 11, 024008
. M R Gleeson, J Guo, J T Sheridan, Optics express. 1922423Gleeson M. R., Guo J., Sheridan J. T., 2011, Optics express, 19, 22423
. H U Käufl, Precision Spectroscopy in Astrophysics. SpringerKäufl H. U., 2008, in , Precision Spectroscopy in Astrophysics. Springer, pp 227-230
. H Kogelnik, Bell Labs Technical Journal. 482909Kogelnik H., 1969, Bell Labs Technical Journal, 48, 2909
. B A Kowalski, R R Mcleod, Journal of Polymer Science Part B: Polymer Physics. 541021Kowalski B. A., McLeod R. R., 2016, Journal of Polymer Science Part B: Polymer Physics, 54, 1021
. M Landoni, R Falomo, A Treves, B Sbarufatti, M Barattini, R Decarli, J Kotilainen, 10.1088/0004-6256/145/4/114AJ. 145114Landoni M., Falomo R., Treves A., Sbarufatti B., Barattini M., Decarli R., Kotilainen J., 2013, AJ, 145, 114
. M Landoni, R Falomo, A Treves, B Sbarufatti, 10.1051/0004-6361/201424232A&A. 570126Landoni M., Falomo R., Treves A., Sbarufatti B., 2014, A&A, 570, A126
. M Landoni, R Falomo, A Treves, R Scarpa, Reverte Payá, D , 10.1088/0004-6256/150/6/181AJ. 150181Landoni M., Falomo R., Treves A., Scarpa R., Reverte Payá D., 2015a, AJ, 150, 181
. M Landoni, R Falomo, A Treves, R Scarpa, D R Payá, The Astronomical Journal. 150181Landoni M., Falomo R., Treves A., Scarpa R., Payá D. R., 2015b, The Astronomical Journal, 150, 181
. M Landoni, A Zanutta, A Bianco, F Tavecchio, G Bonnoli, G Ghisellini, The Astronomical Journal. 15135Landoni M., Zanutta A., Bianco A., Tavecchio F., Bonnoli G., Ghisellini G., 2016a, The Astronomical Journal, 151, 35
. M Landoni, R Falomo, A Treves, R Scarpa, E P Farina, 10.1093/mnras/stv2964MNRAS. 457267Landoni M., Falomo R., Treves A., Scarpa R., Farina E. P., 2016b, MNRAS, 457, 267
. C R Lawrence, J R Zucker, A C S Readhead, S C Unwin, T J Pearson, W Xu, 10.1086/192375ApJS. 107541Lawrence C. R., Zucker J. R., Readhead A. C. S., Unwin S. C., Pearson T. J., Xu W., 1996, ApJS, 107, 541
. J Lawrence, F O'neill, J Sheridan, Optik-International Journal for Light and Electron Optics. 112449Lawrence J., O'Neill F., Sheridan J., 2001, Optik-International Journal for Light and Electron Optics, 112, 449
Instrument Design and Performance for Optical/Infrared Ground-based Telescopes. Le Fèvre, O , 10.1117/12.460959Proc. SPIEVol. 4841. Iye M., Moorwood A. F. M.SPIEVol. 4841Le Fèvre O., et al., 2003, in Iye M., Moorwood A. F. M., eds, Proc. SPIEVol. 4841, Instrument Design and Performance for Optical/Infrared Ground-based Telescopes. pp 1670-1681, doi:10.1117/12.460959
. K.-G Lee, 10.1088/2041-8205/795/1/L12ApJ. 79512Lee K.-G., et al., 2014, ApJ, 795, L12
. H Li, Y Qi, J T Sheridan, 312638Li H., Qi Y., Sheridan J. T., 2014a, JOSA B, 31, 2638
. H Li, Y Qi, J T Sheridan, 312648Li H., Qi Y., Sheridan J. T., 2014b, JOSA B, 31, 2648
. X Liang-Wen, L Shihong, P Bi-Xian, Applied optics. 373678Liang-Wen X., Shihong L., Bi-Xian P., 1998, Applied optics, 37, 3678
. S López, 10.1051/0004-6361/201628161A&A. 59491López S., et al., 2016, A&A, 594, A91
. F Massaro, 10.1007/s10509-016-2926-6Ap&SS. 361337Massaro F., et al., 2016, Ap&SS, 361, 337
I S Mclean, SPIE Astronomical Telescopes+ Instrumentation. McLean I. S., et al., 2012, in SPIE Astronomical Telescopes+ Instrumentation. pp 84460J-84460J
R Mirzoyan, The Astronomer's Telegram. 6999Mirzoyan R., 2015, The Astronomer's Telegram, 6999
. M Moharam, T Gaylord, JOSA. 71811Moharam M., Gaylord T., 1981, JOSA, 71, 811
. M Moharam, L Young, Applied optics. 171757Moharam M., Young L., 1978, Applied optics, 17, 1757
. D A Morozova, 10.1088/0004-6256/148/3/42AJ. 14842Morozova D. A., et al., 2014, AJ, 148, 42
. E Muslimov, N Pavlycheva, G Valyavin, S Fabrika, Astrophysical Bulletin. 71357Muslimov E., Pavlycheva N., Valyavin G., Fabrika S., 2016, As- trophysical Bulletin, 71, 357
. M Ortuño, E Fernández, R Fuentes, S Gallego, I Pascual, A Beléndez, Optical Materials. 35668Ortuño M., Fernández E., Fuentes R., Gallego S., Pascual I., Beléndez A., 2013, Optical Materials, 35, 668
J S Pazder, J C Clemens, SPIE Astronomical Tele-scopes+ Instrumentation. Pazder J. S., Clemens J. C., 2008, in SPIE Astronomical Tele- scopes+ Instrumentation. pp 70184U-70184U
. S Pita, 10.1051/0004-6361/201323071A&A. 56512Pita S., et al., 2014, A&A, 565, A12
. J X Prochaska, M W Lau, J F Hennawi, 10.1088/0004-637X/796/2/140ApJ. 796140Prochaska J. X., Lau M. W., Hennawi J. F., 2014, ApJ, 796, 140
. C M Raiteri, A&A. 35219Raiteri C. M., et al., 1999, A&A, 352, 19
. A Sandrinelli, A Treves, R Falomo, E P Farina, L Foschini, M Landoni, B Sbarufatti, 10.1088/0004-6256/146/6/163AJ. 146163Sandrinelli A., Treves A., Falomo R., Farina E. P., Foschini L., Landoni M., Sbarufatti B., 2013, AJ, 146, 163
. B Sbarufatti, A Treves, R Falomo, J Heidt, J Kotilainen, R Scarpa, 10.1086/427138AJ. 129559Sbarufatti B., Treves A., Falomo R., Heidt J., Kotilainen J., Scarpa R., 2005, AJ, 129, 559
. B Sbarufatti, A Treves, R Falomo, J Heidt, J Kotilainen, R Scarpa, 10.1086/503031AJ. 1321Sbarufatti B., Treves A., Falomo R., Heidt J., Kotilainen J., Scarpa R., 2006, AJ, 132, 1
. M S Shaw, 10.1088/0004-637X/764/2/135ApJ. 764135Shaw M. S., et al., 2013, ApJ, 764, 135
The Messenger. C Snodgrass, I Saviane, L Monaco, P Sinclaire, 13218Snodgrass C., Saviane I., Monaco L., Sinclaire P., 2008, The Mes- senger, 132, 18
. P Spanò, astro-ph/0603763arXiv preprintSpanò P., et al., 2006, arXiv preprint astro-ph/0603763
. M Stickel, J W Fried, H Kuehr, A&AS. 98393Stickel M., Fried J. W., Kuehr H., 1993, A&AS, 98, 393
. J Vernet, 10.1051/0004-6361/201117752A&A. 536105Vernet J., et al., 2011, A&A, 536, A105
S S Vogt, Astronomical Society of the Pacific Conference Series. Bash F. N., Sneden C.2705Vogt S. S., 2002, in Bash F. N., Sneden C., eds, Astronomical Society of the Pacific Conference Series Vol. 270, Astronomical Instrumentation and Astrophysics. p. 5
| [] |
[
"TerraPN: Unstructured terrain navigation through Online Self-Supervised Learning",
"TerraPN: Unstructured terrain navigation through Online Self-Supervised Learning"
] | [
"Adarsh Jagan Sathyamoorthy ",
"Kasun Weerakoon ",
"Tianrui Guan ",
"Jing Liang ",
"Dinesh Manocha "
] | [] | [] | We present TerraPN, a novel method to learn the surface characteristics (texture, bumpiness, deformability, etc.) of complex outdoor terrains for autonomous robot navigation. Our method predicts navigability cost maps for different surfaces using patches of RGB images, odometry, and IMU data. Our method dynamically varies the resolution of the output cost map based on the scene to improve its computational efficiency. We present a novel extension to the Dynamic-Window Approach (DWA-O) to account for a surface's navigability cost while computing robot trajectories. DWA-O also dynamically modulates the robot's acceleration limits based on the variation in the robot-terrain interactions. In terms of perception, our method learns to predict navigability costs in ∼ 20 minutes for five different surfaces, compared to 3-4 hours for previous scene segmentation methods, and leads to a decrease in inference time. In terms of navigation, our method outperforms previous works in terms of vibration costs and generates robot velocities suitable for different surfaces.• A novel method that trains a neural network in an online self-supervised manner to learn a terrain's surface | null | [
"https://arxiv.org/pdf/2202.12873v1.pdf"
] | 247,155,158 | 2202.12873 | 7e15b79dc9a68cb48d9aa5808d34f4caa2fe56b4 |
TerraPN: Unstructured terrain navigation through Online Self-Supervised Learning
Adarsh Jagan Sathyamoorthy
Kasun Weerakoon
Tianrui Guan
Jing Liang
Dinesh Manocha
TerraPN: Unstructured terrain navigation through Online Self-Supervised Learning
We present TerraPN, a novel method to learn the surface characteristics (texture, bumpiness, deformability, etc.) of complex outdoor terrains for autonomous robot navigation. Our method predicts navigability cost maps for different surfaces using patches of RGB images, odometry, and IMU data. Our method dynamically varies the resolution of the output cost map based on the scene to improve its computational efficiency. We present a novel extension to the Dynamic-Window Approach (DWA-O) to account for a surface's navigability cost while computing robot trajectories. DWA-O also dynamically modulates the robot's acceleration limits based on the variation in the robot-terrain interactions. In terms of perception, our method learns to predict navigability costs in ∼ 20 minutes for five different surfaces, compared to 3-4 hours for previous scene segmentation methods, and leads to a decrease in inference time. In terms of navigation, our method outperforms previous works in terms of vibration costs and generates robot velocities suitable for different surfaces.• A novel method that trains a neural network in an online self-supervised manner to learn a terrain's surface
I. INTRODUCTION
Autonomous robots are currently being used for a variety of outdoor applications such as food/grocery delivery, agriculture, surveillance, planetary exploration, etc. Robot navigation methods must account for a terrain's geometric properties such as slope or elevation changes and its surface characteristics such as texture, bumpiness (level of undulations), softness/deformability, etc to compute smooth and efficient robot trajectories.
In addition to a terrain's geometry, its surface properties determine its navigability for a robot. For instance, a surface's texture determines the traction experienced by the robot, its bumpiness determines the vibrations experienced, and deformability determines whether a robot could get stuck (e.g. in sand). Other factors that affect navigability over a terrain involve the robot's properties such as dynamics, inertia, physical dimensions, velocity limits, etc.
Therefore, a precursor to smooth robot navigation on different terrains is perceiving and learning their surface properties. To this end, several works in computer vision, specifically semantic segmentation [1], [2], [3], have demonstrated good terrain classification capabilities on RGB images. However, they rely on large hand-labeled datasets, which do not account for a robot's properties and might misclassify a traversable terrain for a robot as non-traversable. In addition, their classification outputs must be converted into navigability costs to be used for planning and navigation [4], [5].
Self-supervised regression [6], [7] to predict navigability costs using input and label data collected in the real world overcomes the aforementioned limitations. That is, an image (input) can be associated with data vectors collected through [14] (red), TERP [15] (blue), OCR-net (violet), PSP-net (orange). The different shades of TerraPN's trajectory represent our method's different speeds (lighter denotes slower). We observe that our method computes trajectories with low surface navigability costs which navigate the robot in smooth surfaces with higher velocities and bumpy, poor-traction, deformable surfaces with lower velocities. other sensors on the robot (labels) instead of a humanprovided label/annotation. The label vectors generated from real-world sensor data represent the actual ground truth and not a human-perceived ground truth and lead to more accurate characterization of the robot-terrain interaction. Using regression instead of classification would help directly predict a cost useful for navigation. Additionally, training a network using real-world data instead of simulations would avoid any sim-to-real transfer issues.
Existing works on using self-supervised learning (online and offline) for outdoor environments have predominantly focused on unstructured obstacles detection [8], [9], roadway and horizon detection [10], or long-range terrain classification into various categories [11], [12], [13]. Many previous works also train their network offline by first collecting data before training on a powerful GPU [6], [9]. Therefore, understanding and adapting to a terrain in real-time is not possible.
Main Contributions: We present TerraPN, a computationally light online self-supervised learning-based method to compute a surface navigability cost map for efficient robot navigation in outdoor terrains. Using the navigability costmap, we extend the Dynamic Window Approach (DWA) [14] for outdoor surfaces. The novel aspects of our approach are: properties (texture, bumpiness, deformability, etc.) and computes a robot-specific 2D navigability cost map for the terrain. Our method uses RGB image patches (cropped from a full-sized image) and the robot's velocities as inputs and processed 6-DOF IMU measurements and odometry errors as labels. The predicted cost map is a concatenation of n × n patches of costs corresponding to different input RGB patches. Our network trains in ∼ 20 minutes for 5 different surfaces compared to segmentation methods that require 3-4 hours of offline training. Using patches of RGB images leads to a low input dimensionality and makes our method invariant to perspective changes in the camera.
• An algorithm to vary the size of the patches needed to be cropped from the full-sized RGB image to predict the scene's navigability cost map. Our method uses a weak classifier to estimate the number of surfaces in the scene, the regions that predominantly contain a single surface, and crops larger patches from such regions. This leads to a lower number of patches needed for prediction resulting in a reduction in the network's inference time. • An extension to DWA called DWA-O, which accounts for different terrains' surface properties during navigation. DWA-O appropriately modulates the robot's acceleration limits based on a surface's navigability costs and transitions between multiple surfaces. DWA-O also computes velocities with low surface navigability cost for the robot. This leads to smoother trajectories and a reduction in the vibration cost experienced by the robot.
II. RELATED WORKS
In this section, we discuss previous works in computer vision for characterizing a terrain's traversability and methods developed for outdoor navigation.
A. Characterizing Traversability
A few traditional vision works have used methods such as Markov Random fields [16] on fused data from 3D lidar and RGB data and triangular mesh reconstruction [17] from 3D point clouds to analyze surface roughness and traversability. All previous learning-based works in computer vision for traversability prediction fall into a combination of supervised/unsupervised and classification/regression categories (discussed below).
1) Supervised Methods: Works in pixel-wise semantic segmentation classify a terrain into multiple predefined classes such as traversable, non-traversable, obstacle, forbidden, etc. [1], [2], [3]. Fusing a terrain's semantic (visual) and geometric (point cloud) features for better classification has also been studied [3]. These works typically fall under the supervised-classification category and utilize large handlabeled datasets of images [18], [19] to train classifiers. However, manually annotating datasets is time-and laborintensive, not scalable to large amounts of data, and may not be applicable for robots of different sizes, inertias, and dynamics [20]. Such works also assume that terrains that are visually similar have the same traversability [8] without considering the robot's velocities.
2) Self-supervised Methods: Unsupervised learning-based methods overcome the need for such datasets by automating the labeling process by either collecting terrain-interaction data such as forces/torques [6], contact vibrations [21], acoustic data [22], vertical acceleration experienced [23], and stereo depth [11], [12] and associating them with visual features (RGB data) for self-supervision or reinforcement learning [15].
Early work [11], [12] focused on learning long-range (∼ 100m) terrain classification from stereo and RGB image data and 3D lidar points [13] and detecting roadways and the horizon using reverse optical flow [10] in a self-supervised manner. Other works have correlated 3D elevation maps and egocentric RGB images [5] or overhead RGB images [24] for classification.
Few methods have performed self-supervised regression [6], where for each pixel in an image, the corresponding force-torque measurements are predicted post training. [7] presents a method for labeling images with the difference between a robot's actual trajectory and predicted trajectory based on its dynamics model. Ordonez et al. [25] model the interaction between wheeled/tracked robots with pliable outdoor vegetation by mapping RGB data with the resistive forces experienced by the robot. Our method is complementary to these works. We choose regression over terrain classification since it does not limit terrain characterization to a set of classes.
B. Outdoor Navigation
Early works in outdoor navigation proposed using binary classification of obstacles versus free space [26] and potential fields [27] for outdoor collision avoidance. After the advent of deep learning, methods to estimate navigability/energy cost in uneven terrains through imitation learning [28] using egocentric sensor data and a priori environmental information [29] have been proposed.
Several works in deep reinforcement learning (DRL) [30], [31], [32] have proposed end-to-end systems that use data such as elevation maps, depth images, or raw point clouds for perception along with the robot's pose for training navigation networks. Siva et al. [33] propose a method to unify representation and imitation learning to estimate important terrain features for robot adaptation in unstructured environments. Subsequently, [34] addressed navigational setbacks due to wheel slip and reduced tire pressure by learning compensatory behaviors.
BADGR [9] presents an end-to-end DRL-based navigation policy that learns the correlation between different events (such as collisions, bumpiness, and change in position) and the actions performed by the robot. However, such learningbased approaches cannot guarantee any optimality in terms of a navigation metric (minimal cost, path length, dynamic feasibility, etc.). To overcome this limitation, [15] proposed a hybrid model of spatial attention for perception and a dynamically feasible DRL method for navigation. In TerraPN, we use a learning-based approach to develop the perception module and a model-based (extending DWA [14]) approach for navigation. Therefore, TerraPN combines the benefits of accurate characterization of robot-terrain interaction and guaranteed minimal cost dynamically feasible navigation.
III. BACKGROUND
In this section, we define the notations we use, detail how the inputs and labels for our network are collected autonomously, and show how they are processed before being fed into our novel network.
A. Problem Formulation
The proposed formulation has two main components: cost map prediction and navigation. For prediction, a neural network is trained to perform a self-supervised regression task, i.e., it learns a terrain's surface properties by correlating its two inputs (RGB images and the robot's velocity data) with the vibrations and the odometry errors experienced by the robot while traversing it.
Once trained, the network's predicted cost map is used by the navigation component to compute dynamically feasible trajectories with low surface costs. Our method's overall system architecture is shown in Fig. 3. We use i, j for denoting various indices, and x, y to denote positions relative to different coordinate frames. We highlight the frequently used symbols and notation in Table I.
B. Autonomous Data Collection
To generate the input and label data for training the cost map prediction network, we collect the raw sensor data from and RGB camera, robot's odometry, 6-DOF IMU, and 3D lidar autonomously on different surfaces. The robot performs a set of maneuvers in two different speed ranges: 1. slow ([0, If the robot encounters an obstacle, it switches from executing the maneuvers to avoiding a collision using DWA [14]. The maneuvers are designed to cover all the [v, ω] pairs within the robot's velocity limits to emulate all possible terrain interactions while the data is collected. We observe that the variances along the two principal axes clearly differentiate between different surfaces and speed levels.
C. Computing Inputs and Labels
Our network's image input is an n × n patch (I n×n RGB ) that is cropped from the center-bottom of the collected full-sized image of size w × h (I w×h RGB ). To generate the velocity input, the linear and angular velocities for the past n/2 instances from the robot's odometry are obtained and reshaped to a 2 × n/2 vector.
Our label vector consists of IMU and odometry error components. They are robot-specific and implicitly encode the robot's dynamics, inertia, etc., and its interactions with the terrain. To generate the IMU component, we apply Principal Component Analysis (PCA) on the collected 6dimensional IMU data (linear accelerations and angular velocities) to obtain two principal components. From Fig. 2, we observe that the variances (σ P C1 and σ P C2 ) of the data along the principal components help differentiate various surfaces. Additionally, for the same surface, higher velocities lead to higher variances in the data (justifying the need to include velocities as inputs).
To generate the odometry error component of the label vector, the distance traveled by the robot (∆d odom ) and its change in orientation (∆θ odom ) in a time interval ∆t are obtained from the robot's odometry. We obtain the same data (∆d loam , ∆θ loam ) from a 3D lidar-based odometry and mapping system [35]. The distance and orientation change errors are calculated as,
d error = ∆d loam − ∆d odom (1) θ error = ∆θ loam − ∆θ odom .(2)
This component of the label vector differentiates surfaces with high deformability or poor traction where, if the robot's wheels get stuck or slip, ∆d loam ≈ 0 and ∆θ loam ≈ 0, whereas ∆d odom and ∆θ odom would have high values. The final label vector is given by,
l = [σ P C1 σ P C2 d error θ error ] .(3)
IV. SURFACE NAVIGABILITY PREDICTION
In this section, we describe TerraPN's cost map prediction, its novel network architecture, how the output cost map is computed and a novel method to vary the output's resolution.
A. Network Architecture and Online Training
The network (see Fig. 4) predicts the 4 × 1 vector in equation 3 given the image and velocity inputs. The architecture uses a series of 2D convolution with skip connections and batch normalization on the image input, and several layers of fully connected layers with dropout and batch normalization for the velocity input, shown in Fig. 4. In the image stream, after the initial convolution operation with ReLU activation, the image is connected to four residual blocks and one linear layer with batch normalization. Since we have limited data when collecting labels and training the network online, we use residual connection to make sure that the network would not overfit to the collected data given many layers and parameters in the network. Dropout and batch normalization layers also improve generalization capabilities and avoid overfitting.
As the robot autonomously collects sensor data on different surfaces and the inputs and labels are generated and shuffled. The network's training is started and performed online once sufficiently varied data is collected (∼ 6 minutes). The data collection and online training typically completes in 20 − 25 minutes.
B. Navigability Cost Calculation
Based on the vector predicted by the network (l = [σ P C1σP C2derrorθerror ] T ), the navigability cost for a given RGB patch and velocity vector is computed as the weighted norm ofl,
c = lT Wl(4)
Here, W is a diagonal matrix with positive weights. To make navigability cost predictions on a full-sized RGB image I w×h RGB , the image is first resized into new dimensions as follows,
w = w/n · n, h = h/n · n,(5)
where is the nearest integer operator. Next, nonoverlapping n × n patches are cropped along the width and height of the image. This w/n · h/n batch of images is passed as inputs along with a batch containing the input velocity vectors to obtain the navigability cost predictions c ij (i ∈ [1, w/n ] and j ∈ [1, h/n ]) for different regions of the resized image I w ×h RGB . The predicted costs are normalized to be in the range [0, π/2].
Finally, the surface navigability cost map C w ×h is constructed by vertically and horizontally concatenating n × n patches with the values of c ij corresponding to different regions. C w ×h is then resized back to C w×h .
C. Varying Patch Size
To efficiently manage the time required to compute the navigability cost in a full-sized image, we vary the patch size and the total number of patches used for the computation. That is, certain portions of the image would have larger patch sizes 2n, 4n and so on, based on the number of surfaces in the full image.
1) Weak Segmentation: To predict the number of surfaces and differentiate them in a scene, we use the following weak segmentation approach. First, the Sobel edge detector is applied to the grayscale image I w ×h gray of the scene and the histogram of the result is computed. Next, based on the Bayesian information criterion [36] a Gaussian mixture model is fit to the histogram, and the mean of each Gaussian curve is used as a marker/threshold (µ 1 , µ 2 , ..., µ k ) to differentiate the regions of pixels with different intensity levels in the image. Finally, the watershed filter [37] is applied to highlight the regions of different intensities to obtain I w×h W S (See figure 3).
2) Condition for Modulating Resolution: We consider the patches I n×n W S , I 2n×2n
W S , I 4n×4n W S
in the weak segmentation output I w ×h W S and the intensity of pixels within them. If a patch larger than n × n satisfies the following condition in 6, smaller patches are not considered for cost prediction. where num(µ i ) is the number of pixels with intensity greater than or equal to µ i , and ξ is a threshold. This condition ensures that when a large patch has a significant number of pixels with the same intensity, implying the presence of a single surface, smaller patches are not used for cost prediction. The larger patch is resized to n × n before passing into the network.
num(µ i ) n 2 > ξ i ∈ [1, k].(6)
V. OUTDOOR DYNAMIC WINDOW APPROACH
In this section, we briefly provide details about DWA [14] and explain how the computed navigability cost map is used to extend DWA for outdoor terrains with different surfaces.
A. Dynamic Window Approach
DWA uses two major stages for computing dynamically feasible collision-free robot velocities: 1. computing a collision-free, dynamically feasible velocity search space, and 2. choosing a velocity in the search space to maximize an objective function.
In the first stage, all possible velocities in V s = [v ∈ [0, v max ], ω ∈ [−ω max , ω max ], respectively, are considered for the search space V s . Next, all the (v, ω) pairs that prevent a collision in V s are used to form the admissible velocity set V a . Lastly, the velocity pairs that are reachable, accounting for the robot's acceleration limitsv max ,ω max within a short time interval ∆t, are considered to construct a dynamic window set V d . The resulting search space is constructed
as V r = V s ∩ V a ∩ V d .
In the second stage, DWA searches for (v, ω) ∈ V r , which maximizes an objective function. The objective function is a weighted sum of three terms: 1. heading(v, ω), 2. dist(v, ω), and 3. vel(v, ω). Here, heading() measures the progress towards the robot's goal, dist() is the distance to the closest obstacle when executing a certain (v, ω), and vel() measures the forward velocity of the robot and encourages higher velocities.
B. Trajectory Navigability Cost
To adapt DWA to outdoor terrains, the trajectory corresponding to a (v, ω) pair must be associated with a surface navigability cost. The trajectory for a given (v, ω) pair relative to a coordinate frame attached to the robot is calculated as,
x rob i = v cos (ωt i )t i y rob i = v sin (ωt i )t i t i = t 0 + i∆t, i ∈ [0, s num ].(7)
Here, t 0 is the initial time instant and s num is the number of time steps used for extrapolating the trajectory. ] ) using the camera's intrinsic parameters. Since the frames of the cost map C w ×h and I w ×h RGB coincide, the navigability cost for a velocity pair can be computed as,
sur(v, ω) = snum i=0 cost(x img i , y img i ).(8)
Here, cost() is the cost at a given pixel's coordinates.
C. Variable Acceleration Limits
Robot navigation methods consider a constant range of linear ([−v max ,v max ]) and angular ([−ω max ,ω max ]) accelerations. Our formulation varies the linear and angular acceleration limits available for planning depending on the properties of the surface on which the robot is traversing
such thatv ∈ [−v max ,v lim ] andω ∈ [−ω max ,ω lim ].
This is done because, intuitively, the robot accelerating on a smooth surface (e.g., concrete, asphalt) would lead to a low navigability cost. Therefore, the robot can proceed towards its goal faster. Whereas on a bumpy surface or one with poor traction, (e.g., tiled surface, dry leaves), accelerating would lead to high vibration costs and the risk of getting stuck (e.g. sand). We do not limit the maximum deceleration available to the robot since it may have to slow down to avoid obstacles or while moving on a rough surface.
First, we divide the trajectory corresponding to the robot's current (v, ω) and calculate the cost for the second half as follows,
We limit the robot's accelerations using this navigability cost as, v lim = τ ·v max , (10) ω lim = τ ·ω max , (11) τ = cos (C snum 2 :snum ), C snum 2 :snum ∈ [0, π/2].
If C snum 2 +1:snum is low (low-cost surface), the robot is allowed to accelerate towards its goal, while a high C snum 2 +1:snum restricts the robot from speeding up. Considering only the second half of the trajectory also implicitly accounts for transitions between surfaces. Using these acceleration limits, a new dynamic window V d is constructed. The new resultant search space is calculated as V r = V s ∩V a ∩V d .
D. Optimization
Finally, a (v, ω) ∈ V r which maximizes the objective function G(v, ω) = S(α.heading · (1 − δ.sur)) + β.dist+ γ.vel, (13) is chosen. Here, α, β, γ, δ are weights for each component and S is a smoothening function for the weighted sum.
Fig. 1 :
1Our method's cost map prediction (top right) for the robot's perspective (top left) in an unstructured outdoor terrain. The trajectories of our method, TerraPN (green), DWA
vmax 2 ] m/s and [-ωmax 2 , ωmax 2 ] rad/s), and 2. fast ([0, v max ] m/s and [−ω max , ω max ] rad/s). The maneuvers include: 1. moving along a rectangular path, 2. moving in a serpentine trajectory, and 3. random motion.
Fig. 2 :
2Results of the PCA applied on the 6-dimensional IMU data.
Fig. 3 :
3TerraPN's overall system architecture.
Fig. 4 :
4Our novel two-stream network architecture. One stream encodes image patches (green) and the other stream processes the linear and angular velocities (yellow). The feature embeddings from each stream are concatenated and finally passed into a set of linear layers (blue), and the final prediction is the odometry error associated with this image patch.
Fig. 5 :
5Surface cost-map predictions for different scenarios with 2-3 surfaces. We consider resolution modulated non-overlapping patches on the RGB images and feed them into our network with the robot's velocities. We observe that different surfaces are categorized well based on their navigability (dark blue being low cost and therefore better navigable surface and lighter shades for high cost).
Proposition V. 1 .
1For a given environmental configuration, DWA-O generates collision-free, dynamically feasible trajectories with a surface cost lesser than or equal to DWA's trajectory's surface cost.Proof. This result follows from two facts:1. DWA-O's acceleration limits obey [−v max ,v lim ] ⊆ [−v max ,v max ] as τ ∈ [0, 1]and only the velocities in the admissible space V a are considered, and 2. For the same goal, robot and obstacle configuration, heading(), dist() and vel() are fixed. Then maximizing G is equivalent to minimizing sur().
RGB image patch and full-sized image. C w×h Predicted navigability cost map for I w×h RGB . v lim ,ω lim Linear and angular acceleration limits computed by DWA-O.Symbols
Definitions
v, ω,v,ω
Linear and angular velocities and
accelerations respectively.
vmax, ωmax,vmax,ωmax
Robot's maximum linear, angular
velocities and accelerations
respectively.
I n×n
RGB , I w×h
RGB
TABLE I :
IList of symbols used in our approach.
This trajectory is then transformed relative to the camera frame attached to the robot using a homogeneous transformation matrix as [x cam ] . Next, the trajectory is converted to correspond to the image/pixel coordinates of I w ×h RGB ([x imgi
y cam
i
] = H cam
rob · [x rob
i
y rob
i i
y img
i
Ganav: Group-wise attention network for classifying navigable regions in unstructured outdoor environments. T Guan, D Kothandaraman, R Chandra, A J Sathyamoorthy, D Manocha, T. Guan, D. Kothandaraman, R. Chandra, A. J. Sathyamoorthy, and D. Manocha, "Ganav: Group-wise attention network for classifying navigable regions in unstructured outdoor environments," 2021.
T Guan, Z He, D Manocha, L Zhang, arXiv:2109.06250TTM: Terrain Traversability Mapping for Autonomous Excavators. arXiv e-printsT. Guan, Z. He, D. Manocha, and L. Zhang, "TTM: Terrain Traversability Mapping for Autonomous Excavators," arXiv e-prints, p. arXiv:2109.06250, Sept. 2021.
Geometric and visual terrain classification for autonomous mobile navigation. F Schilling, X Chen, J Folkesson, P Jensfelt, 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS. F. Schilling, X. Chen, J. Folkesson, and P. Jensfelt, "Geometric and visual terrain classification for autonomous mobile navigation," in 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2017, pp. 2678-2684.
Real-time semantic mapping for autonomous off-road navigation. D Maturana, P.-W Chou, M Uenoyama, S Scherer, Proceedings of 11th International Conference on Field and Service Robotics (FSR '17). 11th International Conference on Field and Service Robotics (FSR '17)D. Maturana, P.-W. Chou, M. Uenoyama, and S. Scherer, "Real-time semantic mapping for autonomous off-road navigation," in Proceed- ings of 11th International Conference on Field and Service Robotics (FSR '17), September 2017, pp. 335 -350.
Traversability analysis using terrain mapping and online-trained terrain type classifier. H Roncancio, M Becker, A Broggi, S Cattani, 2014 IEEE Intelligent Vehicles Symposium Proceedings. H. Roncancio, M. Becker, A. Broggi, and S. Cattani, "Traversability analysis using terrain mapping and online-trained terrain type classi- fier," in 2014 IEEE Intelligent Vehicles Symposium Proceedings, 2014, pp. 1239-1244.
Where should i walk? predicting terrain properties from images via self-supervised learning. L Wellhausen, A Dosovitskiy, R Ranftl, K Walas, C Cadena, M Hutter, IEEE Robotics and Automation Letters. 42L. Wellhausen, A. Dosovitskiy, R. Ranftl, K. Walas, C. Cadena, and M. Hutter, "Where should i walk? predicting terrain properties from images via self-supervised learning," IEEE Robotics and Automation Letters, vol. 4, no. 2, pp. 1509-1516, 2019.
Complex Terrain Navigation via Model Error Prediction. A Polevoy, C Knuth, K M Popek, K D , arXiv:2111.09768arXiv e-printsA. Polevoy, C. Knuth, K. M. Popek, and K. D. Katyal, "Complex Terrain Navigation via Model Error Prediction," arXiv e-prints, p. arXiv:2111.09768, Nov. 2021.
Traversability classification using unsupervised on-line visual learning for outdoor robot navigation. D Kim, J Sun, S M Oh, J Rehg, A Bobick, Proceedings 2006 IEEE International Conference on Robotics and Automation. 2006 IEEE International Conference on Robotics and AutomationD. Kim, J. Sun, S. M. Oh, J. Rehg, and A. Bobick, "Traversability classification using unsupervised on-line visual learning for outdoor robot navigation," in Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006., 2006, pp. 518-525.
Badgr: An autonomous selfsupervised learning-based navigation system. G Kahn, P Abbeel, S Levine, G. Kahn, P. Abbeel, and S. Levine, "Badgr: An autonomous self- supervised learning-based navigation system," 2020.
Reverse optical flow for self-supervised adaptive autonomous robot navigation. A Lookingbill, J Rogers, D Lieb, J Curry, S Thrun, International Journal of Computer Vision. 74A. Lookingbill, J. Rogers, D. Lieb, J. Curry, and S. Thrun, "Reverse optical flow for self-supervised adaptive autonomous robot naviga- tion," International Journal of Computer Vision, vol. 74, pp. 287-302, 01 2007.
Learning long-range perception using self-supervision from short-range sensors and odometry. M Nava, J Guzzi, R O Chavez-Garcia, L M Gambardella, A Giusti, IEEE Robotics and Automation Letters. 42M. Nava, J. Guzzi, R. O. Chavez-Garcia, L. M. Gambardella, and A. Giusti, "Learning long-range perception using self-supervision from short-range sensors and odometry," IEEE Robotics and Automation Letters, vol. 4, no. 2, pp. 1279-1286, 2019.
Learning terrain segmentation with classifier ensembles for autonomous robot navigation in unstructured environments. M Procopio, J Mulligan, G Grudic, Journal of Field Robotics. 26M. Procopio, J. Mulligan, and G. Grudic, "Learning terrain segmen- tation with classifier ensembles for autonomous robot navigation in unstructured environments," Journal of Field Robotics, vol. 26, pp. 145 -175, 02 2009.
Self-supervised learning to visually detect terrain surfaces for autonomous robots operating in forested terrain. S Zhou, J Xi, M W Mcdaniel, T Nishihata, P Salesses, K Iagnemma, J. Field Robotics. 29S. Zhou, J. Xi, M. W. McDaniel, T. Nishihata, P. Salesses, and K. Iagnemma, "Self-supervised learning to visually detect terrain surfaces for autonomous robots operating in forested terrain," J. Field Robotics, vol. 29, pp. 277-297, 2012.
The dynamic window approach to collision avoidance. D Fox, W Burgard, S Thrun, IEEE Robotics Automation Magazine. 41D. Fox, W. Burgard, and S. Thrun, "The dynamic window approach to collision avoidance," IEEE Robotics Automation Magazine, vol. 4, no. 1, pp. 23-33, March 1997.
Terp: Reliable planning in uneven outdoor environments using deep reinforcement learning. K Weerakoon, A J Sathyamoorthy, U Patel, D Manocha, K. Weerakoon, A. J. Sathyamoorthy, U. Patel, and D. Manocha, "Terp: Reliable planning in uneven outdoor environments using deep reinforcement learning," 2021.
Probabilistic terrain classification in unstructured environments. M Häselich, M Arends, N Wojke, F Neuhaus, D Paulus, Robotics and Autonomous Systems. 612013M. Häselich, M. Arends, N. Wojke, F. Neuhaus, and D. Paulus, "Prob- abilistic terrain classification in unstructured environments," Robotics and Autonomous Systems, vol. 61, pp. 1051-1059, 10 2013.
3d navigation mesh generation for path planning in uneven terrain. S Pütz, T Wiemann, J Sprickerhof, J Hertzberg, 9th IFAC Symposium on Intelligent Autonomous Vehicles IAV 2016. 49S. Pütz, T. Wiemann, J. Sprickerhof, and J. Hertzberg, "3d navigation mesh generation for path planning in uneven terrain," IFAC-PapersOnLine, vol. 49, no. 15, pp. 212-217, 2016, 9th IFAC Symposium on Intelligent Autonomous Vehicles IAV 2016. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S2405896316310102
A rugd dataset for autonomous navigation and visual perception in unstructured outdoor environments. M Wigness, S Eum, J G Rogers, D Han, H Kwon, International Conference on Intelligent Robots and Systems (IROS). M. Wigness, S. Eum, J. G. Rogers, D. Han, and H. Kwon, "A rugd dataset for autonomous navigation and visual perception in unstructured outdoor environments," in International Conference on Intelligent Robots and Systems (IROS), 2019.
Rellis-3d dataset: Data, benchmarks and analysis. P Jiang, P Osteen, M Wigness, S Saripalli, P. Jiang, P. Osteen, M. Wigness, and S. Saripalli, "Rellis-3d dataset: Data, benchmarks and analysis," 2020.
Safe robot navigation via multi-modal anomaly detection. L Wellhausen, R Ranftl, M Hutter, IEEE Robotics and Automation Letters. 52L. Wellhausen, R. Ranftl, and M. Hutter, "Safe robot navigation via multi-modal anomaly detection," IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 1326-1333, 2020.
Self-supervised classification for planetary rover terrain sensing. C A Brooks, K D Iagnemma, 2007 IEEE Aerospace Conference. C. A. Brooks and K. D. Iagnemma, "Self-supervised classification for planetary rover terrain sensing," in 2007 IEEE Aerospace Conference, 2007, pp. 1-9.
A new terrain classification framework using proprioceptive sensors for mobile robots. K Zhao, M Dong, L Gu, Mathematical Problems in Engineering. 2017K. Zhao, M. Dong, and L. Gu, "A new terrain classification frame- work using proprioceptive sensors for mobile robots," Mathematical Problems in Engineering, vol. 2017, pp. 1-14, 09 2017.
Terrain traversability analysis using multi-sensor data correlation by a mobile robot. M A Bekhti, Y Kobayashi, K Matsumura, 2014M. A. Bekhti, Y. Kobayashi, and K. Matsumura, "Terrain traversability analysis using multi-sensor data correlation by a mobile robot," in 2014
IEEE/SICE International Symposium on System Integration. IEEE/SICE International Symposium on System Integration, 2014, pp. 615-620.
Improving robot navigation through self-supervised online learning. B Sofman, E L Ratliff, J A D Bagnell, N Vandapel, A T Stentz, Proceedings of Robotics: Science and Systems (RSS '06). Robotics: Science and Systems (RSS '06)B. Sofman, E. L. Ratliff, J. A. D. Bagnell, N. Vandapel, and A. T. Stentz, "Improving robot navigation through self-supervised online learning," in Proceedings of Robotics: Science and Systems (RSS '06), August 2006.
Characterization and traversal of pliable vegetation for robot navigation. C Ordonez, R Alicea, B Rothrock, K Ladyko, J Nash, R Thakker, S Daftry, M Harper, E Collins, L Matthies, 11C. Ordonez, R. Alicea, B. Rothrock, K. Ladyko, J. Nash, R. Thakker, S. Daftry, M. Harper, E. Collins, and L. Matthies, "Characterization and traversal of pliable vegetation for robot navigation," 11 2018.
An autonomous path planner implemented on the rocky 7 prototype microrover. S Laubach, J Burdick, L Matthies, Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146). 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146)1S. Laubach, J. Burdick, and L. Matthies, "An autonomous path planner implemented on the rocky 7 prototype microrover," in Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146), vol. 1, 1998, pp. 292-297 vol.1.
Potential field navigation of high speed unmanned ground vehicles on uneven terrain. S Shimoda, Y Kuroda, K Iagnemma, Proceedings of the 2005 IEEE International Conference on Robotics and Automation. the 2005 IEEE International Conference on Robotics and AutomationS. Shimoda, Y. Kuroda, and K. Iagnemma, "Potential field navigation of high speed unmanned ground vehicles on uneven terrain," in Proceedings of the 2005 IEEE International Conference on Robotics and Automation, 2005, pp. 2828-2833.
Learning from demonstration for autonomous navigation in complex unstructured terrain. D Silver, J A Bagnell, A Stentz, The International Journal of Robotics Research. 2912D. Silver, J. A. Bagnell, and A. Stentz, "Learning from demonstration for autonomous navigation in complex unstructured terrain," The International Journal of Robotics Research, vol. 29, no. 12, pp. 1565- 1592, 2010.
Energy-efficient path planning algorithm on three-dimensional large-scale terrain maps for mobile robots. K Zakharov, A Saveliev, O Sivchenko, International Conference on Interactive Collaborative Robotics. SpringerK. Zakharov, A. Saveliev, and O. Sivchenko, "Energy-efficient path planning algorithm on three-dimensional large-scale terrain maps for mobile robots," in International Conference on Interactive Collabora- tive Robotics. Springer, 2020, pp. 319-330.
Learning-based methods of perception and navigation for ground vehicles in unstructured environments: a review. D C Guastella, G Muscato, Sensors. 21173D. C. Guastella and G. Muscato, "Learning-based methods of percep- tion and navigation for ground vehicles in unstructured environments: a review," Sensors, vol. 21, no. 1, p. 73, 2021.
Robot navigation of environments with unknown rough terrain using deep reinforcement learning. K Zhang, F Niroui, M Ficocelli, G Nejat, 2018 IEEE International Symposium on Safety, Security, and Rescue Robotics (SSRR). IEEEK. Zhang, F. Niroui, M. Ficocelli, and G. Nejat, "Robot navigation of environments with unknown rough terrain using deep reinforcement learning," in 2018 IEEE International Symposium on Safety, Security, and Rescue Robotics (SSRR). IEEE, 2018, pp. 1-7.
Autonomous navigation in complex environments with deep multimodal fusion network. A Nguyen, N Nguyen, K Tran, E Tjiputra, Q D Tran, 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). A. Nguyen, N. Nguyen, K. Tran, E. Tjiputra, and Q. D. Tran, "Autonomous navigation in complex environments with deep multi- modal fusion network," in 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2020, pp. 5824-5830.
Robot adaptation to unstructured terrains by joint representation and apprenticeship learning. S Siva, M Wigness, J Rogers, H Zhang, S. Siva, M. Wigness, J. Rogers, and H. Zhang, "Robot adaptation to unstructured terrains by joint representation and apprenticeship learning," 7 2019. [Online]. Available: https://www.osti.gov/biblio/ 1560517
Robot adaptation for generating consistent navigational behaviors over unstructured offroad terrain. S Siva, M Wigness, J G Rogers, H Zhang, S. Siva, M. Wigness, J. G. Rogers, and H. Zhang, "Robot adaptation for generating consistent navigational behaviors over unstructured off- road terrain," 2021.
Lego-loam: Lightweight and groundoptimized lidar odometry and mapping on variable terrain. T Shan, B Englot, IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEET. Shan and B. Englot, "Lego-loam: Lightweight and ground- optimized lidar odometry and mapping on variable terrain," in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2018, pp. 4758-4765.
Estimating the dimension of a model. G Schwarz, The annals of statistics. G. Schwarz, "Estimating the dimension of a model," The annals of statistics, pp. 461-464, 1978.
Watersheds in digital spaces: an efficient algorithm based on immersion simulations. L Vincent, P Soille, IEEE Transactions on Pattern Analysis and Machine Intelligence. 136L. Vincent and P. Soille, "Watersheds in digital spaces: an efficient algorithm based on immersion simulations," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 6, pp. 583- 598, 1991.
| [] |
[
"THE ZONOID ALGEBRA, GENERALIZED MIXED VOLUMES, AND RANDOM DETERMINANTS",
"THE ZONOID ALGEBRA, GENERALIZED MIXED VOLUMES, AND RANDOM DETERMINANTS"
] | [
"Paul Breiding ",
"Peter Bürgisser ",
"Antonio Lerario ",
"Léo Mathis "
] | [] | [] | We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordered ring, which we call the zonoid algebra. This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. We also analyze a similar construction based on the complex wedge product, which leads to the new notion of mixed J-volume. These ideas connect to the theory of random determinants. 1 We conclude by pointing out that this work lays out the foundations for the follow-up article [BBML21]. This will aim at developing a probabilistic intersection theory for compact homogeneous spaces, in which the zonoid algebra takes the role played by the Chow ring or the cohomology ring in the classical case. To each subvariety in the space we associate a zonoid in the exterior algebra of the tangent space at a distinguished point. The codimension of the variety equals the degree of the exterior power in which the zonoid lives. Then, the intersection of subvarieties in random position can be identified with the product of their zonoids. This allows to develop further the probabilistic Schubert calculus initiated in [BL20].Outline. In Section 2 we recall basic facts about zonoids, emphasizing how to represent them by random vectors following an approach by Vitale. We discuss the notion of the length of zonoids and discuss some of its properties. Moreover, we define the vector space of virtual zonoids and study its topology. In Section 3 we give the definition of the tensor product map for zonoids, and we discuss its continuity. In Section 4 we introduce and discuss the zonoid algebra, and we prove that any multilinear map induces a unique multilinear map of virtual zonoids. In Section 5 we explain how to relate our construction to intrinsic volumes, mixed volumes and random determinants. Finally, in Section 6 we introduce and study the new notion of mixed J-volume. | 10.1016/j.aim.2022.108361 | [
"https://arxiv.org/pdf/2109.14996v3.pdf"
] | 238,226,967 | 2109.14996 | f36a7e9fd552e220c91c19f172d52de81de2bb27 |
THE ZONOID ALGEBRA, GENERALIZED MIXED VOLUMES, AND RANDOM DETERMINANTS
18 Mar 2022
Paul Breiding
Peter Bürgisser
Antonio Lerario
Léo Mathis
THE ZONOID ALGEBRA, GENERALIZED MIXED VOLUMES, AND RANDOM DETERMINANTS
18 Mar 2022arXiv:2109.14996v3 [math.MG]
We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordered ring, which we call the zonoid algebra. This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. We also analyze a similar construction based on the complex wedge product, which leads to the new notion of mixed J-volume. These ideas connect to the theory of random determinants. 1 We conclude by pointing out that this work lays out the foundations for the follow-up article [BBML21]. This will aim at developing a probabilistic intersection theory for compact homogeneous spaces, in which the zonoid algebra takes the role played by the Chow ring or the cohomology ring in the classical case. To each subvariety in the space we associate a zonoid in the exterior algebra of the tangent space at a distinguished point. The codimension of the variety equals the degree of the exterior power in which the zonoid lives. Then, the intersection of subvarieties in random position can be identified with the product of their zonoids. This allows to develop further the probabilistic Schubert calculus initiated in [BL20].Outline. In Section 2 we recall basic facts about zonoids, emphasizing how to represent them by random vectors following an approach by Vitale. We discuss the notion of the length of zonoids and discuss some of its properties. Moreover, we define the vector space of virtual zonoids and study its topology. In Section 3 we give the definition of the tensor product map for zonoids, and we discuss its continuity. In Section 4 we introduce and discuss the zonoid algebra, and we prove that any multilinear map induces a unique multilinear map of virtual zonoids. In Section 5 we explain how to relate our construction to intrinsic volumes, mixed volumes and random determinants. Finally, in Section 6 we introduce and study the new notion of mixed J-volume.
Introduction
This paper is at the interface of convex geometry and probability. It links the three topics zonoids, mixed volumes and random determinants under a general new framework.
In convex geometry, sums of line segments are called zonotopes and their limits with respect to the Hausdorff metric are called zonoids. Zonoids form a special, yet fundamental class of convex bodies [Sch14]. In functional analysis, probability, combinatorics, and algebraic geometry, it is sometimes possible to reformulate questions in terms of convex bodies, typically involving the notion of mixed volume. In this paper we show that there is a richer multiplicative structure as soon as we confine ourselves to zonoids.
Let V be a Euclidean space of dimension m and Z(V ) denote the set of its centrally symmetric zonoids, which in the following we simply refer to as zonoids. On Z(V ) we have the Minkowski addition and the multiplication with nonnegative reals, and the Hausdorff distance makes it a complete metric space. We study more closely the tensor products of zonoids. More specifically, if V 1 , . . . , V p are Euclidean spaces, we define a Minkowski multilinear map
(1.1) Z(V 1 ) × · · · × Z(V p ) Z(V 1 ⊗ · · · ⊗ V p ),
which is monotone with respect to inclusion and continuous (this is the same tensor product that was defined earlier by Aubrun and Lancien [AL16]). Based on the tensor product, we prove in Theorem 4.1 that each multilinear map between Euclidean spaces induces a multilinear map of the corresponding spaces of zonoids. The resulting maps are again continuous and monotone with respect to inclusion. A main contribution of this paper is to define and study the zonoid algebra. For this, we first pass from the cone of zonoids Z(V ) to a vector space Z(V ), which we call virtual zonoids. In simple terms, Z(V ) consists of formal differences of elements in Z(V ), but this abstract vector space can be naturally embedded into various classical vector spaces, e.g. in the space of measures, or the space of functions on the sphere, see Sections 2.3 and 2.4. In Section 4 we introduce the zonoid algebra over V . As a vector space this is the direct sum of the vector spaces of virtual zonoids in all exterior powers of V :
A(V ) := m k=0 Z(Λ k V ).
The multiplication in A(V ) is the multilinear map on zonoids induced by the wedge products ∧ : Λ k (V ) × Λ l (V ) Λ k+l (V ), (v, w) v ∧ w for k + l ≤ m. Thus, if K ∈ Z(Λ k (V )) and L ∈ Z(Λ l (V )) are zonoids, we can multiply them and get the zonoid K ∧ L ∈ Z(Λ k+l V ). In particular, the product K 1 ∧ · · · ∧ K p of d zonoids K 1 , . . . , K d ∈ Z(V ) is a zonoid in Λ d V .
The zonoid algebra A(V ) has a rich structure: we prove in Theorem 4.5 that it is a graded, commutative partially ordered algebra. While the Minkowski multilinear map (1.1) is continuous, its extension to virtual zonoids requires careful topological considerations, which we discuss in Section 3.1. We also introduce a subalgebra G(V ) (Definition 4.10), which we call Grassmann zonoid algebra, and which plays a crucial role in the probabilistic intersection theory developed in our follow-up work [BBML21].
The connection to the mixed volume enters via the concept of the length ℓ : A(V ) R, which is a monotone, continuous linear functional; see Definition 2.10 and Theorem 2.22. In fact, the length of a zonoid is its first intrinsic volume. More generally, we show in Theorem 5.2 that if K ∈ Z(V ) is a zonoid, then 1 d! ℓ(K ∧d ) equals its d-th intrinsic volume. Moreover, we show in Theorem 5.1 that, if K 1 , . . . , K m are zonoids in V , then their mixed volume satisfies MV(K 1 , . . . , K m ) = 1 m! ℓ(K 1 ∧ · · · ∧ K m ).
For higher degree zonoids, i.e., zonoids in Λ k (V ) with 1 < k < m, the length of their product therefore naturally defines a generalized mixed volume.
For zonoids in C n ∼ = R 2n , we can repeat the same construction using the complex wedge product, seen as a real multilinear map, ∧ C : Λ k (C n ) × Λ l (C n ) Λ k+l (C n ). In Section 6 we define what we call the mixed J-volume (1.2) MV J (K 1 , . . . , K n ) := 1 n! ℓ(K 1 ∧ C · · · ∧ C K n ),
where K 1 , . . . , K n are zonoids in C n ∼ = R 2n . When the K i are real, i.e., contained in R n ⊆ C n , then we retrieve the classical mixed volume MV. Furthermore, we prove in Proposition 6.18 that the J-volume vol J n (K) := 1 n! ℓ(K ∧ C n ) can be extended to polytopes and that it is a weakly continuous, translation invariant valuation. By contrast, using a recent result by Wannerer [Wan20], we prove in Corollary 6.20 that vol J n cannot be extended to a continuous valuation on all convex bodies in C n . We also relate our definition to Kazarnovskii's pseudovolume [Kaz04]; see Definition 6.21.
We apply our theory to the study of the expected absolute value | det[X 1 , . . . , X n ]| of the determinant of random matrices, whose column vectors X 1 , . . . , X m ∈ R m are random vectors. Vitale [Vit91] gave a formula for this expectation in the case where the X i are i.i.d. random vectors. His formula is in terms of the volume of a zonoid: to each random vector X in R m which is integrable, i.e., E X < ∞, there corresponds an associated zonoid K(X); see Definition 2.3. Although X is a random vector, K(X) is deterministic. Vitale's result asserts that E det(M ) = m! · vol m (K(X)), where M = [ X 1 ... Xm ], with i.i.d. random variables X i having the same distribution as X, and where vol m denotes the m-dimensional volume in R m . It is important to emphasize that Vitale's result makes no assumption on the independence of the entries of the random vector X. But the columns must be independent, and identically distributed. If the columns are independent but not identically distributed, we can generalize Vitale's result as follows:
(1.3) E X 1 ,...,Xm are independent det(M ) = ℓ(K(X 1 ) ∧ · · · ∧ K(X m )),
where M = det[X 1 , . . . , X m ].
If the columns are not independent, the situation is far more complicated. Explicit formulas are known for special cases; see, e.g., [Gir90], but general formulas like (1.3) so far were not available. We fill this gap by showing in Theorem 5.4 that, if M = [M 1 , . . . , M p ] is a random m × m matrix partitioned into independent blocks M j = [v j,1 , . . . , v j,d j ] of size m × d j , and if Z j = v j,1 ∧ · · · ∧ v j,d j ∈ Λ d j R m are integrable, then
(1.4) E| det(M )| = ℓ(K(Z 1 ) ∧ · · · ∧ K(Z p )).
This formula links the expected determinants of a matrix with independent blocks to the multiplication in the zonoid algebra A(R m ), which can be studied using methods from convex geometry and commutative algebra [Eis13]. Furthermore, using the complex wedge product, we obtain in Theorem 6.8 a new formula for the expected absolute value of the determinant of a random complex matrix: E| det(M )| = ℓ(K(Z 1 ) ∧ C · · · ∧ C K(Z p )), where the Z j ∈ Λ d j C C n now are complex random vectors. Notice, that if the d j are all equal to one, we obtain the mixed J-volume from (1.2), and if the K(Z j ) are all real zonoids, this specializes to (1.4).
We remark that (1.3) has interesting consequences when combined with the Alexandrov-Fenchel inequality. For instance, if X, Y ∈ R m are independent integrable random vectors, then the expected area of the triangle ∆(X, Y ), whose vertices are the origin and X and Y , satisfies E vol 2 (∆(X, Y )) 2 ≥ E vol 2 (∆(X, X ′ )) · E vol 2 (∆(Y, Y ′ )), where X ∼ X ′ and Y ∼ Y ′ are independent of X, Y , respectively. We show this in (5.6). Another remarkable consequence of our result is that we obtain a simple proof for zonoids of the recent reverse Alexandrov-Fenchel inequality by Böröczky and Hug [BH21]. Namely, if K 1 , . . . , K p ∈ Z(V ) are zonoids and we have integers d 1 + . . . + d p = m = dim(V ), then MV(K 1 [d 1 ], . . . , K p [d p ]) ≤ 1 m! ℓ(K ∧d 1 1 ) · · · ℓ(K ∧dp p ); here, K i [d i ] indicates that K i appears d i -many times. We prove this in Corollary 4.7 and we also characterize when equality holds. We discuss more applications of the Alexandrov-Fenchel inequality to our theory in Section 5.1. G. Aubrun, C. Lancien and Y. Martinez-Maure for insightful discussions. Finally, we thank the anonymous referee for helpful comments.
Zonoids
In this section we collect known facts about convex bodies, for which we refer the reader to Schneider's extensive monograph [Sch14] for more details. We also review and further extend a probabilistic description of zonoids due to Vitale [Vit91], which is a crucial tool in our paper.
Throughout the section, V denotes m-dimensional Euclidean space with inner product , . The corresponding norm of v ∈ V is given by v = v, v 1 2 . The norm defines a topology on V making it a topological vector space. We denote by B(V ) ⊂ V the closed unit ball centered at zero in V and by S(V ) the unit sphere in V . We abbreviate B m := B(R m ) and S m−1 := S(R m ) for V = R m with the standard inner product. We denote by 2.1. Convex bodies and support functions. A subset K ⊂ V is convex, if for all x, y ∈ K also [x, y] ⊂ K. A nonempty compact convex set K ⊂ V is called convex body and K(V ) denotes the set of convex bodies in V . Given two convex bodies K 1 , K 2 ∈ K(V ), we can form their Minkowski sum, K 1 + K 2 := {x + y | x ∈ K 1 , y ∈ K 2 }, and, for λ ≥ 0 the scalar multiple,
(2.1) λK := {λx | x ∈ K}.
In particular, the Minkowski addition turns K(V ) into a commutative monoid. Note that ǫB(V ) is the ball of radius ǫ > 0 centered at zero. Moreover, we denote by K the linear span of K. The dimension of K is defined as the dimension of its affine span (since all convex bodies considered in this paper contain the origin, this will be the the same as the dimension of K ). We set K m := K(R m ) for V = R m with the standard inner product.
The Hausdorff distance between K 1 , K 2 ∈ K(V ) is defined by
(2.2) d H (K 1 , K 2 ) := inf{ǫ > 0 | K 1 ⊂ K 2 + ǫB(V ) and K 2 ⊂ K 1 + ǫB(V )}.
This leads to the norm of a convex body:
(2.3) K := d H (K, {0}) = max x∈K x .
We also call K the radius of K. The Hausdorff distance (2.2) makes K(V ) a complete metric space, see [Sch14, Theorem 1.8.5].
A convex body K ∈ K(V ) can be described by its support function h K : V R defined by:
(2.4) h K (v) := max{ v, x | x ∈ K}.
This function satisfies h(v + w) ≤ h(v) + h(w) and h(λv) = λh(v) for v, w ∈ V and λ ≥ 0.
Functions h : V R satisfying these properties are called sublinear.
All properties of convex bodies can be concisely described in terms of their support functions. We summarize the relevant facts in the next well known proposition. Let C(S(V )) denote the space of real valued, continuous functions on the unit sphere S(V ) of V . We think of C(S(V )) as a Banach space with respect to either the 1-or the ∞-norm. That is, for f ∈ C(S(V )):
(2.5) f 1 = 1 vol m−1 (S(V )) S(V ) |f (x)| λ(dx) and f ∞ = max x∈S(V ) |f (x)|,
where λ is the usual Lebesgue measure on the sphere and vol k denotes the k-dimensional volume with respect to λ. We denote byh K := (h K )| S(V ) the restriction of the support function h K to the unit sphere S(V ).
Proposition 2.1. The map h : (K(V ), d H ) (C(S(V )), · ∞ ), K h K is a Minkowski linear isometric embedding. That is, for K, L ∈ K(V ) and λ ≥ 0, h K+L = h K + h L , h λK = λh K , d H (K, L) = h K − h L ∞ .
In particular, K = h K ∞ . Moreover,
(1) The image of h consists of the sublinear functions on V restricted to S(V ).
(2) The support function h K allows to reconstruct the body K as follows:
K = {x ∈ V | ∀v ∈ V x, v ≤ h K (v)}. (3) K ⊂ L if and only if h K (u) ≤ h L (u) for all u ∈ V . (4) Let W be a Euclidean space and M : V W be linear with adjoint M T : W V . Then M (K) is a convex body with support function h M (K) (v) = h K (M T v).
Proof. The three statements in display are [Sch14, Theorem 1.7.5] and [Sch14, Lemma 1.8.14]. The characterization (1) of support functions as restrictions of sublinear functions is [Sch14, Theorem 1.7.1]. The statement (2) expresses duality and is in the proof of [Sch14, Theorem 1.7.1]. The third statement follows from the second item. The last point is a direct consequence of the definition of the support function in (2.4).
We will be mainly interested in centrally symmetric convex bodies K, characterized by (−1)K = K. They form a closed subset of K(V ). The fourth item in Proposition 2.1 implies that K is centrally symmetric if and only if h
K (−v) = h K (v) for all v ∈ V .
2.2. Zonoids and random convex bodies. A zonotope K in V is defined as the Minkowski sum of a finite number of line segments, i.e., it has the form K = [x 1 , y 1 ] + · · · + [x n , y n ] with x i , y i ∈ V . In general, zonotopes are exactly the polytopes that have a center of symmetry such that all of its 2 dimensional faces also have a center of symmetry; see [Sch14, Theorem 3.5.2]. The centrally symmetric zonotopes (i.e., with the center of symmetry at the origin) are the ones that can be written in the form
K = 1 2 [−x 1 , x 1 ] + · · · + 1 2 [−x n , x n ].
We introduce now the main objects of this paper.
Definition 2.2 (Zonoids). A convex body K is called a zonoid if it is the limit of a sequence of zonotopes with respect to the Hausdorff metric.
Our focus will be on the centrally symmetric zonoids. Those are exactly the limits of centrally symmetric zonotopes. We denote by Z(V ) the space of centrally symmetric zonoids in V . 1 By definition, Z(V ) is a closed subspace of K(V ). It is known [Sch14,Cor. 3.5.7] that Z(V ) is a proper subset of the set of centrally symmetric convex bodies iff dim V > 2. We also note that if a polytope is a zonoid, then it must be a zonotope (e.g., see [BLM89]). We abbreviate Z m := Z(R m ).
To deal with zonoids, we shall extensively use a probabilistic viewpoint going back to Vitale [Vit91], which not only is intuitive, but also allows for a great deal of flexibility.
Let Ω be a probability space. By an integrable random convex body we understand a Borel measurable map Y : Ω K(V ) such that E Y < ∞; see (2.3) for the definition of the norm. The last condition implies that Eh Y (u) is a well defined sublinear function of u, hence it is the support function of a uniquely defined convex body (see Proposition 2.1(1)). We can thus define the expectation of Y to be the convex body EY ⊂ V with the following support function:
(2.6) h EY (v) := Eh Y (v).
Suppose now that X is a random vector in V satisfying E X < ∞. We call such a random vector integrable. Then Y = 1 2 [−X, X] is an integrable, random, centrally symmetric segment. Vitale [Vit91] proved that every zonoid K ⊂ V (including noncentered zonoids) can be written as K = E[0, X] + c for some random integrable vector X ∈ V and a fixed c ∈ V . We can write this as
K = E 1 2 [−X, X] + c + 1 2 EX. If K ∈ Z(V )
is centrally symmetric, then by uniqueness of support functions we have h K (u) = h K (−u) for all u ∈ V , which implies c + 1 2 EX = 0. Thus, we have shown that
(2.7) K(X) := E 1 2 [−X, X]
is a centrally symmetric zonoid, and that every centrally symmetric zonoid arises this way.
Definition 2.3 (Zonoid associated to random variable). If X is an integrable random vector in V , we call K(X) defined by (2.7) the Vitale zonoid associated to the random vector X. We also say that K(X) is represented by X.
In terms of support functions, we can describe K(X) as follows.
(2.8) h K(X) (v) = Eh 1 2 [−X,X] (v) = 1 2 E| v, X |.
(This follows from (2.6) using that
h [−z,z] (v) = | v, z |.) The observation K(X) = K(−X)
shows that the random variable representing a zonoid is not unique. However, it turns out that the random variable representing a zonoid is unique up to sign, if one assumes X to take its values on the unit sphere S(V ). This follows from the measure theoretic interpretation of zonoids, we will discuss this point of view in Section 2.4 below (see also [Sch14,Theorem 3.5.3] and [Bol69, Theorem 5.2]).
We found it of great technical advantage to allow the random vector X to take values outside the unit sphere of V ; the resulting loss in uniqueness is not significant. The next result is a clear indication of the advantage of this viewpoint. It says that the association of a zonoid to an integrable random vector commutes with linear maps. Proving the following proposition using the measure point of view is involved (see Lemma 2.31 below), while the proof using random vectors is almost trivial. This highlights for the first time one of the advantages when working with random vectors.
Proposition 2.4. Let V, W be Euclidean spaces, M : V W be a linear map and X be an integrable random vector in V . Then M (K(X)) = K(M X).
Proof. We have for v ∈ V by (2.8),
h K(M (X)) (v) = 1 2 E| v, M X | = 1 2 E| M T v, X | = h K (M T v) = h M (K) (v),
where last equality is due to Proposition 2.1(4).
Proposition 2.4 also implies that linear images of zonoids are again zonoids. Of course, this also follows directly from the definition.
For instance, we obtain for a standard Gaussian vector X ∈ R m ,
(2.9) K(X) = (2π) − 1 2 B m , where B m denotes the Euclidean unit ball in R m . Indeed, h K(X) (v) = 1 2 E| v, X | = (2π) − 1 2 v , by (2.8), since m i=1 X i v i is distributed as v Z,
where Z ∈ R is standard gaussian and has expected value E|Z| = 2/π. The function h K(X) equals the support function of (2π) − 1 2 B m and hence (2.9) follows.
Remark 2.5. Behind Vitale's result is the following law of large numbers for zonoids [AV75], which provides a geometric way for viewing Vitale zonoids associated to random vectors. Let X be an integrable vector in V and {X k } k≥1 a sequence of independent samples of X. Then, as n ∞, we have in almost sure convergence 1 n n k=1 1
2 [−X k , X k ] K(X).
We next show how to realize the Minkowski sum of two zonoids as the Vitale zonoid associated to a random vector.
Lemma 2.6. Let X, Y be integrable random vectors in V . Then K(Z) = K(X) + K(Y ) for the random variable Z := 2ǫX + 2(1 − ǫ)Y defined with a fair Bernoulli random variable ǫ ∈ {0, 1} that is independent of X and Y .
Proof. By (2.8), we have
h K(Z) (v) = 1 2 E h [−Z,Z] (v) = 1 2 E X,Y E ǫ h 2ǫ[−X,X]+2(1−ǫ)[−Y,Y ] (v).
Expanding the expectation over ǫ, Y ] (v) and taking expectations over X, Y finishes the proof. Lemma 2.6 has a straightforward generalization for adding n random variables. As an application, if x 1 , . . . , x n ∈ V are fixed vectors and Z ∈ V is the random vector taking the value nx i with probability 1/n, we see that K(Z) is the zonotope 1 2 [−x 1 , x 1 ]+ · · · + 1 2 [−x n , x n ]. The next observation states that scaling the random variable X with an independent random function only leads to a rescaling of the resulting zonoid.
E ǫ h ǫ2[−X,X]+(1−ǫ)2[−Y,Y ] (v) = 1 2 h 2[−X,X] (v) + 1 2 h 2[−Y,Y ] (v) = h [−X,X] (v) + h [−Y,
Lemma 2.7. We have K(ρX) = E|ρ| · K(X) if X ∈ V and ρ ∈ R are independent integrable random variables.
Proof. We have h K(ρX) (v) = 1 2 E| v, ρX | = 1 2 E |ρ| · | v, X | = E|ρ| · h K(X) (v),
where the first equality is (2.8) and the last equality is due to the independence of X and ρ. The assertion follows since convex bodies are determined by their support function.
As an application, we compute the zonoid defined by X uniformly distributed on the unit sphere S m−1 in R m . For this, we introduce the following notation:
(2.10) τ m := √ 2π E X , where X ∈ R m is a standard Gaussian random vector.
This number is √ 2π times the expected value of a chi-distributed random variable with m degrees of freedom 2 It has the explicit value τ m :
= √ 2π √ 2 Γ m+1 2 /Γ m 2 .
We can write X := X/ X , where X ∈ R m is standard Gaussian. Then, we have
(2.11) K( X) = 1 τ m √ 2π B m ,
where, as before, B m is the unit ball in R m . This follows from Lemma 2.7 and (2.9), using that X is independent of X and (e.g., see [BC13,§2.2
.3])
In the next example we generalize this from spheres to Grassmannians.
2 In [BL20] the expected value of a chi-distributed random variable with m degrees of freedom is denoted ρm. So, using their notation we have τm = √ 2π ρm. The additional factor of √ 2π makes the formulas in this paper easier to grasp.
Example 2.8. Let ξ 1 , . . . , ξ k ∈ R m be standard, independent Gaussian vectors and consider X := ξ 1 ∧ · · · ∧ ξ k ∈ Λ k (R m ).
Put X := X/ X . By [Jam54, Theorem 8.1], the random variables X and X are independent and X is uniformly distributed on the Grassmannian 3 G(k, m) of oriented k-planes in R m . Moreover, X is distributed as the square root of the determinant of a Wishart matrix 4 . In particular, using [Mui82, Theorem 3.2.15], we get E X = 2
k 2 Γ k m+1 2 /Γ k m 2 , where Γ k (x) := π k(k−1) 4 k j=1 Γ x + 1−j 2
denotes the multivariate Gamma function; see, e.g., [Mui82, Theorem 2.1.12]. Lemma 2.7 tells now that
(2.12) K(X) = 2 k 2 Γ k m+1 2 Γ k n 2 K( X).
Notice that, in the case k = 1, we get E X = τ m / √ 2π (see (2.10)) and we recover (2.11) as a special case of (2.12).
We now show that the expectation of the norm of an integrable random vector X depends only on its zonoid K(X). Recall thath K denotes the restriction of h K to the unit sphere.
Proposition 2.9. Let X be an integrable random vector in R m and K = K(X). Then
E X = √ 2π Eh K (v) = τ m h K 1
where v ∈ R m is a standard Gaussian vector, and the 1-norm is as in (2.5).
Proof. We assume w.l.o.g. that v is independent of X. By rotational invariance and homogeneity, we have x = π/2 E| x, v | for all x ∈ R m . This implies, using the independence of X and v, that
E X = π 2 E X,v | X, v | = π 2 E X,v 2 · h 1 2 [−X,X] (v) = √ 2π · E v h K (v),
which shows the first equality. The second equality follows from the factorization v = v ·ṽ, noting as before that v andṽ, are independent andṽ is uniformly distributed on the unit sphere. We have E v = (2π) − 1 2 τ m and E v h K (v) = h K 1 by (2.5), becauseh K is nonnegative. Putting everything together finishes the proof. Proposition 2.9 shows that the following notion of length is well defined. The length will be investigated more closely in Section 5.1.
Definition 2.10. We define the length of a zonoid K ∈ Z(V ) by
ℓ(K) = E X ,
where X is an integrable random vector representing K.
Example 2.11. We see by Proposition 2.9 that if B m is the unit ball of R m then its length is given by ℓ(B m ) = τ m , where τ m is given by (2.10).
We next show that the length is additive with respect to Minkowski addition. This might be surprising at first sight, given that Definition 2.10 defines the length as the expected value of a norm.
Lemma 2.12. Let K, L ∈ Z(V ) and λ ≥ 0. Then, we have ℓ(K + λL) = ℓ(K) + λℓ(L).
Proof. Let K = K(X) and L = K(Y ) for independent integrable vectors X, Y ∈ V . By Lemma 2.7 we have λL = K(λY ). Using this and Lemma 2.6 we can write the Minkowski sum as K + λL = K(2ǫX + 2λ(1 − ǫ)Y ), where ǫ is a fair Bernoulli random variable ǫ ∈ {0, 1} that is independent of X and Y . This implies ℓ(K + λL) = E 2ǫX + 2λ(1 − ǫ)Y . Taking first the expectation over ǫ yields ℓ(K + λL) = 1 2 E 2X + 1 2 E 2λY = ℓ(K) + λℓ(L). This finishes the proof.
Combining Proposition 2.9 with Proposition 2.1(3) we get the following corollary.
Corollary 2.13 (Monotonicity of the length). Let K, L ⊂ V are zonoids such that K ⊂ L, then ℓ(K) ≤ ℓ(L).
We recall from Proposition 2.1 and Proposition 2.9 that the radius K of a zonoid K can be expressed as the ∞-norm of its support functionh K and that its length ℓ(K) can expressed in terms of its 1-norm. The next result compares these two norms.
Corollary 2.14 (Radius and length). We have
2 K ≤ ℓ(K) ≤ τ m K ,
with inequality on the left hand side iff K is a (centrally symmetric) segment, and equality holding on the right hand side iff K is rotational invariant.
Proof. Recall K = h K ∞ from Proposition 2.1 and ℓ(K) = τ m h K 1 from Proposition 2.9. By definition of the norms we have h K 1 ≤ h K ∞ with equality holding iff h K is constant. The latter means that K is rotationally invariant. This shows the right inequality.
For the left inequality, we write K = K(X). By (2.8) we have 2h K (u) = E| X, u | ≤ E X , if u = 1. This implies 2 h K ∞ ≤ E X = ℓ(K) and equality holds if K is a segment.
The next observation will be useful later.
Lemma 2.15. Let K, L ∈ Z(V ) be zonoids and K , L denote their linear spans. If K, L are represented by the integrable random vectors X, Y , respectively, then
K ⊥ L ⇐⇒ X, Y = 0 almost surely.
Proof. Let π : V L denote the orthogonal projection. By Proposition 2.1 and (2.8), the support function of π(K), for u ∈ L , is given by h π(K) (u) = h K (π(u)) = 1 2 E| π(u), X |. Thus K ⊥ L iff π(K) = 0 iff h π(K) = 0. The latter is equivalent to π(u), X = 0 almost surely, for all u ∈ L . This is easily seen to be equivalent to X, Y = 0 almost surely.
2.3. Virtual zonoids. It is well known that the set of zonoids Z(V ) and convex bodies K(V ) can be interpreted as cones in vector spaces of "virtual zonoids" Z(V ) and "virtual convex bodies" K(V ), respectively (see [BZ88,§25.1]) and [Sch14,§3.5]). This allows to investigate them using tools from linear algebra and functional analysis.
We confine ourselves to zonoids, since only for those we can define a satisfying notion of tensor product, see Section 3. The next result summarizes the situation.
Theorem 2.16. Z(V ) is embedded in an essentially unique way as a subcone of a normed and partially ordered real vector space Z(V ) of virtual zonoids such that any element of Z(V ) can be written as a formal difference K 1 − K 2 of zonoids. The norm and partial order are defined as follows:
K 1 − K 2 = d H (K 1 , K 2 ), 0 ≤ K 1 − K 2 :⇐⇒ h K 2 ≤ h K 1 .
Thus the norm extends the norm of zonoids introduced in (2.3) and the partial order on Z(V ) extends the inclusion of convex bodies (see Proposition 2.1). However, Z(V ) is not complete, so not a Banach space, unless dim V ≤ 1, Proof. We can identify convex bodies with their support functions using Proposition 2.1. Then K(V ) can be seen as a real cone in C(S), which is closed under addition and multiplication with nonnegative scalars (recall that S denotes the unit sphere in V ). We define the space of virtual convex bodies K(V ) as the span of K(V ): it is the subspace of C(S) consisting of differences of support functions. Similarly, we define the space of virtual zonoids Z(V ) as the span of Z(V ), seen as a subset of C(S). Thus Z(V ) ⊂ K(V ) are endowed with the supremum norm (of functions restricted to the unit sphere), which extends the norm of convex bodies defined in (2.3). Moreover, the pointwise order of functions makes them partially ordered vector spaces. For the uniqueness, we refer to the discussion below. The non-completeness of Z(V ) is shown in Proposition 2.24 below.
In Section 2.4 we will see that virtual zonoids can be identified with the space of signed measures on the sphere. Moreover, virtual zonoids also have a geometric realization as hedgehogs, i.e., envelopes of hyperplanes defined by the difference of the corresponding support functions. For this point of view we refer the reader to [MM01].
Despite the efficiency of support functions, used in the proof of Theorem 2.16, it is useful to think of virtual convex bodies more abstractly as formal differences of convex bodies. The abstract point of view reveals the uniqueness of the construction. Let us therefore formulate the above algebraic features in this language.
Given a commutative monoid (M, +), its Grothendieck group ( M, +) can be defined as the group of equivalence classes of pairs (m 1 , m 2 ) ∈ M such that (m 1 , m 2 ) ∼ (n 1 , n 2 ) if and only if m 1 + n 2 = m 2 + n 1 , and with the addition [(m 1 , m 2 )] + [(n 1 , n 2 )] := [(m 1 + n 1 , m 2 + n 2 )]. The Grothendieck group of (M, +) is an essentially unique object characterized by a universal property, see [Lan02]. If the cancellation law holds, then M M, m [(m, 0)] is injective so that M can be seen as a submonoid of M and we write the class [(m 1 , m 2 )] as m 1 − m 2 . If, in addition, there is an action of R + on M satisfying the usual axioms of scalar multiplication in vector spaces, then it is immediate to check that the scalar multiplication extends to R, so that M becomes a real vector space, which we may call the corresponding Grothendieck vector space.
We apply this construction to the monoids of zonoids and convex bodies in a Euclidean space V , endowed with the Minkowski addition and the scalar multiplication given by (2.1). Using support functions, we see that the cancellation law holds in (K(V ), +), see Proposition 2.1 and [Sch14, §3.1].
Definition 2.17 (The vector space of virtual convex bodies). We denote by K(V ) and Z(V ) the Grothendieck vector spaces of K(V ) and Z(V ), respectively. Its elements are called virtual convex bodies and virtual zonoids, respectively.
From the uniqueness of the Grothendieck vector spaces K(V ) and Z(V ) up to canonical isomorphy, it follows that they are isomorphic to the subspaces of C(S) defined via support functions in the proof of Theorem 2.16. We abbreviate K m := K(R m ) and Z m := Z(R m ).
Let us point out that the formal inverse −K of a zonoid K, which is a virtual zonoid, should not be confused with the zonoid (−1)K, which is equal to K.
Remark 2.18. 1. We may define the nonegative cone of K(V ) corresponding to the above defined order by K(V )
+ := K 2 − K 1 | K 1 , K 2 ∈ K(V ), K 1 ≤ K 2 ; similarly, we define Z(V ) + . Then it is clear that K(V ) ⊆ K(V ) + and Z(V ) ⊆ Z(V ) + , however, the inclusions in general are strict [Sch14, §3.2].
Remark 2.19. The elements of K(V ) ∩ Z(V ) are the convex bodies that can be written as a difference of zonoids, they are called generalized zonoids [Sch14,p. 195]. They are the convex bodies whose support function has a representation as in (2.13) below, but with an even signed measure µ. It is known that the generalized zonoids are dense in K(V ); in particular, there exist centrally symmetric convex bodies, which are not generalized zonoids. This is shown in [Sch14, Note 13, p. 206].
If M : V W is a linear map between Euclidean vector spaces then K M (K) gives a morphism Z(V ) Z(W ) of monoids, which commutes with multiplication with nonnegative scalars. It is immediate that this morphism can be extended to a linear map between the corresponding Grothendieck vector spaces.
Definition 2.20. Let M : V W be a linear map between Euclidean vector spaces. We denote by M : Z(V ) Z(W ) the associated linear map defined by
M (K 1 − K 2 ) := M (K 1 ) − M (K 2 ).
We collect some properties of the associated map M .
Proposition 2.21. Let M : V W be a linear map between Euclidean vector spaces. Then the associated linear map M : Z(V ) Z(W ) preserves the order, is continuous, and
M op = M op .
Moreover, M (Z(V )) ⊆ Z(W ), i.e., M maps zonoids to zonoids. Finally, for K ∈ Z(V ),
ℓ(M (K)) ≤ M op ℓ(K).
In particular, the length of zonoids does not increase under an orthogonal projection.
Proof. The preservation of the order follows from the definition and Proposition 2.1. To prove continuity, let Z = K 1 − K 2 ∈ Z(V ). Recall from Theorem 2.16 that the norm defined on
Z(V ) is Z = d H (K 1 , K 2 )
. Using the properties from Proposition 2.1, we have
M (Z) = M (K 1 ) − M (K 2 ) = sup{|h M (K 1 ) (u) − h M (K 2 ) (u) | u ∈ W, u = 1} = sup{|h K 1 (M T u) − h K 2 (M T u)| | u ∈ W, u = 1} ≤ M T op · sup{|h K 1 (v) − h K 2 (v)| | v ∈ W, v = 1} = M T op · d H (K 1 , K 2 ) = M op · K 1 − K 2 = M op · Z .
This proves M op ≤ M op and hence the continuity of the linear map M . On the other hand, let v ∈ V be of norm one such that M op = M v and consider the segment
1 2 [−v, v], of norm 1 2 [−v, v] = v = 1. Then M op ≥ M 1 2 [−v, v] = 1 2 [−M v, M v] = M v = M op ,
This proves the stated equality of operator norms.
Since M maps segments to segments and M is continuous, we have M (Z(V )) ⊆ Z(W ). For the stated upper bound on the length, let X be a integrable random vector representing X. Then M (X) represents the image M (K) and we have by Definition 2.10,
ℓ(M (K)) = E M (X) ≤ M op E X = M op ℓ(X),
which completes the proof.
Our construction may be summarized by saying that we have constructed a covariant functor from the category of Euclidean spaces (with any linear morphisms) to the category of partially ordered and normed vector spaces.
The length of zonoids defined in Definition 2.10 extends to a continuous linear functional.
Theorem 2.22. The length extends to a continuous linear functional ℓ :
Z(V ) R.
Proof. Lemma 2.12 shows that the length on Z(V ) is additive, so we can extend it to a linear functional on Z(V ) by setting ℓ(K − L) := ℓ(K) − ℓ(L). Using Corollary 2.14 we can characterize this as ℓ(
K 1 − K 2 ) = τ m ( h K 1 − h L 1 ). Since bothh K andh L are nonnegative we have h K 1 − h L 1 = h K −h L 1 .
This shows that the length is continuous
We also need the following observation.
Lemma 2.23. If V is one-dimensional, the length ℓ : Z(V ) R induces an isomorphism of additive groups, which preserves the standard norm and the standard order. We shall use this to identify Z(V ) with R and Z(V ) with R + .
Proof. The length is a group homomorphism by Theorem 2.22. Let K = 1 2 [−x, x] be a centrally symmetric zonoid in Z(V ); i.e., a segment. Then, ℓ(K) = x, which shows that ℓ is bijective, hence an isomorphism. This also shows that ℓ preserves the standard norm and the standard order.
We finally show that Z(V ) in general is not a Banach space, as already claimed in Theorem 2.16. Recall that C(S) denotes the Banach space of continuous real functions on the unit sphere S := S(V ) endowed with the supremum norm. We denote by C even (S) its closed subspace of even functions (i.e., f (−v) = f (v)). In the proof of Theorem 2.16 we have constructed the normed vector space of virtual convex bodies K(V ) ⊂ C(S) and the normed vector space Z(V ) ⊂ C even (S) of virtual zonoids. (Recall that the sup-norm corresponds to the Hausdorff metric (2.2).) Unfortunately, these are not Banach spaces. This follows from known facts that we state in the next proposition for the sake of clarity.
Proposition 2.24. The completion of the space of virtual convex bodies K(V ) equals C(S) and the completion of the space of virtual zonoids Z(V ) equals C even (S). Moreover, if dim V > 1, then K(V ) = C(S) and Z(V ) = C even (S), hence K(V ) and Z(V ) are not complete and hence not Banach spaces. In particular, they are real vector spaces of infinite dimension. However,
if dim V = 1, then we have K(V ) ≃ R 2 and Z(V ) ≃ R.
Proof. W.l.o.g. we assume V = R m . It is known [Sch14, Lemma 1.7.8] that every twice continuously differentiable function S m−1 R can be written as a difference h K − rh B m for some K ∈ K m and r > 0. This implies that K m is dense in C(S m−1 ), since continuous function can be uniformly approximated by twice continuously differentiable functions. On the other hand, K m is strictly contained in C(S m−1 ) if m > 1. One one way to see this is that there are continuous, nowhere differentiable functions on S m−1 , while every h ∈ K m is differentiable at least at some point in S m−1 .
The assertion about virtual zonoids is more involved. In [Sch14, Theorem 3.5.4], it is shown that every sufficiently smooth even real function f : S m−1 R has a representation as in Definition 2.25 below, but with a even signed measure µ, which implies that f ∈ Z m . This implies that Z m is dense in C even (S m−1 ). In order to show that Z m is strictly contained in C even (S m−1 ) one can argue as follows. If we had Z m = C even (S m−1 ), then every centrally symmetric convex body would be a virtual zonoid. This contradicts [Sch14, Note 13, p. 206], which states that there exist centrally symmetric convex bodies, which are not generalized zonoids. Finally, K 1 ≃ R 2 and Z 1 ≃ R 1 since K 1 consists of the intervals and Z 1 consist of the symmetric intervals.
We remark that the fact that the cone Z(V ) is a closed subset of the cone K(V ) does not contradict the fact that, by Proposition 2.24, Z(V ) is a dense subset of K(V ).
Virtual zonoids and measures.
There is a correspondence between zonoids and even measures on the sphere, that we discuss now. This point of view is classic when dealing with zonoids, so we include here a review of the principal facts of this correspondence. The approach using measures provides a complimentary viewpoint to our approach using random vectors. This alternative viewpoint will become particularly useful in Section 3.1, where we discuss continuity properties of our constructions.
In what follows we identify the space of even measures on the sphere with the space of measures on the projective space. The space of continuous functions on the projective space is C(P m−1 ). The space of (signed) measures on P m−1 will be denoted by M(P m−1 ). The cone of positive measures will be denoted by M + (P m−1 ). Recall that the weak- * topology on M(P m−1 ) is the coarsest topology on M(P m−1 ) such that for every φ ∈ C(P m−1 ) the linear functional µ P m−1 φ(x)µ(dx) is continuous. For practical purposes, an even measure on the sphere S m−1 will correspond to 1 2 times its pushforward measure on the projective space. Similarly we identify a function on the projective space with the corresponding even function on the sphere; that is, if f :
P m−1 R, then we write f (x) := (f • Π)(x), where Π : R m \ {0} P m−1 is the projection. With these identifications, for all µ ∈ M(P m−1 ) and f ∈ C(P m−1 ), we have P m−1 f (x) µ(dx) = 1 2 S m−1 f (x) µ(dx).
Passing from zonoids to virtual zonoids corresponds to passing from even measures to even signed measures. The main object of the section is the following map.
Definition 2.25. The cosine transform is the linear map H : M(P m−1 ) C(P m−1 ) given for all µ ∈ M(P m−1 ) and x ∈ P m−1 by
(2.13) H(µ)(x) := P m−1 | u, x | µ(du).
The image of the cosine transform will be denoted by H(P m−1 ) and the image of M + (P m−1 ) by H + (P m−1 ).
We could not locate any reference for items (4)-(6) of the next theorem. This is why we include our own proof. Recall that we endow M(P m−1 ) with the weak- * topology and C(P m−1 ) with the topology induced by the ∞-norm.
Theorem 2.26. The cosine transform H : M(P m−1 ) C(P m−1 ) satisfies the following properties.
(1) H is injective.
(2) H(M(P m−1 )) is a dense subspace of C(P m−1 ). (3) There is c = c(m) > 0 such that c H(µ) ∞ ≤ µ(P m−1 ) ≤ H(µ) ∞ for all (nonnega- tive) measures µ ∈ M + (P m−1 ). (4) H is sequentially continuous. (5) The restriction H : M + (P m−1 ) H + (P m−1 ) is a homeomorphism. (6) The inverse H −1 : H(P m−1 ) M(P m−1 )
is not sequentially continuous for m > 1.
Remark 2.27. Our proof for item (4) does not extend to nets. This is why we only prove sequential continuity of H.
Proof of Theorem 2.26. Assertions (1) and (2) are in (the proof of) [Sch14, Theorem 3.5.4]. Assertion (3) follows from Corollary 2.14 stated in our context, and using Remark 2.30 (2) below.
As for assertion (4): because H is linear it suffices to prove sequential continuity at 0. Recall that for the weak- * topology on M(P m−1 ), a sequence of measures (µ i ) converges to 0, if and only if for every φ ∈ C(P m−1 ), the sequence µ i , φ goes to zero, where (2.14) µ, φ :=
P m−1 φ(x) µ(dx). Suppose that (µ i ) converges to 0 in M(P m−1 ) in the weak- * topology. Let h i := H(µ i ), thus h i (v) := x∈P m−1 | v, x | µ i (dx)
, and in particular h i (v) 0 for all v ∈ P m−1 . So we have pointwise convergence of the h i . We are going to show that h i 0 uniformly. Recall that every measure µ has a unique decomposition µ = µ + − µ − (called the Hahn-Jordan decomposition), where µ + , µ − ∈ M + (P m−1 ). We define |µ| := µ + + µ − . The Banach-Steinhaus Theorem (e.g., see [Rud73]) implies that κ := sup i (|µ i |(P m−1 )) < ∞. Therefore, we have for any v ∈ P m−1 ,
|h i (v)| ≤ P m−1 | v, x | |µ i |(dx) ≤ |µ i |(P m−1 ) ≤ κ, and hence sup i h i ∞ ≤ κ. Moreover, for v 1 , v 2 ∈ P m−1 , |h i (v 1 ) − h i (v 2 )| ≤ P m−1 | v 1 − v 2 , x | |µ i |(dx) ≤ κ v 1 − v 2 .
The Arzelà-Ascoli Theorem (e.g., see [Rud73]) implies that (h i ) has a uniformly convergent subsequence (h i j ). Thus h i j 0 uniformly since h i j 0 pointwise. By the same argument we see that any subsequence of (h i ) has a subsequence that uniformly converges to 0. This implies that h i 0 uniformly. Therefore, it follows that the map H is sequentially continuous.
For assertion (5), Bolker [Bol69, Theorem 5.2] showed that H : M + (P m−1 ) C(P m−1 ) is continuous. So we only need to show that the inverse H −1 : H + (P m−1 ) M + (P m−1 ) is continuous. For this it suffices to show that H −1 is sequentially continuous on H + (P m−1 ), because the norm topology on H + (P m−1 ) is first countable and for maps whose domain is first countable topological spaces sequential continuity and continuity are equivalent. To show sequential continuity of (H −1 )| H + (P m−1 ) we take a sequence (h i ) ⊂ H + (P m−1 ) that converges to h. Let (µ i ) ⊂ M + (P m−1 ) be the corresponding sequence so that H(µ i ) = h i and let µ be a measure with H(µ) = h. We have to show that µ i converges to µ. For this, we fix φ ∈ C(P m−1 ) and show that µ i − µ, φ 0. This would imply that µ i − µ 0. Let ε > 0. By assertion (2) there are ξ 1 , . . . , ξ N ∈ P m−1 and c 1 , . . . , c N ∈ R such that the function ψ(
x) := N k=1 c k | x, ξ k | in C(P m−1 ) satisfies ψ − φ ∞ < ε/(2c). We decompose µ i − µ, φ = µ i , φ − ψ + µ i − µ, ψ + µ, ψ − φ . (2.15)
The sequence of real numbers h i ∞ converges to h ∞ and is thus bounded so that there is
c > 0 such that sup i h i ∞ ≤ c and h ∞ ≤ c. An upper bound for the absolute value of third term in (2.15) is | µ, φ − ψ | = | P m−1 (φ(x) − ψ(x)) µ(dx)| ≤ µ(P m−1 ) φ − ψ ∞ (here,
we have used that µ is a measure and not a signed measure). Assertion (4) implies that this is bounded by c φ − ψ ∞ < ε/2. We get the same bound for the first term. The middle term
equals N k=1 c k (h i (ξ k ) − h(ξ k )
) and, by assumption, converges to zero for i ∞. Therefore,
lim sup i | µ i − µ, φ | ≤ ε.
Since ε > 0 was arbitrary, we conclude that indeed µ i − µ, φ 0, which proves assertion (6). Assertion (6) follows from the noncontinuity of the tensor product of zonoids. More precisely, we will prove in Theorem 3.10 below that the map (K, L) h K⊗L is not sequentially continuous. On the other hand, we can write:
h K⊗L = H(T (H −1 (h K ), H −1 (h L ))),
where, for two measures µ 1 , µ 2 the measureT (µ 1 , µ 2 ) is defined in (3.4) below. By Theorem 3.10 (3) below, the mapT is sequentially continuous and, if H −1 were sequentially continuous, then (K, L) h K⊗L would also be sequentially continuous. This contradicts Theorem 3.10.
By Theorem 2.26 (5), the cone H + (P m−1 ) coincides with the cone of support functions of zonoids and consequently H(P m−1 ) coincides with the linear span of support functions of zonoids. Again by Theorem 2.26, since the cone of nonnegative measures can be identified with the cone of zonoids, the vector space of measures can be identified with the vector space of virtual zonoids. In particular, on the space of virtual zonoids Z(V ) we can put two different topologies: the topology T 1 induced by viewing them (through their support functions) as a subspace of continuous functions on P m−1 with the uniform convergence topology induced by the ∞-norm (we denote this topology by T ∞ ), and the topology T 2 induced by viewing them as signed measures with the weak- * topology T weak- * . With this notation, letting V ≃ R m , we have
(2.16) ( Z(V ), T 1 ) ≃ (H(P m−1 ), T ∞ ) and ( Z(V ), T 2 ) ≃ (M(P m−1 ), T weak- * ).
With these identifications, the map H can be seen as an inclusion:
H : ( Z(V ), T 2 ) id − ( Z(V ), T 1 ) ֒ (C(P m−1 ), T ∞ )
. The fact that the inverse of H is not continuous means that the two topologies (2.16) are not the same. What is remarkable, however, is that on the cone of zonoids, they induce the same topology.
Remark 2.28. Next to the weak- * topology, another natural topology on M(P m−1 ) is the one induced by the total variation norm µ := |µ|(P m−1 ), where as in the proof of Theorem 2.26 we define |µ| := µ + + µ − for µ = µ + − µ − being the Hahn-Jordan decomposition of µ. By Remark 2.30 below, for nonnegative measures this coincides with the length. Let us observe that item (3) in Theorem 2.26 does not imply that this topology coincides with either T 1 or T 2 , not even when restricted to Z(V ). As an example, one can consider a sequence of pairwise different segments [−v i , v i ] such that all v i are in the sphere. The corresponding measure is
µ i := 1 2 δ Π(v i ) ,
where the latter is the Dirac-delta measure. Observe, that for the total variation norm we have µ i − µ j = 1 2 . In particular such a sequence cannot converge in the topology induced by the total variation norm, but it can converge in T 1 and T 2 .
Next, we discuss the way of passing from the point of view of random vectors to the point of view of positive measures. Recall that for
x ∈ R m \ {0} and f ∈ C(P m−1 ) we set f (x) := (f • Π)(x), where Π : R m \ {0} P m−1 is the canonical projection.
Proposition 2.29. Let X ∈ R m be an integrable random vector with probability measure ν.
Let ν ′ be the measure such that ν ′ (A) = A x ν(dx) for all measurable sets A ⊂ R m . Then h K(X) = H(µ),
where µ is the push-forward measure of ν ′ under the projection Π :
R m \ {0} P m−1 . Proof. Let u ∈ P m−1 . The support function of K(X) evaluated at u is h K(X) (u) = 1 2 E| u, X |. We write the integral explicitly as E| u, X | = R m | u, x | ν(dx) = R m | u, x/ x | ν ′ (dx) = P m−1 | u, y | µ(dy),
the second equality, because ν ′ is zero where X = 0 and the third equality, because | u, x/ x is constant on preimages of Π.
Remark 2.30. Let X be an integrable vector, K = K(X), and corresponding measure µ.
(1) From Definition 2.10 and Proposition 2.29 it follows that ℓ(K) = 2µ(P m−1 ) = µ(S m−1 ).
(2) If X admits an even measurable density ρ : R m R, then µ admits the densitỹ
ρ(x) := +∞ 0 t m ρ(tx 0 )dt, where x 0 ∈ S m−1 is such that Π(x 0 ) = x.
We close this section by proving that linear maps between spaces of measures are continuous with respect to the weak- * topology. M(P n−1 ) that sends the measure associated to the zonoid K to the measure associated to the zonoid M (K); that is, if H(µ) =h K , then H( M (µ)) =h M (K) . Then, M is sequentially continuous with respect to the weak- * topology.
Proof. Since M is linear, it is enough to show continuity at 0. So, let µ i be a sequence of measures converging to 0, and let ν i := M (µ i ). We have to show that ν i converges to 0. Let K i be the zonoid associated to µ i . Moreover, let us denote the pairing as in (2.14):
µ, φ = P m−1 φ(x) µ(dx) for φ ∈ C(P m−1 ) From Proposition 2.1 (4) we know that, if K is a zonoid, then h M (K) (v) = h K (M T v). Take v ∈ S(W ). Then, we have: h M (K) (v) = M T v ·h K (M T v/ M T v ), if M T v = 0 0, else .
This implies that for any v such that M T v = 0 and for every i we have:
P n−1 | v, y | ν i (dy) = M T v · P m−1 | M T v/ M T v , x | µ i (dx) = P m−1 | M T v, x | µ i (dx),
and P n−1 | v, y | ν i (dy) = 0 otherwise. Since (µ i ) converges to 0 we see that P n−1 | v, y | ν i (dy) converges to 0 for every v ∈ S(W ). Now, we can proceed as we did in (2.15), approximating any continuous function φ ∈ C(P n−1 ) with a linear combination of functions of the form y | v, y |. This concludes the proof.
Tensor product of zonoids
In this section we introduce and study the notion of tensor product of zonoids. The only previous appearance of this notion we are aware of is [AL16, Definition 3.2].
In the whole section, V and W denote Euclidean spaces. We start with the following central definition.
Definition 3.1 (Tensor product of zonoids). Let K be a zonoid in V , L be a zonoid in W and X ∈ V and Y ∈ W be integrable random vectors representing K and L, respectively. We define the tensor product of K and L as
K ⊗ L := K(X ⊗ Y ).
Of course, we need to check that this tensor product does not depend on the choice of random vectors representing the zonoids. This is guaranteed by Lemma 3.2 below. For stating it, we introduce the following notation: for x ∈ V and y ∈ W we define the following linear operators
T x := ·, x ⊗ id W : V ⊗ W W and T y := id V ⊗ ·, y : V ⊗ W V.
Notice that their operator norms satisfy
(3.1) T x op = x and T y op = y .
Lemma 3.2. Let K ∈ Z(V ) and L ∈ Z(W ) be zonoids represented by independent random vectors X ∈ V and Y ∈ W , i.e., K = K(X) and L = K(Y ). Then
(3.2) h K(X⊗Y ) (u) = E Y h K(X) (T Y (u)) = E X h K(Y ) (T X (u))
and K(X ⊗ Y ) ∈ Z(V ⊗ W ) depends only on K and L, and not on the choice of the random vectors X and Y .
Proof of Lemma 3.2. We first show that
∀u ∈ V ⊗ W u, X ⊗ Y = T Y (u), X .
It suffices to check this for simple tensors of the form u = v ⊗ w where v ∈ V and w ∈ W , because the simple vectors generate V ⊗ W . For such u, the definition of the scalar product
in V ⊗ W indeed implies v ⊗ w, X ⊗ Y = v, X w, Y = w, Y v, X = T Y (v ⊗ w), X .
By (2.8) the support function of the zonoid K(X ⊗ Y ) is given by
h K(X⊗Y ) (u) = 1 2 E | u, X ⊗ Y | ,
where the expectation is over the joint distribution of X and Y . Taking first the expectation over X and using the independence of X and Y , we get
h K(X⊗Y ) (u) = E Y h K(X) (T Y (u)).
This shows that the dependence of K(X ⊗ Y ) on X is only through K = K(X). A symmetric argument for Y completes the proof.
Example 3.3 (Tensor product of balls). Let X ∈ R k and Y ∈ R m be independent, standard Gaussian vectors. Then K( √ 2πX) = B k and K( √ 2πY ) = B m by (2.9). Therefore,
B k ⊗ B m = √ 2πK(X) ⊗ √ 2πK(Y ) = 2πK(X ⊗ Y ).
The tensor product K(X ⊗ Y ) was called Segre zonoid in [BL20]. It is a convex body in the space R k ⊗ R m ≃ R k×m , and its support function depends on singular values 5 in the 5 In a recent preprint [SS20], Sanyal and Saunderson have introduced the notion of spectral convex bodies,
i.e., convex bodies in the space of symmetric operators whose support function depends on eigenvalues only.
following sense. Assuming k ≤ m and denoting by sv(M ) ∈ R k the list of singular values of a matrix M ∈ R k×m , we have h B k ⊗B m (M ) = (2π) 1 2 g k (sv(M )), by [BL20, Lemma 5.5], where the function g k : R k R is defined by g k (σ 1 , . . . , σ k ) := E σ 2 1 ξ 2 1 + · · · + σ 2 k ξ 2 k 1 2 , and ξ 1 , . . . , ξ k are independent standard gaussians. Recall from (2.10) the definition of τ m . We obtain B k ⊗ B m = τ k / √ k =: r k from [BL20, Lemma 5.5], which remarkably only depends on k. Thus r k B km is the smallest centered ball containing B k ⊗ B m . In [BL20, Theorem 6.3], it was shown that for fixed k and m ∞, B k ⊗ B m is not much smaller in volume than r k B km : we
have log vol km (B k ⊗ B m ) = log vol km (r k B km ) − O(log m) for m ∞.
The next result shows that the tensor product of zonoids behaves well with respect to Minkowski addition, scalar multiplication, norm, and inclusion.
Proposition 3.4. The tensor product of zonoids is componentwise Minkowski additive and positively homogenous. Moreover, the tensor product is monotonically increasing in each variable; that is, K 1 ⊂ K 2 and L 1 ⊂ L 2 implies K 1 ⊗ L 1 ⊂ K 2 ⊗ L 2 . Finally, the tensor product of zonoids is associative.
Proof. Given K 1 , K 2 ∈ Z(V ), λ 1 , λ 2 ≥ 0, and a random vector Y ∈ W representing L, we use (3.2) to write the support function of (λ 1 K 1 + λ 2 K 2 ) ⊗ L as
h (λ 1 K 1 +λ 2 K 2 )⊗L (u) = E Y h λ 1 K 1 +λ 2 K 2 (T Y (u)) = E Y λ 1 h K 1 (T Y (u)) + E Y λ 2 h K 2 (T Y (u)) = λ 1 h K 1 ⊗L (u) + λ 2 h K 2 ⊗L (u).
For the second factor we argue analogously. Therefore Minkowski additivity and positive homogeneity in each factor follows from Proposition 2.1.
For the monotonicity we assume in addition that K 1 ⊂ K 2 . Proposition 2.1 implies
h K 1 ≤ h K 2 . Again, (3.2) gives h K 1 ⊗L (u) = E Y h K 1 (T Y (u)) and h K 2 ⊗L (u) = E Y h K 2 (T Y (u)).
Therefore, h K 1 ⊗L ≤ h K 2 ⊗L . Again using Proposition 2.1 shows K 1 ⊗ L ⊂ K 2 ⊗ L. For the second factor we argue analogously. Finally, the associativity immediately follows from the one of the usual tensor product.
Example 3.5 (Tensor product of zonotopes). The tensor product of symmetric segments is given by
(3.3) 1 2 [−v 1 , v 1 ] ⊗ · · · ⊗ 1 2 [−v p , v p ] = 1 2 [−v 1 ⊗ · · · ⊗ v p , v 1 ⊗ · · · ⊗ v p ], where v 1 ∈ V 1 , . . . , v p ∈ V p .
(Indeed, just take for X j ∈ V j a random variable taking the constant value v j and note that K(
X j ) = 1 2 [−v j , v j ].)
Together with the biadditivity of the tensor product (Proposition 3.4), we conclude that the tensor product of two zonotopes is
n i=1 1 2 [−x i , x i ] ⊗ m j=1 1 2 [−y j , y j ] = n i=1 m j=1 1 2 [−x i ⊗ y j , x i ⊗ y j ].
We now show that the length is multiplicative with respect to the tensor product and prove an upper bound on the norm of a tensor product of zonoids.
Proposition 3.6. For K ∈ K(V ) and L ∈ K(W ) such that m = dim V ≤ dim W , we have
ℓ(K ⊗ L) = ℓ(K)ℓ(L), K ⊗ L ≤ 2 √ m K L .
Proof. Suppose that K = K(X) and L = K(Y ) with independent random vector X ∈ V and Y ∈ W . Then, by Definition 2.10,
ℓ(K ⊗ L) = E X ⊗ Y = E X ·E Y = ℓ(K)ℓ(L),
showing the first assertion.
For the norm inequality, assume V = R m and W = R n and w.l.o.g. m ≤ n. Recall that the nuclear norm of a matrix M ∈ R m× is defined as the sum of its singular values. The corresponding unit ball B nuc equals the convex hull of the rank one matrices v ⊗ w such that v ∈ R m and w ∈ R n have norm one, e.g., see [Der16]. If we denote by B the unit ball with respect to the Frobenius norm, we get B ⊂ √ mB nuc , where we used that m ≤ n. We obtain
K ⊗ L = max u∈B h K⊗L (u) ≤ √ m max u∈Bnuc h K⊗L (u) = 1 2 √ m max u E| u, X ⊗ Y |. But for u = v ⊗ w with unit vectors v, w, we have E| v ⊗ w, X ⊗ Y | = E| v, X | · E| w, Y ) | = 4h K (v)h L (w) ≤ 4 K · L .
Using the convexity of h K⊗L , implies the second assertion.
It is straightforward to extend the tensor product of zonoids to a bilinear map between spaces of virtual zonoids.
Proposition 3.7 (Tensor product of virtual zonoids). The tensor product of zonoids from Definition 3.1 uniquely extends to a bilinear map T :
Z(V ) × Z(W ) Z(V ⊗ W ).
The resulting tensor product of virtual zonoids is associative.
Proof. The only possible way to define the map T is by setting
(K 1 − K 2 ) ⊗ (L 1 − L 2 ) := K 1 ⊗ L 1 + K 2 ⊗ L 2 − K 1 ⊗ L 2 − K 2 ⊗ L 1 .
Using the biadditivity of the tensor product of zonoids (Proposition 3.4) it is straightforward to check that this is well defined and defines a bilinear map. The associativity follows from the associativity of the tensor product of zonoids.
3.1. Continuity of the tensor product. Here we discuss the continuity of the tensor product map. The main result is that the tensor product is continuous on zonoids, but not on virtual zonoids with the norm topology. It is only separately continuous in each variable, meaning that it is continuous in each component. However, viewing virtual zonoids as measures via the correspondence described in Section 2.4 and endowed with the weak- * topology, the tensor product turns out to be sequentially continuous; see Theorem 3.10 below.
As above we denote by
T : Z(V ) × Z(W ) Z(V ⊗ W )
the tensor product map, T (K, L) = K ⊗ L, and its restriction to zonoids is denoted
T : Z(V ) × Z(W ) Z(V ⊗ W ).
Viewing virtual zonoids as measures, we obtain a bilinear map
T : M(P(V )) × M(P(W )) M(P(V ⊗ W )).
Concretely, this map can be described as follows. Let µ ∈ M(P(V )) and ν ∈ M(P(W )), and let K µ ∈ Z(V ), K ν ∈ Z(W ) be such that h Kµ = H(µ) and h Kν = H(ν), where H is the cosine transform from Definition 2.25. Then T (µ, ν) is the measure on P(V ⊗W ) characterized by H T (µ, ν) = h Kµ⊗Kν .
We now show that the map T has a direct natural characterization. For this, we first recall that the Segre embedding P(V ) × P(W )
P(V ⊗ W ), [v] ⊗ [w] [v ⊗ w]
, is an isomorphism onto its image, which allows to view P(V ) × P(W ) as a subspace of P(V ⊗ W ). Taking the pushforward gives Proof. Since the map T is bilinear we can assume without loss of generality that µ ∈ M(P(V )) and ν ∈ M(P(W )) are probability measures; the general case then follows by homogeneity and linearity. In that case, by Proposition 2.29, K µ = K(X) where X ∈ S(V ) is a random vector of law µ (recall that we identify a measure on P(V ) with the corresponding even measure on the sphere). Similarly, K ν = K(Y ), where Y ∈ S(W ) is a random vector of law ν, that we can assume to be independent of X. By definition, we have K µ ⊗ K ν = K(X ⊗ Y ). The law of X ⊗ Y is the pushforward by the Segre map of the tensor product of measures µ ⊗ ν ∈ M + (P(V ) × P(W )), and this concludes the proof.
One could take Lemma 3.8 as the definition of the tensor product of zonoids, as it may appear simpler. This simplicity however entirely relies on the fact that the tensor product on vectors (the Segre map) sends the product of spheres to the sphere. When later in Section 4, we will deal with multilinear maps that do not have this property, the point of view of random vectors is easier to handle.
To investigate the continuity of T and T , let us prove an inequality.
Lemma 3.9. For K 1 , K 2 ∈ Z(V ) and L ∈ Z(W ) we have with m := dim W
K 1 ⊗ L − K 2 ⊗ L ≤ τ m L K 1 − K 2 ,
where τ m is defined as in (2.10).
Proof. Let u ∈ V ⊗ W be such that K 1 ⊗ L − K 2 ⊗ L = |h K 1 ⊗L (u) − h K 2 ⊗L (u)|. Let L = K(Y ). From (3.2) we get h K 1 ⊗L (u) − h K 2 ⊗L (u) = E Y h K 1 − h K 2 ((T Y (u)), hence, |h K 1 ⊗L (u) − h K 2 ⊗L (u)| ≤ E Y | h K 1 − h K 2 ((T Y (u))| ≤ E Y h K 1 − h K 2 ∞ T Y (u)) ≤ E Y Y · K 1 − K 2 = ℓ(L) · K 1 − K 2
where we used (3.1) for the third inequality. Applying Corollary 2.14 completes the proof.
The main continuity properties are summarized in the following result.
Theorem 3.10. Suppose that dim V, dim W ≥ 2. Then, the tensor product map satisfies the following.
(1) T : Z(V ) × Z(W ) Z(V ⊗ W ) is continuous. More specifically, for K 1 , K 2 ∈ Z(V ) and L 1 , L 2 ∈ Z(W ), we have
d H (K 1 ⊗ L 1 , K 2 ⊗ L 2 ) ≤ (τ m L 1 + τ n K 2 ) ) (d H (K 1 , K 2 ) + d H (L 1 , L 2 )) ,
where n := dim V and m := dim W ;
(2) T : Z(V )× Z(W ) Z(V ⊗W ) with the norm topology on both sides is not sequentially continuous, but separately (i.e., componentwise) continuous;
(3) T : M(P(V )) × M(P(W )) M(P(V ⊗ W )) with the weak- * topology on both sides is sequentially continuous.
Proof. For proving (1), recall that d H (K, L) = K − L . From the multiadditivity of the tensor product and the triangle inequality of the norm, we get
K 1 ⊗ L 1 − K 2 ⊗ L 2 ≤ K 1 ⊗ L 1 − K 2 ⊗ L 1 + K 2 ⊗ L 1 − K 2 ⊗ L 2 .
Combined with Lemma 3.9, this yields
K 1 ⊗ L 1 − K 2 ⊗ L 2 ≤ τ m L 1 K 1 − K 2 + τ n K 2 L 1 − L 2 ,
which proves the first assertion.
As for (2), the separate continuity follows directly from Lemma 3.9. To prove that T is not (sequential) continuous, we begin with a general observation. Let ϕ : E × F G be a bilinear map of real normed vector spaces. Then ϕ is (sequential) continuous if and only if it has finite operator norm:
ϕ op := sup x ≤1, y ≤1 ϕ(x, y) < ∞.
We show now that T has infinite operator norm. It suffices to prove this for V = W = R 2 . Consider the sequence of vectors a n := (n, 1), b n := (n, 0) and the corresponding sequence of segments A n := 1 2 [−a n , a n ], B n := 1 2 [−b n , b n ] in R 2 . This defines the sequence of virtual zonoids A n − B n ∈ Z(R 2 ). It is immediate to check that
A n − B n = d H (A n , B n ) = 1 2 . Consider P n := (A n −B n )⊗(A n −B n ) ∈ Z(R 2 ⊗R 2 )
. It suffices to show that lim n ∞ P n = ∞. For this, we calculate a n ⊗ a n = n 2 n n 1 , b n ⊗ b n = n 2 0 0 0 , a n ⊗ b n = n 2 0 n 0 , b n ⊗ a n = n 2 n 0 0 .
Their inner product with the matrix w n := 1 −n −n 0 is given by a n ⊗ a n , w n = −n 2 , b n ⊗ b n , w n = n 2 , a n ⊗ b n , w n = b n ⊗ a n , w n = 0, Using (2.8), we obtain h An⊗An (w n ) = 1 2 | a n ⊗ a n , w n | = n 2 2 , and similarly h Bn⊗Bn (w n ) = n 2 2 , and h An⊗Bn (w n ) = h Bn⊗An (w n ) = 0. Therefore,
h Pn (w n ) w n = n 2 w n = Ω(n),
which completes the proof of the second item.
For item (3) we recall from Lemma 3.8 that T equals the tensor product of measures composed with the pushforward of the Segre map. The pushforward of a measure under a continuous map is weak- * continuous. Mapping two measures to their product measure is sequentially continuous by [Bil99, Theorem 2.8]. This finishes the proof for the third assertion.
By Proposition 2.24, the completion of the normed vector space Z(V ) is isomorphic to the Banach space of even real valued continous function defined on the unit sphere of V , or equivalently to C(P(V )). One may hope to extend the tensor product map to the completions to obtain a separate continuous bilinear map. We show now that, unfortunately, this is impossible. For proving this, we will rely on Theorem 5.1 below and on [Sch14, Theorem 5.2.2].
Proposition 3.11. There is no bilinear map C(P(V )) × C(P(W )) C(P(V ⊗ W )) that is separate continuous and extends the tensor product map T :
Z(V ) × Z(W ) Z(V ⊗ W ), provided dim V, dim W ≥ 2.
Proof. Suppose by way of contradiction that there is a bilinear map as in the proposition. From this it is straightforward to construct such map ϕ : C(P 1 ) × C(P 1 ) C(P 3 ) for V = W = R 2 . Thus it is enough to prove it for this case. The determinant map R 2 × R 2 R is a bilinear map and hence it factors via a linear map M : R 2 ⊗ R 2 R. By Theorem 5.1 below, the associated linear map M : Z(R 2 ⊗ R 2 ) Z(R) ≃ R gives the mixed volume of the tensor product in the sense that, for K 1 , K 2 ∈ Z(R 2 ),
M (K 1 ⊗ K 2 ) = ℓ(K 1 ∧ K 2 ) = 2 MV(K 1 , K 2 ).
Since M is continuous (Proposition 2.21), we can extend it to a continuous linear map λ : C(P(R 2 ⊗ R 2 )) C(P 0 ) ≃ R by Hahn-Banach. Then, λ • ϕ : C(P 1 ) × C(P 1 ) R is a componentwise continuous bilinear map extending 2 · det : Z(R 2 ) × Z(R 2 ) R. Now recall that C(P 1 ) is identified with the even functions in C(S 1 ).
The existence of λ • ϕ contradicts the fact that there is no separate continuous bilinear function V : C(S 1 ) × C(S 1 ) R that satisfies V (h K ,h L ) = MV(K, L) for K, L ∈ Z(R 2 ). The latter follows by inspecting the proof of [Sch14, Theorem 5.2.2], where the analogous statement is made about bilinear functions V satisfying the above condition for all K, L ∈ K(R 2 ).
Multilinear maps induced on zonoids
In this section we show how to associate with any multilinear map between Euclidean spaces a corresponding multilinear map of the corresponding vector spaces of zonoids. This will allow us to construct the zonoid algebra. The zonoid algebra inherits a duality notion from the Hodge star operator. . We now transfer this construction to multilinear maps via the tensor product. Note that if V 1 , . . . , V p are Euclidean vector spaces, then each Z(V i ) is an partially ordered and normed real vector space. The product space Z(V 1 ) × · · · × Z(V p ) carries a componentwise order and the product topology, where Z(V i ) carries the topology defined by the norm of virtual zonoids from Theorem 2.16.
Theorem 4.1 (Induced multilinear zonoid maps). Let V 1 , . . . , V p and W be Euclidean vector spaces and M : V 1 × · · · × V p W be a multilinear map. There exists a unique multilinear, separate continuous map
M : Z(V 1 ) × · · · × Z(V p ) Z(W ), such that for every v 1 ∈ V 1 , . . . , v p ∈ V p M 1 2 [−v 1 , v 1 ], . . . , 1 2 [−v p , v p ] = 1 2 [−M (v 1 , . . . , v p ), M (v 1 , . . . , v p )]
. Restricting to zonoids, we get a continuous map Z(V 1 ) × · · · × Z(V p ) Z(W ). The map M preserves the componentwise inclusion order of zonoids. If we interpret this map on the level of signed measures, then we obtain a multlinear map M , which is sequentially continuous with respect to the weak- * topology.
Proof. To show existence, we rely on the universal property of tensor product: there is a unique linear map L : V 1 ⊗ · · · ⊗ V p W such that L(v 1 ⊗ · · · ⊗ v p ) = M (v 1 , . . . , v p ). Consider the linear continuous map L : Z(V 1 ⊗ · · · ⊗ V p ) Z(W ) given by Definition 2.20. For (K 1 , . . . , K p ) ∈ Z(V 1 ) × · · · × Z(V p ), we define the map M by
(4.1) M (K 1 , . . . , K p ) := L(K 1 ⊗ · · · ⊗ K p ).
This is the composition of the linear map L with the multilinear tensor product map from Proposition 3.7, therefore it is multilinear.
Restricting to zonoids and using Proposition 3.4, we see M (Z(V 1 ) × · · · × Z(V p )) ⊆ Z(W ). The asserted formula for the image of M on tuples of segments is a direct consequence of (3.3) and the definition of the map M .
Since L is continuous and the tensor product map from Proposition 3.7 is separate continuous, M is separate continuous. Similarly, M | Z(V 1 )×···×Z(Vp) is continuous since the tensor product map on zonoids is continuous (Proposition 3.4).
For the uniqueness of the map M we argue as follows. Since by definition, any zonoid in V i can be approximated by symmetric segments and the values of M are determined on tuples of segments, the componentwise continuity of M determines M on Z(V 1 ) × · · · × Z(V p ). In turn, this determines M by multilinearity.
By Proposition 3.4, the tensor product of zonoids preserves the componentwise order. Moreover, L preserves the order by Proposition 2.21. This implies that M preserves the componentwise order.
The last statement about the sequential continuity with respect to the weak- * topology follows from Lemma 2.31 and Theorem 3.10 (3).
Remark 4.2. The map M : Z(V 1 ) × · · · × Z(V p ) Z(W ) is in general not sequentially continuous for the norm topology. Indeed Theorem 3.10 shows that the tensor product map for two factors is not (sequentially) continuous, and this immediately extends to any number of factors. By contrast, we showed that M is sequentially continuous for the weak- * topology.
Our construction is nicely compatible with the description of zonoids by random vectors.
Corollary 4.3. Let X 1 ∈ V 1 , . . . , X p ∈ V p be integrable and independent random vectors. For a multilinear map M : V 1 × · · · × V p W we have M (K(X 1 ), . . . , K(X p )) = K(M (X 1 , . . . , X p )).
Proof. This follows from the construction (4.1) of the map M and Proposition 2.4.
A recent work that can be interpreted from the point of view of multilinear functions of zonoids is the paper [MM21] by Meroni and Mathis, in which they study so-called fiber bodies. 4.2. Zonoid algebra. We assign to the exterior powers of a Euclidean space the corresponding zonoid vector spaces and apply Theorem 4.1 to the wedge product of the exterior algebra to arrive at a commutative and associative graded algebra, which we call the zonoid algebra of V .
Let V denote a Euclidean vector space of dimension m throughout this section. Consider the d-th exterior power Λ d V of V , defined for d ∈ N, and note that Λ d V = 0 if d > m, This space inherits a Euclidean structure from V , which can be described as follows: if {e 1 , . . . , e m } is an orthonormal basis for V , an orthonormal basis for Λ d V is given by {e i 1 ∧· · ·∧e i d } 1≤i 1 <···<i d ≤m . We form the direct sum of zonoid vector spaces
A(V ) := m d=0 Z(Λ d V ).
The elements of Z(Λ k V ) will be said to have degree d. By Lemma 2.23 we shall identify
Z(Λ 0 V ) = Z(R) ≃ R.
Remark 4.4. Proposition 2.24 implies that A(V ) is an infinite dimensional real vector space, unless dim(V ) ≤ 1.
Via Theorem 4.1, we associate with each wedge product ∧ : Λ d V × Λ e V Λ d+e V a componentwise continuous bilinear map
∧ : Z Λ d V × Z Λ e V Z Λ d+e V ,
and, by extension, a componentwise continuous bilinear map A(V ) × A(V ) A(V ), all denoted by the same symbol. We call this map the wedge product of virtual zonoids. Theorem 4.1 implies that the wedge product of zonoids is a zonoid. We write K ∧d for the wedge of K with itself d many times.
In an analogous way, we define the wedge product of several (virtual) zonoids. We can describe this product explicitly as follows. Assume that X 1 ∈ Λ d 1 V, . . . , X p ∈ Λ dp V are independent random vectors representing the zonoids A j ∈ Z(Λ d j V ). Then Corollary 4.3 implies that
(4.2) A 1 ∧ · · · ∧ A p = K(X 1 ) ∧ · · · ∧ K(X p ) = K(X 1 ∧ · · · ∧ X p ) ∈ Z(Λ d 1 +···+dp V ).
We call A(V ) the zonoid algebra associated with V . This naming is justified by the following theorem.
Theorem 4.5. The wedge product turns A(V ) into a graded, associative and commutative real algebra. The wedge maps of zonoids
Z(Λ d 1 V ) × · · · × Z(Λ dp V ) Z(Λ d 1 +···+dp V ), (A 1 , . . . , A p ) A 1 ∧ · · · ∧ A p
are continuous. These maps preserve the inclusion order of zonoids: if we have A ′ j ⊂ A j for zonoids in Z(Λ d j V ), then A ′ 1 ∧ · · · ∧ A ′ p ⊂ A 1 ∧ · · · ∧ A p . Moreover, the wedge product of zonoids does not increase the length:
ℓ(A 1 ∧ · · · ∧ A p ) ≤ ℓ(A 1 ) · · · ℓ(A p ).
Proof. The associativity follows from the associativity of the wedge product and (4.2). The distributivity is a consequence of Proposition 3.7. The gradedness follows from the definition of A(V ). The multiplicative unit lies in Z(Λ 0 V ) = Z(R) ≃ R. The commutativity of the wedge follows with (4.2) from the known relation X ∧Y = ±Y ∧X and the fact that K(−Z) = K(Z).
The wedge map of zonoids is continuous since it is obtained by composing a continuous linear map with the continuous tensor product of zonoids (see Theorem 3.10). The preservation of the inclusion order follows from Theorem 4.1.
For the length inequality, we use that the antisymmetrization map ⊗ j Λ d j V Λ d 1 +···+dp V is an orthogonal projection. Hence ℓ(A 1 ∧ · · · ∧ A p ) ≤ ℓ(A 1 ⊗ · · · ⊗ A p ) = ℓ(A 1 ) · · · ℓ(A p ) by Proposition 2.21 and Proposition 3.6.
Here is an immediate yet important observation about wedge products of zonoids.
Lemma 4.6. Let K ∈ Z(V ) be a zonoid. Recall that we defined the dimension of K as the dimension of its linear span K . Then, K ∧d = 0 for all d > dim(K).
Proof. Let us write K = K(X). By (4.2) we have K ∧d = K(X 1 ∧ · · · ∧ X d ), where X 1 , . . . , X d are independent copies of X. With probability one we have X ∈ K , and so with probability one the X i are linearly dependent. Hence, X 1 ∧· · ·∧X d = 0 almost surely, so that K ∧d = 0.
In the next section, we will link the length to the mixed and intrinsic volumes. More specificially, Theorem 5.2 show that the jth intrinsic volume V j (K) can be expressed as 1 j! ℓ(K ∧j ). This allows to immediately derive Corollary 4.7, which is a reverse Alexandroy-Fenchel inequality, that was independently found very recently by Böröczky and Hug [BH21].
It is useful to denote by K[d] the zonoid K repeated d times.
Corollary 4.7. Let K 1 , . . . , K p ∈ Z(V ) be zonoids and K 1 , . . . , K p their spans. Assume
d 1 , . . . , d p ∈ N satisfy d 1 + . . . + d p = m. Then m! d 1 ! · · · d p ! MV(K 1 [d 1 ], . . . , K p [d p ]) ≤ V d 1 (K 1 ) · · · V dp (K p ).
Equality holds if and only K 1 . . . , K p are pairwise orthogonal or if K ∧d i i = 0 for at least one i.
By Lemma 4.6 the condition that K ∧d = 0 is equivalent to either K = 0 or d > dim K .
Proof of Corollary 4.7. By Theorem 5.2, the stated inequality can be rephrased as
(4.3) ℓ K ∧d 1 1 ∧ · · · ∧ K ∧dp p ≤ ℓ K ∧d 1 1 · · · ℓ K ∧dp d ,
which is a consequence of the submultiplicativity of the length, see Theorem 4.5.
For analyzing when equality holds, we assume p = 2 to simplify notation. We write K 1 = K(X) and K 2 = K(Y ). Let X 1 , . . . , X d 1 be independent copies of X and Y 1 , . . . , Y d 2 be independent copies of Y . The above inequality (4.3) can be written as an inequality of expectations E X 1 ∧ · · · ∧ X d 1 ∧ Y 1 ∧ · · · ∧ Y d 2 ≤ E X 1 ∧ · · · ∧ X d 1 · E Y 1 ∧ · · · ∧ Y d 2 , and equality holds if and only if X 1 ∧ · · · ∧ X d 1 ∧ Y 1 ∧ · · · ∧ Y d 2 = X 1 ∧ · · · ∧ X d 1 · Y 1 ∧ · · · ∧ Y d 2 almost surely. We have equality if and only if X i , Y j = 0 almost surely for all i, j, or if one of the wedge products is almost surely zero. By Lemma 2.15, the first case is equivalent to K 1 and K 2 being orthogonal, the second case means that K ∧d 1 1 = 0 or K ∧d 2 2 = 0.
Grassmannian zonoids.
Here we describe a particular subalgebra of A(V ), which will play a role in our forthcoming work [BBML21].
Definition 4.8. Let K ∈ Z(Λ k V ). We say that K is a Grassmannian zonoid if there is random vector X ∈ Λ k V such that K = K(X) and such that X is almost surely simple, i.e. X almost surely takes values in the cone of simple vectors {v 1 ∧ · · · ∧ v k | v 1 , . . . , v k ∈ V }. The set of Grassmannian zonoids will be denoted G(k, V ) ⊂ Z(Λ k V ) and its linear span in Z(Λ k V ) will be denoted by G(k, V )
In the correspondence between zonoids and measures on the projective space described in Section 2.4, Grassmannian zonoids have a simple description. Indeed using the definition above and Proposition 2.29 we see that a zonoid K ∈ Z(Λ k V ) is Grassmannian if and only if its corresponding measure is supported on the Grassmannian, considered as the simple vectors in P(Λ k V ) via the Plücker embedding span(v 1 , . . . , v k )
[v 1 ∧ · · · ∧ v k ]. Hence the name Grassmannian zonoids.
Proposition 4.9. The wedge product of two Grassmannian zonoids is a Grassmannian zonoid. Moreover, if they are of the same degree, the sum of two Grassmannian zonoids is a Grassmannian zonoid. Hence G(k, V ) ⊂ Z(Λ k V ) consists only of differences of Grassmannian zonoids and G(k, V ) is a convex cone in G(k, V ).
Proof. Let K = K(X 1 ∧ · · · ∧ X p ) ∈ G(p, V ) and L = K(Y 1 ∧ · · · ∧ Y q ) ∈ G(q, V ) with X 1 ∧ · · · ∧ X p independent of Y 1 ∧ · · · ∧ Y q . Then by definition of the wedge product we have K ∧ L = K(X 1 ∧ · · · ∧ X p ∧ Y 1 ∧ · · · ∧ Y q ) which is Grassmannian. Now suppose p = q then by Lemma 2.6, we have K + L = K(Z) where Z = 2(1 − ǫ)X 1 ∧ · · · ∧ X p + 2ǫY 1 ∧ · · · ∧ Y p where ǫ is a Bernoulli variable of parameter 1 2 independent of the other variables. It is then enough to see that Z ∈ Λ p V is almost surely simple.
Definition 4.10. The Grassmannian zonoid algebra is defined as
G(V ) := d k=0 G(k, V ).
Proposition 4.9 guarantees that G(V ) is a subalgebra of A(V ).
A particular case of Grassmannian zonoids are the decomposable ones. Namely, if we have K 1 , . . . , K k ∈ Z(V ), then K 1 ∧ · · · ∧ K k ∈ G(k, V ) by Proposition 4.9. This corresponds to the case K 1 ∧ · · · ∧ K k = K(X) with X = X 1 ∧ · · · ∧ X k , where X 1 , . . . , X k are independent random vectors. In general, a zonoid in G(k, V ) represented by X = X 1 ∧ · · · ∧ X k does not need to have independent X i . The next proposition says that decomposable Grassmann zonoids span actually a dense subspace.
Proposition 4.11. Finite sums of zonoids of the form K 1 ∧ · · · ∧ K k ∈ G(k, V ) are dense in G(k, V ). Hence the set {K 1 ∧ · · · ∧ K k | K 1 , . . . , K k ∈ Z(V )} spans a dense subspace in the virtual Grassmann zonoids G(k, V ).
Proof. By Remark 2.5, any zonoid in G(k, V ) is the limit of finite sums of segments that are of the form 1 2 [−w, w] with w = x 1 ∧ · · · ∧ x k . It is then enough to see that such segments are decomposable. Indeed we have 1
2 [−w, w] = 1 2 [−x 1 , x 1 ] ∧ · · · ∧ 1 2 [−x k , x k ]
. This last fact gives good hope to extend properties of decomposable zonoids to the Grassmannian ones. For example we conjecture the following.
Conjecture. For any K, L ∈ Z(R m ) and any C ∈ G(m − 2, R m ), we have
ℓ(K ∧ L ∧ C) 2 ≥ ℓ(K ∧ K ∧ C)ℓ(L ∧ L ∧ C).
This would generalize Alexandrov-Fenchel, which corresponds to the case where C is decomposable; i.e., of the form C = K 1 ∧ · · · ∧ K m−2 with K 1 , . . . , K m−2 ∈ Z(R m ), see (5.2) below. 4.4. Hodge duality. The exterior algebra of an oriented Euclidean vector space V comes with the duality given by the Hodge star operation. Let us briefly recall this notion. Upon choosing an orientation of V , Λ m V becomes an oriented one dimensional Euclidean vector space that can be identified with R. More specifically, if e 1 , . . . , e m is an oriented orthonormal basis of V , then e 1 ∧ · · · ∧ e m is the distinguished generator of Λ m V . When d 1 + . . . + d p = m, the wedge product thus induces a multilinear map
Z(Λ d 1 V ) × · · · × Z(Λ dp V ) Z(Λ m V ) ≃ R.
Therefore the wedge product K 1 ∧ · · · ∧ K p of zonoids, which is a segment in Λ m V , can be identified with the real number giving the length of this segment using Lemma 2.23. This remark will play a role in Section 5.1. The Hodge star operation is the isometric linear map Λ d V Λ m−d V, v ⋆v characterized by u, v = u ∧ ⋆v for all u ∈ V . This defines an involution (up to sign) of the exterior algebra ΛV , see [Fla89,p.16]. Via Definition 2.20 we associate with the Hodge star operation the linear isomorphism
Z Λ d V Z Λ m−d V , K ⋆K
that we conveniently denote with the symbol ⋆ as well. If X ∈ Λ k V is an integrable random variable, then by Proposition 2.4
⋆ (K(X)) = K(⋆X).
This shows that, if K ∈ Z Λ k V is a zonoid, then ⋆K is a zonoid as well. The support function of ⋆K satisfies
(4.4) h ⋆K (⋆u) = 1 2 E| ⋆u, ⋆X | = 1 2 E| u, X | = h K (u)
, where we used that ⋆ is isometric for the second equality.
Proposition 4.12. The Hodge star operation on zonoids Z Λ d V Z Λ m−d V is a norm and order preserving linear isomorphism. It also preserves the length of zonoids, so that for all zonoids K ∈ Z Λ d V K = ⋆ K , ℓ(K) = ℓ(⋆K). The Hodge star operation defines a linear involution of the zonoid algebra A(V ).
Proof. Using (4.4) and Proposition 2.1, we obtain
⋆ K = h ⋆K ∞ = h K ∞ = K .
Moreover, using Definition 2.10, we get
ℓ(K) = E X = E ⋆ X = ℓ(⋆K).
The fact that it is an involution on the space of zonoids follows from the fact that the Hodge star is an involution up to sign and that the zonoids are centrally symmetric.
Remark 4.13. ⋆K should not be confused with the polar dual of K ∈ Z Λ d V , which is a convex body living in the same space as K, and in general not a zonoid, see [Bol69].
Remark 4.14. Note that since the Hodge star operation preserves simple vectors, if K ∈ G(k, V ) is a Grassmannian zonoid then ⋆K ∈ G(m − k, V ) is also Grassmannian. In other words, the map ⋆ preserves the subalgebra G(V ) of Grassmannian zonoids.
Let K ∈ Z(V ). The Hodge dual of K ∧(m−1) is a zonoid in Λ 1 V = V . We show now that it is the so called projection body ΠK of K. According to [Sch14, Section 10.9], ΠK is the convex body whose support function is given by u u vol m−1 (π u (K)), where π u denotes the orthogonal projection V u ⊥ .
Proposition 4.15. We have ⋆(K ∧(m−1) ) = (m−1)! 2 ΠK.
Proof. We show that these zonoids have the same support function. Let X ∈ V be a random vector representing K = K(X). By definition,
⋆(K ∧(m−1) ) = K(Y ), where Y := ⋆(X 1 ∧ · · · ∧ X m−1 )
and X 1 , . . . , X m−1 are i.i.d. copies of X. The definition of the Hodge dual yields | ⋆(X 1 ∧ · · · ∧ X m−1 ), u | = |X 1 ∧ · · · ∧ X m−1 ∧ u| and so | Y, u | = |X 1 ∧ · · · ∧ X m−1 ∧ u| = π u (X 1 ) ∧ · · · ∧ π u (X m−1 ) · u . By (2.8), we get h K(Y ) (u) = 1 2 E| Y, u | = 1 2 E π u (X 1 ) ∧ · · · ∧ π u (X m−1 ) · u . Theorem 5.1 from the next section, applied to the space u ⊥ ≃ R m−1 , yields
h K(Y ) (u) = (m − 1)! 2 vol m−1 (π u (K)) u .
This shows that ⋆(K ∧(m−1) ) has the same support function as (m−1)! 2 ΠK.
We close this section by giving an interesting property of Hodge duals. It concerns orthogonal zonoids and will be of relevance in our upcoming work [BBML21].
Corollary 4.16. Let K, L ∈ Λ d V be zonoids and denote by K and L their linear spans.
Then K ∧ ⋆L = 0 ⇐⇒ K ⊥ L .
Proof. Let X and Y be integrable random vectors with K = K(X) and L = K(Y ). Then K ∧ ⋆L = K(X ∧ ⋆Y ). By the definition of the Hodge dual, we have X ∧ ⋆Y = X, Y . Therefore, K ∧ ⋆L = 0 if and only if X, Y = 0 almost surely. By Lemma 2.15, this is equivalent to K ⊥ L .
Mixed volumes and random determinants
In this section we show how the the classical notion of mixed volume fits into our framework. We then apply our theory to study expected absolute determinants of random matrices. Again we denote by V a Euclidean vector space of dimension m.
We recall that the mixed volume is the Minkowski multilinear, translation invariant and continuous map MV : K(V ) m R, defined on an m-tuple K 1 . . . , K m of convex bodies by
MV(K 1 , . . . , K m ) = 1 m! ∂ ∂t 1 · · · ∂ ∂t m vol m (t 1 K 1 + · · · + t m K m ) t 1 =···=tm=0 , see [Sch14, Theorem 5.1.7]. For instance, if K i = 1 2 [−v i , v i ] are segments, then MV( 1 2 [−v 1 , v 1 ], . . . , 1 2 [−v m , v m ]
) equals the volume of the parallelotope spanned by v 1 , . . . , v m , divided by m!. The volume of K is vol m (K) = MV(K, . . . , K), which means that the mixed volume is obtained from the volume function of a single body by polarization.
It is a key insight that the mixed volume of zonoids equals the length of their wedge product, up to a constant factor.
Theorem 5.1 (The wedge product of zonoids). Let det : Z(V ) m Z 1 denote the multilinear map associated to the multilinear determinant map det : V m R via Theorem 4.1. Then, for every K 1 , . . . , K m ∈ Z(V ), we have
det(K 1 , . . . , K m ) = K 1 ∧ · · · ∧ K m and MV(K 1 , . . . , K m ) = 1 m! ℓ(K 1 ∧ · · · ∧ K m ).
Proof. The first identity is immediate from the definition of associated multilinear map det.
For the second identity, we observe that both sides are Minkowski multilinear and continuous maps Z(V ) m R (for the right hand side, use Theorem 2.22). It therefore suffices to verify the identity for segments. So let v 1 , . . . , v m ∈ V . Note that by Theorem 4.1, The previous theorem implies that for K ∈ Z(V )
L := 1 2 [−v 1 , v 1 ] ∧ · · · ∧ 1 2 [−v m , v m ] = 1 2 [− det(v 1 ,(5.1) vol m (K) = 1 m! ℓ(K ∧m ),
where, again, K ∧m = K ∧ · · · ∧ K with m factors.
5.1. Length functional and intrinsic volumes. Recall [KR97,Sch14] that the d-th intrinsic volume V d (K) of a zonoid K is defined as
V d (K) := m d vol m−d (B m−d ) MV(K[d], B m [m − d]).
In the following, we show that the length of a zonoid, introduced in Definition 2.10, is nothing but its first intrinsic volume (see [KR97,Sch14]), and that the higher intrinsic volumes can also be expressed using the length. As before, we denote by B = B(V ) the unit ball in V and B m := B(R m ).
Theorem 5.2. The dth intrinsic volume of a zonoid K ∈ Z(V ) is given by
V d (K) = 1 d! ℓ(K ∧d ),
where, as before, K ∧d is the wedge product of d copies of K. In particular,
V 1 (K) = ℓ(K).
Proof. Suppose X 1 , . . . , X d , Y 1 , . . . , Y m−d are independent random vectors with values in V such that the X i represent K and the Y j are standard Gaussian. Recall that B m = √ 2πK(Y j ). By Theorem 5.1, we can write
MV(K[d], B m [m − d]) = (2π) m−d 2 m! E|X 1 ∧ · · · ∧ X d ∧ Y 1 ∧ · · · ∧ Y m−d |.
We first integrate over the Y j while leaving the X i fixed, thus
MV(K[d], B m [m − d]) = (2π) m−d 2 m! E X i E Y j |X 1 ∧ · · · ∧ X d ∧ Y 1 ∧ · · · ∧ Y m−d |.
By orthogonal invariance of Y := Y 1 ∧ · · · ∧ Y m−d , we can assume that in the inner expectation the space spanned by the X i is the span of a fixed orthonormal frame e 1 , . . . , e d . Then, X 1 ∧ · · · ∧ X d = X 1 ∧ · · · ∧ X d e 1 ∧ · · · ∧ e d and so, using that Y is independent of the X i :
MV(K[d], B m [m − d]) = c m! E X i X 1 ∧ · · · ∧ X d = c m! ℓ(K),
with the constant c := (2π)
m−d 2 E Y e 1 ∧ · · · ∧ e d ∧ Y .
In order to determine this constant, we use that e 1 ∧ · · · ∧ e d ∧ Y = Ỹ 1 ∧ . . . ∧Ỹ d , whereỸ j denotes the orthogonal projection of Y j onto the orthogonal complement R m−d of R d = span{e 1 , . . . , e d }. Since the unit ball B m−d is represented by √ 2πỸ j , we obtain with (5.1),
(2π) m−d 2 E|Ỹ 1 ∧ · · · ∧Ỹ m−d | = ℓ B ∧(m−d) m−d = (m − d)! vol m−d (B m−d ).
We therefore conclude that
MV(K[d], B m [m − d]) = 1 m! vol m−d (B m−d )ℓ(K ∧d ),
which finishes the proof.
An inspection of the proof of Theorem 5.2 reveals the following general insight. for any zonoid K = K 1 ∧ · · · ∧ K d with K 1 , . . . , K d ∈ Z(V ).
The Alexandrov-Fenchel inequality for convex bodies is one of deepest results of the Brunn-Minokowski theory. Via Theorem 5.1, we can express this inequality for zonoids in terms of the length as follows: if K 1 , K 2 , . . . , K m ∈ Z(V ) are zonoids in V , then
(5.2) ℓ(K 1 ∧ K 2 ∧ K) 2 ≥ ℓ(K 1 ∧ K 1 ∧ K) · ℓ(K 2 ∧ K 2 ∧ K), where K = K 3 ∧ · · · ∧ K m ;
see, e.g., [Sch14, Theorems 6.3.1]. More generally, the general Brunn-Minkowski theorem [Sch14, Theorem 6.4.3] implies that for any 1 ≤ d ≤ m:
(5.3) t ℓ(K ∧d t ∧ K d+1 ∧ · · · ∧ K m ) 1 d is concave for t ∈ [0, 1],
where K t := tK 1 + (1 − t)K 2 .
Using Corollary 5.3, we deduce from (5.2) the following special case:
(5.4) ℓ(K ∧ L) 2 ≥ ℓ(K ∧ K) ℓ(L ∧ L)
for all zonoids K, L ∈ Z(V ). Moreover, (5.3) means that
(5.5) t ℓ((tK + (1 − t)L) ∧d ) 1 d is concave for t ∈ [0, 1].
By Corollary 5.3, we can generalize (5.4) by replacing K ∧ L with the wedge product of K ∧ L with any orthogonally invariant zonoid in M ∈ Z(Λ k (V )) such that k + d ≤ m, and get
ℓ(K ∧ L ∧ M ) 2 ≥ ℓ(K ∧ K ∧ M ) ℓ(L ∧ L ∧ M ).
Similarly, in (5.5) we may also take the product of the d-th wedge power with M and obtain that the function ℓ((tK + (1 − t)L) ∧d ∧ M ) 1 d is concave for t ∈ [0, 1].
Random determinants.
The purpose of this section is to generalize a result due to Vitale. In [Vit91] Vitale showed that if X ∈ R m is an integrable random vector and M X is the m × m matrix whose columns are i.i.d. copies of X, then E| det(M X )| = m! vol m (K(X)). We generalize this result to independent blocks that can give different distributions. This is to be compared with Theorem 6.8 below, in which we prove a similar result, but for complex random matrices.
Theorem 5.4 (Expected absolute determinant of independent blocks). Let M = (M 1 , . . . , M p ) be a random m × m matrix partitioned into blocks M j of size m × d j , with d 1 + · · · + d p = m. We denote by v j,1 , . . . , v j,d j the columns of M j and assume that Z j := v j,1 ∧ · · · ∧ v j,d j ∈ Λ d j R m is integrable. If the random vectors Z 1 ∈ Λ d 1 (R m ), . . . , Z p ∈ Λ dp (R m ) are independent, then:
E| det(M )| = ℓ(K(Z 1 ) ∧ · · · ∧ K(Z p )).
In particular, if p = m and d 1 = · · · = d p = 1, then E| det(M )| = m! MV(K(Z 1 ), . . . , K(Z m )).
The formula for independent columns, i.e., where p = m, was already proved by Weil in [Wei76, Theorem 4.2]. In fact, Weil proved a more general version in the case p = m for convex bodies, not just zonoids.
Vitale's theorem corresponds to the special case d j = · · · = d m = 1 and where the matrices M 1 , . . . , M m (which are now column vectors) all have the same distribution: M j ∼ X. In this case, E| det(M )| = m!MV(K(X), . . . , K(X)) = m! vol m (K(X)).
Proof of Theorem 5.4. Let us first observe that | det[v 1 , . . . , v m ]| = v 1 ∧ · · · ∧ v m for vectors v 1 , . . . , v m ∈ R m . Applying this in our case, we get for the absolute determinant of M that | det(M )| = v 1,1 ∧ · · · ∧ v p,ap = Z 1 ∧ · · · ∧ Z p . Taking expectations on both sides, we obtain E| det(M )| = ℓ(K(Z 1 ∧ · · · ∧ Z p )). By the definition of the wedge product, we have K(Z 1 ∧ · · · ∧ Z p ) = K(Z 1 ) ∧ · · · ∧ K(Z p ), which shows the first assertion. The second claim follows from Theorem 5.1. This concludes the proof.
We give two additional examples in which Theorem 5.4 is applied.
Example 5.5. Let Z 1 , . . . , Z n ∈ C n be integrable random vectors and L ∈ C n×n be the random matrix L = (Z 1 , . . . , Z n ). We show how to compute E| det(L)| 2 with Theorem 5.4 (in the next section we will explain how to compute E| det(L)|; see Theorem 6.8). To this end, we decompose Z j = X j + √ −1Y j with real random vectors X j , Y j ∈ R n (possibly dependent),
We put m := 2n and consider the random matrix
M = (M 1 , . . . , M n ), where M j = X j −Y j
Y j X j , which satisfies the hypothesis of Theorem 5.4. Observe that | det(L)| 2 = | det(M )|. If we define the integrable random vector
Q j := X j Y j ∧ −Y j X j ∈ Λ 2 (R m ),
we get by Theorem 5.4 that E| det L| 2 = E| det(M )| = ℓ(K(Q 1 ) ∧ · · · ∧ K(Q n )).
Example 5.6. We interpret here Theorem 5.4 geometrically as follows: we consider a random parallelotope P ⊂ R m spanned by k ≤ m random vectors, and ask for its expected k-dimensional volume. Suppose that the random vectors are v i,j for 1 ≤ i ≤ p and 1 ≤ j ≤ d i with p j=1 d j = k and such that v i 1 ,j 1 and v i 2 ,j 2 are independent, if i 1 = i 2 . Setting Z i := v i,1 ∧ · · · ∧ v i,d i , we therefore have Z 1 ∧ · · · ∧ Z p ∈ Λ k (R m ) and E (vol k (P )) = ℓ(K(Z 1 ) ∧ · · · ∧ K(Z p )).
This shows that the length functional can be used to compute the expected volume of P .
Theorem 5.4 has an interesting consequence when combined with the Alexandrov-Fenchel inequality (5.2) and the general Brunn-Minkowski theorem (5.3).
Corollary 5.7 (Brunn-Minkowski theorem for expected determinants). Let X 1 , . . . , X m be independent integrable random vectors in R m , and let X ′ 1 ∼ X 1 and X ′ 2 ∼ X 2 be independent of X 1 , X 2 , respectively. Then:
E | det[ X 1 X 2 X 3 ... Xm ]| 2 ≥ E | det[ X 1 X ′ 1 X 3 ... Xm ]| · E | det[ X 2 X ′ 2 X 3 ..
. Xm ]|. More generally, for any 1 ≤ d ≤ m, the following function is concave for t ∈ [0, 1]:
t E | det[ X (t) 1 ... X (t) d X d+1 ... Xn ]| 1 d , where X (t) 1 , . . . , X (t)
d are independent copies of tǫ2X 1 + (1 − t)(1 − ǫ)2X 2 and ǫ is a Bernoulli random variable with success probability 1 2 , which is independent of X 1 , X 2 , X d+1 , . . . , X n .
Proof. We set K i := K(X i ) for 1 ≤ i ≤ m. The first inequality is (5.2) combined with Theorem 5.4. For proving the second statement we let K t := tK 1 + (1 − t)K 2 . Then, by Lemma 2.6, K t = K(X (t) ), where X (t) = tǫ2X 1 + (1 − t)(1 − ǫ)2X 2 . We combine Theorem 5.4 with (5.3) to conclude.
We find this a quite remarkable consequence, since it holds under very weak assumptions: only independence and integrability of the X i is assumed. To our best knowledge, this is a new formula that has not been described in the literature.
Remark 5.8. The equality case of Alexandrov-Fenchel for zonoids was described in [Sch88]. From this, one can deduce the equality case for random determinants in Corollary 5.7.
Example 5.9. We can combine (5.4) and (5.5) with Theorem 5.4 to obtain a result about expected volumes of random triangles (this is the case k = 2 in Example 5.6):
(5.6) E vol 2 (∆(X, Y )) 2 ≥ E vol 2 (∆(X, X ′ )) · E vol 2 (∆(Y, Y ′ )),
where X, Y, X ′ , Y ′ are independent vectors with finite expected norm, X ∼ X ′ and Y ∼ Y ′ , and ∆(X, Y ) is the triangle, whose vertices are the origin and X and Y . We also get that the function
t E vol d (P (X (t) 1 , . . . , X (t) d )) 1 d is concave in t ∈ [0, 1],
where the X
Mixed J-volume and random complex determinants
In this section V will be a complex vector space of complex dimension n, that is a real vector space of real dimension 2n together with a real linear endomorphism J : V V such that J 2 = −1. J will be called the complex structure of V . We recall that such a complex structure induces an isomorphism V ∼ = C n under which the automorphism J corresponds to multiplication by i.
Here we will introduce a notion similar to mixed volume for zonoids in V , which is adapted to the complex structure. We call it the mixed J-volume and denote it by MV J . It takes n zonoids in a 2n-dimensional real vector space, while the ordinary mixed volume MV :
Z(V ) 2n
R is instead a function of 2n arguments. Furthermore, MV J is Minkowski additive and positively homogeneous in each argument; see Definition 6.2 below. We have already seen an application of our theory to zonoids in a complex vector space in Example 5.5. This example, however, used real multilinear maps. In this section, we consider complex multilinear maps, which leads to a different notion.
There are some key properties that make the mixed J-volume interesting: (1) it is compatible with the complex structure (see Proposition 6.6 (3)); it allows to formulate a complex version of Vitale's theorem, for computing the expectation of the modulus -and not the modulus squared, as it is done in Example 5.5 -of the determinant of a random n × n matrix with rows which are independent random variables in C n (Theorem 6.8); (3) it can be defined on all polytopes (but does not continuously extends all convex bodies, see Corollary 6.20); and finally (4) it equals the classical mixed volumes when restricted to polytopes in R n ⊂ C n ; see Proposition 6.6 (2).
The extension of the J-volume to polytopes in V ≃ C n is a so called valuation; see Definition 6.17. The proof of the extension of the J-volume to polytopes is especially interesting and uses some particular combinatorial properties of zonotopes, and it connects to similar notions, such as Kazarnovskii's pseudovolume (see Definition 6.21 below).
6.1. The mixed J-volume of zonoids. We endow our complex vector space (V, J) with a hermitian structure φ : V × V C. The associated scalar product is the real part of φ. When V = C n we consider the standard complex structure where J is the multiplication by i = √ −1 and the standard hermitian structure. Moreover, for 0 ≤ k ≤ n we denote by Λ k C (V ) the complex exterior algebra and, given vectors v 1 , . . . , v k ∈ V , we denote by v 1 ∧ C · · · ∧ C v k ∈ Λ k C (V ) their complex exterior product. Note that this construction depends on the choice of the complex structure J, however we prefer the notation with "C" that we find easier to read.
The hermitian structure on V induces an hermitian structure on all the complex exterior powers and, in particular, taking its real part, a real scalar product on each of them. This implies that we have a Euclidean norm on each Λ k C (V ) and consequently we have a length functional ℓ : Z(Λ k C (V )) R; see Definition 2.10. Moreover the complex wedge product is, in particular, a real multilinear map. Therefore we can apply Theorem 4.1 to obtain a well-defined notion of complex product of virtual zonoids.
Definition 6.1. Consider the (real) multilinear map F : V n Λ n C (V ) defined by the complex wedge F (v 1 , . . . , v n ) := v 1 ∧ C · · · ∧ C v n . For any K 1 , . . . , K n ∈ Z(V ), we define:
K 1 ∧ C · · · ∧ C K n := F (K 1 , . . . , K n ).
The next definition uses this construction to define the mixed J-volume.
Definition 6.2 (Mixed J-volume). We define the mixed J-volume MV J : Z(V ) n R to be the R-multilinear map given, for all K 1 , . . . , K n ∈ Z(V ), by:
MV J (K 1 , . . . , K n ) := 1 n! ℓ (K 1 ∧ C · · · ∧ C K n ) .
The J-volume of a zonoid K ∈ Z(V ) is defined to be:
vol J n (K) := MV J (K, . . . , K). Remark 6.3. Notice that, since Λ 2n (V ) ≃ R is of real dimension one, zonoids in Λ 2n (V ) are just segments. By contrast, the top complex exterior power Λ n C (V ) ≃ C is of real dimension two and centered zonoids in this space are more than segments (in fact they are precisely the centrally symmetric convex bodies; see [Sch14, Theorem 3.5.2]). Thus K 1 ∧ C · · · ∧ C K n is a zonoid in Λ n C (V ) ≃ R 2 . Then taking its length loses some information. However, it is easy to see using Definition 6.1, that if one of the K i is invariant under the U (1) action on V , then K 1 ∧ C · · · ∧ C K n is also U (1) invariant and hence must be a disc. We compute the length of a disc in Lemma 6.5 below.
Let us study some of the properties of the mixed J−volume. On some classes of zonoids of the complex space V it behaves particularly well. The first case is when V = C n and all the zonoids are contained in the real n−plane R n ⊂ C n . In that case, we will show that the mixed J−volume is equal to the classical mixed volume (see Proposition 6.6 (2)).
Next, we consider complex discs.
Definition 6.4. Let z ∈ V . We define D z to be the closed centered disc of radius |z| in the complex line Cz ∼ = R 2 .
In order to describe a random vector representing D z , let us introduce the following notation. For θ ∈ R we denote by e θJ : V V the linear operator e θJ := cos(θ)Id + sin(θ)J where Id denotes the identity on V . We then have the following lemma.
Lemma 6.5. Let θ ∈ [0, 2π] be a uniformly distributed random variable and z ∈ V nonzero. Consider the random vector X z ∈ V defined by X z := πe θJ z. Then:
K(X z ) = D z and ℓ(D z ) = π z .
Proof. Since for every θ ∈ [0, 2π] the vector e θJ z belongs to Cz, we have h K(Xz) (u) = 0 for every u ∈ (Cz) ⊥ . This implies that K(X z ) is contained in Cz. It is straightforward to verify that E| e θJ z, z | = z 2 E| cos θ| = 2 π z 2 . This implies for λ ∈ C that h K(Xz) (λz) = 1 2 E| X z , λz | = 1 2 π|λ| E| e θJ z, z | = |λ| z 2 = z · |λz|.
On the other hand, h Dz (λz) = z ·|λz|, hence the first assertion follows. The second statement follows immediately from the fact that X z = π z almost surely.
Proposition 6.6 (Properties of the mixed J-volume). The following properties hold:
(1) The mixed J-volume of zonoids MV J : Z(V ) n R is symmetric, multilinear, and monotonically increasing in each variable.
(2) Suppose V = C n and let K 1 , . . . , K n ∈ Z(R n ) ⊂ Z(C n ). Then:
MV J (K 1 , . . . , K n ) = MV(K 1 , . . . , K n ).
(3) Let T : V V be a C-linear transformation (i.e., such that T J = JT ), and denote by det C (T ) its complex determinant. Then, for all K 1 , . . . , K n ∈ Z(V ),
MV J (T K 1 , . . . , T K n ) = |det C (T )| MV J (K 1 , . . . , K n ).
(4) For every z 1 , . . . , z n ∈ V we have MV J (D z 1 , · · · , D zn ) = π n n! |z 1 ∧ C · · · ∧ C z n | (5) For every θ ∈ R and every K 1 , . . . , K n ∈ Z(V ) we have MV J (e θJ K 1 , K 2 , . . . , K n ) = MV J (K 1 , . . . , K n ).
Proof. Multilinearity of MV J follows from the definition and Theorem 4.1. To see that MV J is symmetric, given zonoids K 1 , . . . , K n in V , let X 1 , . . . , X n ∈ V be independent integrable random vectors such that K j = K(X j ). We have
K 1 ∧ C K 2 ∧ C · · · ∧ C K n = K(X 1 ∧ C X 2 ∧ C · · · ∧ C X n ) = K(−X 2 ∧ C X 1 ∧ C · · · ∧ C X n ) = K(X 2 ∧ C X 1 ∧ C · · · ∧ C X n ) (by Lemma 2.7) = K 2 ∧ C K 1 ∧ C · · · ∧ C K n .
The same argument gives symmetry in each pairs of variables. The fact that the mixed Jvolume is monotonically increasing in each variable is a direct cosequence of the definition, Theorem 4.1 and the monotonicity of the length (Corollary 2.13) .
Let us prove point (2). Let K 1 , . . . , K n ∈ Z(R n ) ⊂ Z(C n ) and let X 1 , . . . , X n ∈ R n be independent random (real) vectors such that K j = K(X j ), 1 ≤ j ≤ n. By Definition 6.2 the mixed J-volume is MV J (K 1 , . . . , K n ) = 1 n! ℓ (K 1 ∧ C · · · ∧ C K n ) = 1 n! E X 1 ∧ C · · · ∧ C X n . Because the X j are real vectors, the span of the X j defines a Lagrangian plane (a plane E that is orthogonal to JE), and so we have X 1 ∧ C · · · ∧ C X n = X 1 ∧ · · · ∧ X n , by Lemma 6.13. We get MV J (K 1 , . . . , K n ) = 1 n! E X 1 ∧ · · · ∧ X n . We conclude from Theorem 5.1 that the latter is equal to MV(K 1 , . . . , K n ). This finishes the proof of point (2).
In order to prove point (3), let K 1 , . . . , K n ⊂ V be zonoids and let again X j ∈ V be random vectors such that K j = K(X j ). Then T K j = K(T X j ) and we have MV J (T K 1 , . . . , T K n ) = 1 n! E|T X 1 ∧ C · · · ∧ C T X n | = 1 n! E|(det C (T ))X 1 ∧ C · · · ∧ C X n | = 1 n! |det C (T )|E|X 1 ∧ C · · · ∧ C X n | = |det C (T )|MV J (K 1 , . . . , K n ).
To show point (4) we use Lemma 6.5 and write, for θ 1 , . . . , θ n ∈ [0, 2π] independent and uniformly distributed:
MV J (D z 1 , . . . , D zn ) = 1 n! ℓ(D z 1 ∧ C · · · ∧ C D zn ) = 1 n! ℓ K(πe Jθ 1 z 1 ) ∧ C · · · ∧ C K(πe Jθn z n ) = 1 n! E (πe Jθ 1 z 1 ) ∧ C · · · ∧ C (πe Jθn z n ) = π n n! |z 1 ∧ C · · · ∧ C z n |,
which is what we wanted.
Finally, to show the last item, let again X j ∈ V be random vectors such that K j = K(X j ). Then e θJ K j = K(e θJ X j ) and we have
MV J (e θJ K 1 , K 2 , . . . , K n ) = 1 n! E|e θJ X 1 ∧ C X 2 ∧ C · · · ∧ C X n | = 1 n! E|X 1 ∧ C · · · ∧ C X n | = MV J (K 1 , . . . , K n ).
This concludes the proof.
Remark 6.7. It is unknown to us if the mixed J-volume satisfies an Alexandrov-Fenchel inequality and we leave this for future work.
6.2. Random complex determinants. Here we state and prove a complex version of Theorem 5.4 which gives a way to describe the expectation of the modulus of the determinant of a random complex matrix with independent blocks (in Example 5.5 we used Theorem 5.4 to compute instead the expectation of the square of the modulus of the determinant). To do that, mimicking the definition of Section 4.2, we associate with each complex wedge product
∧ C : Λ d C V × Λ e C V Λ d+e C V the componentwise continuous bilinear map ∧ C : Z Λ d C V × Z Λ e C V Z Λ d+e C V .
induced from it by Theorem 4.1. We then have the following.
Theorem 6.8. Let M = (M 1 , . . . , M p ) ∈ C n×n be a random complex n × n matrix partitioned into blocks M j of size n × d j , with d 1 + · · · + d p = n. For j = 1, . . . , n, denote by v j,1 , . . . , v j,d j the columns of M j and assume that Z j = v j,1 ∧ · · · ∧ v j,d j ∈ Λ d j C C n is integrable. If the random vectors Z 1 ∈ Λ d 1 C (C n ), . . . , Z p ∈ Λ dp C (C n ) are independent, then: E| det(M )| = ℓ(K(Z 1 ) ∧ C · · · ∧ C K(Z p )).
In particular, if p = n and a 1 = · · · = a p = 1, then E| det(M )| = n! MV J (K(Z 1 ), . . . , K(Z n )).
Proof. Recall from Definition 6.2 that n! MV J (K(Z 1 ), . . . , K(Z n )) = ℓ(K(Z 1 ) ∧ C · · · ∧ C K(Z n )).
Moreover we have ℓ(K(Z 1 ) ∧ C · · · ∧ C K(Z n )) = E|Z 1 ∧ C · · · ∧ C Z n |. The last term is equal to E| det(M )|. This concludes the proof.
Remark 6.9. Notice that, in the case where p = n and d 1 = · · · = d p = 1 and if the random matrix is almost surely real, Theorem 6.8 agrees with Theorem 5.4, since MV and MV J coincide on real zonoids; see Proposition 6.6 (2).
As an application of Theorem 6.8, we compute the J-volume of balls.
Corollary 6.10. The J-volume of the unit ball B 2n ⊂ C n equals:
vol J n (B 2n ) = (4π) n/2 n! n j=1 Γ(j + 1 2 ) Γ(j)
.
Notice that applying Corollary 6.10 when n = 1 we get vol J 1 (B 2 ) = π = vol 2 (B 2 ), but already when n = 2 we get vol J 2 (B 4 ) = 3π 2 4 = π 2 2 = vol 4 (B 4 ). In general vol J n and vol 2n are different, starting by the fact that the first is homogeneous of degree n while the other is of degree 2n.
Proof of Corollary 6.10. Let Z = (z 1 , . . . , z n ) ∈ C n be a random vector filled with independent standard complex Gaussians z j = 1 √ 2 (ξ j,1 +iξ j,2 ), that is, ξ j,1 , ξ j,2 , . . . , ξ n,1 , ξ n,2 are independent standard real Gaussians. We claim that
(6.1) K(Z) = 1 2 √ π B 2n .
To see this, we compute the support function of K(Z). Let u ∈ C n . By definition, we have h K(Z) (u) = 1 2 E| Z, u |. Using the U (n)-invariance of Z, we can then assume that u = u e 1 where e 1 is the first vector of the standard basis of C n . We obtain
h K(Z) (u) = 1 2 E Re(Z T u) = 1 2 √ 2 u E|ξ 1,1 | = 1 2 √ 2 u 2 π = 1 2 √ π u = h 1 2 √ π B 2n (u).
Proposition 2.1 gives (6.1).
Let now M ∈ C n×n be a random complex matrix whose columns are i.i.d. copies of Z. Then, Theorem 6.8 gives E| det(M )| = n! MV J 1 2 √ π B 2n , . . . , 1 2 √ π B 2n = n! (2 √ π) n vol J n (B 2n ).
To conclude the proof, it suffices to verify that E| det(M )| = n j=1 Γ(j + 1 2 )/Γ(j). Note that | det M | = det(W ) 1 2 and W = M M * is a complex Wishart matrix. Following [BC13, p. 83-84], we see that det(W ) is distributed as 1 2 n χ 2 2n · χ 2 2n−2 · · · χ 2 2 , where each χ 2 2j denotes a chi-square distribution with 2j degrees of freedom and the χ 2 2j are independent. Therefore, | det(M )| has the distribution 1 2 n/2 χ 2n · χ 2n−2 · · · χ 2 . Recall from (2.10) that Eχ 2j = √ 2Γ(j+ 1 2 ) Γ(j)
. Using independence, the assertion follows.
6.3. The extension of the J-volume to polytopes. In this section we show that it is possible to extend the notion of J-volume to polytopes in C n ∼ = R 2n . To do so, we develop an alternative formula for the J-volume of zonotopes that makes sense for any polytope (Theorem 6.15). The functional we obtain is a weakly continuous, translation invariant and U (n)-invariant valuation, see Proposition 6.18. However, as we will see, it is not possible to continuously extend the J-volume from polytopes to all convex bodies (see Corollary 6.20). We start by introducing some terminology.
As a first step, we will give in Proposition 6.14 an alternative way of writing the J-volume of zonotopes. This involves the following quantity.
Definition 6.11. For every E ∈ G(n, 2n), we define σ J (E) := |e 1 ∧ · · · ∧ e n ∧ Je 1 ∧ · · · ∧ Je n | ∈ [0, 1]
where {e 1 , . . . , e n } is an orthonormal basis of E.
One can check that this definition does not depend on the choice of an orthonormal basis. Moreover, σ J is invariant under the action of U (n) on G(n, 2n). Note that σ J (E) = 1 if and only if E is Lagrangian, i.e., if E and JE are orthogonal. Moreover σ J (E) = 0 if and only if E contains a complex line.
Remark 6.12. Denoting by θ 1 (E) ≤ · · · ≤ θ ⌊ n 2 ⌋ (E) the Kähler angles of E, introduced in [Tas01], one can easily verify that
σ J (E) = ⌊ n 2 ⌋ j=1 (sin θ i (E)) 2 .
In general, σ J (E) can be computed using the following lemma.
Lemma 6.13. Let z 1 , . . . , z n ∈ C n be R-linearly independent and denote by E ∈ G(n, 2n) its real span. Then, writing z j = x j + iy j with x j , y j ∈ R n , we have
|z 1 ∧ C · · · ∧ C z n | = x 1 y 1 ∧ · · · ∧ x n y n · σ J (E) 1 2 .
Proof. Consider the matrices X = (x 1 , . . . , x n ) ∈ R n×n and Y = (y 1 , . . . , y n ) ∈ R n×n with the columns x j , y j . One can check that det X −Y Y X = | det(X + iY )| 2 , see [BLLP19, Lemma 5]. In particular, we can write
|z 1 ∧ C · · · ∧ C z n | 2 = | det(X + iY )| 2 = det X −Y Y X = x 1 y 1 ∧ · · · ∧ x n y n ∧ −y 1 x 1 ∧ · · · ∧ −y n x n =
x 1 y 1 ∧ · · · ∧ x n y n 2 · |e 1 ∧ · · · ∧ e n ∧ Je 1 ∧ · · · Je n | ,
where {e 1 , . . . , e n } is an orthonormal basis for E, and the last equality follows from:
x 1 y 1 ∧ · · · ∧ x n y n = x 1 y 1 ∧ · · · ∧ x n y n · e 1 ∧ · · · ∧ e n .
The conclusion follows from Definition 6.11.
We denote by G(k, m) the Grassmannian of k-dimensional linear spaces in R m . For a kdimensional face F of a polytope P in R m , we denote by E F ∈ G(k, m) the vector space parallel to the affine span of F . For 0 ≤ k ≤ m we also define
G k (P ) := {E ∈ G(k, m) | there exists a k-dim. face F of P such that E = E F }.
Let E ∈ G k (P ). In the case where P = p j=1 1 2 [−z j , z j ] is a zonotope, then all faces F of P such that E F = E are translates of the following "vectorial" face (see McMullen [McM71])
(6.2) F P (E) := z j ∈E 1 2 [−z j , z j ]
Where the sum runs over all j such that z j ∈ E. In this case, we have the following alternative description of G k (P ):
G k (P ) = {E ∈ G(k, d) | there exist linearly independent z j 1 , . . . , z j k ∈ E}.
The next result gives an explicit expression of the J-volume of a zonotope.
Proposition 6.14. Let P ⊂ C n be a centered zonotope. Then vol J n (P ) = E∈Gn(P ) vol n (F P (E)) · σ J (E) 1 2 .
Proof. We write P = p j=1 1 2 [−z j , z j ]. By definition, we have vol J n (P ) = 1 n! ℓ(P ∧ C n ). Using multilinearity and Theorem 4.1, we can write where w j 1 ,...,jn := z j 1 ∧ C · · · ∧ C z jn . Therefore, using the linearity of the length, vol J n (P ) = 1 n! 1≤j 1 ,...,jn≤p |w j 1 ,...,jn | = j 1 <...<jn |w j 1 ,...,jn |.
p j=1 1 2 [−z j , z j ] ∧ C · · · ∧ C p j=1 1 2 [−z j , z j ] =
We may assume the sum runs only over the j 1 < . . . < j n such that the real span E j 1 ,...,jn of z j 1 , . . . , z jn has dimension n. We write z j = x j + iy j with x j , y j ∈ R n and use Lemma 6.13 to obtain |w j 1 ,...,jn | = x j 1 y j 1 ∧ · · · ∧ x jn y jn · σ J (E j 1 ,...,jn ) 1 2 .
Combining this and exchanging the order of summation, we arrive at
(6.3) vol J n (P ) = E∈Gn(P ) σ J (E) 1 2 · E j 1 ,...,jn =E x j 1 y j 1 ∧ · · · ∧ x jn y jn ,
where for fixed E ∈ G n (P ), the second sum runs over all j 1 < . . . < j n such that E = E j 1 ,...,jn . Shephard's formula [She74,Equation (57)] applied to the zonoid F P (E) (see (6.2)) tells us that E j 1 ,...,jn =E
x j 1 y j 1 ∧ · · · ∧ x jn y jn = vol n (F P (E)).
Substituting this into (6.3) gives the statement.
We now turn to a key result of this section. For this, we need to introduce more notation. Let F be a k-dimensional face of a polytope P ⊂ R m and N P (F ) denote its normal cone. Note that N P (F ) is contained in the orthogonal complement E ⊥ F ≃ R m−k , where we recall that E F denotes the vector space parallel to F . We define the normal angle of P at F as
Θ P (F ) := vol m−k−1 (N P (F ) ∩ S m−1 ) vol m−k−1 (S m−k−1 ) .
The J-volume of zonotopes can now be expressed as follows.
Theorem 6.15. Let P ⊂ C n be a zonotope. Then
vol J n (P ) = F ∈Fn(P ) vol n (F ) · Θ P (F ) · σ J (E F ) 1 2 ,
where F n (P ) denotes the set of n-dimensional faces of P .
Proof. We will prove that the right hand side in this theorem is equal to the right hand side in Proposition 6.14. Let z 1 , . . . , z p ∈ C n be such that P = p j=1 1 2 [−z j , z j ] and let E ∈ G n (P ). As we discussed at the beginning of this subsection (see (6.2)), all the faces F of P such that E F = E are translates of the vectorial face F P (E) = z j ∈E 1 2 [−z j , z j ], we can thus write: It remains to prove that for every E ∈ G n (P ) we have E F =E Θ P (F ) = 1.
To this end, given E ∈ G n (P ), pick a nonzero u ∈ E ⊥ . Then, the set P u := {x ∈ K | h P (u) = u, x } is a face of K and therefore it equals a translate of a vectorial face (see (6.2)). More precisely, there is v u ∈ C n such that:
P u = v u + z j ∈u ⊥ 1 2 [−z j , z j ],
In addition, if F is a face of P such that E F = E, F is a translate of v j ∈E 1 2 [−v j , v j ]. Since E ⊂ u ⊥ , the face P u contains a translate of F . Moreover, dim(F ) = n and it follows that dim(P u ) ≥ n which implies dim(N K (K u )) ≤ n. In other words we proved that if E ∈ G n (K) and u ∈ E ⊥ , then dim(N P (P u )) ≤ n.
We now show that for almost all u ∈ E ⊥ we have dim(N P (P u )) = n. Indeed, for this it is enough to write E ⊥ ⊆ u∈E ⊥ N K (K u ), thus the set {u ∈ E ⊥ | dim(N P (P u )) < n} is contained in a finite union of cones of dimension at most n − 1.
Let now S n−1 E ⊥ be the unit sphere in E ⊥ . Denote by F the set of faces F of P such that E F ⊃ E and dim(N P (F )) < n. Then, by the above reasoning,
{u ∈ S n−1 E ⊥ | dim(P u ) < n} ⊆ F ∈F N P (F ) ∩ S n−1 E ⊥ .
Each set N P (F ) ∩ S n−1 E ⊥ with F ∈ F has dimension at most n − 2. Since the set F is finite, it implies, as above, that {u ∈ S n−1 E ⊥ | dim(P u ) < n} is contained in a finite union of sets of dimension at most n − 2, and in particular it has measure zero in S n−1 E ⊥ . It follows that {u ∈ S n−1 E ⊥ | dim(P u ) = n} ⊂ S n−1 E ⊥ has full measure. Letting u vary in S n−1 E ⊥ the set {P u } exhausts all n-dimensional faces F with E F = E and therefore:
E F =E Θ P (F ) = E F =E vol 2n−1 (N P (F ) ∩ S 2n−1 )
vol n−1 (S n−1 ) = vol n−1 (S n−1 ) vol n−1 (S n−1 ) = 1.
This concludes the proof.
We now note that the formula in Theorem 6.15 still makes sense for polytopes that are not zonotopes. We use this to define the J-volume on polytopes.
Definition 6.16. Let P be polytope in C n . We define its J-volume to be vol J n (P ) := F ∈Fn(P ) vol n (F ) · Θ P (F ) · σ J (E F ) 1 2
where F n (P ) denotes the set of n-dimensional faces of P .
We next study the J-volume in the framework of the theory of valuations on polytopes. Let us first recall the notion of a valuation. We denote by P(V ) the set of polytopes in a finite dimensional real vector space V . for every pair of convex polytopes K, L ∈ P(V ) such that K ∪ L is still a polytope (an analoguous definition applies for valuations on K(V )).
We call ν continuous if it is continuous with respect to the Hausdorff metric. We say that the valuation ν is k-homogeneous if ν(λK) = λ k ν(K) for every K ∈ P(V ) and λ > 0. If a group G acts on P(V ), the valuation ν is said to be G-invariant if ν(gK) = ν(K) for all K ∈ P(V ) and g ∈ G.
Let us also define the notion of weak continuity of a valuation on polytopes. This corresponds to continuity of the valuation on the set of polytopes that share the same given set of normals. More precisely, a valuation ν : P(V )
R is said to be weakly continuous if for every finite set U = {u 1 , . . . , u r } ⊂ V of unit vectors positively spanning V , i.e., i R + u i = V , the function (η 1 , . . . , η r ) ν ({v ∈ V | v, u i ≤ η i , i = 1, . . . , r})
is continuous on the set (η 1 , . . . , η r ) for which the argument of ν is nonempty. One can check that a continuous valuation ν : P(V ) R is weakly continuous. The general form of weakly continuous, translation invariant valuations on P(V ) was described in [McM83].
We now show some properties of the J-volume.
Proposition 6.18. The J-volume has the following properties.
(1) The map vol J n : P(C n ) R is a weakly continuous, translation invariant valuation.
(2) The valuation vol J n is n-homogeneous and U (n)-invariant.
(3) Let P ⊂ R n ⊂ C n be a polytope. Then vol J n (P ) = vol n (P ).
Proof. The first item follows from [McM83, Theorem 1]. The second item follows from the U (n)-invariance of σ J and Definition 6.16. For the third item, if P is of dimension less than n, both volumes are zero and there is nothing to prove. If dim(P ) = n, its only face of dimension n is P itself and E P = R n . Moreover σ J (R n ) = 1. Finally since N P (P ) = (R n ) ⊥ we have Θ P (P ) = 1. The claim follows with Definition 6.16.
The valuation vol J n is a special case of a so called angular valuation, see [Wan20]. Let V be a Euclidean space of (real) dimension m and let ϕ : G(k, m) R be a measurable function. The associated angular valuation ν ϕ on P(V ) is defined by ν ϕ (P ) := F ∈F k (P ) vol k (F ) · Θ P (F ) · ϕ(E F ),
where F k (P ) denotes the set of k-dimensional faces of a polytope P ∈ P(V ). It is known [McM83] that ν ϕ : P(V ) R is a weakly continuous valuation.
The possibility of continuously extending an angular valuation from polytopes to convex bodies was studied by Wannerer. The following is [Wan20, Theorem 1.2]. For its statement, we recall that G(k, m) can be seen as a subset of P(Λ k V ) via the Plücker embedding.
Proposition 6.19. The angular valuation ν ϕ : P(V ) R can be extended to a continuous valuation on K(V ), if and only if ϕ is the restriction to G(k, m) of a homogeneous quadratic polynomial on Λ k V .
If n = 1, then σ J is the function which is constant and equal to 1. The previous proposition implies that in this case we can extend vol J 1 to a continuous valuation on K(V ). If n ≥ 2, however, this is not possible as we will show next.
Corollary 6.20. If n ≥ 2, there is no continuous valuation on K(C n ) that is equal to vol J n on P(C n ). Proof. Using the notation of Proposition 6.19, we have vol J n = ν (σ J ) 1/2 . We identify C n ∼ = R 2n and let J be the standard complex structure on it. Consider the homogeneous quadratic polynomial p : Λ n R 2n Λ 2n R 2n , w w ∧ Jw. From Definition 6.11, σ J (w) = |p(w)| for w ∈ G(n, 2n) (in the Plücker embedding). Suppose there were a homogeneous quadratic polynomial q : Λ n R 2n R such that we have |p(w)| 1 2 = q(w) for all w ∈ G(n, 2n). Let us show that this leads to a contradiction, which will complete the proof by Proposition 6.19. First of all, we notice that q(w) must be a nonnegative polynomial and that we have |p(w)| = q(w) 2 on G(n, 2n).
Let e 1 , . . . , e n ∈ C n be the standard basis, so that |e 1 ∧ · · · ∧ e n ∧ Je 1 ∧ · · · ∧ Je n | = 1. We define the curve w(θ) := (cos(θ)e 1 + sin(θ)Je 2 ) ∧ e 2 ∧ · · · ∧ e n in G(n, 2n) for θ ∈ [0, π]. This curve interpolates between a Lagrangian plane (for θ = 0) and a plane, which contains a complex line (for θ = π). We have that p(w(θ)) = (cos(θ)e 1 + sin(θ)Je 2 ) ∧ e 2 ∧ · · · ∧ e n ∧ (cos(θ)Je 1 − sin(θ)e 2 ) ∧ e 2 ∧ · · · ∧ e n = cos(θ) 2 (e 1 ∧ · · · ∧ e n ∧ Je 1 ∧ · · · ∧ Je n ), and so |p(w(θ))| = cos(θ) 2 . If we have |p(w(θ))| = q(w(θ)) 2 , then q(w(θ)) = cos(θ), because q is nonnegative. Since q is a quadratic form and by the definition of w(θ), there are a, b, c ∈ R such that q(w(θ)) = a cos(θ) 2 + b cos(θ) sin(θ) + c sin(θ) 2 for all θ. Thus, we have an equality of functions a cos(θ) 2 + b cos(θ) sin(θ) + c sin(θ) 2 = cos(θ). It can be checked that such an equality is not possible, so our assumption was wrong and (σ J ) 1/2 cannot be the restriction of the square of a quadratic form to G(n, 2n). Proposition 6.19 implies the assertion.
From the proof, we see that if we remove the square root in Definition 6.16, we could extend the valuation continuously to convex bodies. This leads to the notion of Kazarnovskii's pseudovolume [Kaz04]. We use the expression found in [Ale03] in the proof of his Proposition 3.3.1. The normalization constant can be determined using the fact that it agrees with the classical volume on R n ⊂ C n just like the J-volume.
Definition 6.21. The Kazarnovskii's pseudovolume vol K n is given for any polytope P ⊂ C n by the formula vol K n (P ) = F ∈Fn(P ) vol n (F ) · Θ K (F ) · 1 (ω n ) 2 vol 2n (B(E F ) + JB(E F )),
where F n (P ) denotes the set of n-dimensional faces of P , and as before B(E F ) denotes the unit ball of E F , and ω n := vol n (B(R n )).
In our setting we prove the following, to be compared to Definition 6.16.
Proposition 6.22. For any polytope P ∈ P(C n ) the Kazarnovskii pseudovolume is given by
vol K n (P ) = F ∈Fn(P ) vol n (F ) · Θ K (F ) · σ J (E F ),
where F n (P ) denotes the set of n-dimensional faces of P ;
Proof. We need to prove that for any E ∈ G(n, 2n) we have where we used from Theorem 2.22 that ℓ is linear. Since dim(B(E)) = dim(JB(E)) = n, we see from Lemma 4.6 that (B(E)) ∧j = 0 whenever j > n and that (JB(E)) ∧(2n−j) = 0 whenever j < n. In other words, only the index j = n contributes to the sum and we get (6.5) vol 2n (B(E) + JB(E)) = 1 (n!) 2 ℓ (B(E)) ∧n ∧ (JB(E)) ∧n .
Next let X ∈ E be a random vector such that K(X) = B(E) (for instance, (2.9) shows that we can take X to be √ 2π times a standard Gaussian vector in E) and let X 1 , . . . , X n be i.i.d. copies of X. Let e 1 , . . . , e n be an orthonormal basis of E. Note that we have X 1 ∧ · · · ∧ X n = ± X 1 ∧ · · · ∧ X n e 1 ∧ · · · ∧ e n .
With this in mind, (6.5) gives vol 2n (B(E) + JB(E)) = 1 (n!) 2 E|X 1 ∧ · · · ∧ X n ∧ JX n+1 ∧ · · · ∧ X 2n | = 1 (n!) 2 (E X 1 ∧ · · · ∧ X n ) 2 σ J (E) Again, by Theorem 5.2, we have E X 1 ∧· · ·∧X n = n! vol n (B(E)) = n!ω n and this gives (6.4), which concludes the proof.
Br: Max-Planck-Institute for Mathematics in the Sciences Leipzig, [email protected]. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Projektnummer 445466444. Bü: Supported by the ERC under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 787840).
[x, y] := {(1 − t)x + ty | t ∈ [0, 1]}the segment joining x and y ∈ V .
Lemma 2 . 31 .
231Let M : V W be a linear map between Euclidean vector spaces, m := dim V and n := dim V . Consider the induced linear map M : M(P m−1 )
(3.4)T : M(P(V )) × M(P(W )) M(P(V ⊗ W )),which is a bilinear map.
Lemma 3. 8 .
8The map T : M(P(V )) × M(P(W )) M(P(V ⊗ W )) equals the tensor product of measures composed with the pushforward of the Segre map.
4. 1 .
1Zonoids and multilinear maps. We learned how to associate with a linear map M : V W of Euclidean spaces a continuous, order preserving linear map M : Z(V ) Z(W ) of the corresponding vector spaces of zonoids (Definition 2.20)
. . . , v m ), det(v 1 , . . . , v m )], This is an intervall in Λ m V ≃ R and hence ℓ(L) equals the volume of the parallelotope spanned by v 1 , . . . , v m . On the other hand, this equals MV( 1 2 [−v 1 , v 1 ], . . . , 1 2 [−v m , v m ]), divided by m!. This verifies the identity for segments and completes the proof.
Corollary 5. 3 .
3Let L be a zonoid in Λ e V , which is invariant under the action of the orthogonal group O(V ) (for instance, L = B ∧e ). Further, let d + e ≤ m. Then there exists a constant c d,m (L), only depending on d, m and L, such that ℓ(K ∧ L) = c d,m (L) ℓ(K)
1 ,...,jn , w j 1 ,...,jn ],
F
∈Fn(P ) vol n (F )Θ P (F )σ J (E F ) 1 2 = E∈Gn(P ) vol n (F P (E))σ J (E F ) 1 2 E F =E Θ P (F ).
Definition 6 .
617 (Valuation). A function ν : P(V ) R is called a valuation on P(V ) if ν(K ∪ L) + ν(K ∩ L) = ν(K) + ν(L)
2n (B(E) + JB(E)) = (ω n ) 2 σ J (E). Using Theorem 5.1 we write vol 2n (B(E) + JB(E)) = 1 (2n)! ℓ B(E) + JB(E) ℓ (B(E)) ∧j ∧ (JB(E)) ∧(2n−j)
The standard notation for centrally symmetric zonoids would be Z0(V ), but we omit the subscript in order to avoid unnecessary notation.
Recall that G(k, m) can be identified with the set of unit simple vectors in Λ k (R m ), see[Koz97].4 Recall that, if M = [ξ1, . . . , ξ k ] is a R m×k matrix whose columns are standard independent gaussian vectors in R m , the matrix A := M T M is called a Wishart matrix.
Acknowledgements. We would like to thank A. Bernig for pointing out the definition of Kazarnovskii's pseudovolume and its connection to our work. We also thank A. Khovanskii,
Zonoids and sparsification of quantum measurements. G Aubrun, C Lancien, Positivity20G. Aubrun and C. Lancien. Zonoids and sparsification of quantum measurements. Positivity, 20(1):1-23, 2016.
Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations. S Alesker, J. Differential Geom. 631S. Alesker. Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invari- ant valuations. J. Differential Geom., 63(1):63-95, 2003.
A strong law of large numbers for random compact sets. Z Artstein, R A Vitale, Ann. Probability. 35Z. Artstein and R. A. Vitale. A strong law of large numbers for random compact sets. Ann. Prob- ability, 3(5):879-882, 1975.
Probabilistic intersection theory. P Breiding, P Bürgisser, L Mathis, A Lerario, preparation. P. Breiding, P. Bürgisser, L. Mathis, and A. Lerario. Probabilistic intersection theory. In prepara- tion, 2021.
Condition, volume 349 of Grundlehren der Mathematischen Wissenschaften. P Bürgisser, F Cucker, Fundamental Principles of Mathematical SciencesP. Bürgisser and F. Cucker. Condition, volume 349 of Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences].
The geometry of numerical algorithms. Springer, HeidelbergSpringer, Heidelberg, 2013. The geometry of numerical algorithms.
K Böröczky, D Hug, arXiv:2106.13143Reverse Alexandrov-Fenchel inequalities for zonoids. K. Böröczky and D. Hug. Reverse Alexandrov-Fenchel inequalities for zonoids. arXiv:2106.13143, 2021.
Convergence of probability measures. P Billingsley, Wiley Series in Probability and Statistics. Wiley & Sons2nd editionP. Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics. Wiley & Sons, 2nd edition, 1999.
Probabilistic Schubert calculus. P Bürgisser, A Lerario, J. Reine Angew. Math. 760P. Bürgisser and A. Lerario. Probabilistic Schubert calculus. J. Reine Angew. Math., 760:1-58, 2020.
Random fields and the enumerative geometry of lines on real and complex hypersurfaces. S Basu, A Lerario, E Lundberg, C Peterson, Math. Ann. 3743-4S. Basu, A. Lerario, E. Lundberg, and C. Peterson. Random fields and the enumerative geometry of lines on real and complex hypersurfaces. Math. Ann., 374(3-4):1773-1810, 2019.
Approximation of zonoids by zonotopes. J Bourgain, J Lindenstrauss, V Milman, Acta Math. 1621-2J. Bourgain, J. Lindenstrauss, and V. Milman. Approximation of zonoids by zonotopes. Acta Math., 162(1-2):73-141, 1989.
A class of convex bodies. E D Bolker, Transactions of the American Mathematical Society. 145E. D. Bolker. A class of convex bodies. Transactions of the American Mathematical Society, 145:323- 345, 1969.
Geometric inequalities, volume 285 of Grundlehren der Mathematischen Wissenschaften. Y D Burago, V A Zalgaller, Fundamental Principles of Mathematical SciencesY. D. Burago and V. A. Zalgaller. Geometric inequalities, volume 285 of Grundlehren der Math- ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].
Translated from the Russian by. Springer-Verlag, Springer Series in Soviet Mathematics. A. B. SosinskiȋBerlinSpringer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiȋ, Springer Series in Soviet Mathemat- ics.
On the nuclear norm and the singular value decomposition of tensors. H Derksen, Found. Comput. Math. 163H. Derksen. On the nuclear norm and the singular value decomposition of tensors. Found. Comput. Math., 16(3):779-811, 2016.
Commutative Algebra: With a View Toward Algebraic Geometry. D Eisenbud, SpringerNew YorkGraduate Texts in MathematicsD. Eisenbud. Commutative Algebra: With a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Springer New York, 2013.
Differential forms with applications to the physical sciences. Harley Flanders, Dover Publications, IncNew Yorksecond editionHarley Flanders. Differential forms with applications to the physical sciences. Dover Books on Ad- vanced Mathematics. Dover Publications, Inc., New York, second edition, 1989.
Theory of Random Determinants. V L Girko, SpringerNetherlands1 editionV. L. Girko. Theory of Random Determinants. Springer Netherlands, 1 edition, 1990.
Normal multivariate analysis and the orthogonal group. A T James, Ann. Math. Statistics. 25A. T. James. Normal multivariate analysis and the orthogonal group. Ann. Math. Statistics, 25:40- 75, 1954.
Newton polyhedra" of generalized functions. B Ya, Kazarnovskiȋ, Izv. Ross. Akad. Nauk Ser. Mat. 682B. Ya. Kazarnovskiȋ. "Newton polyhedra" of generalized functions. Izv. Ross. Akad. Nauk Ser. Mat., 68(2):53-70, 2004.
Geometry of real Grassmannian manifolds. S E Kozlov, I, II, III. Zap. Nauchn. Sem. S.-PeterburgS. E. Kozlov. Geometry of real Grassmannian manifolds. I, II, III. Zap. Nauchn. Sem. S.-Peterburg.
. Otdel. Mat. Inst. Steklov. (POMI). 246Geom. i Topol.Otdel. Mat. Inst. Steklov. (POMI), 246(Geom. i Topol. 2):84-107, 108-129, 197-198, 1997.
Introduction to geometric probability. Lezioni Lincee. D A Klain, G.-C Rota, Lincei LecturesD. A. Klain and G.-C. Rota. Introduction to geometric probability. Lezioni Lincee. [Lincei Lectures].
. S Lang, Algebra, Graduate Texts in Mathematics. 211Springer-Verlagthird editionS. Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002.
On zonotopes. P Mcmullen, Trans. Amer. Math. Soc. 159P. McMullen. On zonotopes. Trans. Amer. Math. Soc., 159:91-109, 1971.
Weakly continuous valuations on convex polytopes. P Mcmullen, Arch. Math. (Basel). 416P. McMullen. Weakly continuous valuations on convex polytopes. Arch. Math. (Basel), 41(6):555- 564, 1983.
Hedgehogs and zonoids. Y Martinez-Maure, Adv. Math. 1581Y. Martinez-Maure. Hedgehogs and zonoids. Adv. Math., 158(1):1-17, 2001.
L Mathis, C Meroni, arXiv:2105.12406Fiber convex bodies. L. Mathis and C. Meroni. Fiber convex bodies. arXiv:2105.12406, 2021.
Aspects of multivariate statistical theory. R J Muirhead, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, IncR. J. Muirhead. Aspects of multivariate statistical theory. John Wiley & Sons, Inc., New York, 1982. Wiley Series in Probability and Mathematical Statistics.
W Rudin, Functional analysis. McGraw-Hill Series in Higher Mathematics. New York-Düsseldorf-JohannesburgMcGraw-Hill Book CoW. Rudin. Functional analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.
R Schneider, On the Aleksandrov-Fenchel inequality involving zonoids. Geometriae Dedicata. 27R. Schneider. On the Aleksandrov-Fenchel inequality involving zonoids. Geometriae Dedicata, 27:113-126, 1988.
R Schneider, of Encyclopedia of Mathematics and its Applications. CambridgeCambridge University Press151Convex bodies: the Brunn-Minkowski theory. expanded editionR. Schneider. Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Mathe- matics and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014.
Combinatorial properties of associated zonotopes. G C Shephard, Canadian J. Math. 26G. C. Shephard. Combinatorial properties of associated zonotopes. Canadian J. Math., 26:302-321, 1974.
R Sanyal, J Saunderson, arXiv:2001.04361Spectral polyhedra. R. Sanyal and J. Saunderson. Spectral polyhedra. arXiv:2001.04361, 2020.
Generalization of Kähler angle and integral geometry in complex projective spaces. Hiroyuki Tasaki, Steps in differential geometry. Debrecen; DebrecenHiroyuki Tasaki. Generalization of Kähler angle and integral geometry in complex projective spaces. In Steps in differential geometry (Debrecen, 2000), pages 349-361. Inst. Math. Inform., Debrecen, 2001.
Expected absolute random determinants and zonoids. R A Vitale, Ann. Appl. Probab. 12R. A. Vitale. Expected absolute random determinants and zonoids. Ann. Appl. Probab., 1(2):293- 300, 1991.
On the extendability by continuity of angular valuations on polytopes. T Wannerer, J. Funct. Anal. 279825T. Wannerer. On the extendability by continuity of angular valuations on polytopes. J. Funct. Anal., 279(8):108665, 25, 2020.
Centrally symmetric convex bodies and distributions. W Weil, Israel Journal of Mathematics. 24W. Weil. Centrally symmetric convex bodies and distributions. Israel Journal of Mathematics, 24(3-4), 1976.
| [] |
[
"Role of helicities for the dynamics of turbulent magnetic fields",
"Role of helicities for the dynamics of turbulent magnetic fields"
] | [
"Wolf-Christian Müller \nInstitut für Plasmaphysik\n‡Department of Mathematics\nUniversity of Leeds\nBoltzmannstr. 2D-85748, LS2 9JTGarching bei München, LeedsGermany, UK\n",
"Shiva Kumar Malapaka \nInstitut für Plasmaphysik\n‡Department of Mathematics\nUniversity of Leeds\nBoltzmannstr. 2D-85748, LS2 9JTGarching bei München, LeedsGermany, UK\n",
"† ‡ †max-Planck \nInstitut für Plasmaphysik\n‡Department of Mathematics\nUniversity of Leeds\nBoltzmannstr. 2D-85748, LS2 9JTGarching bei München, LeedsGermany, UK\n"
] | [
"Institut für Plasmaphysik\n‡Department of Mathematics\nUniversity of Leeds\nBoltzmannstr. 2D-85748, LS2 9JTGarching bei München, LeedsGermany, UK",
"Institut für Plasmaphysik\n‡Department of Mathematics\nUniversity of Leeds\nBoltzmannstr. 2D-85748, LS2 9JTGarching bei München, LeedsGermany, UK",
"Institut für Plasmaphysik\n‡Department of Mathematics\nUniversity of Leeds\nBoltzmannstr. 2D-85748, LS2 9JTGarching bei München, LeedsGermany, UK"
] | [
"Geophysical and Astrophysical Fluid Dynamics"
] | Investigations of the inverse cascade of magnetic helicity are conducted with pseudospectral, three-dimensional direct numerical simulations of forced and decaying incompressible magnetohydrodynamic turbulence. The high-resolution simulations which allow for the necessary scale-separation show that the observed self-similar scaling behavior of magnetic helicity and related quantities can only be understood by taking the full nonlinear interplay of velocity and magnetic fluctuations into account. With the help of the eddy-damped quasi-normal Markovian approximation a probably universal relation between kinetic and magnetic helicities is derived that closely resembles the extended definition of the prominent dynamo pseudoscalar α. This unexpected similarity suggests an additional nonlinear quenching mechanism of the current-helicity contribution to α. | 10.1080/03091929.2012.688292 | [
"https://arxiv.org/pdf/1307.4603v2.pdf"
] | 54,539,866 | 1307.4603 | 01fdde7619ae670bde2cbdd81ba448e6c8c34c48 |
Role of helicities for the dynamics of turbulent magnetic fields
20 Sep 2013 May 11, 2014
Wolf-Christian Müller
Institut für Plasmaphysik
‡Department of Mathematics
University of Leeds
Boltzmannstr. 2D-85748, LS2 9JTGarching bei München, LeedsGermany, UK
Shiva Kumar Malapaka
Institut für Plasmaphysik
‡Department of Mathematics
University of Leeds
Boltzmannstr. 2D-85748, LS2 9JTGarching bei München, LeedsGermany, UK
† ‡ †max-Planck
Institut für Plasmaphysik
‡Department of Mathematics
University of Leeds
Boltzmannstr. 2D-85748, LS2 9JTGarching bei München, LeedsGermany, UK
Role of helicities for the dynamics of turbulent magnetic fields
Geophysical and Astrophysical Fluid Dynamics
000020 Sep 2013 May 11, 201410.1080/03091929(This is an Author's Accepted Manuscript of an article published in Geophysical & Astrophysical Fluid Dynamics, First Published online : 01 Jun 2012.10:58 Geophysical and Astrophysical Fluid Dynamics mulmalfinaledpr˙2 Print version: Volume 107, Issue 1-2, 2013 Special Issue: From Mean-Field to Large-Scale Dynamos [copyright Taylor & Francis], available online at:α-effectMagnetic helicityKinetic helicityLarge-scale magnetic structures
Investigations of the inverse cascade of magnetic helicity are conducted with pseudospectral, three-dimensional direct numerical simulations of forced and decaying incompressible magnetohydrodynamic turbulence. The high-resolution simulations which allow for the necessary scale-separation show that the observed self-similar scaling behavior of magnetic helicity and related quantities can only be understood by taking the full nonlinear interplay of velocity and magnetic fluctuations into account. With the help of the eddy-damped quasi-normal Markovian approximation a probably universal relation between kinetic and magnetic helicities is derived that closely resembles the extended definition of the prominent dynamo pseudoscalar α. This unexpected similarity suggests an additional nonlinear quenching mechanism of the current-helicity contribution to α.
Introduction
Understanding large-scale magnetic structure formation in the Universe is one of the challenging problems in modern astrophysics. In this context, mean-field dynamo theory is a prominent approach (Moffatt 1978, Biskamp 2003, Brandenburg and Subramanian 2005. Based on a homogenization formalism, it describes the generation of large-scale magnetic fields by smaller-scale turbulent fluctuations of a magnetofluid. As a result, this classical two-scale closure (Krause and Rädler 1980) yields, next to a turbulent diffusivity, a scalar, α ∼ τ H K , that expresses the nonlinear interaction of large-scale field and smaller-scale turbulence. Here, τ stands for a correlation time of the turbulent fluctuations and H K = (1/2V ) V v · ω dV is the kinetic helicity of the associated velocity field v with V being the volume under consideration and ω = ∇×v defining the vorticity. Statistical closure theory (Pouquet et al. 1976) more specifically the eddy-damped quasi-normal Markovian (EDQNM) approximation, suggests a more complex expression, α ∼ τ (H K − H J ), that introduces the current helicity H J = (1/2V ) V b · j dV with j denoting the electric current density, see also , Subramanian and Brandenburg 2004, Brandenburg and Subramanian 2005. Its name is actually misleading as H J expresses the helicity of the magnetic field and is in this respect a close relative of the kinetic helicity and, furthermore, also proportional to the total resistive dissipation rate of magnetic helicity (see below).
While H K is ideally conserved and is spectrally cascading towards smaller scales in the inertial range of three-dimensional Navier-Stokes turbulence, the current helicity has apparently no comparable significance for turbulent dynamics apart from its meaning for the turbulent dynamo. However, with electric current j = ∇ × b = −∆a, magnetic field b = ∇ × a and magnetic vector potential a (both dimensionless), a link to an ideal invariant of three-dimensional incompressible magnetohydrodynamics (MHD) emerges through the magnetic helicity, H M = (1/2V ) V a · b dV . This quantity characterizing the topology of the magnetic field (Moffatt 1969) is prone to an inverse cascade. The cascade is a robust nonlinear mechanism that creates large scale order out of the chaotic randomness of small-scale magnetic turbulence presupposing a sufficient separation of large and turbulent small scales in the system in combination with a small-scale supply of magnetic helicity.
The present work is motivated by the potential importance of magnetic helicity for the dynamics of largescale dynamo configurations. This is not to be confused with the related issue of the effect of boundary conditions on the magnetic helicity evolution and the consequences for the dynamo process, a topic that has been subject of a number of investigations (see, e.g., Brandenburg 2009, and references therein). In this work, an idealized system, homogeneous incompressible MHD turbulence with triply periodic boundary conditions, is investigated by three-dimensional direct numerical simulations in combination with statistical closure theory.
Model equations and numerical Setup
The dimensionless incompressible MHD equations giving a concise single-fluid description of a plasma are
∂ t ω = ∇ × (v × ω − b × j) + µ n (−1) n/2−1 ∇ n ω + F v + λ∆ −1 ω ,(1a)∂ t b = ∇ × (v × b) + η n (−1) n/2−1 ∇ n b + F b + λ∆ −1 b ,(1b)∇ · v = ∇ · b = 0.(1c)
Relativistic effects are neglected and the mass density is assumed to be unity throughout the system. Other effects such as convection, radiation and rotation are also neglected. Direct numerical simulations are performed by solving the set of model equations by a standard pseudospectral method (Canuto et al. 1988) in combination with leap-frog integration on a cubic box of linear size 2π that is discretized with 1024 collocation points in each spatial dimension. Spherical mode truncation is used for alleviating aliasing errors. By solving the equations in Fourier space, the solenoidality of v and b is maintained algebraically. To observe clear signatures of an inverse cascade of magnetic helicity the system has to contain a source of this quantity at small scales. This is achieved in two different ways resulting in two main configurations: a driven system and a decaying one. In the driven case, the forcing terms F v and F b are delta-correlated random processes acting in a band of wavenumbers 203 ≤ k 0 ≤ 209. They create a small-scale background of fluctuations with adjustable amount of magnetic and kinetic helicity. The results reported in this paper do not change if kinetic helicity injection is finite. The theoretical results presented in the following do not depend on the setup of the forcing as they presuppose an existing self-similar distribution of energies and helicities. For obtaining such spectra in numerical experiments the magnetic source term F b is necessary while a finite momentum source F v speeds up the spectral development significantly. In the decaying case the forcing terms are set to zero and the initial condition represents an ensemble of smooth and random fluctuations of maximum magnetic helicity with respect to the energy content (see below) and a characteristic wavenumber k 0 = 70.
To reduce finite-size effects, the simulations are run for 6.7 (forced) and 9.2 (decaying) large-eddy turnover times of the system, respectively. The time unit is defined using the system size and its total energy. Additionally, a large scale energy sink λ∆ −1 with λ = 0.5 is present for both fields. In the decaying case λ = 0. The hyperdiffusivities µ n and η n are dimensionless dissipation coefficients of order n (always even in these simulations), with n = 8 in both runs. They act like higher-order realizations of viscosity and magnetic diffusivity, respectively. The magnetic hyperdiffusive Prandtl number P r mn = µ n /η n is set to unity.
The initial conditions to these simulations are smooth fluctuations with random phases having a Gaussian energy distribution peaked around k 0 in the decaying and the forced cases. Magnetic and kinetic helicity a) b) of the initial state can be controlled in the same way as for the forcing terms (cf. Biskamp and Müller 2000). The initial/force-supplied ratio of kinetic to magnetic energy is unity with an amplitude of 0.05 in the forced case and an amplitude of unity in the decaying case. Hyperviscosity of order n = 8 is chosen in the simulations to obtain sufficient scale-separation. It is difficult to define an unambiguous Reynolds number owing to the use of hyperviscosity (Malapaka 2009, and the references therein). With the above mentioned simulation set up, the equations are solved both for decaying and forced cases separately and the results obtained are discussed below.
Figure 1. (a) Transmission H M T r =b * · ( v × b) (dotted line) and dissipation H M Di = ηnk 6b * ·j (solid
Simulation results
Using the simulation setup described in the previous section, inverse cascading of magnetic helicity with a clear scale separation between large and small scales is established in both forced and decaying cases for wavenumbers k < k 0 . This is indicated by the spectral flux, figure 1(b) and taken at t = 6.7 and 9.2, respectively as dissipation of magnetic helicity is negligible (see figure 1(a)). The tilde indicates Fourier transformation and * stands for complex conjugate. The inverse flux in the driven case is constant over a significant spectral interval, indicating equilibrium of source and sink, while the temporal decay of the magnetic helicity reservoir in the decaying case is reflected by the associated non constant inverse flux. In both cases the characteristic wavenumber of the H M -source can be identified as the separation between inverse and direct flux regions. The spectral flux of magnetic helicity has been extensively studied in earlier numerical simulations (see, e.g., Brandenburg 2001, Alexakis et al. 2006. These works, however, are lacking the necessary scale separation to observe self-similar scaling laws. The spectrum of magnetic helicity exhibits scaling behavior ∼ k q with q ≈ −3.3 and q ≈ −3.6 (forced and decaying case, respectively) which cannot be explained by the straightforward constant-flux reasoningà la Kolmogorov adopted in Pouquet et al. (1976) to interpret their EDQNM results.
Π H M k = k 0 dk ′ dΩ b * · ( v × b) |k ′ |=k ′ , in both cases depicted in
In fact, the involved dimensional argument (Alfvénic units), [H M k ] = L 4 /T 2 (spectrum), [ε M ] = L 3 /T 3 (spectral flux), in combination with the assumption of spectral self-similarity, H M k ∼ ε a M k b , yields a = 2/3, b = −2, but does not explicitly include the nonlinear interaction of velocity and magnetic fields (see also Biskamp 2003). Here 'L' represents length and 'T ' the time. As a first step in the necessary refinement of the theoretical modeling additional consideration of the kinetic helicity H K k seems appropriate. As a consequence of the inverse spectral transfer of magnetic helicity, all magnetic quantities should inherit the observed spectral inverse transfer property. This is indeed the case for the magnetic energy, the electric current density, and the current helicity. These quantities also show self-similar scaling that however, differs to some degree between the two investigated configurations. It is particularly interesting, that the residual helicity H R = H V − k 2 H M , also shows self-similar scaling with q ≈ −1.4 and q ≈ −1.8 in the forced and decaying cases respectively (see Malapaka 2009, for further details). The interaction of the magnetic field with the velocity in a progressing inverse cascade of magnetic helicity appears to be of importance for a better understanding of the observed scaling laws. At high Reynolds numbers, the process of large-scale magnetic structure formation by the inverse cascade is accompanied by a continuous stirring of the velocity field caused by the expanding magnetic field structure. The magnetic stirring of the MHDfluid leads to a transfer of magnetic to kinetic energy and generates ever larger velocity fluctuations. These also show self-similar scaling, as, for example, reflected by the kinetic helicity spectrum with q ≈ −0.4 (forced case) and q ≈ 0.4 (decaying case).
With regard to the finding (e.g. Alexakis et al. 2006) of the pronounced spectral non-locality of the nonlinear interactions underlying Π H M k a few words about the physical picture of the inverse cascade are in order. The cascading process is realized as a merging of positively-aligned and thus mutually attracting current carrying structures (cf. Biskamp and Bremer 1993). It is not necessary that the structures grow in size as they indeed do in the decaying case, as long as the corresponding current densities increase. This is observed in the simulation with small-scale forcing. As there is no obvious fluid-dynamical constraint on the merging of two current filaments with regard to their size, this picture is consistent with a spectrally non-local inverse cascade of magnetic helicity.
Spectral relationship between kinetic and magnetic helicities
A link between kinetic and magnetic helicities can be constructed with the help of dimensional analysis of the magnetic helicity evolution equation in the EDQNM approximation, a statistical closure model discussed (e.g., in Pouquet et al. 1976). Such an approach was successful earlier, in describing the turbulent residual energy spectrum, (Müller and Grappin 2005) with E k = E M k + E K k , which also turns out to be valid in the present simulations, where E M k and E V k are magnetic and kinetic energies respectively.
E R k = |E M k − E V k | yielding E R k ∼ kE 2 k
Assuming that the most important nonlinearities involve the turbulent velocity and stationarity of the spectral scaling range of H M k , a dynamical equilibrium of turbulent advection and the H M -increasing effect of helical fluctuations is proposed. This can be formulated straightforwardly using the corresponding dimensionally approximated nonlinear terms from the EDQNM model (for a more detailed derivation see Müller et al. 2012), yielding
H K k ∼ E K k /E M k k 2 H M k .(2)
This statement about the spectral dynamics of kinetic and magnetic helicities (or, equivalently, kinetic and current helicities since H J k ∼ k 2 H M k ) is also valid for E K k /E M k = 1. The agreement of relation (2) with the numerical experiments is however significantly improved by a modification (relation (3) below) whose justification is beyond the scope of the presented equilibrium ansatz which basically assumes spectral locality of the inverse cascade
H K k ∼ E K k /E M k 2 H J k .(3)
Relation (3) is a significant improvement over the earlier relations of similar kind (Pouquet et al. 1976, Müller and Malapaka 2010. This is shown in figures 2(a) and 3(a), where Θ = (E K k /E M k ) γ H J k /H K k is shown with γ = 0, 1 and 2 (corresponding to Θ, Θ 1 and Θ 2 ) for the forced and decaying cases respectively. It is remarkable that relation (3) is only fulfilled in wavenumber intervals where the flux of magnetic helicity is spectrally constant. This relation brings back the ratio of energies (kinetic to magnetic) into the picture, which, under the assumption of equipartition of energies was ignored in previous work (Pouquet et al. 1976), while
(E K k /E M k ) γ H J k /H K k . γ = 0 (dash-dot curve) γ = 1 (dashed curve)
, and γ = 2 (solid curve). (b) Magnetic helicity (dash-dot curve), kinetic helicity (dashed curve) and residual helicity (solid curve).
linking the magnetic/current and kinetic helicities. Another interpretation for this expression is the partial Alfvénization of the turbulent flow (Pouquet et al. 2010). Further, it also highlights the influence of kinetic helicity in the inverse cascade of magnetic helicity.
The relation (3) belongs to a class of probably highly universal expressions which are based statistically on the quasi-normal approximation of nonlinear fluxes. It is interesting to note that relation (3) also allows to determine the spectral scaling exponent of magnetic helicity from astronomical current helicity measurements using vector magnetograms (see, e.g., Brandenburg and Subramanian 2005, and references therein) if kinetic and magnetic energy spectra are also measurable or can be estimated with sufficient accuracy.
The modification of the current helicity contribution present in relation (3) suggests a corresponding modification to the residual helicity, H R = H K − H J , and accordingly to the mean-field dynamo α. This, however, has to be taken with care as the present simulations are energetically dominated by the magnetic field although the modifying factor (E K /E M ) 2 should compensate for this. Figures 2(b) and 3(b) allow us to roughly estimate the respective scale-dependent influence of kinetic and magnetic helicity on the modified residual helicity. The spectrum of residual helicity closely follows the spectral kinetic helicity with growing systematic deviations due to the influence of magnetic helicity at large wavenumbers, in both cases. Thus, the modified residual helicity complies with the earlier definitions of α (Krause andRädler 1980, Pouquet et al. 1976) at large scales.
Conclusions
In high-resolution direct numerical simulations of forced and decaying magnetically helical homogeneous MHD turbulence, the nonlinear dynamics of active inverse cascade of magnetic helicity is studied. The
(E K k /E M k ) γ H J k /H K k . γ = 0 (dash-dot curve) γ = 1 (dashed curve)
, and γ = 2 (solid curve). (b) Magnetic helicity (dash-dot curve), kinetic helicity (dashed curve) and residual helicity (solid curve). simulation results, in particular the observed self-similar spectral scaling of magnetic helicity which contradicts an earlier theoretical explanation (Pouquet et al. 1976), motivate the consideration of velocity field characteristics for the nonlinear evolution of this purely magnetic quantity. This is done with the help of statistical closure theory yielding a possibly universal relation between kinetic and current helicities. The relation is corroborated by the numerical results. Its form, H K k − (E K k /E M k ) 2 H J k ∼ constant, closely resembles the extended definition of the pseudo-scalar α ∼ H K k − H J k known from mean-field dynamo theory. The inverse cascade of magnetic helicity is neither a dynamo in itself (as dimensionally H M k ∼ k −1 E M k ) nor even a turbulent cascade in the strict sense, but is a spectral transport process (Müller et al. 2012). It can as a robust and efficient spectral transporter, nevertheless, play a role in the actual realization of turbulent large-scale dynamos like the α-dynamo. In this respect it is interesting that the newly obtained relation includes the squared ratio of kinetic and magnetic energies. This leads to a purely nonlinear quenching of the current helicity contribution to α that has no direct connection to the dynamo-quenching mechanisms considered so far (of order (E M ) −1 ) in the literature which are seemingly consequences of a combination of boundary conditions and the approximate conservation of magnetic helicity. In this context, it is encouraging that Rheinhardt and Brandenburg (2010) for a homogeneous mean flow with Roberts forcing using a test field method observe α-quenching with an (E M ) −2 signature. The comparison with this work assumes equivalence of their imposed mean field with the root-mean-square large-scale magnetic fluctuations in the present simulations.
The present relation (3) needs further investigation as it is an additional possible mechanism for dynamo quenching. This new link between kinetic and magnetic helicity in the inverse cascade of magnetic helicity has to be verified in more complex numerical setups such as mean field dynamos, as well as anisotropic 3D-MHD and isotropic 3D-MHD turbulence with different initial conditions and forcing mechanisms.
line) of magnetic helicity spaceangle-integrated in Fourier space in the forced case (similar for the decaying case, not shown). (b) Spectral flux of magnetic helicity in the forced (top) and decaying cases (bottom), dashed curves: direct flux, solid curves: inverse flux.
Figure 2 .
2Plots of relation (3), and kinetic, magnetic and residual helicities for forced turbulence at t=6.7. (a) Relation (3) Θ =
Figure 3 .
3Plots of relation (3), and kinetic, magnetic and residual helicities for decaying turbulence at t=9.2. (a) Relation (3) Θ =
Acknowledgments SKM thanks B. Despres of JLLL, UPMC, Paris, VI, and CNRS for the financial support they provided to attend the Rädler fest, A. Brandenburg and his NORDITA team for hosting him at this fest, as well as A. Busse currently in Southampton, UK and D. Hughes at University of Leeds, UK for their help and useful discussions. The authors want to thank U. Frisch for useful remarks.
On the inverse cascade of magnetic helicity. A Alexakis, P D Mininni, A Pouquet, Astrophys. J. 640Alexakis, A., Mininni, P.D. and Pouquet, A., On the inverse cascade of magnetic helicity. Astrophys. J. 2006, 640, 335-343.
Dynamics and statistics of inverse cascade processes in 2D magnetohydrodynamic turbulence. D Biskamp, U Bremer, Phys. Rev. Lett. 72NumberBiskamp, D. and Bremer, U., Dynamics and statistics of inverse cascade processes in 2D magnetohydro- dynamic turbulence. Phys. Rev. Lett. 1993, 72, Number 24, 3819-3822.
Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence. D Biskamp, W.-C Müller, Phys. Plasmas. 7Biskamp, D. and Müller, W.-C., Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence. Phys. Plasmas 2000, 7, Number 12, 4889-4900.
Magnetohydrodynamic turbulence. D Biskamp, Cambridge University PressBiskamp, D., Magnetohydrodynamic turbulence. Cambridge University Press 2003.
New dynamical mean-field dynamo theory and closure approach. E G Blackman, G B Field, Phys. Rev. Lett. 265007Blackman, E.G. and Field, G.B., New dynamical mean-field dynamo theory and closure approach. Phys. Rev. Lett. 2002, 89, 265007.
The inverse cascade and nonlinear Alpha-effect in simulations of isotropic helical hydromagnetic turbulence. A Brandenburg, Astrophys. J. 550Brandenburg, A., The inverse cascade and nonlinear Alpha-effect in simulations of isotropic helical hydro- magnetic turbulence. Astrophys. J. 2001, 550, 824-840.
Advances in theory and simulations of large-scale dynamos. A Brandenburg, Space Sci. Rev. 144Brandenburg, A., Advances in theory and simulations of large-scale dynamos. Space Sci. Rev. 2009, 144, 87-104.
Astrophysical magnetic fields and nonlinear dynamo theory. A Brandenburg, K Subramanian, Phys. Rep. 417Brandenburg, A. and Subramanian, K., Astrophysical magnetic fields and nonlinear dynamo theory. Phys. Rep. 2005, 417, 1-209.
C Canuto, M Y Hussaini, A Quarteroni, T A Zang, Spectral Methods in Fluid Mechanics. Berlin: New YorkCanuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A., Spectral Methods in Fluid Mechanics. Berlin: New York, 1988.
Dynamic quenching of the α 2 dynamo. G B Field, E G Blackman, Astrophys. J. 572Field, G.B. and Blackman, E.G., Dynamic quenching of the α 2 dynamo. Astrophys. J. 2002, 572, 685-692.
F Krause, K.-H Rädler, Mean field magnetohydrodynamics and dynamo theory. BerlinAkademieKrause, F. and Rädler K.-H., Mean field magnetohydrodynamics and dynamo theory. Berlin: Akademie, 1980.
A study of magnetic helicity in forced and decaying 3D-MHD turbulence. S K Malapaka, University of BayreuthPhD thesisMalapaka, S.K., A study of magnetic helicity in forced and decaying 3D-MHD turbulence. PhD thesis 2009, University of Bayreuth. http://edoc.mpg.de/display.epl?mode=doc&id=464051&col=33&grp=1311
Magnetic field generation in electrically conducting fluids. H K Moffatt, Cambridge University PressMoffatt, H.K., Magnetic field generation in electrically conducting fluids. Cambridge University Press, 1978.
The degree of knottedness of tangled vortex lines. H K Moffatt, J. Fluid Mech. 35Moffatt, H.K., The degree of knottedness of tangled vortex lines. J. Fluid Mech. 1969, 35, 117-129.
Energy dynamics in magnetohydrodynamic turbulence. W.-C Müller, R Grappin, Phys. Rev. Lett. 114502Müller, W.-C. and Grappin, R., Energy dynamics in magnetohydrodynamic turbulence. Phys. Rev. Lett. 2005, 95, 114502.
Understanding nonlinear cascades in magnetohydrodynamic turbulence by statistical closure theory. W.-C Müller, S K Malapaka, Numerical modeling of space plasma flows ASTRONUM-2009. N.V. Pogorelov, E. Audit and G.P. Zank429Müller, W.-C. and Malapaka, S.K., Understanding nonlinear cascades in magnetohydrodynamic turbulence by statistical closure theory. In Numerical modeling of space plasma flows ASTRONUM-2009, ASP Conference Series Vol. 429, 2010, edited by N.V. Pogorelov, E. Audit and G.P. Zank, 2
Inverse cascade of magnetic helicity in magnetohydrodynamic turbulence. W.-C Müller, S K Malapaka, A Busse, Phys. Rev. E. 8515302Müller, W.-C., Malapaka, S.K. and Busse, A., Inverse cascade of magnetic helicity in magnetohydrody- namic turbulence. Phys. Rev. E 2012, 85, 015302.
Strong MHD helical turbulence and the nonlinear dynamo effect. A Pouquet, U Frisch, J Léorat, J. Fluid Mech. 772Pouquet, A., Frisch, U. and Léorat, J., Strong MHD helical turbulence and the nonlinear dynamo effect. J. Fluid Mech. 1976, 77, part 2, 321-354.
Lack of universality in MHD turbulence, and the possible emergence of a new paradigm. A Pouquet, M.-E Brachet, E Lee, P D Mininni, D Rosenburg, V Uritsky, Astrophysical Dynamics: From Stars to Galaxies, Proceedings IAU Symposium 271. N.H. Brummell, A.S. Brun, M.S. Miesch and Y. PontyPouquet, A., Brachet, M.-E., Lee, E., Mininni, P.D., Rosenburg, D., and Uritsky, V., Lack of universal- ity in MHD turbulence, and the possible emergence of a new paradigm?. In Astrophysical Dynamics: From Stars to Galaxies, Proceedings IAU Symposium 271, 2010, edited by N.H. Brummell, A.S. Brun, M.S. Miesch and Y. Ponty, pp. 304-316.
Test-field method for mean-field coefficients with MHD background. M Rheinhardt, A Brandenburg, Astron. Astrophys. 28Rheinhardt, M. and Brandenburg, A., Test-field method for mean-field coefficients with MHD background. Astron. Astrophys. 2010, 520, A28.
Nonlinear current helicity fluxes in turbulent dynamos and alpha quenching. K Subramanian, A Brandenburg, Phys. Rev. Lett. 20Subramanian, K. and Brandenburg, A., Nonlinear current helicity fluxes in turbulent dynamos and alpha quenching. Phys. Rev. Lett. 2004, 93, 20.
| [] |
[
"Nonperturbative QCD thermodynamics in the external magnetic field",
"Nonperturbative QCD thermodynamics in the external magnetic field"
] | [
"M A Andreichikov \nState Research Center Institute of Theoretical and Experimental Physics\n117218MoscowRussia\n",
"Yu A Simonov \nState Research Center Institute of Theoretical and Experimental Physics\n117218MoscowRussia\n"
] | [
"State Research Center Institute of Theoretical and Experimental Physics\n117218MoscowRussia",
"State Research Center Institute of Theoretical and Experimental Physics\n117218MoscowRussia"
] | [] | The thermodynamics of quarks and gluons strongly depends on the vacuum colormagnetic field, which grows with the temperature T , so that spatial string tension σ s = const g 4 (T )T 2 . We investigate below what happens when one imposes in addition constant magnetic field and discover remarkable structure of the resulting thermodynamic potential. | 10.1140/epjc/s10052-018-5916-8 | [
"https://arxiv.org/pdf/1712.02925v3.pdf"
] | 119,189,325 | 1712.02925 | 9d720f145f9734860741d99f12710746d11c9ac2 |
Nonperturbative QCD thermodynamics in the external magnetic field
20 Dec 2017 October 12, 2018
M A Andreichikov
State Research Center Institute of Theoretical and Experimental Physics
117218MoscowRussia
Yu A Simonov
State Research Center Institute of Theoretical and Experimental Physics
117218MoscowRussia
Nonperturbative QCD thermodynamics in the external magnetic field
20 Dec 2017 October 12, 2018
The thermodynamics of quarks and gluons strongly depends on the vacuum colormagnetic field, which grows with the temperature T , so that spatial string tension σ s = const g 4 (T )T 2 . We investigate below what happens when one imposes in addition constant magnetic field and discover remarkable structure of the resulting thermodynamic potential.
Introduction
The problem of quark gluon thermodynamics in magnetic field is of the high interest in the modern physics, since heavy-ion experiments produce important data on the properties of resulting hadron yields and hadron interactions, which might be influenced by the strong magnetic fields (MF) created during the collision process [1,2,3]. For a recent review on the effects of MF see [4]. On the theoretical side the problem of MF in the quark gluon plasma (qgp) was studied in different aspects, e.g. in the NJL-type models [5] and in the holographic approach [6,7]. Within the nonperturbative QCD the theory of qgp in MF was developed in [8,9,10], where the general form of the thermodynamic potentials was found in MF with zero or nonzero baryon density, summing over all Landau levels including LLL.
In this approach the only nonperturbative interaction, which was taken into account, reduced to the inclusion of Polyakov lines in the resulting expression for the pressure.
The resulting expressions for magnetic susceptibilitiesχ q (T ), obtained in [8], were used in [9,10] to compare with the lattice data from [11,12], and a reasonable agreement was found forχ q (T )with different q = u, d, s, in [9,10] as well as for the sum [10], however somewhat renormalised values of effective quark masses were used.
Recently in [13,14] a new step in the development of the np QCD thermodynamics was made, where the colormagnetic confinement (CC) was included in the dynamics of the qgp. This interaction with the spatial string tension σ s grows with temperature,σ s ∼ g 4 T 2 and is important in the whole region T c < T < 10 GeV. A concise form of the final expression was found in [13,14] in the case of an oscillatory type CC and an approximate one in the realistic case of the linear CC. The resulting behavior both in the SU(3) case, found in [15,16], and in the qgp case [13,14] agrees well with the corresponding lattice data.
It is the purpose of the present paper to extend our previous analysis of the qgp thermodynamics in MF, done in [8,9,10], including the dynamics of CC with the explicit form of the magnetic screening mass m D , generated by CC.
As will bee seen, we propose a simple generalization of the results [8,9,10], where the CC produces the mass M D , entering the final expressions in the combination m 2 q + m 2 D ≡M instead of m q . We check the limiting cases and compare the result with lattice data.
The paper is organized as follows. In the next section the general analysis of the MF effects in thermodynamics is explained, in section 3 the magnetic susceptibility is defined, in section 4 the results are compared to the lattice data, and in the section 5 the summary and prospects are given.
2 General structure of the pressure with and without magnetic field
We start with the quark pressure of a given flavor as expressed via the 3d quark Green's function S 3 (s) in the stochastic field of the colormagnetic confinement (CMC). From the path integral representation [14,15] one obtains
P f q = 4N c √ 4π ∞ 0 ds s 3/2 e −m 2 q s S 3 (s) n=1,2,... (−) (n+1) e − n 2 4T 2 s cosh µn T L n (1) where L = exp − V 1 (∞,T ) 2T
is the quark Polyakov line, and S 3 (s) can be expanded in a series over eigenstates in the CMC on the 2d minimal area surface in 3d space.
S 3 (s) = 1 √ πs ν=0,1,... ψ 2 ν (0)e −m 2 ν s , m 2 0 = m 2 D = 4σ s (T ).(2)
As it was argued in [14,15], in the case of linear CMC one obtains for S 3 (s) an approximate form
S lin 3 (s) ∼ = 1 (4πs) 3/2 e − m 2 D s 4(3)
and pressure can be written as
1 T 4 P f q = N c 4π 2 ∞ n=1 (−) n+1 n 4 L n cosh µn T Φ n (T ),(4)where Φ n (T ) is Φ n (T ) = 8n 2M 2 T 2 K 2 M n T ,M = m 2 q + m 2 D 4 .(5)
On another hand, using the relation from [8] s = nβ 2ω , β = 1/T and the representation
∞ 0 ωdωe − m 2 2ω + ω 2 nβ = 2m 2 K 2 (mnβ),(6)
one obtains as in [8] the pressure of the given quark flavor
P q = f P (f ) q . P (f ) q = N c √ π d 3 p (2π) 3 ∞ n=1 (−) n+1 2 nβ ∞ 0 dω √ ω e − m 2 q +p 2 + m 2 D 4 2ω + ω 2 nβ .(7)
Let us now introduce the magnetic field B along the z axis, so that our system of quarks undergoes the influence of both CMC field and (electro)magnetic field (MF) at the same time.
We consider the influence of the MF only, and write the corresponding equations from [8]
P f q (B) = N c |e q B|T π 2 n ⊥ ,σ ∞ n=1 (−) n+1 n L n cosh µn T ε σ n ⊥ K 1 nε σ n ⊥ T (8) where ε σ n ⊥ = |e q B|(2n ⊥ + 1 −σ) + m 2 q ,σ = e q |e q | σ z .(9)
It is interesting, that in the case µ = 0 one can replace in (7) the exponent (as was suggested in [8])
m 2 q + p 2 2ω + ω 2 → m 2 q + p 2 z + (2n ⊥ + 1 −σ)|e q B| 2ω + ω 2(10)
and using the phase space in MF
V 3 d 3 p (2π) 3 → dp z 2π |e q B| 2π V 3 ,(11)
and the relation
∞ 0 dωe − λ 2 2ω + ω 2 τ = 2λK 1 (λτ ),(12)
one obtains the same Eqs. (8), (9), but with the replacement
m 2 q ⇒ m 2 q + m 2 D 4 = m 2 q + cσ s ,(13)
where c ≃ 1 for T → ∞. Therefore in what follows we shall be using the Eq.(8) with the replacement in (9), m 2 q →M 2 = m 2 q + cσ s .
As was shown in [8], the form (8) can be summed up over n ⊥ , σ to obtain the following result (we consider below for simplicity only the case µ = 0)
P (f ) q (B) = N c |e q B|T π 2 ∞ n=1 (−) n+1 n L n M K 1 nM T + + 2T n |e q B| +M 2 |e q B| K 2 n T |e q B| +M 2 − n|e q B| 12T K 0 n T M2 + |e q B| .
(14) Note, that the first term in (14) appears from the lowest Landau levels (LLL). For these levels it is known from analysis in [17], that the asymptotic quark energy values do not depend on eB and are equal to the √ σ for small quark mass. This agrees with our valuesM = m 2 q + σ s ≈ √ cσ s and supports our expression (14) at least in the high e q B limit, e q B ≫M , when the second and the third term in (14) tend to zero. Hence one obtains in the limit |e q B| ≫M, T
P (f ) q (B) |eqB|→∞ = N c |e q B|T π 2 ∞ n=1 (−) n+1 n L nM K 1 nM T .(15)
Note, that the factor |e q B| appears due to the phase space relation in MF, Eq. (11). Now we turn to the limit of small MF, |e q B| ≪M , T . One obtains from (14) the contribution of the second term only
P (f ) q (B → 0) = 2N c T 2M 2 π 2 ∞ n=1 (−) n+1 n L n K 2 nM T + O((e q B) 2 ).(16)
One can compare (16) with (4), obtained in the case of zero MF, and insertion of (5) in (4) yields the same answer as in (16).
Magnetic susceptibility of the quark matter
Using general expression for the quark pressure (14), one can define a more convenient quantity, the magnetic susceptibilitiesχ (n) q ,χ (2) q ≡χ q ,
P f q (B, T ) − P f q (0, T ) =χ q 2 (e q B) 2 + O((e q B) 4 ).(17)
To this end one expands the Mc Donald functions K n ( √ n 2 + b 2 ) entering in (14)in powers of b, following [8], and one obtains
P f q (B, T ) − P f q (0, T ) = (e q B) 2 N c 2π 2 ∞ n=1 (−) n+1 L n f n (18) f n = ∞ k=0 (−) k k! ne q B 2TM k K k nM T 1 (k + 1)(k + 2) − 1 6 .(19)
As a consequence, one has forχ q
χ q (T ) = N c 3π 2 ∞ n=1 (−) n+1 L n K 0 nM T .(20)
It is possible to sum up the series over n in (18), when one exploits the representation
K 0 nM T = 1 2 ∞ 0 dx x e −n 1 x +M 2 x 4T 2 .(21)
As a result one obtains for the quadratic magnetic susceptibility (ms)
χ q (T ) = N c 3π 2 I q , I q = 1 2 ∞ 0 dx x Le − 1 x +M 2 x 4T 2 1 + Le − 1 x +M 2 x 4T 2 .(22)
The ms in (22) is defined for a given quark flavor q, and the total m.s. for the quark ensemble, e.g. for 2 + 1 species of quarks can be written aŝ
χ q (T ) = q=u,d,s χ q (T ) e q e 2 .(23)
Results and discussions
We have presented above the thermodynamic theory of quarks in the magnetic field, when quarks are affected also by Polyakov line interaction and the CC interaction, generalizing in this way our old results of [8]- [10], where the CC part was absent.
We have included the CC interaction in the energy eigenvalues ε σ n ⊥ , Eq.(9) tentatively via the replacement (13). This substitute can be corroborated in the case of lowest Landau levels withσ = 1, n ⊥ = 0, where magnetic field does not eneter, and the effective quark mass in subject to the CC interaction only. In the general case one can expect possible interference of eB and CC terms, which can spoil the suggested replacement.
To make this first analysis more realistic, we have checked the limits of small and large values of eB. In the first case we have shown the correct correspondence with the eB = 0 result of [13,14], and in the second case of large eB, the leading linear in eB term is just LLL term, which is not influenced by magnetic fields, except for the phase space redefinition. These results enable us to proceed with the analysis and comparisons of obtained equations with lattice data.
We present below an analysis of the MF influence on the quark thermodynamics. It is interesting, that the basic expression for the pressure P (f ) q in (14) has the property, that at small eB < (eB) crit , the dependence of ∆P (B, T ) on eB is quadratic with a good accuracy, according to Eq. (17). At larger eB, eB > (eB) crit , one has the linear dependence of ∆P (B, T ) on eB, given in (15). Fig.1 illustrates this behaviour for thee fixed temperatures (14) and Fig.1 that (eB) crit >M , and actually is around 0.5 GeV 2 for T ≃ 0.2 GeV . We have computed analytically the difference ∆P (T ) = P (T, eB) − P (T, 0) using Eq. (14) and compare with the lattice data from [18] for averaged u, d, s quark ensemble.
One can see in Fig (14) are compared with the lattice data from [18]. We have used in (14) the Polyakov line L(T ) obtained in [19]. One can see a reasonable agreement within the accuracy of the lattice data, which supports the main structure of the theory used in the paper. Detailed analysis for larger intervals of eB and T and for specific quark flavours is possible within the approach and is planned for next publications.
Figure 1 :
1The dependence of ∆P (T, B) on eB for fixed T = 0.15, 0.2, 0.4 GeV . ∆P demonstrates quadratic behaviour ∼ (eB) 2 for eB < 0.5 GeV 2 and linear behaviour ∼ (eB) for > 0.5 GeV 2 according to(14).
Figure 2 :
2The dependence of ∆P (T, B) on T for fixed values of eB = 0.2, 0.4 GeV in comparison with lattice data from [18]. For lattice results, the line thickness corresponds to the estimated error of the calculation. T = 0.15, 0.2, 0.4 GeV . One can see from
.2 the normalized pressure ∆P (B, T ) for T > 0.135 GeV and eB = 0.2, 0.4 GeV 2 for the averaged quark ensemble of u,d,s quarks. In Fig.2 our calculations of ∆P (B, T ) for eB = 0.2, 0.4 GeV 2 and m 2 D = 0.3σ s using
AcknowledgementThis work was done in the framework of the scientific project, supported by the Russian Scientific Fund, grant #16-12-10414.
. V Skokov, A Y Illarionov, V Toneev, arXiv:0907.1396Int. J. Mod. Phys. A. 245925nucl-thV. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod. Phys. A 24, 5925 (2009) [arXiv:0907.1396 [nucl-th]].
. V Voronyuk, V D Toneev, W Cassing, E L Bratkovskaya, V P Konchakovski, S A Voloshin, arXiv:1103.4239Phys. Rev. C. 8354911nucl-thV. Voronyuk, V. D. Toneev, W. Cassing, E. L. Bratkovskaya, V. P. Kon- chakovski and S. A. Voloshin, Phys. Rev. C 83, 054911 (2011) [arXiv:1103.4239 [nucl-th]].
. A Bzdak, V Skokov, arXiv:1111.1949Phys. Lett. B. 710171hep-phA. Bzdak and V. Skokov, Phys. Lett. B 710, 171 (2012) [arXiv:1111.1949 [hep-ph]].
. V A Miransky, I A Shovkovy, arXiv:1503.00732Phys. Rept. 576hep-phV. A. Miransky and I. A. Shovkovy, Phys. Rept. 576, 1 (2015) [arXiv:1503.00732 [hep-ph]].
. A Ayala, L A Hernandez, M Loewe, A Raya, J C Rojas, R Zamora, Phys. Rev. D. 9634007A.Ayala, L.A.Hernandez, M.Loewe, A.Raya, J.C.Rojas, R.Zamora, Phys. Rev. D 96, 034007 (2017).
. J F Fuiri, Iii , L G Yaffe, JHEP. 1507116J.F.Fuiri III, L.G.Yaffe, JHEP 1507, 116 (2015).
. R Oritelli, R Rougemont, S I Finazzo, J Noronha, Phys. Rev. D. 94125019R.Oritelli, R.Rougemont, S.I.Finazzo, and J.Noronha, Phys. Rev. D 94, 125019 (2016).
. V D Orlovsky, Yu A Simonov, arXiv:1311.1087Phys. Rev. D. 8974034V.D. Orlovsky, Yu.A. Simonov, Phys. Rev. D 89, 074034 (2014); [arXiv:1311.1087].
. V D Orlovsky, Yu A Simonov, arXiv:1405.2697JETP Lett. 101423V.D. Orlovsky, Yu.A. Simonov, JETP Lett. 101, 423 (2015);[arXiv:1405.2697].
. V D Orlovsky, Yu A Simonov, arXiv:1406.1056IJMPA. 30155060V.D. Orlovsky, Yu.A. Simonov, IJMPA, 30, 155060 (2015);[arXiv:1406.1056].
. C Bonati, M Elia, M Mariti, F Negro, F Sanfilippo, arXiv:1307.8063Phys. Rev. Lett. 111182001hep-latC. Bonati, M. D'Elia, M. Mariti, F. Negro and F. Sanfilippo, Phys. Rev. Lett. 111, 182001 (2013), arXiv:1307.8063 [hep-lat]
. G S Bali, PRL. 11242301G. S. Bali, et al., PRL 112, 042301 (2014).
. M S Lukashov, Yu A Simonov, arXiv:1703.06666JETP Lett. 105659hep-phM.S. Lukashov and Yu.A. Simonov, JETP Lett. 105, 659 (2017); [arXiv:1703.06666 [hep-ph]].
. M A Andreichikov, M S Lukashov, Yu A Simonov, arXiv:1707.0463M.A. Andreichikov, M.S. Lukashov and Yu.A. Simonov, [arXiv:1707.0463
. N O Agasian, M S Lukashov, Yu A Simonov, arXiv:1701.07959Eur. Phys. J. A. 53138N.O. Agasian, M.S. Lukashov and Yu.A. Simonov, Eur. Phys. J. A 53, 138 (2017) [arXiv:1701.07959].
. N O Agasian, M S Lukashov, Yu A Simonov, arXiv:1610.01472Mod. Phys. Lett. A. 311050222hep-latN.O. Agasian, M.S. Lukashov and Yu.A. Simonov, Mod. Phys. Lett. A 31, 1050222 (2016) [arXiv:1610.01472 [hep-lat]].
. M A Andreichikov, B O Kerbikov, Yu A Simonov, Jhep , arXiv:1610.0688717057M.A. Andreichikov, B.O. Kerbikov, Yu.A. Simonov JHEP, 1705, 007 (2017),[arXiv:1610.06887]
. G S Bali, arXiv:1406.0269JHEP. 08G. S. Bali, et al., JHEP, 08, 177 (2014), [arXiv:1406.0269].
. A Bazavov, N Brambilla, H.-T Ding, P Petreczky, H.-P Schadler, A Vairo, J H Weber, arXiv:1603.06637Phys. Rev. 93114502A. Bazavov, N. Brambilla, H.-T. Ding, P. Petreczky, H.- P. Schadler, A. Vairo, J.H. Weber, Phys. Rev. D93, 114502 (2016), arXiv:1603.06637.
| [] |
[
"Stability conditions of diatomic molecules in Heisenberg's picture: inspired from the stability theory of lasers",
"Stability conditions of diatomic molecules in Heisenberg's picture: inspired from the stability theory of lasers"
] | [
"Jafar Jahanpanah [email protected] \nPhysics Faculty\nKharazmi University\n49 Mofateh Ave15614TehranIran\n",
"Mohsen Esmaeilzadeh \nPhysics Faculty\nKharazmi University\n49 Mofateh Ave15614TehranIran\n"
] | [
"Physics Faculty\nKharazmi University\n49 Mofateh Ave15614TehranIran",
"Physics Faculty\nKharazmi University\n49 Mofateh Ave15614TehranIran"
] | [] | The vibrational motion equations of both homo and hetero-nuclei diatomic molecules are here derived for the first time. A diatomic molecule is first considered as a one dimensional quantum mechanics oscillator. The second and third-order Hamiltonian operators are then formed by substituting the number operator for the quantum number in the corresponding vibrational energy eigenvalues. The expectation values of relative position and linear momentum operators of two oscillating atoms are calculated by solving Heisenberg's equations of motion. Subsequently, the expectation values of potential and kinetics energy operators are evaluated in all different vibrational levels of Morse potential. On the other hand, the stability theory of optical oscillators (lasers) is exploited to determine the stability conditions of an oscillating diatomic molecule. It is peculiarly turned out that the diatomic molecules are exactly dissociated at the energy level in which their equations of motion become unstable. We also determine the minimum oscillation frequency (cut-off frequency) of a diatomic molecule at the dissociation level of Morse potential. At the end, the energy conservation is illustrated for the vibrational motion of a diatomic molecule. | null | [
"https://arxiv.org/pdf/1508.07405v1.pdf"
] | 119,280,690 | 1508.07405 | 58f4211fcaa97c59e8830b5d6ce45ed3aa0be60d |
Stability conditions of diatomic molecules in Heisenberg's picture: inspired from the stability theory of lasers
Jafar Jahanpanah [email protected]
Physics Faculty
Kharazmi University
49 Mofateh Ave15614TehranIran
Mohsen Esmaeilzadeh
Physics Faculty
Kharazmi University
49 Mofateh Ave15614TehranIran
Stability conditions of diatomic molecules in Heisenberg's picture: inspired from the stability theory of lasers
Cut-off frequencydiatomic moleculesLast stable (dissociation) levelStability theory
The vibrational motion equations of both homo and hetero-nuclei diatomic molecules are here derived for the first time. A diatomic molecule is first considered as a one dimensional quantum mechanics oscillator. The second and third-order Hamiltonian operators are then formed by substituting the number operator for the quantum number in the corresponding vibrational energy eigenvalues. The expectation values of relative position and linear momentum operators of two oscillating atoms are calculated by solving Heisenberg's equations of motion. Subsequently, the expectation values of potential and kinetics energy operators are evaluated in all different vibrational levels of Morse potential. On the other hand, the stability theory of optical oscillators (lasers) is exploited to determine the stability conditions of an oscillating diatomic molecule. It is peculiarly turned out that the diatomic molecules are exactly dissociated at the energy level in which their equations of motion become unstable. We also determine the minimum oscillation frequency (cut-off frequency) of a diatomic molecule at the dissociation level of Morse potential. At the end, the energy conservation is illustrated for the vibrational motion of a diatomic molecule.
I. Introduction
The vibrational states of diatomic molecules are recently found to have the modern applications in estimating atomic sizes by exploiting Raman spectroscopy [1], producing high fidelity binary shaped laser pulses for quantum logic gates [2], identifying pseudodiatomic behavior in polyatomic bond dissociation [3], and studying molecular potentials of isolated species [4]. The main aim of this research is to determine the stability conditions of these vibrational states in the theoretical point of you. It is demonstrated that the molecule will be dissociated at the unstable vibrational level.
The diatomic molecules generally consist of three different energy levels of electronic, vibration, and rotation in order of largeness magnitude [5,6]. Each of electronic energy levels has separately consisted of the sublevels of vibrational energies in which two atoms are oscillating under the well-known Morse potential [6]. On the other side, the simultaneous equation of rotational and vibrational motions of a diatomic molecule is formed in quantum mechanics by applying the Morse potential to the well-known Schrödinger equation [7,8]. Then, the eigenvalue equations of rotational and vibrational Hamiltonians are separated from each other by dividing the Schrödinger equation into the azimuthal-polar and radial parts, respectively [9]. Finally, the eigenvalue equation of vibrational Hamiltonian is cumbersomely solved in the spherical (radial) coordinate to determine the vibrational energy eigenvalues of a diatomic molecule up to the second-order nonlinear approximation [7,9,10].
Our priority is to determine the corresponding vibrational Hamiltonian operator of a diatomic molecule in the single space with an operatory constructer independent of any spatial coordinate [11,12]. This goal is achieved by substituting the number operator for the quantum number in the vibrational energy eigenvalue. In this way, the third-order nonlinear . The second and third-order nonlinear coefficients 2 and 3 are here calculated by using the stability theory of optical oscillators (lasers) [13].
Subsequently, we will be able to derive the motion equations of any arbitrary Hermittian operator such as the relative position and linear momentum operators of two oscillating atoms in Heisenberg's picture [14]. The simultaneous solutions of two latter equations give the useful information about the oscillatory behavior of a diatomic molecule in the different quantized energy levels of Morse potential.
We have already exploited the stability theory for evaluating the stability ranges of an electromagnetic wave inside the laser cavity as an optical oscillator [15,16]. The application of stability theory is here extended to determine the stability ranges of an oscillating diatomic molecule as a microscopic material oscillator. The oscillation stability of a diatomic molecule is investigated in the all quantized energy levels of Morse potential until this oscillation tends to infinity at the last stable energy level. It is then turned out that the last stable energy level is the same dissociation level of a diatomic molecule predicted by the different literatures [5,17,18].
The vibrational oscillation of a diatomic molecule will be described by the two different frequencies 1 and 2 ( H and Hydrogen chloride HCL as the two typical important cases of homo and hetero-nuclei diatomic molecules.
2 1 ) in
II. The second-order motion equations of a diatomic molecule
The quantum aspects of microscopic oscillators have always been described by the two general pictures of Schrödinger and Heisenberg [14]. The expectation values of observable quantities (classical variables) are evaluated by the wave function in the former picture and by the corresponding operators in the latter picture. We have here only exploited the more comprehensive picture of Heisenberg to elaborate the oscillating behavior of a diatomic molecule in quantum mechanics.
First consider a diatomic molecule as a quantum microscopic oscillator. The expectation values of the relative position )
( t x and linear momentum ) ( t p operators of two oscillating atoms are thus determined by the solution of their corresponding temporal equations of motion in the Heisenberg's picture as [14]
) ( , ) ( t x H i dt t x d (2.1) and ) ( , ) ( t p H i dt t p d , (2.2) in which 2 h
, and h is the Planck constant. Obviously, the Hamiltonian operator Ĥ of a diatomic molecule is required to deal with the equations (2.1) and (2.2). Therefore, this is compulsory to use the second-order energy eigenvalue of a diatomic molecule in the form [1,10]
2 2 0 2 0 ) 2 / 1 ( 4 ) 2 / 1 ( n D n E e vib ,(2.2 2 0 ) 2 () 2 1 ( ) 2 1 ( H H N D N H e ,(2. x i p dt x d n2 0 . (2.7) in which 0 and 2 1 1 0 n n .
(2.8)
It should be noticed that (2.6) is used to derive the required commutation relations
/ ] , [ 0 p i x H and / ) 2 ( ] , [ 0 2 0 x k i H p i x H in x-p space. Similarly, p i x k dt p d n2 0 , (2.9)
where the commutation relations
x k i p H] , [ 0 and ) 2 ( ] , [ 0 2 0 p i H x k i p H are used.
The second-order imaginary differential equation will finally be derived for the expectation value of position operator )
( t x by substituting (2.7) and (2.9) into the derivative of (2.7) as
0 4 2 2 2 2 x dt x d i dt x d n , (2.10)
which exactly mimics the second-order imaginary differential equation (2.2.17) of Ref [19] associated with the transition coefficient of a single atom to an upper excited level during its interaction with a constant electric field [19]. To our best knowledge, the second-order imaginary differential equation (2.10) which describes the motion of an oscillating diatomic molecule in the different energy levels of Morse potential is here introduced for the first time.
If one lets the dissociation energy of diatomic molecules e D goes to infinity ( e D ), then the Morse potential matches to the Hook potential, and the equation of motion (2.10) is consequently reduced to that of SHO as
0 2 0 2 2 x dt x d ,(2. ) sin( ) cos( ) 0 ( ) ( 0 0 t t x t x , (2.12) in which ) 0 (
x is the initial mean value of position operator. One can calculate the mean potential energy
2 2 0 ) ( 5 . 0 ) ( t x t V ,
the mean kinetic energy ( ), and the total energy
2 ) ( 5 . 0 ) ( t p t K ( dt t x d t p ) ( )cte K V t K t V E ) 0 ( ) 0 ( ) ( ) (
by using the general solution of SHO (2.12). In the meantime, the mean initial values of momentum and position operators are related to each other by the Heisenberg's uncertainty principle
) 0 ( ) 0 ( 0 x p [12].1 ) 0 ( ) 0 (
, which is in complete agreement with the energy conservation.
III. The last stable (dissociation) vibrational level of a diatomic molecule
(Second-order nonlinear approximation) All microscopic oscillators including optical oscillators such as lasers and material oscillators such as diatomic molecules have many common oscillating features. For example, the stability range of the both optical and material oscillators is one of the most important issues which are commonly determined by the stability theory [20].
The general method is to multiply the initial displacement of expectation value of position operator ) 0 ( x by a temporal exponential term t e to probe the later possible situations of oscillating variable
) ( t x in the form [21] t e x t x ) 0 ( ) ( . (3.1)
It is evident that the oscillator is stable for the negative values of parameter , where the next displacements )
( t x are toward the origin (smaller values) at the later times t ( 0 t ).
By contrast, the unstable state of oscillator is corresponding to the positive values of
parameter , where the next displacements ) ( t x are toward the infinity (larger values) at the later times t ( t
). Therefore, the border between two states of stability and instability is determined by the zero values of parameter . The trial solution (3.1) is thus substituted into the second-order imaginary differential equation (2.10) to find a stability equation for the parameter in the form
0 4 2 2 2 n i . (3.2)
We are now looking for the roots of stability equation (3.2) in the forms
2 1 1 2 2 1 0 0 1 1 n i i i i n (3.3) and 2 1 1 2 2 2 0 0 2 2 n i i i i n . (3.4)
The border between the stable and unstable states of a diatomic molecule is then determined by 1 1 0
1 1 stable n (3.5)
and
0 ) 2 ( ) 2 ( 1 ) 2 ( ) 2 ( stable stable n n E E ,(3.9)
where (2.3) and (3.7) should be used.
IV. The second-order solution and cut-off frequency
The second-order imaginary differential equation (2.10) has the general imaginary solution in the form [19]
t i t p t x t x n n n 2 exp ) sin( ) 0 ( ) cos( ) 0 ( ) ( ,(4. ) sin( ) sin( 2 ) 0 ( ) cos( ) cos( 2 ) 0 ( ) ( 2 1 2 1 t t p t t x t x n (4.2) in which ) 1 ( 2 0 1 n n (4.3) and 0 1 0 2 ) 1 ( 1 2 n n ,(4. ) 0 ( ) 0 ( 0 x p .
It is clear that the oscillation will continue until the both frequency components 1 and 2 remain positive. Consequently, the last oscillating levels of 1 n and 2 n are determined by the following conditions Since
1 ) ( 1 1 ) ( max 1 max 2 n n
, the last oscillating level in which the both oscillatory frequencies 1 and 2 never get negative values in (4.5) and (4.6) is equal to
1 1 ) ( max 2 n ,(4.7)
which is in complete agreement with the second-order last stable (dissociation) level (3.7).
The second-order cut-off frequency becomes zero.
V. The third-order nonlinear Hamiltonian and equations of motion
x n D i p n D dt x d e e n) 2 1 ( 24 1 2 1 ) 2 1 ( 3 4 1 3 2 3 2 2 2 0 (5.2) and p n D i x n D k x dt x d i dt x d n ,(5. ) sin( ) sin( 2 ) 0 ( ) cos( ) cos( 2 ) 0 ( ) ( 2 1 2 1 t t p t t x t x n , (5.7)
in which the new third-order oscillating frequencies 1 and 2 are equal to (5.9) where the equality of second and third-order oscillating frequencies
3 2 2 2 0 1 1 1 ) 2 1 ( 3 ) 2 1 ( 3 4 2 n n D e n (5.8) and 1 3 2 2 2 0 2 2 1 ) 2 1 ( 3 ) 2 1 ( 3 4 2 n n D e n ,( n E n H n | | 0 0 ) is
used. It is emphasized that the third-order energy eigenvalue (5.15) is here derived in the single space without solving the complicated radial Schrödinger equation up to third-order approximation in the polar-spherical spatial coordinate [8,23].
There are two different methods to determine the third-order stable (dissociation) level ) 3 ( D n . The first one is to substitute 3 from (5.14) into the following usual equation
0 ) 3 ( ) 3 ( 1 ) 3 ( ) 3 ( D D n n E E . (5.16)
The second one is to substitute 3 from (5.14) into the smaller oscillating frequency 2 (5.9) and looking for the last oscillating level in the following heuristics equation
3 2 ) 3 ( 2 2 ) 3 ( 0 ) 3 ( 4 12 1 D e D off Cut n D n ,(5.19)
where 3 and ) 3 ( D n are given in (5.14) and (5.18), respectively. Clearly, the smaller oscillating frequency 2 (5.9) becomes zero at the third-order last stable (dissociation) energy level (5.18). [18,27] and for the Hydrogen chloride 21 D n [18,28], which are in good agreement with our second and third-order quantum numbers [29].
At the end, we reconfirm the second and third-order solutions (4.2) and (5.7) by considering the general structure of Morse potential in the form [5,27] ).
2 ) ] ) ( [ ( exp 1 ) ) ( ( e e x t x a D t x V , (5.20) 2 2 0 ) ( 5 . 0 ) ) ( ( e x t x t x V ,(5.
VI. Conclusion
The stability theory of lasers is then exploited to determine the last stable level of Morse potential after which the oscillation of a diatomic molecule experiences no turning point toward the equilibrium point of potential well. It is peculiarly turned out that the last stable level is the same dissociation level of a diatomic molecule.
The general solution (4.2) associated with the relative position of two oscillating atoms demonstrate a periodic wave packet consists of two oscillating frequency components 1 and 2 , as illustrated in Figs (2) and (3) To our best knowledge, the comparison idea of material and optical oscillators (the diatomic molecules and lasers) is a heuristic idea which is presented here for the first time.
The stability theory of lasers in quantum optics is exploited to determine the stability range of diatomic molecules in quantum mechanics. In this way, the interested reader is recommended to probe about other common physical features of material and optical oscillators.
Figure Captions
H
is formed for the oscillation motion of a diatomic molecule in term of the linear Hamiltonian operator 0 H of a simple harmonic oscillator (SHO)
key dimensionless parameter for the future applications. The operators T â and â are well recognized as the upper and lower operators on the energy eigenvalues of the is easy to investigate that the operators T â and â play the same respective upper and lower roles on the eigenvalues of the substituting the second-order nonlinear Hamiltonian operator (2.4) into (2.1) as
Figure ( 1
1) illustrates the small oscillations of a diatomic molecule in the linear regime (SHO) in which the temporal variations of potential
now able to compare the second and third-order temporal variations of relative position of two oscillating atoms by considering the simultaneous solutions (4.2) and (5.7). The formation of standing waves in the different energy levels n of Morse potential is completely evident in Figs. 2 (a) and 2(b) for the Hydrogen molecule as a homo-nuclei diatomic molecule, and in Figs. 3 (a) and (b) for the Hydrogen chloride as a hetero-nuclei diatomic molecule. It is seen that the oscillation amplitudes of the both molecules are continuously raised from the basic energy level 0 n toward the upper second and thirdorder that the amplitude of oscillations is suddenly increased by a factor of 100. The similar divergent behavior has also been observed for the fluctuations of electric field amplitude (noise) inside the cavity of class-C lasers at an upper pumping rate determined by the stability theory
.
only valid for the small vibrational oscillations of a diatomic molecule around the equilibrium point e x . Clearly, the Hook potential can be calculated by substituting the general solution of SHO (2.12) into (5.21). The spatial variations of Hook potential and the second and third-order Morse potentials are plotted in Fig. 4 (a) for the Hydrogen molecule, and in Fig. 4(b) for the Hydrogen chloride molecule by choosing the Both figures reveal that the second and third-order solutions (4.2) and (5.7) are completely correct and almost form the whole structure of Morse potential. Therefore, the higher-order terms in nonlinear Hamiltonian operator (5.1) add no more physical insight to the present vibrational features of a diatomic molecule.
A diatomic molecule is here treated as a microscopic material oscillator in quantum mechanics and its behavior is compared with the laser as an optical oscillator in quantumoptics. We have heuristically formed the second-order vibrational Hamiltonian operator in the single space by substituting the number operator for the quantum number in the corresponding energy eigenvalue. This Hamiltonian is then used to derive the coupled equations of motion for the expectation values of relative position and linear momentum operators of two oscillating atoms in the Heisenberg's picture. Finally, the second-order imaginary differential equation (2.10) is derived for the vibrational motion of a diatomic molecule in the different energy levels of Morse potential for the first time. It is interesting that this differential equation is exactly similar to the transitional equation of a single atom interacting with a constant electric field. Meanwhile, it is simplified to that of SHO (2.11) by letting the right branch of Morse potential goes to infinity ( e D
for the respective typical molecules of Hydrogen and Hydrogen chloride. The both frequency components 1 and 2 are reduced by exciting the diatomic molecule to the higher vibrational energy levels. Clearly, the vibrational oscillations of diatomic molecules will be terminated at the last upper oscillating level in which the smaller frequency component 2 has the last positive value equal to zero. It is then turned out that the last upper oscillating level, in which the diatomic molecule only oscillates with the cut-off frequency (4.8), is the same stable (dissociation) level (3.7) determined by the stability theory. The third-order vibrational Hamiltonian operator in the single space is similarly formed in (5.1). The stability theory of lasers is then exploited to determine the third-order important parameters of expansion coefficient 3 in (5.14), the last stable oscillating 5.19). The second and third-order Morse potentials are finally calculated by substituting the corresponding solutions (4.2) and (5.7) into the exact Morse potential (5.20), as illustrated in Figs. 4(a) and 4(b) for the homo and hetero-nuclei molecules of Hydrogen and Hydrogen chloride, respectively.
Fig. 1 .
1The temporal variations of potential and kinetic energies of a regime of linear oscillations (SHO) are illustrated to satisfy the energy conservation in order that their sum is always constant and equal to the initial value
Fig. 2 .
2(a)-The second-order and (b)-third-order expectation values of relative position operator of two oscillating atoms associated with Hydrogen molecule are plotted for the basic level 0 n (red color), and for the typical excited levels 12 n (blue color) and 14 n (green color). The formation of wave packets is evident for the molecule oscillation in the different vibrational energy levels of Morse potential
Fig. 3 .
3(a)-The second-order and (b)-third-order expectation values of relative position operator of two oscillating atoms associated with Hydrogen chloride molecule are plotted for the basic level 0 n (red color), and for the typical excited levels 15 n (blue color) and 20 n (green color). The formation of wave packets is evident for the molecule oscillation in the different vibrational energy levels of Morse potential.
Fig. 4 .
4The Hook potential (green color) and the second (red color) and third (blue color) order Morse potentials are illustrated for (a)-Hydrogen 2 H as a homo-nuclei diatomic molecule, and for (b)-Hydrogen chloride HCL as a hetero-nuclei diatomic molecule.
Fig. 1 Fig
1Fig. 1
order that the both frequencies are simultaneously reduced by exciting the molecule to the higher energy levels of Morse potential. At the last stable (dissociation) energy level, the smaller oscillating frequency 2 tends to zero and the diatomic molecule thus oscillates with the unique frequency will be calculated up to the second and thirdorder approximations. Finally, the second and third-order Morse potentials together with the corresponding last stable (dissociation) energy levels will be calculated for the diatomic molecules of Hydrogen 2off
Cut
1
. The cut-off frequency
off
Cut
is certainly the last stable level for which the non-zero positive values never occur for the both parameters It is immediately turned out that the second-order stable level1
0
2
2
stable
n
.
(3.6)
Since
stable
stable
n
n
2
1
,
stable
n
1
1
and
2
. In other word,
stable
n
1
is corresponding
to the stable conditions
0
1
and
0
2
, whereas the next higher energy level
1
1
2
stable
stable
n
n
is corresponding to the undesirable (unstable) conditions
0
1
and
0
2
.
stable
stable
n
n
1
)
2
(
is the
same second-order dissociation energy level
)
2
(
D
n
of a diatomic molecule given in many
literatures as [17, 18, 22]
1
1
)
2
(
)
2
(
D
stable
n
n
.
(3.7)
One can ensure that the second-order stable energy level
)
2
(
stable
n
is the final oscillating level of
a diatomic molecule by investigating the following relation
Table 1
1summarizes the second and third-order parameters of Hamiltonian expansioncoefficients 2
and 3
, last stable (dissociation) energy levels
Table 1 :
1Thesecond and third-order parameters, and the exact number of dissociation level
for the Hydrogen and Hydrogen chloride molecules
Second -order Parameters
Third-order Parameters
Exact
number
Molecule
2
2
Cut off Hz
(2)
D
n
3
3
Cut off Hz
3
D
n
D
n
2
H
10
-3.09 10
13
4.48 10
17.5
19
-7.69 10
13
8.96 10
16.5
15
HCl
10
-3.55 10
13
2.21 10
23.6
19
-7.42 10
13
4.43 10
22.6
21
. D Wang, W Guo, J Hu, F L Liu, S Chen, Z Du, Tang, Nature Scientific Reports. 31486D. Wang, W. Guo, J. Hu, F. Liu. L. Chen, S. Du, and Z. Tang, Nature Scientific Reports. 3, 1486 (2013).
. R Z Ryan, A Brown, J. Chem. Phys. 13544317R. Z. Ryan and A. Brown, J. Chem. Phys. 135, 044317 (2011).
. J S Murray, A T Labbe, S G Oliva, P Politzer, J. Chem. Phys. 132154308J. S. Murray, A. T. Labbe, S. G. Oliva, and P. Politzer, J. Chem. Phys. 132, 154308 (2010).
. C Miron, C Nicolas, O Travnikova, P Morin, Y Sun, F G Mukhanov, N Kosugi, V Kimberg, Nature Physics. 8135C. Miron, C. Nicolas, O. Travnikova, P. Morin, Y. Sun, F. G. Mukhanov, N. Kosugi, and V. Kimberg, Nature Physics. 8, 135 (2012).
Atoms, Molecules and Photons: An Introduction to Atomic-Molecular and Quantum-Physics. W Demtröder, Springer-VerlagBerlin HeidelbergW. Demtröder, Atoms, Molecules and Photons: An Introduction to Atomic-Molecular and Quantum-Physics. 349-372 (Springer-Verlag Berlin Heidelberg, 2006).
W T Silfvast, Laser Fundamentals. 2 nd EdCambridge University PressW. T. Silfvast, Laser Fundamentals. 135-145 (Cambridge University Press, 2 nd Ed, 2004).
. H Sun, Phys. Let. A. 338309H. Sun, Phys. Let. A. 338, 309 (2005).
. S Miraboutalebi, L Rajaei, J Math Chem. 521119S. Miraboutalebi and L. Rajaei, J Math Chem. 52, 1119 (2014).
. O Bayrak, I Boztosun, J. Phys. A: Math. Gen. 396955O. Bayrak and I. Boztosun, J. Phys. A: Math. Gen. 39, 6955 (2006).
. A Derevianko, E Luc-Koenig, F Masnou-Seeuws, Can. J. Phys. 8767A. Derevianko, E. Luc-Koenig, and F. Masnou-Seeuws, Can. J. Phys. 87, 67 (2009).
. J Recamier, R Jauregui, J. Opt. B: Quantum Semiclass. Opt. 5365J. Recamier and R. Jauregui, J. Opt. B: Quantum Semiclass. Opt. 5, S365 (2003).
S Gasiorowicz, Quantum Physics. John Wiley & SonsS. Gasiorowicz, Quantum Physics (John Wiley & Sons, 3 rd Ed, 2003).
J Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos. Berlin, 3 rd EdSpringerJ. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos (Springer-Berlin, 3 rd Ed, 2014).
Y V Nazarov, J Danon, Advanced Quantum Mechanics. Cambridge University PressY. V. Nazarov and J. Danon, Advanced Quantum Mechanics, 11-13 (Cambridge University Press, 2013).
. J Jahanpanah, A A Rahdar, Laser Phys. 2425001J. Jahanpanah and A. A. Rahdar, Laser Phys. 24, 025001 (2014).
. J Jahanpanah, S Baiat, J. Opt. 16105208J. Jahanpanah and S. Baiat, J. Opt. 16, 105208 (2014).
. C D Yang, H J Weng, Chaos, Solitons & Fractals. 45402C. D. Yang and H. J. Weng, Chaos, Solitons & Fractals. 45, 402 (2012).
. S M Ikhdair, Molecular Physics. 1101415S. M. Ikhdair, Molecular Physics. 110, 1415 (2012).
R Loudon, The Quantum Theory of Light. Oxford University PressR. Loudon, The Quantum Theory of Light (Oxford University Press, 3 rd Ed, 2000).
H M Guido, V Tartwijk, G P , Progress in Quantum Electronics. 2243H. M. Guido, V. Tartwijk, and G. P. Agrawal, Progress in Quantum Electronics. 22, 43 (1998).
. J Jahanpanah, J. Opt. Soc. A. 22680J. Jahanpanah, J. Opt. Soc. A. 22, 680 (2005).
. P G Hajigeorgiou, J. Molecular Spectroscopy. 2861P. G. Hajigeorgiou, J. Molecular Spectroscopy. 286, 1 (2013).
. J Yu, S H Dong, G H Sun, Phys. Let. A. 322290J. Yu, S. H. Dong, and G. H. Sun, Phys. Let. A. 322, 290 (2004).
. K K Irikura, J. Phys. Chem. 36389K. K. Irikura, J. Phys. Chem. 36, 389 (2007).
A K Roy, Results in Physics. 3103A. K. Roy, Results in Physics. 3, 103 (2013).
. P G Hajigeorgiou, J. Molecular Spectroscopy. 263101P. G. Hajigeorgiou, J. Molecular Spectroscopy 263, 101 (2010).
. I N Levine, Physical Chemistry. 747I. N. Levine, Physical Chemistry 747 (The McGraw-Hill Companies, 6 th Ed, 2009).
. X N Zhang, D H Shi, J P Zhang, Z Z Lue, J F Sun, Chin. Phys. B. 1953401X. N. Zhang, D. H. Shi, J. P. Zhang, Z. Z. Lue, and J. F. Sun, Chin. Phys. B. 19, 053401 (2010).
. J Jahanpanah, R Pirzadeh, A Soleimani, IEEE J. Quantum Electron. 511100307J. Jahanpanah, R. Pirzadeh, and A. Soleimani, IEEE J. Quantum Electron. 51, 1100307 (2015).
| [] |
[
"Optimal potentials for temperature ratchets",
"Optimal potentials for temperature ratchets"
] | [
"Florian Berger \nII. Institut für Theoretische Physik\nUniversität Stuttgart\n70550StuttgartGermany\n",
"Tim Schmiedl \nII. Institut für Theoretische Physik\nUniversität Stuttgart\n70550StuttgartGermany\n",
"Udo Seifert \nII. Institut für Theoretische Physik\nUniversität Stuttgart\n70550StuttgartGermany\n"
] | [
"II. Institut für Theoretische Physik\nUniversität Stuttgart\n70550StuttgartGermany",
"II. Institut für Theoretische Physik\nUniversität Stuttgart\n70550StuttgartGermany",
"II. Institut für Theoretische Physik\nUniversität Stuttgart\n70550StuttgartGermany"
] | [] | In a spatially periodic temperature profile, directed transport of an overdamped Brownian particle can be induced along a periodic potential. With a load force applied to the particle, this setup can perform as a heat engine. For a given load, the optimal potential maximizes the current and thus the power output of the heat engine. We calculate the optimal potential for different temperature profiles and show that in the limit of a periodic piecewise constant temperature profile alternating between two temperatures, the optimal potential leads to a divergent current. This divergence, being an effect of both the overdamped limit and the infinite temperature gradient at the interface, would be cut off in any real experiment. | 10.1103/physreve.79.031118 | [
"https://arxiv.org/pdf/0807.4644v2.pdf"
] | 12,292,926 | 0807.4644 | 050bf9fb6ffa833035aec525be003a6d96f86690 |
Optimal potentials for temperature ratchets
2 Apr 2009
Florian Berger
II. Institut für Theoretische Physik
Universität Stuttgart
70550StuttgartGermany
Tim Schmiedl
II. Institut für Theoretische Physik
Universität Stuttgart
70550StuttgartGermany
Udo Seifert
II. Institut für Theoretische Physik
Universität Stuttgart
70550StuttgartGermany
Optimal potentials for temperature ratchets
2 Apr 2009(Dated: April 2, 2009)PACS numbers: 0540-a, 8270Dd
In a spatially periodic temperature profile, directed transport of an overdamped Brownian particle can be induced along a periodic potential. With a load force applied to the particle, this setup can perform as a heat engine. For a given load, the optimal potential maximizes the current and thus the power output of the heat engine. We calculate the optimal potential for different temperature profiles and show that in the limit of a periodic piecewise constant temperature profile alternating between two temperatures, the optimal potential leads to a divergent current. This divergence, being an effect of both the overdamped limit and the infinite temperature gradient at the interface, would be cut off in any real experiment.
I. INTRODUCTION
Noise induced transport occurs in a broad variety of systems both in physics and biology [1]. In a microscopic system embedded in a thermal environment, directed transport generally requires two ingredients: (i) External sources driving the system out of equilibrium and (ii) broken spatial symmetry [1]. The main idea is to impede the thermal motion in one direction in order to obtain a net current in the other direction [2]. This mechanism is called the "ratchet effect". Applications range from particle sorting [3,4] to modeling molecular motors [2, 5,6,7]. Most studies on ratchet motors focus on a given potential landscape and a given driving scheme. For practical purposes, however, optimization of the driving mechanism with respect to a maximal current is an important issue.
For discrete analogues of ratchet motors, where the potential landscape is characterized by only a few parameters, such an optimization has been performed in models for microscopic heat engines [8] and molecular motors [9]. Paradoxical Parrondo games [10] which can be interpreted as discrete analogues of Brownian ratchets [11], have also been optimized recently [12].
For continuous motors, where optimization requires variational calculus, there exist only few results. The optimization of driving schemes has been studied for time-dependently driven ratchet motors [13,14] and for a Brownian heat engine cycling between two heat reservoirs [15]. Potential landscapes have been optimized for the transport across membrane channels [16]. Maximizing the current of flashing ratchets by using a feedback control strategy has been proposed recently [17,18].
In this paper, we focus on a continuous thermal ratchet where transport along a spatially varying timeindependent potential is driven by a periodic spatial temperature profile [19,20,21,22]. The recent generation of temperature gradients on small length scales [23,24,25] may render such molecular heat engines experimentally realizable. Cargoes driven by thermal gradients on a subnanometer scale have already been observed [26].
Thermodynamic efficiencies of such ratchet heat en-gines have been calculated in Refs. [27,28,29]. It has been argued that heat engines should generally be characterized by their performance at maximum power output [30,31]. Recent studies on ratchet heat engines have varied the load force for a constant potential in order to maximize the power output [32,33]. We take a complementary approach and optimize the potential for a given load. The maximization of power output and particle current then is equivalent. So far, this task has been tackled numerically only for a one-parameter class of potentials for a given temperature [34]. To the best of our knowledge, there exists no systematic study on the optimal potential for such a thermal ratchet. Using variational calculus we determine the optimal potential which maximizes the current for a given temperature analytically up to a numerical root search. For a dichotomous temperature profile with infinitely steep gradients as used in most previous studies [29,32,33], the maximal current diverges. This unphysical behaviour is an effect of the idealized assumptions of both an infinite temperature gradient at the interface and the overdamped dynamics. In any realistic system, temperature gradients will be finite which is sufficient to cut off the formal divergence.
II. THE TEMPERATURE RATCHET AND ITS CURRENT
We consider a Brownian particle of mass m moving in a periodic potential V (x) with V (x + L) = V (x) in a viscous fluid with friction coefficient γ. A constant load f is attached to the particle. The surrounding temperature is modeled by T (x) which has the same periodicity L as the potential. A special case is a piecewise constant temperature with a hot and a cold area, see Fig. 1. For a properly chosen potential the particle moves against the load on average. The thermal fluctuations in the hot area can push the particle against the load over the barrier of the potential. As soon as the particle is in the cold area the probability of getting pushed back is smaller because of the weaker thermal fluctuations. In this way the particle drags the load and produces work effectively acting as a heat engine that works between two heat baths. Such a mechanism is not limited to piecewise constant temperatures.
0 L/2 L T c T h T h T c T c T h f V (x)
FIG. 1: A temperature ratchet: a particle with a load f moving in a potential V (x) in two piecewise constant temperature regions T h and Tc.
The time evolution of the position of the particle x(t) is governed by the Langevin equation
mẍ = −γẋ − V ′ (x) + f + g(x)ξ(t),(1)
where time derivatives are denoted by dots and space derivatives by primes. The stochastic term g(x)ξ(t) models the thermal noise from the environment. Its strength
g(x) ≡ k B T (x)γ (2)
depends on the temperature profile T (x) resulting in multiplicative noise.
In the following we want to make two assumptions: the friction dominates the inertia and the noise is uncorrelated Gaussian. When dealing with multiplicative noise, these two limiting procedures do not commute [35,36]. The order of the limiting procedures determines the stochastic interpretation of the term g(x)ξ(t). If we first assume Gaussian white noise
ξ(t)ξ(t ′ ) = 2δ(t − t ′ ),(3)
and then go to the overdamped equation
γẋ = −V ′ (x) + f + g(x)ξ(t)(4)
we end up with a corresponding Fokker-Planck equation
∂ t p(x, t) = −∂ x 1 γ (−V ′ (x) + f ) − 1 γ 2 ∂ x g 2 (x) p(x, t) (5) = −∂ x j(x, t)
in Ito's sense where j(x, t) is the current and p(x, t) the probability distribution.
If we first take the overdamped limit and afterwards assume Gaussian white noise, we end up with a Fokker-Planck equation in Stratonovich's sense. Both equations differ in a drift term which can be absorbed in an effective potential. Thus, the optimal current does not depend on the interpretation. The optimal potential is merely shifted by the drift term. In the following we use the Fokker-Planck equation in Ito's sense.
We next introduce dimensionless units. We express energies in units of k B T 0 where
T 0 ≡ 1 L L 0 T (x) dx.(6)
By introducing a scaled lengthx ≡ x/L and a scaled timet ≡ t/t 0 with t 0 ≡ L 2 γ/k B T 0 , we rewrite the Fokker-Planck equation and identify the dimensionless quantitieŝ
T (x) ≡ T (x) T 0 ,V (x) ≡ V (x) k B T 0 ,f ≡ f L k B T 0 , (x,t) ≡ j(x, t)L 2 γ k B T 0 ,p(x,t) ≡ Lp(x, t).(7)
For ease of notation we drop the hats in the following. The dimensionless potential V (x) and the dimensionless temperature profile T (x) are periodic, V (x + 1) = V (x), T (x + 1) = T (x). In the steady state, the current j is a constant and the Fokker-Planck equation reduces to
j = [−V ′ (x) + f − T ′ (x)]p(x) − T (x)∂ x p(x). (8)
For a periodic temperature we solve this equation under the condition of a periodic p(x). Without loss of generality, we chose V (0) = 0 which results in
p(x) = je φ(x) T (x) 1 1 − e −φ(1) 1 0 e −φ(x ′ ) dx ′ − x 0 e −φ(x ′ ) dx ′ (9) with φ(x) ≡ x 0 −V ′ (x ′ ) + f T (x ′ ) dx ′ .(10)
Using the normalization 1 0 p(x)dx = 1, we obtain the inverse current
j −1 = 1 1 − e −φ(1) 1 0 e −φ(x ′ ) dx ′ 1 0 e φ(x) T (x) dx − 1 0 x 0 e −φ(x ′ ) dx ′ e φ(x) T (x) dx(11)
which has been derived previously in Refs. [20,32].
III. OPTIMIZING THE CURRENT
The inverse current [Eq. (11)] depends on the shape of the potential V (x). Instead of optimizing the current directly with respect to the potential we introduce
B(x) ≡ x 0 e −φ(x ′ ) dx ′(12)
and rewrite the inverse current in a more elegant way as a functional of B(x)
j −1 [B, B ′ ] = 1 0 α − B(x) T (x)B ′ (x) dx (13) with α ≡ B(1) 1 − B ′ (1) .(14)
For the minimization of the functional (13), the function space is constrained by the periodicity of the potential which imposes a constraint on B(x). With the boundary condition V (1) = V (0) = 0 we obtain from Eq. (10) the non-local constraint
1 0 φ ′ T dx = f(15)
which by using Eq. (12) can be transformed into the isoperimetric constraint
1 0 T ′ ln B ′ dx = f + T ln B ′ 1 0 .(16)
In order to minimize the inverse current [Eq. (13)] under the constraint [Eq. (16)] we introduce the effective Lagrangian
L[B(x), B ′ (x), x] ≡ α − B(x) T (x)B ′ (x) + λT ′ (x) ln B ′ (x) (17)
with a Lagrange multiplier λ.
For a unique solution to the corresponding Euler-Langrange equation we have to impose two boundary conditions. The first one, B(0) = 0, arises naturally from the definition of B(x) in Eq. (12). One is tempted to use the condition B ′ (0) = 1 as the second one, but this is not appropriate. In principle, we have to allow the derivative to jump at the boundaries because such jumps do not contribute to the integral of the Lagrangian. In this way the boundary condition fixes the value B ′ (0) = 1, but the value of lim ǫ→0 B ′ (0+|ǫ|), which is the relevant boundary condition for the solution of the Euler-Langrange equation, in fact, remains a free parameter. This feature has been discussed in detail for similar optimization problems [37,38].
For the optimization we proceed in two steps. First we minimize the integral of the Lagrangian 1 0 L dx and then we adjust the remaining parameters to obtain the maximum current. The corresponding Euler-Langrange equation is given by
2B ′2 T +(α−B)(T ′ B ′ +2T B ′′ )+λT 2 B ′ (T ′′ B ′ −T ′ B ′′ ) = 0(18)
with the boundary condition B(0) = 0. Changing variables B(x) = −α exp[I(x)] + α leads to a second order differential equation for I(x) which is integrable. The solution
I ± (x) ≡ x 0 2 dx ′ −λT T ′ ± 4T (c + λT ) + λ 2 T 2 T ′2(19)
still depends on the Lagrange multiplier λ and the new constant c. The solution of the Euler-Lagrange equation
B(x) = −α exp[I ± (x)] + α(20)
leads to the optimal inverse current
j −1 (λ, c) = − 1 0 dx T (x)I ′ − (x) = 1 2 1 0 4 c T + λ + λ 2 T ′2 dx(21)
which depends on the two parameters λ and c but is independent of α. Note that in order to obtain a positive current, we chose the minus sign of the square root in Eq. (19).
In the next step, we optimize the inverse current [Eq. (21)] by adjusting the free parameters λ and c. These parameters are not independent but related by the constraint [Eq. (16)]
n(λ, c) ≡ T ln |I ′ − | 1 0 +f − 1 0 T ′ ln |I ′ − | dx+ 1 0 I ′ − T dx = 0.(22)
In the appendix, we show that in general the optimization problem has a solution for c = 0. Hence, for a given temperature profile the remaining parameter λ can be determined by the constraint [Eq. (22)]. From Eqs. (10,12,20) we derive the optimal potential
V (x) = T ln |I ′ − | x 0 + f x − x 0 T ′ ln |I ′ − | dx ′ + x 0 I ′ − T dx ′(23)
which becomes the basis for the following case studies.
IV. CASE STUDY I: SINUSOIDAL TEMPERATURE PROFILE
In this section we will discuss a sinusoidal temperature profile
T (x) = A sin(2πx) + 1(24)
with the amplitude 0 < A < 1. The external force f is a second parameter. In the following we will study how the optimal potential and the current depend on these two parameters.
The height of the potential is the essential blocking mechanism in the ratchet. The thermal "kicks" from the environment move the particle over this barrier. We expect the largest slope of the optimal potential roughly at the hottest point because there the fluctuations are strong enough to push the particle against a large force. In the colder regions, the potential should decrease. The optimal potential as determined numerically using Eq. (23) indeed fulfills these expectations, see Fig. 2(a). The optimal potential for different amplitudes A with zero external force f = 0 is shown in Fig. 2(b). For amplitudes A → 1, the temperature in the colder area x > 0.5 goes to zero and the optimal potential becomes strongly asymmetric. In this low temperature area the thermal fluctuations are so weak that even a gently declining potential is sufficient to push the particle in one direction. The optimal current is proportional to the absolute value of the amplitude, see Fig. 2(b). This general scaling behavior is not limited to a sinusoidal temperature profile which can be understood as follows. With the periodicity T ′ (0) = T ′ (1), the first term of the constraint [Eq. (22)] vanishes. With c = 0 and f = 0, Eq.
n(λ) = 1 0 T ′ ln λT ′ + 4λ + λ 2 T ′2 − 1 2λ 4λ + λ 2 T ′2 dx = 0 (25) and j −1 (λ) = 1 2 1 0 4λ + λ 2 T ′2 dx.(26)
The derivative of the temperature scales like T ′ (x) ∝ A.
Choosing λ ∝ A −2 , the constraint [Eq. (25)] is independent of A. The current [Eq. (26)] then is a linear function of A. Now we consider the system with a finite external force, f = −0.05. For temperature amplitudes A → 1, effectively corresponding to a large temperature difference, the external force can be neglected and the optimal potential looks like in the case without an external force, see Fig. 2(c). For small A, the ratchet effect induced by the temperature difference is not strong enough against the external force. In this regime, the optimal potential has to prevent the particle from being dragged in the direction of the external force. This is achieved by blocking the particle with a larger barrier, see Fig. 2(c).
B. Optimal potential for different external forces
For stronger external forces, the potential has to compensate this dragging mechanism with a larger barrier in order to obtain a current in the direction opposite to the force. Thus, we expect larger potentials and smaller currents for stronger external forces which is confirmed by our calculations, see Fig. 2(d).
We next calculate the dimensionless power output of the heat engine which is given by
P ≡ −f j.(27)
The power output as a function of the force f is shown in the inset in Fig. 2(d). It exhibits a maximum at intermediate forces where the heat engine thus works in a maximum power regime.
V. CASE STUDY II: PIECEWISE CONSTANT TEMPERATURE
We now consider a piecewise constant temperature
T (x) = 1 + ∆T − 2∆T Θ(x − 1/2)(28)
with a hot and a cold area with temperatures T h ≡ 1+∆T and T c ≡ 1 − ∆T , respectively, where 0 < ∆T < 1. We then face discontinuities in the Fokker-Planck equation.
In this section, we first analyze a continuous approximation to the piecewise constant temperature. The optimal potential then has an complex shape with peaks. We compare these results with a numerical solution for the optimal potential. The continuous temperature profile
T (x) = √ 1 + d sin(2πx) 2 sin 2 (2πx) + d + 1(29)
interpolates between the extreme values d → 0 which corresponds to the piecewise constant temperature [Eq. (28)] with ∆T = 1/2 and d → ∞ where the profile becomes sinusoidal [Eq. 24] with A = 1/2, see Fig. 3(a). The optimal potential for different values of the parameter d is shown in Fig. 3(b). For d ≪ 1, two peaks emerge in the optimal potential at the positions of the temperature discontinuities. In between these two peaks, the potential decreases linearly. The current diverges for d → 0, see inset of Fig. 3(b).
A. Numerical solution for a piecewise constant temperature
We next investigate this complex shape of the potential and the divergent current in more detail by solving the problem numerically on a discrete lattice. The goal is to minimize the inverse current [Eq. (11)] for the piecewise constant temperature [Eq. (28)]. The periodic boundary condition [Eq. (15)] for the potential transforms to the condition (30) considering that φ(0) = 0 by definition [Eq. (10)]. For a numerical solution we discretize φ(x) → φ(x i ) ≡ φ i with i = 0, . . . , N and search for a global minimum of j −1 (φ i ) in this N + 1-dimensional space with a simplex algorithm proposed by Nelder and Mead [39]. The boundary values φ 0 and φ N are given according to Eqs. (10, 30) by
φ(1) = φ(1/2) 2∆T ∆T − 1 + f 1 − ∆Tφ 0 = 0,(31)φ N = φ N/2 2∆T ∆T − 1 + f 1 − ∆T (32)
and φ 1 , . . . , φ N −1 are varied to yield a maximum current. Using Eq. (10), we then calculate the potential V (x) from the optimal φ(x). The numerical solution for the optimal potential depends on the discretization which determines how large the gradient of the temperature and the potential can be, see Fig. 3(c). For finer discretization, the optimal potential shows larger gradients. As a consequence, we find that the current j num (N ) diverges with increasing discretization N → ∞, see Fig. 3(d). This is consistent with the developing divergence visible in Fig. 3(b).
B. Origin of the divergent current
Due to the lattice discretization, the temperature profile can be considered to be linear with gradients with absolute value 1/(2ǫ), see Fig. 4. We calculate the optimal inverse current [Eq. (21)] for such a temperature profile
j −1 (λ, ǫ) = λ 2 + 16λǫ 2 + √ λ(1 − 4ǫ).(33)
The Lagrange multiplier λ and the discretization parameter ǫ are related by the corresponding constraint [Eq. if λ → 0. For an approximated dependence λ(ǫ), we use an iterative method and obtain a sequence of functions
λ 2 4ǫ 2 + 4λ − 1 + 16ǫ 2 λ + 4ǫ − 1 √ λ +f = 0.(34)λ n+1 (ǫ) = 1 (ln |λ n (ǫ)| − 2 ln |2ǫ| − 1 + f ) 2(35)
with λ 0 = 1. For the first members we calculate the current [Eq. (33)] as a function of ǫ. These results are compared to the numerical current, where the discretization 1/N corresponds to 2ǫ, see Fig. 3(d). The divergent behavior of the current obtained from perturbation theory is in good agreement with the divergence from the numerics. The discrepancy is on one hand due to the numerical integration on the lattice in Eq. (11) and on the other hand due to the approximations implicit in the perturbation method. More insight into the origin of the divergent current can be gained from analyzing the probability distribution [Eq. (9)] which can be calculated from the Fokker-Planck equation for a model potential. Guided by the optimal potential obtained from the numerics, see Fig. 3(c), we consider a sawtooth potential with a superimposed finite peak at the temperature discontinuity at x = 0.5. Note that we use this potential only as a case study aiming at a deeper understanding of the divergence of the current. The full optimal potential has a second peak at x = 0 and both the height of the peaks and the linear slopes between the peaks diverge. A finite peak potential is not a δ-function, but can be considered as a result of a limiting process from a triangular shape, see Fig. 6. It is important that the peak arises at the discontinuity of the temperature profile, namely that the rising side of the peak is in one temperature area and the other side is in the other temperature area. Otherwise the peak would not contribute to the integrals in the current. Since the peak has infinitesimally small width, the forward rate is not decreased by a higher peak potential, contrary to a rising barrier with finite width. A finite peak in a constant temperature area would not contribute to integrals and thus would have no effect on mean first passage times. In contrast, for a dichotomous temperature profile and a finite peak at the interface, mean first passage times in-volving a crossing of the peak are affected by its height. The backward mean first passage time increases exponentially with the peak height. In contrast and somewhat counter-intuitively, the forward mean first passage time decreases with increasing peak height due to the strong suppression of peak recrossings from the cold side.
1/2 1/2 T h T c T h T c U U
A finite peak with height U superimposed on a flat potential causes a depletion of the particles on the high temperature side and an accumulation on the other side. With increasing peak height U , this behavior saturates, see Fig. 5(a). In a sawtooth potential, the particles accumulate in front of the barriers, see Fig. 5(b). If we combine both effects by superimposing a finite peak on a sawtooth potential, the depletion can compensate the accumulation, see Fig. 5
(b).
The compensation of the accumulation is the main reason for the divergent current. For a sawtooth potential with large amplitude, the particles would accumulate in front of the barriers. The peaks counteract this effect and allow the particles to overcome the barriers. In between the peaks, the particles (on average) follow the steep potential with large mean velocity. For finer discretization, the peaks get larger and the potential in between steeper corresponding to a stronger force which increases the mean velocity and thus the current.
This behavior is only valid under the assumption of an overdamped dynamics where the particle instantaneously adjusts its velocity according to the Langevin equation (4). In the underdamped case, the particle needs a certain distance to reach the velocity corresponding to the local force. The overdamped limit is a good approximation as long as the relaxation time of the momentum τ R ≡ m/γ is small compared to the (mean) time τ c it takes for the particle to cross a region with basically constant force. Considering the potential above with discretization ǫ, both the peaks and the linear slopes be-tween the peaks should not become too large for the overdamped limit to be appropriate. The peaks constitute the largest slope in the optimal potential and therefore are crucial for the appropriateness of the overdamped description. Considering a peak with height U and width 2ǫL, the relaxation time τ R should be small compared to τ c = ǫL/v s where v s = |V ′ /γ| = U/(ǫLγ) is the (mean) stationary velocity. Therefore, the overdamped limit is approriate for
m γ ≪ L 2 ǫ 2 γ U or m γ 2 L 2 ≪ ǫ 2 U .(36)
For a given value of m/(γ 2 L 2 ), the discretization ǫ thus cannot become too small for the overdamped limit to be still valid. Since the divergence of the current occurs with decreasing values of ǫ, the current presumably does not diverge in any realistic system, where the underlying dynamics is underdamped. This question, however, is hard to decide conclusively, since the current for underdamped dynamics can only be determined numerically for a given potential. The optimization of the potential (with an infinite number of degrees of freedom) is a computationally difficult task to be reserved to future work. For a colloidal particle of radius R ≃ 1µm in a temperature profile with T 0 ≃ 300K and periodicity L ≃ 50µm, we get
m γ 2 L 2 ≃ 2 · 10 −11 1 k B T 0(37)
where we have used Stokes friction γ = 6πηR and the mass m = 4πρR 3 /3 with viscosity η ≃ 10 −3 Pa s and density ρ ≃ 10 3 kg/m 3 . Even for the finest discretization ǫ ≃ 5 · 10 −4 used in our study, we have ǫ 2 /U ≃ 3 · 10 −8 /k B T 0 and condition (36) is fulfilled. The overdamped description thus is valid for the optimal potential for any realistic (finite) temperature gradient. The (large) currents shown in Fig. 3(d) thus are in principle observable in experiments, provided the necessary temperature jump can be generated on the scale of 2ǫ · L ≃ 10 −3 · 50µm ≃ 50nm. A genuine divergence of the current, on the other hand, may be prohibited by the onset of inertia effects.
VI. CONCLUSIONS
Using variational calculus, we have developed a method to calculate the optimal potential which maximizes the current in ratchet heat engines for a continuous temperature profile. In the load free case, we have shown that the maximum current depends linearly on the amplitude of the temperature profile.
In the case of a piecewise constant temperature, the current diverges for the optimal potential consisting of steep linear parts and peaks at the boundaries which induce a long range effect in the probability distribution. However, the underlying assumption of an overdamped dynamic is limited by the slopes of the optimal potential presumably resulting in a bounded current for a particle with finite mass. In addition, a piecewise constant temperature is an artificial model rather than physically feasible. For any future nano-or microfluidic realization, the temperature gradients will be finite and thus the current.
In principle, external potentials for a Brownian particle can be realized by laser traps. However, it might be very difficult to model the optimal potential in every detail. In particular for the dichotomous temperature profile, it is clear that finite peaks must be approximated by a barrier with a finite width in any experiment. In order to estimate the observable velocity, we consider a colloidal particle with radius R trapped in the optimal potential for a sinusoidal temperature profile. In recent experiments, temperature gradients ∆T /L ≃ 10 5 K/m have been generated [25]. In the load free case, the optimal dimensionless current is roughly ≃ 2A, see Fig. 2(b), where A is the scaled temperature amplitude ∆T /T 0 . With Eq. (7) we get a rough estimate for the stationary velocity
v ≃ k B ∆T 3πηRL ≃ 100 R/nm nm s ,(38)
under the assumption of Stokes friction with viscosity η ≃ 10 −3 Pa s. Although such a transport effect is small at a micrometer scale, future realisations at a nanometer scale will yield observable velocities.
Our study can be extended in several directions. In principle, it would be interesting to calculate the efficiency at maximum power for our model and compare it to previous results where only the load force was optimized. However, it is difficult to define efficiencies for continuous temperature profiles. Moreover, Hondou and Sekimoto pointed out in Ref. [40] that heat transfer cannot be treated appropriately within the overdamped Langevin equation. For a similar model of a ratchet heat engine, a heuristic argument was used to propose a potential which leads to a large Peclet number [41]. It would be interesting to see whether our approach can be used to calculate the optimal potential with respect to a maximal Peclet number.
While ratchet heat engines have not been realized experimentally yet, the recent successful generation of large temperature gradients [23,24,25] may facilitate the construction of microscopic heat engines. By using our results, such nanomachines may then be tuned to produce maximum power.
j −1 (λ, c) = 1 2 1 0 4 c T + λ + λ 2 T ′2 dx(39)
with the constraint [Eq. (22)] for λ and c n(λ, c) =
1 0 T ′ ln λT ′ + 4 c T + λ + λ 2 T ′2 dx + 1 0 −2 dx λT ′ + 4( c T + λ) + λ 2 T ′2 + f = 0. (40)
Note that the first term in Eq. (22) vanishes for T ′ (0) = T ′ (1) and T (0) = T (1). By introducing k(λ, c, µ) ≡ j −1 (λ, c) − µ n(λ, c)
with Lagrange multiplier µ we formulate a minimization problem under a constraint. For the optimal parameters c, λ, µ which minimize the optimal inverse current [Eq. (39)] and fulfill the constraint [Eq. (40)], ∂k ∂c , ∂k ∂λ and n(λ, c) have to vanish. These equations cannot be solved analytically. Nevertheless for c = 0 we have ∂k ∂c c=0 = (λ − µ) 1 0 dx λT (x) 4λ + λ 2 T ′2 (x) (42) which vanishes for λ = µ. The partial derivative ∂k ∂λ λ=µ,c=0 = − 1 2
1 0 T ′ (x) dx(43)
also vanishes for a periodic temperature profile. Thereby, we reduce the problem to a one dimensional root search of n(λ, c)| c=0 which can easily be done numerically. Thus, we find one solution but we cannot exclude rigorously that there exist other solutions for this minimization problem. In our case studies, we find (by a comparison to numerical optimization) that the solution with c = 0 indeed is a global minimum. This strongly suggests that it generally is the relevant solution.
B. Perturbation theory for the divergent current
We approximate the dependence λ(ǫ) for a temperature profile as shown in Fig. 4
λ(ǫ) = 1 (ln | λ(ǫ) 4ǫ 2 | − 1 + f ) 2 .(49)
In perturbation theory, iteration is a standard method for algebraic equations [42]. In this way we search for a fixed point which is a solution for the equation. We iterate Eq. (49) λ n+1 (ǫ) = 1 (ln |λ n (ǫ)| − 2 ln |2ǫ| − 1 + f ) 2 (50) and choose
λ 0 = 1(51)
for a good convergence. More iterations lead to λ 1 (ǫ) = 1 (2 ln |2ǫ| + 1 − f ) 2 (52) λ 2 (ǫ) = 1 (2 ln |2 ln |2ǫ| + 1 − f | + 2 ln |2ǫ| + 1 − f ) 2 (53) and so on. Note that this sequence converges slowly but still gives an idea how λ(ǫ) behaves. For each λ we obtain a current from Eq. (33). The convergence of this sequence of currents is shown in Fig. 3(d).
[1] P. Reimann, Phys. Rep. 361, 57 (2002).
[2] R. D. Astumian and M. Bier, Phys. Rev. Lett. 72, 1766
potential V (x) and T (x) = 1 2 sin(2πx) + 1 for load f = −0.05. (b) Optimal potentials V (x) for T (x) = A sin(2πx) + 1 with different amplitudes A and load f = 0. Inset: Optimal current j versus A. (c) Optimal potentials V (x) for T (x) = A sin(2πx) + 1 with different amplitudes A and load f = −0.05. Inset: Optimal current j versus A. (d) Optimal potentials V (x) for T (x) = 1 2 sin(2πx) + 1 and for different loads f . Inset: Optimal current j(f ) and power P (f ) as a function of the force. The power exhibits a maximum Popt ≃ 0.52 at fopt ≃ −1.27. A. Optimal potential for different amplitudes of the temperature profile
FIG. 3 :
3In the appendix, we show with a perturbation method that for ǫ → 0, the constraint [Eq.(34)] can be fulfilled (a) Continuous approximation T (x) [Eq.(29)] to a piecewise constant temperature for different parameters d.In the limit d → ∞, T (x) approaches the sinusoidal profile T (x) = 1 2 sin(2πx) + 1. (b) Optimal potentials V (x) for T (x) from Fig. 3(a) with external force f = −0.05. Inset: current j versus parameter d. (c) Optimal potentials obtained from the numerical minimization on a discrete lattice for a piecewise constant temperature profile T (x) [ Eq. (28)] with ∆T = 1/2 and with an external force f = −0.05 for different discretization N . (d) For an external force f = −0.05, optimal current jnum as obtained from the numerics with discretization 2ǫ = 1/N compared to the perturbative calculation developed in the appendix.
FIG. 4 :
4Temperature profile on the lattice with gradients depending on ǫ.
FIG. 5 :
5(a) Probability distributions for a rising peak with height U in a piecewise constant temperature T (x) [Eq. (28)] with ∆T = 1/2 and load f = −0.05. (b) Sawtooth potential with superimposed peak and its corresponding probability distribution in a piecewise constant temperature T (x) [Eq. (28)] with ∆T = 1/2 and load f = −0.05 for different peak heights.
FIG. 6 :
6Peak as a limiting process. Each side is in one temperature region.
VII. APPENDIX A. Optimization of the current with respect to the remaining free parameters λ and c With I − (x) from Eq. (19), the inverse current [Eq. (21)] reads
in the limit ǫ → 0. The relates λ with ǫ. We define h(ǫ) by For ǫ → 0, the right hand side of Eq. (46) diverges and thus the left hand side must also diverge, yielding h → ∞. We consider the leading terms in Eq. (46) (ln |h| − 1 + f ) 2 .(48)We use Eq. (45) to obtain the correspondingconstraint [Eq. (22)]
ln 1 +
λ
8ǫ 2 +
1
4ǫ
λ 2
4ǫ 2 + 4λ − 1 +
16ǫ 2
λ
+
4ǫ − 1
√
λ
+f = 0
(44)
λ(ǫ) ≡ 4ǫ 2 h(ǫ)
(45)
and from Eq. (44) follows
√
h ln |1+
h
2
+
1
2
h 2 + 4h|−
√
h + 4+2+f
√
h =
1
2ǫ
. (46)
√
h(ln |h| − 1 + f ) =
1
2ǫ
(47)
and rewrite it in a self consistent equation
h =
1
4ǫ 2
AcknowledgmentsWe would like to thank A. Gomez-Marin and R. Finken for inspiring discussions.
. L Faucheux, A Libchaber, J. Chem. Soc. Faraday Trans. 913163L. Faucheux and A. Libchaber, J. Chem. Soc. Faraday Trans. 91, 3163 (1995).
. J Rousselet, L Salome, A Ajdari, J Prost, Nature. 370446J. Rousselet, L. Salome, A. Ajdari, and J. Prost, Nature 370, 446 (1994).
. N Cordova, B Ermentrout, G F Oster, Proc. Natl. Acad. Sci. U.S.A. 89339N. Cordova, B. Ermentrout, and G. F. Oster, Proc. Natl. Acad. Sci. U.S.A. 89, 339 (1992).
. F Jülicher, A Ajdari, J Prost, Rev. Mod. Phys. 691269F. Jülicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69, 1269 (1997).
. R D Astumian, P Hänggi, Physics Today. 551133R. D. Astumian and P. Hänggi, Physics Today 55(11), 33 (2002).
. Z C Tu, J. Phys. A: Math. Gen. 41312003Z. C. Tu, J. Phys. A: Math. Gen. 41, 312003 (2008).
. T Schmiedl, U Seifert, EPL. 8330005T. Schmiedl and U. Seifert, EPL 83, 30005 (2008).
. J M R Parrondo, G P Harmer, D Abbott, Phys. Rev. Lett. 855226J. M. R. Parrondo, G. P. Harmer, and D. Abbott, Phys. Rev. Lett. 85, 5226 (2000).
. P Amengual, A Allison, R Toral, D Abbott, Proc. Roy. Soc. London A. 4602269P. Amengual, A. Allison, R. Toral, and D. Abbott, Proc. Roy. Soc. London A 460, 2269 (2004).
. L Dinis, Phys. Rev. E. 7721124L. Dinis, Phys. Rev. E 77, 021124 (2008).
. M B Tarlie, R D Astumian, Proc. Natl. Acad. Sci. U.S.A. 952039M. B. Tarlie and R. D. Astumian, Proc. Natl. Acad. Sci. U.S.A. 95, 2039 (1998).
. S J Lade, J. Phys. A: Mathematical and Theoretical. 41275103S. J. Lade, J. Phys. A: Mathematical and Theoretical 41, 275103 (2008).
. T Schmiedl, U Seifert, EPL. 81T. Schmiedl and U. Seifert, EPL 81, 20003 (2008).
. S M Bezrukov, A M Berezhkovskii, A Szabo, J. Chem. Phys. 127115101S. M. Bezrukov, A. M. Berezhkovskii, and A. Szabo, J. Chem. Phys. 127, 115101 (2007).
. F J Cao, L Dinis, J M R Parrondo, Phys. Rev. Lett. 9340603F. J. Cao, L. Dinis, and J. M. R. Parrondo, Phys. Rev. Lett. 93, 040603 (2004).
. E M Craig, N J Kuwada, B J Lopez, H Linke, Annalen der Physik. 17115E. M. Craig, N. J. Kuwada, B. J. Lopez, and H. Linke, Annalen der Physik 17, 115 (2008).
R P Feynman, R B Leighton, M Sands, The Feynman Lectures on Physics. Reading, MAAddison-WesleyR. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, MA, 1966).
. M Büttiker, Z. Phys. B. 68161M. Büttiker, Z. Phys. B 68, 161 (1987).
. R Landauer, J. Stat. Phys. 53233R. Landauer, J. Stat. Phys. 53, 233 (1988).
. Y M Blanter, M Büttiker, Phys. Rev. Lett. 814040Y. M. Blanter and M. Büttiker, Phys. Rev. Lett. 81, 4040 (1998).
. S Duhr, D Braun, Phys. Rev. Lett. 96168301S. Duhr and D. Braun, Phys. Rev. Lett. 96, 168301 (2006).
. S Duhr, D Braun, Proc. Natl. Acad. Sci. U.S.A. 10319678S. Duhr and D. Braun, Proc. Natl. Acad. Sci. U.S.A. 103, 19678 (2007).
. F M Weinert, J A Kraus, T Franosch, D Braun, Phys. Rev. Lett. 100164501F. M. Weinert, J. A. Kraus, T. Franosch, and D. Braun, Phys. Rev. Lett. 100, 164501 (2008).
. A Barreiro, R Rurali, E Hernández, J Moser, T Pichler, L Forró, A Bachtold, Science. 320775A. Barreiro, R. Rurali, E. Hernández, J. Moser, T. Pich- ler, L. Forró, and A. Bachtold, Science 320, 775 (2008).
. C Jarzynski, O Mazonka, Phys. Rev. E. 596448C. Jarzynski and O. Mazonka, Phys. Rev. E 59, 6448 (1999).
. M Matsu, S Sasa, Physica A. 276188M. Matsu and S. Sasa, Physica A 276, 188 (2000).
. R Benjamin, R Kawai, Phys. Rev. E. 7751132R. Benjamin and R. Kawai, Phys. Rev. E 77, 051132 (2008).
. F L Curzon, B Ahlborn, Am. J. Phys. 4322F. L. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975).
. C Van Den Broeck, Phys. Rev. Lett. 95190602C. Van den Broeck, Phys. Rev. Lett. 95, 190602 (2005).
. M Asfaw, M Bekele, Eur. Phys. J. B. 38457M. Asfaw and M. Bekele, Eur. Phys. J. B 38, 457 (2004).
. A Gomez-Marin, J M Sancho, Phys. Rev. E. 7462102A. Gomez-Marin and J. M. Sancho, Phys. Rev. E 74, 062102 (2006).
. M Asfaw, Eur. Phys. J. B. 65109M. Asfaw, Eur. Phys. J. B 65, 109 (2008).
. J M Sancho, M S Miguel, D Dürr, J. Stat. Phys. 28291J. M. Sancho, M. S. Miguel, and D. Dürr, J. Stat. Phys. 28, 291 (1982).
. R Kupferman, G A Pavliotis, A M Stuart, Phys. Rev. E. 7036120R. Kupferman, G. A. Pavliotis, and A. M. Stuart, Phys. Rev. E 70, 036120 (2004).
. T Schmiedl, U Seifert, Phys. Rev. Lett. 98108301T. Schmiedl and U. Seifert, Phys. Rev. Lett. 98, 108301 (2007).
. A Gomez-Marin, T Schmiedl, U Seifert, J. Chem. Phys. 12924114A. Gomez-Marin, T. Schmiedl, and U. Seifert, J. Chem. Phys. 129, 024114 (2008).
. J Nelder, R Mead, Computer J. 7308J. Nelder and R. Mead, Computer J. 7, 308 (1965).
. T Hondou, K Sekimoto, Phys. Rev. E. 626021T. Hondou and K. Sekimoto, Phys. Rev. E 62, 6021 (2000).
. B Lindner, L Schimansky-Geier, Phys. Rev. Lett. 89230602B. Lindner and L. Schimansky-Geier, Phys. Rev. Lett. 89, 230602 (2002).
Pertubation Methods (Cambridge texts in applied mathematics. E J Hinch, chap. 1.1E. J. Hinch, Pertubation Methods (Cambridge texts in applied mathematics, 1991), chap. 1.1.
| [] |
[
"Causality and time order -relativistic and probabilistic aspects",
"Causality and time order -relativistic and probabilistic aspects"
] | [
"Michał Eckstein \nInstitute of Theoretical Physics\nJagiellonian University\nul. prof. Stanisława Łojasiewicza 1130-348KrakówPoland\n\nCopernicus Center for Interdisciplinary Studies\nul. Szczepańska 131-016KrakówPoland\n",
"Michael Heller \nCopernicus Center for Interdisciplinary Studies\nul. Szczepańska 131-016KrakówPoland\n\nVatican Observatory\nV-00120Vatican City State\n"
] | [
"Institute of Theoretical Physics\nJagiellonian University\nul. prof. Stanisława Łojasiewicza 1130-348KrakówPoland",
"Copernicus Center for Interdisciplinary Studies\nul. Szczepańska 131-016KrakówPoland",
"Copernicus Center for Interdisciplinary Studies\nul. Szczepańska 131-016KrakówPoland",
"Vatican Observatory\nV-00120Vatican City State"
] | [] | We investigate temporal and causal threads in the fabric of contemporary physical theories with an emphasis on empirical and operationalistic aspects. Building on the axiomatization of general relativity proposed by J. Ehlers, F. Pirani and A. Schild (improved by N. Woodhouse) and the global space-time structure elaborated by R. Penrose, S.W. Hawking, B. Carter and others, we argue that the current way of doing relativistic physics presupposes treating time and causality as primitive concepts, neither of them being 'more primitive' than the other. The decision regarding which concepts to assume as primitive and which statements to regard as axioms depends on the choice of the angle at which we contemplate the whole. This standard approach is based on the presupposition that the concept of a point-like particle is a viable approximation. However, this assumption is not supported by a realistic approach to doing physics and, in particular, by quantum theory. We remove this assumption by analysing the recent works by M. Eckstein and T. Miller. They consider the space P(M ) of probability measures on space-time M such that, for an element µ ∈ P(M ), the number µ(K) specifies the probability of the occurrence of some event associated with the space-time region K and the measure µ. In this way, M is not to be regarded as a collection of space-time events, but rather as a support for corresponding probability measures. As shown by Eckstein and Miller, the space P(M ) inherits the causal order from the underlying space-time and facilitates a rigorous notion of a 'causal evolution of probability measures'. We look at the deductive chains creating temporal and causal structures analysed in these works, in order to highlight their operational (or quasi-operational) aspect. This is impossible without taking into account the relative frequencies and correlations observed in relevant experiments. | null | [
"https://arxiv.org/pdf/2202.07302v1.pdf"
] | 246,863,917 | 2202.07302 | 6ca11e891cfa5096fc7a9c48e48201b26ba19b26 |
Causality and time order -relativistic and probabilistic aspects
February 16, 2022
Michał Eckstein
Institute of Theoretical Physics
Jagiellonian University
ul. prof. Stanisława Łojasiewicza 1130-348KrakówPoland
Copernicus Center for Interdisciplinary Studies
ul. Szczepańska 131-016KrakówPoland
Michael Heller
Copernicus Center for Interdisciplinary Studies
ul. Szczepańska 131-016KrakówPoland
Vatican Observatory
V-00120Vatican City State
Causality and time order -relativistic and probabilistic aspects
February 16, 2022
We investigate temporal and causal threads in the fabric of contemporary physical theories with an emphasis on empirical and operationalistic aspects. Building on the axiomatization of general relativity proposed by J. Ehlers, F. Pirani and A. Schild (improved by N. Woodhouse) and the global space-time structure elaborated by R. Penrose, S.W. Hawking, B. Carter and others, we argue that the current way of doing relativistic physics presupposes treating time and causality as primitive concepts, neither of them being 'more primitive' than the other. The decision regarding which concepts to assume as primitive and which statements to regard as axioms depends on the choice of the angle at which we contemplate the whole. This standard approach is based on the presupposition that the concept of a point-like particle is a viable approximation. However, this assumption is not supported by a realistic approach to doing physics and, in particular, by quantum theory. We remove this assumption by analysing the recent works by M. Eckstein and T. Miller. They consider the space P(M ) of probability measures on space-time M such that, for an element µ ∈ P(M ), the number µ(K) specifies the probability of the occurrence of some event associated with the space-time region K and the measure µ. In this way, M is not to be regarded as a collection of space-time events, but rather as a support for corresponding probability measures. As shown by Eckstein and Miller, the space P(M ) inherits the causal order from the underlying space-time and facilitates a rigorous notion of a 'causal evolution of probability measures'. We look at the deductive chains creating temporal and causal structures analysed in these works, in order to highlight their operational (or quasi-operational) aspect. This is impossible without taking into account the relative frequencies and correlations observed in relevant experiments.
Introduction
It is perhaps commonplace to attribute the relational concept of time to Leibniz. Every philosopher knows that, according to Leibniz, time is but a relation ordering events one after the other. It is less known that at the end of his life Leibniz supplemented his relational conception of time with what later resulted in the causal theory of time 1 . His new idea attempted to identify the nature of relations constituting the time order. When we read Leibniz on ordering relations, we should not ascribe to him our present concept of formal order, since his own aim was to stress the difference between his own understanding and that of his English opponents (Newton and Clarke). In the 'English theory', there exist two classes of entities: instances and events, and the 'natural order' of instances 2 determines the order of events. Events happen at certain instances, but instances are independent of events. In Leibniz's approach, the events are the only class of entities, and time is a derived concept given by relations ordering events. The causal conception of time adds a new idea to Leibniz's philosophy. In his metaphysical-literary style: 'the present is always pregnant with the future' [Leibniz, 1969, p. 557], and as explained by Mehlberg: '. . . if one arranges phenomena in a series such that every term contains the reason for all those which come after it in the series, the causal order of the phenomena so defined will coincide with their temporal order of succession.' [Mehlberg, 1980, p. 46]. Leibniz's idea contrasts with that of Hume who attempted to reduce causal order to temporal order. Whitrow puts this in the following way: in Hume's view 'the only possible test of cause and effect is their 'constant union', the invariable succession of the one after the other' [Whitrow, 1980, p. 323].
The advent of the theory of relativity showed the role played by time and causality in the structure of modern physics in a radically new light and has had a profound impact on our understanding of the world. The aim of the present study is to confront the traditional dispute concerning the mutual relationship between time and causality with what contemporary physical theories import to this issue with an emphasis on the empirical and operationalistic aspects. Our approach consists not so much in analysing separate problems involved in time-causality interaction, but rather in investigating temporal and causal threads in the fabric of contemporary physical theories.
The structural construction of a physical theory is best visible in its axiomatisation. We thus start with an axiomatization of the theory of relativity. Perhaps the earliest such axiomatization is attributed to A. A. Robb [Robb, 1914] who in 1914 proposed an axiomatic system based on the concept of 'conic order', and was able to derive both topological and metric properties of space-time from the 'invariant succession of events'. In the axiomatic systems of both Carnap [Carnap, 1925] and Reichenbach [Reichenbach, 1924] it is the temporal order that is reduced to that of causal order. The same is true for the Mehlberg's system [Mehlberg, 1980] which 1 The note concerning this conception is found in Leibniz's essay entitled 'The metaphysical foundations of mathematics' [Leibniz, 1989] and published posthumously. The present section is based mainly on chapter one of the extensive study by Henry Mehlberg devoted to the causal theory of time [Mehlberg, 1980]. 2 We would today say the order determined by the metric structure of time.
was elaborated within the broader setting of an interdisciplinary study of the causal theory of time 3 . The latter three authors were associated with logical empiricism and, in agreement with its philosophical ideology, aimed at clarifying logically the conceptual situation in the theory of relativity, one which was extensively discussed at that time. In doing so, they eliminated, with the help of their axioms, some 'pathological situations' (such as the existence of closed timelike curves). What they did not take into account was that the later development of physics might need such pathologies, if not for empirical reasons then as auxiliary hypotheses for proving some general theorems. This became evident when in 1949 Kurt Gödel [Gödel, 1949] published his solution to Einstein's field equations with closed timelike curves (soon after more solutions with similar 'time anomalies' were found). A few years later the first general theorem concerning the global structure of space-time was proved stating that every compact space-time must contain closed timelike curves [Bass and Witten, 1957].
In the present study, we take into account both these lines of investigation. We adopt the axiomatization proposed by J. Ehlers, F. Pirani and A. Schild [Elhers et al., 1972] which is considered the most adequate axiomatization of general relativity (in the following, we refer to it as to EPS axiomatization). Its purpose is not so much to 'clarify inter-conceptual relations', as it was in the case with previous axiomatizations, but rather to reveal the structuring of space-time (i.e. showing mutual relations between its various substructures). In Section 2, we briefly present the EPS system with an emphasis on its quasi-operational aspect that we then exploit when discussing the empirical anchoring of space-time physics.
The second thread of investigation, mentioned above, resulted into the so-called global study of space-time structures. Within this approach the causal structure of space-time has been worked out in great detail (see, for instance, [Carter, 1971], [Hawking and Ellis, 1973, Chapter 6]). This approach entered into relativistic physics as a part of its theoretical tool-kit. These tools were used by N. M. J. Woodhouse [Woodhouse, 1973] to improve the EPS axiomatization. In Section 3, we refer to some aspects of this approach, mainly to prepare the conceptual environment for our further analysis.
In Section 4, we draw some conclusions from the above analysis. The current way of doing relativistic physics presupposes treating time and causality as 'primitive concepts'. They are mutually interrelated, but neither of them is 'more primitive' than the other, and the decision which concepts to assume as primitive and which statements to regard as axioms depends on the choice of the angle at which we contemplate the whole.
Everything said above is based on the presupposition that the concept of a point-like space-time event is a viable approximation. However, such an assumption cannot be fulfilled in actual physical experiments. Moreover, in a fundamentally probabilistic theory, such as quantum mechanics, there are no definite events at all. Even in classical theory space and time localization measurements are affected by experimental uncertainties, and in quantum theory statistical predictions apply also to single events. In Section 5, we take this into account by pondering the space P(M ) of all probability measures on space-time M . For any element µ ∈ P(M ) and a space-time region K the number µ(K) is the probability of the occurrence of some event. In this way, M is not to be regarded as a collection of space-time events, but rather as a support for corresponding probability measures.
The question arises: when do 'probabilistic uncertainties' propagate causally in space-time? More precisely, when can the detection statistics µ causally influence the detection statistics ν? To answer this question we briefly present, in Section 6, a generalization of the relativistic causal structure to 'events' spread in space-time and represented by suitable probability measures. This generalisation was recently proposed by Eckstein and Miller [Eckstein and Miller, 2017b]. For two probability measures µ, ν ∈ P(M ), the measure µ is said to causally precede ν, symbolically µ ν, if for every compact set K ⊂ supp µ we have µ(K) ≤ ν(J + (K)), where J + (K) is the causal future of K.
In order to discuss the time-evolution of such measures meaningfully one has to specify a time parameter. Because the measures are inherently non-local objects, one needs to chose a global time-function T : M → R. To this end, one restricts oneself to the class of globally hyperbolic spacetimes which admit well-posed evolution problems with initial data specified on a Cauchy hypersurface S. Then, one considers a family of probability measures supported on the corresponding time-slices, µ t ∈ P(S t ), with S t = {t} × S ⊂ M and t ∈ R. Quite naturally, one says that a time-evolution of measures {µ t } t is causal if µ s µ t for all s ≤ t. Remarkably, as shown by Miller [Miller, 2017, Miller, 2021, there exists an invariant object, call it [σ], which is defined in terms of probability measures on worldlines in a given hyperbolic space-time, and the evolution of measures {µ t } t is causal if and only if such an invariant object exists. Some philosophical aspects of this formalism are discussed in Section 7.
Finally, in Section 8, we collect some comments regarding the analyses carried out in this work. Our guiding idea was to preform all these analyses from an operational or, where this was not possible, 'operational-in-principle' point of view. In the final section we look once again at the deductive chains of our reasoning and the conclusions to which they lead, in order to highlight the operational, or quasi-operational, aspect of them. As we have tried to show, certain fundamental properties of time, space and causality and their interrelationships are enforced by the very method of physics and by the fact that its very essence consists in its empirical approach to the reality under study. It is in the interest of this method that the quasi-operational elements of the method be replaced, as far as possible, by truly operational elements. The basic methodology of experimental physics is based on probabilistic concepts. Therefore, the generalization of the relativistic causal structure to probabilistic measures, presented in the final sections of this work, is to be welcomed as a step in the right direction.
Space-Time Architecture
From the mathematical point of view, a space-time in general relativity is a pair (M, g), where M is a four-dimensional differential manifold, and g a Lorentzian metric defined on it. Furthermore, it is typically assumed that (M, g) is connected and time-oriented [Wald, 1984].
We say that the manifold M carries a Lorentzian structure. This structure not only contains several other mathematical substructures which interact with each other, creating a subtle hierarchical edifice, but also admits, on each of its levels, a physical interpretation, making out of the whole one of the most beautiful models of contemporary physics. We shall briefly describe the 'internal design' of this model. Our analysis will be based on the classic paper by Ehlers, Pirani and Schild [Elhers et al., 1972] in which these authors presented a quasi-operationistic axiomatic system for the Lorentz structure of space-time showing both its mathematical architecture and physical meaning.
The building blocks (primitive concepts) of this axiomatization are: (1) a set M = {p, q, . . .}, the elements of which are called events; and two collections of subsets of M : (2) L = {L 1 , L 2 , . . .}, the elements of which are called histories of light rays or of photons (light rays or photons, for the sake of brevity); (3) P = {P 1 , P 2 , . . .}, the elements of which are called histories of test particles or of observers (particles or observers, for brevity).
The axioms guarantee the following situations. A light is sent from an event p, situated on a particle history P 1 , and received at an event q, situated on a particle history P 2 (a message from p to q). The message can be reflected at q and sent to the event p , situated on the particle history P 1 (an echo on P 1 from P 2 ). By a suitable combination of messages and echoes one can ascribe four coordinates to any event, and construct local coordinate systems. Some mathematical gymnastics, sanctioned by suitable axioms, allows one to organise the set of events, M , into a differential manifold (with the usual manifold topology).
The next set of axioms equips the manifold M with a conformal metric. This is the usual metric (with a Lorentz signature) defined up to a factor of proportionality (conformal factor). This metric allows one to distinguish timelike, null (lightlike) and spacelike histories (curves), and completely determines the geometry of nullcurves.
To determine the geometry of timelike curves one needs additional axioms defining the so-called projective structure. Its task is to determine a distinguished class of timelike curves that physically are interpreted as representing histories of particles (or observers) moving with no acceleration (freely falling particles or observers).
Conformal and projective structures are, in principle, independent and they need to be synchronised. This is done with the help of suitable axioms which enforce null geodesics passing through an event p to form the light cone at p and projective timelike curves to fill in the interior of this light cone. If this is the case, we speak of the Weyl structure.
In a Weyl space (a manifold equipped with the Weyl structure), there is a natural method to define length -the arc length -along any timelike curve. Such a length is interpreted as a time interval measured by a clock carried by the corresponding particle (proper time of the particle). However, proper times of different particles are unrelated; to synchronise them a suitable metric must be introduced on M . This can be done with the help of the following axiom. Let p 1 , p 2 , ... be equidistant events on the history of a freely falling particle P , and let they be correspondingly simultaneous with events q 1 , q 2 , ... on the history of a freely falling particle Q. Simultaneity is understood here in the Einstein sense: two events, p and q, are simultaneous if an observer situated half-way between them sees the light signal emitted by p and q at the same instant as shown by the observer's clock. The metric structure is established if the events q 1 , q 2 , ... on Q are also equidistant. The metric g, constructed in this way, contains in itself the Weyl structure; it is, therefore a Lorentz metric. This completes the construction of the relativistic model of space-time. [Elhers et al., 1972].
It is important to stress that the axioms of general relativity, as formalised in [Elhers et al., 1972], can and have been tested against the empirical data. For example, the axiom on the conformal structure of space-time would be violated if the velocity of photons would depend upon its energy [Amelino-Camelia et al., 1998, Amelino-Camelia, 2013. No such effect has been observed to a high degree of accuracy [Abdo et al., 2009, Perlman et al., 2015, Pan et al., 2020. In the same vein, one could seek deviations from the universality of the projective structure by inspecting the dispersion relations of high-energy massive particles [Amelino-Camelia et al., 2016].
The existence of the Lorentzian structure on top of the Weyl structure could be undermined through the observation of the 'second clock effect' [Elhers et al., 1972]. The latter arises commonly in various modified-gravity theories (see e.g. [De Felice and Tsujikawa, 2010]). Recenly, the second clock effect was constrained using the CERN data on the muon anomalous magnetic moment [Lobo and Romero, 2018].
Eventually, one can question the model of a space-time as a smooth manifold. This is typically done on the basis of some quantum gravity theory [Oriti, 2009]. Although the possible effects of these models are extremely hard to observe, the experimental efforts in this direction are progressing [Amelino-Camelia, 2013]. In particular, some limits on space-time granularity were recently established [Chou et al., 2016].
Causal structure and global time
Every text-book on relativity tells us that causality (the causal structure) is identical with the cone structure of space-time, that is to say with the Weyl structure of the above scheme (synchronised projective and conformal structures). The conformal class of space-time metrics, [g], induces a classification of tangent vectors at any
event p ∈ M . A vector v ∈ T p M is timelike if g(v, v) < 0, spacelike if g(v, v) > 0, null if g(v, v) = 0 and causal if g(v, v) ≤ 0, where g is any representant of the class [g]
. The set of timelike vectors has two components: the 'future' and the 'past'. It is standard to assume [Wald, 1984] that the space-time is time-oriented (and not only time-orientable), which means that a choice of time-direction is fixed (smoothly) for all vectors throughout the tangent bundle T M . A piece-wise smooth curve γ :
I → M , with an affine parameter t ∈ I ⊂ R, is future-directed timelike/causal if γ (t) is future-directed timelike/causal, wherever the vector γ (t) is defined.
The Weyl structure of spacetime induces the chronology and causal relations between the events in M . An event p is said to chronologically (causally) precede an event q, written p q (p q, respectively), if there is a piece-wise smooth future-directed timelike (causal) curve from p to q 4 . Both of them determine channels through which causal influences can propagate rather than actual interactions between cause and effect. In particular, note that a causal piece-wise smooth curve can be composed of timelike and null segments. In terms of the EPS conceptual framework such a curve would correspond to a signal, which is, e.g., initially encoded in a massive particle and then gets transferred -via some interaction -to a photon.
With the help of these relations we define the chronological future I + (p) and the chronological past I − (p) of an event p as the set of all events which chronologically follow (resp. are followed by) p; and analogously, the causal future J + (p) and the causal past J − (p) of p.
If M is a space-time manifold, the tangent space at each of its points (events), T p M has the structure of the Minkowski space. Its causal structure is the familiar light cone structure of special relativity. This structure, essentially unchanged, is inherited by any local neighbourhood (called a normal neighbourhood) of the space-time manifold M . 5 However, globally (outside normal neighbourhoods) causal structure can be very different from that of Minkowski space, sometimes extremely exotic and full of pathologies (see [Carter, 1971, Minguzzi, 2019).
There exists an interaction between the causal structure of space-time and its topology. Sets I + (p) and I − (p) are always open, but sets J + (p) and J − (p) are not always closed. This allows one to define topology 'innate' for causal spaces (i.e. defined entirely in terms of causal relations). It is the so-called Alexandrov topology 6 . This topology is weaker than the usual manifold topology 7 .
This mathematical apparatus proved to be very effective in disentangling various problems related to the structure of space-time. Engaging it into subtleties of spacetime architecture allowed Woodhouse to improve the EPS axiomatization [Woodhouse, 1973]. He was able to derive the differential and causal structures from EPS-like axioms expressed in terms of chronology and causality relations. The main advantage of the Woodhouse approach is that no assumption has to be made concerning paths along which light signals propagate. They are deduced from statements regarding the emission and absorption of light signals. In his approach, one has to assume that there is exactly one history of a particle through each point in each direction in space-time. Physically this means that it is possible to define the history of a freely falling particle through any point in space-time. The above conceptual machinery provides a powerful tool for studying various global aspects of space-time. In what follows, we focus on those of them that elucidate mutual dependencies between time and causality.
The first condition that has to be implemented in order to have something resembling a temporal order is to guarantee the absence of closed timelike and causal curves. The corresponding conditions are called the chronology condition (the absence of closed timelike curves) and the causality condition (the absence of closed causal curves), respectively. The motivation is obvious: with such loops there is no clear distinction between the future and the past. Although the existence of relativistic world models with closed timelike curves (such as the Gödel's famous model [Gödel, 1949]) shows that the idea of 'closed time' is not a logical contradiction, yet the point is that the space-time structure strongly interacting with the rest of physics might be a source of many logical perplexities. Not only in a spacetime with closed timelike curves one might kill one's ancestor to prevent one's birth, but also -more prosaically -in a space-time with causality violations no global Maxwell field could exist that would match a given local field [Geroch and Horowitz, 1979].
There is a rich hierarchy of stronger and stronger conditions 8 which improve temporal and causal properties [Carter, 1971, Hawking and Ellis, 1973, Minguzzi and Sánchez, 2008. Among them there is an important condition, called the strong causality condition, that excludes almost closed causal curves 9 . In such a space-time, the nonexistence of closed timelike curves is satisfied with a certain safety margin. Owing to this margin, the Alexandrov topology improves to the manifold topology.
However, this is not enough. The fact that any measurement can only be done within certain unavoidable error limit, prevents us from measuring the space-time metric exactly. Many 'nearby metrics' are always within the measurement 'box error'. If measurements are to have any meaning at all -and the very existence of physics depends on this -we must postulate a certain stability of measurements. This postulate as regarding causality assumes the form of the stable causality condition; it precludes small perturbations of space-time metric to produce closed causal curves 10 . 8 In fact, the hierarchy is nondenumerable. 9 It states that each neighbourhood of any event in space-time contains a neighbourhood which no causal curve intersects more than once.
10 To define this condition precisely we should consider the space Lor(M ) of all Lorentz metrics on a given space-time manifold M , and equip Lor(M ) with a suitable topolgy. Only then are we able to determine what a small perturbation of a metric means [Hawking and Bondi, 1969].
The condition of stable causality therefore has a certain philosophical significance. When it is not satisfied, measurements of physical quantities become meaningless. As Hawking put it, 'Thus the only properties of space-time that are physically significant are those that are stable in some appropraite topology' [Hawking, 1971].
It is a nice surprise that this condition of 'physical reasonability' meets with another very 'reasonable' property. To Hawking we owe the following theorem: In a space-time M there exists global time, measured by a global time function, if and only if M is stably causal [Hawking and Bondi, 1969]. The history of any clock in the universe (for instance, the history of a vibrating particle) is a timelike curve in space-time M . Indications of such a clock can mathematically be represented by a monotonically increasing function along this curve -the time function for this clock. If there exists a single function T that is a time function for a family of clocks filling the space-time M , such a function T is called a global time function. It measures a global time in the universe. As we can see, there exists a deep connection between global time and the above mentioned measurement stability property. It shows that doing physics automatically requires both: global time and stable causality.
We should notice that the spacelike hypersurfaces T = const give surfaces of simultaneity in the universe, but they are not unique. However, one can 'synchronize the universe' by imposing a yet stronger causality condition. This is done in the following way. First, we define the Cauchy hypersurface of space-time M as a hypersurface S such that every inextendible timelike curve crosses S exactly once. This definition was first introduced in the theory of partial differential equations [Leray, 1953]. The initial data for a given hyperbolic PDE are given on a Cauchy surface. A space-time M is globally hyperbolic if and only if it can be presented as a product manifold M R × S, where R is the range of a global time-function and S a spacelike Cauchy surface in M (see [Geroch, 1970, Bernal andSánchez, 2006]). The global time function T , such that T −1 (t), t ∈ T is a Cauchy surface, is called a Cauchy time function.
Globally hyperbolic space-time is the closest to the Newtonian absolute space we can get within the framework of general relativity. It is deterministic in the sense that initial data given on a Cauchy surface determine, in principle, the entire history of the universe. However, the so-called Cauchy problem in general relativity, in all its mathematical details, is far from being simple and easy (see [Hawking and Ellis, 1973, Chapter 7] or [Ringström, 2009]).
Time and Causality as Primitive Notions
It is clear from the inspection of the axioms of general relativity that they presuppose some primitive notions of time and causality. In the EPS axiomatic system, echoes and messages are operationally defined in terms of coincidences of clock readings and acts of emission or reception of light rays. However, intuition smuggles into this operationistic picture an idea that the received echo is actually caused by the radar signal. It seems, therefore, that some elementary notion of causality underlies all of the EPS system.
The same must be said about the primitive notion of time implicitly employed in the EPS axioms. Indeed, each history of a particle or a photon carries a C 0structure which assures the local homeomorphism with R. This can be interpreted as a local flow of time with no preferred time orientation. On top of it, in order to make sense out of the 'echoes and messages' axioms, one has to assume that the signal is emitted before it is received. This means that each history of a particle or a photon is in fact a (totally) ordered collection of events with a distinguished future-direction. While the choice of a 'future' and a 'past' is purely conventional, fixing a consistent choice for all histories of particles amounts to assuming that the spacetime M is time-oriented. Therefore, the notion of local time flow and some elementary concept of causality are the minimum upon which a further hierarchy of stronger and stronger conditions is built. As we have seen, the stable causality condition plays a special role in this hierarchy. It guarantees both the existence of global time and allows for stable measurement results.
All the structures described above, suitably synchronized with each other, are contained in the Lorentz metric structure. It is a standard result that the Lorentz metric structure exists globally on a space-time manifold M if and only if a nonvanishing direction field exists on M . Moreover, the Lorentz metric can always be chosen in such a way to make this direction field timelike [Geroch, 1971]. Since such a field on a space-time manifold locally always exists, the same is true for the Lorentz metric. The existence of a Lorentz metric is strictly related to the possibility of performing space and time measurements; therefore, it is almost synonymous with the possibility of doing physics. Above, we have identified such a possibility with the existence a local 'topological time' (a C 0 -structure on each history of a test particle); here we have the same condition raised to the metric level. 11 A word of warning is needed at this point. Our conclusions are valid only within the conceptual framework of what we have called relativistic model of space time, and only within its reconstruction as it is presented above. Other axiomatic approaches to the geometry of space-time are possible (see, for, instance, [Andréka et al., 2013, Covarrubias, 1993, Guts, 1995) and they can give rise to different interpretations. 12 However, we should take into account the fact that it is the theory of general relativity that is deeply rooted in this model, and since this theory is very well founded on empirical data, it would be unwise to look for a different model (within the limits of its empirical verifications). We should also emphasise that the EPS axiomatic approach should not be easily replaced by other approaches since it renders justice, and does it very well, to both the 'theoretical practice' of mathematical physicists and the operational demands of experimentalists (it has a strong (quasi)operationistic flavour).
The above analysis shows that temporal and causal properties are strongly coupled with each other, and one cannot say which is logically (or ontologically) prior with respect to the other. They are unified in the Weyl structure to provide a basis for the full dressed concept of time and causality. In this sense, causal theory of time à la Leibniz (causality implies time) is not supported by the relativistic model considered here. The same should be said about an attempt, à la Hume, to reduce causal interactions to merely a temporal succession. It should be taken into account that axiomatic systems can be composed in various ways: various concepts can be selected as primitive and various statements can be accepted as axioms of the system, depending on criteria one adopts. The EPS axiomatic system has an advantage over other axiomatic systems that it is quasi-operational, i.e. its axioms describe some simple empirical procedures, although they do so in a highly idealized way. We could conclude that, from the philosophical point of view, space-time is a rich holistic structure, and the choice of a specific axiomatics corresponds to the choice of the angle at which we contemplate the whole.
Probabilities in space-time
The axiomatics of EPS is based on the assumption that photons and particles are point-like, because their histories L and P are collections of definite events. But the concept of localised particles is not supported by quantum theory [Malament, 1996]. In contrast, quantum mechanics implies that 'statistical predictions do apply to single events' [Peres and Terno, 2004]. Actually, even within the classical theory the space-time location measurements are always affected by experimental uncertainties and, consequently, the events associated with them are somewhat 'spread' in the space-time. How does this fact affect the notions of time and causation?
It is useful to distinguish two levels of conceptualisation in a physical theory: an 'effective' one, which directly relates to the experimental data, and a 'fundamental' one, which aims at providing an explanation for the data. At the effective level we deal with raw data, which correspond to elemental events, such as the click of a detector. The interpretation of these raw data requires a theoretical formalism, which associates them with physical phenomena. It is at this stage when the uncertainties, inflicted by experimental limitations or, possibly, by fundamental randomness inherent in the adopted theory, arise. In consequence, while elemental experimental events are definite, the final outcome acquires a probabilistic form, which involves both systematic and statistical errors of the measured quantities. It applies, in particular, to space-time location measurements.
The concepts of time and causality stemming from the EPS axiomatisation can only be applied at the effective level with definite events. In order to extend them to the 'fundamental' level one needs to take into account the probabilistic nature of experimental outcomes.
A natural universal mathematical structure suitable to grasp the uncertainty of events is provided by the probability measures. Given a space-time M one defines the space P(M ) of all probability measures 13 on M . An element µ ∈ P(M ) is a function, which associates with every (measurable) region of space-time M a probability, i.e. a number from the interval [0, 1]. Given any such region K, the number µ(K) specifies the probability of the occurrence of some event associated with K and µ. For instance, if µ models the response of a detector, which occupies a volume V in space and operates within a time-interval T , then µ(T × V ) gives the probability of a single detector's click. More precisely, the number µ(T × V ) answers the operational question: what is the probability of the signal detection if a suitable detector operating in space-time region T × V is placed. If the detector does click, then we interpret it as registration of the signal to which it was tuned, coming with a space-time label determined by the region K = T × V . If the detector does not click, then it provides us with a definite information that the signal is with certainty somewhere outside of the region K. The latter should be distinguished from the situation in which the detector is not switched on, as the 'unperformed experiments have no results' [Peres, 1978].
Clearly, any physical device has some space-and time-resolution, so that the region K ⊂ M cannot consist of a single point. On the other hand, the measure itself can in principle be localised acutely. Indeed, given any point p ∈ M one can associate with it a Dirac measure δ p ∈ P(M ). The latter would yield 1 whenever the event p lies in the space-time region covered by the detector, p ∈ K, and 0 if this is not the case. The Dirac measures can thus serve to model the classical -localised -particles. Note, however, that even in this case the finite size of the detector induces an uncertainty of ascribing a definite space-time label, as the click only gives the information that the event p happened somewhere within the space-time region K.
We see that the probabilistic nature of experiments coerces a reinterpretation of the relativistic space-time M itself. The points in M should not be seen as actual events, but rather M serves merely as a support-space for the probability measures. The actual events arise at an effective level, as a result of an interaction between a localised detector and a 'particle'. Such an understanding of the measuring process is usually put forward in the context of quantum theory (cf. the discussion in [Haag, 1996, Section I.1] or [Peres and Terno, 2004]), but in fact it is a general operational concept, not relying on how we eventually model the 'particles' (as well as 'detectors' and 'interactions') -cf. [Eckstein et al., 2020]. Consequently, there are no events on the fundamental level in a probabilistic physical theory.
Causality of probabilities
The relativistic causal structure discussed in Section 3 pertained to definite pointlike events. It is fairly straightforward to extend it to a more realistic situation when signal-processing devices are characterised by an extended region K = T × V within the space-time. Indeed, one can simply say that the device operating in K can causally affect the device in C if and only if J + (K) ∩ C = ∅. In other words, if the second device is localised outside of the causal future of the first device, then there can be no causal influence of the first one on the second (see Fig. 2 a)).
Such an idea employs only signal-processing devices and does not refer to the nature of the signals themselves. Moreover, it does not take into account the proba- Figure 2: a) The device localised in region K can causally influence the device in C 1 , but not in C 2 . b) Two pairs of detectors, localised in regions K 1 , K 2 and C 1 , C 2 , respectively, respond to the signal modelled by measures µ and ν, respectively. With the corresponding probabilities of detection, µ(K 1 ) = a, µ(K 2 ) = 1 − a, ν(C 1 ) = b and ν(C 2 ) = 1 − b, we have µ ν if and only if a ≤ b. Intuitively, if a would be greater than b, then the probability would have to 'leak out' of the causal future J + (K 1 ). bilistic nature of the devices' outcomes. Suppose that we have two pairs of localised detectors as in Fig. 2 b). The detectors localised in regions K 1 , K 2 , C 1 and C 2 click with probabilities µ(K 1 ) = a, µ(K 2 ) = 1 − a, ν(C 1 ) = b and ν(C 2 ) = 1 − b, respectively. When can the detection statistics µ causally influence the detection statistics ν?
A rigorous answer to this question is provided within the formalism established recently by Eckstein and Miller [Eckstein and Miller, 2017b]. They proposed the following definition 14 : For two probability measures µ, ν ∈ P(M ) on a given spacetime M we say that µ causally precedes ν, symbolically µ ν, if for every compact 15 set K ⊂ supp µ we have µ(K) ≤ ν(J + (K)). The general intuition behind this definition is that, for any compact region of space-time K ⊂ M the probability cannot 'leak out' of the causal future of K. If we apply this definition to the example depicted in Fig. 2 b), we conclude that the first pair of detectors can causally influence the second pair only if the relevant probabilities satisfy a ≤ b.
Such a notion of causality for probabilities might seem purely formal, but it is in fact coherent with the operational 'no-signalling principle' (see [Popescu, 2014,Brunner et al., 2014 and [Eckstein et al., 2020]). The latter says that no actions or events in a space-time region K can causally influence any measurement statistics outside of the causal future J + (K). For if they did, the information could be transferred superluminally by orchestrating a protocol, in which an observer in K prepares multiple signal carriers and sends them to another observer outside of J + (K), who can read-out the information from his detection statistics. Such a protocol, when 14 Strictly speaking, this definition was presented in another work of Eckstein and Miller [Eckstein and Miller, 2017a], as a refinement of one of the equivalent characterisation of casual precedence for probability measures established in [Eckstein and Miller, 2017b, Theorem 8]. The same definition was independently put forward by Stefan Suhr in [Suhr, 2018].
15 Working solely with compact subsets of M is not a limitation, because µ(X) is determined for any measurable X through the formula µ(X) = sup{µ(K) | K ⊂ X, K compact}, which is valid for Radon measures. Also, on the physical side, one can safely assume that the space-time region associated with any device is compact, i.e. closed and bounded. executed by pair of mutually travelling inertial observers, leads to causal loops and consequent logical paradoxes -see [Eckstein et al., 2020] for a detailed discussion.
It turns out that the causal precedence between probability measures admits another equivalent 16 characterisation: A measure µ causally precedes ν if and only if there exists a joint measure ω ∈ P(M ×M ) such that ω(·×M ) = µ(·), ω(M ×·) = ν(·) and ω(J + ) = 1, where the set J + consists of all pairs of points (p, q) ∈ M × M such that p q. This condition is inspired by the optimal transport theory. It encodes the following intuition (cf. [Eckstein and Miller, 2017a]): 'Each infinitesimal part of a probability distribution must travel along a future-directed casual curve.'
Time-evolution of probabilities
In the previous section we have seen how the causal order of points in a relativistic space-time can be extended to probability measures modelling detection statistics. What about the time-evolution?
Firstly, let us recall that within the relativistic setting there is no preferred time parameter and one needs to treat the space-time as a global holistic structure. Time-evolution arises when an observer adopts a local coordinate chart, for instance by adopting the signals-and-echoes method of EPS. On the other hand, as discussed in Section 3, space-times with sufficiently robust causal structures admit global time functions, T : M → R, which determine the hypersurfaces of simultaneity. However, the choice of a time function is by no means unique. Indeed, even the simplest Minkowski space-time admits a continuous family of equivalent time-slicings associated with different inertial observers. More generally, the choice of a time-function can be associated with a 'global observer', i.e. a collection of observers, parameterised by points on an achronal hypersurface, each of which follows some future-directed time-like curve. In the case of an inertial slicing of the Minkowski space-time, all such observers travel in parallel -in the same direction and with the same speed. In general, this need not be the case and the collection of all admissible time-slicings is vast.
The measures on space-time are inherently non-local objects. They can, in principle, be spread throughout the entire space-time. The latter situation arises, for instance, if one associates measures with the quantum states emerging from quantum field theory [Reeh andSchlieder, 1961, Malament, 2012]. For this reason, if one wants to consider the time-evolution of general measures one needs to adopt the global perspective of time-slicings, rather than the local one based on coordinate charts.
Let us then fix a time-function T : M → R on a given space-time M . In order to guarantee the well-posedness of the time-evolution problems, we shall assume that the space-time M is globally hyperbolic and that the level-sets of T are Cauchy hypersurfaces, which means that they can accommodate initial data for some hyperbolic evolution equation (see e.g. [Ringström, 2009]). Let us emphasise here that this is a limitation only in the cosmic context when we interpret the space-time M as the whole Universe. Locally, any space-time, even one containing closed time-like curves, is always globally hyperbolic in some open neighbourhood of any point [Penrose, 1972]. One can thus safely consider models of phenomena, which are non-local up to some scales. The truly global ones, for instance impelled by quantum field theory, require the (rather standard, see e.g. [Hollands and Wald, 2015]) assumption of global hyperbolicity of the entire Universe.
Having chosen a time-function T we obtain the corresponding splitting M R×S with a Cauchy hypersurface S. Consider now a family of probability measures supported on subsequent time-slices, µ t ∈ P(S t ), with S t := {t}×S. Such objects are thus 'localised in time', but 'delocalised in space'. The time-evolution of measures, for a chosen global time-function T , is the family {µ t } t . Equivalently, one can define it as a map t → µ t ∈ P(M ), with supp µ t ⊂ S t .
For a given time function T the quantity {µ t } t models a time-evolution of probabilities potentially registered by an observer associated with T . Note that the uncertainty of the time-moment of the detection comes solely from the finite timeresolution of the measuring device, characterised by an interval T . The measures themselves are 'localised in time'. This means that, within this framework, there are no a priori limits on the time-resolution of the measurements. Such an assumption is met both in general relativity and in quantum theory, though it might fail in some quantum gravity theories [Hossenfelder, 2013].
Observe that the notion of evolution of measures includes that of a trajectory of point-like particles. Indeed, with any parametrised (piece-wise smooth) curve t → γ(t) ∈ M, for a chosen time-parameter, one can associate the measures µ t = δ γ(t) , which are localised both in time and in space. Recall that such a trajectory of a point-like particle is (future-directed) causal if γ(s) γ(t) for all s ≤ t. In the same vein, one says that a general evolution of measures {µ t } t is causal if µ s µ t for all s ≤ t.
The trajectory of a point-like particle is in fact an observer-independent quantity. Indeed, every such trajectory can be deparametrised, i.e. treated as a worldlinethe collection [γ] of points in M . This fact facilitates a sound interpretation of trajectories: different observers can use different parametrisations, determined by their local frames, of the same moving particle. In other words, the travelling pointlike particle itself is an observer-independent entity, while the description of its motion requires some time-parametrisation. The existence of an invariant objectthe worldline -guarantees that different parameterisations of the trajectory are covariant. That is, there exists a unequivocal prescription allowing two different observers, using different time-parametrisation, to compare the outcomes of their observation of the same moving point-like particle.
As it turns out [Miller, 2017], for a general causal evolution of measures on a globally hyperbolic space-time M there also exists an invariant object, i.e. an object independent of the choice of the time function. In order to unveil it one needs, firstly, to consider the space of all future-directed causal curves on C I T (M ), parametrised in accordance with the global time-function T , for some time-interval I ⊆ R and endow it with a suitable (compact-open) topology, which turns it into a locally compact Polish space 17 . Then, one proves [Miller, 2017, Theorem 1] that an evolution of measures {µ t } t is causal if and only if there exists a measure σ ∈ P(C I T ) such that (ev t ) # σ = µ t for any t ∈ I, where ev is the standard evaluation map: ev t (γ) = γ(t) ∈ M for any curve γ ∈ C I T (M ). Secondly, one shows that if I = R then every such σ is in one-to-one correspondence with a measure [σ] ∈ P(C(M )) on the space of worldlines C(M ) in M . In this way, one arrives at an object [σ]a probability measure on worldlines -which is manifestly invariant.
If the evolution of measures describes the motion of a single point-like particle, µ t = δ γ(t) , then the associated measure [σ] is supported on a single worldline [γ], ie.
[σ] = δ [γ] . However, for a general evolution of measures the object σ, and hence [σ], is not unique. This can be seen from the optimal-transport perspective on the evolution of measures. Indeed, for any two fixed probability distributions on time-slices S s and S t there might exist many causal paths, which transport the infinitesimal portions of probablity from S s to S t . Nevertheless, thanks to the existence of an invariant object, any causal time-evolution of measures is covariant, in the sense that one has a precise prescription on how to translate an evolution of measures as witnessed form the perspective of one global time-function T 1 to the perspective of another T 2 . In particular, any two observers will agree upon the (a)causality of an evolution of measures. For a detailed discussion see [Miller, 2017] and also 18 [Miller, 2021].
The concept of the causal evolution of probability measures is useful to inspect the dynamical equations, in classical, quantum or 'post-quantum' theories, for their compatibility with the causal structure of a relativistic space-time [Eckstein et al., 2020]. In particular, the (normalised) electromagnetic energy density u := 1 2E ε 0 E 2 + 1 µ 0 B 2 , with a finite total energy E = R 3 u(0, x)dx, yields a family of probability measures µ t := u(t, x) dx on R 3 , which evolves causally. The same is true for the probability density ψ † (t, x)ψ(t, x) dx resulting from a wave-function ψ(t, x) evolving according to the Dirac equation, possibly including external gauge potentials [Eckstein and Miller, 2017a]. On the other hand, the evolution of a probability density associated with a quantum wave function driven by the Schrödinger equation with a positive definite Hamiltonian is typically not causal [Eckstein and Miller, 2017a] (see also [Hegerfeldt, 1974,Hegerfeldt, 1985). Such an incompatibility with the relativistic causality can be quantified in terms of the characteristic timeand length-scales [Eckstein and Miller, 2017a] and can be utilised to restrain certain 'post-quantum' theories [Eckstein et al., 2020].
The philosophical picture that emerges from these considerations is the following: With every physical phenomenon one can associate a mathematical object -a probability measure on the space of worldlines in a given space-time M . This object is global, which means that it provides a holistic model for the entire history of a given physical system and thus we have no direct access to it. The time-evolution of this system is an emergent concept, which arises when an observer chooses a global time-function T and studies its local properties by registering the time-evolution of statistics µ t (T × V ) with the help of a device localised in the volume V and having a time-resolution T . A different observer using a different time-function would register different statistics µ t (T × V ), when measuring the same physical system with the same detecting device. Yet, the two observers can consistently compare their results by translating the registered probabilities through the invariant object.
Conclusions
Our analysis shows that some primitive notions of 'time', 'space' and 'causality' are coerced by the very methodology of science. Indeed, in any experimental setup one needs to specify an 'input', which always comes before the 'output'. In the same vein, some primitive notion of space is presupposed by the very fact that any experimental setup is a physical device, which occupies some definite volume and needs to be placed and oriented -e.g. a telescope pointed towards a specific star.
More formally, an experiment is in essence a collection of data {d i }. (Without any loss of generality we can thus assume that d i 's are just bits.) Any such bit is local, that is uniquely associated with an actual event pertaining to the physical world. Regardless of the adopted theoretical framework, we are bound to place the bits in 'space' and 'time'. The very notation b 1 , b 2 informs one that the bit b 1 has been input, acquired or communicated before the bit b 2 . If, for any reason, we are dealing with different data sets, we distinguish them by using different labels, say a and b. This leads to a primitive notion of space: The event a 1 could have taken place simultaneously with b 1 , but at a different 'place'.
Furthermore, we always make some presuppositions about the causal relations among the data. In particular, we always assume that the input data influenced the output, but not the other way round (cf. e.g. Axiom 1 'causality' in [Chiribella et al., 2011]). The same must be said about the basic probabilistic structures. Indeed, any viable analysis of experimental data must be endowed with probabilistic concepts such as relative frequencies of the outcomes and correlations among the data sets. It is clear that the very methodology of physics requires the primitive probabilistic concepts, on top of the space-time-related ones.
Every physical theory is based on some mathematical structures, which per se are purely formal objects. In order to relate theories with actual experiments, one needs a 'minimal' operational interpretation. The latter bridges between the primitive notions of space, time, causation and the rigorous mathematical notions modelling a chosen phenomenon. Moreover, any physical theory fosters probabilistic predictions, which can be verified against empirical data. This means, in particular, that a theory must provide a mechanism explaining the relative frequencies and correlations observed in a relevant experiment, in terms of the modelled physical system.
While the testing of a physical model necessarily requires the use of some primitive notions, a valid physical theory may refine some of these notions and unveil some unexpected connections between them. In this vein, general relativity has revealed, via the Hawking theorem, a deep connection between the existence of a global cosmic time and stability of space-time measurements. On the other hand, quantum theory implies that the measurement outcomes are afflicted by ontic uncertainties, not related to our subjective lack of knowledge about the studied system.
These conclusions clearly hinge upon the adopted primitive notions in the first place, which are necessary for the interpretation of empirical data. Nevertheless, the 'unreasonable effectiveness' of the mathematical-empirical methodology [Wigner, 1960] in modelling natural phenomena strongly suggests that a valid physical theory reveals some ontological aspects, which underlie the primitive concepts of time, causality and probability. It also suggests that the world should be contemplated from a holistic perspective, one in which time, causality and probability are irreducibly related. This calls for the development of a new philosophical discourse, going beyond the classical dichotomy à la Leibniz versus Hume, which would be able grasp the integral connections between the pillars of the methodology of physics.
Figure 1 :
1A diagram illustrating the subsequent layers of axiomatisation of general relativity following
Although Mehlberg's book appeared in 1980, it is based on his works dating back to 1935 and 1937.
For the full account of chronology and causal relations see[Minguzzi and Sánchez, 2008, Minguzzi, 2019].5 Through the so-called exponential mapping. 6 Sets are defined to be open in this topology if they are unions of the sets of the form I + (p) ∩ I − (p).7 Alexandrov topology coincides with manifold topology if the strong causality condition is satisfied; see below.
It is interesting to ask how this condition looks from the global point of view. The answer is that if M is noncompact, such a nonvanishing direction field, and consequently a Lorentz metric, always exists, but if M is compact it exists if and only if the Euler-Poincaré characteristic of M vanishes[Geroch, 1967].12 Different -within certain limits. There is one important constraint: the mathematical structure of space-time must be preserved by all interpretations. One could say that the mathematical structure is 'invariant' with respect to all admissible interpretations.
Technically, one should assume that the measures in P(M ) are Borel, to assure compatibility with the topology of M . Furthermore, since any relativistic space-time is a Polish space, all elements of P(M ) are actually Radon -see[Eckstein and Miller, 2017b] for the details.
The equivalence of these two definitions holds,[Eckstein and Miller, 2017b, Theorem 8], in causally simple space-times, which have slightly weaker causal properties than the globally hyperbolic ones -see e.g.[Minguzzi and Sánchez, 2008].
Polish spaces are separable and completely metrisable, what already makes them suitable for developing much of the abstract probability theory.18 Miller has recently shown that there exists yet another invariant object associated with every causal evolution of measures, which is related to the continuity equation.
AcknowledgementsWe are grateful to Tomasz Miller for his thoughtful comments on the manuscript. M.E. would like to thank Paweł Horodecki for the numerous inspiring discussions on causality and randomness.
Testing Einstein's special relativity with Fermi's short hard γ-ray burst GRB090510. [ References, Abdo, Nature. 462331References [Abdo et al., 2009] Abdo, A. et al. (2009). Testing Einstein's special relativity with Fermi's short hard γ-ray burst GRB090510. Nature, 462:331.
Quantum-spacetime phenomenology. G Amelino-Camelia ; Amelino-Camelia, Living Reviews in Relativity. 516[Amelino-Camelia, 2013] Amelino-Camelia, G. (2013). Quantum-spacetime phe- nomenology. Living Reviews in Relativity, 16(5).
IceCube and GRB neutrinos propagating in quantum spacetime. Amelino-Camelia, Physics Letters B. 761Amelino-Camelia et al., 2016] Amelino-Camelia, G., Barcaroli, L., D'Amico, G., Loret, N., and Rosati, G. (2016). IceCube and GRB neutrinos propagating in quantum spacetime. Physics Letters B, 761:318-325.
Tests of quantum gravity from observations of γ-ray bursts. Amelino-Camelia, Nature. 3936687Amelino-Camelia et al., 1998] Amelino-Camelia, G., Ellis, J., Mavromatos, N., Nanopoulos, D., and Sarkar, S. (1998). Tests of quantum gravity from obser- vations of γ-ray bursts. Nature, 393(6687):763-765.
Andréka, arXiv:1310.1475An axiom system for general relativity complete with respect to Lorentzian manifolds. arXiv preprint[Andréka et al., 2013] Andréka, H., Madarász, J. X., Németi, I., and Székely, G. (2013). An axiom system for general relativity complete with respect to Lorentzian manifolds. arXiv preprint arXiv:1310.1475.
Remark on cosmological models. Witten ; Bass, R W Bass, L Witten, Reviews of Modern Physics. 29[Bass and Witten, 1957] Bass, R. W. and Witten, L. (1957). Remark on cosmolog- ical models. Reviews of Modern Physics, 29:452-453.
Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Sánchez ; Bernal, A Bernal, M Sánchez, Letters in Mathematical Physics. 772[Bernal and Sánchez, 2006] Bernal, A. and Sánchez, M. (2006). Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Letters in Mathematical Physics, 77(2):183-197.
. [ Brunner, Bell nonlocality. Reviews of Modern Physics. 86[Brunner et al., 2014] Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., and Wehner, S. (2014). Bell nonlocality. Reviews of Modern Physics, 86:419-478.
Über die Abhängigkeit der Eigenschaften des Raumes von denen der Zeit. Kant-Studien. R Carnap, 30Carnap, 1925[Carnap, 1925] Carnap, R. (1925). Über die Abhängigkeit der Eigenschaften des Raumes von denen der Zeit. Kant-Studien, 30(1-2):331-345.
Causal structure in space-time. B Carter ; Carter, General Relativity and Gravitation. 14Carter, 1971] Carter, B. (1971). Causal structure in space-time. General Relativity and Gravitation, 1(4):349-391.
Informational derivation of quantum theory. [ Chiribella, Physical Review A. 84112311[Chiribella et al., 2011] Chiribella, G., D'Ariano, G. M., and Perinotti, P. (2011). Informational derivation of quantum theory. Physical Review A, 84(1):012311.
First measurements of high frequency crossspectra from a pair of large Michelson interferometers. [ Chou, Physical Review Letters. 117111102[Chou et al., 2016] Chou, A. S., Gustafson, R., Hogan, C., Kamai, B., Kwon, O., Lanza, R., McCuller, L., Meyer, S. S., Richardson, J., Stoughton, C., Tomlin, R., Waldman, S., and Weiss, R. (2016). First measurements of high frequency cross- spectra from a pair of large Michelson interferometers. Physical Review Letters, 117:111102.
An axiomatization of general relativity. G M Covarrubias ; Covarrubias, International Journal of Theoretical Physics. 3211Covarrubias, 1993] Covarrubias, G. M. (1993). An axiomatization of general rela- tivity. International Journal of Theoretical Physics, 32(11):2135-2154.
f (R) theories. De Felice, Tsujikawa, A De Felice, S Tsujikawa, Living Reviews in Relativity. 1313De Felice and Tsujikawa, 2010] De Felice, A. and Tsujikawa, S. (2010). f (R) the- ories. Living Reviews in Relativity, 13(1):3.
Operational causality in spacetime. Eckstein, Physical Review A. 10142128[Eckstein et al., 2020] Eckstein, M., Horodecki, P., Miller, T., and Horodecki, R. (2020). Operational causality in spacetime. Physical Review A, 101:042128.
Causal evolution of wave packets. Eckstein, Miller, M Eckstein, T Miller, Physical Review A. 9532106[Eckstein and Miller, 2017a] Eckstein, M. and Miller, T. (2017a). Causal evolution of wave packets. Physical Review A, 95:032106.
Causality for nonlocal phenomena. Eckstein, Miller, M Eckstein, T Miller, Annales Henri Poincaré. 18[Eckstein and Miller, 2017b] Eckstein, M. and Miller, T. (2017b). Causality for nonlocal phenomena. Annales Henri Poincaré, 18:3049-3096.
The geometry of free fall and light propagation. Elhers, General Relativity, papers in honour of J. L. Synge. O'Raifeartaigh, L.Oxford-LondonClarendon Press441587Republished in[Elhers et al., 1972] Elhers, J., Pirani, F., and Schild, A. (1972). The geometry of free fall and light propagation. In O'Raifeartaigh, L., editor, General Relativity, papers in honour of J. L. Synge. Clarendon Press, Oxford-London. Republished in: General Relativity and Gravitation 44, 1587 (2012).
Domain of dependence. R Geroch, Journal of Mathematical Physics. 112Geroch, 1970[Geroch, 1970] Geroch, R. (1970). Domain of dependence. Journal of Mathematical Physics, 11(2):437-449.
Space-time structure from a global viewpoint. R Geroch, General Relativity and Cosmology: Proceedings of the 47th International School of Physics "Enrico Fermi. Sachs, R.K.Varena, Italy; New YorkAcademic PressGeroch, 1971[Geroch, 1971] Geroch, R. (1971). Space-time structure from a global viewpoint. In Sachs, R.K., editor, General Relativity and Cosmology: Proceedings of the 47th International School of Physics "Enrico Fermi", Varena, Italy, 30 June -12 July 1969. Academic Press, New York, pages 71-103.
Global structure of spacetimes. Geroch, R Horowitz ; Geroch, G T Horowitz, General Relativity -An Einstein Centenary Survey. Hawking, S. and Israel, W.Cambridge University Press[Geroch and Horowitz, 1979] Geroch, R. and Horowitz, G. T. (1979). Global struc- ture of spacetimes. In Hawking, S. and Israel, W., editors, General Relativity - An Einstein Centenary Survey, pages 212-293. Cambridge University Press.
Topology in general relativity. R P Geroch, Journal of Mathematical Physics. 84Geroch, 1967[Geroch, 1967] Geroch, R. P. (1967). Topology in general relativity. Journal of Mathematical Physics, 8(4):782-786.
An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation Reviews of Modern Physics. K Gödel, 21Gödel, 1949[Gödel, 1949] Gödel, K. (1949). An Example of a New Type of Cosmological So- lutions of Einstein's Field Equations of Gravitation Reviews of Modern Physics, 21:447-450.
Axiomatic causal theory of space-time. A K Guts, Gravitation and Cosmology. 13[Guts, 1995] Guts, A. K. (1995). Axiomatic causal theory of space-time. Gravitation and Cosmology, 1(3):211-215.
R Haag ; Haag, Local Quantum Physics: Fields, Particles, Algebras. Theoretical and Mathematical Physics. Berlin HeidelbergSpringerHaag, 1996] Haag, R. (1996). Local Quantum Physics: Fields, Particles, Algebras. Theoretical and Mathematical Physics. Springer Berlin Heidelberg.
Stable and generic properties in general relativity. S W Hawking ; Hawking, General Relativity and Gravitation. 14Hawking, 1971] Hawking, S. W. (1971). Stable and generic properties in general relativity. General Relativity and Gravitation, 1(4):393-400.
The existence of cosmic time functions. Bondi ; Hawking, S W Hawking, H Bondi, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 308[Hawking and Bondi, 1969] Hawking, S. W. and Bondi, H. (1969). The existence of cosmic time functions. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 308(1494):433-435.
S W Hawking, G F R Ellis, The Large Scale Structure of Space-time. Cambridge University PressHawking and Ellis[Hawking and Ellis, 1973] Hawking, S. W. and Ellis, G. F. R. (1973). The Large Scale Structure of Space-time. Cambridge University Press.
Remark on causality and particle localization. G C Hegerfeldt ; Hegerfeldt, Physical Review D. 103320Hegerfeldt, 1974] Hegerfeldt, G. C. (1974). Remark on causality and particle local- ization Physical Review D, 10:3320.
Violation of causality in relativistic quantum theory?. G C Hegerfeldt, Physical Review Letters. 542395Hegerfeldt, 1985[Hegerfeldt, 1985] Hegerfeldt, G. C. (1985). Violation of causality in relativistic quantum theory? Physical Review Letters, 54:2395.
Quantum fields in curved spacetime. Wald ; Hollands, S Hollands, R M Wald, Physics Reports. 574[Hollands and Wald, 2015] Hollands, S. and Wald, R. M. (2015). Quantum fields in curved spacetime. Physics Reports, 574:1-35.
Minimal length scale scenarios for quantum gravity. S Hossenfelder, Living Reviews in Relativity. 1612Hossenfelder, 2013[Hossenfelder, 2013] Hossenfelder, S. (2013). Minimal length scale scenarios for quantum gravity. Living Reviews in Relativity, 16(1):2.
Hauptschriften zur Grundlegung der Philosophie. G W Leibniz, Meiner1HamburgLeibniz, 1969[Leibniz, 1969] Leibniz, G. W. (1969). Hauptschriften zur Grundlegung der Philoso- phie, volume 1. Meiner, Hamburg.
The metaphysical foundations of mathematics. G W Leibniz ; Leibniz, Philosophical Papers and Letters. Loemker, L. E.SpringerLeibniz, 1989] Leibniz, G. W. (1989). The metaphysical foundations of mathemat- ics. In Loemker, L. E., editor, Philosophical Papers and Letters, pages 666-674. Springer Netherlands, Dordrecht.
J Leray ; Leray, Hyperbolic Differential Equations. Institute for Advanced Study. PrincetonLeray, 1953] Leray, J. (1953). Hyperbolic Differential Equations. Institute for Ad- vanced Study, Princeton.
Experimental constraints on the second clock effect. I Romero ; Lobo, C Romero, Physics Letters B. 783and Romero, 2018] Lobo, I. and Romero, C. (2018). Experimental constraints on the second clock effect. Physics Letters B, 783:306-310.
In defense of dogma: Why there cannot be a relativistic quantum mechanical theory of (localizable) particles. D Malament, Perspectives on Quantum Reality. Clifton, R.KluwerMalament, 1996[Malament, 1996] Malament, D. (1996). In defense of dogma: Why there cannot be a relativistic quantum mechanical theory of (localizable) particles. In Clifton, R., editor, Perspectives on Quantum Reality. Kluwer.
Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. D B Malament, ChicagoMalament, 2012[Malament, 2012] Malament, D. B. (2012). Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. Chicago.
Time, Causality, and the Quantum Theory. H Mehlberg ; Mehlberg, Essay on the Causal Theory of Time. Reidel. Dordrecht-Boston-LondonMehlberg, 1980] Mehlberg, H. (1980). Time, Causality, and the Quantum Theory, volume one: Essay on the Causal Theory of Time. Reidel, Dordrecht-Boston- London.
Polish spaces of causal curves. Journal of Geometry and Physics. 116[Miller, 2017] Miller, T. (2017). Polish spaces of causal curves. Journal of Geometry and Physics, 116:295-315.
Causal evolution of probability measures and continuity equation. T Miller ; Miller, arXiv:2104.02552preprintMiller, 2021] Miller, T. (2021). Causal evolution of probability measures and con- tinuity equation. preprint arXiv:2104.02552.
Lorentzian causality theory. E Minguzzi, Living Reviews in Relativity. 221Minguzzi, 2019[Minguzzi, 2019] Minguzzi, E. (2019). Lorentzian causality theory. Living Reviews in Relativity, 22(1):1-202.
The causal hierarchy of spacetimes. [ Minguzzi, E Sánchez ; Minguzzi, M Sánchez, Recent Developments in Pseudo-Riemannian Geometry. Alekseevsky, D. V. and Baum, H.European Mathematical Society Publishing House[Minguzzi and Sánchez, 2008] Minguzzi, E. and Sánchez, M. (2008). The causal hierarchy of spacetimes. In Alekseevsky, D. V. and Baum, H., editors, Recent Developments in Pseudo-Riemannian Geometry, ESI Lectures in Mathematics and Physics, pages 299-358. European Mathematical Society Publishing House.
Model-independent constraints on Lorentz invariance violation: Implication from updated gamma-ray burst observations. Y Pan, J Qi, S Cao, T Liu, Y Liu, S Geng, Y Lian, Z.-H Zhu, Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter. Oriti, D.CambridgeCambridge University Press890169Pan et al., 2020[Oriti, 2009] Oriti, D., editor (2009). Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter. Cambridge University Press, Cambridge. [Pan et al., 2020] Pan, Y., Qi, J., Cao, S., Liu, T., Liu, Y., Geng, S., Lian, Y., and Zhu, Z.-H. (2020). Model-independent constraints on Lorentz invariance viola- tion: Implication from updated gamma-ray burst observations. The Astrophysical Journal, 890(2):169.
R Penrose ; Penrose, Techniques of Differential Topology in Relativity. 7Penrose, 1972] Penrose, R. (1972). Techniques of Differential Topology in Relativ- ity, volume 7 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM.
Unperformed experiments have no results. A Peres, American Journal of Physics. 467Peres, 1978[Peres, 1978] Peres, A. (1978). Unperformed experiments have no results. American Journal of Physics, 46(7):745-747.
Quantum information and relativity theory. A Peres, D R Terno, Reviews of Modern Physics. 76Peres and Terno[Peres and Terno, 2004] Peres, A. and Terno, D. R. (2004). Quantum information and relativity theory. Reviews of Modern Physics, 76:93-123.
[ Perlman, New constraints on quantum gravity from X-ray and gamma-ray observations. 80510[Perlman et al., 2015] Perlman, E. S., Rappaport, S. A., Christiansen, W. A., Ng, Y. J., DeVore, J., and Pooley, D. (2015). New constraints on quantum gravity from X-ray and gamma-ray observations. The Astrophysical Journal, 805(1):10.
Nonlocality beyond quantum mechanics. S Popescu, Nature Physics. 104264Popescu, 2014[Popescu, 2014] Popescu, S. (2014). Nonlocality beyond quantum mechanics. Nature Physics, 10(4):264.
Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Feldern. H Reeh, S Schlieder, Il Nuovo Cimento. 225Reeh and Schlieder[Reeh and Schlieder, 1961] Reeh, H. and Schlieder, S. (1961). Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Feldern. Il Nuovo Cimento, 22(5):1051- 1068.
Axiomatik der relativistischen. H Reichenbach, F. Vieweg & Sohn Akt.-Ges. 72Reichenbach, 1924[Reichenbach, 1924] Reichenbach, H. (1924). Axiomatik der relativistischen, Raum- Zeit-Lehre, volume 72. F. Vieweg & Sohn Akt.-Ges., Braunschweig.
The Cauchy Problem in General Relativity. H Ringström ; Ringström, ESI Lectures in Mathematics and Physics. 6European Mathematical SocietyRingström, 2009] Ringström, H. (2009). The Cauchy Problem in General Relativity, volume 6 of ESI Lectures in Mathematics and Physics. European Mathematical Society.
A Theory of Time and Space. A A Robb ; Robb, Cambridge University PressRobb, 1914] Robb, A. A. (1914). A Theory of Time and Space. Cambridge Uni- versity Press.
Theory of optimal transport for Lorentzian cost functions. S Suhr ; Suhr, Münster Journal of Mathematics. 11Suhr, 2018] Suhr, S. (2018). Theory of optimal transport for Lorentzian cost func- tions. Münster Journal of Mathematics, 11:13-47.
General Relativity. R M Wald, G J Whitrow, The Natural Philosophy of Time. OxfordClarendon PressWald, 1984. second edition[Wald, 1984] Wald, R.M. (1984). General Relativity. University of Chicago Press. [Whitrow, 1980] Whitrow, G. J. (1980). The Natural Philosophy of Time. Clarendon Press, Oxford, second edition.
The unreasonable effectiveness of mathematics in the natural sciences. E P Wigner ; Wigner, Communications on Pure and Applied Mathematics. 131Wigner, 1960] Wigner, E. P. (1960). The unreasonable effectiveness of mathemat- ics in the natural sciences. Communications on Pure and Applied Mathematics, 13(1):1-14.
The differentiable and causal structures of space-time. N M J Woodhouse, Journal of Mathematical Physics. 144[Woodhouse, 1973] Woodhouse, N. M. J. (1973). The differentiable and causal struc- tures of space-time. Journal of Mathematical Physics, 14(4):495-501.
| [] |
[
"ENDOMORPHISMS OF ABELIAN VARIETIES, CYCLOTOMIC EXTENSIONS AND LIE ALGEBRAS",
"ENDOMORPHISMS OF ABELIAN VARIETIES, CYCLOTOMIC EXTENSIONS AND LIE ALGEBRAS"
] | [
"Yuri G Zarhin "
] | [] | [] | We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero. | 10.1070/sm2010v201n12abeh004132 | [
"https://arxiv.org/pdf/1001.3424v5.pdf"
] | 115,174,631 | 1001.3424 | 54e942e6753206e07a1e4a501e8d9cd8c5eff145 |
ENDOMORPHISMS OF ABELIAN VARIETIES, CYCLOTOMIC EXTENSIONS AND LIE ALGEBRAS
20 Feb 2010
Yuri G Zarhin
ENDOMORPHISMS OF ABELIAN VARIETIES, CYCLOTOMIC EXTENSIONS AND LIE ALGEBRAS
20 Feb 2010
We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.
Introduction
The aim of this note is to extend Faltings' results [4,5] concerning the Tate conjecture on homomorphisms of abelian varieties [15,16] over finitely generated fields K of characteristic zero to their infinite cyclotomic extensions K(ℓ) = K(µ ℓ ∞ ). The possibility of such generarization (in the case of number fields K) was stated (without a detailed proof) in [19,§6,Subsect. 2,; it was pointed out there that this result follows from the theorem of Faltings combined with technique developed in [17] and a theorem of F.A. Bogomolov about homotheties [1,2]. Our main result is the following assertion. (Here Gal(E) stands for the absolute Galois group of E while T ℓ (X) and T ℓ (Y ) are the Tate modules of abelian varieties of X and Y respectively.) Theorem 1.1. Suppose that K is a field that is finitely generated over Q and ℓ is a prime. Let us put E = K(ℓ). If X and Y are abelian varieties over E then the natural embedding of Z ℓ -modules
Hom E (X, Y ) ⊗ Z ℓ ֒→ Hom Gal(E) (T ℓ (X), T ℓ (Y ))
is bijective. Remark 1.2. Replacing K by its suitable finite (sub) extension (of E), we may in the course of the proof of Theorem 1.1 assume that both X and Y are defined over K. Remark 1.3. A.N. Parshin [19, §6, Subsect. 2, pp. 91-92] conjectured that the following analogue of the Mordell conjecture holds true: if K is a number field and C is an absolutely irreducible smooth projective curve over E = K(ℓ) then the set C(E) of its E-rational points is finite if the genus of C is greater than 1. Theorem 1.1 has arisen from attempts to understand which parts of Faltings' proof of the Mordell conjecture [4] can be extended to the case of K(ℓ).
The paper is organized as follows. In Section 2 we discuss the ℓ-adic Lie algebras arising from Tate modules of abelian varieties and their centralizers. In Section 3 we deal with analogues of the Tate conjecture on homomorphisms over arbitrary fields. In Section 4 we prove the main result.
I am grateful to A.N. Parshin for stimulating discussions.
Tate modules, ℓ-adic Lie groups and Lie algebras
Let K be a field,K its algebraic closure and Gal(K) = Aut(K/K) its absolute Galois group. If m is a positive integer that is not divisible by char(K) then we write µ m for the cyclic order m multiplicative subgroup of mth roots of unity in K and K(µ m ) for the corresponding cyclotomic extension of K. If ℓ is a prime different from char(K) then we write K(ℓ) for the "infinite" cyclotomic extension
E = K(µ ℓ ∞ ) = ∪ ∞ i=1 K(µ ℓ i ).
It is well known that the compact Galois group Gal(K(ℓ)/K) is canonically isomorphic to a closed subgroup of Z * ℓ . We write χ ℓ : Gal(K) → Z * ℓ for the corresponding cyclotomic character; its kernel coincides with Gal(K(ℓ)) = Aut(K/K(ℓ)).
We write Z ℓ (1) for the projective limit of the groups µ ℓ j where the transition map µ ℓ j+1 → µ ℓ j is raising to ℓth power. It is well known that Z ℓ (1) is a free Z ℓ -module of rank 1 provided with the natural structure of a Galois module while the defining homomorphism
Gal(K) → Aut Z ℓ (Z ℓ (1)) = Z * ℓ coincides with χ ℓ . Let us consider the 1-dimensional Q ℓ -vector space Q ℓ (1) = Z ℓ (1) ⊗ Z ℓ Q ℓ ,
which carries the structure of a Galois module provided by the same character
χ ℓ : Gal(K) → Z * ℓ ⊂ Q * ℓ = Aut Q ℓ (Q ℓ (1)
). Let A be an abelian variety over K and End K (A) the ring of its K-endomorphisms. If X and Y are abelian varieties over K then we write Hom K (X, Y ) for the group of K-homomorphisms from X to Y . If m is as above then we write A m for the kernel of multiplication by m in A(K). The subgroup A m is a free Z/mZ-module of rank 2dim(A) [6] and a Galois submodule of A(K).
We write T ℓ (A) for the ℓ-adic Tate module of A, which is the projective limit of the groups A ℓ j while the transition map A ℓ j+1 → A ℓ j is multiplication by ℓ [6]. It is well-known that T ℓ (A) is a free Z ℓ -module of rank 2dim(A), the natural map T ℓ (A) → A ℓ j gives rise to the isomorphisms T ℓ (A)/ℓ j = A ℓ j and the Galois actions on A ℓ j give rise to the natural continuous homomorphism (ℓ-adic representation)
ρ ℓ,A = ρ ℓ,A,K : Gal(K) → Aut Z ℓ (T ℓ (A)),
which provides T ℓ (A) with the natural structure of a Gal(K)-module [10]. The image
G ℓ,A,K = ρ ℓ,A (Gal(K)) ⊂ Aut Z ℓ (T ℓ (A))
is a compact ℓ-adic Lie group [10]; clearly, G ℓ,A,K ⊂ 1 + ℓ j End Z ℓ (T ℓ (A)) if and only if Gal(K) acts trivially on T ℓ (A)/ℓ j = A ℓ j , i.e., A ℓ j ⊂ A(K).
Clearly, Gal(K(ℓ)) = Aut(K/K(ℓ)) is a compact normal subgroup of Gal(K). We write G 1 ℓ,A,K for its image ρ ℓ,X (Gal(K(ℓ))), which is a compact normal Lie subgroup of G ℓ,A,K . By definition,
G 1 ℓ,A,K = G ℓ,A,K(ℓ) . Let us consider the 2dim(A)-dimensional Q ℓ -vector space V ℓ (A) = T ℓ (A) ⊗ Z ℓ Q ℓ . One may view T ℓ (A) as the Z ℓ -lattice in V ℓ (A). We have G ℓ,A,K ⊂ Aut Z ℓ (T ℓ (A)) ⊂ Aut Q ℓ (V ℓ (A)),
which provides V ℓ (A) with the natural structure of a Gal(K)-module.
Notice that the Lie algebra of the compact ℓ-adic Lie group
Aut Z ℓ (T ℓ (A)) co- incides with End Q ℓ (V ℓ (A)). The Lie algebra g ℓ,A of G ℓ,A,K is a Q ℓ -linear Lie subalgebra of End Q ℓ (V ℓ (A)). The Lie algebra g 0 ℓ,A of G 1 ℓ,A,K is an ideal in g ℓ,A
. It is known [10] that the Lie algebras g ℓ,A and g 0 ℓ,A will not change if we replace K by its finite algebraic extension.
Let Id be the identity map on V ℓ (A) and let tr :
End Q ℓ (V ℓ (A)) → Q ℓ be the trace map. Let det : Aut Q ℓ (V ℓ (A)) → Q * ℓ be the determinant map. We write sl(V ℓ (A)) for the Lie subalgebra of traceless linear operators in End Q ℓ (V ℓ (A)). Let End g ℓ,A (V ℓ (A)) be the the centralizer of g ℓ,A in End Q ℓ (V ℓ (A)) and End g 0 ℓ,A (V ℓ (A)) be the the centralizer of g 0 ℓ,A in End Q ℓ (V ℓ (A)). Clearly, Q ℓ Id ⊂ End g ℓ,A (V ℓ (A)) ⊂ End g 0 ℓ,A (V ℓ (A)) ⊂ End Q ℓ (V ℓ (A)). Remark 2.1. Since g ℓ,A is the Lie algebra of G ℓ,A,K , End g ℓ,A (V ℓ (A)) ⊃ End G ℓ,A,K (V ℓ (A)) = End Gal(K) (V ℓ (A)). Since g 0 ℓ,A is the Lie algebra of G 1 ℓ,A,K , End g 0 ℓ,A (V ℓ (A)) ⊃ End G 1 ℓ,A,K (V ℓ (A)) = End Gal(K(ℓ)) (V ℓ (A)) = End Gal(E) (V ℓ (A)
). In the following two propositions we assume that A has positive dimension.
Proposition 2.2. (1) g 0 ℓ,A = g ℓ,A ∩ sl(V ℓ (A)) ⊂ sl(V ℓ (A)). (2) Suppose that g ℓ,A contains the homotheties Q ℓ Id. Then g ℓ,A = Q ℓ Id ⊕ g 0 ℓ,A . In particular, the centralizers End g ℓ,A (V ℓ (A)) and End g 0 ℓ,A (V ℓ (A)) do coin- cide.
Proof. It is known [8, Sect. 1.3] that a choice of a polarization on A gives rise to a nondegenerate alternating bilinear form
e l : V ℓ (A) × V ℓ (A) → Q ℓ (1) such that e l (ρ ℓ,A (σ)(x), ρ ℓ,A (σ)(y)) = χ ℓ (σ) · e ℓ (x, y) ∀x, y ∈ V ℓ (A); σ ∈ Gal(K).
By fixing a (non-canonical) isomorphism of Q ℓ -vector spaces
Q ℓ (1) ∼ = Q ℓ ,
we may assume that the alternating form e l takes on values in Q ℓ . We obtain that G ℓ,A,K lies in the corresponding group of symplectic similitudes
Gp(V ℓ (A), e ℓ ) = {s ∈ Aut Q ℓ (V ℓ (A)) | ∃c ∈ Q * ℓ such that e l (sx, sy) = c · e l (x, y) ∀x, y ∈ V ℓ (A)} ⊂ Aut Q ℓ (V ℓ (A))
and χ ℓ coincides with the composition of
ρ ℓ,A : Gal(K) ։ G ℓ,A,K and G ℓ,A,K ⊂ Gp(V ℓ (A), e ℓ ) c → Q * ℓ where the scalar c(s) is defined by e ℓ (sx, sy) = c(s) · e ℓ (x, y) ∀x, y ∈ V ℓ (A); s ∈ Gp(V ℓ (A), e ℓ ).
Clearly, c :
Gp(V ℓ (A), e ℓ )→Q * ℓ , s → c(s) is a homomorphism of ℓ-adic Lie groups and c(s) dim(A) = det(s) ∀s ∈ Gp(V ℓ (A), e ℓ ) (recall that V ℓ (A) is a 2dim(A)-dimensional Q ℓ -vector space).
It is also clear that G 0 ℓ,A,K coincides with the kernel of the homomorphism of ℓ-adic Lie groups c : G ℓ,A,K ⊂ Gp(V ℓ (A), e ℓ ) → Q * ℓ , and therefore g 0 ℓ,A coincides with the kernel of the corresponding tangent map of Lie algebras
g ℓ,A → Q ℓ .
On the other hand, one may easily check that the tangent map is 1 dim(A) tr : g ℓ,A → Q ℓ (because tr is the tangent map to det.) This implies that
g 0 ℓ,A = g ℓ,A ∩ sl(V ℓ (A)
). This proves the first assertion of Proposition. In order to prove the second assertion, notice that Q ℓ Id ∩ sl(V ℓ (A)) = {0} and therefore g ℓ,A contains Q ℓ Id ⊕ g 0 ℓ,A . On the other hand, since g 0 ℓ,A is the kernel of g ℓ,A → Q ℓ , its codimension in g ℓ,A is (at most) 1. This implies that g ℓ,A = Q ℓ Id ⊕ g 0 ℓ,A .
Proposition 2.3. There exists a finite separable algebraic field extension K 0 /K that enjoys the following properties. If K ′ /K 0 is a finite separable algebraic field extension then
End Gal(K ′ ) (V ℓ (A)) = End g ℓ,A (V ℓ (A)).End G0 (V ℓ (A)) = End V0 (V ℓ (A)) = End g ℓ,A (V ℓ (A)).
The preimage of G 0 in Gal(K) is an open subgroup of finite index and therefore coincides with Gal(K 0 ) for a certain finite separable algebraic field extension K 0 of K. It follows that
End Gal(K0) (V ℓ (A)) = End G0 (V ℓ (A)) = End g ℓ,A (V ℓ (A)).
If K ′ /K 0 is a finite separable algebraic field extension then Gal(K ′ ) is a compact subgroup of finite index in Gal(K 0 ) and its image G ′ = ρ ℓ,A (Gal(K ′ )) is a closed subgroup of finite index in G 0 and therefore is open in G 0 and therefore is also open in G ℓ,A,K . As above, V ′ = log(G ′ ) is an open subset of g ℓ,A that contains 0 and G ′ = exp(V ′ ) and
End Gal(K ′ ) (V ℓ (A)) = End G ′ (V ℓ (A)) = End V ′ (V ℓ (A)) = End g ℓ,A (V ℓ (A)).
Homomorphisms of abelian varieties
Throughout this Section, X, Y, Z, A are abelian varieties over K and ℓ is a prime different from char(K). We write Hom K (X, Y ) for the (finitely generated free) commutative group of K-homomorphisms from X to Y . If X = Y = A then Hom K (X, Y ) coincides with the ring End K (A) of K-endomorphisms of A.
There is a natural embedding of Z ℓ -modules i X,Y,K : Hom K (X, Y ) ⊗ Z ℓ ֒→ Hom Gal(K) (T ℓ (X), T ℓ (Y )), whose cokernel is torsion free [6,16].
(X, Y ) ⊗ Z ℓ ֒→ Hom Gal(L) (T ℓ (X), T ℓ (Y ))
is bijective then i X,Y,K is also bijective. This assertion follows easily from the following obvious description of Gal(L/K)-invariants
Hom K (X, Y ) = {Hom L (X, Y )} Gal(L/K) , Hom Gal(K) (T ℓ (X), T ℓ (Y )) = {Hom Gal(L) (T ℓ (X), T ℓ (Y )
)} Gal(L/K) and the Gal(L/K)-equivariance of i X,Y,L .
Extending i X,Y,K by Q ℓ -linearity, we obtain the natural embedding of Q ℓ -vector spacesĩ (1) The map i X,Y,K is bijective if and onlyĩ X,Y,K is bijective.
X,Y,K : Hom K (X, Y ) ⊗ Q ℓ ֒→ Hom Gal(K) (V ℓ (X), V ℓ (Y )),
(2) Let us put Z = X × Y . If the embedding
i Z,Z,K : End K (Z) ⊗ Q ℓ ֒→ End Gal(K) (V ℓ (Z)) is bijective theñ i X,Y,K : Hom K (X, Y ) ⊗ Q ℓ ֒→ Hom Gal(K) (V ℓ (X), V ℓ (Y ))
is also bijective.
Tate [15,16] conjectured and G. Faltings [4,5] proved that this embedding is actually a bijection when K is finitely generated over the field Q of rational numbers. Theorem 3.3. Suppose that K is field and ℓ is a prime that is different from char(K). Let us put E = K(ℓ). Suppose that A is an abelian variety of positive dimension over K such that for all finite separable algebraic field extensions K ′ /K the embeddingĩ
A,A,K ′ : End K ′ (A) ⊗ Q ℓ ֒→ End Gal(K ′ ) (V ℓ (A))
is bijective. If g ℓ,A contains the homotheties Q ℓ Id then the injective maps
i A,A,E : End E (A) ⊗ Q ℓ ֒→ End Gal(E) (V ℓ (A)) and i A,A,E : End E (A) ⊗ Z ℓ ֒→ End Gal(E) (T ℓ (A)) are bijective.
Proof. Let us consider the field K 2 = K(A ℓ 2 ) of definition of all points of A ℓ 2 and put E 2 = K 2 (ℓ). Clearly, K 2 /K and E 2 /E are finite Galois extensions. Applying Remark 3.1 (to E 2 /E instead of L/K) and Lemma 3.2.1 to X = Y = A, we observe that in the course of the proof we may (and will) assume that
K = K 2 = K(A ℓ 2 ),
i.e., A ℓ 2 ⊂ A(K). Since ℓ 2 ≥ 4, it follows from a result of A. Silverberg [14] that all K-endomorphisms of A are defined over K. In particular,
End K (A) = End E (A) = EndK(A).
Using Proposition 2.3, we may replace K by its finite separable algebraic extension in such a way that
End Gal(K ′ ) (V ℓ (A)) = End g ℓ,A (V ℓ (A))
for all finite separable algebraic field extensions K ′ of K.
Let K 0 /K satisfies the conclusion of Proposition 2.3. Replacing K 0 by its normal closure over K, we may and will assume that K 0 /K is a finite Galois extension. Let us put E 0 = K 0 (ℓ). Clearly, E 0 /E is a finite Galois extension and
End E (A) = End K (A) = End K0 (A) = End E0 (A).
By the assumption of Theorem 3.3,
End Gal(K0) (V ℓ (A)) = End K0 (A) ⊗ Q ℓ .
By Proposition 2.3,
End Gal(K0) (V ℓ (A)) = End g ℓ,A (V ℓ (A)).
This implies that
End K0 (A) ⊗ Q ℓ = End g ℓ,A (V ℓ (A)).
By Proposition 2.2, A)). This implies that
End g ℓ,A (V ℓ (A)) = End g 0 ℓ,A (V ℓ (End E0 (A) ⊗ Q ℓ = End K0 (A) ⊗ Q ℓ = End g 0 ℓ,A (V ℓ (A)
). By Remark 2.1 applied to K 0 and E 0 (instead of K and E),
End g 0 ℓ,A (V ℓ (A)) ⊃ End Gal(E0) (V ℓ (A)). So, we get End E0 (A) ⊗ Q ℓ = End g 0 ℓ,A (V ℓ (A)) ⊃ End Gal(E0) (V ℓ (A)) ⊃ End E0 (A) ⊗ Q ℓ , which implies that End E0 (A) ⊗ Q ℓ = End g 0 ℓ,A (V ℓ (A)) = End Gal(E0) (V ℓ (A)) = End E0 (A) ⊗ Q ℓ . In particular, End E0 (A) ⊗ Q ℓ = End Gal(E0) (V ℓ (A)). Now Lemma 3.2.1 applied to X = Y = A and to E 0 (instead of K) implies that End E0 (A) ⊗ Z ℓ = End Gal(E0) (T ℓ (A)).
It follows from Remark 3.1 (applied to E 0 /E instead of L/K) that
End E (A) ⊗ Z ℓ = End Gal(E) (T ℓ (A)).
Again, Lemma 3.2.1 tells us that
End E (A) ⊗ Q ℓ = End Gal(E) (V ℓ (A)).
Lemma 3.4 (of Clifford). Let G be a group and H its normal subgroup. Let W be a vector space of finite positive dimension over a field k. Let ρ : G → Aut k (W ) be a semisimple (completely reducible) linear representation of G. Then the corresponding H-module W is also semisimple.
Proof. Let us split W into a finite direct sum W = ⊕W i of simple G-modules W i . By Theorem (49.2) of [3], the corresponding H-modules W i are semisimple. This implies that the H-module W is a direct sum of semisimple H-modules W i 's and therefore is also semisimple. Proof. Since L/K is Galois, the subgroup Gal(L) of Gal(K) is normal. Now the result follows from Lemma 3.4.
Homotheties, centralizers and semisimplicity
Theorem 4.1 (of Bogomolov). Suppose that K is a field that is finitely generated over Q and ℓ is a prime. Let A be an abelian variety of positive dimension over K, Then g ℓ,A contains the homotheties Q ℓ Id.
Proof. When K is a number field, this assertion was proven by Bogomolov [1,2]. The case of arbitrary finitely generated K is also known [13, . Indeed, there exist a number field F , an abelian variety B over F with dim(A) = dim(B) and an isomorphism of Z ℓ -modules u : T ℓ (A) ∼ = T ℓ (B) such that u −1 G ℓ,B,F u = G ℓ,A,K . Extending u by Q ℓ -linearity, we obtain the isomorphism of Q ℓ -vector spaces V ℓ (A) ∼ = V ℓ (B), which we still denote by u. Clearly, u −1 g ℓ,B u = g ℓ,A . Since F is a number field, g ℓ,B contains all the homotheties, which implies that g ℓ,A also contains all the homotheties. This ends the proof.
Proof of Theorem 1.1. In light of Remark 1.2, we may and will assume that X and Y are defined over K. Let us put A = X × Y . Since K is finitely generated over Q, every finite algebraic extension K ′ of K is also finitely generated over Q. By the theorem of Faltings [4,5], the injectioñ
i A,A,K ′ : End K ′ (A) ⊗ Q ℓ ։ End Gal(K ′ ) (V ℓ (A))
is bijective. Thanks to Theorem 4.1, we know that g ℓ,A contains the homotheties Q ℓ Id. Now the desired result follows from Theorem 3.3 combined with Lemma 3.2.
Corollary 4.2. Suppose that K is a field that is finitely generated over Q and ℓ is a prime. Let E 1 be a field extension of K that lies in K(ℓ). If X and Y are abelian varieties over E 1 then the natural embedding of Z ℓ -modules Hom E1 (X, Y ) ⊗ Z ℓ ֒→ Hom Gal(E1) (T ℓ (X), T ℓ (Y )) is bijective.
Proof. Since K(ℓ)/E 1 is a Galois extension, the result follows from Theorem 1.1 combined with Remark 3.1 applied to K(ℓ)/E 1 . Theorem 4.3. Suppose that K is a field that is finitely generated over Q and ℓ is a prime. Let L/K be a finite or infinite Galois extension. (E.g., L = K(ℓ).) Let A be an abelian variety of positive dimension over K. Then the Gal(L)-module V ℓ (A) is semisimple.
Proof. Faltings [4,5] proved that the Gal(K)-module V ℓ (A) is semisimple. This also covers the case when L/K is a finite (Galois) extension. The case of infinite L/K follows from Faltings' result combined with Proposition 3.5.
Theorem 4.4. Suppose that K is a field that is finitely generated over Q . Let L/K be a finite or infinite Galois extension. Let A be an abelian variety of positive dimension over K. Then the Gal(L)-module A ℓ is semisimple for all but finitely many primes ℓ.
Proof. For all but finitely many primes ℓ the Gal(K)-module A ℓ is semisimple. Indeed, when K is a number field, this assertion is contained in Corollary 5.4.3 on p. 316 of [18] (the proof is based on results of Faltings [4]). The same proof works over arbitrary fields that are finitely generated over Q, provided one replaces the reference to Prop. 3.1 of [4] by the reference to the corollary on p. 211 of [5]. Since L/K is Galois, Gal(L) is a normal subgroup of Gal(K). Now the desired result follows from Lemma 3.4 applied to k = F ℓ , W = A ℓ and G = Gal(K), H = Gal(L).
Proof. (Compare with [ 9 ,
9Prop. 1 and its proof].) Let us choose open neighborhoods V of 0 in g ℓ,A and U of Id in G ℓ,A,K such that the ℓ-adic exponential map exp and logarithm map log establish mutually inverse Q ℓ -analytic isomorphisms between V and U .LetG 0 be an open subgroup of G ℓ,A,K that lies in U . (The existence of such G 0 follows from Corollary 2 in [11, Part II, Ch. 4, Sect. 8, p. 117].) Then V 0 = log(G 0 ) is an open subset of g ℓ,A that contains 0 and G 0 = exp(V 0 ). Clearly, G 0 has finite index in G ℓ,A,K and
Remark 3. 1 .
1Notice that if L/K is a finite or infinite Galois extension and i X,Y,L : Hom L
see [16, Section 1, displayed formula (2) on p. 135]. The following observations are due to J. Tate [16, Sect. 1, Lemma 1 and Lemma 3 and its proof on p. 135].
Proposition 3 . 5 .
35Let L/K be a finite or infinite Galois extension of K. If the Gal(K)-module V ℓ (A) is semisimple then the Gal(L)-module V ℓ (A) is semisimple.
Sect. 1, p. 2] and follows from the number field case with the help of Serre's variant of the Hilbert irreducibility theorem for infinite extensions ([13, Sect. 1], [12, Sect. 10.6], [7, Prop. 1.3 on pp. 163-164])
Sur l'algébricité des représentations ℓ-adiques. F A Bogomolov, C. R. Acad. Sci. Paris Sér. A-B. 290F.A. Bogomolov, Sur l'algébricité des représentations ℓ-adiques. C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A701-A703.
Points of finite order on abelian varieties. F A Bogomolov, Izv. Akad. Nauk SSSR Ser. Mat. 44Math. USSR Izv.F.A. Bogomolov, Points of finite order on abelian varieties. Izv. Akad. Nauk SSSR Ser. Mat. 44:4 (1980), 782-804; Math. USSR Izv. 17 (1981), 55-72.
Representation theory of finite groups and associative algebras. . W Ch, I Curtis, Reiner, Interscience PuvblishersNew York LondonCh. W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras. Interscience Puvblishers, New York London, 1962.
Endlichkeitssätze für abelsche Varietätenüber Zahlkörpern. G Faltings, Invent. Math. 73G. Faltings, Endlichkeitssätze für abelsche Varietätenüber Zahlkörpern. Invent. Math. 73 (1983), 349-366;
. Erratum, 75381Erratum, 75 (1984), 381.
Complements to Mordell. G Faltings, Rational points. G. Faltings, G. WüstholzBraunschweigSecond editionG. Faltings, Complements to Mordell. In: Rational points (G. Faltings, G. Wüstholz, eds.) Second edition. Aspects of Mathematics, E6. Friedr. Vieweg & Sohn, Braunschweig, 1986.
D Mumford, Abelian varieties. Oxford University Presssecond editionD. Mumford, Abelian varieties, second edition. Oxford University Press, 1974.
Abelian varieties -Galois representations and properties of ordinary reduction. R Noot, Compositio Math. 97R. Noot, Abelian varieties -Galois representations and properties of ordinary reduction. Compositio Math. 97 (1995), 161-171.
Sur les groupes des congruence des variétés abéliennes. J.-P Serre, Izv. Akad. Nauk SSSR ser. matem. 28J.-P. Serre, Sur les groupes des congruence des variétés abéliennes. Izv. Akad. Nauk SSSR ser. matem. 28:1 (1964), 3-18;
. I I Oeuvres, Springer-VerlagBerlinOEuvres II, pp. 230-245. Springer-Verlag, Berlin, 1986.
J.-P Serre, Sur les groupes de Galois attachés aux groupes p-divisibles. Proc. Conf. Local Fields (Driebergen, 1966). Berlin; BerlinSpringer-VerlagJ.-P. Serre, Sur les groupes de Galois attachés aux groupes p-divisibles. Proc. Conf. Local Fields (Driebergen, 1966), pp. 118-131, Springer, Berlin, 1967; OEuvres II, pp. 325-338. Springer-Verlag, Berlin, 1986.
Abelian ℓ-adic representations and elliptic curves. J.-P Serre, Addison-WesleyRedwood City2nd editionJ.-P. Serre, Abelian ℓ-adic representations and elliptic curves, 2nd edition. Addison-Wesley, Redwood City, 1989.
J.-P Serre, Lie groups and Lie algebras. Berlin HeidelbergSpringer-Verlag15002nd editionJ.-P. Serre, Lie groups and Lie algebras, 2nd edition. Lecture Notes in Mathematics 1500, Springer-Verlag, Berlin Heidelberg, 1992.
J.-P Serre, Lectures on the Mordell-Weil Theorem. BraunschweigThird editionJ.-P. Serre, Lectures on the Mordell-Weil Theorem, Third edition. Friedr. Viehweg & Sohn, Braunschweig, 1997.
J.-P Serre, Lettresá Ken. BerlinSpringer-VerlagRibet du 1/1/1981 et 29/1/1981. OEuvres IVJ.-P. Serre, Lettresá Ken Ribet du 1/1/1981 et 29/1/1981. OEuvres IV, pp. 1-20. Springer- Verlag, Berlin, 2000.
Fields of definition for homomorphisms of abelian varieties. A Silverberg, J. Pure Applied Algebra. 77A. Silverberg, Fields of definition for homomorphisms of abelian varieties. J. Pure Applied Algebra 77 (1992), 253-262.
Algebraic cycles and poles of zeta functions. J Tate, Arithmetical algebraic geometry. New YorkHarper and RowJ. Tate, Algebraic cycles and poles of zeta functions. In: Arithmetical algebraic geometry, Harper and Row, New York, 1965, 93-110.
Endomorphisms of abelian varieties over finite fields. J Tate, Invent. Math. 2J. Tate, Endomorphisms of abelian varieties over finite fields. Invent. Math. 2 (1966), 134-144.
Torsion of Abelian varieties in finite characteristic. Yu G Zarhin, Mat. Zametki. 22Yu. G. Zarhin, Torsion of Abelian varieties in finite characteristic. Mat. Zametki 22:1 (1977), 1-11;
. Mathematical Notes. 22Mathematical Notes 22 (1978), 493 -498.
A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction. Yu G Zarhin, Invent. Math. 79Yu. G. Zarhin, A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction. Invent. Math. 79 (1985), 309 -321.
Finiteness problems in Diophantine geometry. Yu G Zarhin, A N Parshin, Amer. Math. Soc. Transl. 2arXiv 0912.4325 [math.NTYu. G. Zarhin, A.N. Parshin, Finiteness problems in Diophantine geometry. Amer. Math. Soc. Transl. (2) 143 (1989), 35 -102; arXiv 0912.4325 [math.NT] .
| [] |
[
"DNA Sequence Complexity Reveals Structure Beyond GC Content in Nucleosome Occupancy",
"DNA Sequence Complexity Reveals Structure Beyond GC Content in Nucleosome Occupancy"
] | [
"Hector Zenil \nDepartment of Computer Science\nUniversity of Oxford\nOxfordU.K\n\nCentre for Molecular Medicine\nDepartment of Medicine Solna\nAlgorithmic Dynamics Lab, Unit of Computational Medicine\nSciLife Lab\nKarolinska In-stitute\nStockholmSweden\n\nThe Segal Model\n\n",
"Peter Minary \nDepartment of Computer Science\nUniversity of Oxford\nOxfordU.K\n"
] | [
"Department of Computer Science\nUniversity of Oxford\nOxfordU.K",
"Centre for Molecular Medicine\nDepartment of Medicine Solna\nAlgorithmic Dynamics Lab, Unit of Computational Medicine\nSciLife Lab\nKarolinska In-stitute\nStockholmSweden",
"The Segal Model\n",
"Department of Computer Science\nUniversity of Oxford\nOxfordU.K"
] | [] | We introduce methods that rapidly evaluate a battery of informationtheoretic and algorithmic complexity measures on DNA sequences in application to potential binding sites for nucleosomes. The first application of this new tool demonstrates structure beyond GC content on DNA sequences in the context of nucleosome binding. We tested the measures on well-studied genomic sequences of size 20K and 100K bps. The measures reveal the known in vivo versus in vitro predictive discrepancies, but they also uncover the internal structure of G and C within the nucleosome length, thus disclosing more than simply GC content when one examines alphabet transformations that separate and scramble the GC content signal and the DNA sequence. Most current prediction methods are based upon training (e.g. k-mer discovery), the one here advanced, however, is a training-free approach to investigating informative measures of DNA information content in connection with structural nucleosomic packing. | null | [
"https://arxiv.org/pdf/1708.01751v2.pdf"
] | 35,432,511 | 1708.01751 | 43d6fd0fc3a7593e350361cc9a4fc74d7ac84ee9 |
DNA Sequence Complexity Reveals Structure Beyond GC Content in Nucleosome Occupancy
August 9, 2017
Hector Zenil
Department of Computer Science
University of Oxford
OxfordU.K
Centre for Molecular Medicine
Department of Medicine Solna
Algorithmic Dynamics Lab, Unit of Computational Medicine
SciLife Lab
Karolinska In-stitute
StockholmSweden
The Segal Model
Peter Minary
Department of Computer Science
University of Oxford
OxfordU.K
DNA Sequence Complexity Reveals Structure Beyond GC Content in Nucleosome Occupancy
August 9, 2017Nucleosome positioning/occupancyDNA sequence complex- ityDNA structuregenomic information content
We introduce methods that rapidly evaluate a battery of informationtheoretic and algorithmic complexity measures on DNA sequences in application to potential binding sites for nucleosomes. The first application of this new tool demonstrates structure beyond GC content on DNA sequences in the context of nucleosome binding. We tested the measures on well-studied genomic sequences of size 20K and 100K bps. The measures reveal the known in vivo versus in vitro predictive discrepancies, but they also uncover the internal structure of G and C within the nucleosome length, thus disclosing more than simply GC content when one examines alphabet transformations that separate and scramble the GC content signal and the DNA sequence. Most current prediction methods are based upon training (e.g. k-mer discovery), the one here advanced, however, is a training-free approach to investigating informative measures of DNA information content in connection with structural nucleosomic packing.
The challenge of Predicting Nucleosome Organization
DNA in the cell is organized into a compact form, called chromatin. Nucleosome organization in the cell is referred to as the primary chromatin structure and can depend on the 'suitability' of a sequence for accommodating a nucleosome, which may in turn be influenced by the packing of neighbouring nucleosomes. Depending on the context, nucleosomes can inhibit or facilitate transcription factor binding and are thus a very active area of research. The location of low nucleosomic occupancy is key to understanding active regulatory elements and genetic regulation that is not directly encoded in the genome but rather in a structural layer of information. Structural organization of DNA in the chromosomes is widely known to be heavily driven by GC content, involving a simple count of G and C occurrences in the DNA sequence. Despite its simplicity, uncovering exactly how (and to what extent) GC content drives/affects nucleosome organization is among the central questions in modern molecular biology.
GC content, local and short-range signals carried by DNA sequence 'motifs' or fingertips (k-mer statistical regularities), have been found (Refs. [1] and [2]) to be able to determine a good fraction of the structural (and thus functional) properties of DNA, such as nucleosome occupancy, but the explanatory (and predictive) power of GC content (the G or C count in a sequence) alone and sequence motifs display very significant differences in vivo versus in vitro [3].
Despite intensive analysis of the statistical correspondence between in vitro and in vivo positioning, there is a lack of consensus as to the degree to which the nucleosome landscape is intrinsically specified by the DNA sequence [4], as well as in regards to the apparently profound difference in dependence in vitro versus in vivo. Because the nucleosome landscape is known to be significantly dependent on the DNA sequence, it encodes the structural information of the DNA (particularly demonstrated in vitro). We consider this an opportunity to compare the performance of complexity measures, in order to discover how much of the information encoded in a sequence in the context of the nucleosome landscape can be recovered from information-content versus algorithmic complexity measures. Nucleosome location is thus an ideal test case to probe how informative sequence-based indices of complexity can be in determining a structural (and ultimately functional) property of genomic DNA.
Algorithmic Information Theory in Genomic Sequence Profiling
Previous applications of measures based upon algorithmic complexity include experiments on the evaluation of lossless compression lengths of sets of genomes [5,7] and more recently [13] with interesting results. For example, in a landmark paper in the area, a measure of algorithmic mutual information was introduced to distinguish sequence similarities by way of minimal encodings and lossless compression algorithms in which a mitochondrial phylogenetic tree that conformed to the evolutionary history of known species was reconstructed [6,7]. These approaches have, however, either been purely theoretical or have effectively been reduced to applications of Shannon entropy rather than of algorithmic complexity because, implementations of lossless compression are actually entropy estimators [8,38]. In some other cases, control tests have been missing. For example, in the comparison of the compressibility indices of different genomes [6,7], GC content (counting every G and C in the sequence) can reconstruct the same if not a more accurate phylogenetic tree. This is because two species that are close to each other evolutionarily will have similar GC content (see e.g. [9]). Species close to each other will have similar DNA sequence entropy values, allowing lossless compression algorithms to compress statistical regularities of genomes of related species with similar compression rates. Here we intend to go beyond previous attempts, in breadth as well as depth, using better-grounded algorithmic measures and more biologically relevant test cases.
Current Sequence-based Prediction Methods
While the calculation of GC content is extremely simple, the reasons behind its ability to predict the structural properties of DNA are largely unknown [10,11]. For example, it has been shown that low GC content can explain low occupancy, but high GC content can mean either high or low occupancy [12]. Current algorithms that build upon while probing beyond GC content have been largely influenced by sequence motif ( [14,2]) and dinucleotide models [15]-and to a lesser degree by k-mers [1], DNA sequence motifs that are experimentally isolated and used for their informative value in determining the structural properties of DNA. Table 1 (SI) shows the in vitro nucleosome occupancy dependence on GC content, with a correlation of 0.684 (similar to that reported by Kaplan [3]) for the well-studied 20K bp genomic region (187K -207K) of Yeast Chromosome 14 [16]. Knowledge-based methods dependent on observed sequence motifs [17,18] are computationally cost-efficient alternatives for predicting genome-wide nucleosome occupancy. However, they are trained on experimental statistical data and are not able to predict anything that has not been observed before. They also require context, as it may not be sufficient to consider only short sequence motifs such as dinucleotides [19,3].
More recently, deep machine learning techniques have been applied to DNA accessibility related to chromatin and nucleosome occupancy [17]. However, these techniques require a huge volume of data for training if they are to predict just a small fraction of data with marginally improved accuracy as compared to more traditional approaches based on k-mers, and they have not shed new light on the sequence dependence of occupancy.
Here we test the ability of a general set of measures, statistical and algorithmic, to be informative about nucleosome occupancy and/or about the relationship between the affinity of nucleosomes with certain sequences and their complexities.
Methods
The Dinucleotide Wedge Model
The formulation of models of DNA bending was initially prompted by a recognition that DNA must be bent for packaging into nucleosomes, and that bending would be an informative index of nucleosome occupancy. Various dinucleotide models can account reasonably well for the intrinsic bending observed in different sets of sequences, especially those containing A-tracts [19].
The Wedge model [20] suggests that bending is the result of driving a wedge between adjacent base pairs at various positions in the DNA. The model assumes that bending can be explained by wedge properties attributed solely to an AA dinucleotide (8.7 degrees for each AA). No current model provides a completely accurate explanation of the physical properties of DNA such as bending [21], but the Wedge model (like the more basic Junction model which is less suitable for short sequences and less general [22]) reasonably predicts the bending of many DNA sequences [23]. Although it has been suggested that trinucleotide models may make for greater accuracy in explaining DNA curvature in some of the sequences, dinucleotide models remain the most effective [19].
The Segal Model
Segal et al. established a probabilistic model to characterize the possibility that one DNA sequence is occupied by a nucleosome [16]. They constructed a nucleosome-DNA interaction model and used a hidden Markov model (HMM) to obtain a probability score. The model is based mainly on a 10-bp sequence periodicity that indicates the probability of any base pair being covered by a nucleosome.
All k-nucleotide models, including that of Segal et al., are based upon knowledge-based sequence motifs and are thereby dependent on certain previously learned patterns. They can only account for local curvature and local predictions, not longer range correlations. Perhaps the fact that k-nucleotide models for k > 2 have not been proven to provide a significant advantage over k = 2 has led researchers to disregard longer range signals across DNA sequences involved in both DNA curvature and nucleosome occupancy [24]. To date, these models, including that of Kaplan [3] (which considers up to k = 5) and of Segal et al., are considered the gold standard for comparison purposes.
To study the extent of different signals in the determination of nucleosome occupancy, we applied some basic transformations to the original genomic DNA sequence. The SW transformation substitutes G and C for S (Strong interaction), and A and T for W (for Weak interaction). The RY transformation substitutes A and G for R (for puRines), and C and T for Y (pYrimidines).
Complexity-based Genomic Profiling
In what follows, we generate a function score f c for every complexity measure c (detailed descriptions in the S.I.) by applying each measure to a sliding window of length 147 nucleotides (nts) across a 20K and 100K base pair (bps) DNA sequence from Yeast chromosome 14. At every position of the sliding window, we get a sequence score for every c that is used to compare against in vivo and in vitro experimental occupancy.
The following measures (followed by the name we refer to in parenthesis throughout the text) are here introduced. Among the measures considered are . Middle: Calculated correlation values between the SW DNA transformation, carrying the GC content signal, found highly correlated to the Segal model but poorly explaining in vivo occupancy data. Bottom: The RY DNA transformation, an orthogonal signal to SW (and thus to GC content) whose values report a non-negligible max-min correlation, suggesting that the mixing of AT and GC carries some information about nucleosome occupancy (even if weaker than GC content), with in vivo values showing greatest correlation values unlike SW/GC and thus possibly neglected in predictive models (such as Segal's).
entropy-based ones (see Supplementary Material for exact definitions):
• Shannon entropy (Entropy) with uniform probability distribution.
• Entropy rate with uniform probability distribution. • Coding Theorem Method (CTM) as an estimator of algorithmic randomness by way of algorithmic probability via the algorithmic Coding theorem (see Supplementary Material) relating causal content and classical probability [33,34].
• Logical Depth (LD) as a BDM-based (see below) estimation of logical depth [25], a measure of sophistication that assigns both algorithmically simple and algorithmically random sequences shallow depth, and everything else higher complexity, believed to be related to biological evolution [26,27].
And a hybrid measure of complexity combining local approximations of algorithmic complexity by CTM and global estimations of (block) Shannon entropy (see Supplementary Material for exact definitions):
• The Block Decomposition Method (BDM) that approximates Shannon entropy-up to a logarithmic term-for long sequences, but Kolmogorov-Chaitin complexity otherwise, as in the case of short nucleotides [28].
We list lossless compression under information-theoretic measures and not under algorithmic complexity measures, because implementations of lossless compression algorithms such as Compress and all those based on LempelZivWelch (LZ or LZW) as well as derived algorithms (ZIP, GZIP, PNG, etc.) are actually entropy estimators [38,8,28].
BDM allows us to expand the range of application of both CTM and LD to longer sequences by using Shannon entropy. However, if sequences are divided into short enough subsequences (of 12 nucleotides) we can apply CTM and avoid any trivial connection to Shannon entropy and thus to GC content.
Briefly, to estimate the algorithmic probability [29,30]-on which the measure BDM is based-of a DNA sequence (e.g. the sliding window of length 147 nucleoides or nt), we produce an empirical distribution [33,34] to compare with by running a sample of up to 325 433 427 739 Turing machines with 2 states and 4 symbols (the number of nucleotide types in a DNA sequence) with empty input. If a DNA sequence is algorithmically random, then very few computer programs (Turing machines) will produce it, but if it has a regularity, either statistical or algorithmic, then there is a high probability of its being produced. Producing approximations to algorithmic probability provides approximations to algorithmic complexity by way of the so-called algorithmic Coding Theorem [30,33,34]. Because the procedure is computationally expensive (and ultimately uncomputable) only the full set of strings of up to 12 bits was produced, and thus direct values can be given only to DNA sequences of up to 12 digits (binary for RY and SW and quaternary for full-alphabet DNA sequences).
The tool is available at http://complexitycalculator.com/ where the user can calculate the information content and algorithmic complexity using the methods here introduced on any DNA segment for the purpose of similar of any other investigation of the structure of DNA and beyond. Fig. 1 shows the correlations between in vivo, in vitro, and the Segal model. In contrast, the SW transformation captures GC content, which clearly drives most of the nucleosome occupancy, but the correlation with the RY transformation that loses all GC content is very interesting. While significantly lower, it is existent and indicates a signal not contained in the GC content alone, as verified in Fig. 4.
Results
Complexity-based Indices
In Table 1 (SI), we report the correlation values found between experimental nucleosome occupancy data and ab initio training-free complexity measures. BDM alone explains more than any other index, including GC content in vivo, and unlike all other measures LD is negatively correlated, as theoretically expected [35] and numerically achieved [28], it being a measure that assigns low logical depth to high algorithmic randomness, with high algorithmic randomness implying high entropy (but not the converse).
Surprisingly, entropy alone does not capture all the GC signals, which means that there is more structure in the distributions of Gs and Cs beyond the GC content alone. However, entropy does capture GC content in vivo, suggesting that local nucleotide arrangements (for example, sequence motifs) have a greater impact on in vivo prediction. Compared to entropy, BDM displays a higher correlation with in vivo nucleosome occupancy, thereby suggesting more internal structure than is captured by GC content and Shannon entropy alone.
Model Curvature versus Complexity Indices
The dinucleotide model incorporates knowledge regarding sequence motifs that are known to have specific natural curvature properties and adds to the knowledge and predictive power that GC content alone offers.
Using the Wedge dinucleotide model we first estimated the predicted curvature on a set of 20 artificially generated sequences (Table 4 (SI)) with different statistical properties, in order to identify possibly informative informationtheoretic and algorithmic indices. As shown in Table 2 (SI), we found all measures negatively correlated to the curvature modelled, except for LD, which displays a positive correlation-and the highest in absolute value-compared to all the others. Since BDM negatively correlates with curvature, it is expected that the minima may identify nucleosome positions (see next subsection).
An interesting observation in Table 2 (SI) concerns the correlation values between artificially generated DNA sequences and DNA structural curvature according to the Wedge nucleotide model: all values are negatively correlated, but curvature as predicted by the model positively correlates with LD, in exact inverse fashion vis-a-vis the correlation values reported in Table 1 (SI). This is consonant with the theoretically predicted relation between algorithmic complexity and logical depth [28]. All other measures (except for LD) behave similarly to BDM.
The results in Tables 1 and 2 (SI) imply that for all measures both extrema (min and max for BDM and max and min for LD) may be indicative of high nucleosome occupancy. In the next section we explore whether extrema of these measures are also informative about nucleosome locations.
Nucleosome Dyad and Centre Location Test
Positioning and occupancy of nucleosomes are closely related. Nucleosome positioning is the distribution of individual nucleosomes along the DNA sequence and can be described by the location of a single reference point on the nucleosome, such as its dyad of symmetry [36]. Nucleosome occupancy, on the other hand, is a measure of the probability that a certain DNA region is wrapped onto a histone octamer. Fig. 2 shows the predictive capabilities of algorithmic indices for nucleosome dyad and centre location when nucleosomic regions are placed against a background of (pseudo-) randomly generated DNA sequences with the same average GC content as themselves (∼ 0.5). BDM outperforms all methods in accuracy and strength. When taking the local min/max as potential indicators of nucleosome centres, we find that GC content fails (by design, as the surrounding sequences have the same GC content as the nucleosomic region of interest); lossless compression (Compress) performs well on the second half of the nucleosomes (left panel) but fails for the first half (right panel). Entropy performs as poorly as Compress-not surprisingly, as lossless compression algorithms are Entropy estimators.
The results for BDM and LD suggest that the first 4 nucleosomal DNA sequences, of which 3 are clones, display greater algorithmic randomness (BDM) than the statistically pseudo-randomly generated background (surrounding sequences), while all other nucleosomes are of significantly lower algorithmic randomness (BDM) and mixed (both high and low) structural complexity (LD). The same robust results were obtained after several replications with different pseudo-random backgrounds. Moreover, the signal produced by similar nucleosomes with strong properties [37], such as clones 601, 603 and 605, had similar shapes and convexity. Fig. 3 shows the strength of the BDM signal at indicating the nucleosome centres based on the min/max value of the corresponding functions. The signal- Figure 2: Nucleosome centre prediction (red dots) of 14 nucleosomes on an intercalated background of pseudo-random DNA segments of 147 nts with the same average GC content as the surrounding nucleosomic regions. Values are normalized between 0 and 1 and they were smoothed by taking one data point per 40 and using and interpolating order 2. Experimentally known nucleosome centres (called dyads) are marked with dashed lines. Panels on the right have their nucleotide centre estimated by the centre of the nucleosomic sequence (also dashed lines). Predictions are based on the local min/max values (up to 75 nts to each side) from the actual/estimated dyad/centre. Some red dots may appear to be placed slightly off but this is because there was a local min or max that vanished after the main curve was made smoother for visualization purposes. Figure 3: Signal-to-noise ratio histogram. Distributions of centre predicted values demonstrating how BDM and LD are removed from a normal distribution thus picking a signal, unlike GC content that distributes values normally and performs no better than chance on a pseudo-random background with similar GC content. BDM carries the strongest signal followed by LD skewed in the opposite direction both peaking closer to the nucleosome centres than GC content. On the x-axis are complexity values arranged in bins of 1000 as reported in Fig. 2. to-noise ratio is much stronger for BDM, and is also shifted by LD in the opposite direction (to BDM), as would be consistent with the relation between algorithmic complexity and logical depth (see S.I.).
Both BDM and LD spike at nucleosome positions stronger than GC content on a random DNA background with the same GC content, and perform better than entropy and compression. BDM is informative about every dyad or centre of nucleosomes, with 10 out of the 14 predicted within 1 to 3 bps distance and the rest within a 20 bps range. Unlike all other measures, LD performed better for the first half (left panel) of nucleosome centre locations than for the second half (right panel), suggesting that the nucleosomes of the first half may have greater structural organization. Table 3 (SI) compares distances to the nucleosome centres and error percentages as predicted without any training with BDM, to GC content prediction. The average distance between the predicted and the actual nucleosome centre is calculated to the closest local extreme (minima or maxima) within a window of 41 base pairs or bps (20 bps to each side plus the centre) from the actual centre (the experimentally known dyad or the centre nucleotide when the dyad was not known). In accordance with the results in Table 1 (SI) the maxima of BDM (minima of LD) could be informative about nucleosome positions and for those sequences, whose natural curvature is a fit to the superhelix, the minima BDM (maxima of LD) could also indicate nucleosome locations. This latter finding is supported by results in Table 2 (SI).
Our results suggest that if some measures of complexity peak where GC content is (purposely) tricked, there must be some structure different to GC content along the DNA sequence, either a distribution of GC content within the nucleosome length that is not related simply to the G and C count, or some other signal.
Informative Measures of High and Low Occupancy
To find the most informative measures of complexity c we maximized the potential separation by taking only the sequences with highest (X% high) and lowest (Y% low) nucleosome occupancy. To this end we took as cutoff values 2 and 0.2 respectively, generating 300 disjoint sequences each from a 100K DNA segment for highest and lowest nucleosome occupancy values. The 100K segment starting and ending points are 187K − 40K and 207K + 40K nts in the 14th Yeast chromosome, so 40K nts surrounding the original shorter 20K sequence first explored. In Fig. 4 it was puzzling to find that the Segal model correlates less strongly than GC content alone for in vivo, suggesting that the model assigns greater weight to k-mer information than to GC content for these extreme cases given that we already knew that the Segal model is mostly driven by GC content (Fig. 1 middle). The box plot for the Segal model indicates that the model does not work as well for extreme sequences of high occupancy, with an average of 0.6 where the maximum over the segments on which these nucleosome regions are contained reaches an average correlation of ∼ 0.85 (in terms of occupancy), as shown in Fig. 1 for in vitro data. This means that these high occupancy sequences are on the outer border of the standard deviation in terms of accuracy in the Segal model. While the best model is the one that best separates the highest from the lowest occupancy, and therefore is clearly Segal's model. Except for informationtheoretic indices (entropy and Compress), all algorithmic complexity indices were found to be informative of high and low occupancy. Moreover, all algorithmic complexity measures display a slight reduction in accuracy in vivo versus in vitro, as is consistent with the limitations of current models such as Segal's. All but the Segal model are, however, training-free measures, in the sense that they do not contain any k-mer information related to high or low occupancy and thus are naive indices, yet all algorithmic complexity measures were informative to different extents, with CTM and BDM performing best and LD performing worst, LD displaying inverted values for high and low occupancy as theoretically expected (because LD assigns low LD to high algorithmic complexity) [35]. Also of note is the fact that CTM and BDM applied to the RY transformation were informative of high versus low occupancy, thereby revealing a signal different to GC content that models such as Segal's partially capture in their encoded k-mer information. Interestingly, GC content alone outperforms the Segal model for high occupancy both in vitro and in vivo, but the Segal model outperforms GC content for low occupancy.
Lossless compression was the worst behaved, showing how CTM and BDM outperform what is usually used as an estimator of algorithmic complexity [38, Figure 4: Box plots reporting the informative value of complexity indices in vivo and in vitro for segments of lowest and highest occupancy, providing an overview of the informative value of sequence-dependent complexity measures. The occupancy score is given by the re-scaling of the complexity value f c (yaxis) so that the highest value is 1 and the lowest 0. In the case of the Segal model, f c is the direct score sequence based on the probability assigned by the model [16], with no re-scaling (because it is already scaled from origin with probability values between 0 and 1). Other cases not shown (e.g. entropy rate or Compress on RY or SW) had no significant results. Magenta and pink (bright colours) signify measures of algorithmic complexity; in dark gray colour are information-theoretic based measures. 8,28]. Unlike entropy alone, however, lossless compression does take into consideration sequence repetitions, averaging over all k-mers up to the compression algorithm sliding window length. The results thus indicate that averaging over all sequence motifs-both informative and not-deletes all advantages, thereby justifying specific knowledge-driven k-mer approaches introduced in models such as Segal's.
Conclusions
The Kaplan and Segal models are considered to be the most accurate and the gold standard for predicting in vitro nucleosome occupancy. However, previous approaches including Segal and Kaplan requires extensive (pre-)training. In contrast, all measures considered by our approach are training free.
These training-free measures revealed that there is more structure to nucleosome occupancy than GC content, and potentially to k-mer structure as well (e.g. non-AT-based mers), based on the correlations found in RY transformations indicative of low versus high occupancy.
The nucleotide location test suggests a complexity hierarchy in which natural non-nucleosomic regions are less algorithmically random than nucleosomic regions, which in turn are less algorithmically random than pseudo-randomly generated DNA sequences with GC content equal to the nucleosomic regions. When pseudo-random regions are placed between nucleosomes, we showed that BDM tends to identify nucleosomic regions with a preference for lower algorithmic randomness with relative accuracy, and more consistently than GC content, which showed no pattern and mostly failed, as was expected, when fooled with a background of similar GC content.
We have thus gone beyond previous attempts to connect and apply measures of complexity to structural and functional properties of genomic DNA, specifically in the highly active and open challenge of nucleosome occupancy in molecular biology.
A direction for future research suggested by our work is the exploration of the use of these complexity indices to complement current machine learning approaches for reducing the feature space, by, e.g., determining which kmers are more and less informative, and thereby ensuring better prediction results. Another direction is a more extensive investigation of the possible use of genomic profiling for other types of structural and functional properties of DNA, with a view to contributing to, e.g., HiC techniques or protein encoding/promoter/enhancer region detection, and to furthering our understanding of the effect of extending the alphabet transformation of a sequence to epigenetics.
Supplementary Material 5 Indices of Information and of Algorithmic Complexity
Here we describe alternative measures to explore correlations from an informationtheoretic and algorithmic (hence causal) complexity perspective.
Shannon Entropy
Central to information theory is the concept of Shannon's information entropy, which quantifies the average number of bits needed to store or communicate a message. Shannon's entropy determines that one cannot store (and therefore communicate) a symbol with n different symbols in less than log(n) bits. In this sense, Shannon's entropy determines a lower limit below which no message can be further compressed, not even in principle. Another application (or interpretation) of Shannon's information theory is as a measure for quantifying the uncertainty involved in predicting the value of a random variable.
For an ensemble X(R, p(x i )), the Shannon information content or entropy of X is then given by
H(X) = − n i=1 p(x i ) log 2 p(x i )
where R is the set of possible outcomes (the random variable), n = |R| and p(x i ) is the probability of an outcome in R.
Entropy Rate
The function R gives what is variously denominated as rate or block entropy and is Shannon entropy over blocks or subsequences of X of length b. That is,
R(X) = b=|X| min b=1 H(X b )
If the sequence is not statistically random, then R(X) will reach a low value for some b, and if random, then it will be maximally entropic for all blocks b. R(X) is computationally intractable as a function of sequence size, and typically upper bounds are realistically calculated for a fixed value of b (e.g. a window length). Notice that, as discussed in the main text, having maximal entropy does not by any means imply algorithmic randomness (c.f. 5.3).
Compression algorithms
Two widely used lossless compression algorithms were employed.
Bzip2
Bzip2 is a lossless compression method that uses several layers of compression techniques stacked one on top of the other, including Run-length encoding (RLE), BurrowsWheeler transform (BWT), Move to Front (MTF) transform, and Huffman coding, among other sequential transformations. Bzip2 compresses more effectively than LZW, LZ77 and Deflate, but is considerably slower.
Compress
Compress is a lossless compression algorithm based on the LZW compression algorithm. LempelZivWelch (LZW) is a lossless data compression algorithm created by Abraham Lempel, Jacob Ziv, and Terry Welch, and is considered universal for an infinite sliding window (in practice the sliding window is bounded by memory or choice). It is considered universal in the sense of Shannon entropy, meaning that it approximates the entropy rate of the source (an input in the form of a file/sequence). It is the algorithm of the widely used Unix file compression utility 'Compress', and is currently in the international public domain.
Measures of Algorithmic Complexity
A binary sequence s is said to be random if its Kolmogorov complexity C(s) is at least twice its length. It is a measure of the computational resources needed to specify the object. Formally,
C(s) = min{|p| : T (p) = s}
where p is a program that outputs s running on a universal Turing machine T . A technical inconvenience of C as a function taking s to the length of the shortest program that produces s is its uncomputability. This is usually considered a major problem. The measure was first conceived to define randomness and is today the accepted objective mathematical measure of complexity, among other reasons because it has been proven to be mathematically robust (by virtue of the fact that several independent definitions converge to it).
The invariance theorem guarantees that complexity values will only diverge by a constant (e.g. the length of a compiler, a translation program between T 1 and T 2 ) and will converge at the limit. Formally,
|C(s) T1 − C(s) T2 | < c
Lossless Compression as Approximation to C
Lossless compression is traditionally the method of choice when a measure of algorithmic content related to Kolmogorov-Chaitin complexity C is needed. The Kolmogorov-Chaitin complexity of a sequence s is defined as the length of the shortest computer program p that outputs s running on a reference universal Turing machine T . While lossless compression is equivalent to algorithmic complexity, actual implementations of lossless compression (e.g. Compress) are heavily based upon entropy rate estimations [38,8,28] that mostly deal with statistical repetitions or k-mers of up to a window length size L, such that k ≤ L.
Algorithmic Probability as Approximation to C
Another approach consists in making estimations by way of a related measure, Algorithmic Probability [33,34]. The Algorithmic Probability of a sequence s is the probability that s is produced by a random computer program p when running on a reference Turing machine T . Both algorithmic complexity and Algorithmic Probability rely on T , but invariance theorems for both guarantee that the choice of T is asymptotically negligible.
One way to minimize the impact of the choice of T is to average across a large set of different Turing machines all of the same size. The chief advantage of algorithmic indices is that causal signals in a sequence may escape entropic measures if they do not produce statistical regularities. And it has been the case that increasing the length of k in k-nucleotide models of structural properties of DNA have not returned more than a marginal advantage.
The Algorithmic Probability [29] (also known as Levin's semi-measure [30]) of a sequence s is a measure that describes the expected probability of a random program p running on a universal prefix-free Turing machine T producing s.
Bennett's Logical Depth
Another measure of great interest is logical depth [25]. The logical depth (LD) of a sequence s is the shortest time logged by the shortest programs p i that produce s when running on a universal reference Turing machine. In other words, just as algorithmic complexity is associated with lossless compression, LD can be associated with the shortest time that a Turing machine takes to decompress the sequence s from its shortest computer description. A multiplicative invariance theorem for LD has also been proven [25]. Estimations of Algorithmic Probability and logical depth of DNA sequences were performed as determined in [33,34].
Unlike algorithmic (Kolmogorov-Chaitin) complexity C, logical depth is a measure related to 'structure' rather than randomness. LD can be identified with biological complexity [26,39] and is therefore of great interest when comparing different genomic regions.
Measures Based on Algorithmic Probability and on Logical Depth
The Coding theorem method (or simply CTM) is a method [33,34] rooted in the relation between C(s) and m(s) specified by Algorithmic Probability, that is, between frequency of production of a sequence from a random program and its Kolmogorov complexity as described by Algorithmic Probability. Essentially, it uses the fact that the more frequent a sequence the lower its Kolmogorov complexity, and sequences of lower frequency have higher Kolmogorov complexity. Unlike algorithms for lossless compression, the Algorithmic Probability approach not only produces estimations of C for sequences with statistical regularities, but it is deeply rooted in a computational model of Algorithmic Probability, and therefore, unlike lossless compression, has the potential to identify regularities that are not statistical (e.g. a sequence such as 1234...), that is, sequences with high entropy or no statistical regularities but low algorithmic complexity [38,8].
Let (n, m) be the space of all n-state m-symbol Turing machines, n, m > 1 and s a sequence, then:
D(n, m)(s) = |{T ∈ (n, m) : T produces s}| |{T ∈ (n, m)}| (1)
T is a standard Turing machine as defined in the Busy Beaver problem by Radó [40] with 4 symbols (in preparation for the calculation of the DNA alphabet size). Then using the relation established by the Coding theorem, we have:
CT M (s) = − log 2 (D(n, m)(s))(2)
That is, the more frequently a sequence is produced the lower its Kolmogorov complexity, and vice versa. CTM is an upper bound estimation of Kologorov-Chaitin complexity.
From CTM, a measure of Logical Depth can also be estimated-as the computing time that the shortest Turing machine (i.e. the first in the quasilexicographic order) takes to produce its output s upon halting. CTM thus produces both an empirical distribution of sequences up to a certain size, and an LD estimation based on the same computational model.
Because CTM is computationally very expensive (equivalent to the Busy Beaver problem [40]), only short sequences (currently only up to length k = 12) have associated estimations of their algorithmic complexity. To approximate the complexity of genomic DNA sequences up to length k = 12, we calculated D(5, 4)(s), from which CT M (s) was approximated.
To calculate the Algorithmic Probability of a DNA sequence (e.g. the sliding window of length 147 nt) we produced an empirical Algorithmic Probability The resulting distribution came from 325 378 582 327 non-unique sequences (after removal of those sequences only produced by 5 or fewer machines/programs).
Relation of BDM to Shannon Entropy and GC Content
The Block Decomposition Method (BDM) is a divide-and-conquer method that can be applied to longer sequences on which local approximations of C(s) using CTM can be averaged, thereby extending the range of application of CTM. Formally,
BDM (s, k) = s k log(n) + CTM(r).(3)
The set of subsequences s k is composed of the pairs (r, n), where r is an element of the decomposition of sequence s of size k, and n the multiplicity of each subsequence of length k. BDM (s) is a computable approximation from below to the algorithmic information complexity of s, C(s). BDM approximations to K improve with smaller departures (i.e. longer k-mers) from the Coding Theorem method. When k decreases in size, however, we have shown [28] that BDM approximates the Shannon entropy of s for the chosen k-mer distribution. In this sense, BDM is a hybrid complexity measure that in the 'worst case' behaves like Shannon entropy and in the best approximates K. We have also shown that BDM is robust when instead of partitioning a sequence, overlapping subsequences are used, but this latter method tends to over-fit the value of the resultant complexity of the original sequence that was broken into k-mers. Table 3: Distance in nucleotides to local min/max within a window of 2 tests, around 40 and 140 nts around the centre on a pseudo-randomly generated DNA background with the same GC content as the mean of the GC content of the next contiguous nucleosomic region. Clearly the experiment is designed for GC content to fail, yet BDM predicts the nucleosome position (by its centre) in a high number of cases and with great accuracy, with 10 out of the 14 centres predicted to within a 1 to 3 nt distance, thereby suggesting that there is more structure than GC content. Contrast this to GC content performing no better than chance, with an average fractional distance of 0.538 versus 0.105 for BDM from the predicted centre. Likewise for windows around the centre of 40 [24]. Only the first 6 have known dyads
Figure 1 :
1Top: Correlation values of nucleosome occupancy (measured experimentally from chromosomic Yeast) on a sliding window of length 4K nt for both in vitro and in vivo against different measures/signals: the occupancy predictive Segal model (clearly better for in vitro)
•
Lossless compression (Compress) A set of measures of algorithmic complexity (see Supplementary Material for exact definitions):
theorem beautifully connects C(s) and m(s): C(s) ∼ − log m(s)
Table 1 :
1Spearman correlations between complexity indices with in vivo and in vitro experimental nucleosome occupancy data from position 187 001 bp to 207 000 bp on the 14th Yeast chromosome Turing machines with up to 5 states and 4 symbols (the number of nucleotides in a DNA sequence) with empty input (as required by Algorithmic Probability).in vitro in vivo
in vitro
1
0.5
in vivo
0.5
1
GC content
0.684
0.26
LD
-0.29
-0.23
Entropy
0.588
0.291
BDM
0.483
0.322
Compress
0.215
0.178
distribution from (5, 4) to compare with by running a sample of 325 433 427 739
Table 2 :
2Spearman correlation values of complexity score functions versus the Wedge dinucleotide model prediction of DNA curvature on 20 synthetically generated DNA sequencesGC
Entropy Entropy Compress BZip2 BDM
LD
content
rate (4)
p
0.047
0.051
0.0094
0.0079
0.048
0.0083 0.0019
rho
-0.45
-0.44
-0.57
-0.58
-0.45
-0.57
0.65
nt and 140 nt. All other methods (not included) reported intermediate values between GC content and BDMBDM
601
603
605
5Sr DNA
pGub
chicken β−
globulin
-48
-2
-1
-1
19
25
msat CAG TATA
CA
NoSecs
TGGA
TGA BadSecs
2
1
1
1
14
1
1
1
Table 4 :
4The 20 short artificial DNA sequences generated covering a wide range of patterns and regularities used to find informative measures of DNA curvature.AAAAAAAAAAAA
ATATATATATAT
AAAAAATTTTTT
AAAAAAAAATAA
AAAAAAAACAAT
AAGATCTACACT
ATAGAACGCTCC
ACCTATGAAAGC
TAGGCGGCGGGC
TCGTTCGCGAAT
TGCACGTGTGGA
CTAAACACAATA
CTCTCAGGTCGT
CTCGTGGATATC
CCACGATCCCGT
GGCGGGGGGTGG GGGGGGGCGGGC GGGGGGCCCCCC
GCGCGCGCGCGC GGGGGGGGGGGG
Table 5 :
514 Experimental nucleosome sequences
DNA structural correlation in short and long ranges. C Gu, The Journal of Physical Chemistry B. 11944Gu C et al. (2015) DNA structural correlation in short and long ranges. The Journal of Physical Chemistry B 119(44):13980-13990.
Structured nucleosome fingerprints enable highresolution mapping of chromatin architecture within regulatory regions. A N Schep, Genome research. 2511Schep AN et al. (2015) Structured nucleosome fingerprints enable high- resolution mapping of chromatin architecture within regulatory regions. Genome research 25(11):1757-1770.
The DNA-encoded nucleosome organization of a eukaryotic genome. N Kaplan, Nature. 4587236Kaplan N et al. (2009) The DNA-encoded nucleosome organization of a eukaryotic genome. Nature 458(7236):362-366.
An in vitro-identified high-affinity nucleosomepositioning signal is capable of transiently positioning a nucleosome in vivo. L E Gracey, Epigenetics & chromatin. 311Gracey LE et al. (2010) An in vitro-identified high-affinity nucleosome- positioning signal is capable of transiently positioning a nucleosome in vivo. Epigenetics & chromatin 3(1):1.
Compression and genetic sequence analysis. E Rivals, J P Delahaye, M Dauchet, O Delgrange, Biochimie. 78Rivals E, Delahaye JP, Dauchet M, Delgrange O (1996) Compression and genetic sequence analysis. Biochimie 78:315-322.
The similarity metric. M Li, X Chen, X Li, B Ma, P M Vitányi, IEEE transactions on Information Theory. 5012Li M, Chen X, Li X, Ma B, Vitányi PM (2004) The similarity metric. IEEE transactions on Information Theory 50(12):3250-3264.
Clustering by compression. R Cilibrasi, P M Vitányi, IEEE Transactions on Information theory. 514Cilibrasi R, Vitányi PM (2005) Clustering by compression. IEEE Trans- actions on Information theory 51(4):1523-1545.
Low Algorithmic Complexity Entropy-deceiving. H Zenil, N A Kiani, J ( Tegnér, Graphs Phys Rev E. acceptedZenil H, Kiani, NA, Tegnér, J (2017) Low Algorithmic Complexity Entropy-deceiving Graphs Phys Rev E. (accepted)
Both selective and neutral processes drive GC content evolution in the human genome. U Pozzoli, BMC evolutionary biology. 811Pozzoli U et al. (2008) Both selective and neutral processes drive GC content evolution in the human genome. BMC evolutionary biology 8(1):1.
G+C content dominates intrinsic nucleosome occupancy. D Tillo, T R Hughes, BMC bioinformatics. 1011Tillo D, Hughes TR (2009) G+C content dominates intrinsic nucleosome occupancy. BMC bioinformatics 10(1):1.
GC-content evolution in mammalian genomes: the biased gene conversion hypothesis. N Galtier, G Piganeau, D Mouchiroud, L Duret, Genetics. 1592Galtier N, Piganeau G, Mouchiroud D, Duret L (2001) GC-content evo- lution in mammalian genomes: the biased gene conversion hypothesis. Genetics 159(2):907-911.
Training-free atomistic prediction of nucleosome occupancy. P Minary, M Levitt, Proceedings of the National Academy of Sciences. 11117Minary P, Levitt M (2014) Training-free atomistic prediction of nucle- osome occupancy. Proceedings of the National Academy of Sciences 111(17):6293-6298.
On the Approximation of the Kolmogorov Complexity for DNA Lecture Notes in Computer Science book series. Diogo Pratas, Armando J Pinho, LNCS). 10255Diogo Pratas, Armando J. Pinho (2017) On the Approximation of the Kolmogorov Complexity for DNA Lecture Notes in Computer Science book series (LNCS), vol. 10255:pp 259-266.
Structure-based analysis of DNA sequence patterns guiding nucleosome positioning in vitro. F Cui, V B Zhurkin, Journal of Biomolecular Structure and Dynamics. 276Cui F, Zhurkin VB (2010) Structure-based analysis of DNA sequence pat- terns guiding nucleosome positioning in vitro. Journal of Biomolecular Structure and Dynamics 27(6):821-841.
The pitch of chromatin DNA is reflected in its nucleotide sequence. E N Trifonov, J L Sussman, Proc. Natl. Acad. Sci. USA 77. Natl. Acad. Sci. USA 77Trifonov, E. N., and Sussman, J. L. (1980) The pitch of chromatin DNA is reflected in its nucleotide sequence. Proc. Natl. Acad. Sci. USA 77, pp. 3816-3820.
A genomic code for nucleosome positioning. E Segal, Nature. 4427104Segal E et al. (2006) A genomic code for nucleosome positioning. Nature 442(7104):772-778.
Basset: Learning the regulatory code of the accessible genome with deep convolutional neural networks. D R Kelley, J Snoek, J L Rinn, Genome research. Kelley DR, Snoek J, Rinn JL (2016) Basset: Learning the regulatory code of the accessible genome with deep convolutional neural networks. Genome research.
A high-resolution atlas of nucleosome occupancy in yeast. W Lee, Nature genetics. 3910Lee W et al. (2007) A high-resolution atlas of nucleosome occupancy in yeast. Nature genetics 39(10):1235-1244.
An assessment of three dinucleotide parameters to predict DNA curvature by quantitative comparison with experimental data. A Kanhere, M Bansal, Nucleic acids research. 3110Kanhere A, Bansal M (2003) An assessment of three dinucleotide param- eters to predict DNA curvature by quantitative comparison with experi- mental data. Nucleic acids research 31(10):2647-2658.
Estimation of wedge components in curved DNA. L E Ulanovsky, E N Trifonov, Nature. 3266114Ulanovsky LE, Trifonov EN (1986) Estimation of wedge components in curved DNA. Nature 326(6114):720-722.
Structural details of an adenine tract that does not cause DNA to bend. A M Burkhoff, T D Tullius, Nature. 331Burkhoff AM, Tullius TD (1988) Structural details of an adenine tract that does not cause DNA to bend. Nature 331:455-457.
Intrinsically bent DNA. D M Crothers, T E Haran, J G Nadeau, J. Biol. Chem. 26513Crothers DM, Haran TE, Nadeau JG (1990) Intrinsically bent DNA. J. Biol. Chem 265(13):7093-7096.
DNA structure and function. R R Sinden, ElsevierSinden RR (2012) DNA structure and function. (Elsevier).
Sequencebased prediction of single nucleosome positioning and genome-wide nucleosome occupancy. T Van Der Heijden, J J Van Vugt, C Logie, J Van Noort, Proceedings of the National Academy of Sciences. 10938van der Heijden T, van Vugt JJ, Logie C, van Noort J (2012) Sequence- based prediction of single nucleosome positioning and genome-wide nu- cleosome occupancy. Proceedings of the National Academy of Sciences 109(38):E2514-E2522.
Logical depth and physical complexity. The Universal Turing Machine, A Half-Century Survey. C H Bennett, Bennett CH (1995) Logical depth and physical complexity. The Universal Turing Machine, A Half-Century Survey pp. 207-235.
Dissipation, information, computational complexity and the definition of organization in Santa Fe Institute Studies in the Sciences of Complexity -Proceedings Volume. C H Bennett, Addison-Wesley Publishing Company1Bennett CH (1993) Dissipation, information, computational complexity and the definition of organization in Santa Fe Institute Studies in the Sci- ences of Complexity -Proceedings Volume-. (Addison-Wesley Publishing Company), Vol. 1, pp. 215-215.
The limits of decidable states on open-ended evolution and emergence in ALIFE Conference. S Hernández-Orozco, F Hernández-Quiroz, H Zenil, MIT PressHernández-Orozco S, Hernández-Quiroz F, Zenil H (2016) The limits of decidable states on open-ended evolution and emergence in ALIFE Con- ference. (MIT Press).
A decomposition method for global evaluation of Shannon entropy and local estimations of algorithmic complexity. H Zenil, F Soler-Toscano, N A Kiani, S Hernández-Orozco, A Rueda-Toicen, arXiv:1609.00110arXiv preprintZenil H, Soler-Toscano F, Kiani NA, Hernández-Orozco S, Rueda-Toicen A (2016) A decomposition method for global evaluation of Shannon en- tropy and local estimations of algorithmic complexity. arXiv preprint arXiv:1609.00110.
A formal theory of inductive inference. parts i and ii. R J Solomonoff, Information and control. 71Solomonoff RJ (1964) A formal theory of inductive inference. parts i and ii. Information and control 7(1):1-22 and 224-254.
Laws of information conservation (nongrowth) and aspects of the foundation of probability theory. L A Levin, Problemy Peredachi Informatsii. 103Levin LA (1974) Laws of information conservation (nongrowth) and as- pects of the foundation of probability theory. Problemy Peredachi Infor- matsii 10(3):30-35.
Three approaches to the quantitative definition of information. A N Kolmogorov, International Journal of Computer Mathematics. 21-4Kolmogorov AN (1968) Three approaches to the quantitative definition of information. International Journal of Computer Mathematics 2(1-4):157- 168.
On the length of programs for computing finite binary sequences: statistical considerations. G J Chaitin, Journal of the ACM (JACM). 161Chaitin GJ (1969) On the length of programs for computing finite bi- nary sequences: statistical considerations. Journal of the ACM (JACM) 16(1):145-159.
Numerical evaluation of algorithmic complexity for short strings: A glance into the innermost structure of randomness. J P Delahaye, H Zenil, Applied Mathematics and Computation. 2191Delahaye JP, Zenil H (2012) Numerical evaluation of algorithmic complex- ity for short strings: A glance into the innermost structure of randomness. Applied Mathematics and Computation 219(1):63-77.
Calculating kolmogorov complexity from the output frequency distributions of small Turing machines. F Soler-Toscano, H Zenil, J P Delahaye, N Gauvrit, PloS one. 9596223Soler-Toscano F, Zenil H, Delahaye JP, Gauvrit N (2014) Calculating kolmogorov complexity from the output frequency distributions of small Turing machines. PloS one 9(5):e96223.
. F Soler-Toscano, H Zenil, J P Delahaye, N Gauvrit, Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures Computability. 22Soler-Toscano F, Zenil H, Delahaye JP, Gauvrit N (2013) Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures Computability, vol. 2, no. 2, pp 125-140.
A low resolution structure for the histone core of the nucleosome. A Klug, D Rhodes, J Smith, J Finch, J Thomas, Nature. 2875782Klug A, Rhodes D, Smith J, Finch J, Thomas J (1980) A low resolution structure for the histone core of the nucleosome. Nature 287(5782):509- 516.
A polar barrier to transcription can be circumvented by remodeler-induced nucleosome translocation. D A Gaykalova, 1273Gaykalova DA et al. (2011) A polar barrier to transcription can be cir- cumvented by remodeler-induced nucleosome translocation. Nucleic acids research p. gkq1273.
Algorithmic data analytics, small data matters and correlation versus causation in Computability of the World? Philosophy and Science in the Age of Big Data. H Zenil, Pietsch W, Wernecke J, Ott M.Springer Verlag). In pressZenil H (2013) Algorithmic data analytics, small data matters and cor- relation versus causation in Computability of the World? Philosophy and Science in the Age of Big Data, eds. Pietsch W, Wernecke J, Ott M. (Springer Verlag). In press.
Information increase in biological systems: how does adaptation fit? in Evolutionary systems. J D Collier, SpringerCollier JD (1998) Information increase in biological systems: how does adaptation fit? in Evolutionary systems. (Springer), pp. 129-139.
On non-computable functions. T Rado, Bell System Technical Journal. 413Rado T (1962) On non-computable functions. Bell System Technical Jour- nal 41(3):877-884.
| [] |
[
"Can fermion-boson stars reconcile multi-messenger observations of compact stars?",
"Can fermion-boson stars reconcile multi-messenger observations of compact stars?"
] | [
"Fabrizio Di Giovanni \nDepartamento de Astronomía y Astrofísica\nUniversitat de València\nDr. Moliner 5046100BurjassotValència)Spain\n",
"Nicolas Sanchis-Gual \nDepartamento de Matemática da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications (CIDMA)\nCampus de Santiago3810-183AveiroPortugal\n",
"Pablo Cerdá-Durán \nDepartamento de Astronomía y Astrofísica\nUniversitat de València\nDr. Moliner 5046100BurjassotValència)Spain\n",
"José A Font \nDepartamento de Astronomía y Astrofísica\nUniversitat de València\nDr. Moliner 5046100BurjassotValència)Spain\n\nObservatori Astronòmic\nUniversitat de València\nCatedrático José Beltrán 246980PaternaValència)Spain\n"
] | [
"Departamento de Astronomía y Astrofísica\nUniversitat de València\nDr. Moliner 5046100BurjassotValència)Spain",
"Departamento de Matemática da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications (CIDMA)\nCampus de Santiago3810-183AveiroPortugal",
"Departamento de Astronomía y Astrofísica\nUniversitat de València\nDr. Moliner 5046100BurjassotValència)Spain",
"Departamento de Astronomía y Astrofísica\nUniversitat de València\nDr. Moliner 5046100BurjassotValència)Spain",
"Observatori Astronòmic\nUniversitat de València\nCatedrático José Beltrán 246980PaternaValència)Spain"
] | [] | Mixed fermion-boson stars are stable, horizonless, everywhere regular solutions of the coupled Einstein-(complex, massive) Klein-Gordon-Euler system. While isolated neutron stars and boson stars are uniquely determined by their central energy density, mixed configurations conform an extended parameter space that depends on the combination of the number of fermions and (ultralight) bosons. The wider possibilities offered by fermion-boson stars could help explain the tension in the measurements of neutron star masses and radii reported in recent multi-messenger observations and nuclear-physics experiments. In this work we construct equilibrium configurations of mixed fermion-boson stars with realistic equations of state for the fermionic component and different percentages of bosonic matter. We show that our solutions are in excellent agreement with multimessenger data, including gravitational-wave events GW170817 and GW190814 and X-ray pulsars PSR J0030+0451 and PSR J0740+6620, as well as with nuclear physics constraints from the PREX-2 experiment. | 10.1103/physrevd.105.063005 | [
"https://arxiv.org/pdf/2110.11997v2.pdf"
] | 239,768,286 | 2110.11997 | 79d042ddc943f5e781e7822da026f3fa2a3bd0bd |
Can fermion-boson stars reconcile multi-messenger observations of compact stars?
Fabrizio Di Giovanni
Departamento de Astronomía y Astrofísica
Universitat de València
Dr. Moliner 5046100BurjassotValència)Spain
Nicolas Sanchis-Gual
Departamento de Matemática da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications (CIDMA)
Campus de Santiago3810-183AveiroPortugal
Pablo Cerdá-Durán
Departamento de Astronomía y Astrofísica
Universitat de València
Dr. Moliner 5046100BurjassotValència)Spain
José A Font
Departamento de Astronomía y Astrofísica
Universitat de València
Dr. Moliner 5046100BurjassotValència)Spain
Observatori Astronòmic
Universitat de València
Catedrático José Beltrán 246980PaternaValència)Spain
Can fermion-boson stars reconcile multi-messenger observations of compact stars?
(Dated: October 28, 2021)
Mixed fermion-boson stars are stable, horizonless, everywhere regular solutions of the coupled Einstein-(complex, massive) Klein-Gordon-Euler system. While isolated neutron stars and boson stars are uniquely determined by their central energy density, mixed configurations conform an extended parameter space that depends on the combination of the number of fermions and (ultralight) bosons. The wider possibilities offered by fermion-boson stars could help explain the tension in the measurements of neutron star masses and radii reported in recent multi-messenger observations and nuclear-physics experiments. In this work we construct equilibrium configurations of mixed fermion-boson stars with realistic equations of state for the fermionic component and different percentages of bosonic matter. We show that our solutions are in excellent agreement with multimessenger data, including gravitational-wave events GW170817 and GW190814 and X-ray pulsars PSR J0030+0451 and PSR J0740+6620, as well as with nuclear physics constraints from the PREX-2 experiment.
I. INTRODUCTION
The determination of the equation of state (EoS) of matter at the supernuclear densities attained in neutron star interiors is a long-standing issue in nuclear astrophysics (see [1,2] and references therein). High-precision measurements of the masses and radii of neutron stars are necessary to confidently constrain the EoS. Recent observations in both the electromagnetic channel and the gravitational-wave channel, together with constraints from nuclear physics, are helping shed light on this issue, yet uncertainties remain [3][4][5][6][7][8][9][10][11][12][13].
During the last decade it has been possible to accurately measure the mass of two milisecond pulsars with masses close to 2 M , PSR J1614-2230 [14,15] and PSR J0348+0432 [16]. These results impose a strong lower limit to the maximum mass of neutron stars and have constrained considerably the properties of dense matter [2]. However, only recently it has been possible an accurate joint determination of the mass and the radius of a neutron star. Bayesian inference on pulse-profile modeling of observations from the Neutron Star Interior Composition Explorer (NICER) of the rotation-powered, Xray milisecond pulsar PSR J0030+0451, yielded values of the mass and (circumferential) radius of ∼ 1.4 M and ∼ 13 km, respectively [5,6]. Even more recently, the same teams of researchers have reported the joint determination of the mass and radius of PSR J0740+6620 [10,11], the most massive known neutron star. Combining data from NICER and XMM-Newton [11], and also accounting for radio timing (Shapiro delay) in the case of [10] (see also [15]), these teams have inferred values for the mass and radius of 2.08 M and ∼ 13 km, respectively. The fact that J0740+6620 is about 50% more massive than J0030+0451 while both objects are essentially the same size challenges theoretical models of neutron-star interiors.
Gravitational waves have also been able to put joint constraints on the neutron star mass and radius. The first-ever detection of a binary neutron star merger by the LIGO-Virgo Collaboration (LVC), GW170817 [17], made it possible not only to place constraints on the individual masses of the components of the binary but also on the tidal deformability of neutron stars, which has been used to constrain the neutron star radius [3] (see also [9,18] and references therein).
In addition, the interpretation of the recent LVC detection of the compact binary merger event GW190814 [19] poses quite some difficulties. While the mass of the primary component, 23.2 M , allows to conclusively identify it as a black hole, the mass of the secondary, 2.50 − 2.67 M , raises doubts on the nature of this component, which might be either a black hole or a neutron star. If the latter were the case it would be the most massive neutron star ever observed. A number of recent investigations have tried to explain such a large mass [20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Proposals include the possibility that the secondary were a rapidly-rotating neutron star that collapsed to a spinning black hole before merger [20,21], a neutron star with a stiff high-density EoS or a sufficiently large spin [24] (see also [25]), or a neutron star with exotic degrees of freedom, i.e. a strange quark star, within the scenario in which neutron stars and quark stars coexist [26] (see also [27]). Somewhat more exotic possibilities involving slowly-rotating neutron stars in 4D Einstein-Gauss-Bonnet gravity [31], primordial black holes [29], Thorne-Żytkow objects [30], or dark-matteradmixed neutron stars [32,33], have also been suggested.
The neutron star radius can also be constrained by improving the measurement of nuclear interaction param-eters [1,34]. Very recently the PREX-2 experiment has measured with high accuracy the neutron skin thickness of 208 Pb [35] which constrains the neutron star radius for a 1.4 M neutron star to be larger than 13.25 km [36]. Although compatible with milisecond pulsar radius measurements, this result is in some tension with the gravitational-wave determinations [37]. The combined constraints of the multi-messenger data and PREX-2 measurements have been shown by [38,39] to be compatible with models of hybrid stars with first-order phase transition from nucleonic to quark matter in the core, a result disfavored by the analysis of [40].
Additionally, the nuclear physics modeling of realistic EoS at high densities has lead to the so-called hyperon problem [see e.g. [41] and references therein]. In order to reach the high masses necessary to fulfill the observational constraints on the maximum mass of neutron stars, models have to reach high central densities, at which the appearance of hyperons is expected. However, the presence of hyperons may soften the EoS at those densities and limit the possible values for the maximum mass, making it difficult to reach the ∼ 2 M constraint.
Motivated by these observational and experimental results we put forward in this paper a theoreticallymotivated new model based on mixed fermion-boson stars, i.e. neutron stars that incorporate some amount of bosonic matter. Using this model we are able to construct existence plots (mass-radius equilibrium configurations) compatible with multi-messenger observational data, including gravitational-wave events GW170817 and GW190814, and X-ray pulsars PSR J0030+0451 and PSR J0740+6620. We note that our model shares some similarities with those of [32,33] but also some differences. The study of [32] is only focused on GW190814 and explains the mass of the secondary by admixing neutron stars modelled by stiff EoS with nonannihilating weakly interacting massive particles dark matter. On the other hand, the very recent study of [33] also focuses only on GW190814 and explains the mass of the secondary by resorting to a neutron star admixed with at least 2.0 M dark matter made of axion-like particles. In our study (see below) we employ a complex scalar field while in [33] the authors consider a real field to model QCD axions.
Ultralight bosons form localized, coherently oscillating configurations akin to Bose-Einstein condensates [42,43]. For light-enough bosonic particles, i.e. with a mass µ ∼ 10 −22 eV, these condensates have been proposed to explain large-scale structure formation through darkmatter seeds [44,45]. Heavier bosons lead to much smaller configurations with the typical size and mass of neutron stars -hence the name boson stars [46,47] (see [48,49] and references therein). It is worth mentioning that recent examples have shown the intrinsic degeneracy between the prevailing Kerr black hole solutions of general relativity and boson-star solutions, using both gravitational-wave data [50] and electromagnetic data [51] (see also [52]). Moreover, macroscopic composites of fermions and bosons, dubbed fermion-boson stars, have also been proposed [53][54][55][56][57][58][59]. Such mixed configurations could form from the condensation of some primordial gas containing both types of particles or through episodes of accretion. The dynamical formation of fermion-boson stars through accretion along with their nonlinear stability properties has recently been studied by [55,58,60,61]. In most studies the neutron star is modeled with a polytropic EoS, the only exception being [62] who employed a realistic EoS. Mergers of fermion-boson stars have also been studied by [63].
This paper is organized as follows: Section II briefly describes the theoretical framework to build equilibrium models of fermion-boson stars. (Further details are reported in [60].) Section III contains our main results. Finally in Section IV we discuss our findings and outline possible extensions of this work.
II. FRAMEWORK
In our setup the scalar field is assumed to be only minimally coupled to Einstein's gravity. Therefore, fermions and bosons only interact gravitationally, with the total stress-energy tensor being the sum of both contributions,
T µν = T NS µν +T φ µν , where (using units with c = G = = 1) T NS µν = [ρ(1 + ) + P ]u µ u ν + P g µν ,(1)T φ µν = − 1 2 g µν ∂ αφ ∂ α φ − 1 2 µ 2φ φ − 1 4 λ(φφ) 2 + 1 2 (∂ µφ ∂ ν φ + ∂ µ φ∂ νφ ).(2)
The fermionic part involves the fluid pressure P , restmass density ρ, internal energy , and 4-velocity u µ , with g µν denoting the space-time metric. The bosonic matter is described by the complex scalar field φ (withφ being the complex conjugate) and by the particle mass µ and self-interaction parameter λ.
The equations of motion are obtained from the conservation laws of the stress-energy tensor and of the baryonic particles for the fermionic part
∇ µ T µν NS = 0, (3) ∇ µ (ρu µ ) = 0,(4)
and from the Klein-Gordon equation for the complex scalar field
∇ µ ∇ µ φ = µ 2 φ + λ|φ| 2 φ ,(5)
together with the Einstein equations, G µν = 8πT µν , for the spacetime dynamics. Mixed-star models are built using a static and spherically symmetric metric in Schwarzschild coordinates,
ds 2 = −α(r) 2 dt 2 + a(r) 2 dr 2 + r 2 (dθ 2 + sin θ 2 dϕ 2 ),(6)
written in terms of two geometrical functions a(r) and α(r). A harmonic time dependence ansatz for the scalar field is assumed, φ(r, t) = φ(r)e iωt , where ω is its eigenfrequency. Furthermore we consider a static perfect fluid u µ = (−1/α, 0, 0, 0). In order to construct equilibrium configurations we solve the following set of ordinary differential equations(ODEs), which are obtained from Einstein's equations:
da dr = a 2 1 − a 2 r + 4πr ω 2 α 2 + µ 2 + λ 2 φ 2 a 2 φ 2 +Ψ 2 + 2a 2 ρ(1 + ) ,(7)dα dr = α 2 a 2 − 1 r + 4πr ω 2 α 2 − µ 2 − λ 2 φ 2 a 2 φ 2 +Ψ 2 + 2a 2 P ,(8)dφ dr = Ψ,(9)dΨ dr = − 1 + a 2 − 4πr 2 a 2 (µ 2 φ 2 + λ 2 φ 4 + ρ(1 + ) − P ) Ψ r − ω 2 α 2 − µ 2 − λφ 2 a 2 φ 2 ,(10)dP dr = −[ρ(1 + ) + P ] α α ,(11)
where the prime indicates the derivative with respect to r. The system of equations is closed by the EoS for the nucleonic matter. Previous work on fermion-boson stars [55,58,60,61] assumed a simple polytropic EoS to build equilibrium models and a Γ-law EoS for numerical evolutions to take into account possible shock-heating (thermal) effects. In this work we improve the microphysical treatment of the fermionic part of the models and construct new equilibrium solutions described with realistic, tabulated EoS (see Section III). Despite our models are spherically symmetric we can nevertheless apply them to the X-ray milisecond pulsars J0030+0451 and J0740+6620 since the degree of deformation rotation might induce in these objects is negligible [5,6]. The set of ODEs (7)- (11) is an eigenvalue problem for the frequency of the scalar field ω which depends on two parameters, namely the central value of the rest-mass density, ρ c , and of the scalar field, φ c . As in [60] to obtain the value of the frequency for each solution we employ a two-parameter shooting method to search for the physical solution that fulfills the requirement of vanishing φ at the outer boundary. Once ω is obtained, we use a 4th-order Runge-Kutta integrator to solve the ODEs and reconstruct the radial profiles of all variables.
In order to construct physical initial data we must impose appropriate boundary conditions for the geometric FIG. 1: Gravitational mass vs circumferential fermionic radius for different realistic EoS including the observational constraints (95% confidence levels) from LIGO-Virgo, NICER/XMM-Newton, and mass measurements of two high mass pulsars. We also indicate the PREX-2 1σ lower limit on the radius for a 1.4 M neutron star. Grey curves correspond to all cold EoS compiled by [64] and [65]. We highlight in black the three EoS used for the calculations in this work.
quantities and for both the scalar field and the perfect fluid. We require that the metric functions are regular at the origin. We employ Schwarzschild outer boundary conditions, together with a vanishing scalar field. Explicitly, the boundary conditions read
a(0) = 1, φ(0) = φ c , α(0) = 1, lim r→∞ α(r) = lim r→∞ 1 a(r)
,
Ψ(0) = 0, lim r→∞ φ(r) = 0, ρ(0) = ρ c , P (0) = Kρ Γ c .(12)
The total gravitational mass of the solutions can be defined as
M T = lim r−→∞ r 2 1 − 1 a 2 ,(13)
which coincides with the Arnowitt-Desser-Misner (ADM) mass at infinity. We define the radius of the fermionic part as the radial coordinate at which the fluid pressure vanishes, R f = r(P = 0), which for the Schwarzschild metric coincides with the circumferential radius. As the bosonic component of our mixed stars does not have a hard surface, the radius of this contribution, R b , is evaluated, as customary, as the radius of the sphere containing 99% of bosonic particles. The particle numbers for both bosons and fermions are computed as in [60]. Figure 1 displays the mass-radius relations for a large sample of realistic EoS (grey curves) corresponding to all cold EoS described in [64] and [65]. Those take into account generic nuclear effects while some of them also include hyperons, pion and kaon condensates, and quarks. We compare those results with the observational constraints placed by NICER on PSRJ0030+0451 [5,6], the NICER/XMM-Newton combined analysis of PSRJ0740+6620 [10,11], the constraints set by gravitational-wave event GW170817 [3] (EOS insensitive relations), the mass measurement of two neutron stars with masses close to 2 M , PSR J0348+0432 [16] and PSR J1614-2230 [66], and the lower mass component in the binary merger GW190814 [19] as a possible neutron star with mass ≥ 2.5 M . All constraints are given as 95% (2σ) confidence intervals. Those have been computed using the publicly available posteriors provided by the different groups. Additionally Fig. 1 also shows the 1σ lower limit for the radius of a 1.4 M derived from the PREX-2 measurements of the neutron skin thickness [36] (see, however, the related discussion in [37,67]).
III. RESULTS
For our analysis we select the three EoS highlighted in black in Fig. 1, namely ALF2, which is a hybrid EoS with mixed APR nuclear matter and colour-flavor-locked quark matter [68], MS1b which is a relativistic mean field theory EoS [68], and DD2 [69], which is a finite temperature hadronic EoS which we evaluate at zero temperature and beta equilibrium. The three EoS fulfill the constraints from the recent NICER and XMM-Newton results, the observations of the two high-mass pulsars, as well as the PREX-2 constraints. Of the three, only MS1b would be compatible with the low-mass component of GW190814 being a neutron star. On the other hand, only ALF2 and DD2 are compatible with the results of GW170817, albeit only marginally. This selection of EoS illustrates the current tension that exists between different observational and experimental constraints of the mass and radius of neutron stars. Although it is still possible to find EoS that fit all constraints within the 2σ confidence level (except for GW190814), if these constraints were to tighten in future observations maintaining similar median values, it would pose a serious problem to the modelling of matter at high densities. We explore next the possibility of alleviating some of this tension by considering stars with a bosonic component additional to the fermionic component.
With this aim we build sequences of equilibrium configurations of both, fermion stars described by those three EoS, and of mixed stars with different values of the ratio of the number of bosons to fermions, N b /N f , and particle mass µ. Models are computed for N b /N f = {0.1, 0.2, 0.3} and µ = {0.1, 1.0} in our units, which correspond to µ = {1.34 × 10 −11 , 1.34 × 10 −10 } eV. For all models the self-interaction parameter λ is set to zero (mini-boson stars) and the fermionic matter always dominates over FIG. 2: Total gravitational mass vs circumferential fermionic radius for equilibrium models of neutron stars (black lines) and boson-fermion stars (magenta and cyan lines) for different parameters of the boson-to-fermion ratio N b /N f and particle mass µ. The observational constraints plotted are the same as in Fig. 1 and follow the same colour code. Each panel corresponds to one of the three fermionic EoS described in the text. the bosonic matter, the latter being a small fraction of the total mass. The results are depicted in Figure 2.
For µ = 0.1 (and similarly for smaller values of µ) the size of the bosonic component is larger than the fermionic radius (R b /R f ∼ 10 − 100; see cyan curves in the top panel of Fig. 3). In this case the contribution to the gravitational field of the bosonic component, is relatively flat in the region where fermions are present and therefore the impact in the equilibrium configuration of the fermionic component is small. This results in stars with similar fermionic radii. This is visible in the middle panel of Fig. 3 where we show the radial profiles of ρ and 1 2 µ 2 |φ| 2 for a neutron star and for mixed stars described by the MS1b EoS with the same central value ρ c . However, the additional energy provided by the bosonic component increases the total mass of the system. As a result, in these models (cyan lines in Fig. 2) the mass of the system increases as the ratio N b /N f increases while keeping the radius almost constant.
On the other hand, for µ = 1 (and similarly for larger values of µ) the bosonic component is located in a region similar or smaller than the region occupied by the fermionic component R b /R f ∼ 1 (see magenta curves in the top panel of Fig 3). In those cases, the bosonic component modifies the gravitational field in the neighborhood of the fermionic component to significantly modify the structure of the star by making it more compact. In those cases (magenta lines in Fig. 2) the fermionic radius decreases with increasing values of the ratio N b /N f (see bottom panel of Fig. 3). Additionally, due to this increase in compactness, the maximum mass supported by these models decreases.
IV. DISCUSSION
The additional degrees of freedom provided by the presence of a bosonic component may relieve some of the tension observed in the data in several ways. Leaving aside the question of the existence of ultra-light bosonic fields in nature, the main uncertainty of our model is the astrophysical scenario in which fermion-boson stars could form. A number of theoretical works have tried to address this issue (see e.g. [70] and references therein) in particular in the context of ultra-light bosonic fields as a model for dark matter. In order to broadly assess the impact of bosonic fields we explore here two situations that can be regarded as the two limiting cases in the range of possible models.
1. All stars have a constant bosonic-to-fermionic ratio: the first limiting scenario is the case in which the bosonic field is captured during the formation of the star leading to an approximately universal N b /N f ratio for all fermion-boson stars. In this case, an EoS with relatively low maximum mass for the purely fermionic component, not fulfilling the GW190814 constraint, may produce more massive objects by adding a bosonic component with small values of µ and solve the issue. Examples are DD2 and ALF2. In these two cases, supplementing a 10 − 20% amount of bosonic component rises the maximum mass above 2.5 M , while preserving the good agreement in radius at lower radii. Note that as a general feature of all EoS (see gray lines in Fig. 1) the star radius decreases when the maximum mass decreases, meaning that is difficult to have at the same time high maximum masses and small radii. The bosonic contribution is a way of precisely correcting this feature. Additionally, this procedure can also be used to increase the maximum mass even if in the purely fermionic case this mass is below 2 M , which might be a solution to the so-called hyperon problem [41].
2. Bosonic-to-fermionic ratio changes over time: in the second limiting scenario the bosonic matter is assumed to accrete onto the fermionic star after the latter has formed. In this case the ratio N b /N f would increase over time, being higher for older objects. The set of neutron stars considered in this work can be classified in two categories according to their age. Electromagnetically observed pulsars have typical ages smaller than 10 Gyr; the characteristic age of PSR J0030+0451 is estimated to be 8 Gyr [71], PSR J0740+6620 is in the range 5 − 8.5 Gyr [72], PSR J0348+0432 is 2.6 Gyr [16] and PSR J1614-2230 is 5.2 Gyr [66]. On the other hand typical ages of neutron stars found in mergers of compact binaries, such as those in GW170817 and GW190814, may be significantly larger. The merger time for galactic binary neutron star is expected to be in the range ∼ 0.1 − 1000 Gyr [73] which is consistent with the estimated merger time of the observed double neutron star systems in the Milky Way [74]. These estimates should be valid for the two gravitational-wave sources we consider since the metallicity conditions of the host galaxies is likely to be similar to our galaxy, given the low redshift of the sources. For the specific case of GW170817 it has been estimated that the age of the binary must be higher than 1 Gyr [75]. Therefore, it is plausible for the second class of objects to have accreted a significantly larger amount of bosonic field and thus have a larger ratio of N b /N f than the first class. In this scenario, neutron star radii could be relatively large for young objects with very small amount of bosonic component, fullfilling the constrains set by PSR J0740+6620 and PSR J0348+0432. And at the same time, potentially older objects such as those in GW170817, would have a significant bosonic component and thus smaller radii (see magenta lines in Fig. 2). In this situation the bosonic field would need to have a particle mass of at least µ = 1. On the other hand, the constraint set by GW190814 would be difficult to fulfill if the secondary was a neutron star, because in this scenario all stars should have much smaller maximum masses, but it could still be explained considering that the secondary is a low-mass black hole.
Finally, we have to address some of the caveats of our analysis. The observational constraints for the mass and radius considered here assume as a model that the ob-served object is a neutron star and obtain the posterior distributions according to this model. Therefore, if we change the model by adding a bosonic component, the observational constraints may in principle change as well. For electromagnetic observations of X-ray pulsars, the mass measurement (through Shapiro delay or orbital parameters measurements) relies almost exclusively on the effect of the total gravitational mass, regardless of its composition. The electromagnetic measurement of the radius, on the other hand, determines the size of the observable star, i.e. the fermionic component alone. The distribution of the bosonic field should affect weakly the analysis of NICER and XMM-Newton because the main effect would be to modify the light bending close to the star (see e.g. [5]). For µ = 1 or larger, most of the bosonic field would be confined inside the fermionic radius. Therefore, the metric outside the observable surface would correspond to that of an object with the total mass of the star and the analysis of NICER/XMM-Newton would be perfectly valid. On the other hand, for µ = 0.1 or smaller, most of the bosonic field would be outside the observable surface, and the metric would differ with respect to the one corresponding to the total mass of the system (it would probably be closer to the space-time generated by the fermionic component alone).
In that case the analysis of NICER/XMM-Newton would require corrections.
We also recall that in all of our models the selfinteraction parameter of the bosonic field has been set to zero. It would be interesting to study the effect of self-interactions (λ = 0) on mixed fermion-boson stars with a realistic EoS since a self-interaction potential allows to increase the maximum mass without changing the particle mass µ. We leave this analysis as future work.
Regarding gravitational-wave observations, the measured component gravitational masses would probably be well estimated since the structure of the compact objects appears only at 5PN order in the waveform models for binaries [76]. However, the estimation of the radius, as done in GW170817, may require modifications. This is actually an indirect estimation as the actual parameter measured is the quadrupole tidal deformability. From this, assuming that the object is a neutron star, it is possible to put constrains on the radius [77]. Therefore, to do a proper analysis one should have to either make the relevant corrections to estimate the fermion-boson star (fermionic) radius from the observational constrains on the tidal deformability or to compute the tidal deformability of our mixed stars (in particular the quadrupole Love number) to compare directly with observations. Either of the two analysis is out of the scope of this paper. However, even if we do not perform this analysis we expect that the trends found in our work should at least be qualitatively correct since there is a correlation between the tidal deformability and the radius.
It is also worth noticing that in scalar-tensor theories of gravity, in which the scalar field is not minimally coupled to gravity, neutron star models present significant deviations from general relativity through spontaneous scalarization, leading to neutron stars with significantly larger masses and radii [78][79][80][81]. In this regard, a suitable choice of the scalar field parameters and coupling constants of scalar-tensor theories could effectively reproduce the same mass-radius relations we have discussed in this paper for mixed fermion-boson stars in general relativity. Such potential degeneracy would make difficult to distinguish between the two cases and, thus, between the underlying theory of gravity.
On a similar note, while our model resembles those of [32,33] we have applied it to explain a larger set of observational and experimental data than those authors, who exclusively focused on explaining the secondary component of GW190814 as a potential dark-matter-admixed neutron star. Our findings for GW190814 agree with those of [32,33] which provides an independent consistency check. Since in our model the bosonic component plays the role of dark matter, it is not surprising that any similar dark-matter model would likely fit the data, irrespective of the type of matter considered.
To summarize, we conclude that the addition of a bosonic component to a neutron star leads to mixed configurations with mass-radius relations that are compatible with recent multi-messenger observations of compact stars, both in the electromagnetic channel (PSR J0030+0451 and PSR J0740+6620) and in the gravitational-wave channel (GW170817 and GW190814), as well as with the latest PREX-2 experimental results. The possibility of enlarging the parameter space of neutron stars with different contributions from the bosonic component offers thus a theoretically-motivated approach to reconcile the tension in the data collected by NICER/XMM-Newton and the LIGO-Virgo-KAGRA Collaboration.
FIG. 3 :
3Top panel: ratio of the bosonic and fermionic radius as a function of the total mass, for a subset of the models considered in this work. Models not displayed follow a very similar trend. Middle panel: radial profile of the rest-mass density ρ (solid lines) of an illustrative neutron star model described by the MS1b EoS and of 1 2 µ|φ| 2 (dashed lines) for µ = 0.1 and for different values of N b /N f . Bottom panel: same as middle panel but for µ = 1.0.
AcknowledgmentsWe thank Reed Essick, Nick Stergioulas
. J M Lattimer, 1305.3510Annual Review of Nuclear and Particle Science. 62485J. M. Lattimer, Annual Review of Nuclear and Particle Science 62, 485 (2012), 1305.3510.
. J M Lattimer, M Prakash, 1512.07820Physics Reports. 621J. M. Lattimer and M. Prakash, Physics Reports 621, 127 (2016), 1512.07820.
. B P Abbott, LIGO Scientific Collaboration ; Virgo CollaborationR Abbott, LIGO Scientific Collaboration ; Virgo Collaboration1805.11581Phys. Rev. Lett. 121161101B. P. Abbott, R. Abbott, LIGO Scientific Collaboration, and Virgo Collaboration, Phys. Rev. Lett. 121, 161101 (2018), 1805.11581.
. B Margalit, B D Metzger, 1710.05938Astrophys. J. Lett. 85019B. Margalit and B. D. Metzger, Astrophys. J. Lett. 850, L19 (2017), 1710.05938.
. T E Riley, A L Watts, S Bogdanov, P S Ray, R M Ludlam, S Guillot, Z Arzoumanian, C L Baker, A V Bilous, D Chakrabarty, 1912.05702Astrophys. J. Lett. 88721T. E. Riley, A. L. Watts, S. Bogdanov, P. S. Ray, R. M. Ludlam, S. Guillot, Z. Arzoumanian, C. L. Baker, A. V. Bilous, D. Chakrabarty, et al., Astrophys. J. Lett. 887, L21 (2019), 1912.05702.
. M C Miller, F K Lamb, A J Dittmann, S Bogdanov, Z Arzoumanian, K C Gendreau, S Guillot, A K Harding, W C G Ho, J M Lattimer, 1912.05705Astrophys. J. Lett. 88724M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bog- danov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, A. K. Harding, W. C. G. Ho, J. M. Lattimer, et al., Astrophys. J. Lett. 887, L24 (2019), 1912.05705.
. T Dietrich, M W Coughlin, P T H Pang, M Bulla, J Heinzel, L Issa, I Tews, S Antier, Science. 3701450T. Dietrich, M. W. Coughlin, P. T. H. Pang, M. Bulla, J. Heinzel, L. Issa, I. Tews, and S. Antier, Science 370, 1450 (2020), 2002.11355.
. P Landry, R Essick, K Chatziioannou, 2003.04880Phys. Rev. D. 101123007P. Landry, R. Essick, and K. Chatziioannou, Phys. Rev. D 101, 123007 (2020), 2003.04880.
. K Chatziioannou, 2006.03168General Relativity and Gravitation. 52K. Chatziioannou, General Relativity and Gravitation 52, 109 (2020), 2006.03168.
. T E Riley, A L Watts, P S Ray, S Bogdanov, S Guillot, S M Morsink, A V Bilous, Z Arzoumanian, D Choudhury, J S Deneva, 2105.06980Astrophys. J. Lett. 91827T. E. Riley, A. L. Watts, P. S. Ray, S. Bogdanov, S. Guillot, S. M. Morsink, A. V. Bilous, Z. Arzoumanian, D. Choudhury, J. S. Deneva, et al., Astrophys. J. Lett. 918, L27 (2021), 2105.06980.
. M C Miller, F K Lamb, A J Dittmann, S Bogdanov, Z Arzoumanian, K C Gendreau, S Guillot, W C G Ho, J M Lattimer, M Loewenstein, 2105.06979Astrophys. J. Lett. 91828M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bog- danov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, W. C. G. Ho, J. M. Lattimer, M. Loewenstein, et al., Astrophys. J. Lett. 918, L28 (2021), 2105.06979.
. G Raaijmakers, S K Greif, K Hebeler, T Hinderer, S Nissanke, A Schwenk, T E Riley, A L Watts, J M Lattimer, W C G Ho, 2105.06981Astrophys. J. Lett. 91829G. Raaijmakers, S. K. Greif, K. Hebeler, T. Hinderer, S. Nissanke, A. Schwenk, T. E. Riley, A. L. Watts, J. M. Lattimer, and W. C. G. Ho, Astrophys. J. Lett. 918, L29 (2021), 2105.06981.
. M Breschi, A Perego, S Bernuzzi, W Pozzo, V Nedora, D Radice, D Vescovi, 2101.01201Mon. Not. Roy. Astron. Soc. 5051661M. Breschi, A. Perego, S. Bernuzzi, W. Del Pozzo, V. Ne- dora, D. Radice, and D. Vescovi, Mon. Not. Roy. As- tron. Soc. 505, 1661 (2021), 2101.01201.
. P B Demorest, T Pennucci, S M Ransom, M S E Roberts, J W T Hessels, 1010.5788Nature. 4671081P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, and J. W. T. Hessels, Nature (London) 467, 1081 (2010), 1010.5788.
. E Fonseca, H T Cromartie, T T Pennucci, P S Ray, A Y Kirichenko, S M Ransom, P B Demorest, I H Stairs, Z Arzoumanian, L Guillemot, 2104.00880Astrophys. J. Lett. 91512E. Fonseca, H. T. Cromartie, T. T. Pennucci, P. S. Ray, A. Y. Kirichenko, S. M. Ransom, P. B. Demor- est, I. H. Stairs, Z. Arzoumanian, L. Guillemot, et al., Astrophys. J. Lett. 915, L12 (2021), 2104.00880.
. J Antoniadis, P C C Freire, N Wex, T M Tauris, R S Lynch, M H Van Kerkwijk, M Kramer, C Bassa, V S Dhillon, T Driebe, 1304.6875Science. 340J. Antoniadis, P. C. C. Freire, N. Wex, T. M. Tauris, R. S. Lynch, M. H. van Kerkwijk, M. Kramer, C. Bassa, V. S. Dhillon, T. Driebe, et al., Science 340, 448 (2013), 1304.6875.
. B Abbott, Virgo ; LIGO Scientific1710.05832Phys. Rev. Lett. 119161101B. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev. Lett. 119, 161101 (2017), 1710.05832.
. T Dietrich, T Hinderer, A Samajdar, 2004.02527General Relativity and Gravitation. 53T. Dietrich, T. Hinderer, and A. Samajdar, General Rel- ativity and Gravitation 53, 27 (2021), 2004.02527.
. R Abbott, LIGO Scientific Collaboration ; Virgo CollaborationAstrophys. J. Lett. 89644R. Abbott, LIGO Scientific Collaboration, and Virgo Collaboration, Astrophys. J. Lett. 896, L44 (2020), 2006.12611.
. E R Most, L J Papenfort, L R Weih, L Rezzolla, Mon. Not. Roy. Astron. Soc. 499E. R. Most, L. J. Papenfort, L. R. Weih, and L. Rez- zolla, Mon. Not. Roy. Astron. Soc. 499, L82 (2020), 2006.14601.
. A Nathanail, E R Most, L Rezzolla, 10.3847/2041-8213/abdfc6The Astrophysical Journal Letters. 908A. Nathanail, E. R. Most, and L. Rezzolla, The Astro- physical Journal Letters 908, L28 (2021), URL https: //doi.org/10.3847/2041-8213/abdfc6.
. A Sedrakian, F Weber, J J Li, 2007.09683Phys. Rev. D. 10241301A. Sedrakian, F. Weber, and J. J. Li, Phys. Rev. D 102, 041301 (2020), 2007.09683.
. A Tsokaros, M Ruiz, S L Shapiro, 2007.05526Astrophys. J. 905A. Tsokaros, M. Ruiz, and S. L. Shapiro, Astrophys. J. 905, 48 (2020), 2007.05526.
. B Biswas, R Nandi, P Char, S Bose, N Stergioulas, 2010.02090Mon. Not. Roy. Astron. Soc. 5051600B. Biswas, R. Nandi, P. Char, S. Bose, and N. Ster- gioulas, Mon. Not. Roy. Astron. Soc. 505, 1600 (2021), 2010.02090.
. N.-B Zhang, B.-A Li, 2007.02513Astrophys. J. 902N.-B. Zhang and B.-A. Li, Astrophys. J. 902, 38 (2020), 2007.02513.
. I Bombaci, A Drago, D Logoteta, G Pagliara, I Vidaña, 2010.01509Phys. Rev. Lett. 126162702I. Bombaci, A. Drago, D. Logoteta, G. Pagliara, and I. Vidaña, Phys. Rev. Lett. 126, 162702 (2021), 2010.01509.
. V Dexheimer, R O Gomes, T Klähn, S Han, M Salinas, 2007.08493Phys. Rev. C. 10325808V. Dexheimer, R. O. Gomes, T. Klähn, S. Han, and M. Salinas, Phys. Rev. C 103, 025808 (2021), 2007.08493.
. D A Godzieba, D Radice, S Bernuzzi, Astrophys. J. 908D. A. Godzieba, D. Radice, and S. Bernuzzi, Astrophys. J. 908, 122 (2021), 2007.10999.
. K Vattis, I S Goldstein, S M Koushiappas, Phys. Rev. D. 10261301K. Vattis, I. S. Goldstein, and S. M. Koushiappas, Phys. Rev. D 102, 061301 (2020), 2006.15675.
. I Cholis, K Kritos, D Garfinkle, arXiv:2106.076622106.07662I. Cholis, K. Kritos, and D. Garfinkle, arXiv e-prints arXiv:2106.07662 (2021), 2106.07662.
. C Charmousis, A Lehébel, E Smyrniotis, N Stergioulas, arXiv:2109.011492109.01149arXiv e-printsC. Charmousis, A. Lehébel, E. Smyrniotis, and N. Stergioulas, arXiv e-prints arXiv:2109.01149 (2021), 2109.01149.
. H C Das, A Kumar, S K Patra, arXiv:2109.018532109.01853arXiv e-printsH. C. Das, A. Kumar, and S. K. Patra, arXiv e-prints arXiv:2109.01853 (2021), 2109.01853.
. B K K Lee, M Chu, L.-M Lin, arXiv:2110.055382110.05538arXiv e-printsB. K. K. Lee, M.-c. Chu, and L.-M. Lin, arXiv e-prints arXiv:2110.05538 (2021), 2110.05538.
. C Drischler, R J Furnstahl, J A Melendez, D R Phillips, 2004.07232Phys. Rev. Lett. 125202702C. Drischler, R. J. Furnstahl, J. A. Melendez, and D. R. Phillips, Phys. Rev. Lett. 125, 202702 (2020), 2004.07232.
. D Adhikari, H Albataineh, D Androic, K Aniol, D S Armstrong, T Averett, C Gayoso, S Barcus, V Bellini, R S Beminiwattha, Phys. Rev. Lett. 126D. Adhikari, H. Albataineh, D. Androic, K. Aniol, D. S. Armstrong, T. Averett, C. Ayerbe Gayoso, S. Barcus, V. Bellini, R. S. Beminiwattha, et al., Phys. Rev. Lett. 126, 172502 (2021), 2102.10767.
. B T Reed, F J Fattoyev, C J Horowitz, J Piekarewicz, 2101.03193Phys. Rev. Lett. 126172503B. T. Reed, F. J. Fattoyev, C. J. Horowitz, and J. Piekarewicz, Phys. Rev. Lett. 126, 172503 (2021), 2101.03193.
. R Essick, P Landry, A Schwenk, I Tews, arXiv:2107.055282107.05528arXiv e-printsR. Essick, P. Landry, A. Schwenk, and I. Tews, arXiv e-prints arXiv:2107.05528 (2021), 2107.05528.
. J Li, A Sedrakian, M Alford, arXiv:2108.130712108.13071arXiv e-printsJ. Jie Li, A. Sedrakian, and M. Alford, arXiv e-prints arXiv:2108.13071 (2021), 2108.13071.
. S.-P Tang, J.-L Jiang, M.-Z Han, Y.-Z Fan, D.-M Wei, arXiv:2106.042042106.04204arXiv e-printsS.-P. Tang, J.-L. Jiang, M.-Z. Han, Y.-Z. Fan, and D.-M. Wei, arXiv e-prints arXiv:2106.04204 (2021), 2106.04204.
. P T H Pang, I Tews, M W Coughlin, M Bulla, C Van Den, T Broeck, Dietrich, arXiv:2105.086882105.08688arXiv e-printsP. T. H. Pang, I. Tews, M. W. Coughlin, M. Bulla, C. Van Den Broeck, and T. Dietrich, arXiv e-prints arXiv:2105.08688 (2021), 2105.08688.
. P F Bedaque, A W Steiner, 1412.8686Phys. Rev. C. 9225803P. F. Bedaque and A. W. Steiner, Phys. Rev. C 92, 025803 (2015), 1412.8686.
. S.-J Sin, hep-ph/9205208Phys. Rev. D. 503650S.-J. Sin, Phys. Rev. D 50, 3650 (1994), hep-ph/9205208.
. P.-H Chavanis, T Harko, 1108.3986Phys. Rev. D. 8664011P.-H. Chavanis and T. Harko, Phys. Rev. D 86, 064011 (2012), 1108.3986.
. T Matos, F S Guzman, L A Urena-Lopez, astro-ph/9908152Class. Quant. Grav. 171707T. Matos, F. S. Guzman, and L. A. Urena-Lopez, Class. Quant. Grav. 17, 1707 (2000), astro-ph/9908152.
. W Hu, R Barkana, A Gruzinov, astro-ph/0003365Phys. Rev. Lett. 851158W. Hu, R. Barkana, and A. Gruzinov, Phys. Rev. Lett. 85, 1158 (2000), astro-ph/0003365.
. D J Kaup, Phys. Rev. 1721331D. J. Kaup, Phys. Rev. 172, 1331 (1968).
. R Ruffini, S Bonazzola, Phys. Rev. 1871767R. Ruffini and S. Bonazzola, Phys. Rev. 187, 1767 (1969).
. P Jetzer, Physics Reports. 220163P. Jetzer, Physics Reports 220, 163 (1992).
. S L Liebling, C Palenzuela, Living Reviews in Relativity. 205S. L. Liebling and C. Palenzuela, Living Reviews in Rel- ativity 20, 5 (2017).
. J Bustillo, N Sanchis-Gual, A Torres-Forné, J A Font, A Vajpeyi, R Smith, C Herdeiro, E Radu, S H W Leong, Phys. Rev. Lett. 1265376J. Calderón Bustillo, N. Sanchis-Gual, A. Torres-Forné, J. A. Font, A. Vajpeyi, R. Smith, C. Herdeiro, E. Radu, and S. H. W. Leong, Phys. Rev. Lett. 126, 081101 (2021), 2009.05376.
. C A R Herdeiro, A M Pombo, E Radu, P V P Cunha, N Sanchis, 2102.01703C. A. R. Herdeiro, A. M. Pombo, E. Radu, P. V. P. Cunha, and N. Sanchis-Gual (2021), 2102.01703.
. H Olivares, Z Younsi, C M Fromm, M De Laurentis, O Porth, Y Mizuno, H Falcke, M Kramer, L Rezzolla, 1809.08682Mon. Not. Roy Astron. Soc. 497H. Olivares, Z. Younsi, C. M. Fromm, M. De Lauren- tis, O. Porth, Y. Mizuno, H. Falcke, M. Kramer, and L. Rezzolla, Mon. Not. Roy Astron. Soc. 497, 521 (2020), 1809.08682.
. A Henriques, A R Liddle, R Moorhouse, 0370- 2693Physics Letters B. 251511A. Henriques, A. R. Liddle, and R. Moorhouse, Physics Letters B 251, 511 (1990), ISSN 0370- 2693, URL http://www.sciencedirect.com/science/ article/pii/0370269390907899.
. P Jetzer, Physics Letters B. 24336P. Jetzer, Physics Letters B 243, 36 (1990).
. S Valdez-Alvarado, C Palenzuela, D Alic, L A Urena-López, Physical Review D. 8784040S. Valdez-Alvarado, C. Palenzuela, D. Alic, and L. A. Urena-López, Physical Review D 87, 084040 (2013).
. R Brito, V Cardoso, H Okawa, Physical review letters. 115111301R. Brito, V. Cardoso, and H. Okawa, Physical review letters 115, 111301 (2015).
. R Brito, V Cardoso, C F Macedo, H Okawa, C Palenzuela, Physical Review D. 9344045R. Brito, V. Cardoso, C. F. Macedo, H. Okawa, and C. Palenzuela, Physical Review D 93, 044045 (2016).
. S Valdez-Alvarado, R Becerril, L A Ureña-López, Physical Review D. 10264038S. Valdez-Alvarado, R. Becerril, and L. A. Ureña-López, Physical Review D 102, 064038 (2020).
. A A Roque, L A Ureña-López, arXiv:2109.147472109.14747arXiv e-printsA. A. Roque and L. A. Ureña-López, arXiv e-prints arXiv:2109.14747 (2021), 2109.14747.
. F Di Giovanni, S Fakhry, N Sanchis-Gual, J C Degollado, J A Font, 2006.08583Phys. Rev. D. 10284063F. Di Giovanni, S. Fakhry, N. Sanchis-Gual, J. C. Degol- lado, and J. A. Font, Phys. Rev. D 102, 084063 (2020), 2006.08583.
. F Di Giovanni, S Fakhry, N Sanchis-Gual, J C Degollado, J A Font, 2105.00530Class. Quant. Grav. 38F. Di Giovanni, S. Fakhry, N. Sanchis-Gual, J. C. De- gollado, and J. A. Font, Class. Quant. Grav. 38, 194001 (2021), 2105.00530.
. B Kain, Physical Review D. 10443001B. Kain, Physical Review D 104, 043001 (2021).
. M Bezares, D Viganò, C Palenzuela, https:/link.aps.org/doi/10.1103/PhysRevD.100.044049Phys. Rev. D. 10044049M. Bezares, D. Viganò, and C. Palenzuela, Phys. Rev. D 100, 044049 (2019), URL https://link.aps.org/doi/ 10.1103/PhysRevD.100.044049.
. F Özel, P Freire, 1603.02698Annual Review Astron. Astrophys. 54F.Özel and P. Freire, Annual Review Astron. Astrophys. 54, 401 (2016), 1603.02698.
. Z Arzoumanian, A Brazier, S Burke-Spolaor, S Chamberlin, S Chatterjee, B Christy, J M Cordes, N J Cornish, F Crawford, H Cromartie, 1801.01837Astrophysical J. Suppl. 235Z. Arzoumanian, A. Brazier, S. Burke-Spolaor, S. Cham- berlin, S. Chatterjee, B. Christy, J. M. Cordes, N. J. Cornish, F. Crawford, H. Thankful Cromartie, et al., As- trophysical J. Suppl. 235, 37 (2018), 1801.01837.
. B Biswas, arXiv:2105.028862105.02886arXiv e-printsB. Biswas, arXiv e-prints arXiv:2105.02886 (2021), 2105.02886.
. M Alford, M Braby, M Paris, S Reddy, 10.1086/430902The Astrophysical Journal. 629M. Alford, M. Braby, M. Paris, and S. Reddy, The As- trophysical Journal 629, 969 (2005), URL https://doi. org/10.1086/430902.
. M Hempel, J Schaffner-Bielich, 0911.4073Nuclear Physics A. 837210M. Hempel and J. Schaffner-Bielich, Nuclear Physics A 837, 210 (2010), 0911.4073.
. R Brito, V Cardoso, C F B Macedo, H Okawa, C Palenzuela, 1512.00466Phys. Rev. D. 9344045R. Brito, V. Cardoso, C. F. B. Macedo, H. Okawa, and C. Palenzuela, Phys. Rev. D 93, 044045 (2016), 1512.00466.
. A N Lommen, A Zepka, D C Backer, M Mclaughlin, J M Cordes, Z Arzoumanian, K Xilouris, astro-ph/0008054Astrophys. J. 5451007A. N. Lommen, A. Zepka, D. C. Backer, M. McLaughlin, J. M. Cordes, Z. Arzoumanian, and K. Xilouris, Astro- phys. J. 545, 1007 (2000), astro-ph/0008054.
. D M Beronya, A V Karpova, A Y Kirichenko, S V Zharikov, D A Zyuzin, Y A Shibanov, A Cabrera-Lavers, 1902.11150Mon. Not. R. Astron. Soc. 485D. M. Beronya, A. V. Karpova, A. Y. Kirichenko, S. V. Zharikov, D. A. Zyuzin, Y. A. Shibanov, and A. Cabrera- Lavers, Mon. Not. R. Astron. Soc. 485, 3715 (2019), 1902.11150.
. D R Lorimer, 0811.0762Living Reviews in Relativity. 11D. R. Lorimer, Living Reviews in Relativity 11, 8 (2008), 0811.0762.
. T M Tauris, M Kramer, P C C Freire, N Wex, H T Janka, N Langer, P Podsiadlowski, E Bozzo, S Chaty, M U Kruckow, 1706.09438Astrophys. J. 846T. M. Tauris, M. Kramer, P. C. C. Freire, N. Wex, H. T. Janka, N. Langer, P. Podsiadlowski, E. Bozzo, S. Chaty, M. U. Kruckow, et al., Astrophys. J. 846, 170 (2017), 1706.09438.
. A J Levan, J D Lyman, N R Tanvir, J Hjorth, I Mandel, E R Stanway, D Steeghs, A S Fruchter, E Troja, S L Schrøder, 10.3847/2041-8213/aa905f84828A. J. Levan, J. D. Lyman, N. R. Tanvir, J. Hjorth, I. Mandel, E. R. Stanway, D. Steeghs, A. S. Fruchter, E. Troja, S. L. Schrøder, et al., 848, L28 (2017), URL https://doi.org/10.3847/2041-8213/aa905f.
. L Blanchet, 1310.1528Living Reviews in Relativity. 17L. Blanchet, Living Reviews in Relativity 17, 2 (2014), 1310.1528.
. J S Read, L Baiotti, J D E Creighton, J L Friedman, B Giacomazzo, K Kyutoku, C Markakis, L Rezzolla, M Shibata, K Taniguchi, 1306.4065Phys. Rev. D. 8844042J. S. Read, L. Baiotti, J. D. E. Creighton, J. L. Friedman, B. Giacomazzo, K. Kyutoku, C. Markakis, L. Rezzolla, M. Shibata, and K. Taniguchi, Phys. Rev. D 88, 044042 (2013), 1306.4065.
. T Damour, G Esposito-Farese, Classical and Quantum Gravity. 92093T. Damour and G. Esposito-Farese, Classical and Quan- tum Gravity 9, 2093 (1992).
. T Harada, gr- qc/9801049Phys. Rev. D. 574802T. Harada, Phys. Rev. D 57, 4802 (1998), gr- qc/9801049.
. H Sotani, K D Kokkotas, gr-qc/0409066Phys. Rev. D. 7084026H. Sotani and K. D. Kokkotas, Phys. Rev. D 70, 084026 (2004), gr-qc/0409066.
. J C Degollado, M Salgado, M Alcubierre, Physics Letters B. 808135666J. C. Degollado, M. Salgado, and M. Alcubierre, Physics Letters B 808, 135666 (2020), 2008.10683.
| [] |
[
"Yes-no Bloom filter: A way of representing sets with fewer false positives",
"Yes-no Bloom filter: A way of representing sets with fewer false positives"
] | [
"Laura Carrea ",
"Alexei Vernitski ",
"Martin Reed "
] | [] | [] | The Bloom filter (BF) is a space efficient randomized data structure particularly suitable to represent a set supporting approximate membership queries. BFs have been extensively used in many applications especially in networking due to their simplicity and flexibility. The performances of BFs mainly depends on query overhead, space requirements and false positives. The aim of this paper is to focus on false positives. Inspired by the recent application of the BF in a novel multicast forwarding fabric for information centric networks, this paper proposes the yes-no BF, a new way of representing a set, based on the BF, but with significantly lower false positives and no false negatives. Although it requires slightly more processing at the stage of its formation, it offers the same processing requirements for membership queries as the BF. After introducing the yes-no BF, we show through simulations, that it has better false positive performance than the BF.Index Termsprobabilistic data structure, information representation, information centric networks | null | [
"https://arxiv.org/pdf/1603.01060v1.pdf"
] | 16,308,190 | 1603.01060 | c1f08006866ebc51ad223f77396342509813637e |
Yes-no Bloom filter: A way of representing sets with fewer false positives
Laura Carrea
Alexei Vernitski
Martin Reed
Yes-no Bloom filter: A way of representing sets with fewer false positives
1
The Bloom filter (BF) is a space efficient randomized data structure particularly suitable to represent a set supporting approximate membership queries. BFs have been extensively used in many applications especially in networking due to their simplicity and flexibility. The performances of BFs mainly depends on query overhead, space requirements and false positives. The aim of this paper is to focus on false positives. Inspired by the recent application of the BF in a novel multicast forwarding fabric for information centric networks, this paper proposes the yes-no BF, a new way of representing a set, based on the BF, but with significantly lower false positives and no false negatives. Although it requires slightly more processing at the stage of its formation, it offers the same processing requirements for membership queries as the BF. After introducing the yes-no BF, we show through simulations, that it has better false positive performance than the BF.Index Termsprobabilistic data structure, information representation, information centric networks
In this paper, the yes-no BF is presented. The aim of the yes-no BF is to reduce the number of false positives while keeping its properties of being a time and space efficient data structure. First, the classic BF is revised with a discussion on the false positive probability. Then, the yes-no BF structure in introduced and presented in details and the false positive probability is analytically evaluated. Also, an experimental analysis is carried out in order to evaluate the yes-no BF performance for false positives with respect to the yes-no BF parameters and the characteristics of the set which the structure represents. The average number of false positive occurrences of the yes-no BF is first analysed as a function of the set and data structure parameters and then it is evaluated for the new forwarding mechanism which has been proposed within the framework of the PSIRP/PURSUIT project. However, the data structure is general enough to be used in other applications. The complexity of the processing required at the stage of the formation of the yes-no BF is evaluated in terms of big O notation in comparison with the processing required for the classic BF.
II. THE BLOOM FILTER
The BF is a compressed way of representing a set of elements, with some false positives but no false negatives. A BF is a fixed length Boolean array where a certain number of bits are set to 1. Given a set S of elements to store in the BF, each element of the set S is represented with its own BF, a Boolean array of a fixed length m where up to k bits are set to 1. The bits are set using k hash functions {h 1 , h 2 , ..., h k } applied to each element e i of the set h 1 (e i ), h 2 (e i ), ...h k (e i ) with e i ∈ E, (i = 1, ...n),
where the output of each hash is in the range {1, 2, ..., m}.
Given the BF b e i of each element of the set S, the whole set is represented by a Boolean array, itself a BF, which can be interpreted as a logical disjunction of the BFs of its elements:
b S = b e 1 ∨ b e 1 ... ∨ b en .(2)
Given a set T of elements to be queried, the membership test of an element in T can be performed as a bitwise comparison between the BF of the set and the BF of the element. The comparison is traditionally implemented as a logical conjunction on b S , the BF of the set and I TRUE TABLE OF A LOGICAL CONJUNCTION. be bS be ∧ bS b e , the BF of the element to be queried:
b S ∧ b e = b e ⇒ e ∈ S = b e ⇒ e ∈ S ,(3)
namely if the logical conjunction is equal to the BF of the element to query then the element necessarily belongs to the set otherwise it does not. Another way to interpret the set membership query operation for the BF is to check bitwise if each bit of the BF of the element to query is less than or equal to the corresponding bit in the BF representing the set. This is easy to verify using the truth table of the logical conjunction reported in Table I Because of the probabilistic nature of the BF, the BF can be only queried and its elements can not be deleted. Moreover, the set membership test may give rise to a false positive: i.e. an element appears to belong to the set even though it was not originally included in the BF. Let us consider a BF of m bits. The probability that a specific bit is still 0 after all the elements in S have been hashed with k hash functions is [3]:
p = 1 − 1 m kn(4)
where n = |S| and where it has been assumed that the hash functions are perfectly random.
Given the BF b S of the set S and the set T of elements to be queried, let S the set of elements e such that b e ≤ b E , namely the set of elements that appears to belong to the set S after a membership query. We can call S the set of the positives of the BF and S ⊂ S since the elements in S definitely satisfy the condition. Then, the set F = S \ S is the set of the false positives.
The sets are schematically represented in Fig. 1, where S ∩ F = ∅.
For an element e ∈ S, the probability that it belongs to the BF (namely that all the bits h 1 (e), h 2 (e), ...h k (e) are set to 1) when tested for membership can be computed as [3]
f S = (1 − p) k = 1 − 1 − 1 m kn k ,(5)
where the assumption that if B i and B j are two distinct bits in the Bloom filter, the events B i = 1 and B j = 1 are statistically independent has been made. However, this is not necessary true as shown in [15]. Consequently, (5) is only an approximation of the false positive probability.
In particular, Bose in [15] showed that (5) a strict lower bound for any k ≥ 2 and proposed a calculation of the false positive probability which is not though in closed form. A further approximation of (5) is
f S ≈ 1 − e − kn m k(6)
being e x = lim a→∞ 1 + x a a .
This probability is computed assuming that e ∈ S, namely it is the conditional probability [16]
f s = Pr[S|S c ](7)
where S c = T \ S is the complement of S.
Instead, the probability of a false positive without assuming e ∈ S c can be written as [17] Pr
[F ] = Pr[S \ S] = Pr[S] − Pr[S],(8)
so that the probability of false positive for e ∈ T can be written as
Pr[F ] = Pr[S \ S] = (1 − Pr[S])f S .(11)
If the condition e ∈ S c is considered, namely e ∈ S then Pr[S] = 0 and (11) becomes (7).
The advantages of the BF are that the calculations needed to build it, and test for membership, are easy to program and very fast to perform, since only simple logical operations are required.
A limit of the BF is that its probabilistic nature gives rise to false positives. The aim of this work is to propose a mechanism to limit the number of false positives.
III. THE YES-NO BF
The yes-no BF we propose is a new way of representing a set and it is based on the BF. It is composed by two parts:
• the yes-filter encoding the set;
• the no-filter encoding the elements which generate false-positives.
It aims to offer a smaller number of false positives when compared to the BF and it can be designed to have no false negatives. The main insight behind the yes-no BF is that, in many applications, not only the elements belonging to the set but, more generally, all the elements which will ever be queried are known when the data structure is formed.
Let U be the universe of all elements, S ⊆ U be the set of elements to be encoded in the yes-no BF, and T ⊆ U \ S be the set of all the elements which do not belong to the set S and whose membership in S is likely to be tested. Typically, T is a smaller set than U \ S.
The yes-no BF we propose, consists of a structure that not only has the function to encode the elements of S, it has also some additional features to avoid actively the false positives during the set membership test. Moreover, the yes-no BF aims to preserve the space-time efficiency feature offered by the BF, although the structure requires slightly more processing but only at the stage of its formation. As with the standard BF, we assume that the representation of a set consists, in total, of m bits. These m bits are split into the following parts:
• p bits, to which we shall refer as the yes-filter;
• qr bits, where we shall refer to each of the q bits as to a no-filter.
so that m = p + qr and q << p. We shall refer to the above construction as the yes-no BF and a schematic representation is shown in Fig. 2.
A. Representation of an element in U
Before describing the set construction we will introduce the construction of a single element which will have a yes-filter and a single no-filter repeated r times. Two different sets of hash functions are needed to represent each element e ∈ U with a yes-no BF structure: To construct the yes-filter the element e ∈ S is hashed k times and the bits h i (e) in the yesfilter are set to 1. To construct the no-filter the element e is hashed k times and the bits w j (e)
• H = {h 1 ,
in the no-filter are set to 1. Note that each element e has only one no-filter. The yes-no BF representation of the element e is constructed with its yes-filter and with r repetitions of the no-filter. For convenience, we shall refer to these parts as the yes-filter and the no-filter of e.
As an example, we consider p = 13, q = 2, r = 2 as shown in Fig. 3. The number of hash functions considered are k = 3 and k = 1, and h 1 (e) = 5, h 2 (e) = 2, h 3 (e) = 12 and w 1 (e) = 2.
B. Representation of a set in U
Given the set S of elements to encode in the yes-no BF and the set T ⊆ U \ S of elements whose membership in S may be queried, the yes-no BF representing the set is constructed using the yes-filters of each element of the set S and the no-filters of the elements in T which produce false positives when the yes-filter is queried. In particular, the yes-filter and the no-filters of the set are constructed as described below.
1) The yes-filter: The yes-filter of the set S is formed as a disjunction of the yes-filters of the all the elements e ∈ S, exactly as for a normal BF.
2) The no-filter: The no-filters of the set store the no-filters of the elements which generate false positives when the yes-filter is queried. The yes-no filter reduces false positives compared to the BF by keeping track, in an efficient way, not only of the elements belonging to the set S but also of the elements which generate false positives.
Let us analyse in detail the procedure to construct the no-filters of the set S. Firstly, the set F ⊆ T of elements f which generate false positives is constructed using the membership query procedure described for the BF (see Section II). Namely, the yes-filter of each of the elements in T is checked against the yes-filter of the set. That is, if bit-wise each bit of the yes-filter of an element in T is less than or equal to the yes-filter of the set S then necessarily this element generates false positives and it will be an element of the subset F ⊆ T . Generally, |F | > r, namely the number of false positives is greater than the number of no-filters. Consequently, each of the no-filters of the set S will have to store more than one no-filter.
Each no-filter of the set is a BF which, obviously, may also give rise to false positives. A false positive of a no-filter would mean that the element was not in reality a false positive for the yes-filter even though the no-filter represents it as a false positive of the yes-filter. An element which is not a false positive of the yes-filter may be an element which was added to the yes-filter.
Consequently, a false positive of the no-filter may represent a false negative of the yes-filter. Hence, in the construction it may be required to avoid false negatives in the yes-filter since such false negatives may be more harmful 1 than false positives. The following steps are proposed for the construction of the no-filters of the set S in order to avoid false negatives, but the design can be altered if false negative can be tolerated.
For each element in F , the algorithm will try in turn each no-filter stored in the yes-no filter of the set to test if it does create a false negative. However, if none are suitable, then, this particular false positive cannot be mitigated. Let us see in detail this construction.
• Given the no-filter v f 1 of the first false positive f 1 ∈ F yes and the first no-filter v 1 of the set S, a disjunction v 1 is formed is formed between them to attempt to store the no-filter of the first false positive in the yes-no BF:
v 1 = v f 1 ∨ v 1 .(12)
• Then, to avoid false negatives, the following condition is checked for the no-filter of each element e ∈ S:
if the no-filter v e of each e ∈ S is greater than v 1 , then none of the no-filters of the elements in S is a false positive for the no-filter storing the false positives, namely none of the elements in S can be a false negative. In this case, the first no-filter of S will be set as v 1 .
if the no-filter v e of any e ∈ S is less than or equal to v 1 then it may give raise to false negative. In this case, the second no-filter v 2 of the set S is attempted to be used following the same steps as for the first no-filter. If also the last filter v r satisfy this condition then, f 1 cannot be included in any of the no-filters and necessarily will be a false positive.
This process is repeated for all the element in F and the set R of elements to store in the no-filters is constructed. For convenience, we shall refer to the parts of the yes-no BF of the set S as the yes-filter and the no-filters of the set S with respect to the set T . Fig. 5 shows the sets when false negatives are avoided. In this case, G ∩ S = ∅.
We need to point out that the no-filter for each false positive occurrence can be placed in anyone of the r no-filters of the set S. One would aim to choose the no-filters of the set S in such a way that between them, they minimise the number of false positives. However, this is a complex discrete optimisation problem; it is not unreasonable to conjecture that finding an exact solution to it (that is, the smallest possible number of false positives, given a fixed number of no-filters) is NP-hard. The algorithm described above is a simple and fast greedy algorithm, and we show in the experiments below that its performance provides better false positive rates than the BF. However, in some implementations more time-consuming algorithms may be used, which use the arsenal of operational research methods to further reduce false positive rates of yes-no Bloom filters. Optimisation techniques to form the no-filters in a near-optimal way have been proposed in [19].
C. Membership queries
Given an element t ∈ T , to perform a membership query the following steps are required.
• Firstly, the yes-filter of t is compared with the yes-filter of the set. If the yes-filter of t is greater than the yes-filter of the set, the element t definitely does not belong to the set S. In case that the yes-filter of t is less than or equal to the yes-filter of the set, the element t may be either an element belonging to S, or a false positive. Thus, in this case, the following step is performed. • The no-filters of the set S are checked. If the no-filter of t is less than or equal to any of the no-filters of the set then we can conclude that t was a false positive and therefore, discarded. Otherwise, we assume that t was not a false positive of the yes-filter and that it is an element of S.
To conclude, the yes-no BF is a generalisation of the standard BF: indeed, when r = 0, the yes-no BF turns into the BF of length m. Keeping m fixed, when r > 0, the length of the yes-filter is reduced (compared with the case r = 0), producing an increased number of false positives with high probability; however, at the same time, the no-filters reduce the number of false positives as shown in the next section. The yes-no BF offers no false negatives.
IV. FALSE POSITIVE PROBABILITY OF THE YES-NO BF
Let us consider the case of yes-no BF with no false negatives in Fig. 5. The set S is the set of elements to store in the yes-filter and the set S is the set of all the elements that, when tested, appear to belong to the yes-filter. Consequently, S \ S = F is the set of the false positives of the yes-filter.
The set R is the set of elements that are stored in the no-filter after the formation processing described in Sec. III-B. The set R ⊂ F is the set of elements generating false positives for the yes-filter which are stored in the no-filters. The set R is the set of all the elements that, when tested, appear to belong to the no-filter. Consequently, R \ R = G is the set of the false positives of the no-filter. In this case, G and S are disjoint.
If e ∈ S c then e is discarded by the yes-filter and consequently it is not queried by the nofilters. If e ∈ S than e is queried by the no-filter. The set S can be decomposed in the following 4 sets (see Fig. 5):
• S • R • (R \ R) ∩ S • S \ S \ R
If e ∈ S, e belongs to the set S of elements that were stored in the yes-no BF.
If e ∈ R, then e was a false positive that has been discarded by the no-filters.
If e ∈ (R \ R) ∩ S = G ∩ F then e was a false positive of the no-filter but it was also a false positive of the yes-filter so it was discarded by the no-filter.
If e ∈ S \ S \ R = F \ R then necessarily e is a false positive of the yes-no BF.
We call the set S \ S \ R = F \ R the set E.
Lemma. The probability of the set E can be formulated as
Pr[E] = (1 − Pr[S])f S + (13) −(Pr[S] + (1 − Pr[S])f S )f R − (1 − f R )Pr[R].
Proof:
The set S \ S is the union of the following disjoint sets:
S \ S = [S \ S \ R] ∪ [(R \ R) ∩ S] ∪ R(14)
so that the probability Pr[E] = Pr[S \ S \ R] can be written as
Pr[E] = Pr[S \ S] − Pr[(R \ R) ∩ S] − Pr[R].(15)
Now, Pr[S \ S] can be calculated as in (11):
Pr[S \ S] = (1 − Pr[S])f S .(16)
The probability Pr[(R \ R) ∩ S] can be written as
Pr[(R \ R) ∩ S] = Pr[(R \ R)] − Pr[R \ S](17)
since the set R \ R is the union of the two disjoint sets (R \ R) ∩ S and R \ S as can be seen in Fig. 5. Now, Pr[(R \ R)] can be computed as in (11):
Pr[(R \ R)] = (1 − Pr[R])f R .(18)
while the probability Pr[R \ S] can be computed as
Pr[R \ S] = Pr[R ∩ S c ] = = Pr[S c ] Pr[R|S c ].(19)
Regarding Pr[R|S c ], since S c ⊂ R c , namely if e ∈ S c , then necessarily e ∈ R c , we can write
Pr[R|S c ] = Pr[R|R c ] = f R .(20)
Regarding Pr[S c ] it can be expressed using (10) as:
Pr[S c ] = 1 − Pr[S] = 1 − (Pr[S] + (1 − Pr[S])f S .(21)
Using (21) and (20) we can rewrite (19) as
Pr[R \ S] = f R (1 − (Pr[S] + (1 − Pr[S])f S )).(22)
Using (18) and (22) we can rewrite (17) as
Pr[(R \ R) ∩ S] = (1 − Pr[R])f R + −(1 − (Pr[S] + (1 − Pr[S])f S ))f R = = (Pr[S] − Pr[R] + f S − Pr[S]f S )f R .(23)
From (23) and (16) we can compute the probability of the false positives of the yes-no BF Pr [E] defined in (15) as
Pr[E] = (1 − Pr[S])f S + (24) −(Pr[S] + (1 − Pr[S])f S )f R − (1 − f R )Pr[R]
The probability Pr[E] in (24) is the false positive probability for e ∈ T for the yes-no BF.
It is very easy to verify that the probability of false positive of the yes-no BF is less than the correspondent probability of false positive of a BF of the same size as the yes-filter:
Pr(E) < Pr(F )(25)
where Pr(F ) is defined in (11). This probability has been computed without assuming that e ∈ S c . Instead, with the assumption that e ∈ S c , then the probability of false positives can be written as
Pr[E|S c ] = Pr[E ∩ S c ] Pr[S c ] .(26)Pr[E|S c ] = f S (1 − f R ) − (1 − f R )Pr[R].(27)
If we consider e ∈ S c ∩ R c , assuming that the no-filter has taken care of the false positives, then Pr[R] = 0 and the false positive probability becomes: If only one no-filter is employed, the false positive probability can be explicitly written in function of the yes-no BF parameters. The false positive probability of the yes-filter can be approximated using (6):
Pr[E|S c ∩ R c ] = f S (1 − f R ).f S ≈ 1 − e − kn p k .(29)
The false positive probability f R of the no-filter can be approximated as
f R ≈ 1 − e − k n q k .(30)
The false positive probability of the yes-no BF f E in (28) with one no-filter only could be approximated as
f E = 1 − e − kn p k 1 − 1 − e − k n q k .(31)
However, since the choice of how to fill the no-filters is related to a NP-hard problem (see end of Sec. III-B2) it is not possible to express f R for a number of no-filters. For this reason, since choices need to be made to implement yes-no BFs, simulations has been carried out to study the dependency of the number of false positive from the design parameters. As with the standard BF, one would expect that the parameters defining the structure of the yes-no BF have to be decided upon in advance. The parameters that need to be selected are: m, r, p, q and the values of k and k for the yes-filter and for the no-filters respectively.
F p = t∈T f p = | T | f p(32)
since the probability is the same for each element in T .
The evaluation of the false positive performance has been carried out as function of the cardinality of S and as a function of the yes-no BF parameters. The following values have been used:
• m = 256, the total number of bits of the yes-no BF. This is a typical size for in-packet forwarding applications [11] • n = 30, the number of elements to store Depending on the functionality studied each of these parameter has been varied across a specific range as described in the following. Moreover, for a number of elements lower than around 60 the yes-no BF offers a lower number of false positive occurrences. After this threshold the classic BF performs better on average. This is due to the fact that the number of bits dedicated to store the elements is smaller in the yes-no BF. Just to analyse the actual reduction in false positive occurrences, we compare the performance of the yes-no BF with a BF of the same size as the yes-filter. The yes-no BF offers definitely much less number of false positive occurrences than the classic BF as shown in Fig. 10, following the results of the theoretical analysis of the false positive probability (25). Again, when r = 0 the yes-no Bf becomes the classic BF, but as r increases the number of false positive occurrences decreases. However, the function has a minimum since because the total length of the yes-no BF is kept constant, the length p of the yes-filter will become smaller and smaller generating too many false positives that the yes-no BF will be not be able to cope with.
According to this study, we have chosen appropriate values of these parameters for the application described in Sec. VIII.
VI. ALGORITHM COMPLEXITY
The processing required to form the yes-no BF can be quantified using the big O notation.
Regarding the construction of the yes-filter, the procedure is exactly the same as for the classic BF so we do not consider this in this analysis since we want to quantify how much more processing is required with respect to the classic BF. The complexity comes from the no-filters formation and we analyse the case in which false negatives need to be avoided.
We assume that the operations to form and to query a BF can be parallelised. If the false negatives have to be avoided then an additional processing is required. In the worst case scenario, the time complexity of the algorithm can be quantified as O(r |F ||S|) since the no-filter of each element in S has to be compared with the no-filter in consideration for each false positive and in the worst case scenario this has to be repeated for all the no-filters.
Since the operation involved can be implemented as logical and, or and comparison we can conclude that the processing overhead is only slightly more than for the classic BF at the formation stage.
At the querying stage, the processing required by the yes-no BF is exactly the same as for the BF, since the membership query can be implemented as a logical and and comparison for both the data structure. BF and yes-no BF can answer membership queries in O(1) time.
VII. THE YES-NO BF FOR PACKET FORWARDING
The formulation of the yes-no BF has been inspired by the recent application of BFs in the novel multicast forwarding fabric [11] of the PSIRP/PURSUIT information centric architecture.
However, structures like the BF and the yes-no BF can be used generally for packet forwarding when the path is known. These structures are used as an encoding to identify paths (with one or more destinations) and links between nodes. The yes-no BF representing an element described in Sec. III-A is used to encode a link, where e would be a link descriptor. The yes-no BF representing a set described in Sec. III-B is used to encode the path. Thus, the set of all links is U , the set of links along the path is S, and the set of all the outgoing links from the nodes along the path is T . Fig. 15 shows a schematic representation of a path of multiple destinations
where link belonging to the set S and the links belonging to the set T are differently depicted. The yes-no BF, encoding the path, is placed in the packet header. Note that slightly more processing is required to construct the yes-no BF in comparison with the construction of the BF: the false positive occurrences for the specific path have to be evaluated and the no-filters have to be constructed. At a node, the yes-no BF of each outgoing link is stored. When the packet arrives at the node, the membership query described in Sec. III-C is performed for each of the outgoing links. Note that the membership query can be implemented with a logical and operation and a comparison, exactly like the membership query for the BF. Therefore, we can conclude that the yes-no BF requires a bit more processing at the stage of its formation but the same processing for membership queries as the BF, so that the forwarding operation can be implemented in line speed [11]. A query may result in a false positive, which translates, in our forwarding scheme, into extra traffic in the network. The match is possible along more than one outgoing link per node so that multicast turns out to be the natural communication paradigm.
The formation of the yes-no BF is performed by an entity that has knowledge of the network topology, this could a centralised entity [12].
VIII. EVALUATION OF THE YES-NO BF PERFORMANCE IN REALISTIC TOPOLOGIES
Although the evaluation of the yes-no BF performance could be carried out in artificially generated topologies, we consider realistic topologies and we compare the false positive rates of the yes-no BF and of the BF of the same size. The network topologies used in the simulation are a collection from the Network Topology Zoo containing 261 network topologies [18]. In each of these 261 networks, we select a long (relative to the topology) realistic forwarding path. The allocations. The parameters defining the yes-no BF used in the simulation are listed below:
• The total length of the yes-no BF and of the BF is m = 256 bits.
• The length of the yes-filter is p = 192 bits.
• The number of no-filters are r = 2.
• The length of each no-filter is r = 32 bits.
• The number k of hash functions used for the yes-filter is k = 4.
• The number k of hash functions used for the no-filters is k = 3.
• The number k BF of hash functions for the classic BF is k BF = 6.
These parameters have been chosen according to the analysis performed in Sec. V. The total length, m matches typical lengths used in other studies and is comparable with the length of the IPv6 header address fields. We do not claim that this is the optimal parameter set but the aim of this paper is to show that the yes-no BF offers better false positive performance than the classic BF. Fig. 16 shows the expected number of false positive occurrences of the yes-no BF against the expected number of false positive occurrences of the BF, where it can be easily seen that the yes-no BF offers a lower number of false positive occurrences. For example, the top right point corresponds to a 28-hop path in the network from the TataNld topology [18]. If the BF is used for forwarding along this path, one can expect, approximately, 0.92 false positive occurrences; but if the yes-no BF is used, one can expect, approximately, 0.13 false positives.
The meaning of this is that if we use BFs, packets will almost certainly be diverted to a wrong direction approximately at one point during their travel; however, if we use yes-no BFs, this will only happen approximately to one out of eight possible yes-no BFs of the links in the network.
Note there are many very small networks for which neither the BF nor, of course, the yes-no BF
IX. CONCLUSIONS
We have proposed a novel way of representing sets based on BFs, which we call the yes-no BF. It requires slightly more processing at the stage of its formation and the same processing for membership queries as the classic BF, but offers a considerably smaller number of false positives and no false negatives. We have shown using computational experiments that the yesno BF outperforms the classic BF in the scenario of packet forwarding with in-packet path encoding as introduced in the information-centric architecture PSIRP/PURSUIT. However, the structure is general enough to be used in a wide variety of applications. Moreover, the yes-no BF construction is dynamic and allows a choice of heuristics and optimization algorithms, as demonstrated in [19].
ACKNOWLEDGMENT
, where b e ∧ b S = b e , then b e ≤ b S bitwise.
Fig. 1 .
1Set diagram for the elements included in the BF, the false positives and the elements to be queried.
Fig. 2 .
2Schematic representation of the yes-no BF structure.
8 Fig. 3 .
83An example of the yes-no BF structure for an element e.
Fig. 4 9 Fig. 4 .
494shows the set R of elements that have been included in the no-filters, namely R ⊂ F . Also, it shows the set G of the false positives of the no-filters which may have some elements in S, namely the set G ∩ S is the set of the false negatives. Set diagram of the elements to include in the yes-and no-filters, the false positives of the yes-and the no-filters and the elements to be queried for the case with false negatives.
11 Fig. 5 .
115Set diagram of the elements to include in the yes-and no-filters, the false positives of the yes-and the no-filters and the elements to be queried for the case with no false negatives.
Since
E ⊂ S c , then Pr[E ∩ S c ] = Pr[E]. Consequently, given e ∈ S c , Pr[S c ] = 1, Pr[S] = 0 and the false positive probability in (24) becomes:
( 28 )
28Thus, assuming that e ∈ S c ∩ R c , the false positive probability of the yes-no BF depends on the false positive probability of the yes-filter f S and on the probability 1 − f R for the no-filters, where f R is the probability of false positives of the set of no-filters. It is easy to verify that the probability of false positives of the yes-no BF Pr[E|S c ∩ R c ] is smaller than the probability of false positives of a Bloom filter of the same size of the yes-filter.
V. EVALUATION OF THE YES-NO BF PERFORMANCE WITH RESPECT TO THE YES-NO BF PARAMETERS An experimental evaluation has been carried out in order to study the dependency of the false positive rate from the yes-no BF parameters and from the set S characteristics. The number of false positives for the yes-no BF has been quantified through simulations and plotted against the average number of false positives of the correspondent BF calculated using the approximate formula of the false positive probability f p in (5). Given the false positive probability, the average number F p of false positive occurrences can be computed as
• t = 100 ,
100the number of elements to query • p = 160, the length of the yes-filter • q = 32, the length of a no-filter • so r = 3, the number of no-filters • k = 4, the number of hash functions for the yes-filter • k = 5, the number of hash function for the no-filter
A. Evaluation of false positive performance as a function of the number of hash functions for the yes-no BF For the evaluation of false positive performance as a function of the number of hash functions for the yes-filter the average number of false positive occurrences has been computed for 1 ≤ k ≤ 14. The plot in Fig. 6 shows the average number of false positives of the yes-no BF for 10 4 experiments together with the average number of false positive of the correspondent BF.
Fig. 6 .
6Average number of false positive as a function of the number k of hash functions of the yes-filter. The dotted line represents the average number of false positive of the correspondent BF of size m = 256.The simulations show that the average number of false positive occurrences as a function of the number of hash functions of the yes-filter has a minimum as for the classic BF. Moreover, for small number of hash functions the number of false positive occurrences is much smaller for the yes-no BF while for higher k the BF performs better. This is expected since the yes-filter is smaller in length than the classic BF so as k increases the yes-filter becomes fuller than the BF and the no-filter mechanism which improves the false positive cannot cope any longer. Just to analyse the actual reduction in false positive occurrences, we also compare the performance of the yes-no BF with a BF of the same size as the yes-filter. The no-filters mechanism clearly reduces greatly the average number of fasle positive occurrences, as shown inFig. 7, following the results of the theoretical analysis of the false positive probability (25).
Fig. 7 .
7Average number of false positive as a function of the number k of hash functions of the yes-filter. The dotted line represents the average number of false positive of a BF of the same as the yes-filter m = 160. For the evaluation of false positive performance as a function of the number of hash functions for the no-filter the average number of false positive occurrences has been computed for 1 ≤ k ≤ 14. The plot in Fig. 8 shows the average number of false positives of the yes-no BF for 10 4 experiments together with the average number of false positive of the correspondent BF. The simulations show that the average number of false positive occurrences as a function of the number of hash functions of the no-filter has a minimum as for the classic BF. Changing the number of hash functions of the no-filter does not influence greatly the number of false positive occurrences and its value remains well below the correspondent value for the classic BF since an appropriate value of k has been chosen for the number of hash function of the yes-filter. B. Evaluation of false positive performance as a function of the number of elements to store in the yes-no BF For the evaluation of false positive performance as a function of the number of the elements to store in the yes-no BF the average number of false positive occurrences has been computed for 10 ≤ n ≤ 90. The plot in Fig. 9 shows the average number of false positives of the yes-no BF for 10 4 experiments together with the average number of false positive of the correspondent BF.
Fig. 8 .
8Average number of false positive as a function of the number k of hash functions of the no-filter. The dotted line represents the average number of false positive of the correspondent BF.
Fig. 9 .
9Average number of false positive occurrences as a function of the number n of elements to store in the yes-no BF. The dotted line represents the average number of false positive of the correspondent BF of m = 256 bits. The simulations show that the average number of false positive occurrences as a function of the number of elements to store in the yes-no BF is an increasing function as for the classic BF.
Fig. 10 .
10Average number of false positive occurrences as a function of the number n of elements to store in the yes-no BF. The dotted line represents the average number of false positive of the correspondent BF of m = 160 bits. C. Evaluation of false positive performance as a function of the length of the no-filter For the evaluation of false positive performance as a function of the length of the no-filter the average number of false positive occurrences has been computed for 10 ≤ q < 60. The plot in Fig. 11 shows the statistic of the average number of false positives of the yes-no BF for 10 4 experiments together with the average number of false positive of the correspondent BF. Fig. 12 show the averages together with the average number of false positive occurrences for the classic BF. The simulations show that the average number of false positive occurrences as a function of the length of each no-filters has a minimum value. The number of false positive occurrences is much lower than the correspondent value for the BF since appropriate values has been chosen forFig. 11. Statistics of number of false positive as a function of the length q of the no-filter.
Fig. 12 .
12Average number of false positive occurrences as a function of the length q of the no-filter together with the number of false positive occurrences of the classic BF (dotted line). the other parameters. The length of each no-filter does not have a strong impact on the overall results.D. Evaluation of false positive performance as a function of the number of no-filtersFor the evaluation of false positive performance as a function of the number of no-filters the average number of false positive occurrences has been computed for 1 ≤ r ≤ 9. Firstly, we consider the case in which the length p of the yes-filter is kept fixed and consequently the total length of the yes-no BF changes accordingly. Having p = 160, q = 32 and 1 ≤ r ≤ 9, the total length of the yes-no BF is m = p + qr as discussed in Sec. III. The plot inFig. 13shows the average number of false positives of the yes-no BF for 10 4 experiments together with the average number of false positive of the correspondent BF. The value of m is shown on the upper x-axis.
Fig. 13 .
13Average number of false positive occurrences as a function of the number r of no-filters, keeping the length of the yes-filter fixed, together with the number of false positive occurrences of the classic BF (dotted line). The length of the yes-no BF is reported on the upper x-axis. For r = 0, the yes-no BF becomes a classic BF where m = p = 160 while for greater r the yes-no BF offers a lower number of false positives than the classic BF of the same length m. This is expected since with more no-filters more false positives can be tracked. Secondly, we consider the case in which the length m of the yes-no BF is kept fixed and consequently the length of the yes-filter changes accordingly. Having m = 256, q = 32 andFig. 14 shows the averages together with the average number of false positive occurrences for the classic BF. The value of p is shown on the top x-axis. Fig. 14. Average number of false positive occurrences as a function of the number r of no-filters keeping the total length of the yes-no BF fixed, together with the number of false positive occurrences of the classic BF (dotted line).The length of the yes-filter is reported on the upper x-axis.
First
of all the set F , the set of the elements generating false positives has to be built. The complexity of this operation can be quantified as O(|T |), where |T | is the cardinality of the set T . This is because each element in the set T has to be checked. Given F , if the false negatives can be tolerated then a greedy algorithm can be used to choose in which no-filter of the set S the no-filter of the false positive can be stored. The time complexity of a greedy algorithm can be generally estimated as linear in the number of false positives O(|F |).
Fig. 15 .
15Schematic representation of a path with multiple destinations.
forwarding path is represented as the set S containing all the links along the path, whereas all the other links adjacent to the path, which are obviously not part of the path, are collected in the set T . Second order false positives are not considered since the probability is multiplicative and it can be considered negligible. For each of the elements of the sets S and T we consider 1000 random allocations of the position of the 1s for the yes-filter and the no-filter to simulate the hashing. We build the yes-no BF for all the elements of S and T and calculate the number of false positives as the average over the number of false positive occurrences for the 1000
26 Fig. 16 .
2616produce any false positives. They are not interesting for our comparison. On the diagram, they all are represented by the point with the coordinates 0,0. Furthermore, there are two comparatively large topologies, in which both the BFs and the yes-no BFs produce a relatively large number of The expected number of false positive occurrences of the yes-no BF against the BF.
Fig. 17 .
17The false positive rate of the yes-no BF and of the BF against the number n of links along the path. false positives, and, therefore, perhaps one would want to consider using a different forwarding model.
Fig. 17
17shows the false positive rate against the number of links along the path of the BF and the yes-no BF. The result has been obtained averaging the false positive rate of paths on different topologies having the same length. The false positive rate of the yes-no BF is consistently lower than the false positive rate of the BF.
Fig. 18 .
18The ratio between the false positive rate of the yes-no BF and of the BF as a function of the number of links encoded in the data structure.
Fig. 18
18shows the ratio of the false positive rate of the yes-no BF and of the BF f yn /f BF as a function of the number of links (elements) encoded in the data structure, up to n = 35 links where the yes-no BF offers a false positive rate which only a quarter of the correspondent false positive rate of the BF.
TABLE
since S = S ∪ U and S ∩ U = ∅. The probability of S can be expressed as: Pr[S] = Pr[S] Pr[S|S] + Pr[S c ] Pr[S|S c ].(9)
Since Pr[S|S] = 1, Pr[S|S c ] = f S from (7) and Pr[S c ] = 1 − Pr[S], we obtain
Pr[S] = Pr[S] + (1 − Pr[S])f S
Each hash function in H outputs values in {1, 2, ..p} while each hash function in W outputs values in {1, 2, ..q}, with q << p and qr < p.h 2 , ...h k } for the yes-filter
• W = {w 1 , w 2 , .., w k } for the no-filter
where generally k < k.
for some applications like forwarding in information centric networks.
≤ r ≤ 7, the total length of the yes-filter is p = m − qr as discussed in Sec. III.
Space/time trade-offs in hash coding with allowable errors. B Bloom, Commun. ACM. 137B. Bloom, "Space/time trade-offs in hash coding with allowable errors," Commun. ACM, vol. 13, no.7, pp. 422-426, 1970.
Network applications of Bloom filters: A survey. A Broder, M Mitzenmacher, Internet Mathematics. 14A. Broder and M. Mitzenmacher, "Network applications of Bloom filters: A survey," Internet Mathematics, vol. 1, no.4, pp. 485-509, 2004.
Theory and practice of Bloom filters for distributed systems. S Takoma, C Rothenberg, E Lagerspetz, IEEE Communications Surveys and Tutorials. 141S. Takoma, C. Rothenberg, and E. Lagerspetz, "Theory and practice of Bloom filters for distributed systems," IEEE Communications Surveys and Tutorials, vol. 14(1), pp. 131-155, 2012.
Retouched Bloom filters: Allowing networked applications to trade off selected false positives against false negatives. B Donnet, B Baynat, T Friedman, Lisboa, Portugal: CoNEXTB. Donnet, B. Baynat, and T. Friedman, "Retouched Bloom filters: Allowing networked applications to trade off selected false positives against false negatives." Lisboa, Portugal: CoNEXT 2006, Dec 2006.
The deletable Bloom filter: a new member of the Bloom family. C Rothenberg, C Macapuna, F Verdi, F Magalhaes, IEEE Communications Letters. 146C. Rothenberg, C. Macapuna, F. Verdi, and F. Magalhaes, "The deletable Bloom filter: a new member of the Bloom family," IEEE Communications Letters, vol. 14,no.6, pp. 557-559, 2010.
Summary cache: A scalable wide-area web cache sharing protocol. L Fan, P Cao, J Almeida, A Broder, IEEE/ACM Transaction on Networking. 83L. Fan, P. Cao, J. Almeida, and A. Broder, "Summary cache: A scalable wide-area web cache sharing protocol," IEEE/ACM Transaction on Networking, vol. 8, no.3, pp. 281-293, 2000.
Multilayer compressed counting Bloom filters. D Ficara, S Giordano, G Procissi, F Vitucci, Proceedings of the 27 th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM). the 27 th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM)D. Ficara, S. Giordano, G. Procissi, and F. Vitucci, "Multilayer compressed counting Bloom filters," in Proceedings of the 27 th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), 2008, pp. 311-315.
Scalable Bloom filter. P Almeida, C Baquero, N Preguica, D Hutchison, Information Processing Letters. 1016P. Almeida, C. Baquero, N. Preguica, and D. Hutchison, "Scalable Bloom filter," Information Processing Letters, vol. 101, no.6, pp. 255-261, 2007.
Complement Bloom filter for identifying true positiveness of a Bloom filter. H Lim, J Lee, C Yim, IEEE Communication Letters. 1911H. Lim, J. Lee, and C. Yim, "Complement Bloom filter for identifying true positiveness of a Bloom filter," IEEE Communication Letters, vol. 19, no.11, pp. 1905-1908, 2015.
Optimized hash for network path encoding with minimized false positives. L Carrea, A Vernitski, M Reed, Computer Networks. 58L. Carrea, A. Vernitski, and M. Reed, "Optimized hash for network path encoding with minimized false positives," Computer Networks, vol. 58, pp. 180-191, 2014.
LIPSIN: Line speed publish/subscribe internetworking. P Jokela, A Zahemszky, C E Rothenberg, S Arianfar, P Nikander, ACM SIGCOMM '09. Barcelona, SpainP. Jokela, A. Zahemszky, C. E. Rothenberg, S. Arianfar, and P. Nikander, "LIPSIN: Line speed publish/subscribe inter- networking." Barcelona, Spain: ACM SIGCOMM '09, Aug 2009.
Illustrating a publish/subscribe internet architecture. N Fotiou, G Polyzos, D Trossen, Journal on Telecommunication Systems -Springer. 514N. Fotiou, G. Polyzos, and D. Trossen, "Illustrating a publish/subscribe internet architecture," Journal on Telecommunication Systems -Springer, vol. 51, no. 4, pp. 233-245, 2012.
Designing, implementing and evaluating a new internetworking architecture. D Trossen, J Riihijrvi, P Nikander, P Jokela, J Kjllmand, J Rajahalme, Computer Communications. 3517D. Trossen, J. Riihijrvi, P. Nikander, P. Jokela, J. Kjllmand, and J. Rajahalme, "Designing, implementing and evaluating a new internetworking architecture," Computer Communications, vol. 35, no.17, pp. 2069-2081, 2012.
MPSS: Multiprotocol stateless switching. A Zahemszky, P Jokela, M Sarela, S Ruponen, J Kempf, P Nikander, 13th IEEE Global Internet Symposium. San Diego CA USAA. Zahemszky, P. Jokela, M. Sarela, S. Ruponen, J. Kempf, and P. Nikander, "MPSS: Multiprotocol stateless switching." San Diego CA USA: 13th IEEE Global Internet Symposium 2010, Mar 2010.
On the false positive rate of Bloom filters. P Bose, H Guo, E Kranakis, A Maheshwari, P Morin, J Morrison, M Smid, Y Tang, Inf. Process. Lett. 1084P. Bose, H. Guo, E. Kranakis, A. Maheshwari, P. Morin, J. Morrison, M. Smid, and Y. Tang, "On the false positive rate of Bloom filters," Inf. Process. Lett., vol. 108, no.4, pp. 210-213, 2008.
Less hashing, same performance: Building a better Bloom filter. A Kirsch, M Mitzenmacher, Random Structures and Algorithms. 322A. Kirsch and M. Mitzenmacher, "Less hashing, same performance: Building a better Bloom filter," Random Structures and Algorithms, vol. 32, no.2, pp. 187-218, 2008.
Reducing false positives of a Bloom filter using cross-checking Bloom filters. H Lim, N Lee, J Lee, C Yim, Appl. Math. Inf. Sci. 84H. Lim, N. Lee, J. Lee, and C. Yim, "Reducing false positives of a Bloom filter using cross-checking Bloom filters," Appl. Math. Inf. Sci., vol. 8, no.4, pp. 1865-1877, 2014.
The Internet topology zoo. S Knight, H Nguyen, N Falkner, R Bowden, M Roughan, IEEE Journal on Selected Areas in Communications. 299S. Knight, H. Nguyen, N. Falkner, R. Bowden, and M. Roughan, "The Internet topology zoo," IEEE Journal on Selected Areas in Communications, vol. 29, no.9, pp. 1765-1775, 2011.
An approximate dynamic programming approach for improving accuracy of lossy data compression by Bloom filters. X Yang, A Vernitski, L Carrea, European Journal of Operational Research -IN PRESS. X. Yang, A. Vernitski, and L. Carrea, "An approximate dynamic programming approach for improving accuracy of lossy data compression by Bloom filters," European Journal of Operational Research -IN PRESS.
| [] |
[
"Neutrino and Collider Implications of a Left-Right Extended Zee Model",
"Neutrino and Collider Implications of a Left-Right Extended Zee Model"
] | [
"Sarif Khan \nHarish-Chandra Research Institute\nChhatnag Road211 019Jhunsi, AllahabadIndia\n\nHomi Bhabha National Institute\nTraining School Complex\n400085Anushakti Nagar, MumbaiIndia\n",
"Manimala Mitra \nHomi Bhabha National Institute\nTraining School Complex\n400085Anushakti Nagar, MumbaiIndia\n\nInstitute of Physics\nSachivalaya Marg751005BhubaneswarIndia\n",
"Ayon Patra \nCentre for High Energy Physics\nIndian Institute of Science\nBangalore -560012India\n"
] | [
"Harish-Chandra Research Institute\nChhatnag Road211 019Jhunsi, AllahabadIndia",
"Homi Bhabha National Institute\nTraining School Complex\n400085Anushakti Nagar, MumbaiIndia",
"Homi Bhabha National Institute\nTraining School Complex\n400085Anushakti Nagar, MumbaiIndia",
"Institute of Physics\nSachivalaya Marg751005BhubaneswarIndia",
"Centre for High Energy Physics\nIndian Institute of Science\nBangalore -560012India"
] | [] | We study a simple left-right symmetric (LRS) extension of the Zee model for neutrino mass generation. An extra SU (2) L/R singlet charged scalar helps in generating a loop-induced Majorana mass of the neutrinos. This scenario is quite distinct from other LRS models as the right-handed neutrinos are very light of the order of a few eV to a few MeV. We study the collider signature of the charged scalar at e + e − collider, where a huge enhancement in the production cross-section is possible, resulting in a much stronger signal at the ILC or CLIC experiments. | 10.1103/physrevd.98.115038 | [
"https://arxiv.org/pdf/1805.09844v2.pdf"
] | 119,352,485 | 1805.09844 | eb8ee46d5f80001e174221858b051b26b68ac0b2 |
Neutrino and Collider Implications of a Left-Right Extended Zee Model
Sarif Khan
Harish-Chandra Research Institute
Chhatnag Road211 019Jhunsi, AllahabadIndia
Homi Bhabha National Institute
Training School Complex
400085Anushakti Nagar, MumbaiIndia
Manimala Mitra
Homi Bhabha National Institute
Training School Complex
400085Anushakti Nagar, MumbaiIndia
Institute of Physics
Sachivalaya Marg751005BhubaneswarIndia
Ayon Patra
Centre for High Energy Physics
Indian Institute of Science
Bangalore -560012India
Neutrino and Collider Implications of a Left-Right Extended Zee Model
We study a simple left-right symmetric (LRS) extension of the Zee model for neutrino mass generation. An extra SU (2) L/R singlet charged scalar helps in generating a loop-induced Majorana mass of the neutrinos. This scenario is quite distinct from other LRS models as the right-handed neutrinos are very light of the order of a few eV to a few MeV. We study the collider signature of the charged scalar at e + e − collider, where a huge enhancement in the production cross-section is possible, resulting in a much stronger signal at the ILC or CLIC experiments.
Introduction: The observation of neutrino oscillation leading to the realization that neutrinos are massive is one of the biggest motivations of physics beyond the Standard Model (SM). A large number of models have been suggested to explain neutrino masses and mixings either by the seesaw mechanism [1] or through loop induced processes [2]. The Zee model [3] is one of the simplest such scenarios where neutrino masses are generated at one-loop by extending the SM scalar sector with an extra doublet and a charged singlet scalar field. The charged singlet scalar can mix with other charged scalars while also having non-zero flavor violating couplings with leptons, giving rise to neutrino masses at one-loop. Unfortunately the simplest form of the Zee model was shown to be ruled out by experimental neutrino data [4] however its extensions might still be viable. In this work we study an extended Zee model in a left-right symmetric (LRS) framework [5] to examine its viability from neutrino oscillation data as well as to analyse the possible collider implications for the charged singlet Higgs.
It is quite natural to extend the Zee model in a simple LRS framework to generate the neutrino masses and mixings. LRS models can help explain the origin of parity violation as a spontaneously broken mechanism. It can possibly solve the strong CP problem [6] while also being able to generate the light neutrino masses through seesaw mechanism. Yet the minimal scalar sector in a LRS framework consisting of two doublet and a bidoublet scalar fields can only generate a Dirac mass term for the neutrinos. Inclusion of an extra charged singlet scalar is thus the most economical way to generate neutrino Majorana masses in such a scenario.
The singlet charged scalar can give rise to rich collider phenomenology at lepton colliders. Even though the Large Hadron Collider (LHC) has a very limited capability of observing them, the upcoming lepton colliders will be able to copiously produce these charged particles owing to their large couplings with the leptons. The same coupling also contributes to SM neutrino masses. A detailed study of the pair production and decay of these charged scalars in the upcoming International Linear Collider (ILC) and Compact Linear Collider (CLIC) experiments have been performed in this letter, taking into account neutrino masses and mixings. The final state of two opposite sign leptons and missing energy can be measured quite significantly over the SM background resulting in a possibility to observe such a process even with a very low luminosity at these experiments.
Model and Spectrum: LRS models are simple gauge extensions of the SM with the gauge group being
SU (3) C × SU (2) L × SU (2) R × U (1) B−L .
The charge of a particle is defined as Q = I 3L + I 3R + (B − L)/2. The quarks and leptons consist of left-handed and righthanded doublet fields and hence the right-handed neutrinos are naturally present in this framework.
The minimal Higgs sector consists of
H R (1, 1, 2, 1) = H + R H 0 R , H L (1, 2, 1, 1) = H + L H 0 L , Φ(1, 2, 2, 0) = φ 0 1 φ + 2 φ − 1 φ 0 2 , δ(1, 1, 1, 2) = δ + ,(1)
where the numbers in the brackets denote the quantum numbers under SU (3) C , SU (2) L , SU (2) R , U (1) B−L gauge groups. The model has been proposed in [7], and studied in detail for the LHC and low energy phenomenology.
The Yukawa lagrangian is given as:
L Y = Y q1 ij Q Li ΦQ Rj + Y q2 ij Q Li ΦQ Rj + Y l1 ij l Li Φl Rj + Y l2 ij l Li Φl Rj + λ Lij l T Li iτ 2 l Lj δ + + λ Rij l T Ri iτ 2 l Rj δ + + H.C.(2)
where Y and λ are the Yukawa couplings and
Φ = τ 2 Φ * τ 2 .(3)
The structure of λ L/Rij term is such that the only terms that will survive are the ones with i = j. This is exactly the same that happens in the Zee mechanism of neutrino mass generation. If we expand out any one of the terms involving δ in the Yukawa Lagrangian we will obtain: where ν i and e j are in flavor basis and λ ij = λ ij − λ ji . The Vacuum expectation values (VEV) of the Higgs fields are given as:
L ⊃ i,j ν i e j λ ij ,(4)φ 0 1 = v 1 , φ 0 2 = v 2 , H 0 R = v R , H 0 L = v L ,(5)
with the effective electroweak VEV given as v EW = v 2 1 + v 2 2 + v 2 L . Without loss of generality, one of the bidoublet VEVs can be chosen to be small. Also since v L does not contribute to the top mass, a large v L would automatically require a large top Yukawa coupling resulting in the theory being non-perturbative at quite low scales. The hierarchy in the VEVs thus has been chosen such that
v R >> v 1 > v 2 , v L .(6)
The δ field is responsible for producing the Majorana mass terms in the neutrino mass matrix which are given as [7]:
(M L ν ) αγ = 1 4π 2 λ αβ L m e β 3 i=1 Log M 2 hi m 2 e β × V 5i (Y † l ) βγ V * 2i − ( Y † l ) βγ V * 1i + α ↔ γ , (M R ν ) αγ = 1 4π 2 λ αβ R m e β 3 i=1 Log M 2 hi m 2 e β × V 5i (Y l ) βγ V * 1i − ( Y l ) βγ V * 2i + α ↔ γ . (7)
Here V ij are the charged Higgs boson mixing elements. The neutrino mass matrix would thus be a 6 × 6 matrix given as:
M ν = M L ν M D ν (M D ν ) T M R ν ,(8)
where M L ν and M R ν are generated at one-loop. A simple rotation of the bidoublet fields is performed such that only one of the new fields get a non-zero vev. This along with a redefinition of the couplings gives
M u = Y q v 1 , M d = Y q v 1 , M l = Y l v 1 , M D ν = Y l v 1 . (9)
The redefined coupling Y l , which we have chosen to be diagonal, is entirely determined from the charged lepton masses as can be seen from Eq. 9. Similarly Y q (chosen to be diagonal) and Y q can be determined from the up and down sector quark masses and CKM mixings. For the neutrino sector we first chose Y l to be zero and tried to get the light neutrino masses and mixings from M L ν alone. This approach does not work as there are too few free parameters to fit the experimental neutrino data. We then considered the case with λ L chosen to be zero such that the light neutrino masses arise entirely from M D ν and M R similar to type-I seesaw mechanism. This gives us the correct experimentally observed masses and mixings for the light neutrinos and hence this is the approach we follow for the neutrino sector 1 . The right-handed neutrino masses, generated at the loop-level, are proportional to the square of the lepton Yukawa couplings and hence are naturally small ranging from a few eV to a few MeV. This is quite different from other left-right symmetric models where the right-handed neutrino is naturally heavy as its mass is proportional to the righthanded symmetry breaking scale. For our analysis we have chosen λ R ∼ O(1) which makes the lightest righthanded neutrino M N1 ∼ eV, and the other two M N2,3 ∼ MeV. Since we use a type-I seesaw-like structure for the neutrino mass, the Yukawa couplings Y l are chosen accordingly to satisfy the correct neutrino oscillation parameters. The allowed values for the elements of M D ν obtained by scanning over the allowed parameter space are shown in Fig. 1. Here we have varied the elements of λ R matrix between 0.5 to 1.0 keeping their values quite close by allowing a spread of only 10% from each other. This is done so that any possible hierarchy due to the experimental neutrino data is clearly visible in the M D ν sector. Since the lightest right-handed neutrino mass is quite small, we also make sure that its mixing with the active neutrinos is small (sin θ 10 −2 ). This makes all of their kinematically allowed decays to occur outside the detector.
The scalar sector of the model consists of four CPeven, two CP-odd and three charged Higgs boson states. Two CP-odd and two charged states are eaten up to give masses to the Z R , Z, W R , W gauge bosons respectively. We specifically focus our discussion on the charged Higgs sector, as that is most important for the neutrino masses and the collider analysis of the charged scalar discussed in this letter. The charged Higgs mass-squared matrix is 5×5 which upon diagonalization gives two zero eigenvalues corresponding to the two goldstone bosons absorbed by the W R and W bosons to give them mass. The goldstone bosons primarily consist of H ± R and φ ± 1 states as their corresponding doublet's neutral fields get the large nonzero VEVs. The other three eigenstates give the three physical charged Higgses and are linear combinations of φ ± 2 , H ± L and δ ± . Flavor constraints require the neutral component of the bidoublet field φ 0 2 mass to be heavier than 15 TeV, forcing its charged counterpart to be very heavy as well. So δ ± can primarily mix only with H ± L as φ ± 2 is effectively decoupled owing to its large mass. We will consider two scenarios for our analysis. One where the lightest charged Higgs consists almost entirely of the charged singlet field δ ± and another where the lightest physical state is almost equal admixture of δ ± and H ± L .
Mass
Composition 473.32 0.002φ + 2 + 0.999δ + 1000.7 0.002φ + 2 + 0.999δ + 432.58 0.03 φ 1 − * − 0.006φ + 2 + 0.72H + L + 0.69δ + 1000.9 0.03 φ 1 − * − 0.006φ + 2 + 0.76H + L + 0.65δ + Experimental limits and Collider signature: In Table I we present a list of the various charged Higgs eigenstates considered in this study. We consider two cases with minimal or zero mixing (consisting entirely of δ + ) and two with maximal or half mixing of δ + with H + L . For these benchmark points we study the pair production of charged Higgs states and their decay to a final state of two opposite sign charged leptons and two neutrinos. The most recent experimental bound on this process is from the ALTAS search [8] of two opposite sign leptons and missing energy. They have put a bound of 500 GeV if the final state is coming from pair production of two sleptons. The production cross-section of the charged Higgs in our model is much lower and even a 430 GeV charged Higgs is safe from the LHC bounds 2 . So the benchmark points we consider are allowed by the experimental observations. & E T greater than 2 fb is ruled out while we only get 0.23 fb for M H ± = 450 GeV with similar cuts. The pair-production of the charged Higgs at LHC is through the s-channel process mediated by γ, Z and Z R bosons resulting in a cross-section of a few femtobarns (fb) or less for the mass range considered here. In a lepton collider, there is an additional t-channel process mediated by the neutrinos as shown in Fig. 2. Owing to the large couplings of the charged singlet with the right-handed leptons and the small masses of the righthanded neutrinos in this model, this t-channel process will be the major production channel. We have thus studied the pair production of the charged Higgs at 1 TeV run of the International Linear Collider (ILC) [9] and 3 TeV run by Compact Linear Collider (CLIC) [10]. We include the relevant vertices in FeynRules [11], and use MadGraph [12] for event generation, Pythia [13] for hadronization, and DelPhes [14] for detector simulation. Fig. 3 gives a plot of the pair-production cross-section of the charged singlet Higgs as a function of its mass for four different center-of-mass energies (c.m.energies) at the lepton colliders. The cross-section varies between 0.1-10 pb, for higher c.m.energies. For illustrative purpose, here we consider zero mixing of the singlet scalar and λ Rij ∼ 1. In our analysis, we however consider both the scenarios with and without mixing for four sets of model parameters consistent with neutrino data.
The charged Higgs, once produced, will then decay into a charged lepton and a right-handed neutrino giving a final state of dileptons with opposite charge (l + and l − ) and missing transverse energy (MET). Even the case where the charged Higgs is a mixture of δ ± and H ± L , this is the only kinematically allowed 2-body decay channel with its branching into 3-body decays being almost negligible. This is because H L does not couple to the quarks or leptons and its other physical states (the charged state with H ± L and δ ± orthogonal to the one considered here and the CP-odd and CPeven neutral states coming from H 0 L ) are much heavier. Schematically, the signal looks like
e + e − → H + 1 H − 1 → l + l − E T + X,(10)
where l ± is either one of e ± , µ ± and τ ± or a combination of them, and we consider inclusive di-lepton+missing energy signature. Inside the detector τ lepton will decay leptonically or hadronically and a small portion of it will give opposite sign dilepton increasing the signal strength. As τ decays, eventually we get a final state signal which consist of opposite sign electron (e ± ) or muon (µ ± ) or di-jet. For the chosen final state, the dominant SM backgrounds are e + e − → l + l − Z (→ ν lνl ) (including both the ZZ and virtual photon contributions), e + e − → W + W − → l + l − ν lνl , and e + e − → t(→ b l + ν l )t(→ b l −ν l ). We do not put any veto on the light jet in our analysis. Various kinematic variables have a clear distinction between signal and the backgrounds, that motivates to implement the following set of cuts.
A0 p min T, l > 10 GeV and the pseudo-rapidity |η l | < 2.5 while generating partonic events.
A1 We select only events which contains two opposite sign lepton.
A2 We use cuts on the p T of the hardest lepton p l1 T > 130 GeV and relatively softer cut on the second lepton p l2 T > 60 GeV. A3 To reject background, we implement Z-veto, i.e., we veto events in the di-lepton invariant mass (m ll ) window with |m ll − 91.2| < 10 GeV.
A4 The background (tt) contains b-jets in the final state. However, the signal doesn't have any bjets. Therefore we have used b-veto in the final state to reduce the background without affecting the signal.
A5 We use relatively tight cut on pseudo-rapidity of the leading lepton |η l1 | > 1.
A6 To reduce the background we use cut on the missing energy E T > 80 GeV.
Using these cuts we can reduce the background markedly while keeping the signal at a significant level. Table II gives the background cross-section at 1 Tev ILC and 3 TeV CLIC experiments after putting the above-mentioned cuts. It is easy to see that the backgrounds have become quite small (σ ∼ 1 − 8 fb) and in both cases the dominant one arising from W + W − final state.
The signal cross-sections (σ ∼ 5−53 fb after cut) and their statistical significance (S) over the background are given in Tab. III. The later has been computed using the following expression,
S = 2 × (s + b)ln(1 + s b ) − s .(11)
Clearly the case with no mixing in the Higgs state gives a much larger cross-section, that results in a much better significance of signal over background boosting its chances to be discovered even in the early run of the upcoming lepton colliders operating with higher c.m.energy. This is because δ ± l ∓ R ν R vertex is primarily responsible for the charged Higgs pair-production. The mixing of δ ± with H ± L will only introduce an extra factor of cos 4 θ, where θ is the scalar mixing angle, resulting in a decrease of the cross-section. In particular, we show that only 1 fb −1 luminosity is required for zero-mixing to discover the charged Higgs H ± 1 with mass range 473 GeV -1 TeV. For the relatively less optimistic scenario of half-mixing, 3 fb −1 will be required to claim discovery.
SM Backgrounds
Effective Cross section after applying cuts (fb) Channels
Cross-section (fb) A0 + A1 A2 A3 A4 A5 A6 Conclusion: In this work, we have studied a leftright symmetric extension of Zee model that has quite unique characteristics. The model consists of light right-handed neutrinos of mass from MeV down to eV scale, and an additional charged scalar that can be copiously produced at lepton colliders. The light neutrino mass is generated via a combination of loop-induced processes and seesaw mechanism. We fit the neutrino masses and the observed mixing in this model, and extensively analyse the charged Higgs phenomenology at 1 TeV ILC and 3 TeV CLIC experiments. Owing to the extra interaction of the charged Higgs with the righthanded neutrinos and for moderately large Yukawas, the cross-section at e + e − collider is enormous compared to the LHC. We show that discovery of the charged Higgs with mass between 432-1000 GeV in the di-lepton Signal at e + e − Collider Effective Cross section after cuts (fb) Stat Significance (S) Experiment Mass (GeV) Mixing CS (fb) A0+A1 A2 A3 A4 A5 A6 L = 1 fb −1 L = 3 fb −1 S.K. also acknowledges the cluster computing facility at HRI (http://cluster.hri.res.in). S.K. would also like to thank the Department of Atomic Energy (DAE) Neutrino Project of Harish-Chandra Research Institute. The authors would like to thank Dr. Arnab Dasgupta for very useful discussions at the early stage of this work.
FIG. 1 :
1Scatter plot of neutrino Dirac mass matrix elements M D νij (denoted by M νij in the figure) satisfying the neutrino oscillation data.
FIG. 2 :
2Feynman diagram for the production of H + 1 H − 1 at e + e − collider.
FIG. 3 :
3Pair-production cross section of H + 1 H − 1 at e + e − collider for different center of mass energy.
TABLE I :
ILightest charged Higgs boson H ± 1 .
W + (→ l + ν l ) W − (→ l −ν + (→ l + ν l ) W − (→ l −ν1 TeV ILC
l + l − Z (→ ν lνl )
18.68
10.79
5.99
5.54
5.54
2.30
1.67
l )
126.88
52.72
32.15
32.15
32.15
12.44 7.05
t(→ bl + ν l )t(→bl −ν
l )
13.96
3.10
0.78
0.78
0.1
0.08
0.05
Total Backgrounds
8.77
3 TeV CLIC
l + l − Z (→ ν lνl )
6.33
3.0
2.89
2.86
2.86
0.54
0.44
W l )
13.85
5.45
5.1
5.1
5.1
1.34
1.13
t(→ bl + ν l )t(→bl −ν
l )
1.76
0.05
0.02
0.02
0.005
0.002 0.002
Total Backgrounds
1.57
TABLE II :
IICut-flow table for the obtained cross-sections corresponding to the SM backgrounds.
TABLE III :
IIICut-flow table of signal cross section at 1 TeV ILC and 3 TeV CLIC. + MET will require only 1-3 fb −1 integrated luminosity. Therefore, this model can most economically be tested at the very early run of CLIC or ILC experiments at higher c.m.energies. Acknowledgement: M.M. would like to acknowledge the DST-INSPIRE research grant IFA14-PH-99 and hospitality of CHEP, IISc, Bengaluru, where part of the discussion has been carried out. A.P. is supported by the SERB National Postdoctoral Fellowship [PDF/2016/000202]. S.K. thanks Prof. Sandhya Choubey for discussions.
Even if we keep both Y l and λ L to be non-zero, the value of the elements of λ L matrix satisfying the neutrino constraints come out to be very small to have any observable consequences for our study.
For a set of loose cuts denoted by SF1 in[8], a production cross-section for l + l −
. P Minkowski, Phys. Lett. B. 67421P. Minkowski, Phys. Lett. B 67 (1977) 421;
T Yanagida ; Tsukuba, ; S Glashow ; M. Lévy, No. 79-18proceedings of the Workshop on Unified Theories and Baryon Number in the Universe. A. Sawada, A. Sugamotothe Workshop on Unified Theories and Baryon Number in the UniverseTsukuba; Cargèse; New YorkPlenumKEK ReportQuarks and LeptonsT. Yanagida, proceedings of the Workshop on Uni- fied Theories and Baryon Number in the Universe, Tsukuba, 1979, eds; A. Sawada, A. Sugamoto, KEK Report No. 79-18, Tsukuba; S. Glashow, in Quarks and Leptons, Cargèse 1979, eds; M. Lévy. et al., (Plenum, 1980, New York);
Slansky, proceedings of the Supergravity Stony Brook Workshop. M Gell-Mann, P Ramond, R R Mohapatra, G Senjanović, Phys.Rev.Lett. P. Van Niewenhuizen, D. Freeman44912M. Gell-Mann, P. Ramond, R. Slan- sky, proceedings of the Supergravity Stony Brook Work- shop, New York, 1979, eds. P. Van Niewenhuizen, D. Freeman (North-Holland, Amsterdam). R. Mohapatra, G. Senjanović, Phys.Rev.Lett. 44 (1980) 912.
. E Ma, 10.1103/PhysRevLett.81.1171hep-ph/9805219Phys. Rev. Lett. 811171E. Ma, Phys. Rev. Lett. 81, 1171 (1998) doi:10.1103/PhysRevLett.81.1171 [hep-ph/9805219];
. F Bonnet, M Hirsch, T Ota, W Winter, 10.1007/JHEP07(2012)153arXiv:1204.5862JHEP. 1207153hep-phF. Bonnet, M. Hirsch, T. Ota and W. Winter, JHEP 1207, 153 (2012) doi:10.1007/JHEP07(2012)153 [arXiv:1204.5862 [hep-ph]];
. D Sierra, A Degee, L Dorame, M Hirsch, 10.1007/JHEP03(2015)040arXiv:1411.7038JHEP. 150340hep-phD. Aristizabal Sierra, A. Degee, L. Dorame and M. Hirsch, JHEP 1503, 040 (2015) doi:10.1007/JHEP03(2015)040 [arXiv:1411.7038 [hep-ph]].
. A Zee, 10.1016/0370-2693(80)90349-4,10.1016/0370-2693(80)90193-8Phys. Lett. 93461Phys. Lett.A. Zee, Phys. Lett. 93B, 389 (1980) Erratum: [Phys. Lett. 95B, 461 (1980)]. doi:10.1016/0370- 2693(80)90349-4, 10.1016/0370-2693(80)90193-8
. P H Frampton, M C Oh, T , P. H. Frampton, M. C. Oh and T.
. Yoshikawa, 10.1103/PhysRevD.65.073014hep-ph/0110300Phys. Rev. D. 6573014Yoshikawa, Phys. Rev. D 65, 073014 (2002) doi:10.1103/PhysRevD.65.073014 [hep-ph/0110300].
. Y Koide, 10.1016/S0920-5632(02)01726-7[hep-ph/0201250Nucl. Phys. Proc. Suppl. 111Y. Koide, Nucl. Phys. Proc. Suppl. 111, 294 (2002) doi:10.1016/S0920-5632(02)01726-7 [hep- ph/0201250];
. X G He, 10.1140/epjc/s2004-01669-8[hep-ph/0307172Eur. Phys. J. C. 34371X. G. He, Eur. Phys. J. C 34, 371 (2004) doi:10.1140/epjc/s2004-01669-8 [hep-ph/0307172].
. R N Mohapatra, J C Pati, Phys. Rev. D. 11566R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 566 (1975);
. G Senjanovic, R N Mohapatra, Phys. Rev. D. 121502G. Senjanovic and R. N. Mohapatra, Phys. Rev. D 12, 1502 (1975).
. M A B Beg, H. -S Tsao, Phys. Rev. Lett. 41278M. A. B. Beg and H. -S. Tsao, Phys. Rev. Lett. 41, 278 (1978);
. R N Mohapatra, G Senjanovic, Phys. Lett. B. 79283R. N. Mohapatra and G. Senjanovic, Phys. Lett. B 79, 283 (1978);
. K S Babu, R N Mohapatra, Phys. Rev. D. 411286K. S. Babu and R. N. Mo- hapatra, Phys. Rev. D 41, 1286 (1990);
. S M Barr, D Chang, G Senjanovic, Phys. Rev. Lett. 672765S. M. Barr, D. Chang and G. Senjanovic, Phys. Rev. Lett. 67, 2765 (1991);
. R N Mohapatra, A Rasin, Phys. Rev. Lett. 763490R. N. Mohapatra and A. Rasin, Phys. Rev. Lett. 76, 3490 (1996);
. R Kuchimanchi, Phys. Rev. Lett. 763486R. Kuchimanchi, Phys. Rev. Lett. 76, 3486 (1996);
. R N Mohapatra, A Rasin, G Senjanovic, Phys. Rev. Lett. 794744R. N. Mohapatra, A. Rasin and G. Senjanovic, Phys. Rev. Lett. 79, 4744 (1997);
. K S Babu, B Dutta, R N Mohapatra, Phys. Rev. D. 6516005K. S. Babu, B. Dutta and R. N. Mohapatra, Phys. Rev. D 65, 016005 (2001);
. R Kuchimanchi, Phys. Rev. D. 82116008R. Kuchimanchi, Phys. Rev. D 82, 116008 (2010).
. P Perez, C Murgui, S Ohmer, 10.1103/PhysRevD.94.051701arXiv:1607.00246Phys. Rev. D. 94551701hep-phP. Fileviez Perez, C. Murgui and S. Ohmer, Phys. Rev. D 94, no. 5, 051701 (2016) doi:10.1103/PhysRevD.94.051701 [arXiv:1607.00246 [hep-ph]];
. P , Fileviez Perez, C Murgui, 10.1103/PhysRevD.95.075010arXiv:1701.06801Phys. Rev. D. 95775010hep-phP. Fileviez Perez and C. Mur- gui, Phys. Rev. D 95, no. 7, 075010 (2017) doi:10.1103/PhysRevD.95.075010 [arXiv:1701.06801 [hep-ph]].
. M Aaboud, ATLAS CollaborationarXiv:1803.02762hep-exM. Aaboud et al. [ATLAS Collaboration], arXiv:1803.02762 [hep-ex].
. H Baer, arXiv:1306.6352hep-phH. Baer et al., arXiv:1306.6352 [hep-ph];
. C Adolphsen, arXiv:1306.6353physics.acc-phC. Adolphsen et al., arXiv:1306.6353 [physics.acc-ph];
. C Adolphsen, arXiv:1306.6328physics.acc-phC. Adolphsen et al., arXiv:1306.6328 [physics.acc-ph];
. T Behnke, arXiv:1306.6329physics.ins-detT. Behnke et al., arXiv:1306.6329 [physics.ins-det];
. T Behnke, arXiv:1306.6327physics.acc-phT. Behnke et al., arXiv:1306.6327 [physics.acc-ph].
. H Abramowicz, CLIC Detector and Physics Study CollaborationarXiv:1307.5288hep-exH. Abramowicz et al. [CLIC Detector and Physics Study Collaboration], arXiv:1307.5288 [hep-ex];
. D Dannheim, arXiv:1208.1402hep-exD. Dannheim et al., arXiv:1208.1402 [hep-ex];
. E Accomando, hep-ph/0412251CLIC Physics Working GroupE. Ac- comando et al. [CLIC Physics Working Group], hep-ph/0412251;
. L Linssen, arXiv:1202.5940L. Linssen et al., arXiv:1202.5940; N.
CLICdet: The post-CDR CLIC detector model. , J.-J Alipour Tehrani, B Blaising, D Cure, F Dannheim, K Duarte Ramos, Elsener, Alipour Tehrani, J.-J. Blaising, B. Cure, D. Dannheim, F. Duarte Ramos, K. Elsener et al., CLICdet: The post-CDR CLIC detector model, CLICdp-Note-2017- 001 (Mar, 2017).
. A Alloul, N D Christensen, C Degrande, C Duhr, B Fuks, 10.1016/j.cpc.2014.04.012arXiv:1310.1921Comput. Phys. Commun. 1852250hep-phA. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, Comput. Phys. Commun. 185, 2250 (2014) doi:10.1016/j.cpc.2014.04.012 [arXiv:1310.1921 [hep-ph]].
. J , arXiv:1405.0301JHEP. 140779hep-phJ. Alwall et al., JHEP 1407, 079 (2014) [arXiv:1405.0301 [hep-ph]];
. J Alwall, M Herquet, F Maltoni, O Mattelaer, T Stelzer, arXiv:1106.0522JHEP. 1106128hep-phJ. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, JHEP 1106, 128 (2011) [arXiv:1106.0522 [hep-ph]].
. T Sjostrand, S Mrenna, P Z Skands, hep-ph/0603175JHEP. 060526T. Sjostrand, S. Mrenna and P. Z. Skands, JHEP 0605, 026 (2006) [hep-ph/0603175].
. J De Favereau, DELPHES 3 CollaborationarXiv:1307.6346JHEP. 140257hep-exJ. de Favereau et al. [DELPHES 3 Collaboration], JHEP 1402, 057 (2014) [arXiv:1307.6346 [hep-ex]];
. M Selvaggi, J. Phys. Conf. Ser. 52312033M. Selvaggi, J. Phys. Conf. Ser. 523, 012033 (2014);
. A Mertens, J. Phys. Conf. Ser. 608112045A. Mertens, J. Phys. Conf. Ser. 608, no. 1, 012045 (2015).
| [] |
[
"AGNs at the cosmic dawn: predictions for future surveys from a ΛCDM cosmological model",
"AGNs at the cosmic dawn: predictions for future surveys from a ΛCDM cosmological model"
] | [
"Andrew J Griffin \nInstitute for Computational Cosmology\nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamUK\n",
"Cedric G Lacey \nInstitute for Computational Cosmology\nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamUK\n",
"Violeta Gonzalez-Perez \nInstitute of Cosmology and Gravitation\nUniversity of Portsmouth\nBurnaby RoadPO1 3FXPortsmouthUK\n\nEnergy Lancaster\nLancaster University\nLA1 4YBLancasterUK\n\nDimensions (ASTRO 3D\n\n",
"Claudia Del P Lagos \nInternational Centre for Radio Astronomy Research (ICRAR)\nUniversity of Western Australia\n35 Stirling HwyM468, 6009CrawleyWAAustralia\n\nARC Centre of Excellence for All Sky Astrophysics in\n\n\nCosmic Dawn Center (DAWN)\nCopenhagenDenmark\n",
"Carlton M Baugh \nInstitute for Computational Cosmology\nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamUK\n",
"Nikos Fanidakis \nMax-Planck-Institute for Astronomy\nKönigstugl 17D-69117HeidelbergGermany\n\nBASF\nCarl-Bosch Strasse 3867056LudwigshafenGermany\n"
] | [
"Institute for Computational Cosmology\nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamUK",
"Institute for Computational Cosmology\nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamUK",
"Institute of Cosmology and Gravitation\nUniversity of Portsmouth\nBurnaby RoadPO1 3FXPortsmouthUK",
"Energy Lancaster\nLancaster University\nLA1 4YBLancasterUK",
"Dimensions (ASTRO 3D\n",
"International Centre for Radio Astronomy Research (ICRAR)\nUniversity of Western Australia\n35 Stirling HwyM468, 6009CrawleyWAAustralia",
"ARC Centre of Excellence for All Sky Astrophysics in\n",
"Cosmic Dawn Center (DAWN)\nCopenhagenDenmark",
"Institute for Computational Cosmology\nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamUK",
"Max-Planck-Institute for Astronomy\nKönigstugl 17D-69117HeidelbergGermany",
"BASF\nCarl-Bosch Strasse 3867056LudwigshafenGermany"
] | [
"MNRAS"
] | Telescopes to be launched over the next decade-and-a-half, such as JWST, EUCLID, ATHENA and Lynx, promise to revolutionise the study of the high redshift Universe and greatly advance our understanding of the early stages of galaxy formation. We use a model that follows the evolution of the masses and spins of supermassive black holes (SMBHs) within a semi-analytic model of galaxy formation to make predictions for the Active Galactic Nucleus (AGN) luminosity function at z 7 in the broadband filters of JWST and EUCLID at optical and near-infrared wavelengths, and ATHENA and Lynx at X-ray energies. The predictions of our model are relatively insensitive to the choice of seed black hole mass, except at the lowest luminosities (L bol < 10 43 ergs −1 ) and the highest redshifts (z > 10). We predict that surveys with these different telescopes will select somewhat different samples of SMBHs, with EUCLID unveiling the most massive, highest accretion rate SMBHs, Lynx the least massive, lowest accretion rate SMBHs, and JWST and ATHENA covering objects inbetween. At z = 7, we predict that typical detectable SMBHs will have masses, M BH ∼ 10 5−8 M , and Eddington normalised mass accretion rates,Ṁ /Ṁ Edd ∼ 1 − 3. The SMBHs will be hosted by galaxies of stellar mass M ∼ 10 8−10 M , and dark matter haloes of mass M halo ∼ 10 11−12 M . We predict that the detectable SMBHs at z = 10 will have slightly smaller black holes, accreting at slightly higher Eddington normalised mass accretion rates, in slightly lower mass host galaxies compared to those at z = 7, and reside in haloes of mass M halo ∼ 10 10−11 M . | 10.1093/mnras/staa024 | [
"https://arxiv.org/pdf/1908.02841v2.pdf"
] | 199,511,344 | 1908.02841 | b1993d8f92f2de1b7bb5446957a09f4ff8ea7831 |
AGNs at the cosmic dawn: predictions for future surveys from a ΛCDM cosmological model
2019
Andrew J Griffin
Institute for Computational Cosmology
Department of Physics
University of Durham
South RoadDH1 3LEDurhamUK
Cedric G Lacey
Institute for Computational Cosmology
Department of Physics
University of Durham
South RoadDH1 3LEDurhamUK
Violeta Gonzalez-Perez
Institute of Cosmology and Gravitation
University of Portsmouth
Burnaby RoadPO1 3FXPortsmouthUK
Energy Lancaster
Lancaster University
LA1 4YBLancasterUK
Dimensions (ASTRO 3D
Claudia Del P Lagos
International Centre for Radio Astronomy Research (ICRAR)
University of Western Australia
35 Stirling HwyM468, 6009CrawleyWAAustralia
ARC Centre of Excellence for All Sky Astrophysics in
Cosmic Dawn Center (DAWN)
CopenhagenDenmark
Carlton M Baugh
Institute for Computational Cosmology
Department of Physics
University of Durham
South RoadDH1 3LEDurhamUK
Nikos Fanidakis
Max-Planck-Institute for Astronomy
Königstugl 17D-69117HeidelbergGermany
BASF
Carl-Bosch Strasse 3867056LudwigshafenGermany
AGNs at the cosmic dawn: predictions for future surveys from a ΛCDM cosmological model
MNRAS
0002019Accepted XXX. Received YYY; in original form ZZZPreprint 9th August 2019 Compiled using MNRAS L A T E X style file v3.0galaxies: high-redshift -galaxies: active -quasars: general
Telescopes to be launched over the next decade-and-a-half, such as JWST, EUCLID, ATHENA and Lynx, promise to revolutionise the study of the high redshift Universe and greatly advance our understanding of the early stages of galaxy formation. We use a model that follows the evolution of the masses and spins of supermassive black holes (SMBHs) within a semi-analytic model of galaxy formation to make predictions for the Active Galactic Nucleus (AGN) luminosity function at z 7 in the broadband filters of JWST and EUCLID at optical and near-infrared wavelengths, and ATHENA and Lynx at X-ray energies. The predictions of our model are relatively insensitive to the choice of seed black hole mass, except at the lowest luminosities (L bol < 10 43 ergs −1 ) and the highest redshifts (z > 10). We predict that surveys with these different telescopes will select somewhat different samples of SMBHs, with EUCLID unveiling the most massive, highest accretion rate SMBHs, Lynx the least massive, lowest accretion rate SMBHs, and JWST and ATHENA covering objects inbetween. At z = 7, we predict that typical detectable SMBHs will have masses, M BH ∼ 10 5−8 M , and Eddington normalised mass accretion rates,Ṁ /Ṁ Edd ∼ 1 − 3. The SMBHs will be hosted by galaxies of stellar mass M ∼ 10 8−10 M , and dark matter haloes of mass M halo ∼ 10 11−12 M . We predict that the detectable SMBHs at z = 10 will have slightly smaller black holes, accreting at slightly higher Eddington normalised mass accretion rates, in slightly lower mass host galaxies compared to those at z = 7, and reside in haloes of mass M halo ∼ 10 10−11 M .
formation at low redshift, as they are seen to produce huge X-ray cavities in the hot intracluster gas in galaxy clusters (e.g. Forman et al. 2005;David et al. 2011;Cavagnolo et al. 2011), and AGN feedback is included in theoretical models of galaxy formation to shut off gas cooling in massive haloes and star formation in the largest galaxies (e.g. Di Matteo et al. 2005;Croton et al. 2006;Bower et al. 2006), in order to reproduce the bright end of the galaxy luminosity function. AGNs may also play an important role in galaxy formation at higher redshift, where large-scale outflows driven by AGNs are observed e.g. at z ∼ 2 (Harrison et al. 2012), and at z ∼ 6 (Maiolino et al. 2012;Cicone et al. 2015). X-ray observations have also indicated that faint QSOs may play an important role in reionising the Universe (Giallongo et al. 2015;Onoue et al. 2017;Ricci et al. 2017).
At z ∼ 6, AGNs have been discovered with estimated black hole masses over a billion solar masses (e.g. Willott et al. 2010b;De Rosa et al. 2011;Venemans et al. 2013;Wu et al. 2015). How these SMBHs could grow to such large masses in such a short time is a puzzle. SMBHs grow from seed black holes, which could form from remnants of a first generation of (Population III) stars, or from gas clouds that form supermassive stars that eventually collapse to form a black hole, or from dense star clusters that collapse via stellar dynamical processes (e.g. Volonteri 2010). These seeds are expected to be of mass M seed = 10 − 10 5 M depending on the formation mechanism, with the remnants of Population III stars forming light (∼ 10 − 100M ) seeds, gas cloud collapse forming heavy (∼ 10 4−5 M ) seeds, and star cluster collapse forming seeds of intermediate (∼ 10 3 M ) mass (Volonteri 2010). SMBHs can then grow either by accretion of gas or by merging with other SMBHs. To form the observed high redshift SMBHs by gas accretion, these seeds require sustained accretion near the Eddington rate for several hundred Myr, which may be interrupted by feedback effects.
Fortunately, the next decade-and-a-half promise to be exciting for observing the high redshift Universe. The launch of the James Webb Space Telescope (JWST) in 2021 will pave the way for an increased understanding of the z > 7 Universe (e.g. Gardner et al. 2006;Kalirai 2018). JWST, with its 6.5m diameter mirror, will make observations from the optical to mid-infrared (0.6 µm to 30 µm) to probe the earliest galaxies and the stars contained within them. EU-CLID, also due for launch in 2021, with a 1.2m diameter mirror, is primarily a cosmology mission with the aim of constraining dark energy, but the surveys it will conduct at optical and near-IR wavelengths (0.5-2 µm) will also be useful for detecting high-redshift quasars (Laureijs et al. 2011). While JWST and EUCLID will probe similar wavelength ranges, the specifications of the missions are different. The sensitivity of JWST is better, but EUCLID will survey much larger areas of sky, which will lead to different samples of AGNs being detected by these two missions, as they will sample AGNs with different luminosities and space densities.
The Advanced Telescope for High-ENergy Astrophysics (ATHENA) (Nandra et al. 2013), scheduled for launch in 2031, will observe the high-redshift Universe at X-ray energies (0.5-10 keV). The Lynx X-ray observatory (The Lynx Team 2018), which has a proposed launch date in 2035, will also observe the distant Universe at similar energies (0.2-10 keV). The science objectives of both missions include determining the nature of SMBH seeds and investigating the influence of SMBHs on the formation of the first galaxies. The two missions have different capabilities: ATHENA has a larger field of view and larger effective area (which leads to better instrumental sensitivity) at 6 keV, but a worse angular resolution and lower effective area at 1 keV, compared to Lynx. The improved angular resolution of Lynx results in better sensitivity in practice, as sources that would be affected by source confusion when observed by ATHENA would be unaffected if observed by Lynx. Therefore, the two telescopes will detect different luminosity objects.
We are now entering an era in which the properties of SMBHs in the high redshift Universe (z > 7) during the first billion years of its evolution can be robustly probed. By comparing observations with simulations, we can test theoretical models of galaxy formation, and by comparing to the high redshift Universe, we can test these theoretical models in a regime that up to now is poorly constrained.
In this paper, we present predictions for the AGN population at z 7 for comparison with observations from JWST, EUCLID, ATHENA, and Lynx, using the model for SMBH and AGN evolution presented in Griffin et al. (2019) (hereafter Paper I), which includes a self-consistent treatment of SMBH spin, to predict AGN luminosities. This paper is one of a series of papers exploring SMBH and AGN properties within a physical galaxy formation model based on the ΛCDM model of structure formation. Paper I presented the model for the evolution of SMBH and AGN within the Baugh et al. (2019) recalibration of the Lacey et al. (2016) galform semi-analytical model of galaxy formation, showing a comparison of the predicted SMBH and AGN properties to observations for 0 z 6. Here, we extend the predictions of this model to z 7.
Other theoretical models have also made predictions for the evolution of SMBHs and AGNs through cosmic time, such as semi-analytic models (e.g. Lagos et al. 2008;Marulli et al. 2008;Bonoli et al. 2009;Fanidakis et al. 2012;Hirschmann et al. 2012;Menci et al. 2013;Neistein & Netzer 2014;Enoki et al. 2014;Shirakata et al. 2019), hydrodynamical simulations (e.g. Hirschmann et al. 2014Sijacki et al. 2015;Rosas-Guevara et al. 2016;Weinberger et al. 2018), and empirical models (e.g. Saxena et al. 2017;Weigel et al. 2017). Predictions for z > 7 have also been made using observational models (e.g. Aird et al. 2013) and semi-analytic models (e.g. Ricarte & Natarajan 2018b). In this paper, we are making predictions for AGNs at z 7 from a semianalytic galaxy formation model, which includes more channels of SMBH growth than Ricarte & Natarajan (2018b). A few predictions from our model have also previously been shown in Amarantidis et al. (2019), in which AGN luminosity functions from several different theoretical models are compared.
This paper is structured as follows. In Section 2 we outline the model used. In Section 3 we present predictions for black hole properties, and in Section 4 we present predictions for AGN luminosity functions for z 7. In Section 5 we present predictions for AGNs detectable by future surveys using JWST, EUCLID, ATHENA and Lynx, and in Section 6 we give our conclusions.
METHOD
In this paper, we analyse the properties of SMBHs and AGNs within the galform semi-analytic model of galaxy formation. We briefly outline the galaxy formation model, and the modelling of SMBHs and AGNs, which follow Paper I, apart from one change described below.
The galform galaxy formation model
In this paper, we present predictions using the same galaxy formation model as Paper I, which is the Baugh et al. (2019) recalibration of the Lacey et al. (2016) galform model. galform is a semi-analytic model of galaxy formation, which was introduced in Cole et al. (2000). In galform, galaxies form in dark matter haloes, with the evolution of the dark matter haloes described by halo merger trees. For a recent full description of the model, see Lacey et al. (2016). In the model used here, the halo merger trees are extracted from a cosmological dark matter N-body simulation (Helly et al. 2003). The baryonic exchange between different components (e.g. stars, hot halo gas, cold disc gas, black hole) is modelled by a set of coupled differential equations. Physical processes modelled in galform include i) the merging of dark matter haloes, ii) shock heating and radiative cooling of gas in haloes, iii) collapse of cooled gas onto a rotationally supported disc, iv) a two-phase interstellar medium for the cold gas with star formation from molecular gas, v) feedback from photoionisation, supernovae, and AGNs, vi) the chemical evolution of gas and stars, vii) galaxies merging in haloes due to dynamical friction, viii) bar instabilties in galaxy discs, ix) the evolution of stellar populations, and x) the extinction and reprocessing of stellar radiation by dust. The analytical prescriptions for these processes include a number of free parameters, which are calibrated on a range of observational constraints on galaxy properties.
galform has undergone continual development, with various galform models now in existence, (e.g. Gonzalez-Perez et al. 2014;Lacey et al. 2016). This paper uses the Baugh et al. (2019) recalibration of the Lacey et al. (2016) galform model for the Planck cosmology. This recalibrated model was presented for use with P-Millennium dark matter merger trees . P-Millennium is a high resolution dark matter simulation using the Planck Collaboration et al. (2014) cosmology, with a box of side 800Mpc and a halo mass resolution of 2.12×10 9 h −1 M (corresponding to 20 particles). The Lacey et al. (2016) model matches to a wide range of observational data, both in terms of wavelength (from far-UV luminosity functions to sub-mm number counts), and in terms of redshift, matching a large range of observations from z ∼ 0 to z ∼ 6.
SMBHs and AGNs in galform
SMBHs start out as seed black holes, which we model by adding a seed black hole of mass M seed to each dark matter halo. Unless otherwise stated, the value of M seed adopted is 10h −1 M . SMBHs can then grow via three channels: (i) starbursts triggered by mergers or disc instabilities, which can drive gas to the galaxy centre to be made available for accretion onto the SMBH (ii) 'hot halo' accretion in which gas quiescently accretes from the hot gas atmosphere in the largest haloes and (iii) mergers between SMBHs. Unlike some other models, the gas accretion rate is not assumed to be Eddington-limited.
Building on Fanidakis et al. (2011), in Paper I a model for the evolution of SMBH spin was presented, in which SMBH spin evolves via accretion of gas, or by merging with another SMBH. The SMBH/AGN model involves several free parameters, for which we use the same values as in Paper I. In Paper I, we generally adopted values from previous studies, with two free parameters (η Edd , which controls the suppression of luminosity for super-Eddington accretion rates, and fq which determines the lifetimes of the AGN episodes) calibrated on the observed AGN bolometric luminosity function for 0 z 6.
In the starburst mode, we assume that the SMBH accretion rate is constant over a time:
tacc = fqt bulge (1)
where t bulge is the dynamical timescale of the bulge. In Paper I, we gave the equations for bolometric radiative AGN luminosities in different accretion regimes: i) an Advection Dominated Accretion Flow (ADAF) state accreting via a physically thick, optically thin disc (Narayan & Yi 1994), ii) a thin disc state accreting via a physically thin, optically thick disc (Shakura & Sunyaev 1973), and iii) a super-Eddington state accreting via a slim disc (Abramowicz et al. 1988). We use these same equations in this paper, except for a slightly modified expression for the luminosity in the super-Eddington regime, where for Eddington normalised mass accretion ratesṁ > η Edd (0.1/ (a)), the bolometric luminosity is now given by:
L bol = η Edd 1 + ln ṁ η Edd (a) 0.1 L Edd ,(2)
where (a) is the spin-dependent radiative accretion efficiency for a thin accretion disc, a is the dimensionless spin parameter, η Edd is a free parameter,ṁ =Ṁ /Ṁ Edd is the Eddington normalised mass accretion rate, and L Edd is the Eddington luminosity. The Eddington luminosity is given by:
L Edd = 1.26 × 10 46 MBH 10 8 M ergs −1 ,(3)
and we define the Eddington mass accretion rate by:
M Edd = L Edd 0.1c 2 .(4)
We use a nominal accretion efficiency, = 0.1 in equation (4), so that the Eddington normalised mass accretion rate does not depend on the spin (this is a commonly used convention, cf. Yuan & Narayan 2014). The slight modification to the bolometric luminosities for the super-Eddington regime compared to Paper I ensures that the luminosities vary continuously in the transition from thin disc to super-Eddington accretion rates.
The model calculates luminosities at near-IR to X-ray wavelengths from the bolometric luminosities using the template SED in Marconi et al. (2004). This SED is empirical, where the ratio of luminosities at 2500Å and 2 keV is a function of bolometric luminosity, such that the optical emission dominates at high bolometric luminosities, and the X-ray emission dominates at low bolometric luminosities.
AGNs are understood to be surrounded by a gas and dust torus, which absorbs radiation from the AGN, the absorbed radiation then being re-emitted at longer (IR) wavelengths. To model this obscuration effect, we use several empirical relations for the 'visible fraction' which is the fraction of AGNs that are not obscured by a torus at a given luminosity, redshift, and wavelength (see Section 3.3 of Paper I).
AGN model variants
In Paper I, we showed that the fiducial model overpredicts the rest-frame 1500Å and soft X-ray AGN luminosity functions at z = 6, and so alongside predictions for the fiducial model, we presented two alternative models with slight modifications that provide a better fit to these AGN luminosity functions. The three models and the visible fractions used are as follows (see Paper I for more details):
(i) First, our fiducial model which uses the 'low-z modified Hopkins' (LZMH) visible fraction, which has a functional form that is based on the obscuration model used in Hopkins et al. (2007), but with different coefficients. The visible fraction for rest-frame 1500Å is:
fvis,1500,LZMH = 0.15 L bol 10 46 ergs −1 −0.1 ,(5)
where L bol is the bolometric luminosity. For the rest-frame soft X-ray band (0.5-2 keV), the visible fraction is:
fvis,SX,LZMH = 0.4 L bol 10 46 ergs −1 0.1 .
As in Paper I, in this paper we assume that there is no obscuration for the hard X-ray band (2-10 keV). The coefficients for the visible fraction were derived in Paper I by constructing an observational bolometric luminosity function from the observational optical/UV, soft X-ray, and hard X-ray luminosity functions. The luminosities at these different wavelengths were converted to bolometric luminosities using the Marconi et al. (2004) SED, and the number densities were converted to total number densities using the assumed visible fractions. The coefficients of the visible fractions are then chosen by eye to give the smallest scatter in the resultant bolometric luminosity function.
(ii) The second of the models uses the 'z = 6 modified Hopkins' (Z6MH) visible fraction, which is:
fvis,Z6MH = 0.04.(7)
This value was obtained by selecting coefficients in the power-law expressions for the visible fraction that result in the best agreement with the rest-frame 1500Å and restframe soft X-ray AGN luminosity functions at z = 6.
(iii) The third of the models used in this paper is the 'low accretion efficiency' model, which uses the LZMH visible fraction, but the fraction of mass accreted onto an SMBH in each starburst is lower. This was implemented in the model by decreasing the value of fBH, which represents the fraction of the mass of stars formed in a starburst that is accreted . We also show the black hole mass functions when the gas accretion rate is not allowed to exceed the Eddington mass accretion rate for z = 7 (red dashed line) and z = 10 (blue dashed line). We show the black hole mass function for a seed mass of 10 5 h −1 M , for z = 7 (red dotted line) and at z = 10 (blue dotted line).
on to the SMBH in the form of gas. The modified value is 0.002, compared to 0.005 in the fiducial model. The luminosity suppression for super-Eddington sources was also varied, with the parameter η Edd being increased to 16, compared to a value of 4 in the fiducial model. As for the previous variant of the model, these values were chosen to give agreement with the observed rest-frame 1500Å and rest-frame soft X-ray luminosity functions at z = 6. This low accretion efficiency model predicts fewer objects than the fiducial model.
BLACK HOLE MASS FUNCTION AND ACCRETION RATES
In Figure 1 we show the black hole mass function predicted by the model over the range 6 < z < 15. Black holes build up in the model as a result of galaxies forming in dark matter haloes, which build up hierarchically. In the model, for our simulation volume of (800Mpc) 3 , some SMBHs of mass 10 8 M have already formed by z = 9, but at z = 6 there are no SMBHs with masses above MBH = 3 × 10 8 M . This appears to be in conflict with observations of extremely massive SMBHs at z = 6 (e.g. Willott et al. 2010b;De Rosa et al. 2011;Venemans et al. 2013;Wu et al. 2015), which find estimated masses up to ∼ (0.3 − 1) × 10 10 M . The lack of these objects in this simulation may be because high-redshift surveys probe larger volumes than the volume of the simulation box in this work (e.g. the total survey volume for Bañados et al. (2018a) is of order 10 Gpc 3 compared to the volume of 0.5 Gpc 3 for this simulation), and so are able to detect rarer objects (e.g. Amarantidis et al. 2019). There are also uncertainties in the observational black hole mass es-timates due to the use of observationally calibrated relations to determine black hole masses from observed emission line widths and luminosities. These errors are a mixture of random (these relations have an intrinsic scatter of a factor of about 3 (e.g. Vestergaard & Peterson 2006)), and systematic (these relations are only constrained for certain luminosity ranges in the local Universe).
We also show in Figure 1 the predicted black hole mass function for the case in which gas accretion onto SMBHs in the model is not allowed to exceed the Eddington mass accretion rate (i.e.Ṁ Ṁ Edd ). In our standard model, SMBHs are allowed to accrete mass at super-Eddington accretion rates, and it can be seen that restricting SMBH accretion rates to the Eddington rate results in many fewer high-redshift SMBHs. At z = 7, restricting SMBH accretion in this way causes the number of SMBHs to decrease by about 1 dex at MBH = 10 6−7 M , and by about 1.5 dex at MBH = 10 5 M and 2.5 dex at MBH = 10 8 M . At z = 10, the effect of restricting SMBH growth is even more significant, with the number density of SMBHs decreasing by about 2 dex at MBH = 10 5−7 M . This shows the importance of super-Eddington accretion in building up high-redshift SMBHs in our model.
We also show the black hole mass function at z = 7 and z = 10 when a seed mass, M seed = 10 5 h −1 M is adopted, instead of M seed = 10h −1 M as in the fiducial model. At both of these redshifts, there are a large number of black holes around the seed mass for this case, but at higher masses the black hole mass function converges to the same value as in the fiducial model. This shows how the SMBH masses are relatively unaffected by the choice of seed black hole mass for sufficiently high SMBH mass provided that the gas accretion rate is not Eddington limited.
In Figure 2 we show the number of objects as a function of Eddington normalised mass accretion rate (Ṁ /Ṁ Edd ) predicted by the model at 7 z 15, for SMBHs residing in galaxies with stellar masses above 10 9 M or 10 10 M . At each redshift, the distribution is bimodal, with peaks atṀ /Ṁ Edd ∼ 0.001, andṀ /Ṁ Edd ∼ 1. The peak aṫ M /Ṁ Edd ∼ 1 is produced by AGNs fuelled by starbursts triggered by disc instabilities. The value ofṀ /Ṁ Edd at this peak increases slightly with redshift, which is a result of galaxy bulges having a smaller dynamical timescale at higher redshift, which results in shorter accretion timescales (cf. equation (1)). Galaxies have lower masses at higher redshift, and so the mass of gas transferred in each disc instability episode is typically smaller at higher redshift, and SMBHs are smaller at higher redshift. The former decreaseṡ M /Ṁ Edd , while the latter increasesṀ /Ṁ Edd , and these effects almost cancel out.
The peak atṀ /Ṁ Edd ∼ 0.001 is produced by AGNs fuelled by hot halo accretion. There is also a minor contribution from AGNs fuelled by starbursts triggered by mergers withṀ /Ṁ Edd values in the range 0.1-1. The peak aṫ M /Ṁ Edd ∼ 1 has more objects when the stellar mass cut is 10 9 M , but the peak atṀ /Ṁ Edd ∼ 0.001 has more objects when the stellar mass cut is 10 10 M . This is because AGNs fuelled by starbursts triggered by disc instabilities reside in lower stellar mass galaxies than AGNs fuelled by hot halo accretion. We allow SMBHs to accrete above the Eddington mass accretion rate in our model, and in this figure we see Figure 2. The number density of objects as a function of Eddington normalised mass accretion rate,Ṁ /Ṁ Edd , at z = 7 (red), z = 8 (yellow), z = 9 (light blue), z = 10 (dark blue), z = 12 (purple), and z = 15 (black). Only SMBHs residing in galaxies with stellar masses above M = 10 9 M are shown in the upper panel, whereas this stellar mass threshold is M = 10 10 M for the lower panel.
that there are objects that accrete at super-Eddington rates, but none aboveṀ /Ṁ Edd = 100.
EVOLUTION OF THE AGN BOLOMETRIC LUMINOSITY FUNCTION AT Z > 7
In the left panel of Figure 3, we show the evolution of the AGN bolometric luminosity function for the fiducial model for 7 z 15. As the redshift increases, both the number of objects and the luminosities decrease. By z ≈ 12, there are almost no objects brighter than L bol ∼ 10 46 ergs −1 in our simulated volume of (800Mpc) 3 .
We have investigated the effects of halo mass resolution on our predictions. In Figure A1 we show the bolometric luminosity function for the standard model (with a halo mass resolution of 2.12 × 10 9 h −1 M ) alongside the model with a halo mass resolution of 10 10 h −1 M . This comparison shows The evolution of the bolometric luminosity function for z = 7 (black), z = 8 (red), z = 9 (yellow), z = 10 (green), z = 12 (light blue), z = 15 (purple). The turnover at low luminosity is due to the halo mass resolution. Middle panel: The total AGN bolometric luminosity function at z = 9 (black) split into ADAFs (green), thin discs (purple) and super-Eddington objects (grey). Right panel: The total AGN bolometric luminosity function (black) at z = 9 split into objects fuelled by the hot halo mode (red), by starbursts triggered by mergers (light blue) and by starbursts triggered by disc instabilities (dark blue). Note that the dark blue line is under the black line.
that the turnover in the bolometric luminosity function at low luminosity is due to halo mass resolution. The bolometric luminosity functions are converged for L bol > 10 43 ergs −1 .
In Figure B1, we explore the effect of varying the black hole seed mass on the AGN bolometric luminosity function. We find that the AGN bolometric luminosity function is not sensitive to the choice of seed black hole mass for values in the range M seed = (10−10 5 )h −1 M for L bol > 10 42 ergs −1 at z = 7, and for L bol > 10 43 ergs −1 at z = 12. For luminosities below this, the seed mass does affect the predictions.
In the middle panel of Figure 3 we split the AGN luminosity function at z = 9 into the contributions from ADAFs, thin discs and super-Eddington objects. Paper I showed that at z = 0, the contribution from ADAFs dominates the predicted AGN luminosity function at low luminosities (L bol < 10 44 ergs −1 ), while the contribution from thin discs dominates at intermediate luminosities (10 44 ergs −1 < L bol < 10 46 ergs −1 ) and the contribution from super-Eddington objects dominates at high luminosities (L bol > 10 46 ergs −1 ). As redshift increases, the contribution from ADAFs decreases, and the contribution from thin discs dominates at low luminosities, while the contribution from super-Eddington objects continues to dominate at high luminosities. This trend continues for z > 0, so that by z = 9, the contribution from ADAFs is extremely small. At low luminosities (L bol < 10 45 ergs −1 ), the thin disc contribution just dominates over the contribution from super-Eddington objects, while at higher luminosities super-Eddington objects dominate. This implies that most of the QSOs (with L bol > 10 45 ergs −1 ) that will be detectable by surveys conducted by future telescopes at z = 9 should be accreting above the Eddington rate. This prediction is not straightforward to test, as determining Eddington ratios requires estimations of black hole masses. Black hole masses can be estimated from measurements of emission line widths, or black hole masses and mass accretion rates can be determined by fitting theoretical SED models to multi-wavelength data (e.g. Kubota & Done 2018). The black hole masses estimated using either of these methods will have some model dependencies.
In the right panel of Figure 3 we split the AGN luminosity function at z = 9 by gas fuelling mode, into hot halo mode, and starbursts triggered by galaxy mergers and disc instabilities. The dominant contributor at all luminosities at z = 9 is starbursts triggered by disc instabilities, so we predict that future high-redshift surveys will detect AGNs fuelled by this mechanism. This prediction contrasts with some other theoretical models. Some hydrodynamical simulations predict that gas may be driven into the centres of galaxies by high density cold streams for accretion onto the SMBH (e.g. Khandai et al. 2012;Di Matteo et al. 2017), while some other semi-analytical models simply assume that merger triggered starbursts dominate SMBH growth at highredshift (e.g. Ricarte & Natarajan 2018a).
In Figure 4, we present the number of objects as a function of L/L Edd predicted by the model for z = 7 and z = 10 for black holes with MBH > 10 5 M . The distributions are flat for L/L Edd < 0.1, and peak at L/L Edd ∼ 1. The L/L Edd value of the peak of the distribution slightly increases with redshift. There are no objects with L/L Edd > 10 in our simulated volume at these redshifts, which is a result of there being no objects withṀ /Ṁ Edd > 100 combined with our luminosity suppression for super-Eddington sources (cf. equation (2)). The sharp dip around L/L Edd = 0.01 arises from the thin disc to ADAF transition not being continuous in luminosity.
We also show in Figure 4 the distribution of L/L Edd predicted by the model for 10 7 M < MBH < 10 9 M , alongside the distribution for MBH > 10 5 M . At z = 7, black holes in these two mass ranges have similar distributions of L/L Edd values, while for z = 10, the number of black holes for 10 7 M < MBH < 10 9 M in our simulation is too small to draw any strong conclusion on the form of this distribution.
In Figure 5, we present the AGN bolometric luminosity versus host halo mass for objects in the model, colour-coded by the number density of objects. The objects mostly follow a relation between bolometric luminosity and halo mass, although there are some objects offset from this relation to higher halo masses at z = 7, but not at z = 10. The objects on the main relation are fuelled by starbursts triggered by disc instabilities, whereas the objects offset from the main relation at higher halo masses are fuelled by hot halo mode accretion. The brightest AGNs are not hosted by the most massive haloes at z = 7, but at z = 10 the brightest AGNs are hosted by the most massive haloes.
PREDICTIONS FOR HIGH REDSHIFT SURVEYS WITH FUTURE TELESCOPES
We next employ our model to make predictions for the detection of AGNs at z 7 with the future telescopes described in the Introduction. We use luminosity functions predicted by the model in the different wavelength or energy bands of these telescopes to predict the number of AGNs that should be detectable by surveys with these telescopes. We also describe the typical properties of the SMBHs detectable by the different telescopes. The survey parameters that we assume for JWST 1 , EUCLID 2 , ATHENA 3 , and Lynx 4 are summarised in Table 1.
The number of AGNs detectable in a survey depends on both the flux limit and the survey area. The former affects the ability to detect low luminosity sources and the latter affects the number density of objects down to which 1 https://jwst-docs.stsci.edu/display/JTI/NIRCam+ Sensitivity 2 https://www.euclid-ec.org/?page_id=2581 3 https://www.cosmos.esa.int/documents/400752/507693/ Athena_SciRd_iss1v5.pdf 4 https://wwwastro.msfc.nasa.gov/lynx/docs/ LynxInterimReport.pdf one can probe. From the predicted flux limits of the surveys, luminosity limits can be derived using L = 4πd 2 L f for calculating broadband luminosities (ATHENA and Lynx) and Lν = 4πd 2 L fν /(1 + z) for calculating a luminosity per unit frequency (EUCLID and JWST). Here, f is the flux, fν is the flux per unit frequency and dL is the luminosity distance to the source, L is the luminosity in the rest-frame band or wavelength corresponding to the observed band or wavelength, and Lν is the luminosity per unit frequency in the rest frame corresponding to the observed wavelength and redshift. We use these expressions to calculate luminosity limits (vertical lines) in Figures 6 to 11.
The luminosities shown in Figures 6 to 11 have been k-corrected to a fixed band in the observer frame. Our template SED for this calculation is that of Marconi et al. (2004), for which the ratio of X-ray to optical luminosity varies with bolometric luminosity. To calculate the luminosity in each band we input the bolometric luminosity and the redshift and then integrate the SED over frequency multiplied by the appropriate response function for the filter redshifted into the rest frame of the source. There is a one-to-one relation between bolometric luminosity and luminosity in a particular band.
The number density limit for a survey can be calculated via the following method. The number of objects per log flux per unit solid angle per unit redshift is given by:
d 3 N d(logfν )dzdΩ = d 2 N d(logLν )dV d 2 V dzdΩ ,(8)
where V is the comoving volume, d 2 N/d(logLν )dV is the luminosity function in comoving units, and d 2 V /dzdΩ is the comoving volume per unit solid angle per unit redshift. We define Φ(X) = d 2 N/d(logX)dV so the luminosity function can be written as Φ(Lν ). For there to be an average of at least one object detectable in the survey per log flux per unit redshift, we therefore have the condition:
d 2 N dlogLν dV 1 d 2 V dzdΩ ∆Ω ,(9)
where ∆Ω is the solid angle of sky covered by the survey. This condition allows us to construct the number density limits (horizontal lines) in Figures 6 to 11. Note that this limit is almost independent of redshift over the range 7 z 15, as also seen for the JWST predictions of Cowley et al. (2018) for galaxies. The flux limits and survey areas adopted for the predictions for different telescopes are given in Table 1. These limits then allow us to predict the number of objects detectable by each survey, for the three different model variants, as given in Table C1, and the properties of these objects, for the fiducial model, as given in Tables D1, and D2. In general, the flux limit determines the lower luminosity limit of objects that can be detected, whereas the survey area determines the upper luminosity limit of objects that can be detected. The different flux limits and survey areas of the surveys conducted by the different telescopes therefore provide detections of different populations of AGNs. Table 1. The sensitivities and solid angles covered by the possible surveys by JWST, EUCLID, ATHENA and Lynx. For ATHENA and Lynx, the survey area is assumed is that of a single field of view, whereas for JWST and EUCLID the survey area is assumed to be that of multiple fields of view. The integration time is the total for a survey in that band. For ATHENA and Lynx, the flux limits used are the estimated confusion limits. These flux limits, fν , can be related to apparent AB magnitudes by: m AB = 31.40 − 2.5 log 10 (fν /nJy).
Instrument
Filter ATHENA WFI Soft X-ray 0.5 − 2 keV 2.4 × 10 −17 erg cm −2 s −1 1600 arcmin 2 (FoV) 450 Hard X-ray 2 − 10 keV 1.6 × 10 −16 erg cm −2 s −1 1600 arcmin 2 (FoV) 450
Lynx Soft X-ray 0.5 − 2 keV 7.8 × 10 −20 erg cm −2 s −1 360 arcmin 2 (FoV) 15000 Hard X-ray 2 − 10 keV 1.0 × 10 −19 erg cm −2 s −1 360 arcmin 2 (FoV) 15000
Optical/near-IR surveys with JWST and EUCLID
JWST, planned for launch in 2021, will observe at wavelengths of 0.6-29 µm. It will have instruments for both imaging and spectroscopy, including the NIRCam for optical to near-infrared imaging (0.7-5 µm) and MIRI for midinfrared imaging (5-29 µm). We present predictions for three different NIRCam bands. We do not make predictions for MIRI, because our AGN model does not currently include emission from the dust torus, which would be necessary for modelling AGN emission in the mid-infrared. Figures 6 and 7 show predicted AGN luminosity functions in the observer frame F070W (0.7µm) and F200W (2.0µm) bands respectively. We also find that in the observer frame F444W (4.4µm) band, the predicted luminosity functions are similar to the observer frame F200W band. We present predictions for a survey composed of 1000 fields of view, each with a 10 4 s integration time, giving a total integration time of 10 7 s in each band. Figures 6 and 7 show that the effect of obscuration causes the predicted number of AGNs to be 0.04-0.2 of the predicted number of objects if obscuration is not taken into account. The effect of low accretion efficiency causes the predicted number of objects to be about 0.4 times lower than in the fiducial model if we are assuming the LZMH The horizontal lines indicate the number density limit resulting from a survey area of one field of view (dashed), and the number density limit resulting from 1000 of these fields of view (dotted). The vertical lines show the luminosity limit resulting from the flux limit. The assumed flux limits and survey areas are given in Table 1. Detectable objects are above and to the right of these lines. These luminosities can be converted into absolute AB magnitudes via M AB = 51.59 − 2.5 log(Lν /erg s −1 Hz −1 ). obscuration model. We predict that on average, < 1 AGN per unit z per field of view will be detectable by JWST for a 10 4 s integration, once we allow for obscuration. We give the predicted number of objects for each survey in Table C1. For JWST we are assuming a survey of 1000 fields of view, each with a 10 4 s integration time per band. We predict that 20 − 100 AGNs (depending on which of the three models is used) will be observed at z = 7 in the F070W band, 90−500 in the F200W band and 60−300 in the F444W band. We predict that more objects will be detectable in the F200W band because the assumed flux limit for the F200W band is lower than for the F070W and F444W bands, which translates into a lower limit for the bolometric luminosity and higher number density. Predictions for the number of objects detectable at z = 9, z = 10 and z = 12 are given in Table C1.
From the flux limits in these bands, limits in bolometric luminosity can be calculated. At z = 7, we predict that JWST will detect AGNs with bolometric luminosities in the range (3 × 10 44 − 4 × 10 46 ) ergs −1 (F070W), (6 × 10 43 − 3 × 10 46 ) ergs −1 (F200W), and (1 × 10 44 − 4 × 10 46 ) ergs −1 (F444W). For the assumed survey parameters, we predict that JWST will be able to detect AGNs out to z = 9 for all the optical/near-IR bands, with F200W being more favourable for detecting z > 7 AGNs than F070W and F444W. For F200W, we predict that about 60-90 times fewer AGNs will be detectable at z = 10 than at z = 7. Considering even higher redshift objects, for z > 10 we predict that detection with JWST will become more difficult, as AGNs become extremely rare as well as very faint.
EUCLID, due for launch in 2021, will use its visible and near-IR coverage (0.55-2 µm) of galaxies to probe the nature of dark energy, but these same surveys will also allow detections of high-redshift AGNs. EUCLID will conduct two surveys: a Wide Survey covering 15000 deg 2 of sky and a Deep Survey covering 40 deg 2 in three fields. The mission lifetime of EUCLID will be 6.25 years. The surveys will be conducted in four bands -one visible (VIS) and three near-IR (Y,J,H). We show predictions for the EUCLID VIS (0.55-0.9µm) band and the H (1.5-2µm) band in Figures 8 and 9 respectively. In these figures we show the sensitivity and survey volume limits for both the Deep and Wide surveys. The two surveys are seen to be quite complementary for detecting high redshift AGNs at different luminosities.
At z = 7, we predict that the EUCLID VIS band will detect AGNs with bolometric luminosities L bol = (1×10 45 − 1.2 × 10 47 ) ergs −1 for the Deep Survey, and with L bol = (6×10 45 −2×10 47 ) ergs −1 for the Wide Survey. We therefore predict that the two EUCLID surveys and surveys by JWST will sample different parts of the AGN luminosity function.
At z = 7, we predict that a similar number of AGNs will be detectable in the EUCLID near-IR band compared to the visible band. For the EUCLID Deep survey, we predict that 90 − 400 AGNs will be detectable in the VIS band compared with 100 − 600 in the H band (depending on the model). For the EUCLID Wide survey at z = 7, we predict that (5 − 20) × 10 3 AGNs will be detectable in the VIS band, and (8 − 30) × 10 3 in the H band. At higher redshifts (e.g. z = 10), we predict that the EUCLID H band will detect more AGNs than the VIS band. For AGNs at z = 7, the peak of the observed SED is at 1µm, and so the luminosities in the VIS and H bands are similar, and because the flux limits are also similar, a similar number of AGNs should be detectable. At z = 10, the peak of the observed SED is at 1.3µm, and so the luminosities in the H band are higher, as they are closer to the peak of the AGN SED. Therefore, we predict that the H band will detect more AGNs than the VIS band at z = 10. A similar effect is seen when comparing the JWST F070W and F200W bands. It may be that such observations will reveal that the AGN SED shape at high redshift is different to the Marconi et al. (2004) SED used in this work. According to our model, it will be impossible to detect very high redshift (z = 15) objects with EUCLID, so such investigation may have to wait until surveys after EUCLID. This is because despite the survey area being sufficiently large to probe down to the required number densities, the sensitivity of EUCLID is not sufficient to detect these low luminosity AGNs.
The alternative models featuring a lower visible fraction or lower accretion efficiency predict fewer AGNs than the fiducial model, so observations using EUCLID and JWST may be able to differentiate between these models as well as constraining the form of the AGN SED and thus provide better understanding of the high redshift AGN population.
X-ray surveys with ATHENA and Lynx
Due for launch in 2031, ATHENA will make observations at 0.5-10 keV using two instruments: the X-ray Integral Field Unit (X-IFU) for high resolution spectroscopy and the Wide Field Imager (WFI) with a large field of view for surveys (Nandra et al. 2013). The Lynx X-ray observatory, with a proposed launch date of 2035, will make observations at 0.2-10 keV. Due to the effects of source confusion, Lynx will be able to probe down to lower luminosities than ATHENA as a result of its much better angular resolution.
We have calculated the sensitivity limits due to source confusion for ATHENA and Lynx. Source confusion occurs when multiple sources are separated by angles less than the angular resolution of the telescope and so appear merged together in images. To derive the confusion limits for ATHENA and Lynx, we use the commonly used Condon (1974) 'source density criterion', to obtain the cumulative number count per solid angle at the confusion limit (N (> f conf )), for a given beam solid angle, Ω beam , and number of beams per source N beam :
N (> f conf ) = 1/N beam Ω beam ,(10)
where the beam solid angle is related to the full width half maximum (FWHM) telescope beam width, θFWHM, by Ω beam = πθFWHM/(4(γ − 1) ln 2) for a Gaussian beam pro- Table 1.
file, where γ is the slope of the power law relating differential number count and flux, given by:
d 2 N df dΩ ∝ f −γ .(11)
We use N beam = 30. Having calculated the cumulative number count at the confusion limit from equation (10), we can obtain the flux at the confusion limit by using a model that relates the cumulative number counts to the flux. For this, we use the Lehmer et al. (2012) empirical model, which is a fit to the number counts measured using Chandra assuming a double power law fit for the AGN contribution, and single power law fits for the galaxy and stellar contributions. For the Lynx sensitivities, we are extrapolating the Lehmer et al. (2012) model to 100-1000 times lower fluxes than observed by Chandra. For ATHENA, θFWHM = 5 arcsec, whereas for Lynx, θFWHM = 0.5 arcsec. The γ values that we use are slopes of the differential number counts from Lehmer et al. (2012) at the estimated confusion limits, and are given in Table 2. The fluxes calculated by this procedure are given in Table 1.
In Figure 10, we show predictions for these two telescopes in the soft X-ray (0.5-2 keV) band. Note that the turnover in the luminosity function seen at low luminosities is due to the halo mass resolution of the dark matter simulation (see Section 4). As the luminosity limit for Lynx for z 10 is below the luminosity of this turnover, the predic- tions at low luminosities for z 10 should be viewed as lower limits on the number densities. This figure also shows how Lynx will be transformational in the study of low luminosity AGNs, and will provide unique constraints and tests of our understanding of black hole physics and galaxy formation. This is a result of increased angular resolution of Lynx compared to ATHENA. We do not include obscuration for these soft X-ray predictions because at the redshifts we are considering, the corresponding band in the galaxy rest frame lies at hard Xray energies -a band for which we are assuming no obscuration. We show the fiducial model alongside the low accretion efficiency model (fBH = 0.002 and η Edd = 16) and also a model in which the black holes have a seed mass M seed = 10 5 h −1 M (compared to the default value M seed = 10h −1 M ).
It can be seen how changing the seed black hole mass affects the soft X-ray luminosity function very little at 7 z 9, and only by a small amount for LSX < 10 42 ergs −1 at 10 < z < 15. This analysis suggests that even high Figure 11. As for Figure 10, but for the observer frame hard X-ray band. sensitivity telescopes such as Lynx will struggle to differentiate between different seed masses at 7 z 9 for our model assumptions, but measurements of the number densities of AGNs at low luminosities and very high redshifts (LSX < 10 42 ergs −1 and 10 < z < 15), may be able to exclude models of SMBH seeding that involve high seed masses, although we predict that there will not be a substantial difference in the number densities between these two models.
In Figure 11 we show the predictions for ATHENA and Lynx in the hard X-ray (2-10 keV) band. For our template SED, an AGN emits more energy at hard than at soft X-ray energies, but the minimum luminosity of an object that can be detected is much higher for the hard X-ray band than for the soft X-ray band for ATHENA, while it is only slightly higher for Lynx. This has the effect that for ATHENA, we predcit more AGNs will be detectable in the soft X-ray band compared to the hard X-ray band, whereas for Lynx, we predict that slightly more AGNs will be detectable in the hard X-ray band compared to the soft X-ray band.
For ATHENA, at z = 7 we predict that 30 − 80 AGNs will be detectable per field of view in the soft X-ray band, and 5−20 for the hard X-ray band (cf. Table C1 for the number of objects predicted to be detectable by each survey). At z = 10, we predict that 0 − 2 AGNs will be detectable in the soft X-ray band, and no objects in the hard X-ray band. For Lynx, at z = 7, we predict that about 800 AGNs per field of view will be detectable in the soft X-ray band, and about 800−900 in the hard X-ray band. At z = 10, we predict that about 200 AGNs will be detectable per field of view for both the soft and hard X-ray bands. The low accretion efficiency model predicts fewer AGNs than the fiducial model across all luminosities and redshifts. According to our model, Lynx is the only telescope out of the four studied here that will be able to detect AGNs out to z = 12, with the possibility of detections at z = 15, depending on the model variant.
Properties of detectable AGNs & SMBHs in high-redshift surveys
We show the predictions for SMBH masses, Eddington normalised mass accretion rates, host galaxy stellar masses, and host halo masses for the AGNs detectable by each survey for redshifts 7 z 15 in Figures 12, 13, 14, and 15 respectively. We constructed these plots by generating the number density distributions for each property for AGNs above the luminosity limit for the survey at that redshift, and then selecting the part of the distribution with number density above the survey limit, in the same way as we did for luminosity functions in the preceding sections. We then calculated the median, minimum, and maximum values of these distributions, which are plotted in the figures. We also list the median values of these quantities for z = 7 and z = 10 in Tables D1 and D2. The maximum SMBH masses, Eddington normalised mass accretion rates, galaxy masses, and host halo masses for the EUCLID Wide survey are shown as upward pointing arrows because they are lower limits on the maximum values that EUCLID Wide would detect. This is because the effective survey volume of EUCLID Wide at these redshifts is larger than the volume of the simulation Figure 12. The predicted SMBH masses as a function of redshift for AGNs detectable by the surveys with the different telescopes for the fiducial model. Symbols and errorbars show the median and 0-100 percentiles of the distribution of SMBH masses at z = 7, 8, 9, 10, 12. Left panel: JWST F070W (blue squares), JWST F200W (red circles), and JWST F444W (black squares). Middle panel: EUCLID VIS and H for the Deep survey (blue triangles and red circles), and for the Wide survey (black squares and green pentagons). The maximum SMBH masses for EUCLID Wide are shown as upward pointing arrows because they are lower limits on the maximum SMBH masses that are detectable. Right panel: ATHENA soft and hard X-ray (blue squares and red circles), and Lynx soft and hard X-ray (black squares and green pentagons). In all panels, points for different surveys have been slightly offset in redshift for clarity. Figure 13. The Eddington normalised mass accretion rates as a function of redshift for the AGNs detectable by the surveys with the different telescopes. The lines are as in Figure 12. box, and so there may be massive, rare black holes that the survey would detect, but which are not sampled by our simulation volume.
First we compare the optical/near-IR surveys. Compared to EUCLID Deep, we predict that JWST will probe SMBHs with masses about four times lower, in galaxies with stellar masses about three times lower, and in haloes with masses about two times lower, having Eddington normalised accretion rates about 1.4 times lower. We predict that the two different EUCLID surveys will detect slightly different populations of AGNs, with EUCLID Wide detecting SMBHs with masses about three times higher, in galaxies with stellar masses about two times higher, and in haloes with masses about 1.3 times higher, having Eddington normalised mass accretion rates about two times higher, compared to EUC-LID Deep. Now comparing the X-ray surveys, the properties of objects predicted to be detectable in the two ATHENA bands are similar to those predicted to be detectable by EUCLID Deep, but the ATHENA soft X-ray band is predicted to detect SMBHs with masses about two times lower, in galaxies of stellar mass about two times lower, in host haloes about 1.3 times lower, and having Eddington normalised mass accretion rates about 1.3 times lower, compared to EUCLID Deep. Compared to ATHENA, we predict that Lynx will detect SMBHs with masses about 200 times lower, with galaxy stellar masses about 50 times lower, and in haloes of mass about 10 times lower, with Eddington normalised mass accretion rates about 2 times lower. For each survey, the AGNs detectable at z = 10 have somewhat lower black hole masses, lower host galaxy stellar masses, lower host halo masses, and higher Eddington normalised accretion rates than at z = 7.
Comparing all the distributions of the objects detectable by these surveys at z = 7, we predict that the objects detectable by the Lynx hard X-ray band will have the lowest median black hole mass, stellar mass, halo mass, and Eddington normalised mass accretion rate. On the other hand, we predict that the obects detectable by the VIS band for the EUCLID Wide survey will have the highest median black hole mass, stellar mass, halo mass, and Eddington normalised mass accretion rate.
We predict that Lynx will detect SMBHs that are sub-stantially smaller than in the other surveys, and SMBH host galaxies that are substantially smaller than in the other surveys. Also, Lynx is the only survey that will be able to detect AGNs at z = 7 in the ADAF accretion state (ṁ < 0.01). The much lower black hole, galaxy, and halo masses probed by Lynx compared to the other telescopes are a result of it being able to detect AGN at much lower bolometric luminosities.
While Lynx is predicted here to detect AGNs with smaller black hole masses than the other surveys based on the survey parameters in Table 1, we explored whether AGNs with similarly low mass black holes could be detectable by a similarly long integration time with JWST. We considered a 15Ms integration time survey in the JWST F200W band, for a single field of view (compared to our standard assumption of a 10ks integration time in each of 1000 fields of view), assuming the survey is signal-to-noise limited. We predict that for this long integration time survey, JWST could detect objects at z = 7 down to an AGN bolometric luminosity of L bol = 2.8×10 42 ergs −1 , compared to L bol = 3.8×10 41 ergs −1 for the Lynx soft X-ray band. The smallest black holes at z = 7 that are detectable by this long integration time JWST survey are of mass MBH = 4700M , compared to MBH = 560M for the Lynx soft X-ray band. JWST is therefore in principle as sensitive as Lynx to low luminosity, low SMBH mass AGNs at high redshift. However, this does not account for the 40 times smaller field of view of JWST compared to Lynx, which greatly reduces the survey volume, nor the greater difficulty of separating the light of the AGN from that of the host galaxy in optical/near-IR compared to X-rays.
The largest detectable SMBH is also different for each of these surveys. Surveys with larger survey areas can probe down to lower number densities, and so generally can detect higher mass SMBHs. However, because the black hole mass function decreases fairly steeply at the high mass end, increasing the survey area only slightly increases the mass of the largest SMBH detectable. For halo masses, a larger survey area does not necessarily correspond to detecting larger haloes from the AGNs they contain, because the largest haloes can host lower luminosity objects (see Figure 5). Therefore the maximum halo mass is also affected by the sensitiv-ity limit, as seen for ATHENA and Lynx in the right panel of Figure 15. A similar argument can be applied for stellar masses as seen in Figure 14.
We also explored the effect of halo mass resolution in our simulation on the properties of objects detectable by these surveys (see Section 4). We find that if we degrade the halo mass resolution, as long as the objects have bolometric luminosities above the value at which the luminosity functions converge (i.e. L bol > 10 43 ergs −1 ), the properties of the black holes are the same. The predictions of black hole properties for surveys by JWST, EUCLID and ATHENA are insensitive to this effect, but for Lynx the values given should be regarded as upper limits.
CONCLUSIONS
Recent advances in observational capabilities have opened up studies of the high-redshift Universe, but many uncertainties regarding the early stages of galaxy formation and evolution remain. The origin of supermassive black holes (SMBHs) and their role in the early Universe still remains a mystery. Fortunately the next decade-and-a-half offers us exciting new opportunities to probe the high redshift Universe, especially given the plans for powerful new space-based telescopes such as JWST and EUCLID at optical/near-IR wavelengths, and ATHENA and Lynx at X-ray energies. These will offer us a multiwavelength view of the distant Universe and allow us to characterise physical processes in galaxy formation. The role of SMBHs and their growth in the distant Universe will be probed with much greater accuracy than ever before.
With these potential new developments in mind, we present predictions for AGNs in the high redshift Universe (z 7) using the semi-analytic model of galaxy formation galform. In galform, galaxies (and hence AGNs) form in dark matter haloes, with the evolution of the dark matter haloes described by halo merger trees. Here, the merger trees have been generated from a dark matter N-body simulation. In the model, SMBHs grow by accretion of gas during starbursts triggered either by mergers or disc instabilities, or by accretion of gas from hot gas halos, or by merging with other SMBHs. The evolution of the SMBH spin is also calculated in the model with SMBHs changing spin either by accretion of gas, or by merging with another SMBH. From the SMBH mass accretion rates, AGN bolometric luminosities are then calculated, which when combined with empirical SED and obscuration models can be used to calculate luminosities in different bands. The galform model used here is that presented in Griffin et al. (2019), which showed that the predicted AGN luminosity functions are in good agreement with observational data at 0 z 5.
We present model predictions for the AGN bolometric luminosity function for 7 z 15, finding that it evolves to lower luminosities and lower number densities at higher redshift as a result of hierarchical structure formation. When we split the bolometric luminosity function at these redshifts by accretion disc mode and gas fuelling mode, we find that the dominant accretion disc modes are thin discs at low luminosities (L bol < 10 45 ergs −1 ), and super-Eddington objects at higher luminosities, and the dominant gas fuelling mode at all luminosities is starbursts triggered by disc instabilities.
The model allows SMBHs to grow at mass accretion rates above the Eddington rate, so when we limit the SMBH gas accretion rate to the Eddington rate, the number of SMBHs at high redshift is significantly reduced. We also explore the effect of varying the SMBH seed mass on the bolometric luminosity function. We find that when we use a much larger seed black hole mass (10 5 h −1 M compared to 10h −1 M in the fiducial model), the luminosity functions are relatively unaffected, except for L bol < 10 43 ergs −1 for z > 10.
We then present predictions for JWST, EUCLID, ATHENA, and Lynx, using sensitivities and survey areas for possible surveys with these telescopes. For example, we assume a 1.5 × 10 7 s exposure for Lynx over a survey area of 360 arcmin 2 (1 field of view), whereas we assume a thousand 10 4 s exposures for JWST over a total survey area of 9680 arcmin 2 (1000 fields of view). We find that the different surveys will probe down to different AGN bolometric luminosities and number densities, and hence sample different parts of the AGN population.
We also present predictions for two variants to the fiducial model that provide a better fit to the rest-frame UV and rest-frame soft X-ray luminosity functions of AGNs at z = 6. In these models we vary either the amount of AGN obscuration or the SMBH accretion efficiency (defined here as the fraction of gas accreted onto the SMBH in a starburst). The resulting luminosity functions have lower number densities by factors of about 4 and 2 respectively. AGN obscuration and SMBH accretion efficiency are both uncertainties for the AGN population at high redshift. Comparing these predictions to observations should allow us to better both of these aspects at high redshift.
The properties of the SMBHs and AGNs detectable depend on the survey and wavelength. For our fiducial model, we predict that the AGNs detectable at z = 7 will have median black hole masses that vary from 8 × 10 4 M to 5 × 10 7 M , and median Eddington normalised mass accretion rates that vary from 1 − 3. These AGNs are predicted to reside in host galaxies with median stellar masses that vary from 4 × 10 7 M to 4 × 10 9 M , and in haloes with median masses from 4 × 10 10 M to 3 × 10 11 M . At z = 10, the AGNs detectable are predicted to have black hole masses that vary between 2 × 10 4 M to 4 × 10 7 M , with Eddington normalised mass accretion rates that vary from 1 − 8. The host galaxies of these AGNs are predicted to have masses that vary from 8 × 10 6 M to 1 × 10 9 M , in haloes with masses that very from 2 × 10 10 M to 2 × 10 11 M . The different telescopes will therefore provide different but complementary views on the z > 6 AGN population. For the survey parameters assumed here, Lynx is predicted to detect SMBHs with the lowest masses, in the lowest mass host galaxies and lowest mass host haloes, and so will provide the best opportunity to probe the nature of SMBH seeds. However, a similarly long integration (15Ms) in a single field of view with JWST could in principle detect similarly faint AGN at high redshift.
These future telescopes should therefore be able to detect SMBHs at very high redshift having masses ∼ 10 4 − 10 5 M that are comparable to those of the highest mass seed SMBHs that are envisaged in current scenarios, and put improved constraints on the physical mechanisms by which these seed SMBHs form. Figure A1. The bolometric luminosity function at z = 7 (solid lines), and z = 12 (dotted lines) for the halo mass resolution of 2.12 × 10 9 h −1 M as for the standard model (black lines) and for a halo mass resolution of 10 10 h −1 M (blue lines).
APPENDIX A: EFFECT OF HALO MASS RESOLUTION
In Figure A1 we show the predicted bolometric luminosity function at z = 7 and z = 12 for the fiducial model, which has a halo mass resolution of 2.12 × 10 9 h −1 M , and for a halo mass resolution of 10 10 h −1 M . The figure demonstrates that the turnover seen in the luminosity function at L bol ∼ 10 43 ergs −1 is due to the dark matter simulation only resolving haloes above a certain mass. The two bolometric luminosity functions are converged for L bol 10 43 ergs −1 (depending somewhat on redshift), while the poorer halo mass resolution leads to fewer objects for L bol < 10 43 ergs −1 .
APPENDIX B: THE EFFECT OF THE SMBH SEED MASS
In Figure B1 we show the AGN bolometric luminosity function at z = 7 and z = 12 for three different seed masses (10h −1 M , 10 3 h −1 M , and 10 5 h −1 M ). The luminosity functions for the three different seed masses are consistent with each other within statistical errors for L bol > 10 42 ergs −1 at z = 7, and consistent with each other for L bol > 10 43 ergs −1 at z = 10.
APPENDIX C: NUMBER OF DETECTABLE OBJECTS
In Table C1 we show the number of objects detectable by each survey at z = 7, z = 9, z = 10, and z = 12, with sensitivities and survey areas as in Table 1.
APPENDIX D: PROPERTIES OF DETECTABLE OBJECTS
In Tables D1 and D2 we show the median SMBH masses, Eddington normalised accretion rates, host galaxy stellar Figure B1. The bolometric luminosity function at z = 7 (solid lines), and z = 12 (dashed lines) for seed masses of 10h −1 M (black), 10 3 h −1 M (red) and 10 5 h −1 M (blue). Note that the black lines are underneath the red lines.
masses and host halo masses of AGNs detectable by the future surveys at z = 7 and z = 10. The assumed sensitivities and survey areas are given in Table 1. Table C1. Predictions for the number of AGNs expected to be detectable at different redshifts by the different telescopes, using the sensitivity limits and survey areas given in Table 1. The ranges of values correspond to the three different variants of the model (see Section 2.3): the fiducial model, which uses the LZMH obscuration fraction, the fiducial model using the Z6MH obscuration fraction, and the low accretion efficiency model.
Instrument
Filter z = 7 z = 9 z = 10 z = 12 Table D1. The median SMBH masses, Eddington normalised mass accretion rates, host galaxy stellar masses, and host halo masses of the AGNs predicted to be detectable by JWST, EUCLID, ATHENA, and Lynx at z = 7 for our fiducial model, for the survey parameters given in Table 1.
Instrument Filter M SMBH (M )ṁ =Ṁ /Ṁ Edd M (M ) M halo (M ) JWST F070W
7.2 × 10 6 0.8 1.4 × 10 9 1.9 × 10 11 F200W 2.0 × 10 6 0.7 5.2 × 10 8 1.1 × 10 11 F444W
3.0 × 10 6 0.7 7.1 × 10 8 1.3 × 10 11 EUCLID Deep VIS 1.8 × 10 7 1.1 2.6 × 10 9 2.6 × 10 11 H 1.4 × 10 7 1.0 2.2 × 10 9 2.4 × 10 11 EUCLID Wide VIS 4.6 × 10 7 2.5 4.4 × 10 9 3.4 × 10 11 H 4.0 × 10 7 2.0 4.1 × 10 9 3.3 × 10 11 ATHENA WFI Soft X-ray 8.0 × 10 6 0.8 1.5 × 10 9 1.9 × 10 11 Hard X-ray 2.4 × 10 7 1.3 3.2 × 10 9 2.9 × 10 11
Lynx Soft X-ray 8.9 × 10 4 0.6 4.1 × 10 7 3.7 × 10 10 Hard X-ray 8.2 × 10 4 0.6 3.9 × 10 7 3.6 × 10 10 Table D2. The same as Table D1, but at z = 10. We predict that the ATHENA hard X-ray band will not be able to detect any AGNs at z = 10.
Instrument
Filter
M SMBH (M )ṁ =Ṁ /Ṁ Edd M (M ) M halo (M )
JWST F070W 6.9 × 10 6 2.6 8.3 × 10 8 1.4 × 10 11 F200W 1.8 × 10 6 1.2 3.2 × 10 8 8.6 × 10 10 F444W 2.6 × 10 6 1.4 4.2 × 10 8 1.1 × 10 11 EUCLID Deep VIS 1.4 × 10 7 4.2 1.1 × 10 9 1.6 × 10 11 H 1.1 × 10 7 3.2 1.0 × 10 9 1.5 × 10 11 EUCLID Wide VIS 3.6 × 10 7 8.2 1.4 × 10 9 1.6 × 10 11 H 2.2 × 10 7 7.5 1.4 × 10 9 1.6 × 10 11 ATHENA WFI Soft X-ray 6.0 × 10 6 2.1 7.3 × 10 8 1.3 × 10 11 Hard X-ray ----Lynx Soft X-ray 2.4 × 10 4 1.1 9.8 × 10 6 1.8 × 10 10 Hard X-ray 2.1 × 10 4 1.1 8.4 × 10 6 1.7 × 10 10
Figure 1 .
1The black hole mass function in the fiducial model for z = 6 (pink solid line), z = 7 (red solid line), z = 8 (yellow solid line), z = 9 (light blue solid line), z = 10 (blue solid line), z = 12 (purple solid line), and z = 15 (black solid line)
Figure 3 .
3The predicted AGN bolometric luminosity function for the fiducial model at high redshift. Left panel:
Figure 4 .
4The number density of objects as a function of Eddington normalised luminosity, L/L Edd , predicted by the model at z = 7 (red) and z = 10 (blue), for SMBHs with mass M BH > 10 5 M (solid lines), and for SMBHs with mass 10 7 M < M BH < 10 9 M (dotted lines).
Figure 5 .
5A scatter plot of AGN bolometric luminosity versus host halo mass for AGNs at z = 7 (left panel) and z = 10 (right panel). The colour indicates the number density of objects.
Figure 6 .
6Predictions for the AGN luminosity function in the observer frame JWST NIRCam F070W (0.7µm) band. We show the luminosity function for the fiducial model without obscuration (red dashed) with Poisson errors (orange shading), the fiducial model with the 'low z modified Hopkins' (LZMH) visible fraction (magenta solid), the fiducial model with the 'z = 6 modified Hopkins' (Z6MH) visible fraction (red dotted), and the low accretion efficiency model which uses the 'low z modified Hopkins' visible fraction (blue solid).
Figure 7 .
7As above but for the observer frame JWST NIRCam F200W (2.0µm) band.
Figure 8 .
8Predictions for the AGN luminosity function in the observer frame EUCLID VIS (550-900 nm) band. The dashed lines represent the sensitivity and survey volume limits of the EUCLID Deep survey and the dotted lines represent the sensitivity and survey volume limits of the EUCLID Wide survey.
Figure 9 .
9The same asFigure 8but for the observer frame EUCLID H (1.5-2µm) band.
Figure 10 .
10Predictions for AGN luminosity functions in the observer frame soft X-ray band. Shown are the fiducial model (red solid line), the low accretion efficiency model (blue dotted line), and the fiducial model with seed black hole mass 10 5 h −1 M (black dashed line). We also show the ATHENA (dashed) and Lynx (dotted) luminosity and number density limits (vertical and horizontal lines) for a single field of view and integration down to the estimated confusion limit, as in
Figure 14 .
14The host galaxy stellar masses as a function of redshift for the AGNs detectable by the surveys with the different telescopes. The lines are as inFigure 12.
Figure 15 .
15The host halo masses as a function of redshift for the AGNs detectable by the surveys with the different telescopes. The lines are as inFigure 12.
Table 2 .
2The values of γ used for calculating the confusion limits.Telescope
Soft X-ray Hard X-ray
ATHENA
1.5
1.32
Lynx
2.22
2.29
MNRAS 000, 1-19(2019)
ACKNOWLEDGEMENTSThis work was supported by the Science and Technology facilities Council grants ST/L00075X/1 and ST/P000541/1. AJG acknowledges an STFC studentship funded by STFC grant ST/N50404X/1. CL has received funding from the ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. CL also thanks the MERAC Foundation for a Postdoctoral Research Award. This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grants ST/H008519/1 and ST/K00087X/1, STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure.This paper has been typeset from a T E X/L A T E X file prepared by the author.
. M A Abramowicz, B Czerny, J P Lasota, E Szuszkiewicz, 10.1086/166683ApJ. 332646Abramowicz M. A., Czerny B., Lasota J. P., Szuszkiewicz E., 1988, ApJ, 332, 646
. J Aird, arXiv:1306.2325preprintAird J., et al., 2013, preprint, (arXiv:1306.2325)
. S Amarantidis, 10.1093/mnras/stz551MNRAS. 4852694Amarantidis S., et al., 2019, MNRAS, 485, 2694
. E Bañados, 10.1038/nature25180Nature. 553473Bañados E., et al., 2018a, Nature, 553, 473
. E Bañados, 10.3847/2041-8213/aab61eApJ. 85625Bañados E., et al., 2018b, ApJ, 856, L25
. C M Baugh, 10.1093/mnras/sty3427MNRAS. 4834922Baugh C. M., et al., 2019, MNRAS, 483, 4922
. S Bonoli, F Marulli, V Springel, S D M White, E Branchini, L Moscardini, 10.1111/j.1365-2966.2009.14701.xMNRAS. 396423Bonoli S., Marulli F., Springel V., White S. D. M., Branchini E., Moscardini L., 2009, MNRAS, 396, 423
. R G Bower, A J Benson, R Malbon, J C Helly, C S Frenk, C M Baugh, S Cole, C G Lacey, 10.1111/j.1365-2966.2006.10519.xMNRAS. 370645Bower R. G., Benson A. J., Malbon R., Helly J. C., Frenk C. S., Baugh C. M., Cole S., Lacey C. G., 2006, MNRAS, 370, 645
. K W Cavagnolo, B R Mcnamara, M W Wise, P E J Nulsen, M Brüggen, M Gitti, D A Rafferty, 10.1088/0004-637X/732/2/71ApJ. 73271Cavagnolo K. W., McNamara B. R., Wise M. W., Nulsen P. E. J., Brüggen M., Gitti M., Rafferty D. A., 2011, ApJ, 732, 71
. C Cicone, 10.1051/0004-6361/201424980A&A. 57414Cicone C., et al., 2015, A&A, 574, A14
. S Cole, C G Lacey, C M Baugh, C S Frenk, 10.1046/j.1365-8711.2000.03879.xMNRAS. 319168Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 2000, MNRAS, 319, 168
. J J Condon, 10.1086/152714ApJ. 188279Condon J. J., 1974, ApJ, 188, 279
. W I Cowley, C M Baugh, S Cole, C S Frenk, C G Lacey, 10.1093/mnras/stx2897MNRAS. 4742352Cowley W. I., Baugh C. M., Cole S., Frenk C. S., Lacey C. G., 2018, MNRAS, 474, 2352
. D J Croton, 10.1111/j.1365-2966.2005.09675.xMNRAS. 36511Croton D. J., et al., 2006, MNRAS, 365, 11
. L David, 10.1088/0004-637X/728/2/162ApJ. 728162David L., et al., 2011, ApJ, 728, 162
. De Rosa, G Decarli, R Walter, F Fan, X Jiang, L Kurk, J Pasquali, A Rix, H W , 10.1088/0004-637X/739/2/56ApJ. 73956De Rosa G., Decarli R., Walter F., Fan X., Jiang L., Kurk J., Pasquali A., Rix H. W., 2011, ApJ, 739, 56
. Di Matteo, T Springel, V Hernquist, L , 10.1038/nature03335Nature. 433604Di Matteo T., Springel V., Hernquist L., 2005, Nature, 433, 604
. Di Matteo, T Croft, R A C Feng, Y Waters, D Wilkins, S , 10.1093/mnras/stx319MNRAS. 4674243Di Matteo T., Croft R. A. C., Feng Y., Waters D., Wilkins S., 2017, MNRAS, 467, 4243
. M Enoki, T Ishiyama, M A R Kobayashi, M Nagashima, 10.1088/0004-637X/794/1/69ApJ. 79469Enoki M., Ishiyama T., Kobayashi M. A. R., Nagashima M., 2014, ApJ, 794, 69
. X Fan, 10.1086/324111AJ. 1222833Fan X., et al., 2001, AJ, 122, 2833
. X Fan, 10.1086/368246AJ. 1251649Fan X., et al., 2003, AJ, 125, 1649
. X Fan, AJ. 128515Fan X., et al., 2004, AJ, 128, 515
. N Fanidakis, C M Baugh, A J Benson, R G Bower, S Cole, C Done, C S Frenk, 10.1111/j.1365-2966.2010.17427.xMNRAS. 41053Fanidakis N., Baugh C. M., Benson A. J., Bower R. G., Cole S., Done C., Frenk C. S., 2011, MNRAS, 410, 53
. N Fanidakis, 10.1111/j.1365-2966.2011.19931.xMNRAS. 4192797Fanidakis N., et al., 2012, MNRAS, 419, 2797
. W Forman, 10.1086/429746ApJ. 635894Forman W., et al., 2005, ApJ, 635, 894
. J P Gardner, 10.1007/s11214-006-8315-7Space Sci. Rev. 123485Gardner J. P., et al., 2006, Space Sci. Rev., 123, 485
. E Giallongo, 10.1051/0004-6361/201425334A&A. 57883Giallongo E., et al., 2015, A&A, 578, A83
. V Gonzalez-Perez, C G Lacey, C M Baugh, C D P Lagos, J Helly, D J R Campbell, P D Mitchell, 10.1093/mnras/stt2410MNRAS. 439264Gonzalez-Perez V., Lacey C. G., Baugh C. M., Lagos C. D. P., Helly J., Campbell D. J. R., Mitchell P. D., 2014, MNRAS, 439, 264
. A J Griffin, C G Lacey, V Gonzalez-Perez, C D P Lagos, C M Baugh, N Fanidakis, 10.1093/mnras/stz1216MNRAS. 487198Griffin A. J., Lacey C. G., Gonzalez-Perez V., Lagos C. d. P., Baugh C. M., Fanidakis N., 2019, MNRAS, 487, 198
. C M Harrison, 10.1111/j.1365-2966.2012.21723.xMNRAS. 4261073Harrison C. M., et al., 2012, MNRAS, 426, 1073
. J C Helly, S Cole, C S Frenk, C M Baugh, A Benson, C Lacey, 10.1046/j.1365-8711.2003.06151.xMNRAS. 338903Helly J. C., Cole S., Frenk C. S., Baugh C. M., Benson A., Lacey C., 2003, MNRAS, 338, 903
. M Hirschmann, R S Somerville, T Naab, A Burkert, 10.1111/j.1365-2966.2012.21626.xMNRAS. 426237Hirschmann M., Somerville R. S., Naab T., Burkert A., 2012, MNRAS, 426, 237
. M Hirschmann, K Dolag, A Saro, L Bachmann, S Borgani, A Burkert, 10.1093/mnras/stu1023MNRAS. 4422304Hirschmann M., Dolag K., Saro A., Bachmann L., Borgani S., Burkert A., 2014, MNRAS, 442, 2304
. P F Hopkins, G T Richards, L Hernquist, 10.1086/509629ApJ. 654731Hopkins P. F., Richards G. T., Hernquist L., 2007, ApJ, 654, 731
. L Jiang, 10.1088/0004-6256/138/1/305AJ. 138305Jiang L., et al., 2009, AJ, 138, 305
J Kalirai, 10.1080/00107514.2018.1467648Contemporary Physics. 59251Kalirai J., 2018, Contemporary Physics, 59, 251
. N Khandai, Y Feng, C Degraf, Di Matteo, T Croft, R A C , 10.1111/j.1365-2966.2012.21047.xMNRAS. 4232397Khandai N., Feng Y., DeGraf C., Di Matteo T., Croft R. A. C., 2012, MNRAS, 423, 2397
. A Kubota, C Done, 10.1093/mnras/sty1890MNRAS. 4801247Kubota A., Done C., 2018, MNRAS, 480, 1247
. C G Lacey, 10.1093/mnras/stw1888MNRAS. 4623854Lacey C. G., et al., 2016, MNRAS, 462, 3854
. C D P Lagos, S A Cora, N D Padilla, 10.1111/j.1365-2966.2008.13456.xMNRAS. 388587Lagos C. D. P., Cora S. A., Padilla N. D., 2008, MNRAS, 388, 587
. R Laureijs, 10.1111/j.1365-2966.2007.12040.xarXiv:1110.3193MNRAS. 3791599preprintLaureijs R., et al., 2011, preprint, (arXiv:1110.3193) Lawrence A., et al., 2007, MNRAS, 379, 1599
. B D Lehmer, 10.1088/0004-637X/752/1/46ApJ. 75246Lehmer B. D., et al., 2012, ApJ, 752, 46
. R Maiolino, 10.1111/j.1745-3933.2012.01303.xMNRAS. 42566Maiolino R., et al., 2012, MNRAS, 425, L66
. A Marconi, G Risaliti, R Gilli, L K Hunt, R Maiolino, M Salvati, 10.1111/j.1365-2966.2004.07765.xMNRAS. 351169Marconi A., Risaliti G., Gilli R., Hunt L. K., Maiolino R., Salvati M., 2004, MNRAS, 351, 169
. F Marulli, S Bonoli, E Branchini, L Moscardini, V Springel, 10.1111/j.1365-2966.2008.12988.xMNRAS. 3851846Marulli F., Bonoli S., Branchini E., Moscardini L., Springel V., 2008, MNRAS, 385, 1846
. N Menci, F Fiore, A Lamastra, 10.1088/0004-637X/766/2/110ApJ. 766110Menci N., Fiore F., Lamastra A., 2013, ApJ, 766, 110
. D J Mortlock, 10.1038/nature10159Nature. 474616Mortlock D. J., et al., 2011, Nature, 474, 616
. K Nandra, arXiv:1306.2307ApJ. 42813preprintNandra K., et al., 2013, preprint, (arXiv:1306.2307) Narayan R., Yi I., 1994, ApJ, 428, L13
. E Neistein, H Netzer, 10.1093/mnras/stt2130MNRAS. 4373373Neistein E., Netzer H., 2014, MNRAS, 437, 3373
. M Onoue, 10.3847/2041-8213/aa8cc6ApJ. 84715Onoue M., et al., 2017, ApJ, 847, L15
. 10.1051/0004-6361/201321591A&A. 57116Planck Collaboration et al., 2014, A&A, 571, A16
. A Ricarte, P Natarajan, 10.1093/mnras/stx2851MNRAS. 474Ricarte A., Natarajan P., 2018a, MNRAS, 474, 1995
. A Ricarte, P Natarajan, 10.1093/mnras/sty2448MNRAS. 4813278Ricarte A., Natarajan P., 2018b, MNRAS, 481, 3278
. F Ricci, S Marchesi, F Shankar, La Franca, F Civano, F , 10.1093/mnras/stw2909MNRAS. 4651915Ricci F., Marchesi S., Shankar F., La Franca F., Civano F., 2017, MNRAS, 465, 1915
. Y Rosas-Guevara, R G Bower, J Schaye, S Mcalpine, Dalla Vecchia, C Frenk, C S Schaller, M Theuns, T , 10.1093/mnras/stw1679MNRAS. 462190Rosas-Guevara Y., Bower R. G., Schaye J., McAlpine S., Dalla Vecchia C., Frenk C. S., Schaller M., Theuns T., 2016, MNRAS, 462, 190
. A Saxena, H J A Röttgering, E E Rigby, 10.1093/mnras/stx1150MNRAS. 4694083Saxena A., Röttgering H. J. A., Rigby E. E., 2017, MNRAS, 469, 4083
. A Saxena, 10.1093/mnras/sty1996MNRAS. 4802733Saxena A., et al., 2018, MNRAS, 480, 2733
. N I Shakura, R A Sunyaev, A&A. 24337Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
. H Shirakata, 10.1093/mnras/sty2958MNRAS. 4824846Shirakata H., et al., 2019, MNRAS, 482, 4846
. D Sijacki, M Vogelsberger, S Genel, V Springel, P Torrey, G F Snyder, D Nelson, L Hernquist, 10.1093/mnras/stv1340MNRAS. 452575Sijacki D., Vogelsberger M., Genel S., Springel V., Torrey P., Snyder G. F., Nelson D., Hernquist L., 2015, MNRAS, 452, 575
. arXiv:1809.09642The Lynx. preprintThe Lynx Team 2018, preprint, (arXiv:1809.09642)
. B P Venemans, 10.1088/0004-637X/779/1/24ApJ. 77924Venemans B. P., et al., 2013, ApJ, 779, 24
. M Vestergaard, B M Peterson, 10.1086/500572ApJ. 641689Vestergaard M., Peterson B. M., 2006, ApJ, 641, 689
. M Volonteri, 10.1007/s00159-010-0029-xA&ARv. 18279Volonteri M., 2010, A&ARv, 18, 279
. A K Weigel, K Schawinski, N Caplar, O I Wong, E Treister, B Trakhtenbrot, 10.3847/1538-4357/aa803bApJ. 845134Weigel A. K., Schawinski K., Caplar N., Wong O. I., Treister E., Trakhtenbrot B., 2017, ApJ, 845, 134
. R Weinberger, 10.1093/mnras/sty1733MNRAS. 4794056Weinberger R., et al., 2018, MNRAS, 479, 4056
. C J Willott, 10.1088/0004-6256/139/3/906AJ. 139906Willott C. J., et al., 2010a, AJ, 139, 906
. C J Willott, 10.1088/0004-6256/140/2/546AJ. 140546Willott C. J., et al., 2010b, AJ, 140, 546
. X.-B Wu, 10.1038/nature14241Nature. 518512Wu X.-B., et al., 2015, Nature, 518, 512
. D G York, 10.1086/301513AJ. 1201579York D. G., et al., 2000, AJ, 120, 1579
. F Yuan, R Narayan, 10.1146/annurev-astro-082812-141003ARA&A. 52529Yuan F., Narayan R., 2014, ARA&A, 52, 529
| [] |
[
"Hyper-order baryon number fluctuations at finite temperature and density",
"Hyper-order baryon number fluctuations at finite temperature and density"
] | [
"Wei-Jie Fu \nSchool of Physics\nDalian University of Technology\n116024DalianP.R. China\n",
"Xiaofeng Luo \nInstitute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina\n",
"Jan M Pawlowski \nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 1669120HeidelbergGermany\n\nExtreMe Matter Institute EMMI\nGSI\nPlanckstraße 1D-64291DarmstadtGermany\n",
"Fabian Rennecke \nBrookhaven National Laboratory\nUpton11973NYUSA\n",
"Rui Wen \nSchool of Physics\nDalian University of Technology\n116024DalianP.R. China\n",
"Shi Yin \nSchool of Physics\nDalian University of Technology\n116024DalianP.R. China\n"
] | [
"School of Physics\nDalian University of Technology\n116024DalianP.R. China",
"Institute of Particle Physics\nKey Laboratory of Quark & Lepton Physics (MOE)\nCentral China Normal University\n430079WuhanChina",
"Institut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 1669120HeidelbergGermany",
"ExtreMe Matter Institute EMMI\nGSI\nPlanckstraße 1D-64291DarmstadtGermany",
"Brookhaven National Laboratory\nUpton11973NYUSA",
"School of Physics\nDalian University of Technology\n116024DalianP.R. China",
"School of Physics\nDalian University of Technology\n116024DalianP.R. China"
] | [] | Fluctuations of conserved charges are sensitive to the QCD phase transition and a possible critical endpoint in the phase diagram at finite density. In this work, we compute the baryon number fluctuations up to tenth order at finite temperature and density. This is done in a QCD-assisted effective theory that accurately captures the quantum-and in-medium effects of QCD at low energies. A direct computation at finite density allows us to assess the applicability of expansions around vanishing density. By using different freeze-out scenarios in heavy-ion collisions, we translate these results into baryon number fluctuations as a function of collision energy. We show that a nonmonotonic energy dependence of baryon number fluctuations can arise in the non-critical crossover region of the phase diagram. Our results compare well with recent experimental measurements of the kurtosis and the sixth-order cumulant of the net-proton distribution from the STAR collaboration. They indicate that the experimentally observed non-monotonic energy dependence of fourth-order net-proton fluctuations is highly non-trivial. It could be an experimental signature of an increasingly sharp chiral crossover and may indicate a QCD critical point. The physics implications and necessary upgrades of our analysis are discussed in detail. | 10.1103/physrevd.104.094047 | [
"https://arxiv.org/pdf/2101.06035v2.pdf"
] | 231,627,599 | 2101.06035 | 51d4e35de0295190ffc9ded9fc64f04a952edb16 |
Hyper-order baryon number fluctuations at finite temperature and density
Wei-Jie Fu
School of Physics
Dalian University of Technology
116024DalianP.R. China
Xiaofeng Luo
Institute of Particle Physics
Key Laboratory of Quark & Lepton Physics (MOE)
Central China Normal University
430079WuhanChina
Jan M Pawlowski
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 1669120HeidelbergGermany
ExtreMe Matter Institute EMMI
GSI
Planckstraße 1D-64291DarmstadtGermany
Fabian Rennecke
Brookhaven National Laboratory
Upton11973NYUSA
Rui Wen
School of Physics
Dalian University of Technology
116024DalianP.R. China
Shi Yin
School of Physics
Dalian University of Technology
116024DalianP.R. China
Hyper-order baryon number fluctuations at finite temperature and density
Fluctuations of conserved charges are sensitive to the QCD phase transition and a possible critical endpoint in the phase diagram at finite density. In this work, we compute the baryon number fluctuations up to tenth order at finite temperature and density. This is done in a QCD-assisted effective theory that accurately captures the quantum-and in-medium effects of QCD at low energies. A direct computation at finite density allows us to assess the applicability of expansions around vanishing density. By using different freeze-out scenarios in heavy-ion collisions, we translate these results into baryon number fluctuations as a function of collision energy. We show that a nonmonotonic energy dependence of baryon number fluctuations can arise in the non-critical crossover region of the phase diagram. Our results compare well with recent experimental measurements of the kurtosis and the sixth-order cumulant of the net-proton distribution from the STAR collaboration. They indicate that the experimentally observed non-monotonic energy dependence of fourth-order net-proton fluctuations is highly non-trivial. It could be an experimental signature of an increasingly sharp chiral crossover and may indicate a QCD critical point. The physics implications and necessary upgrades of our analysis are discussed in detail.
I. INTRODUCTION
Some of the most challenging questions of heavy-ion physics are related to the transition from the early, non-equilibrium, state of quarks and gluons to the final hadronic states after chemical freeze out, which is observed in experiments. Unravelling this dynamics necessitates a thorough grasp on the physics in the QCD phase structure close to the confinement-deconfinement and chiral transitions. This regime is strongly correlated with highly non-trivial dynamics. Understanding this part of the phase structure, including the location and dynamics of a potential critical end point (CEP), plays a pivotal role in understanding phases of strongly interacting nuclear matter under extreme conditions. For works on the phase structure of QCD, covering experiment and theory see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13], where theory covers first principles functional approaches and lattice simulations.
Fluctuations of conserved charges are very sensitive to the physics of the strongly correlated regime that governs the transition from the quark-gluon plasma (QGP) to the hadronic phase. They provide detailed information on the underlying dynamics. This includes, but is not limited to, possible experimental signatures of a CEP [4]. It has for example been proposed in [14][15][16] that non-monotonic variations of conserved charge fluctuations as functions of the beam energy can arise from critical physics in the vicinity of a CEP. During the last decade, significant fluctuation measurements have been performed in the first phase of the Beam Energy Scan (BES-I) program at the Relativistic Heavy Ion Collider (RHIC), involving various cumulants of net-proton, netcharge and net-kaon multiplicity distributions [17][18][19][20][21]. Remarkably, very recently the STAR collaboration has reported the first evidence of a non-monotonic variation in the kurtosis (multiplied by the variance) of the netproton number distribution as a function of the collision energy with 3.1 σ significance for central collisions [22]. The measurements have been extended to the sixth-order cumulants of net-proton and net-charge distributions, for preliminary results from STAR see [23,24].
Recent first-principle QCD calculations at finite temperature and density, within both the functional renormalisation group (fRG) and Dyson-Schwinger equations (DSE), show that the transition from the QGP to the hadronic phase is a crossover which becomes sharper with increasing baryon chemical potential, µ B , for µ B /T 4 [7,8,[25][26][27][28]. Beyond this region, a CEP might occur, but quantitative reliability of the theory computations cannot be guaranteed within the present approximations [7,8,25,28]. In addition, critical physics may only be observable in a very small region around the CEP, see e.g. [29]. Since the available RHIC data is limited to µ B /T 3, it is important to understand how conserved charge fluctuations are affected by the increasingly sharp crossover away from a regime with critical scaling.
To address these open questions related to the physics of strong correlations in the QCD phase diagram, we study in detail the T -and µ B -dependence of net-baryon number fluctuations in the range µ B /T 3. We present results for fluctuations up to tenth order (where we refer to everything above fourth order as hyper-order ), including comparisons to available results from RHIC [22][23][24] and predictions for the beam energy dependence of fluctuations where no experimental results are available yet.
To facilitate the comparison between theory and experiment, T and µ B can be mapped onto the beam energy per nucleon, √ s NN , via phenomenological freeze-out curves [30]. While these curves are expected to be close to the QCD crossover at large beam energies (corresponding to small µ B ) [31], they may move away from the transition region at low energies (i.e. larger µ B ) [32]. This can affect the beam energy dependence of particle number fluctuations, and requires a detailed understanding of the physics also outside the critical region. Aside from their phenomenological relevance, netbaryon number fluctuations at finite µ B can also be used to assess the reliability of extrapolations of thermodynamic quantities to finite µ B based on a Taylor expansion at µ B = 0. Such a strategy is commonly used in lattice QCD simulations, where a sign problem prevents direct simulations at finite µ B , see e.g. [9,[33][34][35][36][37][38]. By comparing the results of direct computations at finite µ B to the ones obtained from extrapolations at µ B = 0, we study the range of validity of a Taylor expansion at a given order self-consistently. Understanding the limitations of such an extrapolation is also relevant for phenomenologically constructed equations of state, as, e.g., in [39], where the non-critical physics at finite µ B crucially rely on this extrapolation.
All this is addressed within a QCD-assisted low-energy effective field theory (LEFT) which is described in detail in the next section. We use first-principles QCDresults on the T -dependence of the kurtosis and the µ Bdependence of the chiral phase boundary to map the inmedium scales of the LEFT onto QCD. This improves the reliability of our predictions, in particular at finite µ B . Non-perturbative quantum-, thermal-and density fluctuations are taken into account with the functional renormalisation group (fRG). This work therefore is a significant upgrade of previous work in [40][41][42], where net-baryon number fluctuations up to fourth order have been studied. The present QCD-assisted LEFT approach has various advantages. Most importantly, it is directly embedded in QCD as the relevant low-energy degrees of freedom emerge dynamically from systematically integrating-out the fast partonic modes of QCD [8,[43][44][45][46]. In addition, this approach allows us to capture both critical and non-critical effects in the QCD phase diagram. This entails in particular that our results agree with the results of lattice QCD at small µ B and show the correct universal behaviour of QCD in the vicinity of the CEP, i.e. 3d Ising universality.
Concerning the existence and location of the latter we add that the first-principles results in [7,8,[25][26][27][28] include a CEP in a region of 450 MeV µ B 650 MeV and therefore outside the regime of quantitative reliability of these computations. This suggests that the experimental detection of a CEP requires explorations of the high-µ B in the region with µ B /T c 4. Moreover, the direct experimental measurement of the CEP may be very challenging as it requires very high statistics, and predictions that signal critical dynamics can be further complicated by non-equilibrium effects. In the present work we shall therefore also outline how the location of a CEP could be constrained based on data in the crossover region, with- out the necessity of observing critical scaling. On the theoretical side this asks for first principles QCD studies for µ B /T 4. In turn, a first experimental step towards this goal is the solidification of experimental observation of the non-monotonic energy dependence of fourth-order net-proton fluctuations. Both is safely beyond the scope of the present work. This paper is organised as follows: In Section II we give a brief introduction to the fRG-approach to QCD and low-energy effective theories, including their mutual relationship. Thermodynamics and the hyper-order baryon number fluctuations are discussed in Section III. In Section IV, we first introduce a systematic scale-matching procedure between QCD and the low-energy effective theory. We then present our numerical results and compare them to lattice QCD simulations and experimental measurements. A summary with conclusions is given in Section V. Technical details regarding the flow equations are presented in the appendices.
II. QCD AND EMERGENT LOW-ENERGY EFFECTIVE THEORIES
At low momentum scales the quark-gluon dynamics of QCD successively decouple due to the QCD mass gap and spontaneous chiral symmetry breaking. This decoupling also applies to most dynamical (hadronic) low energy degrees of freedom at even lower energies, finally leaving us with dynamical pions and hence with chiral perturbation theory. Indeed, this successive decoupling is at the root of the success of chiral effective field theory.
The functional renormalisation group approach to QCD with its successive integrating-out of momentum modes is ideally suited to follow and study this decoupling. Diagrammatically, this is already seen within the flow equation for the QCD effective action, depicted in Figure 1. The different lines stand for the full nonperturbative propagators of gluons, ghosts, quarks and emergent low-energy degrees of freedom (hadrons in our case), where the loop momentum q is restricted by the infrared cutoff scale k, q 2 k 2 . In this setup, emergent bound states can be incorporated systematically by dynamical hadronisation [47][48][49][50]. For quantitative QCD applications in the vacuum see [43][44][45][46], for further conceptual developments and the application to the QCD phase structure important for the present work see [8].
The decoupling is apparent in this framework as the propagators carry the mass gaps m gap of gluons and quarks and for cutoff scales k m gap of a given field the respective loop tends towards zero.
More importantly, in this way the emergent low-energy effective theory is naturally embedded in QCD, and its ultraviolet parameters (at Λ 1 GeV) as well as further input may be directly computed from QCD, leading to QCD-assisted low-energy effective theories. We emphasise that this procedure does not lead to a unique LEFT. The dynamical degrees of freedom of QCDassisted LEFTs at Λ 1 GeV depend on the dynamical hadronisation procedure applied within QCD-flows. This setup and the QCD-embedding entail that, provided the relevant quantum, thermal and density fluctuations of low energy QCD are taken into account in the QCDassisted LEFT at hand, all QCD-assisted LEFTs encode the same physics, namely that of low energy QCD.
This leads to an equivalence relation of QCD-assisted Polyakov-loop-enhanced NJL-type LEFTs (PNJL), Polyakov-loop-enhanced QM LEFTs (PQM) and variations including higher meson multiplets and/or diquarks and baryons. We emphasise again that this equivalence relation only holds if low energy quantum, thermal and density fluctuations are taken into account. For more details see in particular [8], and the recent review [51]. Most prominently this embedding has been used for determining the temperature-dependence of the Polyakov loop potential, see [52,53]. This setup was then applied to the computation of fluctuations in [40-42, 54, 55].
In summary this entails, that for sufficiently small momenta k, temperatures T , and also density or quark chemical potential µ q , the gluon (and ghost) loop in Figure 1 decouple from the dynamics, and only provide a non-trivial glue background at finite temperature and chemical potential. The latter is taken into account with the Polyakov loop potential discussed in detail below.
In the present work, we build upon previous investigations of the skewness and kurtosis of baryon number distributions [40][41][42], and baryon-strangeness correlations [56,57] within QCD-assisted LEFTs with the fRG. The present LEFT is an upgrade of those used in the works above, and includes the quantum, thermal and density dynamics of quarks, pions and the sigma mode in a Polyakov loop background. It is a QCD-assisted PQM. As argued above, for low enough chemical potential, this model sufficiently close to QCD, and leads to results that are independent of the LEFT at hand.
For further investigations of fluctuation observables within the fRG-approach to low-energy effective theories see e.g. [58][59][60][61][62], the Dyson-Schwinger approach has been used in e.g. [26,63], for mean-field investigations see e.g. [64][65][66][67][68]. These functional works can be further adjusted and benchmarked with results from lattice QCD simulations [9,[33][34][35][36][37][38], at high temperatures, T T c , and vanishing µ B . In turn, at finite µ B , and in particular for µ B /T c 3, lattice simulations are obstructed by the sign problem.
A. 2-flavour setup
For the physics of fluctuations we are interested in scales below approximately 1 GeV. We restrict ourselves to k 700 MeV and temperatures and quark chemical potentials T, µ q 200 MeV. In this regime the only relevant quarks are the light quarks q = (u, d) and the strange quark s. The latter, while changing the momentum-scale running of the correlation functions, has subleading effects on the form of the fluctuations. Hence, the effect of the momentum-scale running induced by strange fluctuations will be mimicked here by an appropriate scale-matching detailed in Section II B.
We also include the lowest lying hadronic resonances, the pion π = (π ± , π 0 ), and, for symmetry reasons, the scalar resonance σ as effective low energy degrees of freedom. Within QCD flows these fields are emergent low energy degrees of freedom at cutoff scales k 1 GeV, that are taken care of with dynamical hadronisation in e.g. [8]. At the present low energy scales k ≤ 700 MeV, they are fully dynamical, and hence are part of the effective action at the initial cutoff scale. The other members of the lowest lying multiplet as well as further hadronic resonances produce rather subleading contributions to the offshell dynamics and hence are dropped. The mesonic fields are stored in an O(4) scalar field φ = (σ, π) with the corresponding chiral invariant ρ = φ 2 /2.
Quantum, thermal and density fluctuations with scales k Λ = 700 MeV are taken into account within the fRG, whose dynamics is now reduced to the last two loops in Figure 1. The respective effective action of QCD in the low energy regime is approximated by
Γ k = x Z q,kq γ µ ∂ µ − γ 0 (µ + igA 0 ) q + 1 2 Z φ,k (∂ µ φ) 2 + h kq τ 0 σ + τ · π q + V k (ρ, A 0 ) − cσ ,(1)with x = 1/T 0 dx 0 d 3 x and τ = 1/2(1, iγ 5 σ)
. We assume isospin symmetry and the corresponding chemical potential flavour-matrix is given by µ = diag(µ q , µ q ) = 1 3 diag(µ B , µ B ). Z q,k and Z φ,k are the wave function renormalisations for the light quarks and the meson respectively. Further running couplings considered are the Yukawa coupling h k , the scattering between quarks and mesons, as well as the effective potential V k (ρ, A 0 ), that describes the multi-scattering of mesons in the non-trivial glue background present at finite temperature and chemical potential.
The flow equation for the effective action Equation (1), and that for V k , h k , Z φ,q is described in Appendix A and Appendix B. The initial condition for Γ k at the initial cutoff scale k = 700 MeV is described in Appendix C.
The potential V k (ρ, A 0 ) has contributions V glue,k (A 0 ) from offshell glue fluctuations (first two diagrams in Figure 1), and contributions V mat,k (ρ, A 0 ) from the quark loop (third diagram in Figure 1). This leads us to
V k (ρ, A 0 ) =V glue,k (A 0 ) + V mat,k (ρ, A 0 ) ,(2)
The first contribution is typically reformulated in terms of the Polyakov loop L(A 0 ), while the latter is directly computed from the present low energy flow. This allows us to trade the A 0 -dependence for that of the traced Polyakov loop, L(A 0 ),L(A 0 ), see Appendix D, Equation (D2), leading us to to the final form of our potential,
V k (ρ, L,L) = V k (ρ, A 0 ) .(3)
More details can be found in Appendix D.
In conclusion, the QCD-assisted LEFT described above and used in the present work, is a PQM-type model, e.g. [40-42, 52-59, 66, 67, 69-78]. Quantum, thermal and density fluctuations below Λ = 700 MeV are taken account with the functional renormalisation group, and the setup is well-embedded in functional QCD. As argued above, within the present, and analogous, elaborate approximation, the respective results (for fluctuation observables) for all QCD-assisted LEFTs match those of QCD for sufficiently low density. Therefore, we will refer to this model from now on as generic QCD-assisted LEFT.
B. 2 + 1-favour scale-matching in 2-flavour QCD
The current QCD-assisted LEFT setup enables us to compute thermodynamic observables and in particular hyper-order baryon number fluctuations. However, as already briefly discussed in Section II A, we have dropped the dynamics of the strange quark. While we expect sub-dominant effects on hyper-fluctuations, the s-quark influences the momentum running of the correlations in the ultraviolet. Importantly, in [8] it has been observed on the basis of genuine N f = 2 and N f = 2 + 1 flavour computations in QCD, that the latter effect is well approximated by a respective universal scale-matching of the 2-flavour results even in QCD. Such a scale-matching has already led to a quantitative agreement of thermodynamics and kurtosis within the current LEFT setup with lattice results, see [40][41][42]. Thus, the present scale-matching entails that we use information on the T -and µ B -dependence of welldetermined quantities in QCD. This leads to an improved reliability of our results of finite T and µ B , as in-medium effects in the QCD-assisted LEFT are directly connected to in-medium effects in QCD.
2-to 2+1-flavour scale-matching in QCD
Given its relevance for the predictive power of the present LEFT within a QCD scale-matching procedure we briefly recall the respective results in [8]: There, the phase boundaries of 2-and 2+1-flavour QCD have been computed within the fRG approach. These results allow us to evaluate the reliability of even linear scalematchings of temperatures and chemical potentials in 2and 2+1-flavour QCD introduced by
T (N f =2) = c T T (N f =2+1) , µ (N f =2) B = c µ B µ (N f =2+1) B .(4)
With such a linear scale-matching the scaling factors c T , c µ B can be determined by evaluating the relations at a specific temperature and chemical potential. For the thermal scale-matching we naturally take (T, µ B ) = (T c , 0), the crossover temperature at vanishing chemical potential. In [8] the crossover temperatures have been determined with thermal susceptibilities of the renormalised light chiral condensate. Then, the linear rescaling of the 2-flavour chiral crossover temperature to the 2+1-flavour crossover temperature is done with
T (N f =2) c = c QCD T T (N f =2+1) c , c QCD T = 1.1 .(5)
For the matching of the chemical potentials we use the curvature κ of the phase boundary at vanishing µ B = 0,
T c (µ B ) T c = 1 − κ µ B T c 2 + λ µ B T c 4 + · · · .(6)
Adjusting the 2-flavour curvature −κ µ 2 B /T 2 c to the 2+1flavour one leads us to the relation
c QCD µ B = c QCD T κ (N f =2+1) κ (N f =2) 1/2 , c QCD µ B = 0.99 . (7)
The value c QCD µ B ≈ 1 entails that the change in the curvature coefficient κ is balanced by that of the temperature.
The fourth-order expansion coefficient λ is found to be very small in both functional, [8,27,28] as well as lattice computations, [10,79]. Moreover, the results for the phase boundary at finite chemical potential in [7,8,27,28] reveal that the phase boundary is still described well by the leading order expansion with µ 2 B -terms. We estimate, that this prediction is quantitatively reliable within µ B /T 4, using results from [7,8,25,27,28]. This covers the regime studied in the present work.
Applying the two scale-matching relations in Equation (4) with the coefficients Equation (5) and Equation (7) to the 2-and 2+1-flavour data of the QCD phase boundary in [8] leads us to Figure 2. In conclusion, this impressive agreement provides non-trivial support for the scale-matching procedure in QCD.
2-to 2+1-flavour scale-matching in LEFTs
The convincing quantitative accuracy of the linear scale-matching analysis presented for QCD in the last section also sustains its use in the LEFT within the present work. Note however, that we cannot simply Analogously to QCD we choose the chiral crossover temperature at vanishing chemical potential, (T, µ B ) = (T c , 0) for fixing the scale factor c T . Moreover, in the present work we are interested in fluctuations of conserved charges. Hence, instead of the renormalised condensate we use the kurtosis of baryon number fluctuations, or rather R B 42 = χ B 4 /χ B 2 , for the definition see Equation (13) and Equation (14) with Equation (16), Equation (18). This leads us to the following determination of c T : While the temperature-dependence of R B 42 is a prediction of the LEFT, its absolute temperature has to be adjusted. This is done by minimising the χ 2 of the difference between the lattice result and the LEFTprediction as a function of the rescaled absolute temperature c T T c , leading us to
c T = 1.247(12) ,(8)
The respective result for R B 42 is shown in Figure 3 in comparison to the lattice result from [38]. The two curves match quantitatively supporting the predictive power of the LEFT. (14), using the 2+1-flavour lattice results of [38]. This leads to c T = 1.247 (12) in Equation (5). The T /Tc-dependence of R B 42 is a prediction of the QCD-assisted LEFT, and agrees quantitatively with the lattice results.
the art functional approaches: κ = 0.0142(2) in [8], κ = 0.0150 (7) in [27] and κ = 0.0147(5) in [28], the very recent update of [27]. Lattice results are provided with κ = 0.015(4) in [79], κ = 0.0149 (21) in [80], κ = 0.0153 (18) in [10]. Both, functional and lattice results agree within the respective (statistical and systematic) errors with κ ≈ 0.015.
Having adjusted the temperature with results from the WB-collaboration, [80], we use κ = 0.0153(18) from [10] for internal consistency. Note that the results presented here do only change marginally if using one κ in the range κ = (0.0142 − 0.0153). Within the current LEFT we obtain κ LEFT = 0.0193. In comparison, κ LEFT is larger than the 2-flavour QCD result in [8] with κ = 0.0179 (8). This reflects the lack of glue-dynamics in the LEFT. We use this in the relation Equation (7) instead of κ (N f =2) , and arrive at
c µ B = c T κ N f =(2+1) κ LEFT 1/2 = 1.110(66) ,(9)
with the LEFT-c T from Equation (8).
In summary, as our first step towards a quantitative prediction for hyper-order baryon number fluctuations, in this work we do not deal with the strange quark as a dynamical degree of freedom for the moment, but rather take into account its effect on the modification of the momentum-scale running via an appropriate scalematching as shown in Equation (4). The validity of the scale-matching relations between 2-and 2+1-flavour QCD has been well verified in this section by means of results from the first-principle functional QCD in [8]. These relations were applied to the present QCD-assisted LEFT. The scale-matching was done with two observables relevant for the fluctuation physics studied here:
R B
42 as a function of T and the curvature of the phase boundary κ, both at vanishing chemical potential. This led us to the coefficients Equation (8) and Equation (9) in Equation (4). The errors in these coefficients determine the errors of our results in Section IV.
III. THERMODYNAMICS AND HYPER-ORDER BARYON NUMBER FLUCTUATIONS
The thermodynamic potential in the LEFT at finite temperature and baryon chemical potential is readily obtained from the effective action in Equation (1), or rather from its integrated flow: we evaluate the effective action on the solution of the quantum equations of motion (EoMs). In the present work we consider only homoge-
neous (constant) solutions, (σ EoM , A 0,EoM ) with ∂V (ρ, L,L) ∂σ = ∂V (ρ, L,L) ∂L = ∂V (ρ, L,L) ∂L = 0 ,(10)
while the quark fields vanish on the EoMs, q,q = 0. We also note that the assumption of homogeneous solutions has to be taken with a grain of salt for larger chemical potentials with µ B /T 4, see [8]. Such a scenario has been investigated in LEFTs, see e.g. the review [81] and references therein.
With these preparations we are led to the grand potential Ω[T, µ B ] = V k=0 (ρ, L,L), the effective potential, evaluated at vanishing cutoff scale k = 0. It reads
Ω[T, µ B ] =V glue (L,L) + V mat (ρ, L,L) − cσ ,(11)
where the gluonic background field A 0 in Equation (2) has been reformulated in terms of the Polyakov loop L and its complex conjugateL. As mentioned before, the matter sector of the effective potential is integrated out towards the IR limit k = 0, for details see Appendix B. In turn, the glue sector is independent of k, see Appendix D.
The pressure of the system follows directly from the thermodynamic potential,
p = − Ω[T, µ B ] .(12)
The generalised susceptibilities of the baryon number χ B n are defined through the n-th order derivatives of the pressure w.r.t. the baryon chemical potential, to wit,
χ B n = ∂ n ∂(µ B /T ) n p T 4 .(13)
To remove the explicit volume dependence, it is advantageous to consider the ratio between the n-and m-th order susceptibilities, defined by,
R B nm = χ B n χ B m .(14)
The generalised susceptibilities are related to various cumulants of the baryon number distribution, which can be measured in heavy-ion collision experiments through the cumulants of its proxy, i.e., the net proton distribution, see, e.g., [4] for details. For the lowest four orders we get,
χ B 1 = 1 V T 3 N B ,(15)χ B 2 = 1 V T 3 (δN B ) 2 ,(16)χ B 3 = 1 V T 3 (δN B ) 3 ,(17)χ B 4 = 1 V T 3 (δN B ) 4 − 3 (δN B ) 2 2 ,(18)
with · · · denoting the ensemble average and δN B = N B − N B . Thus the mean value of the net baryon number of the system is given by
M = V T 3 χ B 1 , the variance σ 2 = V T 3 χ B 2 , skewness S = χ B 3 /(χ B 2 σ)
, and the kurtosis
κ = χ B 4 /(χ B 2 σ 2 )
, respectively. In this work the emphasis is, however, put on the baryon number fluctuations of orders higher than the fourth, i.e. χ B n>4 , which are named hyper-order baryon number fluctuations. As the low-order ones, the hyperorder susceptibilities are also connected to their respective cumulants, and their relations, taking the fifth through eighth ones for instance, are given as follows,
χ B 5 = 1 V T 3 (δN B ) 5 − 10 (δN B ) 2 (δN B ) 3 , (19) χ B 6 = 1 V T 3 (δN B ) 6 − 15 (δN B ) 4 (δN B ) 2 − 10 (δN B ) 3 2 + 30 (δN B ) 2 3 ,(20)χ B 7 = 1 V T 3 (δN B ) 7 − 21 (δN B ) 5 (δN B ) 2 − 35 (δN B ) 4 (δN B ) 3 + 210 (δN B ) 3 (δN B ) 2 2 ,(21)χ B 8 = 1 V T 3 (δN B ) 8 − 28 (δN B ) 6 (δN B ) 2 − 56 (δN B ) 5 (δN B ) 3 − 35 (δN B ) 4 2 + 420 (δN B ) 4 (δN B ) 2 2 + 560 (δN B ) 3 2 (δN B ) 2 − 630 (δN B ) 2 4 .(22)
Different aspects of hyper-order fluctuations have been studied in mean-field approximations in the past, see e.g. [66,67,82]. However, due to the decisive role of nonperturbative quantum fluctuations for these quantities,
R B 42 = χ B 4 /χ B 2 (left panel), R B 62 = χ B 6 /χ B 2 (middle panel), and R B 82 = χ B 8 /χ B 2 (right panel)
as functions of the temperature at vanishing baryon chemical potential (µB = 0). Results from the QCD-assisted LEFT are compared with lattice results from the HotQCD collaboration [9,36,37] and the Wuppertal-Budapest collaboration (WB) [38]. The inset in the plot of R B 82 shows its zoomed-out view. Our results agree quantitatively with the WB-results , and are qualitatively compatible with the HotQCD results. We also compare to the predictions of a hadron resonance gas (HRG) [30], which predicts only a very mild increase of R B n2 from unity with increasing T .
a treatment beyond mean-field, as in the present work, is necessary for their accurate description. A first study in this direction, discussing hyper-order fluctuations up to χ 8 within a PQM model with the fRG at small µ B /T can be found in [60].
IV. NUMERICAL RESULTS AND DISCUSSIONS
In this section we present and discuss our numerical results for hyper-order fluctuations on the freeze-out curve. At vanishing chemical potential the lower orders are compared to results from lattice calculations. We then discuss the implications of our predictions for the hyperorder baryon number fluctuations for decreasing collision energies (increasing chemical potential) for heavy-ion collision experiments. We start our discussion of the numerical results in our QCD-assisted low-energy effective theory with benchmark results at vanishing chemical potential, µ B = 0. We have already seen in Section II B that the fourth order fluctuations R B 42 , Equation (14), agrees quantitatively with the respective lattice result, see Figure 3 and Figure 4, left panel. We emphasise again that the thermal dependence of R B 42 is a prediction of the present LEFT. Now we also compare the temperature dependence of the hyper-fluctuations R B 62 and R B 82 with the corresponding lattice results in the middle and right panels of Figure 4, respectively. We depict both our numerical results and lattice results from the HotQCD collaboration, [9,36,37], and the Wuppertal-Budapest collaboration, [38]. Note that lattice results in Figure 4 by the Wuppertal-Budapest collaboration in [38], and R B 62 and R B 82 by the HotQCD collaboration in [36] are not continuum extrapolated.
With the increase of the order of fluctuations, the uncertainties of lattice results increase significantly. Moreover, the eighth-order fluctuations R B 82 obtained by the two collaborations show a significant quantitative difference, although their form is qualitatively consistent with each other.
The hyper-order baryon number fluctuations computed in the current setup are in qualitative agreement with both lattice results. However, our results single out the lattice results of the Wuppertal-Budapest collaboration, with which we observe quantitative agreement. This situation is very reminiscent of the pressure prediction in [53]: similarly to the current situation with lattice predictions of hyper-fluctuations, the pressure predictions of the lattice collaborations had not converged yet. A less advanced version of the current QCD-assisted LEFT framework then predicted the correct pressure re- sult. We have also computed the hyper-order fluctuations within a simple hadron resonance gas model [30]. Essentially, they are all constant with only a very minor monotonic increase with T for T 140 MeV, starting from unity at T = 0. This is in quantitative agreement with our findings at low temperatures. In summary, the current setup passes all benchmark tests quantitatively and provides the full temperature-dependence of hyperfluctuations. We have also computed even higher order baryon number fluctuations. In Figure 5 we show our result for the temperature-dependence of the tenth order ratio R B 10 2 = χ B 10 /χ B 2 at vanishing chemical potential, µ B = 0. So far no lattice results for the tenth-order fluctuation are available, and the dependence of R B 10 2 on the temperature in Figure 5 is a prediction by the current QCD-assisted LEFT and awaits confirmation by other calculations, e.g., lattice QCD, in the future.
B. Hyper-order baryon number fluctuations at finite density
With successfully passing the benchmark tests, we proceed with the results for baryon number fluctuations at finite chemical potential. This will allow us to finally compare the theoretical predictions on the freeze-out curve with the experimental measurements in Section IV D.
Equally relevant is the self-consistent evaluation of the reliability of a Taylor expansion in baryon-chemical potential that underlies the extension of lattice results at vanishing chemical potential to µ B = 0. This is particularly important for predictions of the location of the critical end point based on such an expansion. Here we can investigate the reliability range of the Taylor expansion around µ B = 0 by comparison to our direct computation at finite µ B .
First we investigate the temperature-dependence of the baryon number fluctuations for different chemical potentials. This also allows us to discuss the reliability bounds of the current LEFT-setup for increasing chemical potential. In Figure 6 we show the temperature-dependence of the ratios R B 42 , R B 62 and R B 82 for chemical potentials µ B = 0, 100, 160, 200, 300, 400 MeV. First, we note that at small temperatures the thermodynamic properties of the QCD medium are well described by a dilute gas of hadrons, where the net-baryon number follows a Skellam distribution. Thus, all ratios approach unity at vanishing temperature. At very large temperature the system is governed by asymptotically free quarks, where fluctuations approach to the trivial Stefan-Boltzmann limit, and R B n2 goes to zero for all n > 4 at large T . Consequently, the non-trivial behaviour of the fluctuations between these two limiting cases shown in Figure 6 is directly related to the crossover from the hadronic-to the quark-gluon regime of QCD. The magnitude, but also the error of the fluctuations grow with increasing chemical potential. Both effects are more pronounced for higher order fluctuations. The increase in magnitude is directly linked to the sharpening of the chiral crossover with increasing chemical potential, cf. Figure 2. We expect that the current LEFT-setup is gradually loosing its predictive power for fluctuations on the freeze-out curve due to the rapid increase of the computational error for higher-order fluctuations at large baryon chemical potential, e.g., R B 82 with µ B 200 MeV. All results of the subsequent investigations have to be evaluated with this estimate on our systematic error.
For the evaluation of the reliability regime of the Taylor expansion about vanishing chemical potential we consider the Taylor expansion of the pressure in Equation (12) in powers ofμ B ≡ µ B /T aroundμ B = 0. This leads us to
p(µ B ) T 4 = p(0) T 4 + ∞ i=1 χ B 2i (0) (2i)!μ 2i B ,(23)
with the expansion coefficients χ B 2i (0) = χ B 2i (µ B = 0), the hyper-order fluctuations of the baryon charge. In Equation (23) we have suppressed the temperaturedependence of all functions for the sake of readability. Truncating the Taylor expansion in Equation (23) (24). Our results from the QCD-assisted LEFT are compared to those from lattice QCD by the HotQCD collaboration [9] and the Wuppertal-Budapest collaboration [38]. We show the comparison to HotQCD in the inlays, as these results deviate considerably from both ours and the WB results.
tain the expanded baryon number fluctuations,
χ B 2 (µ B ) χ B 2 (0) + χ B 4 (0) 2!μ 2 B + χ B 6 (0) 4!μ 4 B + χ B 8 (0) 6!μ 6 B , χ B 4 (µ B ) χ B 4 (0) + χ B 6 (0) 2!μ 2 B + χ B 8 (0) 4!μ 4 B , χ B 6 (µ B ) χ B 6 (0) + χ B 8 (0) 2!μ 2 B .(24)
In Figure 7 we show the ratios R B 42 = χ B 4 /χ B 2 and R B 62 = χ B 6 /χ B 2 based on the Taylor expansion for two fixed temperatures: T = 155 MeV (close to the crossover temperature T c at µ B = 0) and T = 160 MeV (slightly above T c ). As an input we use χ B 2i (0) (i = 1, 2, 3, 4) from the current setup as well as from the lattice (HotQCD collaboration [9] and Wuppertal-Budapest collaboration [38]), depicted already in For more details about effects of these constraints on the fluctuations and correlations of conserved charges, see the relevant discussions in, e.g., [9,36] in lattice QCD and [56,57] in fRG.
Since we are not restricted by a sign problem within the fRG approach, the χ B n (µ B )'s in Equation (13) can also be computed directly for the current QCD-assisted LEFT without resorting to a Taylor expansion. By comparing this to the results of the Taylor expansion, we can study its range of validity. The results are presented in the left panel of Figure 8, again for T = 155 MeV and T = 160 MeV and with the Taylor expansion up to eights order inμ B .
We observe that the result for R B 42 from the Taylor expansion in Equation (24) We emphasise that this is not an artefact of our model, but rather a generic, physical feature of these fluctuation observables. It reflects the increasingly pronounced non-monotonic temperature dependence of R B n2 due to long-range correlations in the crossover region, as seen in Figure 6. In particular, R B n2 develops distinctive areas around the crossover at larger µ B where its value varies significantly, even including sign changes. By following a line of fixed T close to T c and increasing µ B in the phase diagram, these areas are crossed eventually, leading to the oscillatory behaviour seen in Figure 8. This is also evident in the right plot of Figure 11, where we show the magnitude of R B 42 in the phase diagram. Since these strong fluctuations only occur at larger µ B , the resulting characteristic qualitative features cannot be captured by a (low-order) Taylor expansion at µ B = 0; it is bound to fail at the onset of the oscillatory behaviour, i.e. around µ B /T 1.
It is also interesting to evaluate to what extend a higher-order Taylor expansion can improve its reliability. We therefore include our prediction for R B 10 2 from Figure 5, hence extending the Taylor expansion in Equation (24) to the tenth order. The resulting comparison is shown in right panel of Figure 8 (25) might be increased a bit when terms of orders higher than the tenth one are included in the Taylor expansion. However, as we have discussed above, the full results of the baryon number fluctuations show a non-trivial oscillatory behaviour, which is generic, and stems from the fact that the system crosses over different phases with the increase of µ B , cf. the right panel of Figure 11. For a discussion of the radius of convergence based on mean-field theory we refer to [66]. In general, it is given by the distance to the nearest singularity of the equation of state in the complex µ B /T plane. Hence, possible candidates for this singularity are the CEP, the Roberge-Weiss endpoint at imaginary µ B [83], or the Yang-Lee edge singularity in the complex plane [84]. Evidently, the distance to the CEP is far larger than the radius estimated here. In turn, the closest endpoint at imaginary chemical potential is at |µ B /T | ≤ π. For physical quark masses it is most probably close to |µ B /T | = π. For a discussion within QCD-flows see [85], for lattice results see e.g. [86]. Hence, a particularly intriguing option is the Yang-Lee edge singularity, for a discussion see e.g. [87,88]. How-ever, while the location of the edge singularity has been determined for critical O(N ) theories [89], it is still unknown for QCD.
This interpretation also implies that the results from the Taylor expansion fail to agree even qualitatively with the correct µ B /T -dependence for µ B /T c [µ B /T ] Max (T ), see Figure 8. Interestingly, [µ B /T ] Max (T ) seems to grow for smaller temperatures. Whether or not this holds true requires a more systematic study, which will be considered elsewhere. In conclusion, the extrapolation of fluctuations of conserved charges in the vicinity of the chiral crossover within a Taylor expansion loses its predictive power for µ B 200 MeV, at least at tenth order.
In Figure 9 we show our full results for the temperature-dependence of R B 31 , R B 51 , and R B 71 with different values of baryon chemical potential. A further relevant odd fluctuation observable is R B 32 , depicted in Figure 10. Its experimental analogue, the proton number fluctuation R p 32 has been already measured in Au+Au central (0-5%) collisions at STAR, a comparison will be presented and discussed in Section IV E.
C. Determination of the freeze-out curve
The quantitatively successful benchmark tests analysed in Section IV A, and the evaluation of baryon number fluctuations at finite chemical potential in Section IV B allow us to discuss our main goal: the comparison of theoretical predictions on the baryon number fluctuations with experimental measurements.
A direct comparison between theory and experiment is a very challenging task. This is due to the fact that experimental data are affected by many factors. First, this concerns the acceptance of the detector such as the transverse momentum p T range, rapidity window and the centrality dependence, e.g. [17,19,22,[92][93][94], see [4,91] for more details. Second, the physics setup used in theory and experiment may differ by the presence of volume fluctuations, e.g. [95][96][97], finite volume effects on the location of the chiral phase boundary, e.g. [98][99][100][101][102][103][104][105], the question of global baryon number conservation, e.g. [106][107][108], the inclusion of resonance decays, e.g. [109,110], and others.
All these different effects and experimental restrictions give rise to non-critical contributions to fluctuation observables in experiments, and pinning down their contributions plays a pivotal role in identifying the critical signals in the BES experiment. Additionally, due to critical slowing down, non-equilibrium effects become important in the vicinity of the CEP [111], which necessitates a theoretical description of the dynamics of critical fluctuations. For more details about recent progress on the dynamics of critical fluctuations in QCD, see [112] and references therein. We emphasise, however, that the present results of QCD-assisted LEFT model are well outside the critical region. Therefore they are not subject to critical scaling in the vicinity of the CEP.
In this work we will not take into account the noncritical and dynamical effects discussed above. Instead, we assume that the measured cumulants of the netproton multiplicity distribution at a given collision energy are in one-to-one correspondence to the calculated fluctuations in Equation (13) with single values for T and µ B (with other collision parameters e.g., the centrality and rapidity range fixed). Then, it is suggestive to identify the values of T and µ B with the ones when the chemical freeze-out occurs, viz. T CF and µ B CF . Such an approach for the comparison is usually employed in fluctuation studies of equilibrium QCD matter within functional methods or lattice simulations, see e.g. [9,26,41,42,62].
We adopt the freeze-out temperatures and baryon chemical potentials from [90] and from the STAR experiment [91], which are shown in the left panel of Figure 11 by the blue pentagons and red circles, respectively. They are both obtained from the analysis of hadron yields in the statistical hadron resonance gas model, see the aforementioned references for more details. The freeze-out data in [90] has also been parametrised as functions of the collision energy as follows
µ B CF = a 1 + 0.288 √ s NN ,(26a)
with a = 1307.5 MeV, and
T CF = T (0) CF 1 + exp 2.60 − ln( √ s NN )/0.45 ,(26b)
with T (0) CF = 158.4 MeV. This parametrisation is depicted with the blue dashed line in the left panel of Figure 11. We use the same parametrisation functions in Equation (26) to fit the freeze-out data in STAR experiment, i.e., the red circle points. For this fit we invoke two procedures, called STAR Fit I & II in the following:
For the first one, STAR Fit I, we simply take all 7 data points. The corresponding freeze-out curve is depicted by the red solid line in the left panel of Figure 11.
For the construction of the second one, STAR Fit II, we shall argue that some of the data points are potentially flawed, or rather await a physics explanation, and should be dropped accordingly in a fit based on Equation (26). Accordingly, we drop the first two data points at small chemical potential as well as the last one at the largest available chemical potential µ B ∼ 400. From general considerations we do not expect the freeze-out curve to rise with increasing chemical potential. Moreover, the physically motivated fit formula does not describe signchanges of the curvature of the freeze-out curve. For a respective discussion and possible explanation for the only apparent rise see [113]. The last data point is also not well-described by the fitting procedure described here. This may indicate the onset of a regime with different physics/phases. In this case, Equation (26) would not be an appropriate fit function. It may also indicate the onset of a regime of rapidly worsening systematics. In this case more points are needed in this regime. The freeze-out line of STAR Fit II is depicted by the green dotted line in the left panel of Figure 11. In comparison to STAR Fit I, STAR Fit II is located at slightly lower temperatures, which is more pronounced when µ B 200 MeV. In the right panel of Figure 11, we show the baryon number fluctuation R B 42 in the T − µ B plane. It can be observed that a narrow blue band, indicating the regime of negative R B 42 , develops around the crossover starting at µ B ∼ 250 MeV. The freeze-out curve STAR Fit II is approaching towards the boundary of the blue region firstly at small µ B , and then deviates a bit from it at large µ B . We emphasise that the large chemical potential region, and in particular asymptotically large µ B 500 MeV, is beyond of the reliability bound of the current computation, µ B /T = 4, see Figure 2. For a detailed discussion see Section II B. The determination of the freeze-out curve completes our setup, which enables us to compute hyper-order baryon number fluctuations R B nm along the freeze-out line within the QCD-assisted LEFT. These results are then used to compare with the experimental measurements of cumulants R p nm of the net-proton distribution from STAR experiment.
Before we discuss the numerical results, we also emphasise once more, that it follows from the analysis of Section IV B, that the simple extrapolation with the Taylor expansion about µ B = 0 lacks predictive power for µ B 250 MeV, that is √ s NN 15 GeV, see Equation (25). Moreover, it even lacks predictive power for the qualitative behaviour.
In the left panel of Figure 12 we show the √ s NNor chemical potential dependence of the baryon number fluctuations R B 42 , R B 62 , and R B 82 for the freeze-out lines from Andronic et al. [90] and STAR Fit I. The freeze-out line from Andronic et al. is obtained from an interpolation of the freeze-out data, the grey squares in Figure 11.
In the right panel of Figure 12 we show the same observables for the freeze-out line of STAR Fit II. As discussed in Section IV C, we have singled out the results for this freeze-out curve as the best-informed computation.
In both panels of Figure 12 we also show the experimental measurement of cumulants of the net-proton distributions in the beam energy scan experiments from the STAR collaboration. The fourth-order fluctuations, R p 42 , of the net-proton multiplicity distributions are measured in Au+Au collisions with centrality 0-5%, transverse momentum range 0.4 < p T (GeV/c) < 2.0, and rapidity |y| < 0.5, cf. [22] for more details. Moreover, results for the sixth-order cumulant of the net-proton distribution, R p 62 , are also presented in the middle plot of Figure 12 out curves considered are compatible with the respective experimental measurement of the κσ 2 of net-proton distributions in 0-5% central Au+Au collisions. In particular, the theoretical results feature a non-monotonic √ s NN -dependence: R B 42 first decreases with decreasing beam energy and then increases. The details of this behaviour, in particular how pronounced it is, is highly sensitive to the precise location of the freeze-out. For example, the increase at small √ s NN is larger for smaller freeze-out temperatures. Thus, the weak increase for STAR Fit 1 originates in the slightly larger freeze-out temperature of this freeze-out fit. This shows that even small variations in the freeze-out temperature have a substantial effect on the fluctuations in this regime. The underlying reason is that the freeze-out happens in or close to the crossover region, where the fluctuations vary significantly, see Figure 6. Importantly, this regime cannot be accessed within the extrapolation of the Taylor expansion at least within the current order. This entails that extrapolations based on a Taylor expansion are bound to fail to describe the data in this regime reliably. Consequently this calls for qualitatively improved direct theoretical computations at small beamenergies. This is work in progress and we hope to report on the respective results soon.
Our results for the hyper-order fluctuations R GeV. Another interesting property of the current LEFT setting is that the non-monotonic behaviour of our results for R B n2 at large µ B in Figure 12 does not arise from critical physics: in the LEFT used here, the CEP is at significantly larger µ B 700 MeV. Moreover, it is well established that the critical region is only very small. It is already small within mean-field approximations of lowenergy effective theories, and additionally shrinks considerably if quantum, thermal and density equilibrium fluctuations are taken into account, see [29]. Moreover, this does not change if transport processes are taken into account, see [114].
In the present LEFT the increasing trend at large µ B region originates from two effects: First, fluctuations are enhanced since the chiral crossover becomes sharper with increasing µ B . This leads to a stronger non-monotonic behaviour of R B 42 as a function of T , see Figure 6. This sharpening is also present in the vicinity of a CEP. Second, the freeze-out temperature is shifted away from the pseudo-critical temperature towards small [90] (grey) and the STAR experiment [91] (red). Right panels: The freeze-out curve, STAR Fit II, is obtained from the freeze-out parameters of the STAR experiment [91]. The theoretical error bands show a highly correlated error, and should be interpreted as a family of curves with the same qualitative behaviour as the central curve. For more explanations see Section IV C with Figure 11. STAR data: R p 42 (top) is the kurtosis of the net-proton distributions measured in Au+Au central (0-5%) collisions [22]. R p
62
(middle) is the result on the six-order cumulant of the net-proton distribution at √ sNN=200 GeV, 54.4 GeV and 27 GeV with centrality 0-10% [94].
beam-energies, thereby probing different regimes of the cumulants. However, it should be noted that the uncertainty of our results increases significantly in the low energy region. These uncertainties include an estimate for the systematic error of the QCD-assisted LEFTapproach. This systematic error stems from the uncertainty in the matching of the in-medium scales of the LEFT and QCD, encoded in the coefficients c T and c µ in Equation (4). Moreover, for larger chemical potential the current LEFT lacks the back-reaction of the µ B -dependence of the glue-dynamics. While inherently small, it might still play a rôle. Furthermore, it is found that R B 42 can also be suppressed in the regime of low collision energies due to the effect of global baryon number conservation, cf. [107,108,115] which will be included in our future work.
E. Search for the CEP
The analysis in the previous sections entails that in the present QCD-assisted LEFT the non-monotonic behaviour of baryon-number fluctuations is triggered by the sharpening of the chiral crossover. This is highly nontrivial, since it is evident, e.g., from Figure 6 that neither a system only in the hadronic-nor only in the QGP phase could produce the beam-energy dependencies shown in FIG. 13. Baryon number fluctuation R B 32 as a function of the collision energy in comparison to STAR-data for R p 32 (0-5%) centrality [22]. Left panel: the freeze-out points are those from Andronic et al. [90] (grey) and the STAR experiment [91] (red). Right panel: The freeze-out curve, STAR Fit II, is obtained from the freeze-out parameters of the STAR experiment [91]. The theoretical error bands show a highly correlated error, and should be interpreted as a family of curves with the same qualitative behaviour as the central curve. For more explanations see Section IV C with Figure 11. In conclusion, the agreement of R B 42 , computed in the QCD-assisted LEFT and the measured beam-energy dependence of R p 42 shows, that the latter could be a signature for the presence of a sharpening crossover between these two phases. Whether or not it also signals the onset of the critical region will be subject of a future improved study. In the context of this latter study we also emphasise that the non-universal properties of the LEFT such as the existence and location of the CEP may not quantitatively agree with QCD as the latter regime lies outside the LEFT-regime with quantitative reliability. Still, the present LEFT probably has the same qualitative nonuniversal properties at large chemical potential, and it certainly has the same universal ones.
A non-monotonic energy dependence for the fluctuations is a highly relevant experimental observation, since this behaviour has been proposed as an experimental signature of a CEP [14,16]. The present analysis based on QCD-assisted LEFT model demonstrates that the nonmonotonic behaviour of fluctuations can serve as an indication of a CEP, but is not necessarily a smoking gun signature for it. The latter requires the extraction of critical scaling, or similar definite signatures such as the detection of a first order regime for large µ B , etc. . Still, the non-monotonic behaviour observed in both theory and experiments is a clear signature for interesting strongly correlated physics, whose uncovering requires joint and intensified effort of both, theory and experiment. Of course, whether or not these properties carry over completely to QCD remains to be seen.
Note also that the non-monotonic regime is far away from that covered by a simple extrapolation of the Taylor expansion at µ B = 0. It might be covered by a resummation of the latter, which can already be investigated within the present QCD-assisted LEFT. Constraints on such a resummation should also make use of odd hyper-order fluctuations at finite chemical potential, that are readily computed in the present setup: A prominent and relevant example is R 32 , already measured in the STAR experiment. In Figure 13 we show our predictions for R B 32 computed in the current QCDassisted LEFT on the different freeze-out curves defined in Section IV C, see in particular Figure 11. In this section it also has been argued, that our best-informed freeze-out curve is given by STAR Fit II. The respective results are shown in the right panel of Figure 13 in comparison with the STAR data for R p 32 (0-5% centrality). Indeed, these results show the best compatibility with the experimental data. Moreover, within the respective systematic and statistical errors the theoretical results with the freeze-out curve STAR Fit II and the experimental data agree down to collision energies of √ s NN ≈ 14.5 GeV or µ B ≈ 250 MeV.
Interestingly, below √ s NN ≈ 14.5 GeV the experimental data show a plateau, which is not present in the theoretical prediction. While this is in the large chemical potential regime, in which the LEFT gradually looses its predictive power, also the respective functional first principles QCD computation in [7,8,27,28], based on a grand potential, do not show any sign of new physics in this regime. This suggests that for √ s NN 14.5 GeV at least one of the implicit assumptions underlying the identification of R B nm with R p nm within a grand canonical ensemble with variable baryon charge (density) for given beam energies breaks down. As discussed before, this asks for a re-assessment of the identification The freeze-out curve, STAR Fit II, is obtained from the freezeout parameters of the STAR experiment [91]. The theoretical error bands show a highly correlated error, and should be interpreted as a family of curves with the same qualitative behaviour as the central curve. For more explanations see Section IV C with Figure 11. of baryon and proton number fluctuations, finite volume effects and finite volume fluctuations, the determination of the freeze-out curve for smaller collision energies, the evaluation of non-equilibrium effects such as transport, and finally the use of the grand potential in the theory computations. While highly relevant and interesting, this goes far beyond the scope of the present work and we defer any further investigation to future work.
The above example of R 32 demonstrates very impressively, that the odd (hyper-order) fluctuation observables encode highly relevant physics information which may be difficult or even impossible to extract from the even or-ders. As a first step in this direction, finally aiming at a resummation of the µ B -expansion that allows us to go beyond the validity regime of the Taylor expansion, we also have computed the fluctuation observables R 31 , R 51 , R 71 on the freeze-out curve STAR Fit II in Figure 14. An experimental confirmation of the respective predictions at least for the lower orders would be highly desirable.
The discussion in this section leaves us with the highly exciting possibility of unravelling the location and properties of a potential CEP within a combined experimenttheory analysis: First principle QCD at finite density should provide us with a prediction for the location of the CEP in terms of hyper-order fluctuations, Loc CEP (R nm ). This would allow us to use the experimental data on hyper-order fluctuation observables R p nm as input. We emphasise that this prediction does not necessitate the observation of critical behaviour in the R nm , but utilises the details of the non-monotonicity of the R nm .
In summary, such an analysis does explicitly not rely on the universal property of critical scaling measured in the R nm . Indeed, it uses the non-universal properties of the R nm to predict the non-universal location of the CEP, and hence is far more robust. We hope to report on this matter in the near future.
V. CONCLUSIONS
In this work we have computed baryon number fluctuations up to tenth order with a QCD-assisted lowenergy effective theory. This LEFT incorporates quantum, thermal and density fluctuations from momentum scales less than 700 MeV within the functional renormalisation group approach, and is embedded in QCD, for details see Section II. The quantitative predictability has been benchmarked with a comparison of baryon number fluctuations at µ B = 0 up to the eighth order from the lattice, see Section IV A. Our results are in quantitative agreement with that from the Wuppertal-Budapest collaboration, and are compatible with that of the HotQCD collaboration, as shown in Figure 4.
Our direct computation at finite µ B , presented in Section IV B, has allowed us to assess the range of validity of the Taylor expansion of the free energy of QCD around µ B = 0. Such an expansion is commonly used to extrapolate lattice results to finite density. We have shown that the expansion up to tenth order in µ B /T is only valid for µ B /T 1.5 in the chiral crossover regime, see Figure 8. Beyond this range, the Taylor expansion, at least to this order, fails to even capture the qualitative behaviour of the fourth-and sixth-order baryon number fluctuations. Thus, results for fluctuations at the freezeout curve based on a Taylor expansion around µ B = 0 should be interpreted with great caution for µ B /T 1.5, as relevant physical effects might not be captured by this extrapolation.
The main goal of the current work was the computation of baryon number fluctuations and in particular hyper-order fluctuations along the freeze-out curve at collision energies √ s 7.7 GeV. The respective results are discussed in Section IV D. They have been compared to experimental data of net-proton number cumulants from STAR for different estimates for the freeze-out curve, see Figure 12. Our result for the kurtosis, R B 42 , is in good agreement with the experimental data for collision energies √ s 7.7 GeV. In particular the increasing trend at lower beam energies √ s 19.6 GeV is captured well.
This non-monotonicity is also present in the hyper-order fluctuations R B 62 , R B 82 . We also note that a comprehensive comparison for the higher order cumulants is not possible due to the lack of experimental data. Accordingly, our results in Figure 12 are predictions that await experimental verification.
We have also investigated the twofold origin of the non-monotonicity for √ s 19.6 GeV in the present LEFT: First, for increasing chemical potential the chiral crossover gets sharper. Secondly, for smaller beam energies the freeze-out temperature may move away from the pseudo-critical temperatures. In the current setup both phenomena happen far away from a potential critical end point in the LEFT located at √ s CEP 3 GeV. The latter regime is also safely outside the reliability regime of the current setup, which gradually looses reliability for In summary, the non-monotonicities of hyper-order fluctuations, observed both in experiment and theory, are important signatures for interesting physics in the border regime between quark-gluon plasma and the hadron phase. This of course can include a potential CEP, and in any case deserves further investigation from both exper-iment and theory. In particular, we envisage that experimental data of fluctuation observables and their dependence on collision energy allow us to constrain the onset regime of this strongly correlated physics/CEP. Importantly, such a prediction does not rely on the observation of critical scaling in the hyper-order fluctuations, but is far more robust, for more details see Section IV E. We hope to report on this in the near future. The work is also supported by EMMI, and the BMBF grant 05P18VHFCA. It is part of and supported by the DFG Collaborative Research Centre SFB 1225 (ISOQUANT) and the DFG under Germany's Excellence Strategy EXC -2181/1 -390900948 (the Heidelberg Excellence Cluster STRUCTURES).
Appendix A: The fRG-approach to QCD & LEFTs
The functional renormalisation group or flow equation for QCD provides the evolution of its effective action Γ k with an infrared cutoff scale k. Here we use the setup with dynamical hadronisation, [8,[47][48][49][50]116]. The formulation used here has been developed in [8,[43][44][45][46]. Its current form has been described and further developed in [8], and for further details we refer to this work. The flow equation of the QCD effective action reads,
∂ t Γ k [Φ] = 1 2 Tr G AA,k ∂ t R A,k − Tr G cc,k ∂ t R c,k − Tr G qq,k ∂ t R q,k + 1 2 Tr G φφ,k ∂ t R φ,k .(A1)
In Equation (A1), the Φ = (A, c,c, q,q, φ) is a superfield that comprises all fields. This also includes hadronic (composite) low energy degrees of freedom introduced by dynamical hadronisation. The G's and R's are the propagators and regulators of the different fields, respectively. Diagrammatically it is depicted in Figure 1. For more works on QCD-flows at finite temperature and density see [8, 43-46, 85, 116-121], for reviews on QCD and LEFTs for QCD see [49,51,[122][123][124][125][126][127][128].
For scales k 1 GeV, the gluon decouples from the system due to its confinement-related mass gap. For these momentum scales, the (off-shell) dynamics of QCD is dominated by quarks and the emergent composite hadronic degrees of freedom. In particular, the lowest lying meson multiplet, and specifically the π meson is driving the dynamics. The pion is the pseudo-Goldstone boson of strong chiral symmetry breaking, and hence is the lightest hadron with a mass ∼ 140 MeV in the vacuum.
Consequently, in this regime with k 1 GeV, the flow equation of the QCD effective action in Equation (A1) is reduced to,
∂ t Γ k [Φ] = − Tr G qq,k ∂ t R q,k + 1 2 Tr G φφ,k ∂ t R φ,k ,(A2)
where R q,k and R φ,k are the regulators for the quark and meson fields, respectively. The full propagators in Equation (A2) read,
G qq/φφ,k = 1 Γ (2) k [Φ] + R k qq/φφ ,(A3)
with Γ
(2)
k [Φ] = δ 2 Γ k [Φ]/δΦ 2 .
In this work we employ 3d-flat or Litim regulators [129][130][131],
R φ,k (q 0 , q) = Z φ,k q 2 r B (q 2 /k 2 ) , R q,k (q 0 , q) = Z q,k iγ · qr F (q 2 /k 2 ) ,(A4)
with
r B (x) = 1 x − 1 Θ(1 − x) , r F (x) = 1 √ x − 1 Θ(1 − x) ,(A5)
where Θ(x) denotes the Heaviside step function. Inserting the effective action Equation (1) into the flow equation Equation (A2), we arrive at
∂ t V mat,k (ρ) = k 4 4π 2 N 2 f − 1 l (B,4) 0 (m 2 π,k , η φ,k ; T ) + l (B,4) 0 (m 2 σ,k , η φ,k ; T ) − 4N c N f l (F,4) 0 (m 2 q,k , η q,k ; T, µ) , (A6)
where the threshold functions l (B/F,4) 0 as well as other threshold functions used in the following can be found in e.g., [8,55]. The dimensionless renormalised quark and meson masses read,
m 2 q,k = h 2 k ρ 2k 2 Z 2 q,k ,m 2 π,k = V mat,k (ρ) k 2 Z φ,k ,(A7)m 2 σ,k = V mat,k (ρ) + 2ρV mat,k (ρ) k 2 Z φ,k .(A8)
The anomalous dimensions for the quark and meson fields in Equation (A6) are defined as
η q,k = − ∂ t Z q,k Z q,k , η φ,k = − ∂ t Z φ,k Z φ,k ,(A9)
respectively. Accordingly, the flow equation for the mesonic anomalous dimension is obtained from the (spatial) momentum derivative w.r.t. p 2 of the pion twopoint function, to wit,
η φ,k = − 1 3Z φ,k δ ij ∂ ∂p 2 δ 2 ∂ t Γ k δπ i (−p)δπ j (p) p0=0 p=0 .
(A10)
The approximation Equation (1) to the effective action together with Equation (A10) are based on two approximations: Firstly, in Equation (1) we have dropped the field-dependence of Z φ , which would lead to different Z π and Z σ . In Equation (A10) we have identified Z φ = Z π , and hence also Z σ = Z π . This is motivated by the fact that the meson dynamics are only dominant in the broken regime where the three pions are far lighter than the single sigma mode, which quickly decouples. Hence, the three pions drive the dynamics. Furthermore, in Equation (1) we do not distinguish between spatial and temporal components of Z φ . For finite temperature and density, the Euclidean O(4) rotation symmetry is broken, as the heat bath of density singles out a rest frame. This entails, that η φ,k splits into η ⊥ φ,k and η φ,k , the components transverse and longitudinal to the heat bath/density. We have used the approximation η φ,k = η ⊥ φ,k as we have three spatial directions. The influence of the splitting of η φ,k on the thermodynamics and baryon number fluctuations has been investigated in detail e.g. in [55]. There it has been found that the impact is small, supporting the reliability of the present approximation.
Similarly, the quark anomalous dimension is obtained by projecting the relevant flow onto the vector channel of the 1PI quark-anti-quark correlation function,
η q = 1 4Z q,k × Re ∂ ∂p 2 tr iγ · p − δ 2 δq(p)δq(p) ∂ t Γ k p0,ex p=0 .(A11)
In Equation (A11), the spatial momentum is set to zero, p = 0 as in the mesonic case: vanishing momentum is most relevant to the flow of effective potential in Equation (A6). Note, that the lowest fermionic Matsubara frequency is non-vanishing. We use p 0,ex = 0, its value is further described in Appendix B, based on [40,42]. As is implicit in Equation (A11), the flow of the quark two-point function is complex-valued at non-vanishing chemical potential. This originates in the Silver-Blaze property of QCD at T = 0. For quark correlation functions this entails that they are functions of p 0 − iµ q already before the onset of the baryon density, for a discussion in the present fRG-approach see [40,42,132]. In turn, the couplings are still real (i.e. real functions of the complex variable p 0 −iµ q ) in particular below the density onset. Hence, couplings (i.e. expansion coefficients in a Taylor expansion in momenta) are real. This is readily seen in a resummation of the external frequency of the quark propagator [42]. Without resummation they are obtained from a projection on the real part of the flow, see Equation (A11).
This projection is also used for the Yukawa coupling. Within the present approximation, the flow equation of the (real) Yukawa coupling is given by,
∂ t h k = 1 2σ Re tr − δ 2 δq(p)δq(p) ∂ t Γ k p0,ex p=0 . (A12)
The explicit expressions for the meson and quark anomalous dimensions, as well as the flow of the Yukawa coupling can be found in Appendix B.
Appendix B: Flow equations for V k (ρ), h k , and η φ,q
The flow equation for the effective potential is given in Equation (A6). To resolve its field dependence, we use a Taylor expansion about a k-dependent ρ-value κ k , V mat,k (ρ) = Nv n=0 λ n,k n! (ρ − κ k ) n ,
with the running expansion coefficients λ n,k . Here, N v is the maximal order of Taylor expansion included in the numerics. N v = 5 is adopted in this work, which is large enough to guarantee the convergence of expansion, for more details, see e.g., [55,133]. It is more convenient to rewrite Equation (B1) by means of the renormalised variables, i.e.,
V mat,k (ρ) = Nv n=0λ n,k n! (ρ −κ k ) n ,(B2)
withV mat,k (ρ) = V mat,k (ρ),ρ = Z φ,k ρ,κ k = Z φ,k κ k , and λ n,k = λ n,k /(Z φ,k ) n . Inserting Equation (B2) into the l.h.s. of Equation (A6) leads us to, ∂ n ρ ∂ t ρV mat,k (ρ) ρ=κ k =(∂ t − nη φ,k )λ n,k − (∂ tκk + η φ,kκk )λ n+1,k .
In the present work, we use the EoM of ρ as our expansion point. With Equation (1) this yields,
∂ ∂ρ V mat,k (ρ) −c kσ ρ=κ k = 0 ,(B4)
withσ = Z 1/2 φ,k σ andc k = Z −1/2 φ,k c, with a cutoffindependent c. Another commonly used expansion point is a fixed expansion point, ∂ t κ k = 0. For further details on these two different expansion approaches, and their respective convergence properties see [40,44,55,[133][134][135].
From Equation (B3) and Equation (B4) we get the flow equation for the expansion point,
∂ tκk = −c 2 k λ 3 1,k +c 2 kλ 2,k ∂ρ ∂ t ρV mat,k (ρ) ρ=κ k + η φ,k λ 1,k 2 +κ kλ2,k .(B5)
The meson anomalous dimension in Equation (A10) reads, η φ,k = 1 6π 2 4 k 2κ k (V k (κ k )) 2 BB (2,2) (m 2 π,k ,m 2 σ,k ; T )
+ N ch 2 k F (2) (m 2 q,k ; T, µ)(2η q,k − 3) − 4(η q,k − 2)F (3) (m 2 q,k ; T, µ) ,(B6)
The quark anomalous dimension in Equation (A11) reads, (1,2) (m 2 q,k ,m 2 π,k ; T, µ, p 0,ex ) + FB (1,2) (m 2 q,k ,m 2 σ,k ; T, µ, p 0,ex ) .
η q,k = 1 24π 2 N f (4 − η φ,k )h 2 k × (N 2 f − 1)FB
In the threshold function FB's we have employed p 0,ex = πT for the finite temperature sector and p 0,ex = πT exp{−k/(πT )} for the vacuum sector. This choice guarantees a consistent temperature dependence for all k, which is particularly relevant for the thermodynamics in the low temperature region [40]. This can be resolved by means of a full frequency summation of the quark external leg [42], and the present procedure mimics this physical behaviour. The flow of the Yukawa coupling in Equation (A12) reads,
∂ thk = 1 2 η φ,k + η q,k h k (ρ) +h 3 k 4π 2 N f L(4)
(1,1) (m 2 q,k ,m 2 σ,k , η q,k , η φ,k ; T, µ, p 0,ex )
− (N 2 f − 1)L
(1,1) (m 2 q,k ,m 2 π,k , η q,k , η φ,k ; T, µ, p 0,ex ) . The parameters are fixed with the pion pole mass m π,pol , the mass of the sigma resonance, mσ, and the pion decay constant, fπ ≈σEoM, the expectation value of the sigma-field. The input parameters are those of the initial effective potential VΛ: the meson self-coupling λΛ and the meson mass parameter νΛ.
The pion mass is tuned by cσ, the parameter of explicit chiral symmetry breaking. Finally, the constituent quark mass is fixed via the initial value for the Yukawa coupling, hΛ.
Explicit expressions of all the threshold functions mentioned above, such as BB, F's, FB's, and L can be found in e.g., [8,55]. In summary, the flow equations Equation (A6), Equation (B3), Equation (B5), Equation (B8), supplemented with Equation (B6) and Equation (B7), constitute a closed set of ordinary differential equations, which is evolved from the UV cutoff k = Λ to the IR limit k = 0.
Appendix C: Initial conditions
To solve the flow equation, we need to specify initial conditions. To this end, we choose initial values at a scale k = Λ such that known observables of QCD in the vacuum at k = 0, such as the pion mass and decay constant, are reproduced. The effective potential at the UV cutoff reads,
V mat,Λ (ρ) = λ Λ 2 ρ 2 + ν Λ ρ .(C1)
We initialise the flows at Λ = 700 MeV. The input parameters of the QCD-assisted LEFT are given in Table I.
Appendix D: Glue potential
The dynamics of the glue sector in QCD is partly imprinted in the glue potential V glue,k (A 0 ), see Equation (2). This has been discussed in Section II. This potential is only needed for the determination of the expectation value of the Polyakov loop. Its inherent glue correlation functions are gapped and their backcoupling is suppressed for k 1 GeV. Accordingly, we can simply drop the scale dependence of the glue potential for the present purposes. This leads us to,
where P on the r.h.s. stands for path ordering.
In this work we adopt the parametrisation of the glue potential in [136], which reads Both, the parametrisation of glue potential in Equation (D4), as well as determination of relevant parameters in Table II, is based on lattice results of SU(3) Yang-Mills theory at finite temperature. This potential does not only reproduce the lattice expectation value of the Polyakov loop and the pressure, but also the correct quadratic fluctuations of the Polyakov loop, [136]. These fluctuations, and higher ones, are important for the fluctuation observables discussed here [40,42]. The coefficients in Equation (D4) are temperature-dependent,
x(T ) = x 1 + x 2 /(t r + 1) + x 3 /(t r + 1) 2 1 + x 4 /(t r + 1) + x 5 /(t r + 1) 2 ,
with x = a, c, d, and In Equation (D6) and Equation (D7) we have used the reduced temperature t r = (T − T c )/T c . The parameter values are taken from [136], and are collected in Table II for convenience.
b(T ) = b 1 (t r + 1) −b4 1 − e b2/(tr+1) b 3 .(D7
The parameters in Table II are that of the glue potential in Yang-Mills theory. It has been argued and shown in [52,53,137] that unquenching effects in QCD are well captured by a linear rescaling of the reduced temperature in the regime about T c , very similar to the rescaling discussed in Section II B 2. This leads us to,
(t r ) YM → α (t r ) glue ,(D8)
FIG. 1 .
1Diagrammatic representation of the QCD flow equation. The lines stand for the full propagators of gluon, ghost, quark, and mesons, respectively. The arrows in quark and meson lines indicate the quark number (baryon number) flow. The crossed circles represent the infrared regulators.
FIG. 2 .
2Phase boundaries of 2-and 2+1-flavour QCD from [8] with a 2+1-flavour scale-matching of the 2-flavour data at the crossover temperature and µB = 0. The bands denote the width of the chiral crossover. The scale-matched 2-flavour phase boundary agrees quantitatively with the genuine 2+1flavour one including the location of the critical end point. The dashed line at µB/T = 4 constitutes the reliability bound of the computations in [8] based on the potential emergence of new degrees of freedom discussed in [7, 8, 25]. The dashed lines at µB/T = 2, 3 are reliability estimates of lattice results as well as old ones from functional approaches. take over the above QCD-relations for the present LEFT, which lacks the backcoupling of the glue-dynamics on both large temperature and chemical potential physics. Still, the dominance of the leading order term −κµ 2 B /T 2 c in the model reflects the same property in QCD. This allows us to employ a respective linear scale-matching for µ B /T 4 as studied in the present work.
FIG. 3 .
3For the scale-matching of µ B with the curvature −κµ 2 B /T 2 c we have a plethora of results from state of Matching of the temperature-scale in the QCDassisted 2-flavour LEFT with R B 42 in Equation
FIG. 4. R B 42 = χ B 4 /χ B 2 (left panel), R B 62 = χ B 6 /χ B 2 (middle panel), and R B 82 = χ B 8 /χ B 2 (right panel) as functions of the temperature at vanishing baryon chemical potential (µB = 0). Results from the QCD-assisted LEFT are compared with lattice results from the HotQCD collaboration [9, 36, 37] and the Wuppertal-Budapest collaboration (WB) [38]. The inset in the plot of R B 82 shows its zoomed-out view. Our results agree quantitatively with the WB-results , and are qualitatively compatible with the HotQCD results. We also compare to the predictions of a hadron resonance gas (HRG) [30], which predicts only a very mild increase of R B n2 from unity with increasing T .
FIG
. 5. R B 10 2 = χ B 10 /χ B 2 as a function of the temperature with µB = 0 from the QCD-assisted LEFT.A. Hyper-order baryon number fluctuations at vanishing density: benchmarks and predictions
FIG. 6 .
6R B 42 (left panel), R B 62 (middle panel), and R B 82 (right panel) as functions of the temperature at several values of µB. Insets in each plot show their respective zoomed-out view.
FIG. 7 .
7at the eighth order,μ 8 B , and employing Equation (13), we ob-LEFT T = 155 MeV fRG-LEFT T = 160 MeV HotQCD T = 155 MeV HotQCD T = 160 MeV WB T = 155 MeV WB T = 160 MeV 0.00.10.20.30.40.50.60.R B 42 (left panel) and R B 62 (right panel) as functions of µB/T with T = 155 MeV and T = 160 MeV from the eighthorder Taylor expansion in (µB/T ) 2 around vanishing µB shown in Equation
Figure 4 .
4As expected, the LEFT-results for the µ B -dependence of R B 42 and R B 62 agrees qualitatively with both lattice results. Moreover, it agrees quantitatively with the Wuppertal-Budapest result. Note that constraints, e.g., strangeness neutrality or a fixed ratio of the electric charge to the baryon number density, are not implemented in all the results in Figure 7.
agrees quantitatively with that from the full calculation for µ B /T 1.2 for T = 155 MeV and µ B /T 1.5 for T = 160 MeV. Not surprisingly, this reliability regime is reduced significantly for the hyperorder fluctuation R B 62 to µ B /T 1.2 for T = 160 MeV and µ B /T 0.8 for T = 155 MeV. For larger µ B /T , the fluctuations show a non-trivial oscillatory behaviour that cannot be captured by a (low-order) Taylor expansion.
FIG. 8 .
8. Interestingly, this does not change the compatibility regime for T = 160 MeV significantly. In turn, there are significant changes for LEFT full T = 155 MeV fRG-LEFT full T = 160 MeV fRG-LEFT expansion T = 155 MeV fRG-LEFT expansion T LEFT full T = 155 MeV fRG-LEFT full T = 160 MeV fRG-LEFT expansion T = 155 MeV fRG-LEFT expansion T Comparison between the direct calculation of baryon number fluctuations R B 42 (upper panels) and R B 62 (lower panels) via Equation (13) at finite µB and the Taylor expansion up to χ B 8 (0) in Equation (24) (left panels) and to χ B 10 (0) (right panels). Both calculations are performed within the QCD-assisted LEFT used in the present work. R B 42 , R B 62 are shown as functions of µB/T with T = 155 MeV and T = 160 MeV. T = 155 MeV. While the deviations for R B 42 start to grow at roughly the same µ B /T as for the eights-order expansion, that is µ B /T 1.2, the result for R B 42 stays compatible with the full result for larger values. For R B 62 the reliability regime is essentially doubled: it rises from µ B /T 0.8 to µ B /T 1.5. The analysis above suggests the following picture: We have a temperature-dependent reliability range of the Taylor expansion, T = 155 MeV : [µ B /T ] Max ≈ 1.5 , T = 160 MeV : [µ B /T ] Max ≈ 1.2 . (25) Moreover, the results have already converged for R B 42 , R B 62 for T = 160 MeV as well as for R B 42 for T = 155 MeV within the eighth order and for µ B /T [µ B /T ] Max (T ). In turn, convergence for R B 62 for µ B /T [µ B /T ] Max requires the tenth order Taylor expansion for T = 155 MeV. Note that the values of [µ B /T ] Max in Equation
FIG. 9 .
9R B 31 (left panel), R B 51 (middle panel), and R B 71 (right panel) as functions of the temperature at several values of µB. Insets in each plot show their respective zoomed-out view.
µB = 160 MeV µB = 200 MeV µB = 300 MeV µB = 400 MeV FIG. 10. Baryon number fluctuation R B 32 as a function of the temperature at several values of µB. D. Hyper-order baryon number fluctuations on the freeze-out curve
FIG. 11 .
11, which are obtained at three values of the collision energy, i.e., √ s N N =200 GeV, 54.4 GeV and 27 GeV with centrality 0-10% [94]. The theoretical results for the fourth-order fluctuations R B 42 from the present QCD-assisted LEFT for all freeze-Left panel: chemical freeze-out temperature and baryon chemical potential in the T −µB plane. The blue pentagons and red circles show the freeze-out data from Andronic et al. [90] and STAR experiment [91], respectively. The blue dashed line represents the parametrisation of blue pentagons through Equation (26a) and Equation (26b). The red solid and green dotted lines show the parametrisation of the STAR data based on all the seven data points, and only the four data points in the middle region (100 MeV µB 300 MeV), respectively. The grey squares are obtained by interpolating the blue pentagons. The inlay zooms in the low-µB region. Right panel: Baryon number fluctuations R B 42 in the T −µB plane. The freeze-out curve is the STAR Fit II. The dashed line at µB/T = 4 constitutes the reliability bound of the computations in [8] based on the potential emergence of new degrees of freedom discussed in [7, 8, 25]. The dashed lines at µB/T = 2, 3 are reliability estimates of lattice results as well as old ones from functional approaches, see also Figure 2.
B 62 and R B 82 are shown in the middle and bottom panel of Figure 12. For small chemical potentials or large collision energies both fluctuation observables are negative. Moreover, R B 62 decreases with decreasing √ s NN , while R B 82 increases. The occurrence of non-monotonicities of R B 62 and R B 82 at lower beam energies cannot be shown within the accuracy limits of the current study. For √ s NN =200 GeV, 54.4 GeV and 27 GeV we can compare our results for R B 62 to STAR data within 0-10% centrality [94]. One observes that our results are in agreement with the experimental data within errors at √ s NN =200 GeV and 54.4 GeV, though the central value of STAR data at √ s NN =54.4 GeV is positive. Both the theory and experiment show negative values for the sixth-order fluctuations at the collision energy √ s NN =27
FIG
. 12. QCD-assisted LEFT (fRG-LEFT): Baryon number fluctuations R B 42 (top), R B 62 (middle), and R B 82 (bottom) as functions of the collision energy. Left panels: the freeze-out points are those from Andronic et al.
Figure 12 .
12Figure 12. In conclusion, the agreement of R B 42 , computed in the QCD-assisted LEFT and the measured beam-energy dependence of R p 42 shows, that the latter could be a signature for the presence of a sharpening crossover between these two phases. Whether or not it also signals the onset of the critical region will be subject of a future improved study. In the context of this latter study we also emphasise that the non-universal properties of the LEFT such as the existence and location of the CEP may not quantitatively agree with QCD as the latter regime lies outside the LEFT-regime with quantitative reliability. Still, the present LEFT probably has the same qualitative nonuniversal properties at large chemical potential, and it certainly has the same universal ones. A non-monotonic energy dependence for the fluctuations is a highly relevant experimental observation, since this behaviour has been proposed as an experimental signature of a CEP [14, 16]. The present analysis based on QCD-assisted LEFT model demonstrates that the nonmonotonic behaviour of fluctuations can serve as an indication of a CEP, but is not necessarily a smoking gun signature for it. The latter requires the extraction of critical scaling, or similar definite signatures such as the detection of a first order regime for large µ B , etc. . Still, the non-monotonic behaviour observed in both theory and experiments is a clear signature for interesting strongly correlated physics, whose uncovering requires joint and intensified effort of both, theory and experiment. Of course, whether or not these properties carry over completely to QCD remains to be seen. Note also that the non-monotonic regime is far away from that covered by a simple extrapolation of the Taylor expansion at µ B = 0. It might be covered by
FIG. 14 .
14Baryon number fluctuations R B 31 (top), R B 51 (middle), and R B 71 (bottom) as functions of the collision energy.
√ s 27 GeV. However, its qualitative features may well be present in QCD. The current LEFT-results and its upgrades towards first principle QCD can be compared with the future experimental results of the high statistic data taken from the second phase of RHIC beam energy scan (BES-II, 2019-2021). From the year of 2018 to 2020, the STAR experiment has collected high statistics data of Au+Au collisions at √ s NN = 9.2, 11.5, 14.6, 19.6 and 27 GeV in the collider mode, and √ s NN = 3.0 -7.7 GeV in the fixed target mode. These data give us access to the QCD phase structure for baryon chemical potential up to µ B ≈ 720 MeV. Related further steps in a comprehensive understanding of the physics of fluctuations in a heavy ion collision has been undertaken in Section IV E, where we have presented results for odd fluctuations observables R B 31 , R B 51 , R B 71 as well as R B 32 . While the former observables have not been measured yet, R B 32 agrees quantitatively within the systematic and statistical error with the STAR measurement for R p 32 for collision energies √ s NN 14.5 GeV. For smaller energies, the experimental data show a plateau, that may indicate the loss of one or several of the underlying assumption in the identification of theoretical equilibrium computations of baryon number fluctuations R B nm in a grand canonical ensemble with the experimental results for proton number fluctuations on the freeze-out curve.
ACKNOWLEDGMENTS
We thank Jens Braun, Rob Pisarski, Bernd-Jochen Schaefer and Nu Xu for discussions. The work was supported by the National Key Research and Development Program of China (Grant No. 2020YFE0202002 and 2018YFE0205201) and the National Natural Science Foundation of China under (Grant No. 11775041, 11828501, 11890711 and 11861131009).
. Observables and related initial values for the LEFT couplings at the initial cutoff scale Λ = 700 MeV.
glue . Its dependence on the temporal gluon background field, A 0 , is encoded in the traced Polyakov loop L[A 0 ] and its conjugateL[A 0 ],
+L 3 ) + d(T )(LL) 2 , (D4) with the SU(N c ) Haar measure M H (L,L) = 1 − 6LL + 4(L 3 +L 3 ) − 3(LL) 2 . (D5)
present work we have used α = 0.75 and T glue c = 213 MeV.
) TABLE II. Values of the parameters for the glue potential in Equation (D6) and Equation (D7).1
2
3
4
5
ai -44.14
151.4 -90.0677 2.77173 3.56403
bi -0.32665 -82.9823
3.0
5.85559
ci -50.7961 114.038 -89.4596 3.08718 6.72812
di 27.0885 -56.0859 71.2225 2.9715 6.61433
QCD phase diagram: An Overview. M A Stephanov, arXiv:hep-lat/0701002Proceedings, 24th International Symposium on Lattice Field Theory. 24th International Symposium on Lattice Field TheoryTucson, USA200624hep-latM. A. Stephanov, "QCD phase diagram: An Overview," Proceedings, 24th International Symposium on Lat- tice Field Theory (Lattice 2006): Tucson, USA, July 23-28, 2006, PoS LAT2006, 024 (2006), arXiv:hep- lat/0701002 [hep-lat].
The CBM physics book: Compressed baryonic matter in laboratory experiments. Bengt Friman, Claudia Hohne, Jorn Knoll, Stefan Leupold, Jorgen Randrup, Ralf Rapp, Peter Senger, 10.1007/978-3-642-13293-3Lect. Notes Phys. 814Bengt Friman, Claudia Hohne, Jorn Knoll, Stefan Le- upold, Jorgen Randrup, Ralf Rapp, and Peter Senger, "The CBM physics book: Compressed baryonic mat- ter in laboratory experiments," Lect. Notes Phys. 814, pp.1-980 (2011).
Topical issue on exploring strongly interacting matter at high densities -nica white paper. David Blaschke, Jörg Aichelin, Elena Bratkovskaya, Marek Volker Friese, Jørgen Gazdzicki, Oleg Randrup, Oleg Rogachevsky, Viacheslav Teryaev, Toneev, 10.1140/epja/i2016-16267-xThe European Physical Journal A. 52267David Blaschke, Jörg Aichelin, Elena Bratkovskaya, Volker Friese, Marek Gazdzicki, Jørgen Randrup, Oleg Rogachevsky, Oleg Teryaev, and Viacheslav Toneev, "Topical issue on exploring strongly interacting matter at high densities -nica white paper," The European Physical Journal A 52, 267 (2016).
Search for the QCD Critical Point with Fluctuations of Conserved Quantities in Relativistic Heavy-Ion Collisions at RHIC : An Overview. Xiaofeng Luo, Nu Xu, 10.1007/s41365-017-0257-0arXiv:1701.02105Nucl. Sci. Tech. 28nuclexXiaofeng Luo and Nu Xu, "Search for the QCD Critical Point with Fluctuations of Conserved Quantities in Rel- ativistic Heavy-Ion Collisions at RHIC : An Overview," Nucl. Sci. Tech. 28, 112 (2017), arXiv:1701.02105 [nucl- ex].
. A Dainese, arXiv:1901.04482Physics Beyond Colliders: QCD Working Group Report. QCD Working Grouphep-exA. Dainese et al. (QCD Working Group), "Physics Be- yond Colliders: QCD Working Group Report," (2019), arXiv:1901.04482 [hep-ex].
Mapping the Phases of Quantum Chromodynamics with Beam Energy Scan. Adam Bzdak, Shinichi Esumi, Volker Koch, Jinfeng Liao, Mikhail Stephanov, Nu Xu, 10.1016/j.physrep.2020.01.005arXiv:1906.00936Phys. Rept. 853nucl-thAdam Bzdak, Shinichi Esumi, Volker Koch, Jin- feng Liao, Mikhail Stephanov, and Nu Xu, "Map- ping the Phases of Quantum Chromodynamics with Beam Energy Scan," Phys. Rept. 853, 1-87 (2020), arXiv:1906.00936 [nucl-th].
QCD at finite temperature and chemical potential from Dyson-Schwinger equations. Christian S Fischer, 10.1016/j.ppnp.2019.01.002arXiv:1810.12938Prog. Part. Nucl. Phys. 105hep-phChristian S. Fischer, "QCD at finite temperature and chemical potential from Dyson-Schwinger equa- tions," Prog. Part. Nucl. Phys. 105, 1-60 (2019), arXiv:1810.12938 [hep-ph].
QCD phase structure at finite temperature and density. Wei-Jie Fu, Jan M Pawlowski, Fabian Rennecke, 10.1103/PhysRevD.101.054032arXiv:1909.02991Phys. Rev. D. 10154032hep-phWei-jie Fu, Jan M. Pawlowski, and Fabian Ren- necke, "QCD phase structure at finite temperature and density," Phys. Rev. D 101, 054032 (2020), arXiv:1909.02991 [hep-ph].
Skewness, kurtosis, and the fifth and sixth order cumulants of net baryon-number distributions from lattice QCD confront high-statistics STAR data. A Bazavov, 10.1103/PhysRevD.101.074502arXiv:2001.08530Phys. Rev. D. 10174502hep-latA. Bazavov et al., "Skewness, kurtosis, and the fifth and sixth order cumulants of net baryon-number dis- tributions from lattice QCD confront high-statistics STAR data," Phys. Rev. D 101, 074502 (2020), arXiv:2001.08530 [hep-lat].
QCD Crossover at Finite Chemical Potential from Lattice Simulations. Szabolcs Borsanyi, Zoltan Fodor, Jana N Guenther, Ruben Kara, Sandor D Katz, Paolo Parotto, Attila Pasztor, Claudia Ratti, Kalman K Szabo, 10.1103/PhysRevLett.125.052001arXiv:2002.02821Phys. Rev. Lett. 12552001hep-latSzabolcs Borsanyi, Zoltan Fodor, Jana N. Guenther, Ruben Kara, Sandor D. Katz, Paolo Parotto, Attila Pasztor, Claudia Ratti, and Kalman K. Szabo, "QCD Crossover at Finite Chemical Potential from Lattice Simulations," Phys. Rev. Lett. 125, 052001 (2020), arXiv:2002.02821 [hep-lat].
Conceptual design of the HIRFL-CSR external-target experiment. Liming Lü, Han Yi, Zhigang Xiao, Ming Shao, Song Zhang, Guoqing Xiao, Nu Xu, 10.1007/s11433-016-0342-xSci. China Phys. Mech. Astron. 6012021LiMing Lü, Han Yi, ZhiGang Xiao, Ming Shao, Song Zhang, GuoQing Xiao, and Nu Xu, "Conceptual de- sign of the HIRFL-CSR external-target experiment," Sci. China Phys. Mech. Astron. 60, 012021 (2017).
High Intensity heavy ion Accelerator Facility (HIAF) in China. J C Yang, 10.1016/j.nimb.2013.08.046Proceedings, 16th International Conference on Electromagnetic Isotope Separators and Techniques. 16th International Conference on Electromagnetic Isotope Separators and TechniquesMatsue, Japan317Related to their Applications (EMIS 2012J. C. Yang et al., "High Intensity heavy ion Accelera- tor Facility (HIAF) in China," Proceedings, 16th Inter- national Conference on Electromagnetic Isotope Separa- tors and Techniques Related to their Applications (EMIS 2012): Matsue, Japan, December 2-7, 2012, Nucl. In- strum. Meth. B317, 263-265 (2013).
Studies of extremely dense matter in heavy-ion collisions at J-PARC. H Sako, (j-Parc-Hi) , 10.1016/j.nuclphysa.2018.11.027Nucl. Phys. A. 982H. Sako (J-PARC-HI), "Studies of extremely dense mat- ter in heavy-ion collisions at J-PARC," Nucl. Phys. A 982, 959-962 (2019).
Event-by-event fluctuations in heavy ion collisions and the QCD critical point. Misha A Stephanov, K Rajagopal, Edward V Shuryak, 10.1103/PhysRevD.60.114028arXiv:hep-ph/9903292Phys. Rev. 60114028hep-phMisha A. Stephanov, K. Rajagopal, and Edward V. Shuryak, "Event-by-event fluctuations in heavy ion col- lisions and the QCD critical point," Phys. Rev. D60, 114028 (1999), arXiv:hep-ph/9903292 [hep-ph].
Non-Gaussian fluctuations near the QCD critical point. M A Stephanov, 10.1103/PhysRevLett.102.032301arXiv:0809.3450Phys. Rev. Lett. 10232301hep-phM.A. Stephanov, "Non-Gaussian fluctuations near the QCD critical point," Phys. Rev. Lett. 102, 032301 (2009), arXiv:0809.3450 [hep-ph].
On the sign of kurtosis near the QCD critical point. M A Stephanov, 10.1103/PhysRevLett.107.052301arXiv:1104.1627Phys. Rev. Lett. 10752301hep-phM.A. Stephanov, "On the sign of kurtosis near the QCD critical point," Phys. Rev. Lett. 107, 052301 (2011), arXiv:1104.1627 [hep-ph].
Energy Dependence of Moments of Net-proton Multiplicity Distributions at RHIC. L Adamczyk, STAR10.1103/PhysRevLett.112.032302arXiv:1309.5681Phys. Rev. Lett. 11232302nucl-exL. Adamczyk et al. (STAR), "Energy Dependence of Moments of Net-proton Multiplicity Distributions at RHIC," Phys. Rev. Lett. 112, 032302 (2014), arXiv:1309.5681 [nucl-ex].
Beam energy dependence of moments of the net-charge multiplicity distributions in Au+Au collisions at RHIC. L Adamczyk, STAR10.1103/PhysRevLett.113.092301arXiv:1402.1558Phys. Rev. Lett. 11392301nucl-exL. Adamczyk et al. (STAR), "Beam energy dependence of moments of the net-charge multiplicity distributions in Au+Au collisions at RHIC," Phys. Rev. Lett. 113, 092301 (2014), arXiv:1402.1558 [nucl-ex].
Energy Dependence of Moments of Net-Proton and Net-Charge Multiplicity Distributions at STAR. Xiaofeng Luo, arXiv:1503.02558Proceedings, 9th International Workshop on Critical Point and Onset of Deconfinement (CPOD. 9th International Workshop on Critical Point and Onset of Deconfinement (CPODBielefeld, Germany201419nucl-exXiaofeng Luo (STAR), "Energy Dependence of Mo- ments of Net-Proton and Net-Charge Multiplicity Distributions at STAR," Proceedings, 9th Interna- tional Workshop on Critical Point and Onset of Deconfinement (CPOD 2014): Bielefeld, Germany, November 17-21, 2014, PoS CPOD2014, 019 (2015), arXiv:1503.02558 [nucl-ex].
Collision Energy Dependence of Moments of Net-Kaon Multiplicity Distributions at RHIC. L Adamczyk, STAR10.1016/j.physletb.2018.07.066arXiv:1709.00773Phys. Lett. 785nucl-exL. Adamczyk et al. (STAR), "Collision Energy Depen- dence of Moments of Net-Kaon Multiplicity Distribu- tions at RHIC," Phys. Lett. B785, 551-560 (2018), arXiv:1709.00773 [nucl-ex].
Collision-energy dependence of second-order off-diagonal and diagonal cumulants of net-charge, net-proton, and net-kaon multiplicity distributions in Au + Au collisions. Jaroslav Adam, STAR10.1103/PhysRevC.100.014902arXiv:1903.05370Phys. Rev. C. 10014902nucl-exJaroslav Adam et al. (STAR), "Collision-energy depen- dence of second-order off-diagonal and diagonal cumu- lants of net-charge, net-proton, and net-kaon multiplic- ity distributions in Au + Au collisions," Phys. Rev. C 100, 014902 (2019), arXiv:1903.05370 [nucl-ex].
Net-proton number fluctuations and the Quantum Chromodynamics critical point. J Adam, STARarXiv:2001.02852nucl-exJ. Adam et al. (STAR), "Net-proton number fluc- tuations and the Quantum Chromodynamics critical point," (2020), arXiv:2001.02852 [nucl-ex].
Measurement of the Sixth-Order Cumulant of Net-Proton Distributions in Au+Au Collisions from the STAR Experiment. Toshihiro Nonaka, arXiv:2002.1250528th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions. nucl-exToshihiro Nonaka (STAR), "Measurement of the Sixth- Order Cumulant of Net-Proton Distributions in Au+Au Collisions from the STAR Experiment," in 28th Interna- tional Conference on Ultrarelativistic Nucleus-Nucleus Collisions (2020) arXiv:2002.12505 [nucl-ex].
Measurement of cumulants of conserved charge multiplicity distributions in Au+Au collisions from the STAR experiment. Ashish Pandav, arXiv:2003.12503nucl-exAshish Pandav (STAR), "Measurement of cumu- lants of conserved charge multiplicity distributions in Au+Au collisions from the STAR experiment," (2020), arXiv:2003.12503 [nucl-ex].
Fierz-complete NJL model study III: Emergence from quark-gluon dynamics. Jens Braun, Marc Leonhardt, Martin Pospiech, 10.1103/PhysRevD.101.036004arXiv:1909.06298Phys. Rev. 10136004hep-phJens Braun, Marc Leonhardt, and Martin Pospiech, "Fierz-complete NJL model study III: Emergence from quark-gluon dynamics," Phys. Rev. D101, 036004 (2020), arXiv:1909.06298 [hep-ph].
Baryon number fluctuations in the QCD phase diagram from Dyson-Schwinger equations. Philipp Isserstedt, Michael Buballa, Christian S Fischer, Pascal J Gunkel, 10.1103/PhysRevD.100.074011arXiv:1906.11644Phys. Rev. D. 10074011hep-phPhilipp Isserstedt, Michael Buballa, Christian S. Fis- cher, and Pascal J. Gunkel, "Baryon number fluc- tuations in the QCD phase diagram from Dyson- Schwinger equations," Phys. Rev. D 100, 074011 (2019), arXiv:1906.11644 [hep-ph].
QCD phase structure from functional methods. Fei Gao, Jan M Pawlowski, 10.1103/PhysRevD.102.034027arXiv:2002.07500Phys. Rev. D. 10234027hep-phFei Gao and Jan M. Pawlowski, "QCD phase structure from functional methods," Phys. Rev. D 102, 034027 (2020), arXiv:2002.07500 [hep-ph].
Chiral phase structure and critical end point in QCD. Fei Gao, Jan M Pawlowski, arXiv:2010.13705hep-phFei Gao and Jan M. Pawlowski, "Chiral phase struc- ture and critical end point in QCD," (2020), arXiv:2010.13705 [hep-ph].
Susceptibilities near the QCD (tri)critical point. Bernd-Jochen Schaefer, Jochen Wambach, 10.1103/PhysRevD.75.085015arXiv:hep-ph/0603256Phys. Rev. 7585015hep-phBernd-Jochen Schaefer and Jochen Wambach, "Suscep- tibilities near the QCD (tri)critical point," Phys. Rev. D75, 085015 (2007), arXiv:hep-ph/0603256 [hep-ph].
Particle production in heavy ion collisions. Peter Braun-Munzinger, Krzysztof Redlich, Johanna Stachel, 10.1142/9789812795533_0008arXiv:nucl-th/0304013Quark-Gluon Plasma. Rudolph C. Hwa and Xin-Nian WangPeter Braun-Munzinger, Krzysztof Redlich, and Jo- hanna Stachel, "Particle production in heavy ion col- lisions," in Quark-Gluon Plasma, edited by Rudolph C. Hwa and Xin-Nian Wang (2004) pp. 491-599, arXiv:nucl-th/0304013.
Chemical freezeout and the QCD phase transition temperature. P Braun-Munzinger, J Stachel, Christof Wetterich, 10.1016/j.physletb.2004.05.081arXiv:nucl-th/0311005Phys. Lett. B. 596P. Braun-Munzinger, J. Stachel, and Christof Wet- terich, "Chemical freezeout and the QCD phase tran- sition temperature," Phys. Lett. B 596, 61-69 (2004), arXiv:nucl-th/0311005.
Chemical freeze-out in heavy ion collisions at large baryon densities. Stefan Floerchinger, Christof Wetterich, 10.1016/j.nuclphysa.2012.07.009arXiv:1202.1671Nucl. Phys. A890. 891nucl-thStefan Floerchinger and Christof Wetterich, "Chemi- cal freeze-out in heavy ion collisions at large baryon densities," Nucl. Phys. A890-891, 11-24 (2012), arXiv:1202.1671 [nucl-th].
Freeze-out Conditions in Heavy Ion Collisions from QCD Thermodynamics. A Bazavov, 10.1103/PhysRevLett.109.192302arXiv:1208.1220Phys. Rev. Lett. 109192302hep-latA. Bazavov et al., "Freeze-out Conditions in Heavy Ion Collisions from QCD Thermodynamics," Phys. Rev. Lett. 109, 192302 (2012), arXiv:1208.1220 [hep-lat].
Freeze-out parameters: lattice meets experiment. S Borsanyi, Z Fodor, S D Katz, S Krieg, C Ratti, K K Szabo, 10.1103/PhysRevLett.111.062005arXiv:1305.5161Phys. Rev. Lett. 11162005hep-latS. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, "Freeze-out parameters: lattice meets experiment," Phys. Rev. Lett. 111, 062005 (2013), arXiv:1305.5161 [hep-lat].
Freeze-out parameters from electric charge and baryon number fluctuations: is there consistency?. S Borsanyi, Z Fodor, S D Katz, S Krieg, C Ratti, K K Szabo, 10.1103/PhysRevLett.113.052301arXiv:1403.4576Phys. Rev. Lett. 11352301hep-latS. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, "Freeze-out parameters from elec- tric charge and baryon number fluctuations: is there consistency?" Phys. Rev. Lett. 113, 052301 (2014), arXiv:1403.4576 [hep-lat].
The QCD Equation of State to O(µ 6 B ) from Lattice QCD. A Bazavov, 10.1103/PhysRevD.95.054504arXiv:1701.04325Phys. Rev. D. 9554504hep-latA. Bazavov et al., "The QCD Equation of State to O(µ 6 B ) from Lattice QCD," Phys. Rev. D 95, 054504 (2017), arXiv:1701.04325 [hep-lat].
Skewness and kurtosis of net baryon-number distributions at small values of the baryon chemical potential. A Bazavov, HotQCD10.1103/PhysRevD.96.074510arXiv:1708.04897Phys. Rev. 9674510hep-latA. Bazavov et al. (HotQCD), "Skewness and kurtosis of net baryon-number distributions at small values of the baryon chemical potential," Phys. Rev. D96, 074510 (2017), arXiv:1708.04897 [hep-lat].
Higher order fluctuations and correlations of conserved charges from lattice QCD. Szabolcs Borsanyi, Zoltan Fodor, Jana N Guenther, Sandor K Katz, Kalman K Szabo, Attila Pasztor, Israel Portillo, Claudia Ratti, 10.1007/JHEP10(2018)205arXiv:1805.04445JHEP. 10205heplatSzabolcs Borsanyi, Zoltan Fodor, Jana N. Guenther, Sandor K. Katz, Kalman K. Szabo, Attila Pasztor, Is- rael Portillo, and Claudia Ratti, "Higher order fluctua- tions and correlations of conserved charges from lattice QCD," JHEP 10, 205 (2018), arXiv:1805.04445 [hep- lat].
QCD equation of state matched to lattice data and exhibiting a critical point singularity. Paolo Parotto, Marcus Bluhm, Debora Mroczek, Marlene Nahrgang, Jacquelyn Noronha-Hostler, Krishna Rajagopal, Claudia Ratti, Thomas Schäfer, Mikhail Stephanov, 10.1103/PhysRevC.101.034901arXiv:1805.05249Phys. Rev. C. 10134901hep-phPaolo Parotto, Marcus Bluhm, Debora Mroczek, Mar- lene Nahrgang, Jacquelyn Noronha-Hostler, Krishna Rajagopal, Claudia Ratti, Thomas Schäfer, and Mikhail Stephanov, "QCD equation of state matched to lattice data and exhibiting a critical point singular- ity," Phys. Rev. C 101, 034901 (2020), arXiv:1805.05249 [hep-ph].
Relevance of matter and glue dynamics for baryon number fluctuations. Wei-Jie Fu, Jan M Pawlowski, 10.1103/PhysRevD.92.116006arXiv:1508.06504Phys. Rev. 92116006hepphWei-jie Fu and Jan M. Pawlowski, "Relevance of mat- ter and glue dynamics for baryon number fluctuations," Phys. Rev. D92, 116006 (2015), arXiv:1508.06504 [hep- ph].
Correlating the skewness and kurtosis of baryon number distributions. Wei-Jie Fu, Jan M Pawlowski, 10.1103/PhysRevD.93.091501arXiv:1512.08461Phys. Rev. 9391501hepphWei-jie Fu and Jan M. Pawlowski, "Correlating the skewness and kurtosis of baryon number distributions," Phys. Rev. D93, 091501 (2016), arXiv:1512.08461 [hep- ph].
Baryon number fluctuations at finite temperature and density. Wei-Jie Fu, Jan M Pawlowski, Fabian Rennecke, Bernd-Jochen Schaefer, 10.1103/PhysRevD.94.116020arXiv:1608.04302Phys. Rev. D. 94116020hep-phWei-jie Fu, Jan M. Pawlowski, Fabian Rennecke, and Bernd-Jochen Schaefer, "Baryon number fluctuations at finite temperature and density," Phys. Rev. D 94, 116020 (2016), arXiv:1608.04302 [hep-ph].
Chiral symmetry breaking in continuum QCD. Mario Mitter, Jan M Pawlowski, Nils Strodthoff, 10.1103/PhysRevD.91.054035arXiv:1411.7978Phys. Rev. 9154035hep-phMario Mitter, Jan M. Pawlowski, and Nils Strodthoff, "Chiral symmetry breaking in continuum QCD," Phys. Rev. D91, 054035 (2015), arXiv:1411.7978 [hep-ph].
From Quarks and Gluons to Hadrons: Chiral Symmetry Breaking in Dynamical QCD. Jens Braun, Leonard Fister, Jan M Pawlowski, Fabian Rennecke, 10.1103/PhysRevD.94.034016arXiv:1412.1045Phys. Rev. 9434016hep-phJens Braun, Leonard Fister, Jan M. Pawlowski, and Fabian Rennecke, "From Quarks and Gluons to Hadrons: Chiral Symmetry Breaking in Dy- namical QCD," Phys. Rev. D94, 034016 (2016), arXiv:1412.1045 [hep-ph].
Vacuum structure of vector mesons in QCD. Fabian Rennecke, 10.1103/PhysRevD.92.076012arXiv:1504.03585Phys. Rev. 9276012hep-phFabian Rennecke, "Vacuum structure of vector mesons in QCD," Phys. Rev. D92, 076012 (2015), arXiv:1504.03585 [hep-ph].
Nonperturbative quark, gluon, and meson correlators of unquenched QCD. Anton K Cyrol, Mario Mitter, Jan M Pawlowski, Nils Strodthoff, 10.1103/PhysRevD.97.054006arXiv:1706.06326Phys. Rev. 9754006hep-phAnton K. Cyrol, Mario Mitter, Jan M. Pawlowski, and Nils Strodthoff, "Nonperturbative quark, gluon, and meson correlators of unquenched QCD," Phys. Rev. D97, 054006 (2018), arXiv:1706.06326 [hep-ph].
Renormalization flow of bound states. Holger Gies, Christof Wetterich, 10.1103/PhysRevD.65.065001arXiv:hep-th/0107221Phys. Rev. 6565001hep-thHolger Gies and Christof Wetterich, "Renormalization flow of bound states," Phys. Rev. D65, 065001 (2002), arXiv:hep-th/0107221 [hep-th].
Universality of spontaneous chiral symmetry breaking in gauge theories. Holger Gies, Christof Wetterich, 10.1103/PhysRevD.69.025001arXiv:hep-th/0209183Phys. Rev. 6925001hep-thHolger Gies and Christof Wetterich, "Universality of spontaneous chiral symmetry breaking in gauge the- ories," Phys. Rev. D69, 025001 (2004), arXiv:hep- th/0209183 [hep-th].
Aspects of the functional renormalisation group. Jan M Pawlowski, 10.1016/j.aop.2007.01.007arXiv:hep-th/0512261Annals Phys. 322hep-thJan M. Pawlowski, "Aspects of the functional renormal- isation group," Annals Phys. 322, 2831-2915 (2007), arXiv:hep-th/0512261 [hep-th].
Exact flow equation for composite operators. S Floerchinger, C Wetterich, 10.1016/j.physletb.2009.09.014arXiv:0905.0915Phys. Lett. 680hep-thS. Floerchinger and C. Wetterich, "Exact flow equation for composite operators," Phys. Lett. B680, 371-376 (2009), arXiv:0905.0915 [hep-th].
The nonperturbative functional renormalization group and its applications. N Dupuis, L Canet, A Eichhorn, W Metzner, J M Pawlowski, M Tissier, N Wschebor, arXiv:2006.04853cond-mat.statmechN. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J.M. Pawlowski, M. Tissier, and N. Wschebor, "The nonper- turbative functional renormalization group and its ap- plications," (2020), arXiv:2006.04853 [cond-mat.stat- mech].
Improved Polyakov-loop potential for effective models from functional calculations. Lisa M Haas, Rainer Stiele, Jens Braun, Jan M Pawlowski, Jürgen Schaffner-Bielich, 10.1103/PhysRevD.87.076004arXiv:1302.1993Phys. Rev. 8776004hep-phLisa M. Haas, Rainer Stiele, Jens Braun, Jan M. Pawlowski, and Jürgen Schaffner-Bielich, "Improved Polyakov-loop potential for effective models from func- tional calculations," Phys. Rev. D87, 076004 (2013), arXiv:1302.1993 [hep-ph].
Thermodynamics of QCD at vanishing density. Tina Katharina Herbst, Mario Mitter, Jan M Pawlowski, Bernd-Jochen Schaefer, Rainer Stiele, 10.1016/j.physletb.2014.02.045arXiv:1308.3621Phys. Lett. 731hep-phTina Katharina Herbst, Mario Mitter, Jan M. Pawlowski, Bernd-Jochen Schaefer, and Rainer Stiele, "Thermodynamics of QCD at vanishing density," Phys. Lett. B731, 248-256 (2014), arXiv:1308.3621 [hep-ph].
Baryon number fluctuations in the 2+1 flavor low energy effective model. Rui Wen, Chuang Huang, Wei-Jie Fu, 10.1103/PhysRevD.99.094019arXiv:1809.04233Phys. Rev. D. 9994019hep-phRui Wen, Chuang Huang, and Wei-Jie Fu, "Baryon number fluctuations in the 2+1 flavor low energy effective model," Phys. Rev. D 99, 094019 (2019), arXiv:1809.04233 [hep-ph].
Mesonic dynamics and the QCD phase transition. Rui Shi Yin, Wei-Jie Wen, Fu, 10.1103/PhysRevD.100.094029arXiv:1907.10262Phys. Rev. D. 10094029hep-phShi Yin, Rui Wen, and Wei-jie Fu, "Mesonic dynam- ics and the QCD phase transition," Phys. Rev. D 100, 094029 (2019), arXiv:1907.10262 [hep-ph].
Strangeness Neutrality and QCD Thermodynamics. Wei-Jie Fu, Jan M Pawlowski, Fabian Rennecke, 10.21468/SciPostPhysCore.2.1.002arXiv:1808.00410SciPost Phys. Core. 22hep-phWei-jie Fu, Jan M. Pawlowski, and Fabian Rennecke, "Strangeness Neutrality and QCD Thermodynamics," SciPost Phys. Core 2, 002 (2020), arXiv:1808.00410 [hep-ph].
Strangeness neutrality and baryon-strangeness correlations. Wei-Jie Fu, Jan M Pawlowski, Fabian Rennecke, 10.1103/PhysRevD.100.111501arXiv:1809.01594Phys. Rev. D. 100111501hep-phWei-jie Fu, Jan M. Pawlowski, and Fabian Ren- necke, "Strangeness neutrality and baryon-strangeness correlations," Phys. Rev. D 100, 111501 (2019), arXiv:1809.01594 [hep-ph].
Meson fluctuations and thermodynamics of the Polyakov loop extended quark-meson model. V Skokov, B Stokic, B Friman, K Redlich, 10.1103/PhysRevC.82.015206arXiv:1004.2665Phys. Rev. 8215206hep-phV. Skokov, B. Stokic, B. Friman, and K. Redlich, "Me- son fluctuations and thermodynamics of the Polyakov loop extended quark-meson model," Phys. Rev. C82, 015206 (2010), arXiv:1004.2665 [hep-ph].
Quark number fluctuations in the Polyakov loop-extended quark-meson model at finite baryon density. V Skokov, B Friman, K Redlich, 10.1103/PhysRevC.83.054904arXiv:1008.4570Phys. Rev. 8354904hep-phV. Skokov, B. Friman, and K. Redlich, "Quark number fluctuations in the Polyakov loop-extended quark-meson model at finite baryon density," Phys. Rev. C83, 054904 (2011), arXiv:1008.4570 [hep-ph].
Fluctuations as probe of the QCD phase transition and freeze-out in heavy ion collisions at LHC and RHIC. B Friman, F Karsch, K Redlich, V Skokov, 10.1140/epjc/s10052-011-1694-2arXiv:1103.3511Eur. Phys. J. C. 711694hepphB. Friman, F. Karsch, K. Redlich, and V. Skokov, "Fluctuations as probe of the QCD phase transition and freeze-out in heavy ion collisions at LHC and RHIC," Eur. Phys. J. C 71, 1694 (2011), arXiv:1103.3511 [hep- ph].
Criticality of the net-baryon number probability distribution at finite density. Kenji Morita, Bengt Friman, Krzysztof Redlich, 10.1016/j.physletb.2014.12.037arXiv:1402.5982Phys. Lett. 741hep-phKenji Morita, Bengt Friman, and Krzysztof Redlich, "Criticality of the net-baryon number probability dis- tribution at finite density," Phys. Lett. B741, 178-183 (2015), arXiv:1402.5982 [hep-ph].
Baryon number fluctuations in chiral effective models and their phenomenological implications. Bengt Gabor Andras Almasi, Krzysztof Friman, Redlich, 10.1103/PhysRevD.96.014027arXiv:1703.05947Phys. Rev. D. 9614027hep-phGabor Andras Almasi, Bengt Friman, and Krzysztof Redlich, "Baryon number fluctuations in chiral effective models and their phenomenological implications," Phys. Rev. D 96, 014027 (2017), arXiv:1703.05947 [hep-ph].
Quark number fluctuations at finite temperature and finite chemical potential via the Dyson-Schwinger equation approach. Xian-Yin Xin, Yu-Xin Si-Xue Qin, Liu, 10.1103/PhysRevD.90.076006Phys. Rev. D. 9076006Xian-yin Xin, Si-xue Qin, and Yu-xin Liu, "Quark number fluctuations at finite temperature and finite chemical potential via the Dyson-Schwinger equation approach," Phys. Rev. D 90, 076006 (2014).
Fluctuations and Correlations of Conserved Charges in QCD at Finite Temperature with Effective Models. Wei-Jie Fu, Yu-Xin Liu, Yue-Liang Wu, 10.1103/PhysRevD.81.014028arXiv:0910.5783Phys. Rev. D. 8114028hep-phWei-jie Fu, Yu-xin Liu, and Yue-Liang Wu, "Fluctua- tions and Correlations of Conserved Charges in QCD at Finite Temperature with Effective Models," Phys. Rev. D 81, 014028 (2010), arXiv:0910.5783 [hep-ph].
Fluctuations and Correlations of Conserved Charges near the QCD Critical Point. Wei-Jie Fu, Yue-Liang Wu, 10.1103/PhysRevD.82.074013arXiv:1008.3684Phys. Rev. D. 8274013hep-phWei-jie Fu and Yue-liang Wu, "Fluctuations and Correlations of Conserved Charges near the QCD Critical Point," Phys. Rev. D 82, 074013 (2010), arXiv:1008.3684 [hep-ph].
Towards finite density QCD with Taylor expansions. Frithjof Karsch, Bernd-Jochen Schaefer, Mathias Wagner, Jochen Wambach, 10.1016/j.physletb.2011.03.013arXiv:1009.5211Phys. Lett. B. 698hep-phFrithjof Karsch, Bernd-Jochen Schaefer, Mathias Wag- ner, and Jochen Wambach, "Towards finite density QCD with Taylor expansions," Phys. Lett. B 698, 256- 264 (2011), arXiv:1009.5211 [hep-ph].
QCD critical region and higher moments for three flavor models. B J Schaefer, M Wagner, 10.1103/PhysRevD.85.034027arXiv:1111.6871Phys. Rev. D. 8534027hep-phB.J. Schaefer and M. Wagner, "QCD critical region and higher moments for three flavor models," Phys. Rev. D 85, 034027 (2012), arXiv:1111.6871 [hep-ph].
The kurtosis of net baryon number fluctuations from a realistic Polyakov-Nambu-Jona-Lasinio model along the experimental freeze-out line. Zhibin Li, Kun Xu, Xinyang Wang, Mei Huang, 10.1140/epjc/s10052-019-6703-xarXiv:1801.09215Eur. Phys. J. C. 79245hep-phZhibin Li, Kun Xu, Xinyang Wang, and Mei Huang, "The kurtosis of net baryon number fluctuations from a realistic Polyakov-Nambu-Jona-Lasinio model along the experimental freeze-out line," Eur. Phys. J. C 79, 245 (2019), arXiv:1801.09215 [hep-ph].
Bernd-Jochen Schaefer, Jan M Pawlowski, Jochen Wambach, 10.1103/PhysRevD.76.074023arXiv:0704.3234The Phase Structure of the Polyakov-Quark-Meson Model. 7674023hep-phBernd-Jochen Schaefer, Jan M. Pawlowski, and Jochen Wambach, "The Phase Structure of the Polyakov- Quark-Meson Model," Phys. Rev. D76, 074023 (2007), arXiv:0704.3234 [hep-ph].
Thermodynamics of (2+1)-flavor QCD: Confronting Models with Lattice Studies. Bernd-Jochen Schaefer, Mathias Wagner, Jochen Wambach, 10.1103/PhysRevD.81.074013arXiv:0910.5628Phys. Rev. D. 8174013hep-phBernd-Jochen Schaefer, Mathias Wagner, and Jochen Wambach, "Thermodynamics of (2+1)-flavor QCD: Confronting Models with Lattice Studies," Phys. Rev. D 81, 074013 (2010), arXiv:0910.5628 [hep-ph].
The phase structure of the Polyakovquarkmeson model beyond mean field. Tina Katharina Herbst, Jan M Pawlowski, Bernd-Jochen Schaefer, 10.1016/j.physletb.2010.12.003PhysTina Katharina Herbst, Jan M. Pawlowski, and Bernd-Jochen Schaefer, "The phase structure of the Polyakovquarkmeson model beyond mean field," Phys.
. Lett, 10.1016/j.physletb.2010.12.003arXiv:1008.0081696hep-phLett. B696, 58-67 (2011), arXiv:1008.0081 [hep-ph].
Role of mesonic fluctuations in the Polyakov loop extended quark-meson model at imaginary chemical potential. Kenji Morita, Vladimir Skokov, Bengt Friman, Krzysztof Redlich, 10.1103/PhysRevD.84.074020arXiv:1108.0735Phys. Rev. D. 8474020hep-phKenji Morita, Vladimir Skokov, Bengt Friman, and Krzysztof Redlich, "Role of mesonic fluctuations in the Polyakov loop extended quark-meson model at imag- inary chemical potential," Phys. Rev. D 84, 074020 (2011), arXiv:1108.0735 [hep-ph].
Nonperturbative dynamics and charge fluctuations in effective chiral models. V Skokov, B Friman, K Redlich, 10.1016/j.physletb.2012.01.022arXiv:1108.3231Phys. Lett. B. 708hep-phV. Skokov, B. Friman, and K. Redlich, "Non- perturbative dynamics and charge fluctuations in effec- tive chiral models," Phys. Lett. B 708, 179-185 (2012), arXiv:1108.3231 [hep-ph].
Phase diagram and surface tension in the three-flavor Polyakov-quarkmeson model. W Bruno, Rainer Mintz, Rudnei O Stiele, Juergen Ramos, Schaffner-Bielich, 10.1103/PhysRevD.87.036004arXiv:1212.1184Phys. Rev. D. 8736004hep-phBruno W. Mintz, Rainer Stiele, Rudnei O. Ramos, and Juergen Schaffner-Bielich, "Phase diagram and surface tension in the three-flavor Polyakov-quark- meson model," Phys. Rev. D 87, 036004 (2013), arXiv:1212.1184 [hep-ph].
Phase structure and thermodynamics of QCD. Tina K Herbst, Jan M Pawlowski, Bernd-Jochen Schaefer, 10.1103/PhysRevD.88.014007arXiv:1302.1426Phys. Rev. 8814007hep-phTina K. Herbst, Jan M. Pawlowski, and Bernd- Jochen Schaefer, "Phase structure and thermody- namics of QCD," Phys. Rev. D88, 014007 (2013), arXiv:1302.1426 [hep-ph].
Phase diagram and nucleation in the Polyakov-loop-extended Quark-Meson truncation of QCD with the unquenched Polyakov-loop potential. Rainer Stiele, Juergen Schaffner-Bielich, 10.1103/PhysRevD.93.094014arXiv:1601.05731Phys. Rev. D. 9394014hep-phRainer Stiele and Juergen Schaffner-Bielich, "Phase di- agram and nucleation in the Polyakov-loop-extended Quark-Meson truncation of QCD with the unquenched Polyakov-loop potential," Phys. Rev. D 93, 094014 (2016), arXiv:1601.05731 [hep-ph].
Baryon number probability distribution at finite temperature. Rui Ke-Xin Sun, Wei-Jie Wen, Fu, 10.1103/PhysRevD.98.074028arXiv:1805.12025Phys. Rev. 9874028hep-phKe-xin Sun, Rui Wen, and Wei-jie Fu, "Baryon number probability distribution at finite temperature," Phys. Rev. D98, 074028 (2018), arXiv:1805.12025 [hep-ph].
Quark and Polyakov-loop correlations in effective models at zero and nonvanishing density. Hubert Hansen, Rainer Stiele, Pedro Costa, 10.1103/PhysRevD.101.094001arXiv:1904.08965Phys. Rev. D. 10194001hep-phHubert Hansen, Rainer Stiele, and Pedro Costa, "Quark and Polyakov-loop correlations in effective mod- els at zero and nonvanishing density," Phys. Rev. D 101, 094001 (2020), arXiv:1904.08965 [hep-ph].
Chiral crossover in QCD at zero and non-zero chemical potentials. A Bazavov, HotQCD10.1016/j.physletb.2019.05.013arXiv:1812.08235Phys. Lett. 795hep-latA. Bazavov et al. (HotQCD), "Chiral crossover in QCD at zero and non-zero chemical potentials," Phys. Lett. B795, 15-21 (2019), arXiv:1812.08235 [hep-lat].
The QCD phase diagram from analytic continuation. R Bellwied, S Borsanyi, Z Fodor, J Guenther, S D Katz, C Ratti, K K Szabo, 10.1016/j.physletb.2015.11.011arXiv:1507.07510Phys. Lett. 751hep-latR. Bellwied, S. Borsanyi, Z. Fodor, J. Guenther, S. D. Katz, C. Ratti, and K. K. Szabo, "The QCD phase diagram from analytic continuation," Phys. Lett. B751, 559-564 (2015), arXiv:1507.07510 [hep-lat].
Inhomogeneous chiral condensates. Michael Buballa, Stefano Carignano, 10.1016/j.ppnp.2014.11.001arXiv:1406.1367Prog. Part. Nucl. Phys. 81hep-phMichael Buballa and Stefano Carignano, "Inhomoge- neous chiral condensates," Prog. Part. Nucl. Phys. 81, 39-96 (2015), arXiv:1406.1367 [hep-ph].
On the efficient computation of high-order derivatives for implicitly defined functions. Mathias Wagner, Andrea Walther, Bernd-Jochen Schaefer, 10.1016/j.cpc.2009.12.008arXiv:0912.2208Comput. Phys. Commun. 181hep-phMathias Wagner, Andrea Walther, and Bernd-Jochen Schaefer, "On the efficient computation of high-order derivatives for implicitly defined functions," Comput. Phys. Commun. 181, 756-764 (2010), arXiv:0912.2208 [hep-ph].
Gauge Theories With Imaginary Chemical Potential and the Phases of QCD. Andre Roberge, Nathan Weiss, 10.1016/0550-3213(86)90582-1Nucl. Phys. 275Andre Roberge and Nathan Weiss, "Gauge Theories With Imaginary Chemical Potential and the Phases of QCD," Nucl. Phys. B275, 734-745 (1986).
Statistical theory of equations of state and phase transitions. 1. Theory of condensation. Chen-Ning Yang, T D Lee, 10.1103/PhysRev.87.404Phys. Rev. 87Chen-Ning Yang and T.D. Lee, "Statistical theory of equations of state and phase transitions. 1. Theory of condensation," Phys. Rev. 87, 404-409 (1952).
Phase Structure of Two-Flavor QCD at Finite Chemical Potential. Jens Braun, Lisa M Haas, Florian Marhauser, Jan M Pawlowski, 10.1103/PhysRevLett.106.022002arXiv:0908.0008Phys. Rev. Lett. 10622002hep-phJens Braun, Lisa M. Haas, Florian Marhauser, and Jan M. Pawlowski, "Phase Structure of Two-Flavor QCD at Finite Chemical Potential," Phys. Rev. Lett. 106, 022002 (2011), arXiv:0908.0008 [hep-ph].
Nature of the Roberge-Weiss transition in N f = 2 QCD with Wilson fermions. Owe Philipsen, Christopher Pinke, 10.1103/PhysRevD.89.094504arXiv:1402.0838Phys. Rev. D. 8994504hep-latOwe Philipsen and Christopher Pinke, "Nature of the Roberge-Weiss transition in N f = 2 QCD with Wilson fermions," Phys. Rev. D 89, 094504 (2014), arXiv:1402.0838 [hep-lat].
QCD critical point and complex chemical potential singularities. M A Stephanov, 10.1103/PhysRevD.73.094508arXiv:hep-lat/0603014Phys. Rev. D. 7394508M.A. Stephanov, "QCD critical point and complex chemical potential singularities," Phys. Rev. D 73, 094508 (2006), arXiv:hep-lat/0603014.
Universality driven analytic structure of QCD crossover: radius of convergence in baryon chemical potential. Swagato Mukherjee, Vladimir Skokov, arXiv:1909.04639hep-phSwagato Mukherjee and Vladimir Skokov, "Universal- ity driven analytic structure of QCD crossover: radius of convergence in baryon chemical potential," (2019), arXiv:1909.04639 [hep-ph].
Universal Location of the Yang-Lee Edge Singularity in O(N ) Theories. Andrew Connelly, Gregory Johnson, Fabian Rennecke, Vladimir Skokov, 10.1103/PhysRevLett.125.191602arXiv:2006.12541Phys. Rev. Lett. 125191602condmat.stat-mechAndrew Connelly, Gregory Johnson, Fabian Rennecke, and Vladimir Skokov, "Universal Location of the Yang- Lee Edge Singularity in O(N ) Theories," Phys. Rev. Lett. 125, 191602 (2020), arXiv:2006.12541 [cond- mat.stat-mech].
Decoding the phase structure of QCD via particle production at high energy. Anton Andronic, Peter Braun-Munzinger, Krzysztof Redlich, Johanna Stachel, 10.1038/s41586-018-0491-6arXiv:1710.09425Nature. 561nucl-thAnton Andronic, Peter Braun-Munzinger, Krzysztof Redlich, and Johanna Stachel, "Decoding the phase structure of QCD via particle production at high en- ergy," Nature 561, 321-330 (2018), arXiv:1710.09425 [nucl-th].
Bulk Properties of the Medium Produced in Relativistic Heavy-Ion Collisions from the Beam Energy Scan Program. L Adamczyk, STAR10.1103/PhysRevC.96.044904arXiv:1701.07065Phys. Rev. 9644904nucl-exL. Adamczyk et al. (STAR), "Bulk Properties of the Medium Produced in Relativistic Heavy-Ion Collisions from the Beam Energy Scan Program," Phys. Rev. C96, 044904 (2017), arXiv:1701.07065 [nucl-ex].
Cumulants and correlation functions versus the QCD phase diagram. Adam Bzdak, Volker Koch, Nils Strodthoff, 10.1103/PhysRevC.95.054906arXiv:1607.07375Phys. Rev. C. 9554906nucl-thAdam Bzdak, Volker Koch, and Nils Strodthoff, "Cu- mulants and correlation functions versus the QCD phase diagram," Phys. Rev. C 95, 054906 (2017), arXiv:1607.07375 [nucl-th].
Proton Cumulants and Correlation Functions in Au + Au Collisions at √ sNN=7.7-200 GeV from UrQMD Model. Shu He, Xiaofeng Luo, 10.1016/j.physletb.2017.10.030arXiv:1704.00423Phys. Lett. B. 774nucl-exShu He and Xiaofeng Luo, "Proton Cumulants and Cor- relation Functions in Au + Au Collisions at √ sNN=7.7- 200 GeV from UrQMD Model," Phys. Lett. B 774, 623- 629 (2017), arXiv:1704.00423 [nucl-ex].
Measurement of the sixth-order cumulant of net-proton multiplicity distributions in Au+Au collisions at √ sNN = 27. Mohamed Abdallah, STAR200Mohamed Abdallah et al. (STAR), "Measurement of the sixth-order cumulant of net-proton multiplicity distribu- tions in Au+Au collisions at √ sNN = 27, 54.4, and 200
. Rhic Gev At, arXiv:2105.14698nucl-exGeV at RHIC," (2021), arXiv:2105.14698 [nucl-ex].
Volume fluctuation and auto-correlation effects in the moment analysis of net-proton multiplicity distributions in heavy-ion collisions. Xiaofeng Luo, Ji Xu, Bedangadas Mohanty, Nu Xu, 10.1088/0954-3899/40/10/105104arXiv:1302.2332J. Phys. G. 40105104nucl-exXiaofeng Luo, Ji Xu, Bedangadas Mohanty, and Nu Xu, "Volume fluctuation and auto-correlation effects in the moment analysis of net-proton multiplicity distributions in heavy-ion collisions," J. Phys. G 40, 105104 (2013), arXiv:1302.2332 [nucl-ex].
Effect of centrality selection on higher-order cumulants of netproton multiplicity distributions in relativistic heavyion collisions. Arghya Chatterjee, Yu Zhang, Jingdong Zeng, Xiaofeng Nihar Ranjan Sahoo, Luo, 10.1103/PhysRevC.101.034902arXiv:1910.08004Phys. Rev. C. 10134902nucl-exArghya Chatterjee, Yu Zhang, Jingdong Zeng, Ni- har Ranjan Sahoo, and Xiaofeng Luo, "Effect of centrality selection on higher-order cumulants of net- proton multiplicity distributions in relativistic heavy- ion collisions," Phys. Rev. C 101, 034902 (2020), arXiv:1910.08004 [nucl-ex].
Effects of centrality fluctuation and deuteron formation on proton number cumulant in Au+Au collisions at √ sNN = 3 GeV from JAM model. Arghya Chatterjee, Yu Zhang, Hui Liu, Ruiqin Wang, Shu He, Xiaofeng Luo, arXiv:2009.03755nucl-exArghya Chatterjee, Yu Zhang, Hui Liu, Ruiqin Wang, Shu He, and Xiaofeng Luo, "Effects of centrality fluctu- ation and deuteron formation on proton number cumu- lant in Au+Au collisions at √ sNN = 3 GeV from JAM model," (2020), arXiv:2009.03755 [nucl-ex].
On the scaling behavior of the chiral phase transition in QCD in finite and infinite volume. Jens Braun, Bertram Klein, Piotr Piasecki, 10.1140/epjc/s10052-011-1576-7arXiv:1008.2155Eur. Phys. J. C. 711576hep-phJens Braun, Bertram Klein, and Piotr Piasecki, "On the scaling behavior of the chiral phase transition in QCD in finite and infinite volume," Eur. Phys. J. C 71, 1576 (2011), arXiv:1008.2155 [hep-ph].
On the Phase Structure of QCD in a Finite Volume. Jens Braun, Bertram Klein, Bernd-Jochen Schaefer, 10.1016/j.physletb.2012.05.053arXiv:1110.0849Phys. Lett. 713hep-phJens Braun, Bertram Klein, and Bernd-Jochen Schae- fer, "On the Phase Structure of QCD in a Fi- nite Volume," Phys. Lett. B713, 216-223 (2012), arXiv:1110.0849 [hep-ph].
Effect of fluctuations on the QCD critical point in a finite volume. Ralf-Arno Tripolt, Jens Braun, Bertram Klein, Bernd-Jochen Schaefer, 10.1103/PhysRevD.90.054012arXiv:1308.0164Phys. Rev. 9054012hep-phRalf-Arno Tripolt, Jens Braun, Bertram Klein, and Bernd-Jochen Schaefer, "Effect of fluctuations on the QCD critical point in a finite volume," Phys. Rev. D90, 054012 (2014), arXiv:1308.0164 [hep-ph].
Volume dependence of baryon number cumulants and their ratios. Gabor Almasi, Robert Pisarski, Vladimir Skokov, 10.1103/PhysRevD.95.056015arXiv:1612.04416Phys. Rev. 9556015hep-phGabor Almasi, Robert Pisarski, and Vladimir Skokov, "Volume dependence of baryon number cumulants and their ratios," Phys. Rev. D95, 056015 (2017), arXiv:1612.04416 [hep-ph].
Modeling Finite-Volume Effects and Chiral Symmetry Breaking in Two-Flavor QCD Thermodynamics. Bertram Klein, 10.1016/j.physrep.2017.09.002arXiv:1710.05357Phys. Rept. hep-phBertram Klein, "Modeling Finite-Volume Effects and Chiral Symmetry Breaking in Two-Flavor QCD Ther- modynamics," Phys. Rept. 707-708, 1-51 (2017), arXiv:1710.05357 [hep-ph].
Finite volume effects on the chiral phase transition from DysonSchwinger equations of QCD. Bo-Lin Li, Zhu-Fang Cui, Bo-Wen Zhou, Sun An, Li-Ping Zhang, Hong-Shi Zong, 10.1016/j.nuclphysb.2018.11.015arXiv:1711.04914Nucl. Phys. 938hep-phBo-Lin Li, Zhu-Fang Cui, Bo-Wen Zhou, Sun An, Li- Ping Zhang, and Hong-Shi Zong, "Finite volume effects on the chiral phase transition from DysonSchwinger equations of QCD," Nucl. Phys. B938, 298-306 (2019), arXiv:1711.04914 [hep-ph].
Finite volume effects on QCD susceptibilities with a chiral chemical potential. Run-Lin Liu, Meng-Yun Lai, Chao Shi, Hong-Shi Zong, 10.1103/PhysRevD.102.014014Phys. Rev. D. 10214014Run-Lin Liu, Meng-Yun Lai, Chao Shi, and Hong-Shi Zong, "Finite volume effects on QCD susceptibilities with a chiral chemical potential," Phys. Rev. D 102, 014014 (2020).
Finite Size Effects on the Chiral Phase Transition of Quantum Chromodynamics. Daize Shen-Song Wan, Bonan Li, Marco Zhang, Ruggieri, arXiv:2012.05734hep-phShen-Song Wan, Daize Li, Bonan Zhang, and Marco Ruggieri, "Finite Size Effects on the Chiral Phase Transition of Quantum Chromodynamics," (2020), arXiv:2012.05734 [hep-ph].
Effects of Nuclear Potential on the Cumulants of Net-Proton and Net-Baryon Multiplicity Distributions in Au+Au Collisions at √ sNN = 5 GeV. Shu He, Xiaofeng Luo, Yasushi Nara, Shinichi Esumi, Nu Xu, 10.1016/j.physletb.2016.09.053PhysShu He, Xiaofeng Luo, Yasushi Nara, ShinIchi Esumi, and Nu Xu, "Effects of Nuclear Potential on the Cumu- lants of Net-Proton and Net-Baryon Multiplicity Distri- butions in Au+Au Collisions at √ sNN = 5 GeV," Phys.
. 10.1016/j.physletb.2016.09.053arXiv:1607.06376Lett. B. 762nuclexLett. B 762, 296-300 (2016), arXiv:1607.06376 [nucl- ex].
Bridging the gap between event-by-event fluctuation measurements and theory predictions in relativistic nuclear collisions. P Braun-Munzinger, A Rustamov, J Stachel, 10.1016/j.nuclphysa.2017.01.011arXiv:1612.00702Nucl. Phys. A. 960nucl-thP. Braun-Munzinger, A. Rustamov, and J. Stachel, "Bridging the gap between event-by-event fluctuation measurements and theory predictions in relativistic nu- clear collisions," Nucl. Phys. A 960, 114-130 (2017), arXiv:1612.00702 [nucl-th].
Connecting fluctuation measurements in heavy-ion collisions with the grand-canonical susceptibilities. Volodymyr Vovchenko, Oleh Savchuk, Roman V Poberezhnyuk, Mark I Gorenstein, Volker Koch, 10.1016/j.physletb.2020.135868arXiv:2003.13905Phys. Lett. B. 811135868hep-phVolodymyr Vovchenko, Oleh Savchuk, Roman V. Poberezhnyuk, Mark I. Gorenstein, and Volker Koch, "Connecting fluctuation measurements in heavy-ion col- lisions with the grand-canonical susceptibilities," Phys. Lett. B 811, 135868 (2020), arXiv:2003.13905 [hep-ph].
Impact of resonance regeneration and decay on the net-proton fluctuations in a hadron resonance gas. Marlene Nahrgang, Marcus Bluhm, Paolo Alba, Rene Bellwied, Claudia Ratti, 10.1140/epjc/s10052-015-3775-0arXiv:1402.1238Eur. Phys. J. C. 75573hep-phMarlene Nahrgang, Marcus Bluhm, Paolo Alba, Rene Bellwied, and Claudia Ratti, "Impact of resonance re- generation and decay on the net-proton fluctuations in a hadron resonance gas," Eur. Phys. J. C 75, 573 (2015), arXiv:1402.1238 [hep-ph].
Effects of resonance weak decay and hadronic rescattering on the proton number fluctuations in Au+Au collisions at √ sNN = 5 from a microscopic hadronic transport (JAM) model. Yu Zhang, Shu He, Hui Liu, Zhenzhen Yang, Xiaofeng Luo, 10.1103/PhysRevC.101.034909arXiv:1905.01095Phys. Rev. C. 10134909nucl-exYu Zhang, Shu He, Hui Liu, Zhenzhen Yang, and Xiaofeng Luo, "Effects of resonance weak decay and hadronic rescattering on the proton number fluctuations in Au+Au collisions at √ sNN = 5 from a microscopic hadronic transport (JAM) model," Phys. Rev. C 101, 034909 (2020), arXiv:1905.01095 [nucl-ex].
Slowing outof-equilibrium near the QCD critical point. Boris Berdnikov, Krishna Rajagopal, 10.1103/PhysRevD.61.105017arXiv:hep-ph/9912274Phys. Rev. D. 61105017Boris Berdnikov and Krishna Rajagopal, "Slowing out- of-equilibrium near the QCD critical point," Phys. Rev. D 61, 105017 (2000), arXiv:hep-ph/9912274.
Dynamics of critical fluctuations: Theory -phenomenology -heavy-ion collisions. Marcus Bluhm, 10.1016/j.nuclphysa.2020.122016arXiv:2001.08831Nucl. Phys. A. 1003122016nuclthMarcus Bluhm et al., "Dynamics of critical fluctuations: Theory -phenomenology -heavy-ion collisions," Nucl. Phys. A 1003, 122016 (2020), arXiv:2001.08831 [nucl- th].
. Marcus Bluhm, Marlene Nahrgang, Jan M , Marcus Bluhm, Marlene Nahrgang, and Jan M.
Locating the freeze-out curve in heavy-ion collisions. Pawlowski, arXiv:2004.08608nucl-thPawlowski, "Locating the freeze-out curve in heavy-ion collisions," (2020), arXiv:2004.08608 [nucl-th].
Time-evolution of fluctuations as signal of the phase transition dynamics in a QCD-assisted transport approach. M Bluhm, Y Jiang, M Nahrgang, J M Pawlowski, F Rennecke, N Wink, 10.1016/j.nuclphysa.2018.09.058arXiv:1808.01377Proceedings, 27th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter. 27th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark MatterVenice, Italy982hep-phM. Bluhm, Y. Jiang, M. Nahrgang, J. M. Pawlowski, F. Rennecke, and N. Wink, "Time-evolution of fluc- tuations as signal of the phase transition dynamics in a QCD-assisted transport approach," Proceedings, 27th International Conference on Ultrarelativistic Nucleus- Nucleus Collisions (Quark Matter 2018): Venice, Italy, May 14-19, 2018, Nucl. Phys. A982, 871-874 (2019), arXiv:1808.01377 [hep-ph].
Relativistic nuclear collisions: Establishing a non-critical baseline for fluctuation measurements. P Braun-Munzinger, B Friman, K Redlich, A Rustamov, J Stachel, 10.1016/j.nuclphysa.2021.122141arXiv:2007.02463Nucl. Phys. A. 1008122141nucl-thP. Braun-Munzinger, B. Friman, K. Redlich, A. Rus- tamov, and J. Stachel, "Relativistic nuclear colli- sions: Establishing a non-critical baseline for fluctuation measurements," Nucl. Phys. A 1008, 122141 (2021), arXiv:2007.02463 [nucl-th].
The QCD Phase Boundary from Quark-Gluon Dynamics. Jens Braun, 10.1140/epjc/s10052-009-1136-6arXiv:0810.1727Eur. Phys. J. 64hep-phJens Braun, "The QCD Phase Boundary from Quark- Gluon Dynamics," Eur. Phys. J. C64, 459-482 (2009), arXiv:0810.1727 [hep-ph].
Quark Confinement from Color Confinement. Jens Braun, Holger Gies, Jan M Pawlowski, 10.1016/j.physletb.2010.01.009arXiv:0708.2413Phys.Lett. 684hep-thJens Braun, Holger Gies, and Jan M. Pawlowski, "Quark Confinement from Color Confinement," Phys.Lett. B684, 262-267 (2010), arXiv:0708.2413 [hep-th].
Landau gauge Yang-Mills correlation functions. Anton K Cyrol, Leonard Fister, Mario Mitter, Jan M Pawlowski, Nils Strodthoff, 10.1103/PhysRevD.94.054005arXiv:1605.01856Phys. Rev. 9454005hep-phAnton K. Cyrol, Leonard Fister, Mario Mitter, Jan M. Pawlowski, and Nils Strodthoff, "Landau gauge Yang- Mills correlation functions," Phys. Rev. D94, 054005 (2016), arXiv:1605.01856 [hep-ph].
Nonperturbative finite-temperature Yang-Mills theory. Anton K Cyrol, Mario Mitter, Jan M Pawlowski, Nils Strodthoff, 10.1103/PhysRevD.97.054015arXiv:1708.03482Phys. Rev. 9754015hep-phAnton K. Cyrol, Mario Mitter, Jan M. Pawlowski, and Nils Strodthoff, "Nonperturbative finite-temperature Yang-Mills theory," Phys. Rev. D97, 054015 (2018), arXiv:1708.03482 [hep-ph].
Chiral susceptibility in ( 2+1 )-flavor QCD. Jens Braun, Wei-Jie Fu, Jan M Pawlowski, Fabian Rennecke, Daniel Rosenblüh, Shi Yin, 10.1103/PhysRevD.102.056010arXiv:2003.13112Phys. Rev. D. 10256010hep-phJens Braun, Wei-jie Fu, Jan M. Pawlowski, Fabian Ren- necke, Daniel Rosenblüh, and Shi Yin, "Chiral suscepti- bility in ( 2+1 )-flavor QCD," Phys. Rev. D 102, 056010 (2020), arXiv:2003.13112 [hep-ph].
Chiral and effective U (1)A symmetry restoration in QCD. Jens Braun, Marc Leonhardt, Jan M Pawlowski, Daniel Rosenblüh, arXiv:2012.06231hep-phJens Braun, Marc Leonhardt, Jan M. Pawlowski, and Daniel Rosenblüh, "Chiral and effective U (1)A sym- metry restoration in QCD," (2020), arXiv:2012.06231 [hep-ph].
On gauge invariant Wilsonian flows. F Daniel, Jan M Litim, Pawlowski, arXiv:hep-th/9901063The exact renormalization group. Proceedings, Workshop. Faro, Portugalhep-thDaniel F. Litim and Jan M. Pawlowski, "On gauge invariant Wilsonian flows," in The exact renormaliza- tion group. Proceedings, Workshop, Faro, Portugal, September 10-12, 1998 (1998) pp. 168-185, arXiv:hep- th/9901063 [hep-th].
Nonperturbative renormalization flow in quantum field theory and statistical physics. Juergen Berges, Nikolaos Tetradis, Christof Wetterich, 10.1016/S0370-1573(01)00098-9arXiv:hep-ph/0005122Phys. Rept. 363hep-phJuergen Berges, Nikolaos Tetradis, and Christof Wet- terich, "Nonperturbative renormalization flow in quan- tum field theory and statistical physics," Phys. Rept. 363, 223-386 (2002), arXiv:hep-ph/0005122 [hep-ph].
Renormalization group approach towards the QCD phase diagram. Bernd-Jochen Schaefer, Jochen Wambach, 10.1134/S1063779608070083arXiv:hep-ph/0611191Helmholtz International Summer School on Dense Matter in Heavy Ion Collisions and Astrophysics. Dubna, Russia39hep-phBernd-Jochen Schaefer and Jochen Wambach, "Renor- malization group approach towards the QCD phase diagram," Helmholtz International Summer School on Dense Matter in Heavy Ion Collisions and Astro- physics Dubna, Russia, August 21-September 1, 2006, Phys. Part. Nucl. 39, 1025-1032 (2008), arXiv:hep- ph/0611191 [hep-ph].
Renormalization group and effective field theory approaches to many-body systems. Holger Gies, 10.1007/978-3-642-27320-9_6arXiv:hep-ph/0611146Lect. Notes Phys. 852Introduction to the functional RG and applications to gauge theories. hep-phHolger Gies, "Introduction to the functional RG and applications to gauge theories," Renormalization group and effective field theory approaches to many-body sys- tems, Lect. Notes Phys. 852, 287-348 (2012), arXiv:hep- ph/0611146 [hep-ph].
Fundamentals of the Exact Renormalization Group. J Oliver, Rosten, 10.1016/j.physrep.2011.12.003arXiv:1003.1366Phys. Rept. 511hep-thOliver J. Rosten, "Fundamentals of the Exact Renor- malization Group," Phys. Rept. 511, 177-272 (2012), arXiv:1003.1366 [hep-th].
Fermion Interactions and Universal Behavior in Strongly Interacting Theories. Jens Braun, 10.1088/0954-3899/39/3/033001arXiv:1108.4449J. Phys. 3933001hep-phJens Braun, "Fermion Interactions and Universal Be- havior in Strongly Interacting Theories," J. Phys. G39, 033001 (2012), arXiv:1108.4449 [hep-ph].
Equation of state and phase diagram of strongly interacting matter. Jan M Pawlowski, 10.1016/j.nuclphysa.2014.09.074Proceedings, 24th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (Quark Matter. 24th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (Quark MatterDarmstadt, Germany931Jan M. Pawlowski, "Equation of state and phase dia- gram of strongly interacting matter," Proceedings, 24th International Conference on Ultra-Relativistic Nucleus- Nucleus Collisions (Quark Matter 2014): Darmstadt, Germany, May 19-24, 2014, Nucl. Phys. A931, 113- 124 (2014).
Optimization of the exact renormalization group. Daniel F Litim, 10.1016/S0370-2693(00)00748-6arXiv:hep-th/0005245Phys. Lett. 486hep-thDaniel F. Litim, "Optimization of the exact renor- malization group," Phys. Lett. B486, 92-99 (2000), arXiv:hep-th/0005245 [hep-th].
Optimized renormalization group flows. Daniel F Litim, 10.1103/PhysRevD.64.105007arXiv:hep-th/0103195Phys. Rev. 64105007hep-thDaniel F. Litim, "Optimized renormalization group flows," Phys. Rev. D64, 105007 (2001), arXiv:hep- th/0103195 [hep-th].
Nonperturbative thermal flows and resummations. F Daniel, Jan M Litim, Pawlowski, 10.1088/1126-6708/2006/11/026arXiv:hep-th/0609122JHEP. 1126Daniel F. Litim and Jan M. Pawlowski, "Non- perturbative thermal flows and resummations," JHEP 11, 026 (2006), arXiv:hep-th/0609122.
The Phase Diagram of QC2D from Functional Methods. Naseemuddin Khan, Jan M Pawlowski, Fabian Rennecke, Michael M Scherer, arXiv:1512.03673hep-phNaseemuddin Khan, Jan M. Pawlowski, Fabian Ren- necke, and Michael M. Scherer, "The Phase Dia- gram of QC2D from Functional Methods," (2015), arXiv:1512.03673 [hep-ph].
Higher order quark-mesonic scattering processes and the phase structure of QCD. Jan M Pawlowski, Fabian Rennecke, 10.1103/PhysRevD.90.076002arXiv:1403.1179Phys. Rev. 9076002hep-phJan M. Pawlowski and Fabian Rennecke, "Higher or- der quark-mesonic scattering processes and the phase structure of QCD," Phys. Rev. D90, 076002 (2014), arXiv:1403.1179 [hep-ph].
The Chiral Phase Transition of QCD. Fabian Rennecke, 10.11588/heidok.00019205Ph.D. thesis, U. Heidelberg (mainFabian Rennecke, The Chiral Phase Transition of QCD., Ph.D. thesis, U. Heidelberg (main) (2015).
Fluctuation-induced modifications of the phase structure in (2+1)-flavor QCD. Fabian Rennecke, Bernd-Jochen Schaefer, 10.1103/PhysRevD.96.016009arXiv:1610.08748Phys. Rev. 9616009hep-phFabian Rennecke and Bernd-Jochen Schaefer, "Fluctuation-induced modifications of the phase structure in (2+1)-flavor QCD," Phys. Rev. D96, 016009 (2017), arXiv:1610.08748 [hep-ph].
Polyakov loop fluctuations in SU(3) lattice gauge theory and an effective gluon potential. Pok Man Lo, Bengt Friman, Olaf Kaczmarek, Krzysztof Redlich, Chihiro Sasaki, 10.1103/PhysRevD.88.074502arXiv:1307.5958Phys. Rev. 8874502hep-latPok Man Lo, Bengt Friman, Olaf Kaczmarek, Krzysztof Redlich, and Chihiro Sasaki, "Polyakov loop fluc- tuations in SU(3) lattice gauge theory and an effec- tive gluon potential," Phys. Rev. D88, 074502 (2013), arXiv:1307.5958 [hep-lat].
The QCD phase diagram: Results and challenges. Jan M Pawlowski, 10.1063/1.3574945arXiv:1012.5075Proceedings, 9th Conference on Quark Confinement and the Hadron Spectrum. 9th Conference on Quark Confinement and the Hadron SpectrumMadrid, Spain1343hep-phJan M. Pawlowski, "The QCD phase diagram: Re- sults and challenges," Proceedings, 9th Conference on Quark Confinement and the Hadron Spectrum: Madrid, Spain, 30 Aug-3 Sep 2010, AIP Conf. Proc. 1343, 75-80 (2011), arXiv:1012.5075 [hep-ph].
| [] |
[
"Stellar population synthesis of galaxies with chemical evolution model",
"Stellar population synthesis of galaxies with chemical evolution model"
] | [
"M Boquien ",
"E Lusso ",
"C Gruppioni ",
"P Tissera ",
"Shiyin Shen \nKey Laboratory for Research in Galaxies and Cosmology\nShanghai Astronomical Observatory\nChinese Academy of Sciences\n80 Nandan Road200030ShanghaiChina\n\nKey Lab for Astrophysics\n200234ShanghaiChina\n",
"Jun Yin \nKey Laboratory for Research in Galaxies and Cosmology\nShanghai Astronomical Observatory\nChinese Academy of Sciences\n80 Nandan Road200030ShanghaiChina\n"
] | [
"Key Laboratory for Research in Galaxies and Cosmology\nShanghai Astronomical Observatory\nChinese Academy of Sciences\n80 Nandan Road200030ShanghaiChina",
"Key Lab for Astrophysics\n200234ShanghaiChina",
"Key Laboratory for Research in Galaxies and Cosmology\nShanghai Astronomical Observatory\nChinese Academy of Sciences\n80 Nandan Road200030ShanghaiChina"
] | [
"Challenges in Panchromatic Galaxy Modelling with Next Genera-tion Facilities Proceedings IAU Symposium No"
] | The derivation of accurate stellar populations of galaxies is a non-trivial task because of the well-known age-metallicity degeneracy. We aim to break this degeneracy by invoking a chemical evolution model(CEM) for isolated disk galaxy, where its metallicity enrichment history(MEH) is modelled to be tightly linked to its star formation history(SFH). Our CEM has been successfully tested on several local group dwarf galaxies whose SFHs and MEHs have been both independently measured from deep color-magnitude diagrams of individual stars. By introducing the CEM into the stellar population fitting algorithm as a prior, we expect that the SFH of galaxies could be better constrained. | 10.1017/s1743921319002758 | [
"https://arxiv.org/pdf/1812.05812v1.pdf"
] | 119,332,395 | 1812.05812 | 6160783ebb39899dc4616aca5b44d04b871ec3fa |
Stellar population synthesis of galaxies with chemical evolution model
2019
M Boquien
E Lusso
C Gruppioni
P Tissera
Shiyin Shen
Key Laboratory for Research in Galaxies and Cosmology
Shanghai Astronomical Observatory
Chinese Academy of Sciences
80 Nandan Road200030ShanghaiChina
Key Lab for Astrophysics
200234ShanghaiChina
Jun Yin
Key Laboratory for Research in Galaxies and Cosmology
Shanghai Astronomical Observatory
Chinese Academy of Sciences
80 Nandan Road200030ShanghaiChina
Stellar population synthesis of galaxies with chemical evolution model
Challenges in Panchromatic Galaxy Modelling with Next Genera-tion Facilities Proceedings IAU Symposium No
3412019galaxies: stellar contentgalaxies: evolutiongalaxies: abundances
The derivation of accurate stellar populations of galaxies is a non-trivial task because of the well-known age-metallicity degeneracy. We aim to break this degeneracy by invoking a chemical evolution model(CEM) for isolated disk galaxy, where its metallicity enrichment history(MEH) is modelled to be tightly linked to its star formation history(SFH). Our CEM has been successfully tested on several local group dwarf galaxies whose SFHs and MEHs have been both independently measured from deep color-magnitude diagrams of individual stars. By introducing the CEM into the stellar population fitting algorithm as a prior, we expect that the SFH of galaxies could be better constrained.
Introduction
Star formation history(SFH), i.e. the amount of stars formed in galaxies as function of time, is one of the elements that describes galaxy evolution. Stars are formed from cool phase gas and then evolved stars return metals into the interstellar medium(ISM). ISM cools and new generation of stars formed. Among this circle, SFH is the key factor that links the gas cooling and metallicity enrichment history(MEH) of galaxies.
However, recovering SFH of galaxies is difficult (Conroy 2013). For very nearby galaxy, the deep color-magnitude diagram(CMD) reaching the turn-off stars is known as the only direct and most reliable method that can recover both of its detailed SFH and MEH. For galaxy at larger distance with only integrated stellar light observed, stellar population synthesis methods have been developed and used to fit its spectral energy distribution(SED) . In an idealized case, the SED of a galaxy can be viewed as a composition of single stellar populations(SSPs) with different ages and metallicities. However, because of the very similarities of the old SSPs(t > 1 Gyr) and the age-metallicity degeneracy among SSPs, the recovering of the detailed weights of each SSPs, especially for those with age older than 1 Gyr, is very difficult. In reality, the observables are further complicated by many other details, e.g. stellar kinematics, ionized gas emissions, central AGNs, dust attenuation and emission etc. Therefore, typically, only the first order description of the SFH, e.g. the average age or the fraction of young stellar population(t < 1 Gyr), rather than its detailed shape, could be reliably derived from the observed SEDs of galaxies.
Recovering detailed SFH
In popular SED fitting algorithms(e.g. STARLIGHT, Cid Fernandes et al. 2005), the ages and metallicties of SSPs are considered as independent parameters. To recover a detailed SFH and MEH of a galaxy, a library of SSPs with different ages and metallicities is first built, then the best combinations of these SSPs are searched in the huge library 2 Shiyin Shen & Jun Yin space. However, because of the age-metallicity degeneracy, the recovering of the SSP weights from SED is an ill-posed problem.
To show this problem more intuitively, we make a mock galaxy spectrum and test how good a full spectrum fitting can recover its SFH and MEH. We take the SFH and MEH of the LMC bar region, which are derived from deep CMD fitting (Ruiz-Lara et al. 2015) and are shown as the solid circles in Fig. 1, to build the mock spectrum. We use SSPs from MILES library (Falcón-Barroso et al., 2011) to generate the mock spectrum in the wavelength range 3600 − 7500Å with resolution R = 2000 and S/N=50 per pixel. For simplicity, we neither consider the kinematics of stellar populations nor reshape the spectrum with dust attenuation.
For this idealized case, whether can we recover the input SFH and MEH accurately using a full spectrum fitting? We take the same MILES SSP library and use a Monte Carlo Markov Chain to probe the full age and metallicity space and search for the best solution. The resulted SFH and MEH, converted from the likelihood distributions of SSP weights, are shown as the red squares in each panel of Fig. 1. As can be seen, the full spectrum fitting algorithm can not give a good constraint on either MEH or SFH.
Is an accurate MEH knowledge helpful to recover the accurate SFH? To test this idea, we set the metallicities of the different age SSPs to follow the input MEH(the blue circles in the bottom panel of Fig. 1). We then make the full spectrum fitting again. Now, we only need to probe the weights of the SSPs in age space. The resulted SFH is shown as the green crosses in the top panel of Fig. 1. As can be seen, when MEH is known as a prior, the SFH is recovered with much better accuracy. Fig. 1 shows that the prior information on the MEH, which breaks the age-metallicity degeneracy, is a key factor to recover the accurate SFH of galaxies. Actually, the importance of the prior information on SFH or MEH have already been explored by many SED fitting codes. For example, the code STECKMAP (Ocvirk et al. 2006) allows a penalization of the best fitting with an assumed MEH. However, a realistic MEH needs physical justification. That is the chemical evolution model we will discuss next.
Chemical evolution model
We consider galaxy as a pool of stars and gas. At any given time, there are both inflow and outflow of gas into the pool. The inflow gas is primordial, i.e. with zero metallicity. Stars formed from gas, and died stars return enriched gas into the pool. Some of the enriched gas makes up of the outflows.
We write the time dependent star formation rate(SFR), gas inflow and outflow rate as Ψ(t), f in (t) and f out (t) respectively. Stars are born from gas pool following the initial mass function(IMF).We assume that the massive stars(M > 1M ⊙ ) die immediately and return gas into the pool, while the low mass stars(M < 1M ⊙ ) have infinite lifetime. For the classical Salpter IMF, the gas return fraction R is ∼ 0.3. Thus, the change rate of gas in pool is
dM gas (t) dt = −(1 − R)Ψ(t) + f in (t) − f out (t) . (3.1)
On the other hand, the surface star formation rate density of a galaxy is known to be tightly correlated with its surface gas density Σ gas , i.e. the well-known Kennicutt-Schmidt relation, Ψ Σ ∝ Σ 1.4 gas (Kennicutt, 1998). For gas outflow, we assume it is driven by supernova explosion so that it is proportional to SFR and inversely correlated with the potential well of a galaxy,
f out (t) = η 0.5 + v vir 70 km s −1 −3 · Ψ(t) (3.2)
where v vir is the circular velocity of galaxy halo, and η is the wind efficiency. The evolution of metallicity Z(t) is a balance between the star formation and metal outflow, which is written as
d(ZM gas ) dt = −Z(1 − R)Ψ(t) + y(1 − R)Ψ(t) − Zf out , (3.3)
where y is the yield and we take y = 0.1. With above equations (3.1 to 3.3), once with structure parameters(to convert gas mass to surface density), for any given Ψ(t) of a galaxy, we can predict both its Z(t) and gas inflow/outflow histories.
Test on LCID galaxies
We test above CEM with LCID galaxies (Gallart, 2007), whose detailed SFHs and MEHs have been obtained using deep CMDs from HST. As an example, we take the SFHs of three different type LCID galaxies, Tucana(dSph), LGS-3(dTran), IC 1613(dIrr), and predict their Z(t) from the above CEM. We calculate their average surface densities inside the half-light radii and use their stellar masses to estimate the circular velocities. The wind efficacy is set to be η = 0.4 in all the cases. The results are shown in Fig. 2. As can be seen, for all three different galaxies, our CEM reproduces their MEHs from SFHs quite well.
Conclusion
Encouraged by Fig. 2, we believe that our CEM can be used to break the age-metallicity degeneracy in stellar population synthesis studies. Specifically, we may start from the SED fitting with both SFH and MEH free. New MEH then is predicted from preliminary SFH using CEM. With several iterations, a self-consistent SFH and MEH would be finally obtained. We expect that this algorithm would reduce the uncertainties of the final SFH estimation, and is proper for isolated disk galaxies.
Figure 1 .
1SFH(top panel) and MEH(bottom panel) of a mock galaxy. The solid circles represent the input SFH and MEH, whereas the red squares show the SFH and MEH recovered from the mock spectrum with no prior information. The green crosses in the top panel show the recovered SFH when its MEH is assumed to be the blue circles in the bottom panel.
Figure 2 .
2SFH(left panels) and MEH(right panels) of 3 LCID galaxies. The black crosses show the best estimations of SFHs and MEHs from LCID project,whereas the yellow triangles show one of their Monte-Carlo realization(indicating the uncertainty, Dan Weisz, private communication.). The solid lines in the left panels are the continuous SFHs used as the input of CEM, while the model predicted MEHs are shown as the solid curves with corresponding colors in right panels.
. Cid Fernandes, R Mateus, A Sodré, L , MNRAS. 358363Cid Fernandes R., Mateus A., Sodré L., et al., 2005, MNRAS, 358, 363
. C Conroy, ARA&A. 51393Conroy C. 2013, ARA&A , 51, 393
. J Falcón-Barroso, P Sánchez-Blázquez, A Vazdekis, A&A. 53295Falcón-Barroso J., Sánchez-Blázquez P., Vazdekis, A. et al., 2011, A&A, 532, A95
C Gallart, Proceedings of the International Astronomical Union. the International Astronomical UnionGallart C. 2007, Proceedings of the International Astronomical Union, 2
. R C Kennicutt, ARA&A. 36189Kennicutt R. C. 1998, ARA&A, 36, 189
. T Ruiz-Lara, I Pérez, C Gallart, D Alloin, M Monelli, Koleva, A&A. 58360Ruiz-Lara T., Pérez I., Gallart C., Alloin D., Monelli M., Koleva et al., 2015, A&A, 583, A60
. P Ocvirk, C Pichon, A Lanon, E Thiébaut, MNRAS. 36574Ocvirk P., Pichon C., Lanon A., Thiébaut E., 2006, MNRAS, 365, 74
| [] |
[
"Einstein-Podolsky-Rosen paradox in twin images",
"Einstein-Podolsky-Rosen paradox in twin images"
] | [
"Paul-Antoine Moreau \nDépartement d'Optique\nInstitut FEMTO-ST\nUniversité de Franche-Comté\nCNRS\nBesançonFrance\n",
"Fabrice Devaux \nDépartement d'Optique\nInstitut FEMTO-ST\nUniversité de Franche-Comté\nCNRS\nBesançonFrance\n",
"Eric Lantz \nDépartement d'Optique\nInstitut FEMTO-ST\nUniversité de Franche-Comté\nCNRS\nBesançonFrance\n"
] | [
"Département d'Optique\nInstitut FEMTO-ST\nUniversité de Franche-Comté\nCNRS\nBesançonFrance",
"Département d'Optique\nInstitut FEMTO-ST\nUniversité de Franche-Comté\nCNRS\nBesançonFrance",
"Département d'Optique\nInstitut FEMTO-ST\nUniversité de Franche-Comté\nCNRS\nBesançonFrance"
] | [] | Spatially entangled twin photons provide both promising resources for modern quantum information protocols, because of the high dimensionality of transverse entanglement 1,2 , and a test of the Einstein-Podolsky-Rosen (EPR) paradox 3 in its original form of position versus impulsion. Usually, photons in temporal coincidence are selected and their positions recorded, resulting in a priori assumptions on their spatio-temporal behavior 4 . Here, we record on two separate electron-multiplying charge coupled devices (EMCCD) cameras twin images of the entire flux of spontaneous down-conversion. This ensures a strict equivalence between the subsystems corresponding to the detection of either position (image or near-field plane) or momentum (Fourier or far-field plane) 5 . We report the highest degree of paradox ever reported and show that this degree corresponds to the number of independent degrees of freedom 6,7 or resolution cells 8 , of the images.In 1935, Einstein, Podolsky and Rosen (EPR) showed that quantum mechanics predicts that entangled particles could have both perfectly correlated positions and momenta, in contradiction with the so-called local realism where two distant particles should be treated as two different systems. Though the original intention of EPR was to show that quantum mechanics is not complete, the standard present view is that entangled particles do experience nonlocal correlations 9,10 . It can be shown that the spatial extent of these correlations corresponds to the size of a spatial unit of information, or mode, offering the possibility of detecting high dimensional entanglement in an image with a sufficient number of resolution cells 2,4 . However, in most experiments the use of single photon detectors and coincidence counting leads to the detection of a very few part of selected photons, generating a sampling loophole in fundamental demonstrations. High sensitivity array detectors have been used outside the single photoncounting regime in order to witness the quantum feature of light, showing the possibility of achieving larger signal-tonoise ratio than in classical imaging 11,12 . However, the EPR paradox is intimately connected to the particle character of light and its detection should involve single photon imaging, possible either with intensified charge coupled devices (ICCD) or, more recently, EMCCDs 13 . ICCDs exhibit a lower noise but have also a lower quantum efficiency than EMC-CDs and a more extended spatial impulse response. ICCD are therefore convenient to isolate pairs of entangled photons 14 , as shown in a recent experiment: an ICCD triggered by a single photon detector was used to detect heralded photons in various spatial modes 15 .On the other hand, because of their higher quantum efficiency EMCCDs allow efficient detection of quantum correla-tions in images, as demonstrated some years ago by measuring sub-shot-noise correlations in far-field images of spontaneous parametric down-conversion (SPDC)16,17. More recently, two experiments intended to achieve the demonstration of an EPR paradox in couples of near field and far-field images recorded with in an EMCCD. The first experiment, in our group, involved the detection of twin images on a single camera, by separating in the near-field the cross-polarized photons with a polarizing beam-splitter, inducing some overlap of the nearfield images and a rather small resolution in the far-field because of walk-off. The results exhibited a low degree of paradox, far from the theoretical values, though highly significant and in accordance with the full-field requirements 18 . The second experiment 19 exhibited also both near field (position) and far field (momentum) correlations, with a much lower product of the spatial extents. However only one beam of particles was involved, because of type-I phase matching, making the results questionable if intended as a demonstration of an EPR paradox. Moreover, the results were obtained for only one dimension because of smearing effects and detection of correlations between adjacent pixels.In the present experiment, the use of two cameras allows a separation of the twin images without any further optical component, thanks to walk-off, and a perfect identity of the subsystems but the position of the imaging systems, composed on each arm of a lens and a camera. Before describing our experimental results, let us recall that an EPR paradox arises when correlations violate an inequality corresponding to the Heisenberg uncertainty principle if applied to a single particle 1 or 2, but expressed in terms of conditional variances 5,21 :(1)where ρ i is the transverse position of photon i (i = 1, 2) at the center of the crystal and p i its transverse momentum. In order to make the demonstration consistent, the statistical average made to estimate the variances should be evaluated on the same system in the near and the far field. By using two EM-CCD cameras that detect photons in the whole SPDC field, we ensured this consistency. By approximating the phase matching function of SPDC to a Gaussian, the wave function of the biphoton can be written 22 : | 10.1103/physrevlett.113.160401 | [
"https://arxiv.org/pdf/1404.3028v2.pdf"
] | 7,811,563 | 1404.3028 | c3e2e02894ae683d0d71c8f61fa1e3e652074b32 |
Einstein-Podolsky-Rosen paradox in twin images
24 Nov 2014
Paul-Antoine Moreau
Département d'Optique
Institut FEMTO-ST
Université de Franche-Comté
CNRS
BesançonFrance
Fabrice Devaux
Département d'Optique
Institut FEMTO-ST
Université de Franche-Comté
CNRS
BesançonFrance
Eric Lantz
Département d'Optique
Institut FEMTO-ST
Université de Franche-Comté
CNRS
BesançonFrance
Einstein-Podolsky-Rosen paradox in twin images
24 Nov 2014(Dated: November 25, 2014)
Spatially entangled twin photons provide both promising resources for modern quantum information protocols, because of the high dimensionality of transverse entanglement 1,2 , and a test of the Einstein-Podolsky-Rosen (EPR) paradox 3 in its original form of position versus impulsion. Usually, photons in temporal coincidence are selected and their positions recorded, resulting in a priori assumptions on their spatio-temporal behavior 4 . Here, we record on two separate electron-multiplying charge coupled devices (EMCCD) cameras twin images of the entire flux of spontaneous down-conversion. This ensures a strict equivalence between the subsystems corresponding to the detection of either position (image or near-field plane) or momentum (Fourier or far-field plane) 5 . We report the highest degree of paradox ever reported and show that this degree corresponds to the number of independent degrees of freedom 6,7 or resolution cells 8 , of the images.In 1935, Einstein, Podolsky and Rosen (EPR) showed that quantum mechanics predicts that entangled particles could have both perfectly correlated positions and momenta, in contradiction with the so-called local realism where two distant particles should be treated as two different systems. Though the original intention of EPR was to show that quantum mechanics is not complete, the standard present view is that entangled particles do experience nonlocal correlations 9,10 . It can be shown that the spatial extent of these correlations corresponds to the size of a spatial unit of information, or mode, offering the possibility of detecting high dimensional entanglement in an image with a sufficient number of resolution cells 2,4 . However, in most experiments the use of single photon detectors and coincidence counting leads to the detection of a very few part of selected photons, generating a sampling loophole in fundamental demonstrations. High sensitivity array detectors have been used outside the single photoncounting regime in order to witness the quantum feature of light, showing the possibility of achieving larger signal-tonoise ratio than in classical imaging 11,12 . However, the EPR paradox is intimately connected to the particle character of light and its detection should involve single photon imaging, possible either with intensified charge coupled devices (ICCD) or, more recently, EMCCDs 13 . ICCDs exhibit a lower noise but have also a lower quantum efficiency than EMC-CDs and a more extended spatial impulse response. ICCD are therefore convenient to isolate pairs of entangled photons 14 , as shown in a recent experiment: an ICCD triggered by a single photon detector was used to detect heralded photons in various spatial modes 15 .On the other hand, because of their higher quantum efficiency EMCCDs allow efficient detection of quantum correla-tions in images, as demonstrated some years ago by measuring sub-shot-noise correlations in far-field images of spontaneous parametric down-conversion (SPDC)16,17. More recently, two experiments intended to achieve the demonstration of an EPR paradox in couples of near field and far-field images recorded with in an EMCCD. The first experiment, in our group, involved the detection of twin images on a single camera, by separating in the near-field the cross-polarized photons with a polarizing beam-splitter, inducing some overlap of the nearfield images and a rather small resolution in the far-field because of walk-off. The results exhibited a low degree of paradox, far from the theoretical values, though highly significant and in accordance with the full-field requirements 18 . The second experiment 19 exhibited also both near field (position) and far field (momentum) correlations, with a much lower product of the spatial extents. However only one beam of particles was involved, because of type-I phase matching, making the results questionable if intended as a demonstration of an EPR paradox. Moreover, the results were obtained for only one dimension because of smearing effects and detection of correlations between adjacent pixels.In the present experiment, the use of two cameras allows a separation of the twin images without any further optical component, thanks to walk-off, and a perfect identity of the subsystems but the position of the imaging systems, composed on each arm of a lens and a camera. Before describing our experimental results, let us recall that an EPR paradox arises when correlations violate an inequality corresponding to the Heisenberg uncertainty principle if applied to a single particle 1 or 2, but expressed in terms of conditional variances 5,21 :(1)where ρ i is the transverse position of photon i (i = 1, 2) at the center of the crystal and p i its transverse momentum. In order to make the demonstration consistent, the statistical average made to estimate the variances should be evaluated on the same system in the near and the far field. By using two EM-CCD cameras that detect photons in the whole SPDC field, we ensured this consistency. By approximating the phase matching function of SPDC to a Gaussian, the wave function of the biphoton can be written 22 :
Spatially entangled twin photons provide both promising resources for modern quantum information protocols, because of the high dimensionality of transverse entanglement 1,2 , and a test of the Einstein-Podolsky-Rosen (EPR) paradox 3 in its original form of position versus impulsion. Usually, photons in temporal coincidence are selected and their positions recorded, resulting in a priori assumptions on their spatio-temporal behavior 4 . Here, we record on two separate electron-multiplying charge coupled devices (EMCCD) cameras twin images of the entire flux of spontaneous down-conversion. This ensures a strict equivalence between the subsystems corresponding to the detection of either position (image or near-field plane) or momentum (Fourier or far-field plane) 5 . We report the highest degree of paradox ever reported and show that this degree corresponds to the number of independent degrees of freedom 6,7
or resolution cells 8 , of the images.
In 1935, Einstein, Podolsky and Rosen (EPR) showed that quantum mechanics predicts that entangled particles could have both perfectly correlated positions and momenta, in contradiction with the so-called local realism where two distant particles should be treated as two different systems. Though the original intention of EPR was to show that quantum mechanics is not complete, the standard present view is that entangled particles do experience nonlocal correlations 9,10 . It can be shown that the spatial extent of these correlations corresponds to the size of a spatial unit of information, or mode, offering the possibility of detecting high dimensional entanglement in an image with a sufficient number of resolution cells 2,4 . However, in most experiments the use of single photon detectors and coincidence counting leads to the detection of a very few part of selected photons, generating a sampling loophole in fundamental demonstrations. High sensitivity array detectors have been used outside the single photoncounting regime in order to witness the quantum feature of light, showing the possibility of achieving larger signal-tonoise ratio than in classical imaging 11,12 . However, the EPR paradox is intimately connected to the particle character of light and its detection should involve single photon imaging, possible either with intensified charge coupled devices (ICCD) or, more recently, EMCCDs 13 . ICCDs exhibit a lower noise but have also a lower quantum efficiency than EMC-CDs and a more extended spatial impulse response. ICCD are therefore convenient to isolate pairs of entangled photons 14 , as shown in a recent experiment: an ICCD triggered by a single photon detector was used to detect heralded photons in various spatial modes 15 .
On the other hand, because of their higher quantum efficiency EMCCDs allow efficient detection of quantum correla-tions in images, as demonstrated some years ago by measuring sub-shot-noise correlations in far-field images of spontaneous parametric down-conversion (SPDC) 16,17 . More recently, two experiments intended to achieve the demonstration of an EPR paradox in couples of near field and far-field images recorded with in an EMCCD. The first experiment, in our group, involved the detection of twin images on a single camera, by separating in the near-field the cross-polarized photons with a polarizing beam-splitter, inducing some overlap of the nearfield images and a rather small resolution in the far-field because of walk-off. The results exhibited a low degree of paradox, far from the theoretical values, though highly significant and in accordance with the full-field requirements 18 . The second experiment 19 exhibited also both near field (position) and far field (momentum) correlations, with a much lower product of the spatial extents. However only one beam of particles was involved, because of type-I phase matching, making the results questionable if intended as a demonstration of an EPR paradox. Moreover, the results were obtained for only one dimension because of smearing effects and detection of correlations between adjacent pixels.
In the present experiment, the use of two cameras allows a separation of the twin images without any further optical component, thanks to walk-off, and a perfect identity of the subsystems but the position of the imaging systems, composed on each arm of a lens and a camera. Before describing our experimental results, let us recall that an EPR paradox arises when correlations violate an inequality corresponding to the Heisenberg uncertainty principle if applied to a single particle 1 or 2, but expressed in terms of conditional variances 5,21 :
∆ 2 (ρ 1 − ρ 2 ) ∆ 2 (p 1 + p 2 ) ≥h 2 4
(1)
where ρ i is the transverse position of photon i (i = 1, 2) at the center of the crystal and p i its transverse momentum. In order to make the demonstration consistent, the statistical average made to estimate the variances should be evaluated on the same system in the near and the far field. By using two EM-CCD cameras that detect photons in the whole SPDC field, we ensured this consistency. By approximating the phase matching function of SPDC to a Gaussian, the wave function of the biphoton can be written 22 : where N is a normalization constant, ρ i = (x i , y i ), p i = (p xi , p yi ), σ P the standard deviation of the gaussian pump beam, and σ φ the standard deviation, defined in the near-field, of the Fourier transform of the phase matching function defined in the far-field. In our experimental conditions where σ P >> σ φ , these equations show that the product of conditional variances is equal to:
Ψ(ρ 1 , ρ 2 ) = N exp − |ρ 1 + ρ 2 | 2 4σ 2 p exp − |ρ 1 − ρ 2 | 2 4σ 2 φ (2) Ψ(p 1 , p 2 ) = 1 N π 2 exp −σ 2 P |p 1 + p 2 | 2 4h 2 exp −σ 2 φ |p 1 − p 2 | 2 4h 2(3)∆ 2 (ρ 1 − ρ 2 ) ∆ 2 (p 1 + p 2 ) =h 2 σ 2 φ σ 2 p =h 2 4V(4)
where V is defined by this equation as the degree of paradox. Using results of Law and Eberly 6 , it can be shown that V is also the Schmidt number of the entanglement, i.e. the whole dimensionality of the biphoton in the two-dimensional transverse space supposed isotropic. For an unidimensional system, V becomes the square of the Schmidt number 23 .
The experimental setup is shown in Fig.1. Pump pulses at 355 nm provided by a 27 mW laser illuminated a 0.6-mm long β barium borate (BBO) nonlinear crystal cut for type-II phase matching. The signal and idler photons were separated by means of two mirrors and sent to two independent imaging systems. The far-field image of the SPDC was formed on the EMCCDs placed in the focal plane of two 120-mm lenses, Fig.1a. In the near-field configuration, Fig.1b, the plane of the BBO crystal was imaged on the EMCCDs with a transversal magnification M = 2.47 ± 0.01. Note that only the positions of the lenses and cameras are different in the two configurations. The back-illuminated EMCCD cameras (Andor iXon3) have a quantum efficiency greater than 90% in the visible range. The detector area is formed by 512 × 512 pixels, with a pixel size of s pix = 16 × 16µm 2 . We used a readout rate of 10 MHz at 14 bits, and the cameras were cooled to −100 • C. An image corresponds to the summation of 100 laser shots, i.e an exposure time of 0.1s and a dead-time between two successive images of about the same value, in order to allow a perfect synchronization between both cameras. Measurements were performed for a crystal orientation corresponding to noncritical phase matching at degeneracy, i.e., collinear orientation of the signal and idler Poynting vectors in the crystal 24 . Photon pairs emitted around the degeneracy were selected by means of narrow-band interference filters centered at 710 nm (∆λ = 4nm). The photon-counting regime was ensured by adjusting the exposure time in such a way that the mean fluency of SPDC was between 0.1 and 0.2 photon per pixel in order to minimize the whole number of false detections 13 . The mean number of photons per spatiotemporal mode was less than 10 −3 , in good agreement with the hypothesis of pure spontaneous parametric downconversion, without any stimulated amplification. Following a method previously reported 13 , we applied a thresholding procedure on the image to convert the gray scales into binary values that correspond to 0 or 1 photon. The conditional probability distributions calculated using 35000 images are shown in Fig.2. The correlation profiles agree with the theoretical expectations (2) and (3)
with σ p >> σ φ .
We have shown 20 that the conditional variances ∆ 2 (ρ 1 − ρ 2 ) and ∆ 2 (p 1 + p 2 ) correspond to the width of the normalized cross-correlation of photo-detection images. The experimental values obtained by fitting the normalized crosscorrelations presented in Fig.3 are reported in Tab.I, for the two orthogonal directions of the transverse plane x and y. Using the measured values given in Tab. 1, we find the following product of conditional variances:
∆ 2 (x1 − x2) 299 ± 14 µm 2 ∆ 2 (y1 − y2) 168 ± 7 µm 2 ∆ 2 (px1 − px2) (9.70 ± 0.1) · 10 −6h2 µm −2 ∆ 2 (py1 − py2) (2.53 ± 0.04) · 10 −6h2 µm −2∆ 2 (x 1 − x 2 )∆ 2 (p x1 + p x2 ) = (2.9 ± 0.2) · 10 −3h2 (5) ∆ 2 (y 1 − y 2 )∆ 2 (p y1 + p y2 ) = (4.2 ± 0.2) · 10 −4h2 (6)
Those results clearly violate inequality (1), thus exhibiting an EPR paradox in the two transverse dimensions. The variance product of the experimental values is more than 1,200 standard deviations under the classical limit along x and more than 12,000 standard deviations under this limit along y. Moreover, the results are in rather good agreement with the theoretical expectations 8.6 · 10 −4h2 on x and 2.6 · 10 −4h2 on y obtained by a numerical computation that takes into account the effect of the width of the interference filter. This effect shown in Fig.2e explains the anisotropy in Fig.3a, i.e an enlargement in the x direction for the large values of p x1 . By using Eq.(4), we find along x a degree of paradox of 86 ± 5 and along y of 595 ± 40. To the best of our knowledge, this degree of 595 is the highest ever reported for an EPR paradox, whatever the considered domain. We show in Fig. 4 that the minimum number of images that allows a safe assessment of the correlation peaks in both spaces is of the order of 20. Indeed a quantum correlation peak is evidenced if it cannot be confounded, with high probability, with random fluctuations of the background noise. Without any a priori assumption on the position of the peak, this is ensured with a confidence of 99% if the magnitude of the true peak is greater than 4,5 standard deviations, for an image of 64 × 64 pixels obtained by summing the correlations on groups of 8x8 pixels. This grouping is performed in order to adapt the size of the effective pixel to the size of the correlation peak. In Fig. 4, we have defined the signal-to-noise-ratio (SNR) as the magnitude of the correlation peak divided by the standard deviation, after grouping, of the correlation image outside the peak area. The minimum number of images necessary to demonstrate entanglement is only two in the farfield, where deterministic distortions appear to be smaller than in the image plane.
Finally, we have verified that the images exhibit a sub-shotnoise statistics in both the near-field and the far-field: r n = 0.9975±0.0004 and r f = 0.9959±0.0003, where r is defined by :
r = ∆ 2 (N 1 − N 2 ) N 1 + N 2(7)
that is, the variance of the photon number difference N 1 (ρ) − N 2 (ρ) (and N 1 (p) − N 2 (−p) in far field) normalized to be expressed in shot noise units. These experimental results are under the classical limit 1 respectively by more than 5 and 10 standard deviation, witnessing the quantum, i.e particle like, character of the correlations 16 . Note that smaller values of r can be obtained by grouping the pixels 20 , in accordance with the fact that the quantum correlation peak extends on several pixels.
To conclude, we have demonstrated a two dimensional EPR paradox in the closest form of its original proposal by recording the behavior of light in couples of twin images. The quantum character of these images has been doubly demon-strated firstly by full-field measurement of a high degree of EPR paradox for both transverse directions and secondly by demonstrating sub-shot noise character in both the near-field and the far-field. Reliable results can be obtained with 20 images, i.e. an acquisition time of 4 seconds and a computation time that scales also in seconds since cross correlations are computed using FFT algorithms. This should be compared to days for raster scanning, or hours for compressivesensing 1 . Because of the experimental anisotropy, the dimensionality of entanglement, or Schmidt number K, can be assessed as the square root of the product of the paradox degrees in each direction: K = √ 594 × 85 = 225. Such highdimensionality spatial entanglement has applications in numerous fields of quantum optics, like quantum cryptography 25 or quantum computation 26 .
FIG. 1 :FIG. 2 :
12Experimental setups used to imaging correlations. (a), measurement of momentum correlations with the cameras in the focal plane. Inserts: sums of 700 far-field images; px1 = −px2, py1 = −py2 are the coordinates of twin pixels. (b), cameras in the crystal image plane and sums of 700 near-field images with twin pixels in x1 = x2, y1 = y2. Joint probabilities versus the transverse spatial coordinates Color scales are expressed in coincidence counts over 35000 pairs of images, corrected from the mean corresponding to accidental coincidences. (a,b): near-field. (c,d): far-field. (e):In the far field, the correlations arise in a coherence area that is larger for momenta the most distant from the pump direction, i.e. for the largest values of x1, x being the direction along which the two fluorescence beams are separated from the walk-off. This observation is explained by the finite bandwidth of the interference filter20 , that leads to the detection of twin photons out of the degeneracy, λ1 = λ2.
FIG. 3 :
3Normalized cross-correlation functions in position and momentum :The cross-correlation is calculated over 700 images in the far-field (a,c) and image plane (b,d). In (c) and (d) are presented cross-correlation of images that do not share any pump pulses.
FIG. 4 :
4Normalized cross-correlation function versus the number of images: left images: correlation computed on the physical pixels (only the central part is presented). Smaller right images: correlation computed after grouping 8 × 8 pixels .(a,b), far-field. (c,d), near field .
TABLE I :
IInferred variances.
Variances
Measured values
images SNR = 13.3 20 images SNR = 1.8
Author ContributionsE.L. and F.D. designed the experiment. P.A.M. optimized the experimental set-up and performed the experimental work under the supervision of F.D. P.A.M. made the data treatment and the numerical modeling, with a participation of E.L. P.A.M. and E.L. wrote the initial manuscript text. All authors discussed the results and substantially contributed to the manuscript.Additional informationThe authors declare no competing financial interests.
Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera. G A Howland, J C Howell, Physical review X. 311013Howland, G. A. & Howell, J. C. Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera. Physical review X 3, 011013 (2013).
Quantum mutual information capacity for high-dimensional entangled states. P B Dixon, G A Howland, J Schneeloch, J C Howell, Phys. Rev. Lett. 108143603Dixon, P. B., Howland, G. A., Schneeloch, J. & Howell, J. C. Quantum mutual information capacity for high-dimensional en- tangled states. Phys. Rev. Lett. 108, 143603 (2012).
Can quantummechanical description of physical reality be considered complete?. A Einstein, B Podolsky, N Rosen, Phys. Rev. 47Einstein, A., Podolsky, B. & Rosen, N. Can quantum- mechanical description of physical reality be considered com- plete? Phys. Rev. 47, 777-780 (1935).
Realization of the Einstein-Podolsky-Rosen paradox using momentum-and position-entangled photons from spontaneous parametric down conversion. J C Howell, R S Bennink, S J Bentley, R W Boyd, Phys. Rev. Lett. 92210403Howell, J. C., Bennink, R. S., Bentley, S. J. & Boyd, R. W. Realization of the Einstein-Podolsky-Rosen paradox using momentum-and position-entangled photons from spontaneous parametric down conversion. Phys. Rev. Lett. 92, 210403 (2004).
Colloquium : The Einstein-Podolsky-Rosen paradox: from concepts to applications. M D Reid, Rev. Mod. Phys. 81Reid, M. D. et al. Colloquium : The Einstein-Podolsky-Rosen paradox: from concepts to applications. Rev. Mod. Phys. 81, 1727-1751 (2009).
Analysis and interpretation of high transverse entanglement in otical parametric down conversion. C Law, J Eberly, Phys. Rev. Lett. 12127903Law, C. & Eberly, J. Analysis and interpretation of high trans- verse entanglement in otical parametric down conversion. Phys. Rev. Lett. 12, 127903 (2004).
Effect of spatial filtering on the Schmidt decomposition of entangled photons. M Van Exter, A Aiello, S Oemrawsingh, G Nienhuis, J P Woerdman, Phys. Rev. A. 7412309van Exter, M., Aiello, A., Oemrawsingh, S., Nienhuis, G. & Woerdman, J. P. Effect of spatial filtering on the Schmidt de- composition of entangled photons. Phys. Rev. A 74, 012309 (2006).
Transfer function of spatial frequencies in parametric amplification : experimental analysis and application to picosecond spatial filtering. F Devaux, E Lantz, Optics Communications. 114Devaux, F. & Lantz, E. Transfer function of spatial frequencies in parametric amplification : experimental analysis and appli- cation to picosecond spatial filtering. Optics Communications 114, 295-300 (1995).
On the Einstein-Podolsky-Rosen paradox. J S Bell, Physics. 1Bell, J. S. On the Einstein-Podolsky-Rosen paradox. Physics 1, 195-200 (1964).
Experimental tests of realistic local theories via Bell's theorem. A Aspect, P Grangier, G Roger, Phys. Rev. Lett. 47Aspect, A., Grangier, P. & Roger, G. Experimental tests of realistic local theories via Bell's theorem. Phys. Rev. Lett. 47, 460-463 (1981).
Experimental realization of sub-shot-noise quantum imaging. G Brida, M Genovese, I R Berchera, Nat Photon. 4Brida, G., Genovese, M. & Berchera, I. R. Experimental re- alization of sub-shot-noise quantum imaging. Nat Photon 4, 227-230 (2010).
Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion. O Jedrkiewicz, Phys. Rev. Lett. 93243601Jedrkiewicz, O. et al. Detection of sub-shot-noise spatial cor- relation in high-gain parametric down conversion. Phys. Rev. Lett. 93, 243601 (2004).
Multiimaging and Bayesian estimation for photon counting with EMCCDs. E Lantz, J.-L Blanchet, L Furfaro, F Devaux, Monthly Notices of the Royal Astronomical Society. 386Lantz, E., Blanchet, J.-L., Furfaro, L. & Devaux, F. Multi- imaging and Bayesian estimation for photon counting with EMCCDs. Monthly Notices of the Royal Astronomical Soci- ety 386, 2262-2270 (2008).
Two-dimensional wave vector correlations in spontaneous parametric downconversion explored with an intensified ccd camera. S S R Oemrawsingh, W J Van Drunen, E R Eliel, J P Woerdman, J. Opt. Soc. Am. B. 19Oemrawsingh, S. S. R., van Drunen, W. J., Eliel, E. R. & Wo- erdman, J. P. Two-dimensional wave vector correlations in spontaneous parametric downconversion explored with an in- tensified ccd camera. J. Opt. Soc. Am. B 19, 2391-2395 (2002).
Real-time imaging of quantum entanglement. R Fickler, M Krenn, R Lapkiewicz, S Ramelow, A Zeilinger, Sci. Rep. 3Fickler, R., Krenn, M., Lapkiewicz, R., Ramelow, S. & Zeilinger, A. Real-time imaging of quantum entanglement. Sci. Rep. 3, -(2013).
Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime. J.-L Blanchet, F Devaux, L Furfaro, E Lantz, Phys. Rev. Lett. 101233604Blanchet, J.-L., Devaux, F., Furfaro, L. & Lantz, E. Measure- ment of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime. Phys. Rev. Lett. 101, 233604 (2008).
Purely spatial coincidences of twin photons in parametric spontaneous down-conversion. J.-L Blanchet, F Devaux, L Furfaro, E Lantz, Phys. Rev. A. 8143825Blanchet, J.-L., Devaux, F., Furfaro, L. & Lantz, E. Purely spatial coincidences of twin photons in parametric spontaneous down-conversion. Phys. Rev. A 81, 043825 (2010).
Realization of the purely spatial Einstein-Podolsky-Rosen paradox in full-field images of spontaneous parametric downconversion. P.-A Moreau, J Mougin-Sisini, F Devaux, E Lantz, Phys. Rev. A. 8610101Moreau, P.-A., Mougin-Sisini, J., Devaux, F. & Lantz, E. Re- alization of the purely spatial Einstein-Podolsky-Rosen para- dox in full-field images of spontaneous parametric down- conversion. Phys. Rev. A 86, 010101 (2012).
Imaging high-dimensional spatial entanglement with a camera. M Edgar, Nat Commun. 3984Edgar, M. et al. Imaging high-dimensional spatial entanglement with a camera. Nat Commun 3, 984-(2012).
Towards the evidence of a purely spatial Einstein-Podolsky-Rosen paradox in images: measurement scheme and first experimental results. F Devaux, J Mougin-Sisini, P.-A Moreau, E Lantz, The European Physical Journal D. 66Devaux, F., Mougin-Sisini, J., Moreau, P.-A. & Lantz, E. To- wards the evidence of a purely spatial Einstein-Podolsky-Rosen paradox in images: measurement scheme and first experimental results. The European Physical Journal D 66, 1-6 (2012).
Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification. M D Reid, Phys. Rev. A. 40Reid, M. D. Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification. Phys. Rev. A 40, 913-923 (1989).
Propagation of transverse intensity correlations of a two-photon state. D S Tasca, S P Walborn, P H Souto Ribeiro, F Toscano, P Pellat-Finet, Phys. Rev. A. 7933801Tasca, D. S., Walborn, S. P., Souto Ribeiro, P. H., Toscano, F. & Pellat-Finet, P. Propagation of transverse intensity correlations of a two-photon state. Phys. Rev. A 79, 033801 (2009).
Gaussian modelling and Schmidt modes of SPDC biphoton states. M Fedorov, Y M Mikhailova, P A Volkov, J. Phys. B. 42175503Fedorov, M., Mikhailova, Y. M. & Volkov, P. A. Gaussian mod- elling and Schmidt modes of SPDC biphoton states. J. Phys. B 42, 175503 (2009).
The phase-mismatch vector and resolution in image parametric amplification. E Lantz, F Devaux, Journal of Optics A. 2Lantz, E. & Devaux, F. The phase-mismatch vector and resolu- tion in image parametric amplification. Journal of Optics A 2, 362-364 (2000).
One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering. C Branciard, E G Cavalcanti, S P Walborn, V Scarani, H M Wiseman, Phys. Rev. A. 8510301Branciard, C., Cavalcanti, E. G., Walborn, S. P., Scarani, V. & Wiseman, H. M. One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steer- ing. Phys. Rev. A 85, 010301 (2012).
Continuous-variable quantum computation with spatial degrees of freedom of photons. D S Tasca, R M Gomes, F Toscano, P H Souto Ribeiro, S P Walborn, Phys. Rev. A. 8352325Tasca, D. S., Gomes, R. M., Toscano, F., Souto Ribeiro, P. H. & Walborn, S. P. Continuous-variable quantum computation with spatial degrees of freedom of photons. Phys. Rev. A 83, 052325 (2011).
| [] |
[
"Reverse Mathematics and Algebraic Field Extensions",
"Reverse Mathematics and Algebraic Field Extensions"
] | [
"François G Dorais ",
"Jeffry Hirst ",
"Paul Shafer "
] | [] | [] | This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section §2, we show that WKL 0 is equivalent to the ability to extend F -automorphisms of field extensions to automorphisms of F , the algebraic closure of F . Section §3 explores finitary conditions for embeddability. Normal and Galois extensions are discussed in section §4, and the Galois correspondence theorems for infinite field extensions are treated in section §5.Reverse mathematics is a foundational program in which mathematical theorems are analyzed using a hierarchy of subsystems of second order arithmetic. This paper uses three such subsystems. The base system RCA 0 includes Σ 0 1 -IND (induction for Σ 0 1 formulas) and set comprehension for ∆ 0 1 definable subsets of N. The stronger system WKL 0 appends König's theorem restricted to binary trees (subtrees of 2 <N ). The even stronger system ACA 0 adds comprehension for arithmetically definable subsets of N. For a detailed formulation of these subsystems and related analysis of many mathematical theorems, see Simpson's book[16].Reverse mathematics of countable algebra, including topics from group theory, ring theory, and field theory, can be found in the paper of Friedman, Simpson, and Smith [4]. Further discussion appears throughout Simpson's book[16]. A field is a set of natural numbers with operations and constants satisfying the field axioms. Field embeddings and isomorphisms can be defined as sets of (codes for) ordered pairs of field elements. Polynomials can be encoded by finite strings of coefficients, so polynomial rings are sets of (codes for) finite strings, with related ring operations. For details pertaining to any of these definitions, see either of the references above.Our study of fields begins in the next section with the definition of an algebraic field extension. To simplify the exposition in sections §1 through §3, we restrict our discussion to characteristic 0 fields. Consequently, in these sections all irreducible polynomials are separable. We indicate how to extend results of earlier sections to fields of other characteristics in section §6. | 10.3233/com-13021 | [
"https://arxiv.org/pdf/1209.4944v2.pdf"
] | 11,031,541 | 1209.4944 | a351f765cb328ac924b58a2f7a0699f4acb87f51 |
Reverse Mathematics and Algebraic Field Extensions
May 2013 September 3, 2012
François G Dorais
Jeffry Hirst
Paul Shafer
Reverse Mathematics and Algebraic Field Extensions
May 2013 September 3, 2012(Revised May 10, 2013)
This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section §2, we show that WKL 0 is equivalent to the ability to extend F -automorphisms of field extensions to automorphisms of F , the algebraic closure of F . Section §3 explores finitary conditions for embeddability. Normal and Galois extensions are discussed in section §4, and the Galois correspondence theorems for infinite field extensions are treated in section §5.Reverse mathematics is a foundational program in which mathematical theorems are analyzed using a hierarchy of subsystems of second order arithmetic. This paper uses three such subsystems. The base system RCA 0 includes Σ 0 1 -IND (induction for Σ 0 1 formulas) and set comprehension for ∆ 0 1 definable subsets of N. The stronger system WKL 0 appends König's theorem restricted to binary trees (subtrees of 2 <N ). The even stronger system ACA 0 adds comprehension for arithmetically definable subsets of N. For a detailed formulation of these subsystems and related analysis of many mathematical theorems, see Simpson's book[16].Reverse mathematics of countable algebra, including topics from group theory, ring theory, and field theory, can be found in the paper of Friedman, Simpson, and Smith [4]. Further discussion appears throughout Simpson's book[16]. A field is a set of natural numbers with operations and constants satisfying the field axioms. Field embeddings and isomorphisms can be defined as sets of (codes for) ordered pairs of field elements. Polynomials can be encoded by finite strings of coefficients, so polynomial rings are sets of (codes for) finite strings, with related ring operations. For details pertaining to any of these definitions, see either of the references above.Our study of fields begins in the next section with the definition of an algebraic field extension. To simplify the exposition in sections §1 through §3, we restrict our discussion to characteristic 0 fields. Consequently, in these sections all irreducible polynomials are separable. We indicate how to extend results of earlier sections to fields of other characteristics in section §6.
Algebraic extensions and algebraic closures
We provide a definition of algebraic field extension in the context of second order arithmetic and give a few examples of fields and extensions which RCA 0 proves exist. Our definition of an algebraic extension extends the definition of algebraic closure in Simpson's book [16,Definition II.9.2]. The definition uses the following notational shorthand. Given a field F , a ∈ F , f (x) = i∈I c i x i a polynomial in F [x], and ϕ a field embedding of F , we write ϕ(f ) = i∈I ϕ(c i )x i and ϕ(f )(a) = i∈I ϕ(c i )a i . Definition 1. (RCA 0 ) An algebraic extension of a countable field F is a pair K, ϕ where K is a countable field, ϕ is an embedding of F into K, and for every a ∈ K there is a nonzero f (x) ∈ F [x] such that ϕ(f )(a) = 0. When appropriate, we drop the mention of ϕ and denote the extension by K alone.
If K is an algebraic extension of F that is algebraically closed, we say K is an algebraic closure of F , and often write F for K.
RCA 0 can prove the existence of algebraic closures, as shown in Theorem 2.5 of Friedman, Simpson, and Smith [4]. However, the notation F in the preceding definition is somewhat misleading, since RCA 0 does not prove the uniqueness of algebraic closures up to isomorphism. To be specific, Theorem 3.3 of Friedman, Simpson, and Smith [4] shows that the statement "for every field F , the algebraic closure of F is unique up to isomorphism" is equivalent to WKL 0 . As for other algebraic extensions, we often drop ϕ and simply denote an algebraic closure by F .
In order to describe the images of fields under embeddings, Friedman, Simpson, and Smith [4] introduce the notion of a Σ 0 1 -subfield.
Definition 2. (RCA 0 ) Suppose K is a countable field. A Σ 0 1 formula θ(x) defines a Σ 0 1subfield of K if Lemma 3. (RCA 0 ) If K, ϕ is an algebraic extension of F and θ(x) defines a Σ 0 1 -F -subfield of K, then there is an algebraic extension G, ψ of F and an embedding τ of G into K such that 1. (∀x ∈ F )(ϕ(x) = τ (ψ(x))), and 2. (∀x ∈ K)(θ(x) ↔ (∃y ∈ G)(τ (y) = x)).
Proof. If the subfield defined by θ is finite, then the theorem is trivial. Let K, ϕ be an algebraic extension of F and suppose θ defines an infinite Σ 0 1 -F -subfield of K. Since θ is a Σ 0 1 formula, RCA 0 proves the existence of an injective function τ : N → K that enumerates all those elements of K for which θ holds. Without loss of generality, we may assume that τ (0) = 0 K and τ (1) = 1 K . Define field operations + and · on N by i + j = τ −1 (τ (i) + τ (j)) and i · j = τ −1 (τ (i) · τ (j)). Let G denote N with these operations. Define ψ : F → G by letting ψ(x) = τ −1 (ϕ(x)) for each x ∈ F . Since θ defines a Σ 0 1 -F -subfield of K, RCA 0 proves that ψ and the field operations of G all exist and are all total. Routine verifications show that G, ψ and τ satisfy the conclusions of the theorem.
In later constructions, it is convenient to have ready access to familiar field extensions of Q. Working in RCA 0 , we can fix a representation of Q, for example that in Theorem II.4.2 of Simpson [16]. By Theorem 2.5 of Friedman, Simpson, and Smith [4], we can find Q, an algebraic closure of Q. As a concrete example of a specific extension, we can locate the first element of Q satisfying x 2 − 2 = 0, and denote it by √ 2. The collection of terms of the form q 0 + q 1 √ 2 with q 0 , q 1 ∈ Q is a Σ 0 1 -subfield of Q. By Lemma 3, RCA 0 proves that there is an algebraic extension of Q that is isomorphic to this Σ 0 1 -subfield; we denote it by Q( √ 2). In the minimal model of RCA 0 consisting of ω and the computable sets, this field is a computable presentation of Q( √ 2); in this case, an algebraist might say it is Q( √ 2). Similarly, for any sequence α i | i ∈ N of elements in Q, RCA 0 proves the existence of the algebraic extension Q(α i | i ∈ N). If we like, we can apply Theorem 2.12 of [4], take the algebraic closure of the real closure of Q, and adjoin a real (or non-real) cube root of 2 to Q in the same fashion. Similar constructions can be carried out over other base fields.
Besides proving the existence of all these field extensions, RCA 0 can prove many useful results about them. The following two examples play an important role in the next section. Proof. This is a formalization of the main theorem in the paper of Roth [15]. His argument is essentially an application of Π 0 1 -IND, which is provable in RCA 0 by Corollary II.3.10 of Simpson [16]. For a sketch of a generalization of this result to fields other than Q, see Lemma 33 in section §6.
Lemma 5. (RCA 0 ) Let p 1 , . . . , p n and q 1 , . . . , q r be disjoint lists of distinct primes. Then √ q 1 / ∈ Q( √ p 1 , . . . , √ p n , √ q 1 q 2 , . . . , √ q 1 q r ).
Proof. Suppose p 1 , . . . , p n and q 1 , . . . , q r are as specified and the lemma fails. Write √ q 1 as a linear combination of products of elements of { √ p 1 , . . . , √ p n , √ q 1 q 2 , . . . , √ q 1 q r } with coefficients in Q. Separating the summands in which √ q 1 appears an even number of times from those in which it appears an odd number of times, we may write √ q 1 = α + β √ q 1 where α and β are elements of F = Q( √ p 1 , . . . , √ p n , √ q 2 , . . . , √ q r ) and β contains some √ q i for 2 ≤ i ≤ r. Since β = 1 implies √ q 1 ∈ F , contradicting Lemma 4, we must have β = 1. Since β contains some √ q i for 2 ≤ i ≤ r, we can separate and solve for
√ q i , showing that √ q i ∈ Q( √ p 1 , . . . , √ p n , √ q 2 , . . . , √ q i−1 , √ q i+1 , . . . , √ q r ), again contradicting Lemma 4.
Thus, the lemma must hold.
Extensions of isomorphisms
We analyze the strength required to extend an isomorphism between two fields to an isomorphism between their algebraic closures. If K and J are isomorphic fields, then the isomorphism extends to an isomorphism of K and J. This type of extension can be used to show that if F is not algebraically closed, then there is an automorphism of F that fixes F but is not the identity. As F is not algebraically closed, there is a irreducible polynomial in F [x] with distinct roots α and β in F . The fields F (α) and F (β) are isomorphic by an isomorphism that fixes F and sends α to β, and this isomorphism extends to an automorphism of F that fixes F but is not the identity. We show that in general WKL 0 is required to extend an isomorphism between two fields to their algebraic closures and to produce a nonidentity automorphism of F that fixes F when F is not algebraically closed.
Definition 6. (RCA 0 ) Suppose K, ϕ and J, ψ are algebraic extensions of F . We say K is embeddable in J over F (and write K F J) if there is an embedding τ : K → J such that for all x ∈ F , τ (ϕ(x)) = ψ(x). We also say that τ fixes F and call τ an F -embedding. If τ is also bijective, we say K is isomorphic to J over F , write K ∼ = F J, and call τ an F -isomorphism.
Informally, when K, ϕ and J, ψ are algebraic extensions of F , one identifies F both with its image in K under ϕ and also with its image in J under ψ. Given such identifications, if τ fixes F as in the preceding definition, then F is in the domain of τ and for all x ∈ F , τ (x) = x. In the formal setting, the preceding definition describes the relationship between K and J without asserting that F is a subset of K or J. Similarly in the following definition, the phrases "θ extends τ " and "θ restricts to τ " do not imply that F is a subset of K or that G is a subset of H. Definition 7. (RCA 0 ) Suppose τ : F → G is a field embedding, K, ϕ is an extension of F , H, ψ is an extension of G, and θ : K → H satisfies θ(ϕ(v)) = ψ(τ (v)) for all v ∈ F . Then we say θ extends τ , θ is an extension of τ , θ restricts to τ , and τ is a restriction of θ.
Using the preceding definitions, we can formalize the following version of Theorem 1.8 of Hungerford [8] and prove it in RCA 0 . Theorem 8. (RCA 0 ) If τ is an isomorphism from a field F onto a field G and α ∈ F is a root of an irreducible polynomial p(x) of F [x], then for any root β of τ (p)(x) in G, there is an isomorphism of F (α) onto G(β) which extends τ . In particular, taking F = G, we have that if p(x) is an irreducible polynomial over F with roots α and β then F (α) ∼ = F F (β).
Proof. Suppose F , G, τ , p, α, β are as in the hypothesis of the theorem, and let F (α), ϕ and G(β), ψ be the associated algebraic extensions. In order to define a map θ : F (α) → G(β) extending τ , we need to characterize a typical element of F (α). Recall that F (α) is isomorphic to a Σ 0 1 -F -subfield of F containing α, so let α e ∈ F (α) be the pre-image of α under this isomorphism. Define β e ∈ G(β) similarly. Then for every element γ ∈ F (α) we can uniformly find polynomials q(x) and r(x) in F [x] such that γ = ϕ(q)(αe) ϕ(r)(αe) . For any such γ, define θ(γ) = ψ(τ (q))(βe) ψ(τ (r))(βe) . Using the fact that p(x) is irreducible over F , one can prove that if ϕ(r)(α e ) = 0, then ψ(τ (r))(β e ) = 0. Thus θ(x) is well-defined. The subset of F (α) × G(β) defining θ exists by ∆ 0 1 comprehension. Verification of the remaining properties of θ can be proved without further uses of comprehension or induction. In particular, the proof that θ is single-valued relies on the fact that p(x) is irreducible over F . The proofs that θ preserves operations and is onto G(β) rely on the fact that τ is an isomorphism of F onto G. Given that θ is single-valued and that the isomorphisms map multiplicative identities to multiplicative identities, one can prove that θ extends τ .
Ordinarily, one can iterate Hungerford's theorem to create automorphisms of algebraic closures. Proving the existence of such extensions inherently demands greater logical strength than Hungerford's theorem alone, as shown by the following result. Other results related to iteration of Hungerford's theorem appear as Theorems 18 and 19 in section §3. 2. Let F be a field with algebraic extensions K and K ′ . If ϕ is an isomorphism witnessing K ∼ = F K ′ , then ϕ extends to an isomorphism witnessing K ∼ = F K ′ . In the case when K = K ′ , ϕ extends to an F -automorphism of K.
3. Let F be a field with an algebraic closure F . If α ∈ F and ϕ :
F (α) → F (α) is an F -automorphism of F (α), then ϕ extends to an F -automorphism of F . Furthermore, if K is a subset of K fixed by its embedding, then (2) is provable in RCA 0 . Similarly, if F (α) is a subset of F fixed by its embedding, then (3) is provable in RCA 0 .
Proof. We will work in RCA 0 throughout. To prove that (1) implies (2), assume WKL 0 and let F , K, K ′ , and ϕ be as in the hypothesis of (2). Let K, τ and K ′ , τ ′ be algebraic closures of K and K ′ . Then K ′ , τ ′ • ϕ is an algebraic closure of K. By Theorem 3.3 of Friedman, Simpson, and Smith [4], WKL 0 implies the uniqueness of algebraic closures. (This theorem also appears as Lemma IV.5.1 in Simpson [16] in a formulation that serves our purposes particularly well.) Thus there is an isomorphism ψ : K → K ′ such that for all x ∈ K, ψ(τ (x)) = τ ′ (ϕ(x)). By Definition 7, ψ extends ϕ. Since ϕ fixes F , so does ψ. Thus ψ witnesses K ∼ = F K ′ . Since (3) is a restriction of (2), we can complete the proof of the theorem by showing that (3) implies WKL 0 . It suffices to use (3) to separate the ranges of two injections with no common values. Let f and g be injections such that for all i and j, f (i) = g(j). Without loss of generality, we may assume that 0 is not in the range of either function. Let p i denote the i th prime, where 2 is the 0 th prime. By Lemma 3 the field F = Q( p f (i) , 2p g(i) | i ∈ N) exists. By Lemma 5, 2 / ∈ F . On the other hand, we may chose F = Q, so 2 ∈ F . Define ϕ on F ( 2) by ϕ(a + b 2) = a − b 2. Note that every value of F ( 2) can be written uniquely in the form a + b 2. By (3), ϕ can be extended to an automorphism ϕ of F that fixes F . By recursive comprehension, the set S = {i | ϕ( p i ) = p i } exists. For any i,
p f (i) ∈ F , so f (i) ∈ S. Also, 2p g(i) ∈ F , so ϕ( 2p g(i) ) = 2p g(i) = 2 p g(i)
. Since ϕ is a homomorphism, ϕ( 2p g(i) ) = ϕ( 2)ϕ( p g(i) ) = − 2ϕ( p g(i) ). Thus ϕ( p g(i) ) = − p g(i) , so g(i) / ∈ S. Thus S is the desired separating set. This completes the proof of the equivalence results.
To prove the final two sentences of the theorem, consider item (2) and suppose K is a subset of K. By Lemma 2.7 and Lemma 2.8 of Friedman, Simpson, and Smith [4], given any finite extension of K, we can uniformly find all the irreducible polynomials of the extension. In particular, we can locate the first such polynomial in some enumeration of all the polynomials in K[x]. Let p i i∈N and r ij j≤j i i∈N be sequences such that for each i, p i is the first irreducible polynomial of K(r tj | t < i ∧ j ≤ j t )[x], and r ij j≤j i are the roots of p i in K. Let r ′ 0j j≤j 0 be the roots of ϕ(p 0 ). Any k in K(r 0j | j ≤ j 0 ) can be written as q(r 00 , . . . , r 0j 0 ) for some q ∈ K[x 0 , . . . , x j 0 ]. Define ϕ * (k) = ϕ(q)(r ′ 00 , . . . , r ′ 0j 0 ). In general, if ϕ * is defined on K(r tj | t < i ∧ j ≤ j t ), let r ′ ij j≤j i be the roots of ϕ * (p i ) and for k ∈ K(r tj | t < i ∧ j ≤ j t )(r ij | j ≤ j i ), let ϕ(k) = ϕ * (q)(r ′ i0 , . . . , r ′ ij i ). Routine arguments verify that ϕ * witnesses K ∼ = F K ′ and extends ϕ. As noted before, item (3) is a special case of item (2), so RCA 0 also suffices to prove (3) when F (α) ⊂ F .
In section 5 of their paper [12], Metakides and Nerode construct a computably presented field F in an extension K such that the only computable F -automorphism of K is the identity. Their proof gradually constructs F while diagonalizing to avoid computable nontrivial automorphisms. The reversal of the following theorem may be viewed as the construction of a computably presented field such that every nontrivial F -automorphism of F encodes a separating set for computably inseparable computably enumerable sets. 2. Let K be a proper algebraic extension of F and let K be an algebraic closure of K.
Then there are at least two F -embeddings of K into K.
Let K, ψ be an algebraic extension of F . Suppose that every irreducible polynomial over F that has a root in K splits into linear factors in K. (This is called NOR1
in Definition 20.) If α ∈ K and α is not in the range of ψ, then there is an Fautomorphism ϕ of K such that ϕ(α) = α.
4. If F is not algebraically closed, then there is an F -automorphism of F that is not the identity.
Proof. To see that (1) implies (2), assume WKL 0 and let K, ψ be an algebraic extension of F and let K, τ be an algebraic closure of K. Let α be an element of K that is not in the range of ψ. By the separability of F , the minimal polynomial of α in F [x] has a root β ∈ K such that τ (α) = β. By Theorem 8, there is an isomorphism ϕ of F (α) onto F (β).
Using WKL 0 , we can apply item (2) of Theorem 9 and extend ϕ to an F -automorphism of K. Restricting this extended map to K yields an F -embedding of K into K which is distinct from τ . Since F -embeddings must map any roots of a polynomial over F to roots of the same polynomial, adding the splitting hypothesis to (3) insures that the F -embedding of (2) is also an automorphism on K. Thus (2) implies (3). Since F satisfies the splitting hypothesis of (3) and the automorphism of (3) is not the identity, (3) implies (4). It remains only to show that (4) implies WKL 0 .
As in the proof of the reversal of Theorem 9, it suffices to use (4) to separate the ranges of injections f and g satisfying 0 = f (s) = g(t) = 0 for all s and t. As a notational convenience, we identify the ordered pair (i, j) with its integer code (i + j) 2 + i. (This coding of pairs is described in Section II.2 of Simpson's book [16].) Enumerate the polynomials in Q[x], with x 2 − 2 occurring first in the ordering. Because we will be working with finite extensions of Q, Lemma 2.8 of Friedman, Simpson, and Smith [4] shows that RCA 0 suffices to determine which polynomials are irreducible over any of these extensions. Their Lemma 2.6 [4] proves the existence of primitive elements in RCA 0 . Define sequences v i i∈N of algebraic numbers and d i i∈N of degrees of polynomials as follows. If i = (j, 0) for some j, let g(x) be the next irreducible polynomial which does not split into linear factors over
Q(v k | k < i). Let G be the splitting field of g(x) over Q(v k | k < i). Let v i be a primitive element for G over Q(v k | k < i), and let d i be the degree of v i over Q(v k | k < i). Since x 2 − 2 is the first polynomial and (0, 0) = 0, v 0 = √ 2 (or some other primitive element for Q( √ 2)
) and d i = 2. If i = (j, n) and n > 0, let d j be the degree of v j over Q and let p be the first
prime such that x d j − p is irreducible over Q(v k | k < i). Let v i = p 1/d j v j and let d i be the degree of v i over Q(v k | k < i). Note that the degree of v j v i over Q(v k | k < i) is d j and d j ≤ d i .
By Lemma 2.6 and Lemma 2.8 of Friedman, Simpson, and Smith [4], the sequences v i i∈N and d i i∈N can be constructed in RCA 0 . By our construction, for each i the set of
products { j<i v e j j | ∀j(0 ≤ e j < d j )} is a vector space basis for Q(v k | k < i) over Q. Also, {1, v i , . . . , v d i −1 i } is a basis for Q(v k | k ≤ i) over Q(v k | k < i).
These claims can be proved in RCA 0 by imitating the proof of Proposition 1.2 in Lang [11].
In order to apply (4), use Lemma 3 and let
F = Q(v (i,f (j)) , v (i,g(j)) · v i | i, j ∈ N) and F = Q.
Assume for a moment that Q is a nontrivial extension; details are given below. Applying (4), there is a nontrivial F -automorphism ϕ of Q. If ϕ fixed every v i , then ϕ would be the identity on Q, so we can fix some i such that ϕ(
v i ) = v i . Since ϕ fixes F , for every j ∈ N, ϕ(v (i,f (j)) ) = v (i,f (j)) , and ϕ(v (i,g(j)) · v i ) = v (i,g(j)) · v i . Since ϕ(v (i,g(j)) ) = v (i,g(j)) implies ϕ(v i ) = v i , we must have ϕ(v (i,g(j)) ) = v (i,g(j)) . By ∆ 0 1 comprehension, the separating set {k | ϕ(v (i,k) ) = v k } exists.
To complete the proof of the reversal and the proof of the theorem, it remains only to show that the field F defined above is a proper subfield of Q. Suppose by way of contradiction that √ 2 ∈ F . Since F is generated by elements of the bases we constructed, we may write √ 2 as a linear combination of products of generators of F . We will use j 0 and j 1 to denote components of the pair encoded by j, so j = (j 0 , j 1 ). Let
√ 2 = i∈I q i j∈J i (v j 0 v (j 0 ,g(j 1 )) ) e j k∈K i v e k (k 0 ,f (k 1 ))
where I, J i , and K i denote finite sets of integers, 0 < e j < d j 0 , and 0 < e k < d (k 0 ,f (k 1 )) . For a sufficiently large value of i, all the products on the right are elements of the basis
B i = { j<i v e j j | ∀j(0 ≤ e j < d j )} for Q(v k | k < i) over Q, as is v 0 = √ 2.
By linear independence of B i , there must be some i 0 and some q ∈ Q such that:
√ 2 = q j∈J i 0 (v j 0 v (j 0 ,g(j 1 )) ) e j k∈K i 0 v e k (k 0 ,f (k 1 ))
Let s be the largest subscript appearing on a v in this product. Since g is nonzero, g(j 1 ) > 0, so by the definition of the pairing function we have j 0 < (j 0 , g(j 1 )). Thus s is of the form (j 0 , g(j 1 )) or (k 0 , f (k 1 )). Since the ranges of f and g are disjoint, only one of these may hold. Thus for some 0 < e < d s , v e s ∈ Q(v i | i < s), contradicting our construction of F . This shows that √ 2 / ∈ F and completes the proof.
As noted before the presentation of the preceding theorem, it has an immediate corollary in computable field theory.
Corollary 11. Given any pair of disjoint computably enumerable sets, there is a computable field F that is not algebraically closed and has a computable algebraic closure F such that any nontrivial F -automorphism of F computes a separating set for the computably enumerable sets. In particular, if the computably enumerable sets are computably inseparable, then any nontrivial F -automorphism is noncomputable. Additionally, every computable field F that is not algebraically closed has a computable algebraic closure F , and any such closure has a nontrivial F -automorphism ϕ such that ϕ ′ ≤ T 0 ′ .
Proof. To prove the first part of the corollary, imitate the construction from Theorem 10, using computable enumerations of the disjoint c.e. sets as the functions with disjoint ranges. To prove the last sentence, note that Theorem VIII.2.17 of [16] proves the existence of a model of WKL 0 consisting of only low sets. This model contains all the computable fields, an algebraic closure of each one, and by Theorem 10, the desired nontrivial automorphism. One could avoid the discussion of models by applying the Jockusch/Soare low basis theorem, Theorem 2.1 of [9], to a computably bounded computable tree constructed as in the proof of Theorem 9.
The constructions of this section can be used to find computable binary trees whose infinite paths can be matched in a degree preserving fashion with the F -automorphisms of K for appropriately chosen fields F and K. Since the degree of K over F is either finite or countable, the number of F -automorphisms of K is either finite or the continuum. Many computable binary trees have countably many infinite paths. Thus, given an arbitrary computable binary tree, we cannot expect to be able to construct fields so that the automorphisms match the infinite paths. This is reminiscent of the argument for why Remmel's result on 3-colorings of graphs [14] does not extend to 2-colorings. It would be nice to know if some analog of Remmel's result holds in an algebraic setting.
Question 12.
Is there a nice characterization of those computable binary trees whose infinite paths can be matched via a degree preserving bijection to the F -automorphisms of K for some computable extension K of a computable field F ? How does this class of trees compare with similar classes for automorphisms of other computable algebraic structures?
Extensions of embeddings
Informally, if J and K are algebraic extensions of F , and both F (j) F K for every j ∈ J and F (k) F J for every k ∈ K, then J is F -isomorphic to K. The proof that J ∼ = F K can be carried out in two steps: First prove that J F K and K F J and second deduce the existence of the isomorphism. This second step can be carried out in RCA 0 .
Theorem 13. (RCA 0 ) If J F K and K F J, then J ∼ = F K.
Proof. Suppose J, ψ and K, ϕ are algebraic extensions of F , θ : J → K embeds J into K, and τ : K → J embeds K into J. We need only show that θ is onto. Fix k 0 ∈ K. Let p ∈ F [x] be the minimal polynomial for k 0 over F and let k 0 , . . . , k n be the roots of ϕ(p) in K. Let j 0 , . . . , j m be the roots of ψ(p) in J. Since θ maps j 0 , . . . , j m one-to-one into k 0 , . . . , k n and τ maps k 0 , . . . , k n one-to-one into j 0 , . . . , j m , by the finite pigeonhole principle (which is provable in RCA 0 ) we must have that m = n and k 0 is in the range of θ.
In light of Theorem 13, our next goal is to formulate existence theorems for embeddings. Of course, in any embedding K F J, each element k ∈ K must map to a root in J of its irreducible polynomial. The next two definitions describe functions that are helpful for bounding the search for acceptable images of roots. Eventually, we will prove embedding existence theorems with bounds (Theorem 18) and without bounds (Theorem 19).
Definition 14. (RCA 0 ) Suppose K, ϕ is an algebraic extension of F . A function r : F [x] → K <N is a root modulus for K over F if for every p ∈ F [x]
, r(p) is (a code for) the finite set of all the roots of ϕ(p) in K. We code finite sets as in Theorem 11.2.5 of Simpson [16], so the integer code for the set is always greater than the maximum element. Thus r(p) is also an upper bound on the roots of ϕ(p) in K.
Definition 15. (RCA 0 ) Suppose K, ϕ and J, ψ are algebraic extensions of F . An F embedding bound of K into J is a function f : K → J <N such that for each k ∈ K, f (k) contains all the roots in J of the minimal polynomial of k over F . Equivalently, for k ∈ K and j ∈ J, if ∀p ∈ F [x](ψ(p)(j) = 0 → ϕ(p)(k) = 0) then j ∈ f (k)
. By our choice of coding, f (k) is also an upper bound on the roots in J of the minimal polynomial of k over F .
Suppose K and J are fields, f is an F embedding bound, and p is the minimal polynomial of k over F . Under our definition, f (k) may contain a finite number of elements that are not roots of ψ(p) in J. Also, f (k) might be empty if K is not embeddable into J. The next two theorems explore relationships between root moduli and F embedding bounds. The first theorem shows that a root modulus can act as a sort of universal F embedding bound.
Lemma 16. (RCA 0 ) Suppose J is an algebraic extension of F . J has a root modulus over F if and only if for every algebraic extension K of F , there is an F embedding bound of K into J. If there is an F embedding bound of F into J, then J has a root modulus.
Proof. Suppose J, ψ is an algebraic extension of F . First, let r be a root modulus for J and let K, ϕ be an extension of F . For each k ∈ K, let p k be the first polynomial in some enumeration of F [x] such that ϕ(p k )(k) = 0. Define f : K → J <N by f (k) = r(p k ). For k ∈ K, the minimal polynomial of k over F divides p k , so all of its roots are in f (k). Thus f is an F embedding bound of K into J.
Since F is an algebraic extension of F , the remaining implication of the second sentence follows from the third sentence. To prove the third sentence, suppose f is an F embedding bound of F into J. Given any polynomial p ∈ F [x], let q 0 , . . . , q n be a list of all the roots of p(x) in F , and define r(
p) = {j ∈ f (q 0 ) ∪ f (q 1 ) ∪ · · · ∪ f (q n ) | ϕ(p)(j) = 0}
. RCA 0 proves that r exists and is a root modulus for J.
General assertions of the existence of F embedding bounds and root moduli require additional set comprehension. Proof. Working in RCA 0 , we begin by proving the equivalence of (1) and (2). To prove that (1) implies (2), suppose J, ψ is an algebraic extension of F . Since the finite set of all roots of ψ(p) in J is uniformly arithmetically definable using p as a parameter, ACA 0 proves the existence of a root modulus for J.
To prove that (2) implies (1), let g : N → N be an injection. ACA 0 follows from the existence of the range of g. Let F = Q. Let p i denote the i th prime and consider Q( √ p g(i) | i ∈ N) as a Σ 0 1 -subfield of some algebraic closure Q of the rationals. We can find J, ψ , a field extension of Q, such that Q(
√ p g(i) | i ∈ N)
is an isomorphic image of J in Q. Apply
(2) to find a root modulus for J. Note that for every natural number k,
∃t(g(t) = k) ↔ r(x 2 − p k ) = ∅.
Since r(x 2 − p k ) is a code for a finite set, {k | r(x 2 − p k ) = ∅} exists by ∆ 0 1 -comprehension. Thus RCA 0 and (2) suffice to prove the existence of the range of g.
Now we turn to the equivalence of (1) and (3). Since (1) implies (2), by Lemma 16, (1) also implies (3). To prove that (3) implies (1), let g, F = Q, and J, ψ be as in the preceding paragraph. Let K, ϕ be a field extension of F such that Q(
√ p i | i ∈ N) is an isomorphic image of K in Q; let τ be that isomorphism. Apply (3) to find f : K → J <N , an F embedding bound of K into J.
Note that for every natural number k,
∃t(g(t) = k) ↔ ∃a(a ∈ f (τ −1 ( √ p k )) ∧ ψ(a) 2 = p k ).
Since f (τ −1 ( √ p k )) is a finite set, the range of g exists by ∆ 0 1 -comprehension, completing the proof.
Despite the fact that root moduli and embedding bounds are not interchangeable, they both can serve to formulate bounded versions of an embedding theorem.
K into J. If F (k) F J for all k ∈ K, then K F J.
3. Suppose K and J are algebraic extensions of F and r J is a root modulus of J over F .
If F (k) F J for all k ∈ K, then K F J.
Proof. To prove that WKL 0 implies (2), let K, J, F , and f K be as in (2) and suppose F (k) F J for all k ∈ K. Consider the formula θ(ϕ, k) that asserts:
• ϕ is a subset of K × J.
• ϕ preserves field operations.
• ϕ is one-to-one.
• If k ∈ K, then there is some j ∈ f K (k) such that (k, j) ∈ ϕ.
Because f K (k) is always finite, θ(ϕ, k) is a Π 0 1 formula. For any n, we can find a primitive element k 0 for F (k | k ∈ K ∧ k < n). Any ϕ witnessing F (k 0 ) F J will also witness ∃ϕ∀k < n θ(ϕ, k). By Lemma VIII.2.4.1 of Simpson [16], WKL 0 proves ∃ϕ∀k θ(ϕ, k). Any ϕ satisfying this formula F -embeds K into J.
The proof that (2) implies (3) is immediate from Lemma 16. To prove that (3) implies (1), note that given two algebraic closures of a field, RCA 0 can prove the existence of the root moduli and embeddings as in (3). The conclusion of (3) shows that each algebraic closure is embeddable in the other. By Theorem 13, the algebraic closures are F -isomorphic. This implies WKL 0 by Theorem 3.3 of Friedman, Simpson, and Smith [4].
The construction used by Miller and Shlapentokh [13] to prove their Proposition 4.3 can be used as an interesting alternative proof that (2) implies (1) in the preceding theorem. The fields in their construction have computable embedding bounds, but do not have computable root moduli.
In the absence of root moduli and embedding bounds, the theorem is much stronger.
Theorem 19. (RCA 0 ) The following are equivalent:
1. ACA 0 . 2. Suppose K and J are algebraic extensions of F . If F (k) F J for all k ∈ K then K F J.
Proof. To show that ACA 0 implies (2), it suffices to note that given K and J as in (2), a root moduli for K over F is arithmetically definable. Since ACA 0 implies WKL 0 , we may apply Theorem 18 to find the desired isomorphism.
To prove the converse, let g : N → N be an injection. We prove that the range of g exists. First, extend Q to a real closure, then extend the real closure to an algebraic closure Q. Since the algebraic closure is a finite separable extension of the real closure, the image of the real closure exists inside the algebraic closure by Friedman, Simpson, and Smith The field Q({2 1/pn , ζ pn | ∃m(g(m) = n)}) is a subfield of both Q({ζ pn | ∃m(g(m) = n)} ∪ {2 1/pm | m ∈ N}) and Q({ζ pn | ∃m(g(m) = n)} ∪ {ζ pm 2 1/pm | m ∈ N}), so we define maps ψ K : F → K and ψ J : F → J by ψ K = τ −1 K • τ F and ψ J = τ −1 J • τ F which witness that K and J are both algebraic extensions of F .
To see that F (k) F J for all k ∈ K, fix a k ∈ K and let M be such that τ (k) ∈ Q({ζ pn | ∃m(g(m) = n)} ∪ {2 1/pm | m < M}). By bounded Π 0 1 comprehension, let X = {n < M | ¬∃m(g(m) = n)}. Then k ∈ F (τ −1 K (2 1/pn ) | n ∈ X), which embeds into J by extending ψ J so that ψ J (2 1/pn ) = ζ pn 2 1/pn for each n ∈ X.
By (2), let ϕ be an F -embedding of K into J. Let X be the set of numbers n such that τ J (ϕ(τ −1 K (2 1/pn ))) ∈ Q is real. We show that X is the range of g. Suppose n = g(m) for some m. Then τ −1 F (2 1/pn ) exists and τ −1 K (2 1/pn ) = ψ K (τ −1 F (2 1/pn )). Thus ϕ(τ −1 K (2 1/pn )) = ϕ(ψ K (τ −1 F (2 1/pn ))), and the fact that ϕ is an F -embedding means that ϕ(ψ K (τ −1 F (2 1/pn ))) = ψ J (τ −1 F (2 1/pn )) = τ −1 J (2 1/pn ). All together, this gives τ J (ϕ(τ −1 K (2 1/pn ))) = τ J (τ −1 J (2 1/pn )) = 2 1/pn , which is real. On the other hand, if there is no m such that n = g(m), then the only root of x pn − 2 in Q({ζ pn | ∃m(g(m) = n)} ∪ {ζ pm 2 1/pm | m ∈ N}) is ζ pn 2 1/pn , and τ J (ϕ(τ −1 K (2 1/pn ))) must be a root of x pn − 2. Thus τ J (ϕ(τ −1 K (2 1/pn ))) = ζ pn 2 1/pn , which is not real.
Normal extensions and Galois extensions
The field theory literature contains a variety of definitions of normal algebraic extensions. For example, Lang [11] lists three versions corresponding to NOR1, NOR2, and NOR3 in the following definition. We add a fourth version to the list that makes use of the notion of restriction presented in Definition 7. While algebraists view these as equivalent definitions, this section shows that the equivalence proofs vary in logical strength.
Definition 20. (RCA 0 ) Let K, ψ be an algebraic extension of F . For 1 ≤ i ≤ 4, we say K is a NORi-normal extension of F if the condition NORi in the list below holds. NOR1: If p(x) ∈ F [x] is irreducible and ψ(p)(x) has a root in K, then ψ(p)(x) splits into linear factors in K.
NOR2: There is a sequence of polynomials over F such that the image under ψ of each polynomial in the sequence splits into linear factors in K, and K is generated by the roots of these polynomials. That is, K is the splitting field of the images under ψ of some sequence of polynomials over F .
NOR3: If ϕ : K → K is an F -embedding, then ϕ is an F -automorphism of K.
NOR4: If ϕ : K → K is an F -automorphism, then ϕ restricts to an F -automorphism of K.
Lang [11] defines Galois extensions as normal separable extensions. In light of the preceding list, this yields four reasonable definitions. Before addressing the equivalence of the various definitions, we append the following definition from Hungerford [8].
Definition 21. (RCA 0 ) A Galois extension of the field F is an algebraic extension K of F such that the only elements of K that are fixed by all F -automorphisms of K are the elements of F . To parallel our NORi notation, we will say that Galois extensions have the property GAL.
Usage of the terms "normal" and "Galois" is far from standardized. Emil Artin uses "normal" for GAL in his Galois Theory [1], as does Irving Kaplansky in Fields and Rings [10]. Artin and Kaplansky do not use the term "Galois" in this sense. David Hilbert uses "Galoisscher" for NOR3 in Theorie der algebraischen Zahlenkörper [6]. Normal doesn't appear in Hilbert's index. Zariski and Samuel use "normal" for NOR1, pointing out the equivalence with NOR2, in their Commutative Algebra [19]. They only use "Galois" in the context of finite fields.
Theorem 22. (RCA 0 ) For every field F and every algebraic extension K of F we have:
GAL → NOR1 ↔ NOR2 → NOR3 → NOR4
Moreover, if F is a subset of K fixed by its embedding and K is a subset of K fixed by its embedding, then the four versions of normal are equivalent. If the previous conditions hold and K is separable, then all five conditions are equivalent.
Proof. We will work in RCA 0 throughout. NOR1 can be deduced from NOR2 by a straightforward formalization of the proof of the last theorem in section §6.5 of Van der Waerden's text [18]. We now turn to the left to right implications.
To see that GAL implies NOR1, let K be a Galois extension of F . Suppose p(x) is a monic irreducible polynomial over F and that ψ(p)(x) has a root in K. Let α 1 , . . . , α k be all the roots of ψ(p)(x) in K. Consider the polynomial q(x) = (x − α 1 ) · · · (x − α k ). Every F -automorphism ϕ of K must permute the set {α 1 , . . . , α k } and thus the coefficients of q(x) are all fixed by ϕ. Since K is a Galois extension of F , it follows that q(x) = ψ(r)(x) for some r(x) ∈ F [x]. Since r(x) divides p(x) and p(x) is monic irreducible, it follows that p(x) = r(x) and hence that ψ(p)(x) (which is q(x)) factors completely in K.
To see that NOR1 implies NOR2, let p n n∈N be an enumeration of all the elements of F [x] whose images under ψ are finite products of linear terms in K[x]. This list consists of all those polynomials over F whose images under ψ split completely in K. Since NOR1 holds, the splitting field of the images under ψ of this sequence of polynomials is a subfield of K. Also, if a ∈ K, then the minimal polynomial of a is ψ(p n ) for some n. Thus, K is equal to the splitting field of the images under ψ of the sequence of polynomials.
To see that NOR2 implies NOR3, suppose NOR2 holds. Let K, τ be an algebraic closure of K, and let ϕ : K → K be an F -embedding. If p(x) ∈ F [x] is a defining polynomial of K and α is any root of ψ(p), then there must be a root β of ψ(p) such that ϕ(α) = τ (β). Since every element of K is expressible as a sum of products of these roots, ϕ must map K into the image of K in K under τ . Thus we can find an automorphism ϕ * : K → K such that for all k ∈ K, ϕ(k) = τ (ϕ * (k)). Since ϕ fixes F , so does the restriction ϕ * .
To see that NOR3 implies NOR4, suppose that ϕ is an F -automorphism of K. Then the restriction of ϕ to K is an F -embedding of K into K. By NOR3, this restriction is an F -automorphism of K, as desired.
To prove the penultimate sentence of the theorem, we will work in RCA 0 , assume that F ⊂ K ⊂ K, and prove that the negation of NOR1 implies the negation of NOR4. Let p be a polynomial irreducible over F that does not split in K but has a root α in K. Let β be a root of p not lying in K. By Theorem 8 there is an F -isomorphism ϕ : F (α) → F (β). By the last sentence of Theorem 9, ϕ extends to an F -automorphism of K. The restriction of ϕ to K maps α to β, so it is not an F -automorphism of K. Thus, NOR4 fails as desired.
To prove the final sentence of the theorem, we continue working in RCA 0 . Assume that F ⊂ K ⊂ K and NOR4 holds. Suppose α ∈ K \ F . Let p be the minimal polynomial of α over F and apply the separability of F to find a root β of p that is not equal to α. By Theorem 8 there is an F -isomorphism ϕ : F (α) → F (β). By the last sentence of Theorem 9, ϕ extends to an F -automorphism of K. By NOR4, this restricts to an F -automorphism of K that moves α. So K is a Galois extension of F . Each converse omitted from the preceding theorem is equivalent to WKL 0 .
Theorem 23. (RCA 0 ) The following are equivalent:
1. WKL 0 .
2. For every field F and every algebraic extension K of F , NOR4 → NOR1.
For every field F and every algebraic extension
K of F , NOR4 → NOR3.
For every field F and every algebraic extension
K of F , NOR3 → NOR1.
For every field F and every separable algebraic extension
K of F , NOR1 → GAL.
In light of Theorem 22, the equivalences hold with NOR1 replaced by NOR2.
Proof. To prove that (1) implies (2), we will use WKL 0 and ¬NOR1 to deduce ¬NOR4. Let K, ψ be an algebraic extension of F . On the basis of ¬NOR1, let p(x) be an irreducible polynomial in F [x] such that α ∈ K is a root of ψ(p)(x) and ψ(p)(x) does not split completely over K. Let q(x) be a nonlinear irreducible factor of ψ(p)(x) in K[x], and let β be a root of q(x). By Theorem 8, F (α) ∼ = F F (β). Using WKL 0 , we can apply Theorem 9 and extend this isomorphism to an F -automorphism of K. Since this automorphism does not restrict to an automorphism of K, we have ¬NOR4.
By Theorem 22, RCA 0 proves NOR1 → NOR3. Thus RCA 0 proves that (2) implies (3). Before dealing with (4), we will prove that (3) implies (1). Our plan is to assume the contrapositive of (3), that is that ¬NOR3 → ¬NOR4, and construct a separating set for the ranges of disjoint injections. Let f and g be disjoint injections and without loss of generality, assume that 0 is not in either of their ranges. Suppose Q is an algebraic closure of a real closure of Q in which the positive roots and the elements 4 √ 2, − 4 √ 2, i 4 √ 2, and −i 4 √ 2 have been designated. Using the notation for primes from the reversal of Theorem 9, define F = Q( p f (i) , 2p g(i) | i ∈ N) and consider F ( 4 √ 2). RCA 0 proves that the usual F -isomorphism from F ( 4 √ 2) to F (i 4 √ 2) exists and that it is an embedding of F ( 4 √ 2) into F which is not an automorphism of F ( 4 √ 2). Since ¬NOR3 holds, we may apply ¬NOR4 to find an F -automorphism of ψ of F which maps some element of F ( 4 √ 2) to an element not in F ( 4 √ 2). Thus ψ( 4 √ 2) = ±i 4 √ 2 and so ψ(
√ 2) = − √ 2.
As in the reversal of Theorem 9, S = {i | ψ( √ p i ) = √ p i } is a separating set for the ranges of f and g.
Consider item (4). Since Theorem 22 shows NOR3 → NOR4 and by (2), WKL 0 implies that NOR4 → NOR1, WKL 0 implies (4). To prove the converse, we will use ¬NOR1 → ¬NOR3 to find a separating set for the ranges of disjoint injections with nonzero ranges. Let f , g, and F be as in the preceding paragraph and let K = F ( 4 √ 2). The polynomial x 4 − 2 has a root in K, but x 4 − 2 does not split in K, since i 4 √ 2 is not in K. Since ¬NOR1 holds for F and K, by the contrapositive of (4), ¬NOR3 holds. Let ψ : K → K be an F -embedding which maps some element of K outside K. Then ψ( 4
√ 2) = ±i 4 √ 2, so ψ( √ 2) = − √ 2 and S = {i|ψ( √ p i ) = √ p i } is a separating set.
The equivalence of WKL 0 and (5) is immediate from part (3) of Theorem 10, using terminology from Definition 21.
We conclude this section by recasting Theorem 18 using normal field extensions. The resulting formulation avoids root moduli, but is interestingly weaker than the unbounded statement in Theorem 19. 2. Suppose that J and K are NOR1 algebraic extensions of F . If F (k) F J for all k ∈ K then K F J.
Moreover, the equivalence holds if NOR1 is replaced by NOR2, NOR3, or NOR4. If K and J are separable extensions, then the equivalence holds if NOR1 is replaced by GAL.
Proof. The proof follows from two simple observations. Given NOR1 field extensions as in (2), RCA 0 can prove the existence of F -embedding bounds of J into K and of K into J. The forward implication follows immediately from Theorem 18. The proof of the reversal of Theorem 18 also proves this reversal, since every algebraic closure of F satisfies NOR1.
Galois correspondence theorems
Lemma 2.11 of Friedman, Simpson, and Smith [4] shows that Galois correspondence for field extensions of finite degree is provable in RCA 0 . In this section, we analyze Galois correspondence for infinite extensions. If E, ψ is an algebraic extension of F and K, ϕ is an algebraic extension of E, then K, ϕ • ψ is an algebraic extension of F . In this case we say E is an intermediate extension between F and K. By Lemma 3, every Σ 0 1 -F -subfield of K is the isomorphic image of an intermediate extension field between F and K.
Theorem 25. (RCA 0 ) The following are equivalent:
1. WKL 0
If K is a Galois extension of F and E is an intermediate extension, then K is a Galois extension of E.
Proof. By Theorem 22, if K is a Galois extension of F , then it is a NOR2-normal extension.
It is easy to see that if K is a NOR2-normal extension of F and E is an intermediate extension, then K is necessarily a NOR2-normal extension of E. Therefore, (1) implies (2) by Theorem 23. The fact that (2) implies (1) follows from the reversal of Theorem 10. The field F constructed there is strictly intermediate between Q and Q. It is not hard to see that Q is a Galois extension of Q. By (2), Q is a Galois extension of F , so there must be a F -automorphism of Q that is not the identity. As in the proof of Theorem 10, this automorphism encodes the desired separating set.
We now turn to the group-theoretic aspects of Galois theory. The group Sym of permutations of N has a topology which makes it into a complete separable metric space with respect to the distance d(ϕ, ψ) = inf{2 −n : (∀i < n)(ϕ(i) = ψ(i) ∧ ϕ −1 (i) = ψ −1 (i))}.
Note that composition and inversion are both continuous operations with respect to this topology. Furthermore, Sym is easily understood even in RCA 0 with the usual representation of complete metric spaces in subsystems of second-order arithmetic. See section II.5 of Simpson's book [16].
If F is a subfield of K, the class Aut(K/F ) of F -automorphisms of K corresponds to a closed subgroup of Sym. Indeed, if ϕ is a permutation of K which is not an F -automorphism, then there is a finite initial segment of ϕ that cannot be extended to an F -automorphism of K. The Galois correspondence says that there is an inclusion-reversing correspondence between intermediate fields F ⊂ E ⊂ K and closed subgroups of Aut(K/F ); this correspondence is provable in WKL 0 .
Theorem 26. (WKL 0 ) (Galois Correspondence.) Suppose K is a Galois extension of F .
• For every intermediate extension E between F and K, K is a Galois extension of E,
and Aut(K/E) is a closed subgroup of Aut(K/F ).
• For every closed subgroup H of Aut(K/F ), there is an intermediate extension E such that K is a Galois extension of E, and H = Aut(K/E).
Proof. The first part of the theorem is immediate from Theorem 25, but the second part requires proof. The first observation is that Aut(K/F ) is a bounded subgroup of Sym. Indeed, since K is a normal extension of F , for every k ∈ K, we can effectively find a polynomial p k (x) ∈ F [x] such that p k (k) = 0 and p k (x) splits completely in K. Consequently, RCA 0 proves the existence of an F embedding bound, b : K → K <N . Any F -automorphism of K must send k to some element of b(k). By the last sentence of Definition 15, if ϕ is an F -automorphism of K, then ϕ(k) ≤ b(k) for all k ∈ K.
Applying ∆ 0 1 -comprehension, we can prove the existence of a b-bounded tree of initial segments of elements of Aut(K/F ). Briefly, given an enumeration k i i∈N of K, place σ in the tree if for all i, j < lh(σ) we have (1) σ(i) ≤ b(k i ), (2) if j witnesses that k i ∈ F then σ(i) = k i , and (3) σ preserves field operations. A closed subgroup H of Aut(K/F ) corresponds to branches through a b-bounded subtree T H . By WKL 0 , an element k n of K is fixed by every automorphism in H if and only if there is a level m > n such that every element of T H ∩ N m fixes n. Since T H is b-bounded, this is a Σ 0 1 definition of the fixed field K H . By Lemma 3, there is an isomorphic intermediate extension E, τ . By Theorem 25, K is a Galois extension of E.
It remains to see that H = Aut(K/E). The inclusion H ⊂ Aut(K/E) is clear, so suppose that ψ is an E-automorphism of K. We need to show that every initial segment of ψ is in the tree T H . Let p(x) be a polynomial in E(x) such that τ (p) splits in K and the roots of τ (p) include k 0 , . . . , k n−1 . Let L be the splitting field of τ (p). Then ψ restricts to an E-automorphism ψ of L. Every element ϕ of H also restricts to an E-automorphism ϕ of L and these restrictions form a group H of automorphisms of L. Furthermore, E is the subfield of L fixed by H since E is the subfield of K fixed by H. It follows from finite Galois theory that H = Aut(L/E) [4,Lemma 2.11], which means that ψ = ϕ for some ϕ ∈ H. Since k 0 , k 1 , . . . , k n−1 ∈ L, it follows that ψ(m) = ϕ(m) for all m < n and hence that the initial segment of ψ with length n belongs to T H .
We already saw in Theorem 23 that the first part of the Galois correspondence requires WKL 0 (though Aut(K/E) is always a closed subgroup of Aut(K/F )). In the second part of the correspondence theorem, E is essentially the fixed field for H, and the fixed field associated with a closed subgroup of Aut(K/F ) is difficult to define in subsystems weaker than WKL 0 .
Although Aut(K/F ) is always a closed subgroup of Sym, this does not mean that Aut(K/F ) is a complete separable metric space like Sym. Indeed, Aut(K/F ) could fail to have a countable dense subset. The following definitions are related to those of Brown [2].
Definition 27. (RCA 0 ) Let F be a subfield of K.
• We say Aut(K/F ) is separably closed if there is a sequence ϕ i i∈N of elements of Aut(K/F ) such that for every ψ ∈ Aut(K/F ) and every n ∈ N, there is an i ∈ N such that d(ϕ i , ψ) ≤ 2 −n .
• We say Aut(K/F ) is separably closed and totally bounded if there is a sequence ϕ i i∈N of elements of Aut(K/F ) and a function b : N → N such that for every ψ ∈ Aut(K/F ) and every n ∈ N, there is an i ≤ b(n) such that d(ϕ i , ψ) ≤ 2 −n .
When Aut(K/F ) is separably closed, this group can also be understood using the usual representation of complete metric spaces in second-order arithmetic. However, this is not always the case unless we assume ACA 0 (in which case every closed subgroup of Sym is separably closed).
Lemma 28. (RCA 0 ) Suppose K is a Galois extension of F . Then the following are equivalent:
1. Aut(K/F ) is separably closed and totally bounded.
2. Aut(K/F ) is separably closed.
3. F is a subset of K fixed by its embedding.
Proof. It is clear that (1) implies (2).
To see that (2) implies (3), suppose that ϕ i i∈N enumerates a dense set of elements of Aut(K/F ). We claim that α ∈ F ↔ (∀i)(ϕ i (α) = α).
Since the displayed formula is Π 0 1 , this shows that F is a ∆ 0 1 subset of K. Since ϕ i i∈N consists of elements of Aut(K/F ), the forward implication is clear. For the converse, suppose α is an element of K that is not in F . Then, since K is a Galois extension of F , there is an Fautomorphism ϕ of K such that ϕ(α) = α. By density, there is an i such that ϕ i (α) = ϕ(α) and so ϕ i (α) = α.
To see that (3) implies (1), assume that F is a set. Given the first n elements of K, by Lemma 2.8 of Friedman, Simpson, and Smith [4] we can find polynomials irreducible over F corresponding to each element and the roots of these polynomials in K. From these construct the finite list of all possible related initial segments of F -automorphisms of K. Emulating the construction at the end of the proof of Theorem 9, we can extend these to F -automorphisms of K. For every ψ ∈ Aut(K/F ) there will be a ϕ in this collection such that d(ψ, ϕ) ≤ 2 −n . This construction can be carried out uniformly, yielding the sequence and function witnessing that Aut(K/F ) is separably closed and totally bounded.
Theorem 29. (RCA 0 ) (Strong Galois Correspondence.) Suppose K is a Galois extension of F .
• For every set E which is a field that contains F and is contained in K, K is a Galois extension of E, and Aut(K/E) is a separably closed and totally bounded subgroup of Aut(K/F ).
• For every separably closed and totally bounded subgroup H of Aut(K/F ), the collection E of elements fixed by H is a set contained in K, K is a Galois extension of E, and H = Aut(K/E).
Proof. The first part of the theorem follows from the last sentence of Theorem 22 and Lemma 28. For the second part of the theorem, suppose that ϕ i i∈N and b : N → N witness that H is separably closed and totally bounded. Then, the subfield E of K fixed by H can be defined by the bounded formula (∀i ≤ b(k))(ϕ i (k) = k), which therefore exists by ∆ 0 1 -comprehension. It remains to see that H = Aut(K/E). The inclusion H ⊂ Aut(K/E) is clear, so suppose that ψ is an E-automorphism of K. Pick n elements {k 0 , . . . , k n−1 } of K. Let L be the normal closure of E(k 0 , . . . , k n−1 ). (That is, L is the splitting field for the minimal polynomials of k 0 , . . . , k n−1 .) For each i < n, let p i ∈ F [x] be a polynomial with root k i that splits into linear factors in K, and let m be the largest root of these polynomials. Now ψ restricts to an E-automorphism ψ of L. Every ϕ i also restricts to an E-automorphism ϕ i of L and the first b(m) + 1 such restrictions actually form a group H = {ϕ 0 , . . . , ϕ b(m) } of automorphisms of L. Furthermore, E is the subfield of L fixed by H since E is the subfield of K fixed by H. It follows from finite Galois theory that H = Aut(L/E) [4,Lemma 2.11], which means that ψ = ϕ i for some i ≤ b(m). Since k 0 , k 1 , . . . , k n−1 ∈ L, it follows that d(ϕ i , ψ) ≤ 2 −n . Since this holds for every n ∈ N we see that ϕ ∈ H.
Galois theory also says that if K is a Galois extension of F and L is an intermediate field, then L is a Galois extension of F if and only if Aut(K/L) is a normal subgroup of Aut(K/F ), in which case Aut(L/F ) is isomorphic to the quotient group Aut(K/F )/ Aut(K/L). To analyze this, we first prove a variant of Theorem 23 in RCA 0 . Proof. Theorem 22 shows that (1) implies (2) and that (2) implies (3). The proof that (3) implies (4) is analogous to the proof that NOR2 implies NOR3 in Theorem 22. The proof that (4) implies (5) is analogous to the proof that NOR3 implies NOR4 in Theorem 22. Since K is a Galois extension of F it follows immediately that (5) implies (1).
The next theorem uses the following terminology. If G is a class that is a group and N is a subclass that is also a group, we say that N is a normal subgroup of G if for all ϕ ∈ N and ψ ∈ G, ψϕψ −1 is in N.
Theorem 31. (RCA 0 ) Let K be a Galois extension of F and let L be an intermediate extension.
1. If L is a Galois extension of F then Aut(K/L) is a normal subgroup of Aut(K/F ). Proof. For the first statement, suppose ϕ is an element of Aut(K/L) and ψ is an element of Aut(K/F ). Then ψ −1 is also in Aut(K/F ). Consider ψϕψ −1 and let x ∈ L. Since L is a Galois extension of F , by part (5) of Theorem 30, ψ −1 (x) ∈ L. Thus ϕ(ψ −1 (x)) = ψ −1 (x) and ψ(ϕψ −1 (x)) = x. Thus ψϕψ −1 ∈ Aut(K/L) and so Aut(K/L) is a normal subgroup of Aut(K/F ). For the second statement, a simple algebraic computation shows that if ϕ is an Fautomorphism of K, then Aut(K/ϕ[L]) = ϕ Aut(K/L)ϕ −1 . If Aut(K/L) is a normal subgroup of Aut(K/F ) then ϕ Aut(K/L)ϕ −1 = Aut(K/L). Assuming that K is Galois over L, it follows that L = ϕ[L] and hence that ϕ restricts to an automorphism of L. By part (5) of Theorem 30, it follows that L is a Galois extension of F .
If
The last statement follows from the previous two and Theorem 29 which shows that K is necessarily a Galois extension of L.
Informally, if L is an intermediate Galois extension of F , then the restriction map from K to L takes each element of Aut(K/F ) and restricts its domain to create an automorphism of L. Consequently, the restriction map as described in part (5) of Theorem 30 is a homomorphism from Aut(K/F ) to Aut(L/F ) whose kernel is Aut(K/L). However, the homomorphism from Aut(K/F ) to Aut(L/F ) needs to be surjective in order to conclude that Aut(L/F ) is isomorphic to the quotient of Aut(K/F ) by Aut(K/L), which we can't really talk about in second-order arithmetic other than via the First Isomorphism Theorem. If L is a subset of K fixed by its embedding, then (2) is provable in RCA 0 .
Proof. Note that (2) simply states that any F -automorphism of L can be extended to an F -automorphism of K. The proof is similar to that of Theorem 9.
Other characteristics
Results in sections §2 and §3 can be extended to fields of finite characteristic. In many cases, separability conditions must be appended to the hypotheses. Additionally, when the characteristic is specified in the result, any reversal must reflect this. The final result of this section, based on Theorem 10, illustrates the adaptation process. Many of the reversals in previous sections involve extensions of Q. Adaptation of these arguments relies on the following observation. Let p be a prime and let GF(p n ) denote the field of integers mod p n . The field of rational functions GF(p n )(x) is an infinite field of characteristic p and is the quotient field of the Euclidean ring GF(p n )[x]. Because GF(p n ) is finite, RCA 0 can prove the existence of the set of monic irreducible polynomials of GF(p n )[x]. These irreducible polynomials can play the role the prime numbers in our prior constructions. For example, we have the following versions of Lemma 4.
Lemma 33. (RCA 0 ) Let R be a Euclidean ring with quotient ring Q of characteristic not equal to 2. Let p 1 , . . . , p n and q 1 , . . . , q r be disjoint lists of distinct primes (irreducible elements). Then
√ q 1 . . . q r / ∈ Q( √ p 1 , . . . , √ p n ) and √ q 1 / ∈ Q( √ p 1 , . . . , √ p n , √ q 1 q 2 , . . . , √ q 1 q r ).
Proof. We will work in RCA 0 . Fix R. Note that the first conjunct of the conclusion can be written as: for every n, for every list of ps, for every list of qs, for every quotient of Q-linear combinations of products of roots of ps, the square of the linear combination is not equal to the product of the qs. Since this conjunct can be expressed as a Π 0 1 formula, we can proceed to prove it in RCA 0 by induction on n.
For the base case, suppose by way of contradiction that √ q 1 . . . q r ∈ Q. Let √ q 1 . . . q r = r 0 r 1 where r 0 , r 1 ∈ Q and gcd(r 0 , r 1 ) = 1. Thus r 2 1 q 1 . . . q r = r 2 0 . Since q 1 is prime and q 1 |r 2 0 , we have q 1 |r 0 . So r 2 0 = q 2m 1 r 2 where m ≥ 1 and gcd(q 1 , r 2 ) = 1. Since q 2 1 |r 2 1 q 1 . . . q r and q 1 , . . . , q r are distinct primes, q 1 |r 2 1 . Thus q 1 |r 1 and so r 2 1 q 1 . . . q r = q 2k+1 1 r 3 where k ≥ 1 and gcd(q 1 , r 3 ) = 1. Summarizing, q 2k+1 1 r 3 = q 2m 1 r 2 where q 1 | r 3 and q 1 | r 2 , a contradiction. For the induction step, suppose the lemma is true for n−1. Fix distinct primes p 1 , . . . , p n . Let F 0 = Q( √ p 1 , . . . , √ p n−1 ). Let q 1 , . . . , q r be a list of distinct primes disjoint from p 1 , . . . , p n . Suppose by way of contradiction that √ q 1 . . . q r ∈ F 0 ( √ p n ). Then we may write √ q 1 . . . q r = α + β √ p n where α, β ∈ F 0 . Squaring yields q 1 . . . q r = α 2 + β 2 p n + 2αβ √ p n . Consider three cases: (1) If αβ = 0 then √ p n ∈ F 0 , contradicting the induction hypothesis.
(2) If β = 0 then √ q 1 . . . q r = α ∈ F 0 , contradicting the induction hypothesis. (3) If α = 0 then √ q 1 . . . q r = β √ p n so √ q 1 . . . q r p n = p n β ∈ F 0 , contradicting the induction hypothesis.
This completes the induction proof of the first conjunct of the conclusion of the lemma. The remaining conjunct is proved by the same argument as Lemma 5. ∀i 0 ≤ ε i ≤ 2} is linearly independent over GF(4)(x). Consequently, if Q = GF(4)(x) and p 1 , . . . , p n and q 1 , . . . , q r are disjoint lists of distinct primes, then 3 √ q 1 . . . q r / ∈ Q( 3 √ p 1 , . . . , 3 √ p n ) and 3 √ q 1 / ∈ Q( 3 √ p 1 , . . . , 3 √ p n , 3 √ q 1 q 2 , . . . , 3 √ q 1 q r ).
Proof. A straightforward algebraic argument proves that A is pairwise linearly independent over GF(4)(x). The first sentence of the lemma follows from Theorem 1.3 of Carr and O'Sullivan [3], substituting GF(4)(x) for their K, K for L, and A (as in the statement) for A. This instance of their theorem can be proved in RCA 0 . The remainder of the lemma can be proved in much the same fashion as Lemma 33.
Theorem 35. (RCA 0 ) Let p be a prime or 0. The following are equivalent:
1. WKL 0 .
2. Let F be an infinite field of characteristic p and let K be an algebraic extension of F that includes a separable element α / ∈ F . Then there is an F -embedding of K into K that is not the identity.
Proof. To prove that (1) implies (2), assume WKL 0 . Since α is separable, it is a root of a polynomial p(x) ∈ F [x] with no repeated roots. Since α / ∈ F , the degree of p(x) is greater than 1. Let β = α be another root of this polynomial. Imitate the proof of Theorem 10. Since the proof of Theorem 9 does not rely on the characteristic of F , it can be used to complete the proof.
Next, we will prove the reversal for characteristic 0, and then adapt the argument for other characteristics. Let f and g be injections such that ∀s∀t(0 = f (s) = g(t) = 0). As in the proof of Theorem 10, let (i, j) denote both the ordered pair and the integer code for that ordered pair. Let p i denote the i th prime. Define the fields F and K by:
F = Q( √ p (i,f (j)) , √ p i p (i,g(j)) | i, j ∈ N) K = Q( √ p i | i ∈ N)
By Lemma 5, √ 2 is not an element of F , so K is a nontrivial extension of F . Suppose ϕ is a nontrivial F -embedding of K into K. Then for some prime p i , ϕ( √ p i ) = √ p i . For this i and any j, ϕ( √ p (i,g(j)) ) = √ p (i,g(j)) and ϕ( √ p (i,f (j)) ) = √ p (i,f (j)) . The separating set S = {k | ϕ( √ p (i,k) ) = √ p (i,k) } exists by ∆ 1 0 comprehension using the parameter ϕ. Since S includes the range of f and avoids the range of g, this proves WKL 0 . Now suppose p is an odd prime and (2) holds for fields of characteristic p. Our goal is to adapt the previous construction to the characteristic p setting. Let {p i | i ∈ N} be a list of distinct irreducible monic polynomials in GF(p)[x]. These will play the role that the prime numbers played in the preceding argument. For each p i , the polynomial z 2 − p i and its derivative have no common roots, so z 2 − p i is separable. Let r i denote a root of z 2 − p i . Given disjoint injections f and g that never take the value 0, define the fields F and K by F = GF(p)(x)(r (i,f (j)) , r i r (i,g(j)) | i, j ∈ N) K = GF(p)(x)(r i | i ∈ N) By Lemma 33, K is a nontrivial extension of F . To complete the proof, use a nontrivial F -embedding of K to find a separating set for the ranges of f and g.
To carry out the reversal for characteristic 2, modify the previous argument by using GF(4)(x), z 3 − p i , and Lemma 34.
Some of the reversals in previous sections use algorithms for factoring polynomials over Q. One can find factoring algorithms for the characteristic p fields used in this section by adapting work of Stoltenberg-Hansen and Tucker [17].
Lemma 4 .
4(RCA 0 ) Let p 1 , . . . , p n and q 1 , . . . , q r be disjoint lists of distinct primes. Then √ q 1 . . . q r / ∈ Q( √ p 1 , . . . , √ p n ).
Theorem 9 .
9(RCA 0 ) The following are equivalent:1. WKL 0 .
Theorem 10 .
10(RCA 0 ) The following are equivalent:1. WKL 0 .
Theorem 17 .
17(RCA 0 ) The following are equivalent:1. ACA 0 .2. If J is an algebraic extension of F , then J has a root modulus.3. If K and J are algebraic extensions of F , then there is an F embedding bound of K into J.
Theorem 18 .
18(RCA 0 ) The following are equivalent:1. WKL 0 .2. Suppose K and J are algebraic extensions of F and f K is an F embedding bound of
[ 4 ]
4Lemma 2.6. This allows us to distinguish the real elements of Q from the complex elements of Q. Fix an enumeration of Q, let p m m∈N enumerate the odd primes, and for each m > 0, let ζ m ∈ Q be the first enumerated primitive m th root of unity. Thefields Q({2 1/pn , ζ pn | ∃m(g(m) = n)}), Q({ζ pn | ∃m(g(m) = n)} ∪ {2 1/pm | m ∈ N}), and Q({ζ pn | ∃m(g(m) = n)} ∪ {ζ pm 2 1/pm | m ∈ N}) are all Σ 0 1 -Q-subfields of Q.By Lemma 3, let F , K, and J, be algebraic extensions of Q together with embeddings τ F , τ K , and τ J of F , K, and J, respectively, into Q such that ran(τ F ) = Q({2 1/pn , ζ pn | ∃m(g(m) = n)}); ran(τ K ) = Q({ζ pn | ∃m(g(m) = n)} ∪ {2 1/pm | m ∈ N}); ran(τ J ) = Q({ζ pn | ∃m(g(m) = n)} ∪ {ζ pm 2 1/pm | m ∈ N}).
Theorem 24 .
24(RCA 0 ) The following are equivalent. 1. WKL 0 .
Theorem 30 .
30(RCA 0 ) Let K be a Galois extension of F and let L be an intermediate extension. The following are equivalent: 1. L is a Galois extension of F . 2. L is a NOR1-normal extension of F . 3. L is a NOR2-normal extension of F . 4. If ϕ : L → K is an F -embedding, then ϕ is an F -automorphism of L. (This is a variant of NOR3.) 5. Every F -automorphism of K restricts to an F -automorphism of L. (This is a variant of NOR4 and uses the notion of restriction from Definition 7.)
K is a Galois extension of L and Aut(K/L) is a normal subgroup of Aut(K/F ) then L is a Galois extension of F . 3. If L is also a subset of K, then Aut(K/L) is a normal subgroup of Aut(K/F ) if and only if L is a Galois extension of F .
Theorem 32 .
32(RCA 0 ) The following are equivalent:1. WKL 02. If K is a Galois extension of F and L is an intermediate extension of F , then therestriction map is a surjective homomorphism from Aut(K/F ) onto Aut(L/F ) whose kernel is Aut(K/L).
Lemma 34 .
34(RCA 0 ) Let {p i | i ≤ n} be a sequence of distinct irreducible elements of GF(4)[x]. For each i ≤ n, let r i be a solution of x 3 − p i = 0. Then the set A = { i≤n r ε i i |
AcknowledgementsThe authors would like to thank Bill Cook for useful discussions, and the referees for their helpful comments and suggestions. Portions of Jeffry Hirst's work were supported by a grant (ID#20800) from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. Paul Shafer's work was funded in part by an FWO Pegasus Long Postdoctoral Fellowship.Bibliography
Emil Artin, Galois theory. Arthur N. Milgram. MR1616156Mineola, NYDover Publications Inc2nd ed.. 98k:12001Emil Artin, Galois theory, 2nd ed., Dover Publications Inc., Mineola, NY, 1998. Edited and with a supplemental chapter by Arthur N. Milgram. MR1616156 (98k:12001)
Notions of closed subsets of a complete separable metric space in weak subsystems of second-order arithmetic, Logic and computation. Douglas K Brown, DOI10.1090/conm/106/1057814.MR1057814(91i:03108Contemp. Math. 106Amer. Math. SocDouglas K. Brown, Notions of closed subsets of a complete separable metric space in weak subsystems of second-order arithmetic, Logic and computation (Pittsburgh, PA, 1987), Contemp. Math., vol. 106, Amer. Math. Soc., Providence, RI, 1990, pp. 39-50. DOI 10.1090/conm/106/1057814. MR1057814 (91i:03108)
On the linear independence of roots. Richard Carr, O' Cormac, Sullivan, DOI10.1142/S1793042109002018.MR2499028Int. J. Number Theory. 5111041Richard Carr and Cormac O'Sullivan, On the linear independence of roots, Int. J. Number Theory 5 (2009), no. 1, 161-171, DOI 10.1142/S1793042109002018. MR2499028 (2010c:11041)
Countable algebra and set existence axioms. M Harvey, Stephen G Friedman, Rick L Simpson, Smith, DOI10.1016/0168-0072(83)90012-X.MR725732Ann. Pure Appl. Logic. 2523157Harvey M. Friedman, Stephen G. Simpson, and Rick L. Smith, Countable algebra and set existence ax- ioms, Ann. Pure Appl. Logic 25 (1983), no. 2, 141-181, DOI 10.1016/0168-0072(83)90012-X. MR725732 (85i:03157)
Countable algebra and set existence axioms. Addendum To, Ann. Pure Appl. Logic. 252, Addendum to: "Countable algebra and set existence axioms" [Ann. Pure Appl. Logic 25 (1983), no. 2, 141-181;
. DOI10.1016/0168-0072(85)90020-X.MR790391Ann. Pure Appl. Logic. 2833141MR0725732 (85i:03157)MR0725732 (85i:03157)], Ann. Pure Appl. Logic 28 (1985), no. 3, 319-320, DOI 10.1016/0168-0072(85)90020-X. MR790391 (87f:03141)
Die Theorie der algebraischen Zahlkörper. David Hilbert, Jahresber. Deutsch. Math.-Verein. 4David Hilbert, Die Theorie der algebraischen Zahlkörper, Jahresber. Deutsch. Math.-Verein 4 (1897), 175-546.
Jeffry L Hirst, DOI10.4064/ba55-4-2.MR2369116Representations of reals in reverse mathematics. 553015Jeffry L. Hirst, Representations of reals in reverse mathematics, Bull. Pol. Acad. Sci. Math. 55 (2007), no. 4, 303-316, DOI 10.4064/ba55-4-2. MR2369116 (2009j:03015)
Abstract Algebra. Thomas Hungerford, Graduate Texts in Mathematics. New York, NYSpringer Science+Business MediaThomas Hungerford, Abstract Algebra, Graduate Texts in Mathematics, Springer Science+Business Media, New York, NY, 1980.
Soare, Π 0 1 classes and degrees of theories. Carl G JockuschJr, I Robert, Trans. Amer. Math. Soc. 1734775MR0316227Carl G. Jockusch Jr. and Robert I. Soare, Π 0 1 classes and degrees of theories, Trans. Amer. Math. Soc. 173 (1972), 33-56. MR0316227 (47 #4775)
Fields and rings. Irving Kaplansky, Chicago Lectures in Mathematics. University of Chicago PressReprint of the second (1972) edition. MR1324341 (96a:12001Irving Kaplansky, Fields and rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995. Reprint of the second (1972) edition. MR1324341 (96a:12001)
Serge Lang, Algebra , Graduate Texts in Mathematics. New YorkSpringer-Verlag21133rd ed.Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR1878556 (2003e:00003)
Effective content of field theory. G Metakides, A Nerode, DOI10.1016/0003-4843(79)90011-1.MR556895(82b:03082Ann. Math. Logic. 173G. Metakides and A. Nerode, Effective content of field theory, Ann. Math. Logic 17 (1979), no. 3, 289-320, DOI 10.1016/0003-4843(79)90011-1. MR556895 (82b:03082)
Computable categoricity for algebraic fields with spitting algorithms. R Miller, A Shlapentokh, PreprintR. Miller and A. Shlapentokh, Computable categoricity for algebraic fields with spitting algorithms (November 7, 2011). Preprint, http://arxiv.org/pdf/1111.1205.pdf.
Graph colorings and recursively bounded Π 0 1 -classes. J B Remmel, DOI10.1016/0168-0072(86)90051-5.MR863333Ann. Pure Appl. Logic. 3223065J. B. Remmel, Graph colorings and recursively bounded Π 0 1 -classes, Ann. Pure Appl. Logic 32 (1986), no. 2, 185-194, DOI 10.1016/0168-0072(86)90051-5. MR863333 (87m:03065)
Classroom Notes: On Extensions of Q by Square Roots. R L Roth, DOI10.2307/2316910.MR1536291Amer. Math. Monthly. 784R. L. Roth, Classroom Notes: On Extensions of Q by Square Roots, Amer. Math. Monthly 78 (1971), no. 4, 392-393, DOI 10.2307/2316910. MR1536291
Subsystems of second order arithmetic. Stephen Simpson, Perspectives in Logic. CambridgeCambridge University Press30732nd ed.Stephen Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009. MR2517689 (2010e:03073)
Computable rings and fields, Handbook of computability theory. V Stoltenberg-Hansen, J V Tucker, DOI10.1016/S0049-237XStud. Logic Found. Math. 140993100to appear in printV. Stoltenberg-Hansen and J. V. Tucker, Computable rings and fields, Handbook of computability theory, Stud. Logic Found. Math., vol. 140, North-Holland, Amsterdam, 1999, pp. 363-447, DOI 10.1016/S0049- 237X(99)80028-7, (to appear in print). MR1720739 (2000g:03100)
Based in part on lectures by E. Artin and E. Noether; Translated from the seventh German edition by. B L Van Der Waerden, Fred Blum and John R. Schulenberger. MR1080172Springer-VerlagINew York91h:00009aB. L. van der Waerden, Algebra. Vol. I, Springer-Verlag, New York, 1991. Based in part on lectures by E. Artin and E. Noether; Translated from the seventh German edition by Fred Blum and John R. Schulenberger. MR1080172 (91h:00009a)
. Oscar Zariski, Pierre Samuel, Commutative algebra. IISpringer-VerlagReprint of the 1960 editionOscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Springer-Verlag, New York, 1975. Reprint of the 1960 edition;
. Graduate Texts in Mathematics. 2910706Graduate Texts in Mathematics, Vol. 29. MR0389876 (52 #10706)
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"Trapped fermionic clouds distorted from the trap shape due to many-body effects"
] | [
"Masudul Haque \nMax Planck Institute for Physics of Complex Systems\nDresdenGermany\n\nInstitute for Theoretical Physics\nUtrecht University\nLeuvenlaan 43584 CEUtrechtThe Netherlands\n",
"H T C Stoof \nInstitute for Theoretical Physics\nUtrecht University\nLeuvenlaan 43584 CEUtrechtThe Netherlands\n"
] | [
"Max Planck Institute for Physics of Complex Systems\nDresdenGermany",
"Institute for Theoretical Physics\nUtrecht University\nLeuvenlaan 43584 CEUtrechtThe Netherlands",
"Institute for Theoretical Physics\nUtrecht University\nLeuvenlaan 43584 CEUtrechtThe Netherlands"
] | [] | We present a general approach for describing trapped Fermi gases, when the cloud shape is distorted with respect to the trap shape. Our approach provides a consistent way to explore physics beyond the local density approximation, if this is necessary due to the distortion. We illustrate this by analyzing in detail experimentally observed distortions in a trapped imbalanced Fermi mixture. In particular, we demonstrate in that case dramatic deviations from ellipsoidal cloud shapes arising from the competition between surface and bulk energies. | 10.1103/physrevlett.98.260406 | [
"https://arxiv.org/pdf/cond-mat/0701464v2.pdf"
] | 656,163 | cond-mat/0701464 | 8abad50de8f55d7fbeca2464f45f4584c2b24897 |
Trapped fermionic clouds distorted from the trap shape due to many-body effects
19 Jul 2007 (Dated: May 22, 2018)
Masudul Haque
Max Planck Institute for Physics of Complex Systems
DresdenGermany
Institute for Theoretical Physics
Utrecht University
Leuvenlaan 43584 CEUtrechtThe Netherlands
H T C Stoof
Institute for Theoretical Physics
Utrecht University
Leuvenlaan 43584 CEUtrechtThe Netherlands
Trapped fermionic clouds distorted from the trap shape due to many-body effects
19 Jul 2007 (Dated: May 22, 2018)
We present a general approach for describing trapped Fermi gases, when the cloud shape is distorted with respect to the trap shape. Our approach provides a consistent way to explore physics beyond the local density approximation, if this is necessary due to the distortion. We illustrate this by analyzing in detail experimentally observed distortions in a trapped imbalanced Fermi mixture. In particular, we demonstrate in that case dramatic deviations from ellipsoidal cloud shapes arising from the competition between surface and bulk energies.
Introduction. -The cooling of trapped fermionic atoms to degeneracy has opened up a fascinating new arena for studying many-fermion physics [1]. In particular, the opportunity to access the BEC-BCS crossover using Feshbach resonances has led to a large effort in the experimental study of fermionic pairing using trapped atomic gases [2]. Among other topics, pairing in polarized Fermi gases is now being intensely explored, experimentally in Refs. [3,4,5,6] and theoretically in, e.g., Refs. [7,8,9,10,11,12]. Apart from possible new insights for solid-state and nuclear physics, the field of trapped fermionic atoms comes with its own set of new concerns, such as the role of the trapping potential. In this Letter, we therefore provide results on the relationship between the trap shape and the shape of the gas cloud.
For a large class of trapped many-body systems, the local density approximation (LDA) gives an accurate description of the spatial distributions of atomic and energy densities. However, motivated by recent experiments [4,6], we propose that a number of mechanisms may lead to the intriguing situation that the combination of manybody physics and the external trapping potential causes the shape of the gas cloud to be distorted compared to the shape determined by the trap alone. In such cases, the simple LDA procedure in terms of a local chemical potential is no longer appropriate, and we need to consider the angular structure of the local Fermi surface. We have therefore formulated an extension of the LDA that allows for a distinction between the different directions. We give below some examples where one observes or expects such spatial distortions. We note however that there might be many other situations, beyond what we anticipate here, where such distortions take place. Our formalism is thus of very general applicability and interest.
As a first scenario for spatial distortions, consider a many-body system that is phase separated in an elongated trap due to the presence of a first-order transition. The interface between the two phases then carries a surface tension. To minimize the surface energy, the interface shape will differ from the trap shape. In fact, such a system has recently been experimentally realized using a trapped polarized Fermi mixture of resonantlyinteracting atoms [4,6]. At sufficiently low temperatures this mixture phase separates into an unpolarized superfluid core and an outer shell of normal polarized fermions [4,7,8,9,10,11]. When the trap is elongated enough and the particle number is small enough, the surface tension causes significant deviation of the core aspect ratio from the trap aspect ratio [6,12]. In the following we will analyze this situation in detail.
Second, Pomeranchuk instabilities are shape deformation instabilities of a Fermi surface [13], which have attracted intense interest in the strongly correlated electrons community [14,15]. A d-wave Pomeranchuk transition leads to a "nematic" Fermi liquid with an elliptical Fermi surface. Other than their possible realization in solids, it is hoped that such a nematic liquid may be realized in atomic dipolar Fermi gases since a longrange interaction can cause a Pomeranchuk instability if the short-range part is suppressed [15]. A Fermi surface deformation that is constant in space causes no distortion in coordinate space. However, in an external trap, the Fermi surface deformation would vary from point to point, leading to a distortion of the gas cloud in coordinate space as well. To calculate particle and energy densities in such a case, would again require an anisotropic generalization of the LDA, of the kind we provide here.
Finally, one possible mechanism for pairing of polarized fermions is via opposite deformation of the two Fermi surfaces [16]. A prolate (oblate) shape for the majority (minority) Fermi surface leads to an equatorial region in momentum space along which opposite-momenta fermions can pair. With the extended parameter space expected to be accessible with trapped atoms in the future, it is likely that this pairing mechanism is energetically favorable in some parameter regime [16]. In the case of such pairing in a trap, we expect both Fermi surfaces to have ellipticities (of opposite sign) that vary with position. As a result, the majority and minority clouds are expected to be distorted in opposite directions relative to the trap. Again, a treatment of the type presented in this Letter is needed to describe this situation.
Anisotropic generalization of local density approxima-tion-The LDA encodes the inhomogeneity of the gas completely in terms of a local chemical potential at every point in space, µ(r) = µ − V tr (r). Here µ is the true chemical potential of the gas and V tr (r) is the trapping potential. The shape of the gas cloud is determined by an equipotential surface of the trap. For harmonic traps, this surface is an ellipsoid.
The key to treating distorted systems is to generalize the definition of the local chemical potential µ(r) = µ− V tr (r) in a way that discriminates between directions. For definiteness, we will consider axially symmetric traps, V tr (r) = V tr (r, z), where r and z are the radial and axial length variables of a cylindrical coordinate system, respectively. For each position in the trap, we need to determine the momentum-space location of the deformed Fermi surface, in both the radial and axial directions. We denote by R(z) and Z(r) the edge of a reference ideal Fermi cloud, that would be obtained by hypothetically turning off the interactions but retaining the distortion. Our prescriptions for the location of the Fermi surface at position r in the radial and axial directions are
µ R (r) = µ R (r, z) = V tr (R(z), z) − V tr (r, z) , (1a) µ Z (r) = µ Z (r, z) = V tr (r, Z(r)) − V tr (r, z) . (1b)
We have written the Fermi surface locations in energy units, i.e., µ R and µ Z are the free-particle energies corresponding to the Fermi momenta in the radial and axial directions. They are, of course, not components of the chemical potential. Eqs. (1) are obtained by semiclassical considerations similar to the WKB approximation, which is one way to derive the LDA. Here, we have applied the semiclassical approximation separately to the r and z directions ( Fig. 1a) as appropriate for an axially symmetric trap, where the single-particle Schrödinger equation separates in these two directions.
After obtaining the locations of the local Fermi surface in the radial and axial directions, it is for many purposes sufficient to use an ellipsoidal Fermi surface with µ R and µ Z as the principal axes. At the center of the trap, the same semiclassical argument can be carried out in any direction and the local Fermi surface has exactly the same shape as the edge of the ideal gas reference cloud. The cloud shape is generally near ellipsoidal but may deviate from this shape either due to anharmonicity of the trap or due to many-body effects, as we will see. Although the shape of the Fermi surface away from the trap center is not fixed by similar arguments, it is physically reasonable to use the same shape as in the trap center.
The local non-spherical Fermi surfaces obtained using this prescription can now be used to calculate densities. For example, for an elliptically distorted normal gas in a harmonic trap, the density is given in terms of the volume of the local Fermi surface by n(r) = (2m) 3/2 /6π 2 µ R (r) µ Z (r). This density vanishes at the surface defined by R(z) or Z(r). For a superfluid, on the other hand, it may be appropriate to use the BCS equations generalized to anisotropic Fermi surfaces, i.e., the momentum integrand in the density profile
n(r) = ∞ 0 k 2 dk 2π 2 1 −1 dx 1 − ǫ k − ǫ F (x) [ǫ k − ǫ F (x)] 2 + |∆| 2
now has angular dependence. Here x = cos θ, ǫ k = 2 k 2 /2m with m the atomic mass, ∆ is the gap, and ǫ F (x) is the deformed Fermi surface with principal axes µ R and µ Z . Similar modifications can be made for the BCS energy equation and the BCS gap equation.
Application to polarized resonantly-interacting Fermi gases -We now demonstrate our procedure by elaborating on a particular case, i.e., we analyze the experiment reported in Ref. [6] using the formalism presented above. Figure 2 of Ref. [6] shows that the majority aspect ratio does not change much with polarization. We therefore describe the majority species in state |↑ with the usual LDA. On the other hand, the boundary of the minority atom cloud in state |↓ is distorted due to surface tension. We thus treat the minority according to the approach described above. At each position within the minority cloud, we calculate the two parameters for the Fermi surface of the minority species, using as an input the boundary of the minority cloud, which coincides with the superfluid core if any intermediate partially-polarized normal shell can be neglected.
For a choice of µ ↑ , R ↓ ≡ R ↓ (0), and Z ↓ ≡ Z ↓ (0), the particle densities n ↑ (r) = n ↓ (r) and the energy density e(r) at any point inside the core can be calculated from the local Fermi surfaces. The majority Fermi surface is given by µ ↑ (r) = µ ↑ − V tr (r). The minority Fermi surface is deformed, with µ R↓ (r) and µ Z↓ (r) calculated using Eqs. (1). Since at resonance the gap is larger than the maximum difference between the Fermi surfaces, the minority momentum distribution is not anisotropic even though µ R↓ = µ Z↓ , unlike the case of Refs. [16]. For the fully polarized normal gas outside the core, n ↑ (r) and e(r) are calculated by the usual LDA.
To calculate the densities and superfluid gap in the superfluid core, we could use generalized BCS equations as described above. However, with one additional approximation, we can also make use of Monte Carlo results [17] for the resonant situation near which the experiments of Refs. [4,6] are performed. The additional approximation is to use at every position an effective chemical potential µ ↓ (r) = [µ R↓ (r)] 2/3 [µ Z↓ (r)] 1/3 for the minority species as well. The superfluid properties are then given by the universal unitarity results [17] in terms of the average chemical potential µ(r) = [µ ↓ (r) + µ ↑ (r)]/2.
In Fig. 1b, we show the resulting axial density profiles, and compare with experimental data for two different polarizations P = (N ↑ − N ↓ )/(N ↑ + N ↓ ). The calculation requires three parameters (µ ↑ , R ↓ , Z ↓ ) as input. Unlike the calculation in Ref. [10], the required input quantities are not immediately fixed by the trap parameters and particle numbers (N ↑ ,N ↓ ) alone. An additional energy calculation is required for the aspect ratio of the core, as explained later. The experimental majority axial density profiles have pronounced peaks at the center. This is reproduced well by the calculations shown in Fig. 1 using Monte Carlo unitarity parameters, but not if we use the BCS equations to describe the core. Note that these central peaks are not prominent in the first experiment by Partridge et al. [4], nor in the experiment by Shin et al. [5], suggesting that thermal fluctuations play a stronger role in those cases. In all calculations reported in this Letter, anharmonicities of the experimental trap shape are included unless explicitly stated otherwise.
Energy minimization -At unitarity, the surface energy density associated with the superfluid-normal interface is expected to be given by the chemical potentials through universal constants. Defining µ = (µ ↑ + µ ↓ )/2 and h = (µ ↑ − µ ↓ )/2, we first note that a quantity with the dimension of a surface energy density constructed out of µ alone must be proportional to (m/ 2 )µ 2 . The dependence on h may be included as a function of the dimensionless ratio h/µ. The surface tension thus has the form σ = η s (m/ 2 )µ 2 f s (h/µ) . Here η s is a dimensionless number expected to be of order 1 and f s is a dimensionless function. Noting in addition that the position of the interface is fixed by the first-order transition determined by a universal value of h/µ [17], we infer that f s (h/µ) is a constant along the surface we are interested in. We thus absorb it into the universal constant η s .
We can use the observed core aspect ratios Z ↓ /R ↓ extracted from the axial density data to determine η s , because the correct value of η s must produce a total energy minimum at the correct aspect ratio. Denoting by n N (µ) and e N (n) the particle and energy-density functions of a homogeneous two-component ideal Fermi gas, the bulk energy density in the superfluid core is given by e(r) = (1 + β)e N (n(r)) + V tr (r)n(r), where n(r) = n N (µ(r)/(1 + β)) is the density profile in the core, β ≃ −0.585 is a universal constant [17], and µ(r) is calculated as described previously. The energy density in the outer shell is given simply by the single-component ideal Fermi gas expression. Integrating over the core and the outer shell gives the total bulk energy E b . The total energy is E b + η s I s , where I s is the surface integral over (m/ 2 )µ 2 (r). In Fig. 2a, this quantity is shown for several values of η s , for fixed populations. The axial data in an experimental shot for these populations is fit well with Z ↓ /R ↓ ≃ 21.5 and a minimum is located at this core aspect ratio for η s ≃ 0.59. Carrying out this procedure with several experimental axial density profiles, we obtain the estimate η s = 0.60 ± 0.15. The error bar leads to uncertainties in the calculated aspect ratios that is about 2% for the smallest deformations and about 10% for the largest deformations.
Variation of core deformation -Having determined the universal constant η s , we can compare our theory to the experimental core aspect ratio versus polarization curve in Figure 2 of Ref. [6]. The aspect ratio actually depends on the total particle number in addition to the polarization. In the experiment, there are shot-to-shot variations of N ↑ and a systematic bias towards larger N ↑ at smaller P . We model this bias as N ↑ = (200 − 80P ) × 10 3 . Most of the shots are scattered around this line. Our results are shown in Fig. 2b, together with the experimental data.
In Fig. 2c, we display the core deformation as a function of total particle number. Surface effects are less important in the thermodynamic limit N → ∞ and the difference between core and trap aspect ratios vanishes as N −1/3 in this limit [12]. The lower curve is for a harmonic trap matching the trap of Ref. [6] at the center. The upper curve presents our expected result for the experiments of Shin et al. [5]. The predicted deformation of about 10% appears to be sufficiently large to have been observable in these experiments, which is consistent with the interpretation of Gubbels et al. [18] that no phase separation has occurred in this case. Deviation from ellipsoidal shape -Examination of the high-polarization data of Figure 1 of Ref. [6] reveals a peculiar feature. The minority absorption images, other than having a lower aspect ratio than the trap, also have shapes that are not quite ellipsoidal and tend to being somewhat cylindrical. The axial density profiles reveal the same geometric effect in terms of flatter peaks than would be expected for an ellipsoidal shape. Our formalism is well-suited for studying this effect, since the core boundary simply translates into distortions of local Fermi surfaces that can be used to calculate consistent density and energy profiles. We parameterize non-elliptical shapes by describing the core surface by the equation (r/R) γ + (z/Z) γ = 1. Using a γ equal to 2 gives an ellipsoid and infinite γ gives a cylinder. Comparing axial density profiles obtained with various values of γ, as in Figs. 3a and 3b, shows that the experimental features, e.g. the flat tops, are best reproduced with γ between 3 and 6, with the best γ tending to increase with increasing polarization, up to about P ≃ 0.6. However, a precise determination of γ is difficult from these fits alone.
Our formalism also allows us to extract γ from an energy calculation. Minimizing the total (bulk plus surface) energy, under the constraint of constant particle numbers, produces the observed deviation from ellipsoidal shape, with γ > 2 and γ increasing with P for moderate polarizations (Fig. 3d). The energy calculation also predicts a decrease of γ at larger P . This prediction is difficult to verify from the currently available axial den-sity data because of the large fluctuations of (N ↑ , N ↓ ) from the line N ↑ /10 3 = 200 − 80P .
Conclusions -To summarize, we have presented an anisotropic extension of the local density approximation in order to deal with situations where the aspect ratio of trapped fermionic gases differs from the trap aspect ratio due to many-body effects. Our procedure allows for a consistent calculation of the density and energy profiles. We have demonstrated this via a detailed analysis of experiments with phase-separated polarized fermions.
In particular, we have uncovered and explained a striking new effect concerning the shape of the superfluid core, which takes an unexpected non-ellipsoidal form. Our treatment dramatically highlights the utility of our formulation: we know of no other way of calculating the density distributions for an arbitrary cloud shape.
online.) (a) Calculation of the local anisotropic Fermi surface at position r (full dot). Points 1 and 2 with coordinates (±R(z), z) are the classical turning points for the motion in the r-direction. Points 3 and 4, i.e., (r, ±Z(r)), are the classical turning points for motion in the z direction. (b) Fits to the axial density profiles for both species using our approach. The top and bottom panels show two experimental shots, at polarizations P ≃ 0.5 and P ≃ 0.65, respectively.
FIG. 2 :
2(Color online.) (a) Total energy curves for fixed (N ↑ , N ↓ ) = (194, 92.7) × 10 3 , shown for ηs values of 0.30, 0.59 and 0.90 from bottom to top. (b) Core aspect ratio (A.R.) as a function of polarization for N ↑ /10 3 = 200 − 80P and calculated with ηs = 0.6. (c) Core A.R. versus particle number, for fixed polarization P = 0.49. The regions relevant to experiments [3, 4, 5, 6] are indicated by shaded ellipses.
FIG. 3 :
3(Color online.) Experimental axial density curves fitted with core shapes (a) γ = 2 and (b) γ = 8. Top panels show species axial densities, bottom panels show difference. The value γ = 2 misses the "flat" features, while γ = 8 overemphasizes them. (c) Surface shapes for γ = 2, 4, and 9, respectively. (d) Calculated value of γ for various polarizations, along the line N ↑ /10 3 = 200 − 80P .
Acknowledgments -We benefited greatly from discussions and collaboration with the experimental group at Rice University, in particular Wenhui Li and Randy Hulet.
. B Demarco, D S Jin, Science. 2851703B. DeMarco and D. S. Jin, Science 285, 1703 (1999);
. A G Truscott, Science. 2912570A. G. Truscott et. al., Science 291, 2570 (2001);
. F Schreck, Phys. Rev. Lett. 8780403F. Schreck et. al., Phys. Rev. Lett. 87, 080403 (2001);
. C A Regal, M Greiner, D S Jin, Phys. Rev. Lett. 9240403C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004);
. M W Zwierlein, Phys. Rev. M. W. Zwierlein et al., Phys. Rev.
. Lett, 92120403Lett. 92, 120403 (2004);
. M Bartenstein, Phys. Rev. Lett. 92120401M. Bartenstein et al., Phys. Rev. Lett. 92, 120401 (2004).
. M W Zwierlein, A Schirotzek, C H Schunck, W Ketterle, Science. 311492M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Science 311, 492 (2006).
. G B Partridge, W Li, R I Kamar, Y A Liao, R G Hulet, Science. 311503G. B. Partridge, W. Li, R. I. Kamar, Y. A. Liao, and R. G. Hulet, Science 311, 503 (2006).
. Y Shin, M W Zwierlein, C H Schunck, A Schirotzek, W Ketterle, Phys. Rev. Lett. 9730401Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle, Phys. Rev. Lett. 97 030401 (2006).
. G B Partridge, W Li, Y A Liao, R G Hulet, M Haque, H T C Stoof, Phys. Rev. Lett. 97190407G. B. Partridge, W. Li, Y. A. Liao, and R. G. Hulet, M. Haque and H. T. C. Stoof, Phys. Rev. Lett. 97, 190407 (2006).
. D E Sheehy, L Radzihovsky, Phys. Rev. Lett. 9660401D. E. Sheehy and L. Radzihovsky, Phys. Rev. Lett. 96, 060401 (2006).
. H Caldas, Phys. Rev. A. 6963602H. Caldas, Phys. Rev. A 69, 063602 (2004).
. F Chevy, Phys. Rev. Lett. 96130401F. Chevy, Phys. Rev. Lett. 96, 130401 (2006).
. M Haque, H T C Stoof, Phys. Rev. A. 7411602M. Haque and H. T. C. Stoof, Phys. Rev. A 74, 011602(R) (2006).
. T N Silva, E J Mueller, Phys. Rev. A. 7351602T. N. De Silva and E. J. Mueller, Phys. Rev. A 73, 051602(R) (2006).
. T N Silva, E J Mueller, Phys. Rev. Lett. 9770402T. N. De Silva and E. J. Mueller, Phys. Rev. Lett. 97, 070402 (2006).
. I Ia, Pomeranchuk, JETP. 35524I. Ia. Pomeranchuk, JETP 35, 524 (1958).
. V Oganesyan, S A Kivelson, E Fradkin, ; W Metzner, D Rohe, S Andergassen, Phys. Rev. Lett. 6466402Phys. Rev. BV. Oganesyan, S. A. Kivelson, and E. Fradkin, Phys. Rev. B 64, 195109 (2001). W. Metzner, D. Rohe, and S. Andergassen; Phys. Rev. Lett. 91, 066402 (2003).
. J Quintanilla, A J Schofield, Phys. Rev. B. 74115126J. Quintanilla and A. J. Schofield, Phys. Rev. B 74, 115126 (2006).
. H Müther, A Sedrakian, Phys. Rev. Lett. 88252503H. Müther and A. Sedrakian, Phys. Rev. Lett. 88, 252503 (2002);
. A Sedrakian, J Mur-Petit, A Polls, H Müther, Phys. Rev. A. 7213613A. Sedrakian,1 J. Mur-Petit, A. Polls, and H. Müther, Phys. Rev. A 72, 013613 (2005).
. J Carlson, S Reddy, Phys. Rev. Lett. 9560401J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401 (2005).
. K B Gubbels, M W J Romans, H T C Stoof, Phys. Rev. Lett. 97210402K. B. Gubbels, M. W. J. Romans, and H. T. C. Stoof, Phys. Rev. Lett. 97, 210402 (2006).
| [] |
[
"Higher geometry for non-geometric T-duals",
"Higher geometry for non-geometric T-duals"
] | [
"Thomas Nikolaus ",
"Konrad Waldorf "
] | [] | [] | We investigate topological T-duality in the framework of non-abelian gerbes and higher gauge groups. We show that this framework admits the gluing of locally defined T-duals, in situations where no globally defined ("geometric") T-duals exist. The gluing results into new, higher-geometrical objects that can be used to study non-geometric T-duals, alternatively to other approaches like non-commutative geometry. | 10.1007/s00220-019-03496-3 | [
"https://arxiv.org/pdf/1804.00677v2.pdf"
] | 59,151,208 | 1804.00677 | ed24c649085abd982869545795326982fa4d0c33 |
Higher geometry for non-geometric T-duals
May 2019
Thomas Nikolaus
Konrad Waldorf
Higher geometry for non-geometric T-duals
6May 2019topological T-dualitynon-abelian gerbesT-foldscategorical groups
We investigate topological T-duality in the framework of non-abelian gerbes and higher gauge groups. We show that this framework admits the gluing of locally defined T-duals, in situations where no globally defined ("geometric") T-duals exist. The gluing results into new, higher-geometrical objects that can be used to study non-geometric T-duals, alternatively to other approaches like non-commutative geometry.
Introduction
A topological T-background is a principal torus bundle π : E → X equipped with a bundle gerbe G over E, see [BEM04b,BHM04,BEM04a,BS05,BRS06]. Topological T-backgrounds take care for the underlying topology of a structure that is considered in the Lagrangian approach to string theory. There, the torus bundle is equipped with a metric and a dilaton field, and the bundle gerbe is equipped with a connection ("B-field"). Motivated by an observation of Buscher about a duality between these string theoretical structures [Bus87], topological T-backgrounds have been invented in order to investigate the quite difficult underlying topological aspects of this duality. In [BRS06] two T-backgrounds (E 1 , G 1 ) and (E 2 , G 2 ) are defined to be T-dual if the pullbacks p * 1 G 1 and p * 2 G 2 of the two bundle gerbes to the correspondence space
E 1 × X E 2 p 2 " " ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ p 1 | | ③ ③ ③ ③ ③ ③ ③ ③ E 1 π 1 " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ E 2 π 2 | | ② ② ② ② ② ② ② ② X
are isomorphic, and if this isomorphism satisfies a certain local condition relating it with the Poincaré bundle. The relevance of this notion of T-duality is supported by string-theoretical considerations, see [FSS18,Remark 6.3]. One of the basic questions in this setting is to decide, if a given T-background has T-duals, and how the possibly many T-duals can be parameterized. We remark that T-duality is a symmetric relation, but neither reflexive nor transitive.
Other formulations of T-duality have been given in the setting of non-commutative geometry [MR05,MR06b,MR06a] and algebraic geometry [BSST07], and equivalences between these three approaches have been established in [Sch07,BSST07].
Topological T-backgrounds can be grouped into different classes in the following way. The Serre spectral sequence associated to the torus bundle comes with a filtration π * H 3 (X, Z) = F 3 ⊆ F 2 ⊆ F 1 ⊆ F 0 = H 3 (E, Z), and we classify T-backgrounds by the greatest n such that the Dixmier-Douady class [G] ∈ H 3 (E, Z) is in F n . A result of Bunke-Rumpf-Schick [BRS06] is that a T-Background (E, G) admits T-duals if and only if it is F 2 . Further, possible choices are related by a certain action of the additive group so(n, Z) of skew-symmetric matrices B ∈ Z n×n , where n is the dimension of the torus. The group so(n, Z) appears there as a subgroup of the group O(n, n, Z), which acts on the set of (equivalence classes of) Tduality correspondences in such a way that both "legs" become mixed. The subgroup so(n, Z) preserves the left leg (E, G), and transforms one T-dual right leg into another. The results of [BRS06] have been obtained by constructing classifying spaces for F 2 T-backgrounds and Tduality correspondences. The actions of O(n, n, Z) and its subgroup so(n, Z) are implemented on the latter space as an action up to homotopy by homotopy equivalences.
If a T-background is only F 1 , then it does not have any T-duals; these are then called "mysteriously missing" [MR06a] or "non-geometric" T-duals [Hul07]. The approach via noncommutative geometry allows to define them as bundles of non-commutative tori [MR05,MR06b,MR06a]. In this paper, we propose a new, 2-stack-theoretical ansatz, which allows to remain in ordinary (commutative) but higher-categorical geometry. The basic idea behind our approach is quite simple: every F 1 background is locally F 2 , and so has locally defined T-duals. On overlaps, these are related by the so(n, Z)-action of Bunke-Rumpf-Schick. The main result of this article is to fabricate a framework in which this gluing can be performed, resulting in a globally defined, new object that we call a half-geometric T-duality correspondence.
Half-geometric T-duality correspondences are called "half-geometric" because they have a well-defined "geometric" left leg, but no "geometric" right leg anymore. It can be seen as a baby version of what Hull calls a T-fold (in the so-called doubled geometry perspective): upon choosing a "polarization", a half-geometric T-duality correspondence splits into locally defined T-duality correspondences between the geometric left leg and an only locally defined geometric right leg. Additionally, it provides a consistent set of so(n, Z) matrices that perform the gluing. Our main result about half-geometric T-duality correspondences is that (up to isomorphism) every F 1 T-background is the left leg of a unique half-geometric T-duality correspondence.
T-folds should be regarded and studied as generalized (non-geometric) backgrounds for string theory, and we believe that understanding the underlying topology is a necessary prerequisite. Since currently no definition of a T-fold in general topology is available, this has to be seen as a programme for the future. Only two subclasses of T-folds are currently well-defined: the first subclass consists of T-duality correspondences, and the result of Bunke-Rumpf-Schick states that these correspond to the (geometric) F 2 T-backgrounds. The second subclass consists of our new half-geometric T-duality correspondences. It contains the first subclass, and our main result states that it corresponds to the (geometric) F 1 T-backgrounds. We plan to continue this programme in the future and to define and study more general classes of T-folds.
The framework we develop consists of 2-stacks that are represented by appropriate Lie 2groups. This is fully analogous to the situation that a stack of principal bundles is represented by a Lie group. Using 2-stacks is essential for performing our gluing constructions. In principle, the reader is free to choose any model for such 2-stacks, for instance non-abelian bundle gerbes [ACJ05,NW13a], principal 2-bundles [Woc11, SP11,Wal17], or any ∞-categorical model [Lur09,NSS15]. For our calculations we use a simple and very explicit cocycle model for the non-abelian cohomology groups that classify our 2-stacks. Our work is based on two new strict Lie 2-groups, which we construct in Sections 2.2 and 3.2:
• A strict (Fréchet) Lie 2-group TB F2 , see Section 2.2. It represents a 2-stack T-BG F2 , whose objects are precisely all F 2 T-backgrounds (Proposition 2.2.3).
• A strict Lie 2-group TD, see Section 3.2. It is a categorical torus in the sense of Ganter [Gan18], and represents a 2-stack T-Corr of T-duality correspondences. Its truncation to a set-valued presheaf is precisely the presheaf of T-duality triples of [BRS06] (Propositions 3.3.2 and 3.2.5).
These two Lie 2-groups have two important features. The first is that they admit Lie 2-group homomorphisms ℓeℓe : TD → TB F2 and riℓe : TD → TB F2 (1.1) that represent the projections to the left leg and the right leg of a T-duality correspondence, respectively. The second feature is that they admit a strict so(n, Z)-action by Lie 2-group homomorphisms. This is an important improvement of the action of [BRS06] on classifying spaces which is only an action up to homotopy, i.e., in the homotopy category. Taking the semi-direct products for these actions, we obtain new Lie 2-groups:
• A Lie 2-group TB F1 := TB F2 ⋉ so(n, Z)
of which we prove that it represents the 2-stack T-BG F1 of F 1 T-backgrounds (Proposition 2.3.1).
• A Lie 2-group TD 1 2 -geo := TD ⋉ so(n, Z)
that represents a 2-stack which does not correspond to any classical geometric objects, and which is by definition the 2-stack of half-geometric T-duality correspondences.
We show that the left leg projection ℓeℓe is so(n, Z)-equivariant, and that it hence induces a Lie 2-group homomorphism ℓeℓe so(n,Z) : TD
1 2 -geo → TB F1 .
Our main result about existence and uniqueness of half-geometric T-duality correspondences is that this homomorphism induces an isomorphism in non-abelian cohomology (Theorem 4.2.2).
On a technical level, two innovations in the theory of Lie 2-groups are developed in this article. The first is the notion of a crossed intertwiner between crossed modules of Lie groups (Definition A.1.1). Crossed modules are the same as strict Lie 2-groups, and crossed intertwiners are in between strict homomorphisms and fully general homomorphisms ("butterflies"). The notion of crossed intertwiners is designed exactly so that the implementations needed in this article can be performed; for instance, the two Lie 2-group homomorphisms in Eq. (1.1) are crossed intertwiners. The second innovation is the notion of a semi-direct product for a discrete group acting by crossed intertwiners on a strict Lie 2-group (Appendix A.4). We also explore various new aspects of non-abelian cohomology in relation to crossed intertwiners and semi-direct products, for example the exact sequence of Proposition A.4.3.
In the background of our construction lures the higher automorphism group AUT(TD) of TD. In an upcoming paper we compute this automorphism group in collaboration with Nora Ganter. We will show there that π 0 (AUT(TD n )) = O ± (n, n, Z), a group that was already mentioned in [MR06a], which contains the split-orthogonal group O(n, n, Z) as a subgroup of index two. We will see that AUT(TD n ) splits over the subgroup so(n, Z). The action of so(n, Z) on TD n we describe here turns out to be the action of AUT(TD n ) induced along this splitting, and so embeds our approach into an even more abstract theory. Our plan is to provide new classes of T-folds within this theory.
This article is organized in the following way. In Section 2 we discuss T-backgrounds and introduce the two Lie 2-groups that represent F 2 and F 1 T-backgrounds. In Section 3 we construct the Lie 2-group TD and put it into relation to ordinary T-duality. In Section 4 we introduce the Lie 2-group TD 1 2 -geo that represents half-geometric T-duality correspondences, and prove our main results. In Appendix A we develop our inventions in the theory of Lie 2-groups, and in Appendix B we summarize all facts about the Poincaré bundle that we need.
Acknowledgements.
We would like to thank Ulrich Bunke, Nora Ganter, Ruben Minasian, Ingo Runkel, Urs Schreiber, Richard Szabo, Christoph Schweigert and Peter Teichner for many valuable discussions. KW was supported by the German Research Foundation under project code WA 3300/1-1.
Higher geometry for topological T-backgrounds
In this section we study three bicategories of T-backgrounds. The bicategorical framework is a necessary prerequisite for applying 2-stack-theoretical methods, as the gluing property only holds in that setting. As an important improvement of the theory of T-backgrounds, we introduce Lie 2-groups that represent the corresponding 2-stacks (see Definition A.3.1 for the precise meaning). We will later obtain most statements by only considering the representing 2-groups.
T-backgrounds as 2-stacks
Let X be a smooth manifold. We recall [Ste00,Wal07] that U(1)-bundle gerbes over X form a bigroupoid, with the set of 1-isomorphism classes of objects in bijection to H 3 (X, Z). Let n > 0 be a fixed integer; all definitions are relative to this integer (often without writing it explicitly). We recall the definition of a (topological) T-background; their bi-groupoidal structure will be essential in this article. We denote by T n := U(1) × ... × U(1) the n-torus.
Definition 2.1.1. (a) A T-background over X is a principal T n -bundle π : E → X together with a U(1)-bundle gerbe G over E. (b) A 1-morphism (E, G) → (E ′ , G ′ ) is a pair (f, B)
of an isomorphism f : E → E ′ of principal T n -bundles over X and a bundle gerbe 1-morphism B :
G → f * G ′ over E. (c) A 2-morphism (E, G) (f,B 1 ) ( ( (f,B 2 ) 6 6 (E ′ , G ′ )
is a bundle gerbe 2-morphism β : B 1 ⇒ B 2 over E. Vertical composition is the vertical composition of bundle gerbe 2-morphisms, and horizontal composition is given by
(E, G) (f 1 ,B 1 ) ( ( (f 1 ,B ′ 1 ) 6 6 β 1 (E ′ , G ′ ) (f 2 ,B 2 ) ( ( (f 2 ,B ′ 2 ) 6 6 β 2 (E ′′ , G ′′ ) = (E, G) (f 2 •f 1 ,f * 1 B 2 •B 1 )
) )
(f 2 •f 1 ,f * 1 B ′ 2 •B ′ 1 ) 6 6 f * 1 β 2 • β 1 (E ′′ , G ′′ ).
T-backgrounds over X form a bigroupoid T-BG(X), and the assignment X → T-BG(X) is a presheaf of bigroupoids over smooth manifolds. The following statement is straightforward to deduce from the fact that principal bundles and bundle gerbes form (2-)stacks over the site of smooth manifolds (i.e. they satisfy descent with respect to surjective submersions).
Proposition 2.1.2. The presheaf T-BG is a 2-stack.
The grouping of T-backgrounds into classes depending on the class of the bundle gerbe will be done in a different, but equivalent way, compared to Section 1. A T-background (E, G) is called F 1 if G is fibre-wise trivializable, i.e. for every point x ∈ X the restriction G| Ex of G to the fibre E x is trivializable. For dimensional reasons, every T-background with torus dimension n ≤ 2 is F 1 . The condition of being fibre-wise trivial extends for free to neighborhoods of points:
Lemma 2.1.3. A T-background (E, G) is F 1 if and only if every point x ∈ X has an open neighborhood x ∈ U ⊆ X such that the restriction of G to the preimage E U := π −1 (U ) is trivializable. Proof. Choose a contractible open neighborhood x ∈ U ⊆ X that supports a local trivializa- tion φ : U × T n → E U . Since U is contractible, we have φ * G ∼ = pr * T n H for a bundle gerbe H on T n . Since (E, G) is F 1 , we know that φ * G| {x}×T n = (pr * T n H)| {x}×T n is trivializable. Pulling back along T n → {x} × T n shows that H is trivializable.
We recall that the cohomology H k (E, Z) of the total space of a fibre bundle has a filtration
H k (E, Z) ⊇ F 1 H k (E, Z) ⊇ ... ⊇ F k H k (E, Z) = π * H k (X, Z).
(2.1.1)
Here, ξ ∈ H k (E, Z) is in F i H k (E, Z)
if the following condition is satisfied for every continuous map f : C → X defined on an (i − 1)-dimensional CW complex C: iff : f * E → E denotes the induced map, thenf * ξ = 0. Obviously, we have the following result.
Lemma 2.1.4. A T-background is F 1 if and only if [G] ∈ F 1 H 3 (E, Z).
We let T-BG F1 (X) be the full sub-bicategory of T-BG(X) on the F 1 T-backgrounds. Consider the full sub-bicategory F 1 (X) ⊆ T-BG F1 (X) on the single object (X × T n , I), where I denotes the trivial bundle gerbe. Then, X → F 1 (X) is a sub-presheaf of T-BG F1 . By Proposition 2.1.2 one can form the closure F 1 ⊆ T-BG of F 1 under descent, i.e., F 1 is a sub-2-stack of T-BG that 2-stackifies F 1 . For example, one can use the 2-stackification F → F + described in [NS11], and then choose an equivalence between F + 1 and a sub-2-stack F 1 of T-BG, which exists since T-BG is a 2-stack.
Proposition 2.1.5. We have F 1 = T-BG F1 . In particular, T-BG F1 is a 2-stack.
Proof. It is clear from the definition of T-BG F1 that F 1 ⊆ T-BG F1 . Let (E, G) be some object in T-BG F1 (X). By Lemma 2.1.3 there exists an open cover U = {U i } i∈I over which (E, G) is isomorphic to the single object (X ×T n , I). Thus, the descent object (E, G)| U ∈ Des T-BG F1 (U ) is isomorphic to an object x ∈ Des F 1 (U ). This shows that (E, G) is in F 1 (X).
Next we consider the following sub-bicategory F 2 (X) ⊆ T-BG(X).
(a) Its only object is (X × T n , I).
(b) The 1-morphisms are of the form (f, id I ).
(c) The 2-morphisms are all 2-morphisms in T-BG(X).
The assignment X → F 2 (X) forms a sub-presheaf F 2 ⊆ T-BG. Its closure in T-BG under descent is by definition the 2-stack T-BG F2 := F 2 . A T-background over X is called F 2 if it is in T-BG F2 (X). By construction, F 2 ⊆ F 1 , so that we have 2-stack inclusions
T-BG F2 ⊆ T-BG F1 ⊆ T-BG
of which the first is full only on the level of 2-morphisms, and the second is full. This has to be taken with care: two F 2 T-backgrounds can be isomorphic as T-backgrounds without being isomorphic as F 2 T-backgrounds.
Lemma 2.1.6. A T-background is F 2 if and only if [G] ∈ F 2 H 3 (E, Z).
Proof. Let (E, G) be an F 2 T-background. Let C be a 1-dimensional CW complex and f : C → X be continuous. Let {U i } i∈I be a cover of X by open sets over which (E, G) trivializes. For dimensional reasons, the pullback cover {f −1 (U i )} i∈I of C can be refined to an open cover with no non-trivial 3-fold intersections. Descend data of f * (E, G) with respect to this cover has only trivial 2-morphisms. In other words, the bundle gerbef * G is obtained by gluing trivial gerbes I along identity 1-morphisms id I and identity 2-morphisms. This gives the trivial bundle gerbe, hence [f * G] = 0.
Conversely, since F 2 H 3 ⊆ F 1 H 3 , we know that (E, G) is locally trivializable. Thus, there exists an open cover {U i } i∈I and 1-morphisms (U i ×T n , I) ∼ = (E, G)| U i . We form an open cover V i := π −1 (U i ) of E, such that the bundle gerbe G is isomorphic to one whose surjective submersion is the disjoint union of the open sets V i . In particular, there are principal U(1)-bundles P ij over V i ∩ V j . By construction, we have diffeomorphisms φ ij :
V i ∩ V j → (U i ∩ U j ) × T n .
Assuming that all double intersections U i ∩ U j are contractible, and we can assume that P ij ∼ = φ * ij pr * P B ij , for a matrix B ij ∈ so(n, Z) and pr : (U i ∩ U j ) × T n → T n the projection, see Appendix B. We remark that the two inclusions V i ∩ V j → V i , V j correspond under the diffeomorphisms φ ij to the maps pr 1 (x, a) := (x, a) and pr 2 (x, a) = (x, ag ij (x)), respectively, where g ij are the transition functions of the E.
Next we consider a triangulation of X subordinate to the open cover {U i } i∈I , take the dual of that, discard all simplices of dimension ≥ 2, and let C be the remaining 1-dimensional CW-complex with its inclusion f :
C → X. Letf : f * E → E the induced map. By assumption,f * G is trivializable. Consider the open setsṼ i :=f −1 (V i ) that cover f * E, so that f * G has the U(1)-bundlesP ij :=f * P ij ∼ =f * φ * ij pr * P B ij . Thatf * G is trivializable means that there exist principal U(1)-bundles Q i overṼ i with bundle isomorphismsP ij ⊗ pr * 2 Q j ∼ = pr * 1 Q i overṼ i ∩Ṽ j . We have a diffeomorphismφ i :Ṽ i → f −1 (U i ) × T n , where f −1 (U i ) is star-shaped and hence contractible. Thus, we have Q i ∼ =φ * i pr * P C i , for matrices C i ∈ so(n, Z). Evaluating above bundle isomorphism at a point (x, a) with f (x) ∈ U i ∩ U j , we obtain an isomorphism P ij | (x,a) ⊗ Q j | x,ag ij (x) ∼ = Q i | (x,a) .
In terms of the principal bundles over T n , this means
P B ij | a ⊗ P C j | ag ij (x) ∼ = P C i | a .
Using the equivariance of the Poincaré bundle described in Appendix B, this leads to B ij + C j = C i . With this statement about matrices, we go back into the original situation over X, where it means that the bundle gerbe G is isomorphic to one with respect to the open cover V i , all whose principal U(1)-bundles are all trivial. This shows that the given T-background is F 2 .
Remark 2.1.7. For completeness, we remark that a T-background (E, G) is called F 3 if G admits a T n -equivariant structure. This is equivalent to the statement that G is the pullback of a bundle gerbe over X, which is in turn equivalent to [G] ∈ F 3 H 3 (E, Z).
A 2-group that represents F 2 T-backgrounds
We define a strict (Fréchet) Lie 2-group TB F2' n that represents the 2-stack T-BG F2 in the sense of Definition A.3.1, see Appendix A for more information.
Remark 2.2.1. We fix the following notations, which will be used throughout this article.
• For a matrix A ∈ R n×n and v, w ∈ R n we write, as usual,
v|A|w := n i,j=1 A ij v i w j ∈ R.
Suppose B ∈ R n×n is skew-symmetric, i.e. B ∈ so(n, R). We let B low ∈ R n×n be the unique lower-triangular nilpotent matrix with B = B low − B tr low . We write v|B|w low := v|B low |w . Note that v|B|w = v|B|w low − w|B|v low .
We use the notation v|A| for the linear form w → v|A|w , and similarly v|B| low for w → v|B|w low .
• For the abelian groups T n , in particular T 1 = U(1), we use additive notation. If B ∈ so(n, Z) and v ∈ Z n , then v|B| and v|B| low induce well-defined group homomorphisms T n → U(1).
• If α ∈ C ∞ (T n , U(1)) and b ∈ T n , then we define b α ∈ C ∞ (T n , U(1)) by b α(a) := α(a − b), which is an action of T n on C ∞ (T n , U(1)). Note that a v|B| = v|B| − v|B|a .
We define the Lie 2-group TB F2' n as a crossed module (G, H, t, α) in the following way, see Appendix A. We put G := T n , H := C ∞ (T n , U(1)) with the point-wise Lie group structure and the usual Fréchet manifold structure, t : H → G is defined by t(τ ) := 0 and α : G×H → H is defined by α(g, τ ) := g τ . We instantly find
π 0 (TB F2' n ) = T n and π 1 (TB F2' n ) = C ∞ (T n , U(1)), (2.2.1)
with π 0 acting on π 1 by (g, τ ) → g τ .
Remark 2.2.2. There is a slightly bigger but weakly equivalent 2-group denoted by TB F2 n . It fits better to our T-duality 2-group of Section 3 and will therefore be used later. For TB F2 n we put G := R n , H := C ∞ (T n , U(1)) × Z n with the direct product group structure, t : H → G is defined by t(τ, m) := m and α : G × H → H is defined by α(g, (τ, m)) := ( g τ, m). There is a strict intertwiner (see Appendix A.3)
TB F2
n → TB F2' n defined by reduction mod Z n , R n → T n , and the group homomorphism
Z n × C ∞ (T n , U(1)) → C ∞ (T n , U(1)) : (τ, m) → τ .
It induces identities on π 0 and π 1 and preserves the π 0 action on π 1 ; it is hence a weak equivalence.
We will often suppress the index n from the notation of both Lie 2-groups. In order to establish a relation between the 2-group TB F2' and F 2 T-backgrounds we consider the presheaf BTB F2' of smooth BTB F2' -valued functions, and show that its 2-stackification (BTB F2' ) + is isomorphic to T-BG F2 . We describe BΓ-valued functions for a general Lie 2-group Γ in Appendix A.3; reducing it to the present situation we obtain over a smooth manifold X the following bicategory BTB F2' (X):
• It has one object.
• The 1-morphisms are smooth maps g : X → T n , the composition is point-wise addition.
• There are only 2-morphisms from a 1-morphism to itself; these are all smooth maps τ : X → C ∞ (T n , U(1)). The vertical composition is the point-wise addition, and the horizontal composition is *
g 1 g 1 B B τ 1 * g 2 g 2 B B τ 2 * = * g 1 +g 2 g 1 +g 2 @ @ τ 2 + g 2 τ 1 * .
We consider the following strict 2-functor
F 2 : BTB F2' (X) → F 2 (X).
It associates to the single object the T-background with the trivial T n -bundle X × T n over X and the trivial gerbe I over X × T n . To a 1-morphism g it associates the 1-morphism consisting of the bundle morphism f g : X × T n → X × T n : (x, a) → (x, g(x) + a), and the identity 1-morphism id I between I and f * g I = I. This respects strictly the composition. To a 2-morphism τ : g ⇒ g it associates the bundle gerbe 2-morphism β τ : id I ⇒ id I over X × T n which is induced by the smooth map β τ :
X × T n → U(1) : (x, a) → −τ (x)(g(x) + a). It is clear that β τ 1 +τ 2 = β τ 1 + β τ 2 = β τ 2 • β τ 1 ,
i.e. the vertical composition is respected. It remains to show that the horizontal composition is respected, i.e. β τ 2 • β τ 1 = β τ 2 + g 2τ 1 , (2.2.2) where τ 1 : g 1 ⇒ g 1 and τ 2 : g 2 ⇒ g 2 . On the left we have horizontal composition in T-BG(X), so that it is f * g 1 β τ 2 • β τ 1 in terms of the composition bundle gerbe 2-morphisms. We have
(f * g 1 β τ 2 • β τ 1 )(x, a) = β τ 2 (x, g 1 + a) + β τ 1 (x, a) = −τ 2 (x)(g 1 + g 2 + a) − τ 1 (x)(g 1 + a) = −(τ 2 + g 2 τ 1 )(x)(g 1 + g 2 + a) = β τ 2 + g 2τ 1 (x, a).
This proves Eq. (2.2.2) and shows that F 2 is a 2-functor.
Proposition 2.2.3. The 2-functor F 2 induces an equivalence (BTB F2' ) + ∼ = T-BG F2 .
Proof. We show that the 2-functor F 2 is an isomorphism of presheaves. Then, it becomes an isomorphism between the 2-stackifications; this is the claim. The 2-functor is bijective on the level of objects; in particular it is essentially surjective. It is also bijective on the level of 1-morphisms, since the automorphisms of the trivial T n -bundle over X are exactly the smooth T n -valued functions on X. Finally, it is bijective on the level of 2-morphisms, since the automorphisms of id I are exactly the smooth U(1)-valued functions on X × T n .
g ij : U i ∩ U j → T n and τ ijk : U i ∩ U j ∩ U k → C ∞ (T n , U(1))
satisfying the cocycle conditions
g ik = g jk + g ij (2.2.3) τ ikl + g kl τ ijk = τ ijl + τ jkl . (2.2.4)
Two TB F2' n -cocycles (g, τ ) and (g ′ , τ ′ ) are equivalent if there exist smooth maps
h i : U i → T n and ǫ ij : U i ∩ U j → C ∞ (T n , U(1)) such that g ′ ij + h i = h j + g ij (2.2.5) τ ′ ijk + g ′ jk ǫ ij + ǫ jk = ǫ ik + h k τ ijk . (2.2.6)
Remark 2.2.5. Similarly, a TB F2 -cocycle consists of numbers m ijk ∈ Z n and smooth maps a ij : U i ∩ U j → R n and τ ijk satisfying a ik = a jk + a ij + m ijk (2.2.7)
τ ikl + a kl τ ijk = τ ijl + τ jkl .
(2.2.8)
Here we have identified smooth maps U i ∩ U j ∩ U k → Z n with elements in Z n , since we can refine any open cover by one whose open sets and all finite intersections are either empty or connected. Note that Eq. (2.2.7) implies m ikl + m ijk = m ijl + m jkl .
(2.2.9)
Two TB F2 -cocycles (a, τ, m) and (a ′ , τ ′ , m ′ ) are equivalent if there exist numbers z ij ∈ Z n and smooth maps p i : U i → R n and ǫ ij :
U i ∩ U j → C ∞ (T n , U(1)) satisfying a ′ ij + p i = z ij + p j + a ij (2.2.10) τ ′ ijk + a ′ jk ǫ ij + ǫ jk = ǫ ik + p k τ ijk . (2.2.11)
Note that Eq. (2.2.10) implies
m ′ ijk + z ij + z jk = z ik + m ijk .
(2.2.12)
Remark 2.2.6. Cocycles for TB F2' or TB F2 where τ ijk takes values in the constant U(1)-valued functions U(1) ⊆ C ∞ (T n , U(1)) correspond to F 3 T-backgrounds.
A 2-group that represents F 1 T-backgrounds
We show in this section that F 1 T-backgrounds are obtained by letting the (additive) group so(n, Z) of skew-symmetric n × n matrices with integer entries act on the 2-group TB F2 constructed in the previous section. In this context so(n, Z) appears as the group H 2 (T n , Z) of isomorphism classes of principal U(1)-bundles over T n ; a group isomorphism is induced by an assignment B → P B of a principal U(1)-bundle P B to a matrix B ∈ so(n, Z), which we described in Appendix B.
We define for each B ∈ so(n, Z) a crossed intertwiner
F B : TB F2 n → TB F2 n ,
in the notation of Definition A.1.1 as the triple F B = (id R n , f B , η B ), with smooth maps
f B : C ∞ (T n , U(1)) × Z n → C ∞ (T n , U(1)) × Z n and η B : R n × R n → C ∞ (T n , U(1))
given by For B 1 , B 2 ∈ so(n, Z) we compute the composition of the corresponding crossed intertwiners using the formula of the Eq. (A.1.1):
F B 2 • F B 1 = (id, f B 2 , η B 2 ) • (id, f B 1 , η B 1 ) = (id, f B 2 • f B 1 , η B 2 + η B 1 ) = (id, f B 1 +B 2 , η B 1 +B 2 ) = F B 2 +B 1 .
Thus, we have an action of so(n, Z) on TB F2 by crossed intertwiners in the sense of Definition A.4.1. As explained in Appendix A.4 we can now form a semi-strict Lie 2-group TB F1 := TB F2 ⋉ so(n, Z), whose two invariants are π 0 (TB F1 ) = T n × so(n, Z) and π 1 (TB F1 ) = C ∞ (T n , U(1)).
(2.3.1)
Since semi-strict 2-groups have no description as crossed modules, we can only describe TB F1 as a monoidal category, where the monoidal structure is the multiplication. Reducing the general theory of Appendix A.4 to the present situation, we obtain the following. The objects of TB F1 are pairs (a, B) with a ∈ R n and B ∈ so(n, Z), and multiplication is the direct product group structure, (a 2 , B 2 ) · (a 1 , B 1 ) = (a 2 + a 1 , B 2 + B 1 ).
The morphisms are tuples (τ, m, a, B) with source (a, B) and target (m + a, B). The composition is (τ 2 , m 2 , m 1 + a 1 , B) • (τ 1 , m 1 , a 1 , B) := (τ 2 + τ 1 , m 1 + m 2 , a 1 , B).
Multiplication is given by (τ 2 , m 2 , a 2 , B 2 ) · (τ 1 , m 1 , a 1 , B 1 ) = (τ 2 − a 1 |B 2 |m 1 low + a 2 τ 1 − a 2 m 1 |B 2 |, m 2 + m 1 , a 2 + a 1 , B 2 + B 1 ).
The semi-strictness of this 2-group corresponds to the fact that this multiplication is not strictly associative; instead, it has an associator that satisfies a pentagon axiom. The associator is given by the formula λ((a 3 , B 3 ), (a 2 , B 2 ), (a 1 , B 1 )) = (− a 1 |B 3 |a 2 low , 0, a 3 + a 2 + a 1 , B 3 + B 2 + B 1 ).
Next we establish a relation between the 2-group TB F1 and F 1 T-backgrounds. We consider the presheaf BTB F1 of smooth BTB F1 -valued functions. Consulting Appendix A.3, the bicategory BTB F1 (X) over a smooth manifold X is the following:
• It has one object.
• The morphisms are pairs (a, B) of a smooth map a : X → R n and a skew-symmetric matrix B ∈ so(n, Z). The composition is (point-wise) addition.
• There are only 2-morphisms between (a, B) and (a ′ , B ′ ) if a ′ − a ∈ Z n and B ′ = B; in this case a 2-morphism is a pair (τ, m) with τ : X → C ∞ (T n , U(1)) and m ∈ Z n such that a ′ − a = m. The vertical composition is (pointwise) addition, and the horizontal composition is *
(a 1 ,B 1 ) (a 1 +m 1 ,B 1 ) B B (τ 1 , m 1 ) * (a 2 ,B 2 ) (a 2 +m 2 ,B 2 ) B B (τ 2 , m 2 ) * = * (a 1 +a 2 ,B 1 +B 2 ) & & (a 1 +a 2 +m 1 +m 2 ,B 1 +B 2 ) 8 8 τ 2 + a 2 τ 1 − a 1 |B 2 |m 1 low − a 2 m 1 |B 2 | * .
• The associator
λ (a 3 ,B 3 ),(a 2 ,B 2 ),(a 1 ,B 1 ) : ((a 3 , B 3 ) • (a 2 , B 2 )) • (a 1 , B 1 ) ⇒ (a 3 , B 3 ) • ((a 2 , B 2 ) • (a 1 , B 1 ))
is the pair (− a 1 |B 3 |a 2 low , 0).
Next we construct a 2-functor
F 1 : BTB F1 (X) → F 1 .
It associates to the single object the T-background (X × T n , I). To a 1-morphism (a, B) it associates the 1-morphism (f a , B B ), consisting of the bundle morphism
f a : X × T n → X × T n : (x, b) → (x, a(x) + b) (2.3.2)
and the following 1-morphism B B : I → f * a I = I over X × T n . We recall that the groupoid Aut(G) of automorphisms of a bundle gerbe G over a smooth manifold M is a module category over the monoidal groupoid Bun U(1) (M ) of principal U(1)-bundles over M in terms of a functor
Aut(G) × Bun U(1) (M ) → Aut(G) : (A, P ) → A ⊗ P . (2.3.3)
The 1-morphism B B we want to construct is obtained by letting the principal U(1)-bundle pr * T n P B over X × T n act on the identity, where P B is the matrix-depending version of the Poincaré bundle defined in Appendix B. Thus,
B B := id ⊗ pr * T n P B . To a 2-morphism (τ, m, a, B) : (a, B) ⇒ (a + m, B)
it assigns the 2-morphism β τ,m,a,B : (f a , B B ) ⇒ (f a , B B ) over X × T n induced by acting on id B B with an automorphism of the trivial U(1)-bundle over X × T n , given the smooth map
κ τ,m,a,B : X × T n → U(1) : (x, b) → −τ (x)(a(x) + b),
i.e., β τ,m,a,B := id B B ⊗ κ τ,n,a,B . It is straightforward to check that the vertical composition is respected. The horizontal composition is not strictly preserved:
we have f a 2 +a 1 = f a 2 • f a 1 but B B 2 +B 1 = f * a 1 B B 2 • B B 1 . A compositor c (a 1 ,B 1 ),(a 2 ,B 2 ) : (f a 2 , B B 2 ) • (f a 1 , B B 1 ) ⇒ (f a 2 +a 1 , B B 2 +B 1 )
is induced over {x} × T n from the isomorphism
P B 1 ⊗ r * x 1 a 1 (x) P B 2 id⊗R B 2 (a 1 (x)) −1 / / P B 1 ⊗ P B 2 = P B 2 +B 1 of U(1)-bundles over T n , where the bundle morphismR B is explained in Appendix B.
We have to show that this compositor satisfies a pentagon diagram. This diagram can be reduced to the following condition about the equivariance of the Poincaré bundle:
P B 1 ⊗ r * a 1 P B 2 ⊗ r * a 2 +a 1 P B 3 id⊗id⊗r * a 1R B 3 (a 2 ) −1
u } r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r
id⊗R B 2 (a 1 ) −1 ⊗id 3 A ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ P B 1 ⊗ r * a 1 P B 2 ⊗ r * a 1 P B 3 id⊗R B 2 (a 1 ) −1 ⊗R B 3 (a 1 ) −1 " 0 ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ P B 1 ⊗ P B 2 ⊗ r * a 2 +a 1 P B 3 id⊗id⊗R B 3 (a 1 +a 2 ) −1 Ð Ø ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ P B 1 ⊗ P B 2 ⊗ P B 3 a 1 |B 3 |a 2 low C Q P B 1 ⊗ P B 2 ⊗ P B 3
Splitting this diagram into the three tensor factors, the only nontrivial diagram is the third tensor factor, where it becomes
r * a 2 +a 1 P B 3 r * a 1R B 3 (a 2 ) −1 R B 3 (a 1 +a 2 ) −1 C Q P B 3 r * a 1 P B 3R B 3 (a 1 ) −1 C Q P B 3 a 1 |B 3 |a 2 low u
The commutativity of this diagram is precisely Eq. (B.2); this finishes the definition of the compositor. Now it remains to show that the horizontal composition is respected relative to the compositor. Employing the horizontal composition in T-BG(X) and the definitions of the 2-functor on 1-morphisms and of the compositor, this condition becomes the commutativity of the diagram
P B 1 ⊗ r * x 1 a 1 P B 2 κ τ 2 ,m 2 ,a 1 +a 2 ,B 2 +κ τ 1 ,m 1 ,a 1 ,B 1 C Q id⊗R B 2 (a 1 ) −1 P B 1 ⊗ r * x 1 a 1 P B 2 id⊗R B 2 (a 1 +m 1 ) −1 P B 1 ⊗ P B 2 κ τ 2 + α 2τ 1 − a 1 |B 2 |m 1 low − a 2 m 1 |B 2 |,m 2 +m 1 ,a 2 +a 1 ,B 2 +B 1 C Q P B 1 ⊗ P B 2 (2.3.4)
It can be checked in a straightforward way using the definition of κ and Eq. (B.4). This finishes the construction of the 2-functor F 1 .
Proposition 2.3.1. The 2-functor F 1 induces an equivalence (TB F1 ) + ∼ = T-BG F1 .
Proof. We show that the 2-functor F 1 is an isomorphism of presheaves for each contractible open set U ⊆ X. Then, it becomes an isomorphism between the stackifications; this is the claim. The 2-functor is bijective on the level of objects; in particular it is essentially surjective. The Hom-functor
F 1 : Hom BTB F1 (U ) ( * , * ) → Hom F 1 (U × T n , U × T n ) = Hom T-BG(U ) (U × T n , U × T n )
is clearly fully faithful, and we claim that it is essentially surjective. Indeed, if (f, B) is some automorphism of U × T n , then f = f a for some smooth map a : U → R n since U is simplyconnected, and B = pr * T n P B for some matrix B ∈ so(n, Z), as the cohomology of U × T n has no contributions from U since U is 2-connected.
Remark 2.3.2. In Appendix A.4 we describe cocycles for semi-strict Lie 2-groups. A TB F1 ncocycle with respect to an open cover {U i } i∈I is a quadruple (B, a, m, τ ) consisting of matrices B ij ∈ so(n, Z), numbers m ijk ∈ Z n , and smooth maps
a ij : U i ∩ U j → R n and τ ijk : U i ∩ U j ∩ U k → C ∞ (T n , U(1))
subject to the relations
B ik = B jk + B ij (2.3.5) a ik = a jk + a ij + m ijk (2.3.6) τ ikl + a kl τ ijk − a kl m ijk |B kl | = a ik |B kl |m ijk low + a ij |B kl |a jk low + τ ijl + τ jkl . (2.3.7)
Note that the Eq. (2.3.6) implies
m ikl + m ijk = m ijl + m jkl . (2.3.8) Two TB F1 -cocycles (B, a, m, τ ) and (B ′ , a ′ , m ′ , τ ′ ) are equivalent if there exist matrices C i ∈ so(n, Z)
, numbers z ij ∈ Z and smooth maps
p i : U i → R n and ǫ ij : U i ∩ U j → C ∞ (T n , U(1)) such that C j + B ij = B ′ ij + C i (2.3.9) z ij + p j + a ij = a ′ ij + p i (2.3.10) and τ ′ ijk + p i |B ′ jk |a ′ ij low − p j + a ij |B ′ jk |z ij low + a ′ jk ǫ ij − a ′ jk z ij |B ′ jk | − a ij |B ′ jk |p j low + ǫ jk = ǫ ik − a ik |C k |m ijk low + p k τ ijk − p k m ijk |C k | − a ij |C k |a jk low (2.3.11)
Note that Eq. (2.3.10) implies
m ′ ijk + z ij + z jk = z ik + m ijk . (2.3.12)
We remark that the subclass of F 2 backgrounds is represented by cocycles with B ij = 0; in that case, above equations are precisely the cocycle conditions for the 2-group TB F2 n .
Remark 2.3.3. The sequence
TB F2 → TB F1 → so(n, Z) dis ,
of semi-strict Lie 2-groups and semi-strict homomorphisms induces by Proposition A.4.3 the following exact sequence in cohomology:
H 1 (X, TB F2 )/so(n, Z) → H 1 (X, TB F1 ) → H 1 (X, so(n, Z)) → 0.
Exactness implies the following results:
(a) Every F 1 T-background (E, G) has an underlying so(n, Z)-principal bundle, and if this bundle is trivializable, then (E, G) is isomorphic (as F 1 T-backgrounds) to an F 2 Tbackground.
(b) Since a principal so(n, Z)-bundle is nothing but a collection of 1 2 n(n − 1) principal Zbundles, every such bundle is trivializable if H 1 (X, Z) = 0, for instance if X is connected and simply connected. Hence, every F 1 T-background over a connected and simply-connected smooth manifold X is in fact F 2 ; in particular, it is T-dualizable (Theorem 3.3.3). This question has been considered in [BHM], also see Section 4.4.1.
(c) The map H 1 (X, TB F2 ) → H 1 (X, TB F1 ) is indeed not injective, corresponding to the fact that the inclusion T-BG F2 ֒→ T-BG F1 is not full. In fact, two F 2 T-backgrounds are isomorphic as F 1 T-backgrounds if and only if they are related by the so(n, Z)-action (the "only if" holds only for connected X).
T-backgrounds with trivial torus bundle
In this section we investigate F 1 T-backgrounds with trivial torus bundle. For this purpose we consider a sequence
so(n, Z) dis × BZ n × BU(1) I / / TB F1 T / / T n dis (2.4.1)
of semi-strict homomorphisms. The homomorphism T sends an object (a, B) of TB F1 to a ∈ T n reduced mod Z n , and a morphism (τ, m, a, B) to the identity morphism of a. The homomorphism I sends an object (B, * , * ) to the object (0, B) of TB F1 and an endomor-
phism (B, m, t) of (B, * , * ) to the endomorphism (τ t,m , 0, 0, B) of (0, B) in TB F1 , where τ t,m (c) = t + mc.
Lemma 2.4.1. The following sequence is exact:
H 1 (X, so(n, Z)) × H 2 (X, Z n ) × H 3 (X, Z) I * / / H 1 (X, TB F1 ) T * / / H 2 (X, Z n ) / / 0.
Here we have used the usual identifications between non-abelian cohomology and ordinary cohomology, see Remark A.3.3.
Proof. The homomorphism T •I is trivial, and T * is obviously surjective. Suppose (B, a, m, τ ) is a TB F1 -cocycle whose class vanishes under T . Thus, there exists b i :
U i → T n such that a ij = b j −b i . We can assume that b i lift to smooth maps p i : U i → R n ; then we obtain z ij ∈ Z n such that a ij = p j − p i + z ij . These establish an equivalence between (B ij , a ij , m ijk , τ ijk ) and (B ij , 0, 0, τ ′ ijk ), where τ ′ ijk : U i ∩ U j ∩ U k → C ∞ (T n , U(1))
is a 3-cocycle. For 1 ≤ p ≤ n we consider the group homomorphism w p : C ∞ (T n , U(1)) → Z that extracts the winding number of a map around the p-th torus component. We also consider the evaluation at 0, which is a Lie group homomorphism ev 0 : C ∞ (T n , U(1)) → U(1). The composition of τ ′ ijk with w p is locally constant and thus a 3-cocycle m ijk ∈ Z n . The composition with ev 0 is a 3-cocycle t ijk :
U i ∩ U j ∩ U k → U(1). We claim that τ ′ ijk is equivalent to the 3- cocycle τ t ijk ,m ijk , so that (B ij , 0, 0, τ ′ ijk ) is in the image of I. Indeed, by definition of a winding number, (x, a) → τ ′ ijk (x)(a) − m ijk a lifts to a smooth mapτ ijk : U i ∩ U j ∩ U k → C ∞ (T n , R).
The liftτ ijk satisfies the cocycle condition only up to a constant ǫ ijkl ∈ Z; hence, the smooth maps β ijk :
U i ∩ U j ∩ U k → C ∞ (T n , R) defined by β ijk (x)(a) :=τ ijk (x, a) −τ ijk (x, 0) do form a cocycle. SinceȞ 2 (X, R) = 0, there exists e ij : U i ∩ U j → R with coboundary β ijk .
Pushing to U(1)-valued maps, e ij establishes an equivalence betweenτ ijk and t ijk ; hence between τ ′ ijk and τ t ijk ,m ijk .
The sequence Eq. (2.4.1) restricts over the F 2 T-backgrounds, ending up in a diagram of semi-strict Lie 2-groups and semi-strict homomorphisms:
BZ n × BU(1) / / TB F2 / / T n dis so(n, Z) dis × BZ n × BU(1) I / / TB F1 T / / T n dis . (2.4.2)
Concerning the geometric counterparts of the homomorphisms I and T , it is clear that T represents the 2-functor
T-BG F1 (X) → Bun T n (X) : (E, G) → E
that takes an F 1 T-background to its underlying torus bundle. Concerning the strict homomorphism I, we describe the 2-functor
Bun so(n,Z) (X) × Grb Z n (X) × Grb U(1) (X) → T-BG (2.4.3)
represented by I. We start by treating the first factor and assume that we have an so(n, Z)bundle Z over X. We construct the following bundle gerbe R so(n,Z) (Z) over X × T n . Its surjective submersion is Z × T n → X × T n . Its 2-fold fibre product Z [2] × T n is equipped with a smooth (and hence locally constant) map B : Z [2] → so(n, Z), since Z is a principal bundle, and it is equipped with the projection pr T n to T n . We consider the principal U(1)bundle P := pr * T n P B over Z [2] × T n , and over the triple fibre product the bundle gerbe product induced by the equality pr * 12 B + pr * 23 B = pr * 12 B over Z [3] . For the second factor, let H be a Z n -bundle gerbe over X. We assume that H is defined over a surjective submersion π : Y → X, with a principal Z n -bundle P over Y [2] and a bundle gerbe product µ over Y [3] . We define the following U(1)-bundle gerbe R Z (H) over X × T n . Its surjective submersion isỸ := Y × T n → X × T n . We consider the map τ : (Y [2] × T n ) × Z n → U(1) : (y 1 , y 2 , a, m) → am, which is fibrewise a group homomorphism. The principal U(1)-bundle of R Z (H) is the parameter-dependent bundle extension τ * (P ). Similarly, we extend the bundle gerbe product τ * (µ).
For the third factor, we simply pull back a U(1)-bundle gerbe G over X to X × T n . Putting the three pieces together, the 2-functor Eq. (2.4.3) is defined by
(Z, H, G) → (X × T n , R so(n,Z) (Z) ⊗ R Z (H) ⊗ pr * X G).
In the following two lemmas we compute the Dixmier-Douady classes of the bundle gerbes R so(n,Z) (Z) and R Z (H), in order to see of which type the resulting T-backgrounds are.
Lemma 2.4.2. Let Z be a principal so(n, Z)-bundle over X. For 1 ≤ p, q ≤ n we have a principal Z-bundle Z pq := (pr pq ) * (Z) with a corresponding class [Z pq ] ∈ H 1 (X, Z). We have DD(R so(n,Z) (Z)) = 1≤q<p≤n pr * p γ ∪ pr * q γ ∪ pr * X [Z pq ] ∈ H 3 (X × T n , Z),
where pr p : X × T n → S 1 is the projection to the p-th component of T n , and γ ∈ H 1 (S 1 , Z) is a generator.
Proof. We observe that pr * p γ ∪ pr * q γ is the first Chern class of pr * pq P, and that the bundle gerbe
R so(n,Z) (Z) is of the form R so(n,Z) (Z) = 1≤q≤p≤n pr * pq R so(2,Z) (Z pq ),
where pr pq : X × T n → X × T 2 projects to the two indexed components. We choose an open cover {U i } i∈I of X × T n such that Z pq admits sections, leading to transition matrices z ij ∈ Z. Then, the bundle gerbe R so(2,Z) (Z pq ) becomes isomorphic to one with principal U(1)-bundle P ij := pr * T 2 P z ij over U i ∩ U j , and with the bundle gerbe product µ ijk induced from the cocycle condition z ij +z jk = z ik . We can additionally assume that there exist sections s i : U i → pr * T 2 P, leading to transition functions g ij : U i ∩ U j → U(1). Then, P ij has the section s z ij i , and the calculation
µ ijk (s z ij i ⊗ s z jk j ) = µ ijk (s z ij i ⊗ s z jk i ) · g z jk ij = s z ik i · g z jk ij
show that R so(2,Z) (Z pq ) is classified by the 3-cocycle η ijk := g z jk ij . We compute cup products in Z-valuedČech cohomology, where the cup product of a k-cocycle α i 0 ,...,i k with an l-cocycle β i 0 ,...,i l is given by
α i 0 ,...,i k · β i k ,...,i k+l , see [Bry93, Section 1.3]. We choose liftsg ij : U i ∩ U j → R of g ij .
Then, q ijk := (δg) ijk is a Z-valued 3-cocycle that corresponds to g ij under the connecting homomorphism of the exponential sequence. Likewise,η ijk :=g ij z jk is a lift of η ijk , and p ijkl = (δη) ijkl corresponds to η ijk . We calculate
p ijkl =η ikl +η ijk −η jkl −η ijl =g ik z kl +g ij z jk −g jk z kl −g ij z jl = (g ik −g jk −g ij )z kl = q ijk z kl .(X, Z n ). Let [H] p ∈ H 2 (X, Z) denote its p-th component. Then, DD(R Z (H)) = n p=1 [H] p ∪ pr * p γ ∈ H 3 (X × T n , Z).
Proof. We can assume that we have an open cover {U i } i∈I on which the Z n -bundle gerbe H is given by a constant 3-cocycle m ijk ∈ Z n . Then, R Z (H) has the surjective submersion
Y := i∈I U i × T n → X × T n , and the cocycle U i ∩ U j ∩ U k × T n : (x, a) → m ijk a.
In order represent its Dixmier-Douady class by a Z-valued 4-cocycle, we consider the surjective submersion
Y ′ := i∈I U i × R n → X × T n so that the cocycle m ijk a lifts to an R-valued map U i ∩ U j ∩ U k × R n × T n R n × T n R n → R defined by (x, a 1 , a 2 , a 3 ) → m ijk a 1 . Its coboundary is m ijk a 1 + m ikl a 1 − m jkl a 2 − m ijl a 1 = m jkl (a 1 − a 2 ), (2.4.4)
which is a Z-valued 4-cocycle representing the Dixmier-Douady class of G m . In order to compute the cup product (w p ) * (H) ∪ pr * p γ we have to represent both classes by Z-valuedČech cocycles. By construction, the class of (w p ) * (H) is represented by the Z-valuedČech 3-cocycle m p,ijk , the p-th component of m ijk . To represent the class pr * p γ by a Z-valued 2-cocycle we consider again the surjective submersion Y ′ . Then, the smooth map pr p : X × T n → U(1), whose homotopy class represents pr * p γ, lifts to the real valued projection pr p : Y ′ → R. Then, the Z-valued 2-cocycle we are looking for is
pr p,ij : U i ∩ U j × R n × T n R n → Z : (x, a 1 , a 2 ) → a p,1 − a p,2 . Thus, our cup product (w p ) * (H) ∪ pr * p γ is represented by the Z-valuedČech 4-cocycle U i ∩ U j ∩ U k ∩ U l × R n → Z : (x, a) → m p,ijk · (a p,1 − a p,2 ).
Summation over p shows the coincidence with Eq. (2.4.4).
Summarizing above constructions and results, we have discussed a 2-functor
Bun so(n,Z) (X) × Grb Z n (X) × Grb U(1) (X) → T-BG(X)
that constructs an F 1 T-background with trivial torus bundle, from a principal so(n, Z)bundle, a Z n -bundle gerbe, and a U(1)-bundle gerbe over X. By Lemma 2.4.1, these Tbackgrounds are, up to isomorphism, all F 1 T-backgrounds with trivial torus bundle. We remark that the filtration F k = F k H 3 (E, Z) of Eq. (2.1.1) in case of the trivial torus bundle E = X × T n can be expressed in terms of the Künneth formula by
F 2 \ F 3 ∼ = H 2 (X, Z) × H 1 (T n , Z) , F 1 \ F 2 ∼ = H 1 (X, Z) × H 2 (T n , Z).
With Lemmas 2.4.2 and 2.4.3 we can read off to which steps in the filtration the given structure contributes: the U(1)-bundle gerbe G contributes to F 3 , the Z n -bundle gerbe H contributes to F 2 , and the so(n, Z)-bundle Z contributes to F 1 .
Example 2.4.4. We consider X = S 1 and n = 2, and the trivial T 2 -bundle E := T 3 = S 1 × T 2 over S 1 . We consider Z := R (under the identification so(2, Z) ∼ = Z), and the corresponding bundle gerbe G := R so(n,Z) (Z) over E. We have [Z 12 ] = γ ∈ H 1 (S 1 , Z) and obtain from Lemma 2.4.2:
DD(G) = pr * 1 γ ∪ pr * 2 γ ∪ pr * 3 γ ∈ H 3 (T 3 , Z), i.e.
G represents the canonical class of T 3 . This explicit example of a T-background has been described in [MR06a]. It is interesting because it is not an F 2 T-background, and hence is not T-dualizable in the classical sense, see [BRS06] and Theorem 3.3.3. We will see that it gives rise to a half-geometric T-duality transformation in the formalism introduced in this paper, see Example 4.2.6.
Higher geometry for topological T-duality
In this section we discuss a bicategory of T-duality correspondences, and introduce a strict Lie 2-group TD that represents a 2-stack of T-duality correspondences. We also discuss the relation to T-duality triples of [BS05,BRS06].
T-duality-correspondences as 2-stacks
Let X be a smooth manifold.
Definition 3.1.1.
(a) A correspondence over X consists of T-backgrounds (E, G) and (Ê, G) over X, and of a bundle gerbe isomorphism D :
pr * 1 G → pr * 2 G over E × XÊ . (b) A 1-morphism ((E, G), (Ê, G), D) → ((E ′ , G ′ ), (Ê ′ , G ′ ), D ′ ) consists of 1-morphisms (f, B) : (E, G) → (E ′ , G ′ ) and (f , B) : (Ê, G) → (Ê ′ , G ′ )
between the T-backgrounds and of a bundle gerbe 2-morphism
pr * 1 G pr * 1 B D / / pr * 2 G ζ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ s { ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ pr * 2 B pr * 1 f * G ′ (f,f ) * D ′ / / pr * 2f * G ′ over E × XÊ .
Composition is the composition of T-background 1-morphisms together with the stacking 2-morphisms.
(c) A 2-morphism consists of 2-morphisms β 1 : (f, B) ⇒ (f, B ′ ) and β 2 : (f , B) ⇒ (f , B ′ ) of T-backgrounds such that pr * 1 G pr * 1 B ′ & & pr * 1 B D / / pr * 2 G ζ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ s { ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ pr * 2 B pr * 1 f * G ′ (f,f ) * D ′ / / pr * 2f * G ′ pr * 1 β 1 k s = pr * 1 G pr * 1 B ′ D / / pr * 2 G pr * 2 B x x ζ ′ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ s { ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ pr * 2 B ′ pr * 1 f * G ′ (f,f ) * D ′ / / pr * 2f * G ′ pr * 2 β 2 k s over E × XÊ .
Horizontal and vertical composition are those of 2-morphisms between T-backgrounds.
Correspondences over X form a bigroupoid Corr(X), and the assignment X → Corr(X) is a presheaf of bigroupoids over smooth manifolds. Since principal bundles and bundle gerbes form (2-)stacks over smooth manifolds, we have the following. Next we want to define a sub-2-stack T-Corr ⊆ Corr of T-duality correspondences. We define the correspondence T X := ((X × T n , I), (X × T n , I), id I ⊗ pr * P n ) over any smooth manifold X, where id I ⊗ pr * P n denotes the action of the pullback of the n-fold Poincaré bundle P n over T n × T n (see Appendix B) on the identity 1-morphism id I between trivial bundle gerbes over X × T n × T n , see Eq. (2.3.3). The correspondence T X is the prototypical T-duality correspondence; general T-duality correspondences are obtained via gluing of T X in a certain way. The gluing has to be performed along particular automorphisms of T X , which we describe next. We define for a pair (a,â) of smooth maps a,â : X → R n a 1-morphism A a,â := ((f a , id I ), (fâ, id I ), ζ a,â ) :
T X → T X ,
where f a : X × T n → X × T n is defined in Eq. (2.3.2) and ζ a,â is a 2-isomorphism ζ a,â : id I ⊗ pr * P n ⇒ (f a , fâ) * (id I ⊗ pr * P n ) = id I ⊗ pr * (r a,â ) * P n , which we define in the following way. We consider the point-wise scalar product of a andâ as a smooth map aâ : X → U(1), and regard it as an automorphism of the trivial U(1)-bundle over X. We act with this automorphism on the identity 1-morphism id I , obtaining another 1-morphism that we denote again by aâ : I → I. Next we consider the bundle morphism R(a,â) : P n → r * a,â P n over T 2n from Appendix B. Putting the pieces together, we define ζ a,â := aâ ⊗R(a,â). The composition of two 1-morphisms of the form A a,â is in general not of this form. We denote by A the class of automorphisms of T X generated by the automorphisms A a,â , i.e. A consists of all possible finite compositions of automorphisms of the form A a,â and their inverses. The class A is exactly the class of automorphisms along which we allow to glue. Unfortunately, at current time, no better description of the class A is known to us.
We let T (X) be the sub-bicategory of Corr(X) with the single object T X , all 1-morphisms of the class A, and all 2-morphisms. The assignment X → T (X) is a sub-presheaf of Corr(X), and we let T-Corr := T ⊆ Corr be its closure under descent, which exists due to Proposition 3.1.2. (b) If C is a T-duality correspondence, then it obviously satisfies the so-called Poincaré condition P(x) for every point x ∈ X, namely that there exists a 1-isomorphism C| {x} ∼ = T {x} [BRS06]. Spelling out what this means, there exist T n -equivariant maps t : T n → E| x andt : T n →Ê| x together with trivializations T : t * G → I and T :t * G → I such that the bundle gerbe isomorphism
I pr * 1 T −1 / / pr * 1 t * G (t,t) * D / / pr * 2t * G pr * 2 T / / I
between trivial gerbes over T n × T n is given (up to a 2-isomorphism over T n × T n ) by acting with the Poincaré bundle P n on the identity id I . Conversely, it follows from Proposition 3.3.2 below that a correspondence that satisfies the Poincaré condition P(x) for all x ∈ X is a T-duality correspondence.
(c) The 2-functor C → C ∨ restricts to T-duality correspondences, and thus induces a 2-functor T-Corr(X) op → T-Corr(X). In particular, T-duality correspondence is a symmetric relation on h 0 (T-BG(X)). In general, it is neither reflexive nor transitive.
(d) By construction, the assignment X → T-Corr(X) is a 2-stack; this is an important advantage of our definition over the T-duality triples of [BRS06].
A 2-group that represents T-duality correspondences
For x, y ∈ R 2n we define the notation
[x, y] := n i=1 x n+i y i ,
which is a bilinear form on R 2n with matrix
F := 0 0 E n 0 .
Often we write x = a ⊕â for a,â ∈ R n , so that [x 1 , x 2 ] = [a 1 ⊕â 1 , a 2 ⊕â 2 ] =â 1 a 2 .
We consider the categorical torus associated to the bilinear form [−, −] in the sense of Ganter [Gan18]. This is a strict Lie 2-group TD n , defined as a crossed module (G, H, t, α) with G := R 2n and H := Z 2n × U(1). The group homomorphism t : H → G is projection and inclusion, t(m, t) := m. The action α : G × H → H is given by
α(x, (z, t)) := (z, t − [x, z]).
We have π 0 (TD n ) = T 2n and π 1 (TD n ) = U(1) (3.2.1)
Since the projection Ob(TD n ) → π 0 (TD n ) is a surjective submersion, TD n is smoothly separable in the sense of [NW13b]. Note that the induced action of R 2n on U := ker(t) = U(1) is trivial. This shows that we have a central extension
BU(1) → TD n → T 2n dis
in the sense of [SP11]. We will often write just TD instead of TD n .
We recall that central extensions of a Lie group K by the Lie 2-group BU(1) are classified by H 4 (BK, Z) [SP11]. In our case, K = T 2n , we consider the Poincaré class
poi n := n i=1 pr * i c 1 ∪ pr * i+n c 1 ∈ H 4 (BT 2n , Z),
where c 1 ∈ H 2 (BU(1), Z) is the universal first Chern class, and pr i : T 2n → U(1) is the projection to the i-th factor.
Proposition 3.2.1. The central extension TD n is classified by poi n ∈ H 4 (BT 2n , Z).
Proof. Since H 4 (BT 2n , Z) has no torsion, it suffices to compare the images in real cohomology. Chern-Weil theory provides an algebra homomorphism Sym k ((R 2n ) * ) ∼ = H 2k (BT 2n , R) for all k > 0, and Ganter proves [Gan18, Theorem 4.1] that the class of TD n corresponds to
F :=F +F tr = 0 E n E n 0 ∈ Sym 2 ((R 2n ) * ).
It is well-known that c 1 ∈ H 2 (BU(1), R) corresponds to id ∈ Sym 1 (R * ) = R * , so that
pr * i c 1 ∈ Sym 1 ((R 2n ) * ) = (R 2n ) * is pr i : R 2n → R. It remains to notice that n i=1 pr i · pr n+i = F ,
which is a straightforward calculation. ] that the multiplicative bundle gerbe associated to TD n is the trivial bundle gerbe over T 2n , with the multiplicative structure given by a principal U(1)-bundle L over T 2n × T 2n which descends from the surjective submersion
Z := R 2n × R 2n → T 2n × T 2n and theČech 2-cocycle α : Z × T 2n ×T 2n Z → U(1) defined by α((x 1 + z 1 , x 2 + z 2 ), (x 1 , x 2 )) = [x 1 , z 2 ].
In order to identify this bundle, we consider Z ′ := R 2n → T 2n and the commutative diagram
Z pr 23 / / Z ′ T 2n × T 2n pr 23 / / T 2n ,
where pr 23 ((a 1 ⊕â 1 ), (a 2 ⊕â 2 )) := (â 1 ⊕ a 2 ). We consider for Z ′ → T 2n theČech 2-cocycle
α ′ : Z ′ × T 2n Z ′ → U(1) defined by α ′ (x + z, x) := [z, x].
Comparing with Remark B.3, this is the cocycle of the n-fold Poincaré bundle P n . It is straightforward to check that α = (pr 23 × pr 23 ) * α ′ . In other words, L ∼ = pr * 23 P n . Thus, the Lie 2-group TD n corresponds to the trivial gerbe over T 2n with multiplicative structure given by pr * 23 P n over T 2n × T 2n .
Remark 3.2.3. A TD n -cocycle with respect to an open cover {U i } i∈I is a tuple (a,â, m,m, t) consisting of numbers m ijk ,m ijk ∈ Z n and smooth maps
a ij ,â ij : U i ∩ U j → R n , t ijk : U i ∩ U j ∩ U k → U(1)
satisfying the following cocycle conditions:
a ik = m ijk + a jk + a iĵ a ik =m ijk +â jk +â ij m ikl + m ijk = m ijl + m jkl m ikl +m ijk =m ijl +m jkl t ikl + t ijk − m ijkâkl = t ijl + t jkl
Two TD-cocycles (a,â, m,m, t) and (a ′ ,â ′ , m ′ ,m ′ , t ′ ) are equivalent if there exist numbers z ij ,ẑ ij ∈ Z n , smooth maps p i ,p i : U i → R n and e ij :
U i ∩ U j → U(1) such that a ′ ij + p i = z ij + p j + a iĵ a ′ ij +p i =ẑ ij +p j +â ij m ′ ijk + z ij + z jk = z ik + m ijk m ′ ijk +ẑ ij +ẑ jk =ẑ ik +m ijk t ′ ijk + e ij −â ′ jk z ij + e jk = e ik + t ijk −p k m ijk
We consider the presheaf BTD of smooth BTD-valued functions. Over a smooth manifold X the bicategory BTD(X) is the following:
• It has one object.
• The 1-morphisms are pairs (a,â) of smooth maps a,â : X → R n ; the composition is pointwise addition.
• A 2-morphism from (a,â) to (a ′ ,â ′ ) is a triple (t, m,m) consisting of m,m ∈ Z n and a smooth map t : X → U(1), such that a ′ = a + m andâ ′ =â +m. Vertical composition is (pointwise) addition, and horizontal composition is * (a 1 ,â 1 ) (a 1 +m 1 ,â 1 +m 1 )
? ?
(t 1 , m 1 ,m 1 ) * (a 2 ,â 2 ) (a 2 +m 2 ,â 2 +m 2 )
? ?
(t 2 , m 2 ,m 2 ) * = * (a 1 +a 2 ,â 1 +â 2 ) % % (a 1 +a 2 +m 1 +m 2 ,â 1 +â 2 +m 1 +m 2 ) 9 9
(t 2 + t 1 − m 1â2 , m 1 + m 2 ,m 1 +m 2 ) * .
We define a 2-functor
C : BTD(X) → T (X),
where T is the presheaf defined in Section 3.1. It sends the single object to the correspondence T X . A 1-morphism (a,â) is sent to the 1-morphism A a,â . It is straightforward to check this identity using Appendix B. In order to complete the definition of the 2-functor C, we need to provide an associator and to check the axioms. The vertical composition is respected since the following equations hold:
α t 2 ,m 2 ,m 2 + α t 1 ,m 1 ,m 1 = α t 2 +t 1 ,m 2 +m 1 ,m 2 +m 1 β t 2 ,m 2 ,m 2 + β t 1 ,m 1 ,m 1 = β t 2 +t 1 ,m 2 +m 1 ,m 2 +m 1
The associator c (a 1 ,â 1 ),(a 2 ,â 2 ) : ((f a 2 , id I ), (fâ 2 , id I ), ζ a 2 ,â 2 ) • ((f a 1 , id I ), (fâ 1 , id I ), ζ a 1 ,â 1 )
⇒ ((f a 1 +a 2 , id I ), (fâ 1 +â 2 , id I ), ζ a 1 +a 2 ,â 1 +â 2 )
is defined as follows. The horizontal composition on the left is ((f a 1 +a 2 , id I ), (fâ 1 +â 2 , id I ), (a 2â2 + a 1â1 +â 1 a 2 ) ⊗ pr * R (a 2 + a 1 ,â 2 +â 1 )).
We set c (a 1 ,â 1 ),(a 2 ,â 2 ) := (id id I ⊗â 2 a 1 , id id I ). The condition for 2-morphisms is (â 2 a 1 +a 2â2 +a 1â1 +â 1 a 2 )⊗pr * R (a 2 +a 1 ,â 2 +â 1 ) = (a 1 +a 2 )(â 1 +â 2 )⊗pr * R (a 2 +a 1 ,â 2 +â 1 ), and obviously satisfied. Proof. Both categories have a single object. On the level of 1-morphisms, it is obviously essentially surjective. Now consider two 1-morphisms (a,â) and (a ′ ,â ′ ). The set of 2morphisms between (a,â) and (a ′ ,â ′ ) in BTD(X) are triples (t, m,m) with t : X → U(1) and m,m ∈ Z n such that a ′ = a + m andâ ′ =â +m. The set of 2-morphisms between C(a,â) = ((f a , id), (fâ, id), ζ a,â ) and C(a ′ ,â ′ ) = ((f a+m , id), (fâ +m , id), ζ a+m,â+m ) consists of pairs (α, β) of smooth maps α, β : X × T n → U(1) such that Eq. (3.2.2) is satisfied, α(x, c) = mb(x) − cm + md + β(x, d). Since m andm are uniquely determined, an equality α t,m,m = α t ′ ,m,m implies already t = t ′ . Thus, our map is injective. Conversely, given (α, β), we define t(x) := −mb(x)−β(x, 0). Then we get from Eq. (3.2.3)
α t,m,m (x, c) = mb(x) + β(x, 0) −mc = α(x, c) and β t,m,m (x, c) = mb(x) + β(x, 0) − mc − mb(x) = α(x, 0) − mb(x) − mc = β(x, c).
This shows that our map is also surjective.
Since T-Corr was the 2-stackification of T , we obtain from Lemma 3.2.4:
Proposition 3.2.5. The 2-functor C induces an isomorphism BTD + ∼ = T-Corr.
T-duality triples
In [BRS06] a category of "T-duality triples" was defined. In this section we relate that definition to our notion of T-duality correspondences and the Lie 2-group TD.
The paper [BRS06] is written with respect to an arbitrary 1-categorical model for U(1)gerbes. In order to compare this with our definitions, we choose the 1-truncation h 1 (Grb (X)) as a model. Then, we have the following definitions: [D]
/ / pr * 2 G pr * 2 [ B]
pr * 1 f * G ′ (c) A triple is called T-duality triple, if the two T-backgrounds are F 2 , and if its restriction to any point x ∈ X is isomorphic to T {x} . The category Trip(X) of T-duality triples is the full subcategory on these.
Lemma 3.3.1. The geometric realization |BTD| is a classifying space for T-duality triples.
Proof. By [BRS06, Thm. 2.14], the classifying space for T-duality triples is the homotopy fibre of the map BT 2n → K(Z, 4) whose homotopy class is poi n ∈ H 4 (BT 2n , Z). On the other hand, a central extension BU(1) → Γ → K dis induces a fibre sequence of classifying spaces, so that |BΓ| is the homotopy fibre of the map BK → |BBU(1)| ≃ K (Z, 4). The classification of central extensions in [SP11] exhibits its class in H 4 (BK, Z) as the homotopy class of this map. Now, Proposition 3.2.1 shows the claim.
There is an obvious functor h 1 (T-Corr(X)) → Trip(X) (3.3.1) obtained by reducing every 1-morphism to its 2-isomorphism class. On the level of objects, this is well-defined because of Remark 3.1.4 (a). On the level of morphisms, every 1-morphism in Corr(X) yields a morphism of triples, and it is straightforward to see that 2-isomorphism 1-morphisms result in the same morphism. In general, the functor Eq.
.1) induces a bijection
h 0 (T-Corr(X)) ∼ = h 0 (Trip(X)).
In particular, the notion of T-dualizability given in Definition 3.1.3 coincides with the one of [BRS06].
Thus, all results of [BRS06] can be transferred to our setting. For instance, we have the following important result [BRS06, Theorem 2.23]:
Theorem 3.3.3. A T-background is T-dualizable if and only if it is F 2 .
A cohomological formulation of this theorem is presented below as Theorem 3.4.5. Even later we extend this result to half-geometric T-duality correspondences and F 1 T-backgrounds, see Theorem 4.2.2. Theorem 3.3.3 can then be deduced as a special case. Proposition 3.4.1. The crossed intertwiner fℓip represents the 2-functor C → C ∨ , in the sense that the diagram
Implementations of legs and the flip
BTD(X) C / / B(fℓip) T-Corr(X) () ∨ BTD(X) C / / T-Corr(X)
is strictly commutative.
Remark 3.4.2. We remark that fℓip is not strictly involutive: fℓip 2 = fℓip • fℓip is the crossed intertwiner (id, id,η) withη(a 1 ⊕â 1 , a 2 ⊕â) = a 1â2 +â 1 a 2 . As an element of the automorphism 2-group of TD discussed in an upcoming paper, it is only coherently isomorphic to the identity.
We define homomorphisms 2) we collect the a ij : U i ∩ U j → R n and m ijk ∈ Z n as they are, and put
τ ijk (x)(a) := t ijk (x) + (a − a ik (x))m ijk − a ij (x)â jk (x). (3.4.3)
We have the following result:
Theorem 3.4.5. The left leg projection ℓeℓe * : H 1 (X, TD) → H 1 (X, TB F2 ) is a bijection.
Theorem 3.4.5 can be proved by reducing the proof of our main result, Theorem 4.2.2, to the case that all occurring so(n, Z)-matrices are zero. Since we never use Theorem 3.4.5 directly, we will not write this out.
Remark 3.4.6. Theorem 3.4.5 implies that the classifying spaces |BTD| and |BTB F2 | are equivalent. Indeed, we have [X, |BTB F2 |] ∼ = H 1 (X, TB F2 ) ∼ = H 1 (X, TD) ∼ = [X, |BTD|] for all X; using Theorem 3.4.5 and the bijection of Remark A.3.2. Thus, the Yoneda lemma implies the equivalence. An alternative proof of Theorem 3.4.5 would be to prove the equivalence between the classifying spaces |BTD| and |BTB F2 | directly; this is the strategy pursued in [BRS06]. We remark that the Lie 2-groups TD and TB F2 are not isomorphic, since they have different homotopy types, see Eqs. (2.2.1) and (3.2.1).
Remark 3.4.7. The results of this section are related to the results of [BRS06]) in the following way. The bare existence of the map ℓeℓe * shows the "only if"-part of Theorem 3.3.3 (corresponding to [BRS06, Theorem 2.23]). Surjectivity in Theorem 3.4.5 provides the "if"-part. Another result [BRS06, Theorem 2.24 (2)] is that two T-duality triples with isomorphic left legs are related under the action of so(n, Z) on triples defined in [BRS06]; in particular, they do not have to be isomorphic. This is not a contradiction to the injectivity of Theorem 3.4.5 because in [BRS06] the left leg projection is the composite
H 1 (X, TD) ℓeℓe * / / H 1 (X, TB F2 ) / / H 1 (X, TB F1 ),
of which the second map is not injective (Remark 2.3.3).
Remark 3.4.8. We obtain a well-defined, canonical map
H 1 (X, TB F2 ) (ℓeℓe * ) −1 / / H 1 (X, TD) riℓe * / / H 1 (X, TB F2 ).
The existence of this map might be confusing, as it looks like a "T-duality transformation" for arbitrary F 2 T-backgrounds, whereas the results of [BRS06] imply that such a transformation exist only for n = 1 (since then so(n, Z) = {0}). The point is, again, the non-injectivity of the map H 1 (X, TB F2 ) → H 1 (X, TB F1 ), which prevents to define a T-duality transformation on H 1 (X, TB F1 ).
Half-geometric T-duality
In this section, we introduce and study the central objects of this article: half-geometric T-duality correspondences.
Half-geometric T-duality correspondences
We define an action of so(n, Z) on the 2-group TD n by crossed intertwiners in the sense of Definition A.4.1. A matrix B ∈ so(n, Z) acts by a crossed intertwiner We show in a separate paper that the automorphism 2-group of TD is a (non-central, nonsplitting) extension of the split-orthogonal group O(n, n, Z) ⊆ GL(2n, Z). This extension splits over the subgroup so(n, Z). Above action is induced from the action by automorphism via the inclusion B → e B .
We define the half-geometric T-duality 2-group as the semi-direct product where so(n, Z) acts on T 2n by multiplication with e B ⊆ GL(2n, Z).
The Lie 2-group TD 1 2 -geo represents a 2-stack over smooth manifolds that we call the 2-stack of half-geometric T-duality correspondences. The reader is free to pick any model for this 2-stack. The simplest possibility is to take (TD 1 2 -geo ) + , which leads to a cocycle description carried out below in Remark 4.1.2. Other possibilities are to take non-abelian a tuple (B, a,â, m,m, t) consisting of matrices B ij ∈ so(n, Z), numbers m ijk ,m ijk ∈ Z n , and smooth maps a ij ,â ij : U i ∩ U j → R n and t ijk : Also note that for B ij = 0 we obtain precisely the cocycles for TD. Two TD 1 2 -geo -cocycles (B, a,â, m,m, t) and (B ′ , a ′ ,â ′ , m ′ ,m ′ , t ′ ) are equivalent if there exist C i ∈ so(n, Z), numbers z ij ,ẑ ij ∈ Z n , and smooth maps p i ,p i : U i → R n , e ij : U i ∩ U j → U(1) such that:
U i ∩ U j ∩ U k → U(1) satisfying B ik = B jk + B ij (4.1.2) a ik = m ijk + a jk + a ij (4.1.3) a ik =m ijk +â jk +â ij + B jk a ij (4.1.4) t ikl + t ijk =â kl m ijk + m ijk |B kl |a ik low + a jk |B kl |a ij low + t ijl + t jkl .C j + B ij = B ′ ij + C i (4.1.7) z ij + p j + a ij = a ′ ij + p i (4.1.8) z ij +p j + C j a ij +â ij =â ′ ij + B ′ ij p i +p i (4.1.9) t ′ ijk + a ′ ij |B ′ jk |p i low − z ij |B ′ jk |p j + a ij low + e ij −â ′ jk z ij − p j |B ′ jk |a ij low + e jk = e ik − m ijk |C k |a ik low + t ijk −p k m ijk − a jk |C k |a ij low
(4.1.10)
Note that Eq. (4.1.8) implies
m ′ ijk + z ij + z jk = z ik + m ijk (4.1.11)
Also note that Eq. (4.1.9) implieŝ
m ′ ijk +ẑ ij +ẑ jk =m ijk +ẑ ik + B ′ jk z ij + C k m ijk .
(4.1.12)
The left leg of a half-geometric correspondence
We recall from Remark 3.4.4 that the left leg of a T-duality correspondence is represented by a crossed intertwiner ℓeℓe = (φ, f, η) : TD n → TB F2 , where φ(a ⊕â) = a and f (m ⊕m, t) = (τ t,m , m) and η(a ⊕â, a ′ ⊕â ′ )(c) =âa ′ . We have actions of so(n, Z) on both 2-groups by crossed intertwiners (id R n , f B , η B ) and (φ e B , f e B , η e B ), respectively (defined in Sections 2.3 and 4.1). We have the following key lemma.
Lemma 4.2.1. ℓeℓe is strictly so(n, Z)-equivariant in the sense of Definition A.5.1.
Proof. We check for each B ∈ so(n, Z) the three conditions for equivariance listed in Remark A.5.2:
(a) Equivariance of φ: (id R n • φ)(a ⊕â) = a = φ(a ⊕ (Ba +â)) = (φ • φ e B )(a ⊕â). (b) Equivariance of f : (f B • f )(m ⊕m, t) = f B (τ t,m , m) = (τ t,m − m|B|, m) = (τ t,Bn+m , m) = f (m ⊕ (Bm +m), t) = (f • f e B )(m ⊕m, t).
(c) Compatibility of η: a 1 ⊕â 1 ), φ e B (a 2 ⊕â 2 )) + f (η e B (a 1 ⊕â 1 , a 2 ⊕â 2 )).
η B (φ(a 1 ⊕â 1 ), φ(a 2 ⊕â 2 )) + f B (η(a 1 ⊕â 1 , a 2 ⊕â 2 )) = η B (a 1 , a 2 ) + f B (0,â 1 a 2 ) = (0 ⊕ 0, a 2 |B|a 1 low ) + (0 ⊕ 0,â 1 a 2 ) = (0 ⊕ 0, (− a 1 |B|a 2 +â 1 a 2 ) + f (0 ⊕ 0, a 1 |B|a 2 low ) = η(a 1 ⊕ (Ba 1 +â 1 ), a 2 ⊕ (Ba 2 +â 2 )) + f (0 ⊕ 0, a 1 |B|a 2 low ) = η(φ e B (
Due to the equivariance ℓeℓe induces a semi-strict homomorphism ℓeℓe so(n,Z) : TD 1 2 -geo → TB F1 , see Appendix A.5. As described there, it induces in cohomology following map (ℓeℓe so(n,Z) ) * .
Consider a TD 1 2 -geo -cocycle (B, a,â, m,m, t) with respect to an open cover {U i } i∈I as in Remark 4.1.2. Its image under (ℓeℓe so(n,Z) ) * is the TB F1 -cocycle (B, a, m, τ ) consisting of the matrices B ij , the smooth maps a ij , the numbers m ijk , and the smooth maps
τ ijk : U i ∩ U j ∩ U k → C ∞ (T n , U(1)) defined by τ ijk (x)(a) := t ijk +m ijk (a − a ik (x)) − a ij (x)â jk (x). (4.2.1)
The following theorem is the main result of this article, and it will be proved in Section 4.3. .3.1) and (4.1.1).
Remark 4.2.4. In a local version of T-folds, surjectivity was proved by Hull on the level of differential forms, i.e. with curvature 3-forms H ∈ Ω 3 (E) instead of bundle gerbes [Hul07]. In that context, Hull proves that the existence of T-duals of a T-background (E, G) requires that ι X ι Y H = 0, where X, Y are vertical vector fields on E; this is a local, infinitesimal version of the F 2 condition. Hull explains that admitting "non-geometric T-duals" allows to replace it by the weaker condition ι X ι Y ι Z H = 0; this is a local, infinitesimal version of the F 1 condition. Thus, Theorem 4.2.2 shows that our half-geometric T-duality correspondences realize Hull's non-geometric T-duals.
Remark 4.2.5. We consider the half-geometric T-duality correspondence C B whose TD 1 2 -geo -cocycle is trivial except for the matrices B ij , which then form a 2-cocycle B ij : U i ∩ U j → so(n, Z). This realizes the splitting homomorphism so(n, Z) dis → TD n ⋉ so(n, Z) = TD 1 2 -geo .
We observe that the left leg of C B is the F 1 T-background represented by a cocycle (B, 0, 0, 0), whose only data are the given matrices B ij . It is the image of the cocycle B under the 2-functor I constructed in Section 2.4. In terms of its geometric version, if Z is a principal so(n, Z)-bundle over X classified by [B] ∈ H 1 (X, so(n, Z)), then the F 1 T-background (X × T n , R so(n,Z) (Z)) is the left leg of the half-geometric T-duality correspondence C B .
Example 4.2.6. We consider over X = S 1 the so(2, Z)-principal bundle Z whose components are Z 11 = Z 22 = S 1 × Z and Z 12 = R → S 1 , and Z 21 = R ∨ → S 1 . If B is a classifying cocycle, and C B the corresponding half-geometric T-duality correspondence, then C B is the "non-geometric T-dual" of the F 1 T-background (E, G) of Example 2.4.4, with E = T 3 and G representing the canonical class in H 3 (T 3 , Z).
Proof of the main result
In this section we prove Theorem 4.2.2. Our proof is an explicit calculation on the level of cocycles, and is performed in three steps.
Step 1: Construction of a pre-image candidate. We start with a TB F1 -cocycle (B, a, m, τ ) with respect to an open cover U = {U i } i∈I , i.e. matrices B ij ∈ so(n, Z), numbers m ijk ∈ Z n and smooth maps a ij : U i ∩ U j → R n and τ ijk :
U i ∩ U j ∩ U k → C ∞ (T n , U(1))
satisfying Eqs. (2.3.5) to (2.3.7). Next we construct a TD 1 2 -geo -cocycle (with respect to a refinement of the open cover U ). We definem ijk ∈ Z n to be the n winding numbers of τ ijk (x) : T n → U(1), which is independent of x ∈ U i ∩ U j ∩ U k . The cocycle condition for τ Eq. (2.3.7) implies the necessary condition of Eq. (4.1.6), namelŷ
m ikl +m ijk + B kl m ijk =m ijl +m jkl , (4.3.1) since the i-th winding number of − m ijk |B kl | is the i-th component of B kl m ijk . We define smooth mapsm ijk : U i ∩ U j ∩ U k → R bym ijk (x) :=m ijk + B jk a ij (x)
. It is straightforward to check using Eq. (4.3.1) thatm ijk is aČech cocycle, i.e.
[m] ∈Ȟ 2 (M, R n ) = 0. Thus, after passing to a refinement of U we can assume that there exist smooth mapsâ ij :
U i ∩ U j → R n satisfyingâ ik =m ijk +â jk +â ij + B jk a ij ;
this is cocycle condition Eq. (4.1.4) for TD 1 2 -geo -cocycles. On the other hand, by definition of a winding number, there exist smooth maps
τ ijk : (U i ∩ U j ∩ U k ) × T n → R such that τ ijk (x)(a) =τ ijk (x, a) + am ijk in U(1). The cocycle condition for τ implies τ ijl (x, a) + am ijl +τ jkl (x, a) + am jkl =τ ijk (x, a − a kl (x)) − a kl (x)m ijk + am ijk +τ ikl (x, a) + am ikl − m ijk |B kl |a − a kl (x) − a ij (x)|B kl |a jk (x) low − a ik (x)|B kl |m ijk low + ǫ ijkl
for a uniquely defined ǫ ijkl ∈ Z. Subtracting a times Eq. (4.3.1) we get
τ ijl (x, a) +τ jkl (x, a) −τ ijk (x, a − a kl (x)) −τ ikl (x, a) + a kl (x)m ijk − m ijk |B kl |a kl (x) + a ij (x)|B kl |a jk (x) low + a ik (x)|B kl |m ijk low = ǫ ijkl
Since T n is connected, we see that the expression
δ ijkl (x) :=τ ijl (x, a) +τ jkl (x, a) −τ ijk (x, a − a kl (x)) −τ ikl (x, a) ∈ R (4.3.2)
is independent of a, and
ǫ ijkl = δ ijkl (x) + a kl (x)m ijk − m ijk |B kl |a kl (x) + a ij (x)|B kl |a jk (x) low + a ik (x)|B kl |m ijk low . (4.3.3)
Using the independence of a, one can now check that δ ijkl is aČech cocycle. Thus, [δ] ∈Ȟ 3 (M, R) = 0, and (after again passing to a refinement) there exist smooth maps ω ijk : U i ∩ U j ∩ U k → R with δω = δ. Substituting this in Eq. (4.3.3), and pushing to U(1)-valued maps, we get (δω) ijkl (x) + a kl (x)m ijk − m ijk |B kl |a kl (x) + a ij (x)|B kl |a jk (x) low + a ik (x)|B kl |m ijk low = 0.
We define
t ijk (x) := −ω ijk (x) + a ij (x)â jk (x) +m ijk a ik (x).
A tedious but straightforward calculation shows that t ijk satisfies the cocycle condition Eq. (4.1.5) for TD 1 2 -geo -cocycles. Thus, we have obtained a TD 1 2 -geo -cocycle (B, a,â, m,m, t).
Step 2: Check that the candidate is a pre-image. The left leg of our TD 1 2 -geo -cocycle (B, a,â, m,m, t) is given by B ij , a ij , m ijk , and
τ ′ ijk (x, a) := t ijk (x) + (a − a ik (x))m ijk − a ij (x)â jk (x) = −ω ijk (x) + am ijk .
We prove that the TB F1 n -cocycles (B ij , a ij , m ijk , τ ijk ) and (B ij , a ij , m ijk , τ ′ ijk ) are equivalent. For this it suffices to provide ǫ ij :
U i ∩ U j → C ∞ (T n , U(1)) such that τ ′ ijk + a jk ǫ ij + ǫ jk = ǫ ik + τ ijk .
since this is Eq. (2.3.11) for C i = 0, p i = 0 and z ij = 0. In order to construct ǫ ij , we consider the smooth maps β ijk :
U i ∩ U j ∩ U k → C ∞ (T n , R) defined by β ijk (x)(a) :=τ ijk (x, a) + ω ijk (x).
These satisfy the following condition:
β ikl + a kl β ijk − β ijl − β jkl Eq. (4.3.2) ↓ = −δ ijkl + (δω) ijkl = 0 (4.3.4)
It also satisfies (after pushing to C ∞ (T n , U(1))):
β ijk (x)(a) =τ ijk (x, a) + ω ijk (x) = τ ijk (x)(a) − am ijk + ω ijk (x) = τ ijk (x)(a) − τ ′ ijk (x)(a).
Let {ψ i } i∈I be a partition of unity subordinate to our open cover. We define
ǫ ij (x)(a) := h∈I ψ h (x)β hij (x)(a).
Then we obtain
ǫ ij (x)(a − a jk ) + ǫ jk (x)(a) − ǫ ik (x)(a) = h∈I ψ h (x) (β hij (x)(a − a jk ) + β hjk (x)(a) − β hik (x)(a)) Eq. (4.3.4) ↓ = h∈I ψ h (x)β ijk (x, a) = β ijk (x, a).
This shows the claimed equivalence.
Step 3: Injectivity of the left leg. We suppose that we have two TD 1 2 -geo -cocycles (B, a,â, m,m, t) and (B ′ , a ′ ,â ′ , m ′ ,m ′ , t ′ ) whose left legs (B, a, m, τ ) and (B ′ , a ′ , m ′ , τ ′ ) are equivalent. Thus, there exist matrices C i ∈ so(n, Z), numbers z ij ∈ Z n and smooth maps
p i : U i → R n and ǫ ij : U i ∩ U j → C ∞ (T n , U(1))
such that the cocycle conditions Eqs. (2.3.9) to (2.3.12) are satisfied. Expressing Eq. (2.3.11) in terms of t ijk and t ′ ijk using the definition of left legs, we obtain
t ′ ijk (x) − t ijk (x) −m ′ ijk a ′ ik (x) +m ijk (a ik (x) + p k (x)) − a ′ ij (x)â ′ jk (x) + a ij (x)â jk (x) + p i (x)|B ′ jk |a ′ ij (x) low − p j (x) + a ij (x)|B ′ jk |z ij low + z ij |B ′ jk |a ′ jk (x) − a ij (x)|B ′ jk |p j (x) low + a ik (x)|C k |m ijk low − m ijk |C k |p k (x) + a ij (x)|C k |a jk (x) low = ǫ ik (x, a) − ǫ ij (x, a − a ′ jk (x)) − ǫ jk (x, a) + (m ijk −m ′ ijk )a − m ijk |C k |a + z ij |B ′ jk |a . (4.3.5)
In order to show the equivalence between the TD 1 2 -geo -cocycles (B, a,â, m,m, t) and (B ′ , a ′ ,â ′ , m ′ ,m ′ , t ′ ), our goal is to find numbersẑ ij ∈ Z n , and smooth mapsp i : U i → R n and e ij : U i ∩ U j → U(1) such that the remaining required cocycle conditions Eqs. (4.1.9) and (4.1.10) are satisfied. We letẑ ij ∈ Z n be the n winding numbers of ǫ ij (x) : T n → U(1). Eq. (4.3.5) implies
0 =ẑ ik −ẑ ij −ẑ jk +m ijk −m ′ ijk + C k m ijk − B ′ jk z ij ; (4.3.6)
this is necessary for Eq. (4.1.12). We consider β ij :=ẑ ij + C j a ij +â ij −â ′ ij − B ′ ij p i , which is by Eq. (4.3.6) an R n -valuedČech 1-cocycle, and chosep i : U i → R n such that β ij =p i −p j ; this gives Eq. (4.1.9). By definition of a winding number, there exist smooth mapsǫ ij : (U i ∩U j )×T n → R such that ǫ ij (x)(a) =ǫ ij (x, a)+aw ij . Substituting in Eq. (4.3.5) and subtracting Eq. (4.3.6) we get
t ′ ijk (x) − t ijk (x) −m ′ ijk a ′ ik (x) +m ijk (a ik (x) + p k (x)) − a ′ ij (x)â ′ jk (x) + a ij (x)â jk (x) + p i (x)|B ′ jk |a ′ ij (x) low − p j (x) + a ij (x)|B ′ jk |z ij low + z ij |B ′ jk |a ′ jk (x) − a ij (x)|B ′ jk |p j (x) low + a ik (x)|C k |m ijk low − m ijk |C k |p k (x) + a ij (x)|C k |a jk (x) low − a ′ jk (x)ẑ ij = ǫ ijk +ǫ ik (x, a) −ǫ ij (x, a − a ′ jk (x)) −ǫ jk (x, a).
as an equation in R, for a uniquely determined constant ǫ ijk ∈ Z. This means that the expression ρ ijk (x, a) :=ǫ ik (x, a) −ǫ ij (x, a − a ′ jk (x)) −ǫ jk (x, a) is independent of a. It is straightforward to check that ρ ijk is aČech cocycle, so that [ρ] ∈Ȟ 2 (X, R) = 0. Thus, there exist e ′ ij :
U i ∩ U j → R n such that ρ ijk = (δe ′ ) ijk . Now we have t ′ ijk (x) − t ijk (x) −m ′ ijk a ′ ik (x) +m ijk (a ik (x) + p k (x)) − a ′ ij (x)â ′ jk (x) + a ij (x)â jk (x) + p i (x)|B ′ jk |a ′ ij (x) low − p j (x) + a ij (x)|B ′ jk |z ij low + z ij |B ′ jk |a ′ jk (x) − a ij (x)|B ′ jk |p j (x) low + a ik (x)|C k |m ijk low − m ijk |C k |p k (x) + a ij (x)|C k |a jk (x) low − a ′ jk (x)ẑ ij = e ′ ik − e ′ ij − e ′ jk .
We consider η ijk := (p i − p j )(p k −p j ). It is easy to check that (δη) ijkl = 0. After subtracting η ijk from ρ ijk , we can assume that (δe ′ ) ijk = ρ ijk − η ijk . Finally, we consider
e ij := e ′ ij −ẑ ij a ′ ij − a ijpj +â ij p i + p i |C j |a ij ∈ U(1).
It is again tedious, but straightforward to check that
(δe) ijk = t ′ ijk − t ijk − z ijâ ′ jk + m ijkpk + a ′ ij |B ′ jk |p i low − p j |B ′ jk |a ij low + m ijk |C k |a ik low + a jk |C k |a ij low − z ij |B ′ jk |p j + a ij low
i.e., the remaining cocycle condition Eq. (4.1.10) is satisfied, and we have proved the equivalence of the two TD In particular, every half-geometric T-duality correspondence C has an underlying principal so(n, Z)-bundle p * (C), and if that bundle is trivializable, then the half-geometric T-duality correspondence is isomorphic to a (geometric) T-duality correspondence. We have the following natural definition:
Definition 4.4.1.1. A polarization of a half-geometric T-duality correspondence C over X is a section σ of the underlying so(n, Z)-bundle p * (C).
One can easily verify on the level of cocycles, that a choice of a polarization σ of C determines a T-duality correspondence C geo σ together with an isomorphism C ∼ = C geo σ . Alternatively, this follows from the fact that the sequence TD → TD 1 2 -geo → so(n, Z) is a fibre sequence; see [NW13b]. In particular, R(C geo σ ) is T-dual to L(C geo σ ), and the F 2 T-background L(C geo σ ) is isomorphic as T-backgrounds to the left leg of C.
Polarizations exist always locally. If U ⊆ X is connected, then -since so(n, Z) is discrete -two polarizations σ 1 and σ 2 over U differ by a uniquely defined matrix D ∈ so(n, Z), via σ 2 = σ 1 · D. The corresponding T-duality correspondences C geo σ 1 and C geo σ 2 differ then by the action of precisely this D ∈ so(n, Z).
Assume H 1 (X, Z) = 0, for example when X is connected and simply-connected. We have seen in Remark 2.3.3 that every F 1 T-background is isomorphic to an F 2 T-background, and hence T-dualizable. Correspondingly, every half-geometric T-duality correspondence is isomorphic to a (geometric) T-duality correspondence, under the choice of a global polarization.
Inclusion of ordinary T-duality
The semi-strict homomorphism i : TD → TD 1 2 -geo allows to consider ordinary T-duality correspondences as half-geometric T-duality correspondences. We have the following result:
Proposition 4.4.2.1. Let C 1 and C 2 be two T-duality correspondences over a smooth manifold X. Then, the following two statements are equivalent:
(a) C 1 and C 2 are isomorphic as half-geometric T-duality correspondences.
(b) The left legs L(C 1 ) and L(C 2 ) are isomorphic as T-backgrounds.
(c) For each connected component of X there exists a matrix B ∈ so(n, Z) such that (F e B ) * (C 1 ) and C 2 are isomorphic as T-duality correspondences.
In (c), F e B : TD → TD is the action of so(n, Z) on TD defined in Section 4.1.
Proof. By construction, the diagram Remark 4.4.2.2. The equivalence of (b) and (c) was already proved in [BRS06].
Half-geometric T-duality correspondences with trivial torus bundle
In order to investigate half-geometric T-duality correspondences whose left legs have trivial torus bundles, we consider the following sequence of Lie 2-group homomorphisms:
so(n, Z) dis × R n / /Z n × BU(1)Ĩ / / TD 1 2 -geoT / / T n dis ,(4.H 1 (X, so(n, Z)) × H 1 (X, R n / /Z n ) × H 3 (X, Z)Ĩ * / / H 1 (X, TD 1 2 -geo )T * / / H 2 (X, Z n ) / / 0
Here we have used the usual identifications between non-abelian cohomology and ordinary cohomology, see Remark A.3.3.
Proof. We compare with the sequence of Lemma 2.4.1. There is an obvious strict intertwiner ζ : R n / /Z n → BZ n , making the diagram so(n, Z) dis × R n / /Z n × BU(1)Ĩ / / id×ζ×id TD 1 2 -geo ℓeℓe so(n,Z) so(n, Z) dis × BZ n × BU(1)
I / / TB F1
commutative. It is easy to check that ζ induces a bijection
H 1 (X, R n / /Z n ) ∼ = H 1 (X, BZ n ) = H 2 (X, Z n ).
Together with Theorem 4.2.2 we have the claim.
Restricting to ordinary T-duality correspondences, we obtain a diagram The homomorphismĨ lifts the 2-functor I defined in Section 2.4 along the left leg projection, in the sense that we have the following commutative diagram, which combines the results of this section and Section 2.4:
R n / /Z n × BU(1) ζ×id t t ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ / / TD ℓeℓe | | ② ② ② ② ② ② / / T n dis ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ so(n, Z) dis × R n / /Z n × BU(1)Ĩ / / id×ζ×id TD 1 2 -geo / / ℓeℓe so(n,Z) T n dis BZ n × BU(1) / / t t ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ TB F2 { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ / / T n dis so(n, Z) dis × BZ n × BU(1) I / / TB F1 / / T n dis ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧
In Section 2.4 we have described a geometric counterpart of the homomorphism I, the 2-functor of Eq. (2.4.3). The 2-functorĨ has in general no such counterpart in classical geometry, since the resulting half-geometric T-duality correspondences are only half -geometric. Geometrically accessible is only the restriction ofĨ to R n / /Z n × BU(1), which assigns a Tduality correspondence ((E, G), (Ê, G), D) to a R n / /Z n -bundle gerbe H and a U(1)-bundle gerbe G over X. For completeness, let us describe this correspondence. Without explaining in more detail what a R n / /Z n -bundle gerbe is, we remark that it induces a Z n -bundle gerbe H := (ζ) * ( H), and, via the strict intertwiner R n / /Z n → T n dis , a T n -bundle E over X. The two characteristic classes coincide:
DD(H) = c 1 (E) ∈ H 2 (X, Z n ).
The left leg of this correspondence is the F 2 T-background (E, G) = (X × T n , R Z (H) ⊗ pr * X G), see Section 2.4. For the right leg, we notice that the composition
R n / /Z n × BU(1)Ĩ / / TD riℓe ′ / / TB F2'
sends an object (b, * ) ∈ R n to (0, b) in TD and then to b ∈ T n , and a morphism (m, b, t) ∈ Z n ×R n ×U(1) to (0, b, 0, m, t) in TD and then to (b, t) ∈ T n ×C ∞ (T n , U(1)), where t is regarded as a constant map. Thus, the right leg is the F 3 T-background (Ê, G) = (E, pr * X G), see Remark 2.2.6. It remains to construct the 1-isomorphism D : pr * 1 G → pr * 2 G over E × XÊ . We identify the latter correspondence space canonically with T n × E, so that D becomes a 1-isomorphism D : (id × pr X ) * R Z (H) ⊗ pr * X G → pr * X G over T n × E. This is equivalent to specifying a trivialization of (id × pr X ) * R Z (H).
Twisted K-theory
The twisted K-theory K • (C) of a half-geometric T-duality correspondence C is by definition the twisted K-theory of its left leg (E, G), i.e. the G-twisted K-theory of the manifold E, i.e.,
K • (C) := K • (E, G).
We discuss the local situation. Consider an open set U ⊆ X over which C admits a polarization σ, with associated T-duality correspondence C geo σ and an isomorphism A σ : C| U → C geo σ . We denote the restriction of the left leg (E, G) to U by (E U , G U ). The isomorphism A induces an isomorphism A σ : (E U , G U ) → L(C geo σ ), and
K • (C) = K • (E, G) res / / K • (E U , G U ) Aσ / / K • (L(C geo σ )) T / / K •−n (R(C geo σ )),
where T denotes the T-duality transformation for T-duality correspondences, defined in [BHM04] (for twisted de Rham cohomology) and (in full generality) in [BS05]. If another polarization σ ′ over U is chosen, we see that there is a canonical isomorphism
K • (R(C geo σ )) ∼ = K • (R(C geo σ ′ ))
between the locally defined twisted K-theories of the locally defined right legs. In an upcoming paper we will discuss more general versions of T-folds, and their twisted K-theory.
A Lie 2-groups
In this appendix we collect required definitions and results in the context of Lie 2-groups, and provide a number of complimentary new results.
A.1 Crossed modules and crossed intertwiners
A crossed module is a quadruple Γ = (G, H, t, α) with Lie groups G and H, a Lie group homomorphism t : H → G and a smooth action α : G×H → H by Lie group homomorphisms, such that α(t(h), h ′ ) = hh ′ h −1 and t(α(g, h)) = gt(h)g −1 for all g ∈ G and h, h ′ ∈ H. We write U := Ker(t), which is a central Lie subgroup of H and invariant under the action of G on H.
Essential for this article is the choice of an appropriate class of homomorphisms between crossed modules, which we call "crossed intertwiners". They are weaker than the obvious notion of a "strict intertwiner" (namely, a pair of Lie group homomorphisms respecting all structure) but stricter than weak equivalences (also known as "butterflies").
Definition A.1.1. Let Γ = (G, H, t, α) and Γ ′ = (G ′ , H ′ , t ′ , α ′ ) be crossed modules. A crossed intertwiner F : Γ → Γ ′ is a triple F = (φ, f, η) consisting of Lie group homomorphisms φ : G → G ′ and f : H → H ′ , and of a smooth map η : G × G → U ′ satisfying the following axioms for all h, h ′ ∈ H and g, g ′ , g ′′ ∈ G:
(CI1) φ(t(h)) = t(f (h)). (CI2) η(t(h), t(h ′ )) = 1. (CI3) η(g, t(h)g −1 ) · f (α(g, h)) = α ′ (φ(g), η(t(h)g −1 , g)) · α ′ (φ(g), f (h)). (CI4) η(g, g ′ ) · η(gg ′ , g ′′ ) = α ′ (φ(g), η(g ′ , g ′′ )) · η(g, g ′ g ′′ ).
We remark that these axioms imply the following:
• f (u) ∈ U ′ for all u ∈ U .
• η(g, 1) = 1 = η(1, g).
• η(g, g −1 ) = α ′ (φ(g), η(g −1 , g)).
A crossed intertwiner (φ, f, η) is called strict intertwiner if η = 1. The composition of crossed intertwiners is defined by
(φ 2 , f 2 , η 2 ) • (φ 1 , f 1 , η 1 ) := (φ 2 • φ 1 , f 2 • f 1 , η 2 • (φ 1 × φ 1 ) · f 2 • η 1 ). (A.1.1)
It is straightforward but a bit tedious to show that the composition is again a crossed intertwiner,whereas it is easy to check that composition is associative. The identity crossed intertwiner is (id G , id H , 1). The invertible crossed intertwiners from a crossed module Γ to itself form a group Aut CI (Γ), which we will use in Appendix A.4 in order to define group actions on crossed modules.
Example A.1.2.
• If A is an abelian Lie group, then BA := ({e}, A, t, α) is a crossed module in a unique way. If A ′ is another Lie group, then a crossed intertwiner BA → BA ′ is exactly the same as a Lie group homomorphism A → A ′ .
• If G is any Lie group, then G dis := (G, { * }, t, α) is a crossed module in a unique way. If G ′ is another Lie group, then a crossed intertwiner G dis → G ′ dis is exactly the same as a Lie group homomorphism φ : G → G ′ .
A.2 Semi-strict Lie 2-groups
Crossed modules of Lie groups correspond to strict Lie 2-groups. We need a more general class of Lie 2-groups.
Definition A.2.1. A semi-strict Lie 2-group is a Lie groupoid Γ together with smooth functors m : Γ × Γ → Γ and i : Γ → Γ, a distinguished element 1 ∈ Ob(Γ), and a smooth natural transformation ("associator")
Γ × Γ × Γ id×m m×id / / Γ × Γ λ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ s { ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ m Γ × Γ m / / Γ such that:
(a) 1 is a strict unit with respect to m, i.e. m(γ, id 1 ) = γ = m(id 1 , γ) for all γ ∈ Mor(Γ).
(b) i provides strict inverses for m, i.e. m(γ, i(γ)) = id 1 = m(i(γ), γ) for all γ ∈ Mor(Γ).
(c) λ is coherent over four copies of Γ, all components are endomorphisms (i.e. s • λ = t • λ), and λ(1, 1, 1) = id 1 .
The manifold Ob(Γ) is a Lie group with multiplication m, unit 1, and inversion i; likewise, the set π 0 Γ of isomorphism classes of objects is a group. The set π 1 Γ of automorphisms of 1 ∈ Ob(Γ) forms a group under composition and multiplication; these two group structures commute and are hence abelian. Semi-strict Lie 2-groups are (coherent) Lie 2-groups in the sense of [BL04] and have been considered in [JSW15]. A semi-strict Lie 2-group is called strict Lie 2-group if the associator is trivial, i.e. λ(g 1 , g 2 , g 3 ) = id g 1 g 2 g 3 .
Definition A.2.2. Let Γ and Γ ′ be semi-strict Lie 2-groups. A semi-strict homomorphism between Γ and Γ ′ is a smooth functor F : Γ → Γ ′ together with a natural transformation ("multiplicator")
Γ × Γ m / / F ×F Γ χ ② ② ② ② ② ② ② ② ② ② x Ð ② ② ② ② ② ② F Γ ′ × Γ ′ m ′ / / Γ ′
satisfying the following conditions:
(a) its components are endomorphisms, i.e. s(χ(g 1 , g 2 )) = t(χ(g 1 , g 2 )) for all g 1 , g 2 ∈ Ob(Γ).
(b) it respects the units: χ(1, 1) = id 1 .
(c) it is compatible with the associators λ and λ ′ in the sense that for all g 1 , g 2 , g 3 ∈ Ob(Γ) we have λ ′ (F (g 3 ), F (g 2 ), F (g 1 )) • (χ(g 3 , g 2 ) · id F (g 1 ) ) • χ(g 3 g 2 , g 1 ) = (id F (g 3 ) · χ(g 2 , g 1 )) • χ(g 3 , g 2 g 1 ) • F (λ(g 3 , g 2 , g 1 ))
We remark that a semi-strict homomorphism F induces a group homomorphism on the level of objects. Semi-strict homomorphisms have an associative composition given by the composition of functors and the "stacking" of the multiplicators. A semi-strict homomorphism is called strict homomorphism if the multiplicator χ is trivial.
We consider crossed modules and crossed intertwiners as special cases of semi-strict Lie 2-groups and semi-strict homomorphisms. A crossed module Γ = (G, H, t, α) defines the Lie groupoid (we denote it by the same letter) Γ, with Ob(Γ) := G and Mor(Γ) = H ⋉ α G with source (h, g) → g and target (h, g) → t(h)g, and the composition is induced from the group structure of H. The functors m : Γ × Γ → Γ and i : Γ → Γ are defined using the Lie group structures on G and H. The associator is trivial; thereby, Γ is a strict Lie 2-group. It is well-known that every strict Lie 2-group is of this form.
Next we consider a crossed intertwiner F = (φ, f, η) : Γ → Γ ′ between crossed modules Γ = (G, H, t, α) and Γ ′ = (G ′ , H ′ , t ′ , α ′ ), and identify Γ and Γ ′ with their associated strict Lie 2-groups. We define a corresponding semi-strict homomorphism (denoted by the same letter), based on the smooth functor with the following assignments to objects g ∈ Ob(Γ) and morphisms (h, g) ∈ Mor(Γ):
F (g) := φ(g) and F (h, g) := (η(t(h), g) −1 · f (h), φ(g)).
The smooth map η defines a multiplicator χ for F with component map χ(g, g ′ ) := (η(g, g ′ ), φ(gg ′ )).
It is straightforward though again tedious to show that it satisfies all conditions of Definition A.2.2. Thus, (F, χ) is a semi-strict homomorphism. It is obvious that strict intertwiners induce strict homomorphisms.Further, the smooth functor associated to a composition of crossed intertwiners is the composition of the separate functors, and the "stacking" of the corresponding multiplicators χ 1 and χ 2 is precisely the multiplicator of the composition. Summarizing, we have defined a functor from the category of crossed modules and crossed intertwiners to the category of strict Lie 2-groups and semi-strict homomorphisms.
A.3 Non-abelian cohomology for semi-strict Lie 2-groups
To a semi-strict Lie 2-group Γ we associate a presheaf of bicategories BΓ := C ∞ (−, BΓ) of smooth BΓ-valued functions. Explicitly, the bicategory BΓ(X) associated to a smooth manifold X is the following:
• It has just one object.
• The 1-morphisms are all smooth maps g : X → Ob(Γ); composition is the pointwise multiplication.
• The 2-morphisms between g 1 and g 2 are all smooth maps h : X → Mor(Γ) such that s • h = g 1 and t • h = g 2 . Vertical composition is the pointwise composition in Γ, and
(5) s(ǫ ij ) = h j · g ij and t(ǫ ij ) = g ′ ij · h i . (6) (γ ′ ijk · id h i ) • λ(g ′ jk , g ′ ij , h i ) −1 • (id g ′ jk · ǫ ij ) • λ(g ′ jk , h j , g ij ) • (ǫ jk · id g ij ) = ǫ ik • (id h k · γ ijk ) • λ(h k , g jk , g ij ).
A semi-strict homomorphism F : Γ → Γ ′ induces a map
F * : H 1 (X, Γ) → H 1 (X, Γ ′ )
in non-abelian cohomology. It can be described on the level of Γ-cocycles in the following way. If (g, γ) is a Γ-cocycle with respect to an open cover {U i } i∈I then the corresponding Γ ′ -cocycle (g ′ , γ ′ ) := F * (g, γ) is given by γ ′ ijk := F (γ ijk ) • χ(g jk , g ij ) −1 and g ′ ij := F (g ij ). If the Lie 2-group Γ is strict, one can reduce and reformulate Γ-cocycles in terms of the corresponding crossed module (G, H, t, α), resulting in the usual cocycles for nonabelian cohomology. Concerning a cocycle (g, γ), we keep the functions g ij as they are, and write γ ijk = (a ijk , g jk g ij ) under the decomposition Mor(Γ) = H ⋉ G, for smooth maps a ijk : U i ∩ U j ∩ U k → H satisfying conditions equivalent to (1) to (3): a iij = a ijj = 1, t(a ijk )g jk g ij = g ik and a ikl · α(g kl , a ijk ) = a ijl · a jkl .
Similarly, for an equivalence between cocycles, we write ǫ ij = (e ij , h j g ij ) for smooth maps e ij : U i ∩ U j → H satisfying conditions equivalent to (4) to (6): e ii = 1 and t(e ij )h j g ij = g ′ ij h i and a ′ ijk α(g ′ jk , e ij )e jk = e ik α(h k , a ijk ). (A.3.1) Now let F : Γ → Γ ′ be a crossed intertwiner between crossed modules Γ and Γ ′ , with F = (φ, f, η). Passing to the associated semi-strict homomorphism, using its induced map in cohomology, and reformulating in terms of crossed modules, we obtain the following. If (g, a) is a Γ-cocycle, then the corresponding Γ ′ -cocycle (g ′ , a ′ ) := F * (g, a) is given by g ′ ij = φ(g ij ) and a ′ ijk := η(t(a ijk ), g jk g ij ) −1 · f (a ijk ) · η(g jk , g ij ) −1 .
A.4 Semi-direct products
Let U be a (discrete) group, and let Γ be a crossed module.
Definition A.4.1. An action of U on Γ by crossed intertwiners is a group homomorphism
ϕ : U → Aut CI (Γ).
The crossed intertwiner ϕ(u) associated to u ∈ U will be denoted by F u = (φ u , f u , η u ), and as before we may consider Γ as a strict Lie 2-group, F u as a smooth functor F u : Γ → Γ with some multiplicator χ u .
Given an action of U on Γ by crossed intertwiners, we define a semi-strict Lie 2-group Γ ⋉ ϕ U , called the semi-direct product of Γ with U . Its underlying Lie groupoid is Γ × U dis .
We equip it with a multiplication functor m : (Γ × U dis ) × (Γ × U dis ) → Γ × U dis defined on objects and morphisms by (g 2 , u 2 ) · (g 1 , u 1 ) := (g 2 F u 2 (g 1 ), u 2 u 1 ) and (γ 2 , u 2 ) · (γ 1 , u 1 ) := (γ 2 · F u 2 (γ 1 ), u 2 u 1 ).
It is straightforward to check that this is a functor. The associator λ for the multiplication m is given by λ ((g 3 , u 3 ), (g 2 , u 2 ), (g 1 , u 1 )) := (id g 3 · χ u 3 (g 2 , F u 2 (g 1 )) −1 , u 3 u 2 u 1 ).
The inversion functor i : Γ → Γ is defined by
i(g, u) := (F u −1 (g −1 ), u −1 ) and i(γ, u) := (F u −1 (γ −1 ), u −1 ).
It is again straightforward, though again a bit tedious, to check that all conditions for semistrict Lie 2-groups are satisfied. Its invariants are
π 0 (Γ ⋉ ϕ U ) = π 0 (Γ) ⋉ U and π 1 (Γ ⋉ ϕ U ) = π 1 (Γ),
where the action of π 0 on π 1 is induced from the one of Γ.
We investigate how the (Γ ⋉ ϕ U )-cocycles look like, reducing the Γ-cocycles of Appendix A.3 to the present situation. For an open cover {U i } i∈I a cocycle is a triple (u, g, γ) consisting of smooth maps
u ij : U i ∩ U j → U , g ij : U i ∩ U j → Ob(Γ) and γ ijk : U i ∩ U j ∩ U k → Mor(Γ)
such that the following conditions are satisfied:
(1) g ii = 1, u ii = 1, γ iij = γ ijj = id g ij .
(2) u jk · u ij = u ik and s(γ ijk ) = g jk · φ u jk (g ij ) and t(γ ijk ) = g ik (3) γ ikl • (id g kl · F u kl (γ ijk )) = γ ijl • (γ jkl · F u jl (id g ij )) • (id g kl · χ u kl (g jk , F u jk (g ij ))).
Two (Γ ⋉ ϕ U )-cocycles (u, g, γ) and (u ′ , g ′ , γ ′ ) are equivalent, if there exist smooth maps
h i : U i → Mor(Γ), v i : U i → U and ǫ ij : U i ∩ U j → Mor(Γ) such that (4) ǫ ii = id h i . (5) v j · u ij = u ′ ij · v i and s(ǫ ij ) = h j · φ v j (g ij ) and t(ǫ ij ) = g ′ ij · φ u ′ ij (h i ). (6) (γ ′ ijk · F u ′ ik (id h i )) • (id g ′ jk · χ u ′ jk (g ′ ij , F u ′ ij (h i ))) • (id g ′ jk · F u ′ jk (ǫ ij )) •(id g ′ jk · χ u ′ jk (h j , F v j (g ij )) −1 ) • (ǫ jk · F v k u jk (id g ij )) = ǫ ik • (id h k · F v k (γ ijk )) • (id h k ·η v k (g jk , F u jk (g ij )) −1 ).
We formulate these results in terms of the crossed module Γ = (G, H, t, α) and of the crossed intertwiner F u = (φ u , f u , η u ) associated to u ∈ U . For a cocycle (u, g, γ), we start by writing γ ijk = (a ijk , g jk · φ u jk (g ij )) for functions a ijk : U i ∩ U j ∩ U k → H, and get the following conditions:
(1') g ii = 1, u ii = 1, a iij = a ijj = 1.
(2') u jk · u ij = u ik and t(a ijk ) · g jk · φ u jk (g ij ) = g ik (3') a ikl · α(g kl , η u kl (g ik φ u jk (g ij ) −1 g −1 jk , g jk φ u jk (g ij )) −1 f u kl (a ijk )) = a ijl · a jkl · α(g kl , η u kl (g jk , φ u jk (g ij ))).
For an equivalence (ǫ, h, v) we write ǫ ij = (e ij , h j · φ v j (g ij )) with e ij : U i ∩ U j → H, and get (4') e ii = 1.
(5') v j · u ij = u ′ ij · v i and t(e ij ) · h j · φ v j (g ij ) = g ′ ij · φ u ′ ij (h i ). (6') a ′ ijk · α(g ′ jk , η u ′ jk (g ′ ij , φ u ′ ij (h i ))) · α(g ′ jk , η u ′ jk (t(e ij ), h j φ v j (g ij ))) −1 · α(g ′ jk , f u ′ jk (e ij )) · α(g ′ jk , η u ′ jk (h j , φ v j (g ij )) −1 ) · e jk = e ik · α(h k , η v k (t(a ijk ), g jk φ u jk (g ij ))) −1 · α(h k , f v k (a ijk )) · α(h k , η v k (g jk , φ u jk (g ij )) −1 )
Finally, we notice that the semi-direct product fits into a sequence
Γ i / / Γ ⋉ ϕ U p / / U dis (A.4.1)
of semi-strict Lie 2-groups and semi-strict homomorphisms defined in the obvious way. We want to investigate the induced sequence in non-abelian cohomology. For u ∈ U the crossed intertwiner F u : Γ → Γ induced a map (F u ) * : H 1 (X, Γ) → H 1 (X, Γ), forming an action of U on the set H 1 (X, Γ).
Lemma A.4.2. Consider x 1 , x 2 ∈ H 1 (X, Γ). Then, i * (x 1 ) = i * (x 2 ) if there exists u ∈ U with (F u ) * (x 1 ) = x 2 .
If X is connected, then "only if " holds, too.
Proof. We show first the "only if"-part under the assumption that X is connected. Let (g, a) and (g,ã) be Γ-cocycles with respect to some open cover of X, such that i * (g, a) = (1, g, a) and i * (g,ã) = (1,g,ã) are equivalent. Thus, after a possible refinement of the open cover, there exists equivalence data (e, h, v) satisfying (4') to (6'). Reduced to the case u ij = u ′ ij = 1, these are (4') e ii = 1.
(5') v j = v i and t(e ij ) · h j · φ v j (g ij ) =g ij · h i . (6')ã ijk · α(g jk , e ij ) · e jk = e ik · α(h k , η v k (t(a ijk ), g jk g ij ) −1 · f v k (a ijk ) · η v k (g jk , g ij ) −1 )
The first part of (5'), together with the fact that X is connected, shows that there exists u ∈ U with u = v i for all i ∈ I. We have (F u ) * (g, a) = (g ′ , a ′ ) with g ′ ij := φ u (g ij ) and a ′ ijk := η u (t(a ijk ), g jk g ij ) −1 · f u (a ijk ) · η u (g jk , g ij ) −1 . (A.4.2)
Under these definitions, the second part of (5') and (6') become exactly Eq. (A.3.1), showing that (g, a) and (g ′′ , a ′′ ) are equivalent.
In order to show the "if"-part, let (g, a) be a Γ-cocycle with respect to some open cover of X. Let u ∈ U with associated crossed intertwiner F u = (φ u , f u , η u ). We have (F u ) * (g, a) = (g ′ , a ′ ) with g ′ ij and a ′ ijk defined exactly as in Eq. (A.4.2), and i * (g ′ , a ′ ) = (1, g ′ , a ′ ). We have to show that the (Γ ⋉ ϕ U )-cocycles (1, g, a) and (1, g ′ , a ′ ) are equivalent. Indeed, we employ equivalence data (e, h, v) with e ij := 1, h i := 1 and v i := u. Then, (4') is trivial, (5') is satisfied since v i = u = v j and φ u (g ij ) = g ′ ij and (6') is precisely above definition of a ′ ijk .
The "if"-part of Lemma A.4.2 shows that the map i * : H 1 (X, Γ) → H 1 (X, Γ ⋉ ϕ U ) is constant on the U -orbits, i.e. it descents into the (set-theoretic) quotient H 1 (X, Γ)/U . Now we are in position to formulate all properties of the sequence induced by Eq. (A.4.1) in nonabelian cohomology. is an exact sequence of pointed sets, for all smooth manifolds X. If X is connected, then the first map is injective.
Proof. First of all, p * is surjective, since a givenČech 2-cocycle u ij : U i ∩ U j → U can be lifted to a (Γ ⋉ ϕ U )-cocycle (u, g, a) by putting g ij = 1 and a ijk = id. Second, the composition p • i : Γ → U dis is the trivial functor. In order to see exactness at H 1 (X, Γ ⋉ ϕ U ), let (u, g, a) be a (Γ ⋉ ϕ U )-cocycle with respect to an open cover {U i } i∈I of X. If we assume that p * (u, g, a) = u = 0, then there exist v i : U i → U such that u ij = v −1 j · v i . We define data (1, 1, v i ) for an equivalence between (u, g, a) and another (Γ ⋉ ϕ U )-cocycle (u ′ ij , g ′ ij , a ′ ijk ), which we define such that (4') to (6') are satisfied. We get u ′ ij = 1. By inspection, we observe that any (Γ ⋉ ϕ U )-cocycle (u ′ , g ′ , a ′ ) with u ′ = 1 yields a Γ-cocycle (g ′ , a ′ ) such that i * (g ′ , a ′ ) = (1, g ′ , a ′ ). Finally, the injectivity of i * is the "only if"-part of Lemma A.4.2.
A.5 Equivariant crossed intertwiners
We suppose that F : Γ → Γ ′ is a crossed intertwiner between two crossed modules Γ and Γ ′ , on which a discrete group U acts by crossed intertwiners. The crossed intertwiners of the actions are denoted by F u : Γ → Γ and F ′ u : Γ ′ → Γ ′ , respectively.
Definition A.5.1. The crossed intertwiner F is called strictly U -equivariant if
F ′ u • F = F • F u for all u ∈ U .
Remark A.5.2. We write F = (φ, f, η) as well as F u = (φ u , f u , η u ) and F ′ u = (φ ′ u , f ′ u , η ′ u ). Then, the commutativity of the diagram splits into three conditions, namely
φ ′ u • φ = φ • φ u and f ′ u • f = f • f u (A.5.1)
as well as η ′ u (φ(g 1 ), φ(g 2 )) · f ′ u (η(g 1 , g 2 )) = η(φ u (g 1 ), φ u (g 2 )) · f (η u (g 1 , g 2 )) (A.5.2) for all g 1 , g 2 ∈ G.
A strictly U -equivariant crossed intertwiner F induces a semi-strict homomorphism
F U : Γ ⋉ ϕ U → Γ ′ ⋉ ϕ U
between the semi-direct products. Indeed, as a functor it is defined by F U := F × id U dis , and its multiplicator is defined by χ((g 2 , u 2 ), (g 1 , u 1 )) := (χ(g 2 , F u 2 (g 1 )), u 2 u 1 ),
where χ is the multiplicator of F . Checking that all conditions for a multiplicator are satisfied is again tedious but straightforward.
We go one step further and consider the induced map in cohomology,
(F U ) * : H 1 (X, Γ ⋉ ϕ U ) → H 1 (X, Γ ′ ⋉ ϕ U ).
We describe this map at the level of (Γ ⋉ U )-cocycle (u, g, a) in the formulation with the crossed module Γ = (G, H, t, α) as in Appendix A.4. Thus, (u, g, a) consists of smooth maps
u ij : U i ∩ U j → U , g ij : U i ∩ U j → G and a ijk : U i ∩ U j ∩ U k → H
satisfying (1') to (3'). The map induced by a general semi-strict homomorphism was described in Appendix A.2; here it reduces to g ′ ij := φ(g ij ), u ′ ij := u ij , and α ′ ijk := η(t(a ijk ), g jk φ u jk (g ij )) −1 · f (a ijk ) · η(g jk , φ u jk (g ij )) −1 .
B The Poincaré bundle
In this section we recall and introduce required facts about the Poincaré bundle. We work with writing U(1) = R/Z additively. Basically, the Poincaré bundle is the following principal U(1)-bundle P over T 2 = U(1) × U(1). Its total space is P := (R × R × U(1)) / ∼ with (a,â, t) ∼ (a + n,â + m, nâ + t) for all n, m ∈ Z and t ∈ U(1). The bundle projection is (a,â, t) → (a,â), and the U(1)-action is (a,â, t) · s := (a,â, t + s).
Remark B.1. The Poincaré bundle P carries a canonical connection, which descends from the 1-formω ∈ Ω 1 (R × R × U(1)) defined byω := adâ − dt. The curvature of ω is pr * 1 θ ∧ pr * 2 θ ∈ Ω 2 (T 2 ), where θ ∈ Ω 1 (U(1)) is the Maurer-Cartan form. Since H * (T 2 , Z) is torsion free, this shows that the first Chern class of P is pr 1 ∪ pr 2 ∈ H 2 (T 2 , Z), where pr i : T 2 → S 1 is the projection, whose homotopy class is an element of [T 2 , S 1 ] = H 1 (T 2 , Z).
The Poincaré bundle has quite difficult (non)-equivariance effects, which we shall explore in the following. We let R 2 act on T 2 by addition, and let r x,x : T 2 → T 2 denote the action of (x,x) ∈ R 2 . It lifts to P in terms of a bundle isomorphism R x,x : P → P defined by R x,x (a,â, t) := (x + a,x +â, t + ax). Equivalently, we can regard R x,x as a bundle isomorphismR x,x : P → r *
x,x P over the identity on T 2 . The lifts R x,x do not define an action of R 2 on P. Indeed, we find R x ′ ,x ′ (R x,x (a,â, t)) = R x ′ +x,x ′ +x (a,â, t) · xx ′ .
In order to treat this "error", we define ν : R 2 × R 2 → U(1) by ν((x ′ ,x ′ ), (x,x)) :=x ′ x, so that R x ′ ,x ′ • R x,x = R x ′ +x,x ′ +x · ν((x ′ ,x ′ ), (x,x)).
Equivalently, we have
r * x,xRx ′ ,x ′ •R x,x =R x ′ +x,x ′ +x · ν((x ′ ,x ′ )
, (x,x)).
Next, we restrict to Z 2 ⊆ R 2 . For (m,m) ∈ Z 2 the bundle morphism R m,m covers the identity on T 2 , and it is given by multiplication with the smooth map f m,m : T 2 → U(1) : (a,â) → am − mâ.
Since the restriction of ν to Z 2 × Z 2 vanishes, this is a genuine action of Z 2 on P.
Next we generalize the Poincaré bundle to n-fold tori. Let B ∈ so(n, Z) be a skewsymmetric matrix. We define the principal U(1)-bundle P B := 1≤j<i≤n pr * ij P ⊗B ij over T n , where pr ij : T n → T 2 denotes the projection to the two indexed factors. Braiding of tensor factors gives a canonical isomorphism P B 1 +B 2 ∼ = P B 1 ⊗P B 2 , and we have P 0 = T n ×U(1).
Hence, assigning to B the first Chern class of P B is group homomorphism so(n, Z) → H 2 (T n , Z) : B → c 1 (P B ).
Via the Künneth formula it is easy to see that it is an isomorphism.
For a = (a 1 , ..., a n ) ∈ R n we define a map R B (a) : P B → P B tensor-factor-wise as R a i ,a j : P → P; this is a U(1)-equivariant smooth map that covers the action of R n on T n by addition. We have from the definitions Remark B.2. The Poincaré bundle P has a smooth section χ : R 2 → P : (a,â) → (a,â, 0) along the projection R 2 → T 2 , whose transition function is ((a + m,â +m), (a,â)) → mâ.
The pullback of P B along R n → T n has an induced section whose transition function is R n × T n R n → U(1) : (a + m, a) → m|B|a low .
This clarifies how P B can be obtained from a local gluing process.
Finally, we reduce the previous consideration to the matrix B n := 0 −E n E n 0 ∈ so(2n, Z).
The corresponding principal U(1)-bundle over T 2n is called the n-fold Poincaré bundle and denoted by P n := P Bn . Note that for n = 1 we get P 1 = pr * 21 P ∼ = P ∨ . We writẽ R(a,â) : P n → r * a,â P n for the bundle isomorphismR B (a,â). We have from Eq. (B.2) r * a,âR (a ′ ⊕â ′ ) •R(a ⊕â) =R((a ′ + a) ⊕ (â ′ ⊕â)) ·âa pr * i+n θ ∧ pr * i θ ∈ Ω 2 (T 2n ), and the transition function of Remark B.2 reduces to (x + z, x) → [z, x] in the notation of Section 3.2, where x ∈ R 2n and z ∈ Z 2n .
f
B (τ, m) := (τ − m|B|, m) and η B (a, a ′ ) := a ′ |B|a low , where η B (a, a ′ ) is considered as a constant U(1)-valued map. It is straightforward to check the axioms of a crossed intertwiner: (CI1) and (CI2) are trivial, (CI3) follows from the skewsymmetry of B, and (CI4) follows from the bilinearity and R-invariance of η B .
The presheaf Corr is a 2-stack.If C = ((E, G), (Ê, G), D) is a correspondence, then the two T-backgrounds L(C) := (E, G) and R(C) := (Ê, G) are called the left leg and the right leg, respectively. Projecting to left and right legs forms 2-functors L, R : Corr(X) → T-BG(X). There is another 2-functor () ∨ : Corr(X) op → Corr(X), which takes C to the correspondenceC ∨ := ((Ê, G), (E, G), s * D −1 ),where s :Ê × X E → E × XÊ exchanges components.
Definition 3.1. 3 .
3A correspondence C over X is called T-duality correspondence if it is in T-Corr(X). A T-background (E, G) is called T-dualizable, if there exists a T-duality correspondence C and a 1-isomorphism L(C) ∼ = (E, G).Remark 3.1.4. We have the following consequences of this definition:(a) The legs of a T-duality correspondence are F 2 T-backgrounds.
Remark 3.2. 2 .
2The Lie 2-group TD n can be directly related to the n-fold Poincaré bundle, via multiplicative gerbes. Multiplicative gerbes over T 2n are also classified by H 4 (BT 2n , Z) and in fact equivalent as a bicategory to categorical central extensions of T 2n by BU(1) [Wal12, Thm. 3.2.5]. Ganter shows [Gan18, Prop. 2.4
A 2-morphism (t, m,m) : (a,â) ⇒ (a + m,â +m) is sent to the 2-morphism (α t,m,m , β t,m,m ) whose 2morphisms α t,m,m : id I ⇒ id I and β t,m,m : id I ⇒ id I over X × T n are given by acting with the U(1)-valued functions α t,m,m (x, c) := −t(x) −mc and β t,m,m (x, c) := −t(x) − m(c +â(x)), considered as automorphisms of the trivial U(1)-bundle, on the identity 2-morphism X × T n . X × T n ) × X (X × T n ). This boils down to an identity pr * 1 α t,m,m · aâ ·R(a,â) = (a + m)(â +m) ·R(a + m,â +m) · pr * 2 β t,m,m .(3.2.2)
Lemma 3.2. 4 .
4The 2-functor C : BTD(X) → T (X) is an equivalence.
to show that the map (t, m,m) → (α t,m,m , β t,m,m ) is a bijection, where α t,m,m (x, c) := −t(x) −mc and β t,m,m (x, c) := −t(x) − m(c +â(x)).
(a) A triple over X consists of F 2 T-backgrounds (E, G) and (Ê, G) over X, and of a 2isomorphism class of 1-isomorphism [D] : pr * 1 G → pr * 2 G over E × XÊ . (b) A morphism between triples ((E, G), (Ê, G), [D]) and ((E ′ , G ′ ), (Ê ′ , G ′ ), [D ′ ]) consists of two 1-morphisms (f, [B]) : (E, G) → (E ′ , G ′ ) and (f , [ B]) : (Ê, G) → (Ê ′ , G ′ ) of Tbackgrounds, where the gerbe 1-morphisms are taken up to 2-isomorphisms,
(f,f ) * [D ′ ] / / pr * 2f * G ′is commutative in the 1-truncation h 1 (Grb (E × XÊ )), see[BRS06, Def. 4.5].
. 2 .
2The functor of Eq. (3.3
We define a crossed intertwiner fℓip : TD n → TD n as the triple (φ, f, η) with φ(a ⊕â) :=â ⊕ a , f (m ⊕m, t) := (m ⊕ m, t) and η(x, x ′ ) := (0, [x, x ′ ]).
. 4 .
4ℓeℓe ′ : TD n → TB F2' n and riℓe ′ : TD n → TB F2' n that will produce the left leg and right leg of a T-duality correspondence. The crossed intertwiner riℓe ′ is strict; it consists of the group homomorphisms φ(a ⊕â) :=â and f (m ⊕m, t)(c) := t + mc, whereas η := 0. We define ℓeℓe ′ := riℓe ′ • fℓip. This givesφ(a ⊕â) = a , f (m ⊕m, t)(c) = t +mc and η(a ⊕â, a ′ ⊕â ′ )(c) =âa ′ .By inspection of the definitions, we see the following:Proposition 3.4.3. The crossed intertwiners ℓeℓe ′ and riℓe ′ represent the left leg and the right leg projections of a T-duality correspondence, in the sense that the diagrams The homomorphisms ℓeℓe ′ and riℓe ′ can be lifted into the bigger 2-group TB F2 n . Indeed, riℓe ′ can be lifted to a strict intertwiner riℓe : TD n → TBF2 n given by φ(a ⊕â) :=â and f (m ⊕m, t) := (τ t,m ,m), where τ t,m (c) = t + mc. The left leg ℓeℓe : TD n → TB F2 n is again defined by ℓeℓe := riℓe • fℓip, resulting in φ(a ⊕â) = a , f (m ⊕m, t) = (τ t,m , m) and η(a ⊕â, a ′ ⊕â ′ )(c) =âa ′ . (3.4.2) We identify the induced map ℓeℓe * : H 1 (X, TD) → H 1 (X, TB F2 ) on the level of cocycles. Suppose (a,â, m,m, t) is a TD-cocycle with respect to an open cover {U i } i∈I . Applying the general construction of Appendix A.3 to Eq. (3.4.
F
e B = (φ e B , f e B , η e B ) : TD n → TD n , which we define by: φ e B (a ⊕â) := (a ⊕ (Ba +â)) f e B (m ⊕m, t) := (m ⊕ (Bm +m), t) η e B (a ⊕â, b ⊕b) := a|B|b low .Remark 4.1.1. The notation e B is used in order to distinguish the so(n, Z)-action on TD from the so(n, Z)-action on TB F2 , which was introduced in Section 2.3 in terms of crossed intertwiners (φ B , f B , η B ) : TB F2 → TB F2 . In fact, e B is our notation for the inclusion so(n, Z) → O(n, n, Z) : B → e B
TD 1 2
1-geo := TD n ⋉ so(n, Z), see Appendix A.4. It has the following invariants: geo ) = T 2n ⋉ so(n, Z) and π 1
+m ijk + B kl m ijk =m ijl +m jkl . (4.1.6)
Theorem 4 .2. 2 .
42The left leg projection of a half-geometric T-duality correspondence,(ℓeℓe so(n,Z) ) * : H 1 (X, TD 1 2 -geo ) → H 1 (X, TB F1 ),is a bijection. In other words, up to isomorphism, every F 1 T-background is the left leg of a unique half-geometric T-duality correspondence.
A.4.3 we have for all smooth manifolds an exact sequence H 1 (X, TD)/so(n, Z) → H 1 (X, TD 1 2 -geo ) → H 1 (X, so(n, Z)) → 0.
/
/ H 1 (X, TB F1 ).Commutativity shows that (a) implies (b). The injectivity in Theorem 4.2.2 shows the converse implication. By Proposition A.4.3 i induces a well-defined map H 1 (X, TD)/so(n, Z) → H 1 (X, TD 1 2 -geo ), whose existence shows that (c) implies (a). Over each connected component this map is injective, which shows that (a) implies (c).
= T • ℓeℓe so(n,Z) , with T : TB F1 → T n dis defined in Section 2.4. In other words, T projects to the underlying torus bundle of the left leg. By R n / /Z n we have denoted the crossed module (R n , Z n , t, α), where t : Z n → R n is the inclusion, and α is the trivial action. The homomorphismĨ is defined as follows.It sends an object (B, b, * ) to (0, b, B), and a morphism (B, m, b, t) : (B, b, t) → (B, b + m, t) to (0, b, 0, m, t, B).
The following sequence induced by Eq. (4.4.3.1) in cohomology is exact:
(n, Z) dis × R n / /Z n × BU(1)Ĩ / / TD 1 2 -geoT / / T n dis in which all horizontal and vertical sequences induce exact sequences in the sense of Lemma 4.4.3.1 and Proposition A.4.3. This diagram describes ordinary and half-geometric T-duality correspondences whose left legs have trivial torus bundles.
.
Suppose a group U acts on a crossed module by crossed intertwiners.
R
B (a ′ ) • R B (a) = R B (a ′ + a) · a|B|a ′ low . (B.1)If we denote byR B (a) : P B → r * a P B the corresponding bundle morphism over the identity of T n , then we can rewrite Eq. (B.1) asr * aRB (a ′ ) •R B (a) =R B (a ′ + a) · a|B|a ′ low . (B.2)Concerning the restriction to integers m ∈ Z n , we note that R B (m) acts factor-wise as R mi ,m j , i.e. by multiplication with the smooth map f m i ,m j : T 2 → U(1). Thus, R B (m) is multiplication with the map f m : T n → U(1) : a → a|B|m (B.3) In particular, we obtain from Eq. (B.2) and Eq. (B.3) R B (a + m) =R B (a) · (|B|m − m|B|a low ). (B.4)
a + n) ⊕ (â +m)) =R(a ⊕â) · η m,m,a , with a smooth map η m,m,a : T 2n → U(1) defined by η m,m,a (x, y) := x ⊕ y|B|m ⊕m − m ⊕m|B|a ⊕â low = −xm + my − am. Remark B.3. The connection of Remark B.1 induces a connection on the n-fold Poincaré bundle P n of curvature n i=1
This shows that [p] = [q] ∪ [z]. Since [z] = [Z pq ] and [q] = c 1 (P), this gives the claim. Lemma 2.4.3. Let H be a Z n -bundle gerbe over X classified by a class [H] ∈ H 2
TD 1 2
1-geo -bundle gerbes[ACJ05,NW13a], principal TD 1 2 -geo -2-bundles [Woc11, SP11, Wal17], or principal ∞-bundles [NSS15]. Remark 4.1.2. According to Appendix A.4, a TD 1 2 -geo -cocycle with respect to an open cover {U i } i∈I is
-geo | and |BTB F1 | are equivalent, see Remark 3.4.6. Showing this equivalence directly would provide an alternative proof of Theorem 4.2.2. The Lie 2-groups TD 1 2 -geo and TB F1 are, however, not equivalent, since they have different homotopy types, see Eqs. (2Remark 4.2.3. Theorem 4.2.2 implies that the classifying spaces |BTD
1
2
horizontal composition is pointwise multiplication. Definition A.3.1. Let F be a 2-stack over smooth manifolds, and Γ be a semi-strict Lie 2group. We say that F is represented by Γ, if there exists an isomorphism of 2-stacks BΓ + ∼ = F.Here, BΓ + denotes the 2-stackification, which can be performed e.g. with a construction described in[NS11]. The objects of BΓ + (X) are called Γ-cocycles, and the non-abelian cohomology of X with values in smooth Γ-valued functions is by definition the set of equivalence classes of Γ-cocycles, i.e., H 1 (X, Γ) := h 0 (BΓ + (X)).Remark A.3.2. For strict Lie 2-groups Γ, there is a classifying space for the 0-truncation of a 2-stack F represented by a semi-strict Lie 2-group Γ: one can use a certain geometric realization |Γ| such thatwhere the last bijection was shown in[BS09]. More precisely, in [BS09, Theorem 1] it was shown for well-pointed strict topological 2-groups that |Γ| represents the continuous nonabelian cohomology, and in[NW13a,Prop. 4.1] we have proved for Lie 2-groups (which are automatically well-pointed) that continuous and smooth non-abelian cohomologies coincide.Remark A.3.3.(a) For a Lie group G and the strict Lie 2-group Γ = G dis we have H 1 (X, G dis ) =Ȟ 1 (X, G), theČech cohomology with values in the sheaf of smooth G-valued functions. In particular, if G is discrete, this is the ordinary cohomology H 1 (X, G).(b) For an abelian Lie group A and strict Lie 2-group Γ = BA we have H 1 (X, BA) =Ȟ 2 (X, A). In particular, H 1 (X, BU(1)) =Ȟ 2 (X, U(1)), which is isomorphic to H 3 (X, Z).The Γ-cocycles for semi-strict Lie 2-groups have been worked out in[JSW15]. With respect to an open cover {U i } i∈I , they are pairs (g, γ) consisting of smooth mapssuch that the following conditions are satisfied:(1) g ii = 1, γ iij = γ ijj = id g ij .(2) s(γ ijk ) = g jk · g ij and t(γ ijk ) = g ik .(3) γ ikl • (id g kl · γ ijk ) • λ(g kl , g jk , g ij ) = γ ijl • (γ jkl · id g ij ).Here we have abbreviated the multiplication m of Γ by "·". Two Γ-cocycles (g, γ) and (g ′ , γ ′ ) are equivalent, if there exist smooth maps h i : U i → Ob(Γ) and ǫ ij : U i ∩ U j → Mor(Γ) such that (4) ǫ ii = id h i .
Nonabelian bundle gerbes, their differential geometry and gauge theory. P Aschieri, L Cantini, B Jurco, arxiv:hep-th/0312154Commun. Math. Phys. 254P. Aschieri, L. Cantini, and B. Jurco, "Nonabelian bundle gerbes, their differential geometry and gauge theory". Commun. Math. Phys., 254:367-400, 2005. [arxiv:hep-th/0312154]
T-Duality: Topology Change from H-flux. P Bouwknegt, J Evslin, V Mathai, arxiv:/hep-th/0306062Commun. Math. Phys. 2492P. Bouwknegt, J. Evslin, and V. Mathai, "T-Duality: Topology Change from H-flux". Commun. Math. Phys., 249(2):383-415, 2004. [arxiv:/hep-th/0306062]
Topology and H-flux of T-dual manifolds. P Bouwknegt, J Evslin, V Mathai, Phys. Rev. Lett. 9218181601P. Bouwknegt, J. Evslin, and V. Mathai, "Topology and H-flux of T-dual manifolds". Phys. Rev. Lett., 92(18):181601, 2004.
T-duality, Gerbes and Loop Spaces. D M Belov, C M Hull, R Minasian, arxiv:0710.5151PreprintD. M. Belov, C. M. Hull, and R. Minasian, "T-duality, Gerbes and Loop Spaces". Preprint. [arxiv:0710.5151]
T-duality for principal torus bundles. P Bouwknegt, K Hannabuss, V Mathai, arxiv:/hep-th/0312284J. High Energy Phys. 18P. Bouwknegt, K. Hannabuss, and V. Mathai, "T-duality for principal torus bundles". J. High Energy Phys., 2004:018, 2004. [arxiv:/hep-th/0312284]
Higher-dimensional algebra V: 2-groups. J C Baez, A D Lauda, arxiv:math/0307200Theory Appl. Categ. 12J. C. Baez and A. D. Lauda, "Higher-dimensional algebra V: 2-groups". Theory Appl. Categ., 12:423-491, 2004. [arxiv:math/0307200]
U Bunke, P Rumpf, T Schick, arxiv:/math/0501487The topology of T-duality for T n -bundles. 18U. Bunke, P. Rumpf, and T. Schick, "The topology of T-duality for T n -bundles". Rev. Math. Phys., 18(10):1103-1154, 2006. [arxiv:/math/0501487]
Loop spaces, characteristic classes and geometric quantization. J.-L Brylinski, Number 107 in Progr. Math. Birkhäuser. J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization. Number 107 in Progr. Math. Birkhäuser, 1993.
On the topology of T-duality. U Bunke, T Schick, arxiv:math/0405132Rev. Math. Phys. 1717U. Bunke and T. Schick, "On the topology of T-duality". Rev. Math. Phys., 17(17):77-112, 2005. [arxiv:math/0405132]
The classifying space of a topological 2-group. J C Baez, D Stevenson, arxiv:0801.3843Algebraic Topology. N. BaasSpringer4J. C. Baez and D. Stevenson, "The classifying space of a topological 2-group". In N. Baas, editor, Algebraic Topology, volume 4 of Abel Symposia, pages 1-31. Springer, 2009. [arxiv:0801.3843]
Duality for topological abelian group stacks and T-duality. U Bunke, T Schick, M Spitzweck, A Thom, arxiv:/math/0701428K-theory and noncommutative geometry. Enseign. Math., II. Sér.U. Bunke, T. Schick, M. Spitzweck, and A. Thom, "Duality for topological abelian group stacks and T-duality". In K-theory and noncommutative geometry, EMS Series of Congress Reports (ECR), pages 227-347. Enseign. Math., II. Sér., 2007. [arxiv:/math/0701428]
A symmetry of the string background field equations. T Buscher, Phys. Lett. B. 1941T. Buscher, "A symmetry of the string background field equations". Phys. Lett. B, 194(1):59-62, 1987.
T-Duality from super Lie n-algebra cocycles for super p-branes. D Fiorenza, H Sati, U Schreiber, Adv. Theor. Math. Phys. 225to appearD. Fiorenza, H. Sati, and U. Schreiber, "T-Duality from super Lie n-algebra cocycles for super p-branes". Adv. Theor. Math. Phys., 22(5), 2018, to appear.
Categorical tori. N Ganter, SIGMAarxiv:1406.7046v2N. Ganter, "Categorical tori". SIGMA, 2018. [arxiv:1406.7046v2]
Global aspects of T-duality, gauged sigma models and T-folds. C Hull, arxiv:/hep-th/0604178J. High Energy Phys. 57C. Hull, "Global aspects of T-duality, gauged sigma models and T-folds". J. High Energy Phys., page 057, 2007. [arxiv:/hep-th/0604178]
Semistrict Higher Gauge Theory. B Jurco, C Saemann, M Wolf, arxiv:1403.7185J. High Energy Phys. 0487B. Jurco, C. Saemann, and M. Wolf, "Semistrict Higher Gauge Theory". J. High Energy Phys., 04:087, 2015. [arxiv:1403.7185]
J Lurie, Higher topos theory. Number 109 in Annals of Mathematics Studies. Princeton University PressJ. Lurie, Higher topos theory. Number 109 in Annals of Mathematics Studies. Princeton University Press, 2009.
T-duality for torus bundles with H-fluxes via noncommutative topology. V Mathai, J Rosenberg, arxiv:/hep-th/0401168Commun. Math. Phys. 2533V. Mathai and J. Rosenberg, "T-duality for torus bundles with H-fluxes via noncommutative topology". Commun. Math. Phys., 253(3):705-721, 2005. [arxiv:/hep-th/0401168]
On mysteriously missing T-duals, H-flux and the T-duality group. V Mathai, J Rosenberg, arxiv:/hep-th/0409073Differential geometry and physics. 10World Sci. Publ.V. Mathai and J. Rosenberg, "On mysteriously missing T-duals, H-flux and the T-duality group". In Differential geometry and physics, volume 10 of Nankai Tracts Math., pages 350-358. World Sci. Publ., 2006. [arxiv:/hep-th/0409073]
T-duality for torus bundles with H-fluxes via noncommutative topology. II. The high-dimensional case and the T-duality group. V Mathai, J Rosenberg, arxiv:/hep-th/0508084Adv. Theor. Math. Phys. 101V. Mathai and J. Rosenberg, "T-duality for torus bundles with H-fluxes via noncommutative topology. II. The high-dimensional case and the T-duality group". Adv. Theor. Math. Phys., 10(1):123-158, 2006. [arxiv:/hep-th/0508084]
Equivariance in higher geometry. T Nikolaus, C Schweigert, arxiv:1004.4558Adv. Math. 2264T. Nikolaus and C. Schweigert, "Equivariance in higher geometry". Adv. Math., 226(4):3367-3408, 2011. [arxiv:1004.4558]
Principal infinity-bundles -General theory. T Nikolaus, U Schreiber, D Stevenson, arxiv:1207.0248J. Homotopy Relat. Struct. 104T. Nikolaus, U. Schreiber, and D. Stevenson, "Principal infinity-bundles -General theory". J. Homotopy Relat. Struct., 10(4):749-801, 2015. [arxiv:1207.0248]
Four equivalent versions of non-abelian gerbes. T Nikolaus, K Waldorf, arxiv:1103.4815Pacific J. Math. 2642T. Nikolaus and K. Waldorf, "Four equivalent versions of non-abelian gerbes". Pacific J. Math., 264(2):355-420, 2013. [arxiv:1103.4815]
Lifting problems and transgression for non-abelian gerbes. T Nikolaus, K Waldorf, arxiv:1112.4702Adv. Math. 242T. Nikolaus and K. Waldorf, "Lifting problems and transgression for non-abelian gerbes". Adv. Math., 242:50-79, 2013. [arxiv:1112.4702]
Die lokale Struktur von T-Dualitätstripeln. A Schneider, arxiv:/0712.0260Universität GöttingenPhD thesisA. Schneider, Die lokale Struktur von T-Dualitätstripeln. PhD thesis, Universität Göttingen, 2007. [arxiv:/0712.0260]
Central extensions of smooth 2-groups and a finite-dimensional string 2-group. C Schommer-Pries, arxiv:0911.2483Geom. Topol. 15C. Schommer-Pries, "Central extensions of smooth 2-groups and a finite-dimensional string 2-group". Geom. Topol., 15:609-676, 2011. [arxiv:0911.2483]
The geometry of bundle gerbes. D Stevenson, arxiv:math.DG/0004117University of AdelaidePhD thesisD. Stevenson, The geometry of bundle gerbes. PhD thesis, University of Adelaide, 2000. [arxiv:math.DG/0004117]
More morphisms between bundle gerbes. K Waldorf, arxiv:math.CT/0702652Theory Appl. Categ. 189K. Waldorf, "More morphisms between bundle gerbes". Theory Appl. Categ., 18(9):240-273, 2007. [arxiv:math.CT/0702652]
A construction of string 2-group models using a transgression-regression technique. K Waldorf, arxiv:1201.5052Analysis, Geometry and Quantum Field Theory. C. L. Aldana, M. Braverman, B. Iochum, and C. Neira-JiménezAMS584K. Waldorf, "A construction of string 2-group models using a transgression-regression technique". In C. L. Aldana, M. Braverman, B. Iochum, and C. Neira-Jiménez, editors, Analysis, Geometry and Quantum Field Theory, volume 584 of Contemp. Math., pages 99-115. AMS, 2012. [arxiv:1201.5052]
A global perspective to connections on principal 2-bundles. K Waldorf, arxiv:1608.00401Forum Math. 304K. Waldorf, "A global perspective to connections on principal 2-bundles". Forum Math., 30(4):809-843, 2017. [arxiv:1608.00401]
Principal 2-bundles and their gauge 2-groups. C Wockel, arxiv:0803.3692Forum Math. 23C. Wockel, "Principal 2-bundles and their gauge 2-groups". Forum Math., 23:565-610, 2011. [arxiv:0803.3692]
Thomas Nikolaus a ([email protected]). Thomas Nikolaus a ([email protected])
. D-17487 Greifswald, D-17487 Greifswald
| [] |
[
"Join the Shortest Queue with Many Servers. The Heavy Traffic Asymptotics",
"Join the Shortest Queue with Many Servers. The Heavy Traffic Asymptotics"
] | [
"Patrick Eschenfeldt \nMIT Operations Research Center\nMIT Sloan School of Management and Operations Research Center\n\n",
"David Gamarnik \nMIT Operations Research Center\nMIT Sloan School of Management and Operations Research Center\n\n"
] | [
"MIT Operations Research Center\nMIT Sloan School of Management and Operations Research Center\n",
"MIT Operations Research Center\nMIT Sloan School of Management and Operations Research Center\n"
] | [] | We consider queueing systems with n parallel queues under a Join the Shortest Queue (JSQ) policy in the Halfin-Whitt heavy traffic regime. Because queues with at least two customers form only when all queues have at least one customer and we expect the number of waiting customers to be of the order O( √ n), we restrict our attention to a truncated system that rejects arrivals creating queues longer than two. We provide simulation results supporting this intuition. We use the martingale method to prove that a scaled process counting the number of idle servers and queues of length 2 weakly converges to a two-dimensional reflected Ornstein-Uhlenbeck process. This limiting system is comparable to that of the traditional Halfin-Whitt model, but there are key differences in the queueing behavior of the JSQ model. In particular, it is possible for the system to have both idle servers and waiting customers at the same time. | 10.1287/moor.2017.0887 | [
"https://arxiv.org/pdf/1502.00999v2.pdf"
] | 52,274,612 | 1502.00999 | 665cbaf533fb14f2392d7bd42f442490b74df206 |
Join the Shortest Queue with Many Servers. The Heavy Traffic Asymptotics
Patrick Eschenfeldt
MIT Operations Research Center
MIT Sloan School of Management and Operations Research Center
David Gamarnik
MIT Operations Research Center
MIT Sloan School of Management and Operations Research Center
Join the Shortest Queue with Many Servers. The Heavy Traffic Asymptotics
We consider queueing systems with n parallel queues under a Join the Shortest Queue (JSQ) policy in the Halfin-Whitt heavy traffic regime. Because queues with at least two customers form only when all queues have at least one customer and we expect the number of waiting customers to be of the order O( √ n), we restrict our attention to a truncated system that rejects arrivals creating queues longer than two. We provide simulation results supporting this intuition. We use the martingale method to prove that a scaled process counting the number of idle servers and queues of length 2 weakly converges to a two-dimensional reflected Ornstein-Uhlenbeck process. This limiting system is comparable to that of the traditional Halfin-Whitt model, but there are key differences in the queueing behavior of the JSQ model. In particular, it is possible for the system to have both idle servers and waiting customers at the same time.
1 Introduction.
In this paper we consider queueing systems with many parallel servers under a heavy-traffic regime where the workload scales with the number of servers. Such systems are well understood when a global queue is maintained [10], but in many practical situations it may be advantageous to instead maintain parallel queues. Even if a global queue is itself not problematic, it may be necessary to keep queued customers close to the server who will eventually serve them. Consider for example an airport setting with arriving passengers who need to have their passports checked with one of a large number of passport controllers. In this situation having only a global queue can lead to significant walk times between the front of the queue and the server, leaving servers idle while they wait for their next customer. This idle time can be avoided by routing customers to individual queues for each server before earlier customers finish service.
At the same time, a parallel scheme will necessarily allow servers to idle if their own queue is empty, even if customers are waiting in another queue, thus sacrificing some efficiency. We will study a particular example of a parallel system and demonstrate that under certain (Halfin-Whitt) heavy traffic conditions for this system, the number of idle servers is on the order O ( √ n), where n is the number of servers.
Specifically, this result will be for a parallel queueing system in which each arriving customer is immediately routed to the queue containing the smallest number of customers, namely the Join the Shortest Queue (JSQ) policy. We consider the system in heavy traffic by allowing the arrival rate λ n to depend on n. In particular we consider the Halfin-Whitt regime [10], defined by letting the quantity (1−λ n ) √ n having a non-degenerate limit, which we will denote β > 0. Note that JSQ is a logical first step for understanding the tradeoffs involved in maintaining parallel queues, because Winston [20] proved that among policies immediately assigning customers to one of n < ∞ parallel queues, JSQ is optimal in the case of Poisson arrivals and exponential service times. That is, it maximizes, with respect to stochastic order, the number of customers served in a given time interval. Weber [18] extended this result to the more general class of service times with non-decreasing hazard rate, with no assumptions on the arrival process. We will consider Poisson arrivals and exponential service times, and denote this system M/M/n-JSQ, distinguishing it from the traditional M/M/n system which maintains a global queue.
Our main result describes the behavior of processes counting the number of idle servers and queues with a customer waiting to enter service. These processes will be analyzed in a truncated variant of the M/M/n-JSQ system in which no queue may have length longer than 2, and thus each server can have at most one customer waiting to enter service. We use this simplified system because longer queues form only when all queues are length 2. We conjecture that in steady state the number of queues with length 2 will be on the order O ( √ n), so with high probability longer queues will not exist. While we are not able to verify this conjecture, our intuition is supported by simulation results. In particular, we simulate the system which starts with no queues of length 3 or longer and a small number of queues with length 2. In these simulations, longer queues do not appear. See Figure 1 for a simulated sample path. Figure 1: A sample path of the M/M/n-JSQ system scaled by √ n. Simulated with n = 10 5 , β = 2.0, started with 3β √ n idle servers, 3β √ n queues of length two and the remaining servers with a single customer in service. Plot shows idle servers (−X 1 ) and queues of length two (X 2 ) as a function of time.
Thus we believe that the truncated system, in which arrivals are rejected if they would create a queue of length 3, will be an accurate representation of the number of idle servers and waiting customers in the original system. Therefore in this paper we consider a sequence of truncated M/M/n-JSQ systems in the Halfin-Whitt regime and prove that this sequence converges weakly to a diffusion process. This diffusion process, which is a multidimensional reflected Ornstein-Uhlenbeck process, will be defined in terms of a stochastic integral equation which we prove has a unique solution. This existence and uniqueness result is stated in Theorem 1 and the weak convergence result is stated in Theorem 2, which is our main result.
A feature of interest in this queueing system is the waiting time experienced by arriving customers. We conjecture that in steady state the waiting time W of a typical customer behaves as follows: P(W = 0) = 1 − O(1/ √ n), and conditioned on W > 0 (which thus occurs with probability O(1/ √ n), it is exponential with parameter 1. Our heuristic reasoning is as follows. Because customers do not jockey between queues, when the customer joins the server with another customer in service, the waiting time of this arriving customer is at least of constant order. Notice, on the other hand, that an arriving customer immediately enters service if there are any idle servers in the system. In this case the incurred waiting time is zero. Thus the expected waiting time can be characterized by the fraction of arrivals which occur when there are no idle servers. In the limiting system this will be the local time at zero of a reflected diffusion process, which is zero in the limit. Based on that we conjecture that in steady state the probability of a customer having to wait will be of the order O (1/ √ n), in which case the conditioned waiting time equals unity, because it corresponds to time to finish servicing the preceding customer (service times are exponential and normalized to unity). Simulation results support this intuition. In particular, the fraction of arrivals which find no idle server in simulated sample paths multiplied by √ n tends to a constant as n increases. This conjecture if true would imply that the expected waiting time of the truncated JSQ system is on the same order as that of the M/M/n system, so the inefficiency of allowing servers to idle while customers wait does not result in an order of magnitude change in the expected waiting time.
Related literature.
The JSQ model was initially studied in the special case of 2 queues by Haight [8]. Kingman [12] proved stability results along with considering the stationary distribution of the system, and Flatto and McKean [5] also examine the stationary distribution. Further work on the n = 2 case includes bounds on the distribution of the number of people in the system by Halfin [9].
Foschini and Salz [6] consider diffusion limits for the heavy traffic case of the M/M/2-JSQ system, first proving that the queue-length processes for the two queues are identical in the limit and then deriving the limiting distribution. The limiting behavior of the waiting time is the same as the standard M/M/2 system in heavy traffic. Their results extend to the case of k parallel queues, but they do not consider the case where the number of queues grows as the traffic intensity increases. Zhang and Wang [11] and Zhang and Hsu [21] look at a similar problem but drop the assumption of Poisson arrivals and exponential service times, deriving functional central limit theorems for the heavy traffic JSQ system with s servers.
In contrast to much early work on JSQ systems which consider only a fixed number of servers, we are interested in the asymptotics as the number of servers n increases. Our first observation is that for fixed λ < 1 as n increases the probability of any customer arriving to find all servers busy will decrease to zero. In this case the JSQ nature of the system becomes irrelevant as customers will be assigned to an idle server immediately upon arrival. In particular we see that the limiting behavior of the system will essentially be that of the well known M/M/∞ system, and thus it is of interest to consider this model in heavy traffic with λ approaching unity. There has been some work on models similar to ours, most notably Tezcan [16], who considers a variant of the JSQ system with multiple pools of servers who each have their own queue. He uses a state-space collapse argument based on a framework of Dai and Tezcan [2] to prove diffusion limits under the Halfin-Whitt heavy traffic regime. In this case that regime has the number of servers and traffic intensity increasing together in the limit, but the number of pools of servers is fixed so the number of queues is also fixed. Therefore our model is similar to Tezcan's but is not a special case of it. The state-space collapse argument implies that in the limit the system can be fully described by the total number of people in the system (rather than the queue lengths in the individual pools) and the diffusion limit of that process is very similar to the original Halfin and Whitt result [10].
Another branch of analysis of JSQ-like queueing systems has focused on the "supermarket model" in which arriving customers join the shortest queue from among d randomly selected queues rather than from the entire system. It was proved independently by Mitzenmacher [13] and Vvedenskaya, Dobrushin, and Karpelevich [17] that this system achieves an exponential improvement in expected waiting time over a system with n independent M/M/1 queues. Versions of this system where d depends on n are particularly closely related to our JSQ model, which essentially sets d = n. Brightwell and Luczak [1] give a set of d and λ values depending on n for which they prove the steady-state system is usually in a particular state with most queues having the same (known) length. Their conditions require (1 − λ) −1 > d, which excludes the d = n, (1 − λ) √ n → β case considered in this paper. Dieker and Suk [4] prove fluid and diffusion limits for queue length processes when when d increases to infinity at a rate slower than n and with fixed λ < 1.
The remainder of the paper is laid out as follows: Section 2 will define the model and state our main result. Section 3 will prove Theorem 1, verifying that the integral representation of the limiting system is well defined. This result will also be the key to proving convergence via a continuous mapping theorem (CMT) argument. Section 4 will construct a representation of the system as a combination of martingales and reflecting processes. In Section 5 we will establish the convergence properties of these martingales, and then apply the CMT to translate the convergence of martingales to convergence of the scaled queue length processes. This section will conclude our proof of Theorem 2. We will conclude in Section 6 with a brief discussion of the implications of Theorem 2 and possible extensions.
We use ⇒ to denote weak convergence, 1{A} to denote the indicator function for the event A, (x) + = max(x, 0). We let R + = R + ∪ {∞} represent the extended positive real line. Most processes in this paper will live in the space D = D([0, ∞), R) of right continuous functions with left limits mapping [0, ∞) into R. We also consider D k = D([0, ∞), R k ) for k ≥ 2, which we will treat as the product space D × D × · · · × D (see, e.g., [19] §3.3). We will denote the uniform norm ||x|| t = sup 0≤s≤t |x(s)| for x ∈ D and the max norm
||(x 1 , . . . , x k )|| t = max 1≤i≤k ||x i || t for x ∈ D k .
Similarly we will use the max norm
|b| = max 1≤i≤k |b i | for b ∈ R k .
2 The model, simulations, and the main result.
We consider a queueing system with n servers where each server maintains a unique queue, with service proceeding according to a first-in-first-out discipline. Service time is exponentially distributed at each server, with the rate fixed at 1. Arrivals occur in a single stream, as a Poisson process with rate λ n n, where 0 < λ n < 1 and
lim n→∞ √ n(1 − λ n ) = β (2.1)
for fixed β > 0. Upon arrival, each customer is routed to the server with the shortest queue. In the event of a tie, one of the options is selected uniformly at random. The state of the system will be represented primarily via the process Q n (t) = (Q n 1 (t), Q n 2 (t), . . .), with Q n k (t) representing the number of queues with at least k customers (including any customer in service) at time t ≥ 0. We note that for the system as described we have
n ≥ Q n 1 (t) ≥ Q n 2 (t) ≥ · · · ≥ 0 ∀t ≥ 0, (2.2)
and that we can recover the number of queues with exactly k customers in service via the quantity Q n k (t) − Q n k+1 (t), including the number of idle servers n − Q n 1 (t). To state our weak convergence results, we also introduce a scaled version X n (t) of this process defined as
X n 1 (t) = Q n 1 (t) − n √ n and X n k (t) = Q n k (t) √ n . (2.
3)
The k = 1 case is treated differently because the number of queues with length 1 behaves differently than the number of queues of all larger lengths, as we are about to see. We expect, both through an intuitive consideration of the system and through simulations, that outside of the effect of the starting state the relevant behavior will be captured by only considering queues of length at most two. Specifically, we expect the number of idle servers and number of queues with length two is order Θ ( √ n) while queues with length three or more will be present only if they are present in the initial condition. This is because such long queues only form when all n queues have length 2, but the number of queues with length 2 is Θ ( √ n). See Figure 2 to see a sample path of an untruncated system which starts with order Θ ( √ n) queues of length 4. Note that the number of queues length 3 and length 4 decrease monotonically. Considering all possible queue lengths k ≥ 3 complicates notation and technical aspects of proofs, so we will exclusively consider a modification of the system in which long queues are excluded by design. This will allow us to demonstrate weak convergence to a particular diffusion process.
Formally, any arrivals while all servers have a queue of length two are rejected. In this modified system we need only consider Q n 1 and Q n 2 (and their scaled counterparts). We note that (Q n 1 , Q n 2 ) is a member of the function space D 2 . Before we state our primary result, we will state the theorem that demonstrates the existence and uniqueness of the diffusion limit we will prove.
Our diffusion limit will be the solution to a system of integral equations, so we first prove that the system has a unique solution. Furthermore, we prove that the system defines a continuous map from R + × R 2 × D 2 to D 2 × D 2 with respect to appropriate topologies. This continuity, along with the further fact that the function maps continuous functions to continuous functions, allows us to use the CMT to prove weak convergence once we show the weak convergence of the arguments. Because our limiting system has continuous sample paths, we equip D with the topology of uniform convergence over bounded intervals.
Theorem 1. Given B ∈ R + , b ∈ R 2 , and y ∈ D 2 , consider x 1 (t) = b 1 + y 1 (t) + t 0 (−x 1 (s) + x 2 (s))ds − u 1 (t), (2.4) x 2 (t) = b 2 + y 2 (t) + t 0 (−x 2 (s))ds + u 1 (t) − u 2 (t), (2.5) x 1 (t) ≤ 0, 0 ≤ x 2 (t) ≤ B, t ≥ 0, (2.6)
with u 1 and u 2 nondecreasing nonnegative functions in D such that
∞ 0 1{x 1 (t) < 0}du 1 (t) = 0, ∞ 0 1{x 2 (t) < B}du 2 (t) = 0. Then (2.4)-(2.6) has a unique solution (x, u) ∈ D 2 × D 2 so that there is a well defined function (f, g) : R + × R 2 × D 2 → D 2 × D 2 mapping (B, b, y) into x = f (B, b, y) and u = g(B, b, y). Furthermore, the function (f, g) is continuous on R + × R 2 × D 2 .
Finally, if y is continuous, then so are x and u.
We will prove this theorem in Section 3. One implication of Theorem 1 is that the limiting system we find in our main result is well defined because it is an application of the function (f, g) with specific arguments (B, b, y). Note in particular that we will have B = ∞, which implies u 2 = 0. Our main result is the following: Theorem 2. In the sequence of truncated JSQ models described above, suppose that
X n k (0) ⇒ X k (0) in R as n → ∞, k = 1, 2. (2.7) Then X n k ⇒ X k in D as n → ∞, k = 1, 2, where X 1 ≤ 0 and X 2 ≥ 0 are unique solutions in D for the stochastic integral equations X 1 (t) = X 1 (0) + √ 2W (t) − βt + t 0 (−X 1 (s) + X 2 (s)) ds − U 1 (t), (2.8) X 2 (t) = X 2 (0) + U 1 (t) + t 0 −X 2 (s)ds, (2.9)
for t ≥ 0 where W is a standard Brownian motion and U 1 is the unique nondecreasing nonnegative process in D satisfying
∞ 0 1{X 1 (t) < 0}dU 1 (t) = 0.
(2.10)
We note that condition (2.7) does place significant but not unreasonable restrictions on the starting state of the finite systems Q n . In particular, Q n 1 (0) − n = O ( √ n) so the number of customers initially in service must be sufficiently near n. Similarly, (2.7) requires Q n
2 (0) = O ( √ n).
3 Integral representation.
We will now prove Theorem 1, showing that the representation of the limiting system in Theorem 2 is a valid and unique representation. We will also show that it defines a continuous map from
R + ×R 2 ×D 2 to D 2 ×D 2 .
The continuity of the map in the topology of uniform convergence over bounded intervals will allow us to use the continuous mapping theorem (CMT) to demonstrate the convergence X n k ⇒ X k once we write X n k in the appropriate integral form. Note that by using R + in the domain of this map we allow the upper barrier B for the function x 2 to take the value ∞, which corresponds to there being no upper barrier on the √ n scale.
The reflection map.
In several places we will make use of the well known one-dimensional reflection map for an upper barrier. Given upper barrier κ ∈ R + , we let (φ κ , ψ κ ) : D → D 2 be the one-sided reflection map with upper barrier at κ (see, e.g., [19] §5.2 and §13.5). In particular for x ∈ D with x(0) ≤ κ we have z = ψ κ (y) ≥ 0, z nondecreasing,
x = φ κ (y) = y − z ≤ κ, and ∞ 0 1{x < κ}dz = 0.
Recall that these functions can be defined explicitly by
ψ κ (x)(t) = sup 0≤s≤t (x(s) − κ) + (3.1) and φ κ (x)(t) = x(t) − ψ κ (x)(t). (3.2)
We will also make use of a slight variant of the usual Lipschitz condition for these functions to allow for different values of κ. In particular, for x, x ∈ D, κ, κ ∈ R, and t ≥ 0 we have
||ψ κ (x) − ψ κ (x )|| t ≤ ||x − x || t + |κ − κ |, (3.3) ||φ κ (x) − φ κ (x )|| t ≤ 2 ||x − x || t + |κ − κ |. (3.4)
These follow straightforwardly from (3.1) and (3.2). Note that for κ = κ we recover the usual Lipschitz constants of 1 for ψ κ and 2 for φ κ . We also define a trivial reflection map for κ = ∞ by letting (φ ∞ , ψ ∞ ) = (e, 0) where e is the identity map. That is, the reflection map leaves the argument unchanged and the regulator is identically zero. We prove the following:
Lemma 1. The function (φ, ψ) : R + × D → D 2 defined by (3.1)-(3.
2) for finite κ and by (φ ∞ , ψ ∞ ) = (e, 0) for κ = ∞ is continuous with respect to the product topology when R + is equipped with the order topology and D is equipped with the topology of uniform convergence over bounded intervals.
Proof. By (3.3)-(3.4) the function is continuous at any finite κ ∈ R + . For x ∈ D and x κ ∈ D such that x κ → x as κ → ∞,
lim κ→∞ ||ψ κ (x κ )|| t = lim κ→∞ sup 0≤s≤t |ψ κ (x κ )| = lim κ→∞ sup 0≤s≤t (x κ (s) − κ) + = sup 0≤s≤t lim κ→∞ (x κ (s) − κ) + = 0,
where we have made use of the fact that ||x|| t < ∞. Therefore ψ κ (x κ ) → ψ ∞ (x) and by (3.2) we conclude φ κ (x κ ) → φ ∞ (x). Thus the function is continuous at κ = ∞, completing the proof.
With these facts about the reflection map in hand, we will now prove a result similar to Theorem 1 for a related system:
Lemma 2. Given B ∈ R + , b ∈ R 2 and y ∈ D 2 , consider w 1 (t) = b 1 + y 1 (t) + t 0 (−φ 0 (w 1 (s)) + φ B (w 2 (s))) ds, (3.5) w 2 (t) = b 2 + y 2 (t) + ψ 0 (w 1 (t)) + t 0 (−φ B (w 2 (s))) ds ≥ 0. (3.6)
Then (3.5)-(3.6) has a unique solution w ∈ D 2 so that there is a well defined function ξ : ξ(B, b, y). Furthermore, the function ξ is continuous with respect to the product topology, with R + equipped with the order topology and D equipped with the topology of uniform convergence over bounded intervals. Finally, if y is continuous, then so is w.
R + × R 2 × D 2 → D 2 mapping (B, b, y) into w =
Before proceeding with the proof we introduce a version of Gronwall's inequality first proved by Greene [7] and proved in the form we use by Das [3]:
Lemma 3 (Gronwall's inequality). Let K 1 and K 2 be nonnegative constants, let h i be real constants, and let f, g be continuous nonnegative functions for all t ≥ 0 such that
f (t) ≤ K 1 + h 1 t 0 f (s)ds + h 2 t 0 g(s)ds, g(t) ≤ K 2 + h 3 t 0 f (s)ds + h 4 t 0 g(s)ds for all t ≥ 0. Then f (t) ≤ M e ht and g(t) ≤ M e ht for all t ≥ 0 where M = K 1 + K 2 and h = max{h 1 + h 3 , h 2 + h 4 }. In particular, if K 1 , K 2 = 0, then f (t), g(t) = 0 for all t.
Proof of Lemma 2. We will show existence via a contraction mapping argument. First we will show that for t ≥ 0 there exists a solutionw = (w 1 ,w 2 ) to the system of integral equations
w 1 (t) = b 1 + y 1 (t) + t 0 −φ 0 (w 1 (s)) + φ B w 2 (s) + ψ 0 (w 1 (s)) ds, (3.7) w 2 (t) = b 2 + y 2 (t) + t 0 −φ B w 2 (s) + ψ 0 (w 1 (s)) ds ≥ 0. (3.8)
Once we have such a solution, it follows immediately that w = (w 1 , w 2 ) = (w 1 ,w 2 + ψ 0 (w 1 )) is a solution to (3.5)-(3.6). We first show that the map defined by the right hand side of (3.7)-(3.8) is a contraction for small enough t. We define T :
D 2 → D 2 by T (w) 1 (t) = b 1 + y 1 (t) + t 0 −φ 0 (w 1 (s)) + φ B w 2 (s) + ψ 0 (w 1 (s)) ds, (3.9) T (w) 2 (t) = b 2 + y 2 (t) + t 0 −φ B w 2 (s) + ψ 0 (w 1 (s)) ds. (3.10) Forw,ṽ ∈ D 2 we have ||T (w) 1 − T (ṽ) 1 || t ≤ t 0 ||−φ 0 (w 1 ) + φ 0 (ṽ 1 )|| s ds + t 0 ||φ B (w 2 + ψ 0 (w 1 )) − φ B (ṽ 2 + ψ 0 (ṽ 1 ))|| s ds ≤ 2 t 0 ||w 1 −ṽ 1 || s ds + 2 t 0 ||w 2 + ψ 0 (w 1 ) −ṽ 2 − ψ 0 (ṽ 1 )|| s ds ≤ 2t ||w 1 −ṽ 1 || t + 2t ||w 2 −ṽ 2 || t + t 0 ||w 1 −ṽ 1 || s ds ≤ 2t ||w 1 −ṽ 1 || t + 2t ||w 2 −ṽ 2 || t + t ||w 1 −ṽ 1 || t ≤ 5t ||w −ṽ|| t and ||w 2 −ṽ 2 || t ≤ t 0 ||−φ B (w 2 + ψ 0 (w 1 )) + φ B (ṽ 2 + ψ 0 (ṽ 1 ))|| s ds ≤ 2t ||w 2 + ψ 0 (w 1 ) −ṽ 2 − ψ 0 (ṽ 1 )|| t ≤ 2t ||w 2 −ṽ 2 || t + t ||w 1 −ṽ 1 || t ≤ 3t ||w −ṽ|| t .
We therefore conclude that
||T (w) − T (ṽ)|| t ≤ 5t ||w −ṽ|| t ,
so for t 0 < 1 5 , T is a contraction on D([0, t 0 ], R 2 ). Therefore by the contraction mapping principle (see, e.g., [15, p.220]), T has a unique fixed pointw on D([0, t 0 ], R 2 ) such that T (w) =w. This fixed point solves (3.7)-(3.8) for t ∈ [0, t 0 ]. Now we extend the fixed point argument to t ∈ [t 0 , 2t 0 ], [2t 0 , 3t 0 ], . . . and repeat to find a solutionw to (3.7)-(3.8) for t ≥ 0. As noted above, this provides a solution w to (3.5)-(3.6).
To prove uniqueness of this solution, suppose w and w are two solutions to (3.5)-(3.6). We consider
||w 1 − w 1 || t ≤ t 0 ||−φ 0 (w 1 ) + φ 0 (w 1 ) + φ B (w 2 ) − φ B (w 2 )|| s ds ≤ 2 t 0 (||w 1 − w 1 || s + ||w 2 − w 2 || s ) ds (3.11)
and
||w 2 − w 2 || t ≤ ||ψ 0 (w 1 ) − ψ 0 (w 1 )|| t + t 0 ||φ B (w 2 ) − φ B (w 2 )|| s ds ≤ ||w 1 − w 1 || t + 2 t 0 ||w 2 − w 2 || s ds. (3.12)
To match the form of Gronwall's inequality (Lemma 3) we rewrite (3.12) as
||w 2 − w 2 || t − ||w 1 − w 1 || t ≤ 2 t 0 ||w 2 − w 2 || s ds
and note that the right hand side is nonnegative so the inequality remains true as
(||w 2 − w 2 || t − ||w 1 − w 1 || t ) + ≤ 2 t 0 ||w 2 − w 2 || s ds (3.13)
We now define
u 1 (t) = ||w 1 − w 1 || t , u 2 (t) = (||w 2 − w 2 || t − ||w 1 − w 1 || t ) + and note ||w 2 − w 2 || s ≤ u 2 (s) + u 1 (s) s ≥ 0. (3.14)
Then (3.11), (3.13), and (3.14) imply
u 1 (t) ≤ 4 t 0 u 1 (s)ds + 2 t 0 u 2 (s)ds,||w 1 − w 1 || t = ||w 2 − w 2 || t = 0
for all t ≥ 0 and therefore the solution w is unique. We now establish the continuity of ξ. Suppose (B n , b n , y n ) → (B, b, y) as n → ∞.
Fix > 0 and suppose w n and w satisfy (3.5)-(3.6) for (B n , b n , y n ) and (B, b, y), respectively. Choose N such that for all n ≥ N |b n − b| + ||y n − y|| t + ||φ B n (w 2 ) − φ B (w 2 )|| t < δ for some δ > 0 which is yet to be determined. Note that such an N exists by Lemma 1 and the assumption B n → B. We have
||w n 1 − w 1 || t ≤ |b n − b| + ||y n − y|| t + t 0 ||−φ 0 (w n 1 ) + φ 0 (w 1 ) + φ B n (w n 2 ) − φ B (w 2 )|| s ds ≤ δ + t 0 2 ||w n 1 − w 1 || s + ||φ B n (w n 2 ) − φ B n (w 2 )|| s + ||φ B n (w 2 ) − φ B (w 2 )|| s ds ≤ δ + t 0 (2 ||w n 1 − w 1 || s + 2 ||w n 2 − w 2 || s + δ) ds ≤ δ(1 + t) + 2 t 0 (||w n 1 − w 1 || s + ||w n 2 − w 2 || s ) ds (3.15)
and
||w n 2 − w 2 || t ≤ δ(1 + t) + ||w n 1 − w 1 || t + 2 t 0 ||w n 2 − w 2 || s ds. (3.16)
As in the uniqueness argument above, we will apply Gronwall's inequality, with functions
u 1 (t) = ||w n 1 − w 1 || t and u 2 (t) = (||w n 2 − w 2 || t − ||w n 1 − w 1 || t ) + .
Then we have
u 1 (t) ≤ δ(1 + t) + 4 t 0 u 1 (s)ds + 2 t 0 u 2 (s)ds, u 2 (t) ≤ δ(1 + t) + 2 t 0 u 1 (s)ds + 2 t 0 u 2 (s)ds,
so Gronwall's inequality implies u 1 (t) ≤ 2δ(1 + t)e 6t and u 2 (t) ≤ 2δ(1 + t)e 6t and we have ||w n 1 − w 1 || t ≤ 2δ(1 + t)e 6t and ||w n 2 − w 2 || t ≤ 4δ(1 + t)e 6t .
We choose δ = 1 4+4t e −6t to establish the desired continuity. For the proof of continuity of w we note
|w 1 (t + s) − w 1 (t)| ≤ |y 1 (t + s) − y 1 (t)| + t+s t |φ B (w 2 (z)) − φ 0 (w 1 (z))|dz and |w 2 (t + s) − w 2 (t)| ≤ |y 2 (t + s) − y 2 (t)| + |w 1 (t + s) − w 1 (t)| + t+s t |φ B (w 2 (z))|dz
The boundedness of w 1 and w 2 proved in Lemma 7 imply that w 1 and w 2 are continuous if y 1 and y 2 are continuous.
We are now prepared to prove Theorem 1.
Proof of Theorem 1. Our key insight is to see that a solution is found by setting x 1 = φ 0 (w 1 ), u 1 = ψ 0 (w 1 ), x 2 = φ B (w 2 ), and u 2 = ψ B (w 2 ) where (w 1 , w 2 ) is the unique solution defined by Lemma 2. To see that it is unique, note that the conditions on u 1 and u 2 imply that they can be written as ψ 0 (z 1 ) and ψ B (z 2 ) for some functions z 1 , z 2 ∈ D. Then x 1 and x 2 are φ 0 (z 1 ) and φ B (z 2 ) for the same z 1 and z 2 . Then (2.4)-(2.6) imply that z = (z 1 , z 2 ) must be a solution of (3.5)-(3.6). By Lemma 2 this solution is unique. In particular, this solution is
x 1 = f 1 (b, y) = (φ 0 • ξ 1 )(b, y), u 1 = g 1 (b, y) = (ψ 0 • ξ 1 )(b, y), x 2 = f 2 (b, y) = (φ B • ξ 2 )(b, y), u 2 = g 2 (b, y) = (ψ B • ξ 2 )(b, y).
The reflection maps (φ 0 , ψ 0 ) and (φ B , ψ B ) are continuous in the uniform topology and also preserve continuity. Since ξ also has these properties by Lemma 2, we conclude that (f, g) are continuous and preserve continuity.
Martingale representation.
We will now construct the process X n (t) and show that it has the integral form in Theorem 1. Our representation will be similar to the first martingale representation of [14]; in particular it will rely upon random time changes of rate-1 Poisson processes.
Random time change.
We let A, D 1 , and D 2 be rate-1 Poisson processes and write
Q n 1 (t) = Q n 1 (0) + A (λ n nt) − D 1 t 0 (Q n 1 (s) − Q n 2 (s)) ds − U n 1 (t), (4.1) Q n 2 (t) = Q n 2 (0) + U n 1 (t) − D 2 t 0 Q n 2 (s)ds − U n 2 (t), (4.2)
where U n 1 (t) is the number of arrivals in [0, t] when every server has at least one customer, and U n 2 (t) is the number of arrivals in [0, t] when every server has at least one customer and exactly B √ n servers have two customers. Note that for this definition to make sense we must have B ≤ √ n. Formally, we define
U n 1 (t) = t 0 1 {Q n 1 (s) = n} dA(λ n ns), (4.3) U n 2 (t) = t 0 1 Q n 1 (s) = n, Q n 2 (s) = B √ n dA(λ n ns). (4.4)
We can understand (4.1) term-by-term: first we record the initial state of the system with Q n 1 (0), then arrivals are counted at their full rate λ n n. The D 1 term represents departures, which occur as a Poisson process with rate equal to the number of customers in service. Since Q 1 includes queues of length 1 and length 2, however, Q 1 will only decrease when a customer departs a queue and leaves the server empty. Therefore the instantaneous rate at time s in the D 1 term is Q n 1 (s) − Q n 2 (s), the number of queues of length exactly 1 at time s. Through the first three terms of (4.1) we have recorded what the value of Q 1 would be if it were not constrained to be at most n, so the final term will represent this barrier. The process U n 1 records any arrival which would increase Q 1 above n, balancing the overcounting we get from A(λ n nt).
We can understand (4.2) in much the same way, with the key difference being in the arrival process. Since arriving customers will always join the shortest available queue, the number of length 2 queues will increase only when all servers are busy. Such arrivals are exactly recorded by U n 1 , so this will be the process we use to record potential increases to Q n 2 . The process U n 2 provides the upper barrier B √ n on Q n 2 . We note that this regulating process must incorporate information about Q n 1 as well as Q n 2 because of the way increases to Q n 2 rely on the value of Q n 1 . Therefore the arrivals that need to be "rejected" from the Q n 2 process are precisely those that occur when Q n 1 (s) = n and Q n 2 (s) = B √ n, since the uncompensated result of such an arrival would be Q n 2 (s) > B √ n. We will see in Section 4.2, however, that this dependence on Q n 1 can be hidden through properties of the one-dimensional reflection map.
As in [14, Lemma 2.1], we can verify that this construction is well defined and generates an element of D 2 by conditioning on the starting state Q n (0) and processes A, D 1 , D 2 then constructing recursively.
Martingales.
Because our approach to (4.1)-(4.2) will be to apply the functional central limit theorem (FCLT) for Poisson processes, we will now rewrite the time changes of Poisson processes as time changes of scaled Poisson processes. To that end, we define scaled martingales
M n,1 1 (t) = 1 √ n A (λ n nt) − λ n √ nt, M n,2 1 (t) = 1 √ n D 1 t 0 (Q n 1 (s) − Q n 2 (s)) ds − 1 √ n t 0 (Q n 1 (s) − Q n 2 (s)) ds, M n,2 2 (t) = 1 √ n D 2 t 0 Q n 2 (s)ds − 1 √ n t 0 Q n 2 (s)ds,
and also define
V n 1 (t) = U n 1 (t) √ n and V n 2 (t) = U n 2 (t) √ n .
Then we have
X n 1 (t) = Q n 1 (t) − n √ n = Q n 1 (0) − n √ n + 1 √ n A (λ n nt) − 1 √ n D 1 t 0 (Q n 1 (s) − Q n 2 (s))ds − U n 1 (t) √ n = X n 1 (0) + M n,1 1 (t) + λ n √ nt − V n 1 (t) − M n,2 1 (t) − 1 √ n t 0 (Q n 1 (s) − Q n 2 (s)) ds = X n 1 (0) + M n,1 1 (t) − M n,2 1 (t) + λ n √ nt − V n 1 (t) − √ nt − t 0 Q n 1 (s) − n √ n − Q n 2 (s) √ n ds = X n 1 (0) + M n,1 1 (t) − M n,2 1 (t) − (1 − λ n ) √ nt (4.5) − t 0 (X n 1 (s) − X n 2 (s))ds − V n 1 (t) and X n 2 (t) = X n 2 (0) + V n 1 (t) − M n,2 2 (t) − t 0 X n 2 (s)ds − V n 2 (t). (4.6)
Via an argument exactly analogous to that of in §7.1 of [14] leading to Theorem 7.2 we obtain that M n, 1 1 , M n,2 1 , and M n,2 2 are square-integrable martingales with respect to an appropriate filtration. We note for later use that this argument also supplies the predictable quadratic variations M n,1
1 (t) = λ n t,
(4.7)
M n,2
1 (t) = 1 n t 0 (Q n 1 (s) − Q n 2 (s))ds, (4.8) M n,2 2 (t) = 1 n t 0 Q n 2 (s)ds. (4.9)
At this point we can also note that (4.5)-(4.6) put X n (t) in the integral form of Theorem 1. The only potential differences are the processes V n , which are not described in exactly the same way. We see, however, that by (4.3) we have Because our definition of U n 2 uses information about Q n 1 this is the natural way to write a condition about the relationship between increases in U n 2 and the values of Q n . We notice however that U n 2 can increase only when we have both Q n 1 (s) = n and Q n 2 (s) = B √ n. In particular this means that it cannot increase when Q n 2 (s) < B √ n regardless of the value of Q n 1 (s), so we also have the condition
0 = ∞ 0 1{Q n 1 (s) < n}dU n 1 (s) = ∞ 0 1{X n 1 (s) < 0}dU n 1 (s) = ∞ 0 1{X n 1 (s) < 0}dV n 1 (s).0 = ∞ 0 1{Q n 2 (s) < B √ n}dU n 2 (t) = ∞ 0 1{X n 2 (s) < B}dV n 2 (t). (4.11)
Note that here we are "hiding" the dependence of U n 2 on Q n 1 . Intuitively we are able to do this because the dependence on Q n 1 comes entirely through the fact that increases to Q n 2 are from U n 1 , which is captured in the integral representation itself. U n 2 is acting only as a regulator for the reflecting upper barrier on Q n 2 , so it can be described as a simple reflecting barrier. Based on (4.5)-(4.6) and (4.10)-(4.11) we have X n and X represented according to Theorem 1 so to apply the CMT it remains to prove the convergence of the martingale pieces of (4.5)-(4.6). where W 1 and W 2 are independent standard Brownian motions.
To prove this lemma we will rely upon the CMT and the FCLT for Poisson processes ( [14,Theorem 4.2]), which we state for our purposes as This result is a special case of the FCLT for a renewal process, which is discussed in [19]. More directly it can be derived via an application of the martingale FCLT, as discussed in §8 of [14].
To apply Lemma 5 we will define random and deterministic time changes such that the martingales M n,i k can be written as a composition of a time change and the scaled Poisson processes M C,n . Specifically, let To apply the CMT with the composition map •, we need to determine the limits of the time changes (5.2)-(5.4). First we note that (2.1) implies λ n → 1, which in turn implies Φ A,n ⇒ e as n → ∞, (5.5) where e is the identity function in D.
Φ A,n (t) = λ n t, (5.2) Φ D1,n (t) = 1 n t 0 Q n 1 (s)ds − 1 n t 0 Q n 2 (s)ds,(5.
Next we note that the second term of (5.3) is precisely Φ D2,n , so
Φ D1,n ⇒ f − g as n → ∞,
where g is the limit of Φ D2,n and f is the limit ofΦ D1,n with Φ D1,n (t) = 1 n t 0 Q n 1 (s)ds.
To find f and g we will first show fluid limits for Q n 1 and Q n 2 .
Lemma 6. Let Ψ n i for i = 1, 2 be defined by
Ψ n i (t) = Q n i (t) n , t ≥ 0. Then Ψ n 1 ⇒ ω and Ψ n 2 ⇒ 0 as n → ∞ (5.6)
where ω(t) = 1 for t ≥ 0.
The proof of this lemma is found in Section 5.1. To use Lemma 6 we define a continuous function h : D → D by = (W 1 , W 2 , 0).
Fluid limit.
We will prove Lemma 6 by showing that X n 1 is stochastically bounded in D. Namely, we will prove that the sequence of real-valued random variables ||X n 1 || t is tight for every t > 0. For a more complete discussion of stochastic boundedness as we use it here see §5 of [14].
The stochastic boundedness of X n 1 will follow from the stochastic boundedness of M n,1 1 , M n,2 1 , and M n,2 2 . To see this, we prove Lemma 7. Given (B n , X n i (0), Y n i ) a random element of R + ×R×D for each n ≥ 1 and i = 1, 2, recall that Theorem 1 implies that the system
X n 1 (t) = X n 1 (0) + Y n 1 (t) + t 0 (−X n 1 (s) + X n 2 (s))ds − V n 1 (t), X n 2 (t) = X n 2 (0) + Y n 2 (t) + t 0 (−X n 2 (s))ds + V n 1 (t) − V n 2 (t) ≥ 0, 0 = ∞ 0 1{X n 1 (t) < 0}dV n 1 (t), 0 = ∞ 0 1{X n 2 (t) < B n }dV n 2 (t),
has a unique solution (X n , V n ). If the sequences (X n (0), n ≥ 1) and (Y n i , n ≥ 1) are stochastically bounded for i = 1, 2, then the sequence (X n , n ≥ 1) is stochastically bounded in D.
Note that we do not require boundedness for B n .
Proof. We fix t > 0. We will establish the bound
||X n || t ≤ 8e 6t (|X n (0)| + ||Y n || t ) ,(5.9)
from which the result follows.
To show (5.9), we will prove a similar bound for the unreflected process W n defined by Lemma 2. Then (5.9) will follow from the Lipschitz continuity of the reflection maps φ 0 and φ Bn .
Just as in Theorem 1 and Lemma 2 we write X n 1 (t) = φ 0 (W n 1 (t)) and X n 2 (t) = φ B n (W n 2 (t)) where W n 1 (t) and W n 2 (t) satisfy
W n 1 (t) = X n 1 (0) + Y n 1 (t) + t 0 (−φ 0 (W n 1 (s)) + φ B n (W n 2 (s)))ds, (5.10)
W n 2 (t) = X n 2 (0) + Y n 2 (t) + t 0 (−φ B n (W n 2 (s)))ds + ψ 0 (W n 1 (t)) .
(5.11)
We now use Gronwall's inequality as stated in Lemma 3. Using the Lipschitz property for φ 0 ,φ Bn , and ψ 0 we have for t ≥ 0
||W n 1 || t ≤ |X n 1 (0)| + ||Y n 1 || t + 2 t 0 (||W n 2 || s + ||W n 1 || s ) ds, ||W n 2 || t ≤ |X n 2 (0)| + ||Y n 2 || t + ||ψ 0 (W n 1 )|| t + t 0 ||W n 2 || s ds.
Now we note that we have ||ψ 0 (W n 1 )|| t ≤ ||W n 1 || t . We define
u 1 (t) = ||W n 1 || t and u 2 (t) = (||W n 2 || t − ||W n 1 || t ) + .
Finally we note ||W n 2 || t ≤ u 2 (t) + u 1 (t), (5.12) so we can write the inequalities Let |X n (0)| + ||Y n || t = K. Then Lemma 3 implies u 1 (t) ≤ 2Ke 6t and u 2 (t) ≤ 2Ke 6t .
u 1 (t) ≤ |X n 1 (0)| + ||Y n 1 || t + 4
From (5.12) and the definitions of u 1 and u 2 we obtain ||W n 1 || t ≤ 2Ke 6t and ||W n 2 || t ≤ 4Ke 6t .
Since φ 0 and φ B n are Lipschitz continuous with constant 2 this implies ||X n 1 || t ≤ 4Ke 6t and ||X n 2 || t ≤ 8Ke 6t , which proves (5.9).
Note that this proof also provides the boundedness of w that we use in the proof of continuity of w in Lemma 2, and that it does not use any of the continuity properties proved using that boundedness.
In our application of Lemma 7 we will have Y n 1 = M n,1 1 −M n,2 1 −(1−λ n ) √ nt and Y n 2 = M n,2 2 , so it remains to prove that each martingale M n,i k is stochastically bounded. To prove the stochastic boundedness of these martingales we will use the following lemma from [14]: Lemma 8 ([14] Lemma 5.8). Suppose that, for each n ≥ 1, M n is a square integrable martingale with predictable quadratic variation M n . If the sequence of random variables M n (T ) is stochastically bounded in R for each T > 0, then the sequence of stochastic processes M n is stochastically bounded in D.
We now prove that the predictable quadratic variations of M n,i k are stochastically bounded. In the case of M n,1 1 this is immediate since by (4.7) the quadratic variation is deterministic. For M n,2 1 we refer to (4.8) and apply crude bounds to see It suffices to show stochastic boundedness of each term in the sum. For Q n 1 (0) this follows from assumption (2.7).
For A(λ n nt) we note λ n → 1 so by the strong law of large numbers (SLLN) for Poisson processes we have A(λ n nt) n → e(t)
with probability 1, which implies stochastic boundedness, so we conclude that M n,2 1 is stochastically bounded.
For M n,2 2 we have M n,2 2 (t) ≤ t n (Q n 2 (0) + U n 1 (t)) ≤ t n (Q n 2 (0) + A(λ n nt)) ,
and stochastic boundedness follows. We now return to the proof of Lemma 6:
Proof of Lemma 6. We have that
Significant questions also remain about the steady state behavior of our system. In particular, we do not characterize the distribution of the steady state of the limiting system or show that the steady state of the n-th system converges to the steady state of the limiting diffusion process (interchange of limits).
Questions of the steady state are closely related to the waiting time for the system. Because customers immediately enter service if there are any idle servers and otherwise wait a constant order amount of time for the previous customer in their queue to finish service, the waiting time can be characterized by the amount of time when the system has no idle servers. We conjecture that this time is on the order O (1/ √ n), which would lead to an expected waiting time of the same order. This is also the order of the expected waiting time in the M/M/n system, so a proof of this conjecture would demonstrate that the JSQ system has a minimal loss of efficiency as measured by expected waiting time.
Finally it is always of interest to analyze our system for general interarrival and, especially, general service times distribution. We conjecture that the qualitative behaviour established in this paper in the transient domain and the conjectures above regarding the steady-state behavior and the interchange of steady-state limits remain true in this case as well.
Figure 2 :
2A simulated sample path of an untruncated M/M/n-JSQ system, showing the scaled number of idle servers (a) and queues of length at least two (X 2 ), three (X 3 ), four (X 4 ), and five (X 5 ). Simulated with n = 10 5 , β = 2.0.
the definition of u 1 and (3.14) we have
Lemma 5 .
5(FCLT for independent Poisson processes) If A, D 1 , and D 2 are independent rate-1 Poisson processes andM C,n (t) = C(nt) − nt √ n for C = A, D 1 , D 2 , then (M A,n , M D1,n , M D2,n ) ⇒ (W 1 , W 2 , W 3 ) in D 3 as n → ∞where W 1 , W 2 , and W 3 are independent standard Brownian motions.
=
M A,n • Φ A,n , M n,2 1 = M D1,n • Φ D1,n , M n,2 2 = M D2,n • Φ D2,n .
t ≥ 0. TheΦ D1,n = h • Ψ n so by the CMT and Lemma 6 we know f = h • ω. Namely, f = e is the identity function in D. Therefore we havẽ Φ D1,n ⇒ e as n → ∞.
g = 0 on D. We conclude that Φ D1,n ⇒ e as n → ∞,(5.7)and Φ D2,n ⇒ 0 as n → ∞.(5.8) Therefore once we establish Lemma 6 we can prove Lemma 4: Proof of Lemma 4. We apply the CMT with Lemma 5 and the limits (5.5),(5.7), and (5.M A,n • Φ A,n , M D1,n • Φ D1,n , M D2,n • Φ D2,n ) ⇒ (W 1 • e, W 2 • e, W 3 • 0)
(t) ≤ |X n 2 (0)| + ||Y n 2 || t + t 0 u 1 (s)ds + t 0 u 2 (s)ds.
+ A(λ n nt)) .
Martingale convergence.We will now prove the convergence of M n,i k to Brownian motions. In particular we prove
Acknowledgments.This work was supported by NSF grant CMMI-1335155.We have B n → ∞. Note that this, along with the proof of Theorem 1 from Lemma 2 and the definition of ψ ∞ , implies that the process u 2 in the limiting system will be identically zero, as it is in the statement of Theorem 2.By assumption we have X n k (0) ⇒ X k (0), so in the limiting system we have b 1 = X 1 (0) and b 2 = X 2 (0). Next we have by (2.1) and (5.1)where W is a standard Brownian motion and d = indicates equivalence in distribution. Another application of (5.1) implies −M n,2 2 ⇒ 0 so in the limiting system we have y 1 (t) = √ 2W (t) − βt and y 2 (t) = 0. The CMT then implies X n k ⇒ X k in D as n → ∞ where X k is described by (2.8)-(2.10).Open questions.Theorem 2 proves that the behavior of the truncated M/M/n-JSQ system in the Halfin-Whitt regime is best understood on the order of O ( √ n). In particular, the numbers of idle servers and waiting customers will both be O( √ n). Several open questions remain. The conjecture that this truncated system is a good model for the original untruncated system remains to be proven. Thus an extension of the result for the original model may be of interest.
The supermarket model with arrival rate tending to one. G Brightwell, M Luczak, ArXiv e-printsG. Brightwell and M. Luczak. The supermarket model with arrival rate tending to one. ArXiv e-prints, January 2012.
State space collapse in many-server diffusion limits of parallel server systems. J G Dai, Tolga Tezcan, Mathematics of Operations Research. 362J. G. Dai and Tolga Tezcan. State space collapse in many-server diffusion limits of parallel server systems. Mathematics of Operations Research, 36(2):pp. 271-320, 2011.
A note on an inequality due to Greene. K M Das, Proc. Amer. Math. Soc. 773K. M. Das. A note on an inequality due to Greene. Proc. Amer. Math. Soc., 77(3):424-425, 1979.
Randomized longest-queue-first scheduling for large-scale buffered systems. A B Dieker, T Suk, ArXiv e-printsA. B. Dieker and T. Suk. Randomized longest-queue-first scheduling for large-scale buffered systems. ArXiv e-prints, June 2013.
Two queues in parallel. L Flatto, H P Mckean, Communications on Pure and Applied Mathematics. 302L. Flatto and H. P. McKean. Two queues in parallel. Communications on Pure and Applied Mathematics, 30(2):255-263, 1977.
A basic dynamic routing problem and diffusion. G J Foschini, J Salz, IEEE Transactions on. 263CommunicationsG.J. Foschini and J. Salz. A basic dynamic routing problem and diffusion. Communications, IEEE Transactions on, 26(3):320-327, Mar 1978.
An inequality for a class of integral systems. David E Greene, Proceedings of the American Mathematical Society. 621David E. Greene. An inequality for a class of integral systems. Proceedings of the American Mathematical Society, 62(1):pp. 101-104, 1977.
Two queues in parallel. A Frank, Haight, Biometrika. 453-4Frank A. Haight. Two queues in parallel. Biometrika, 45(3-4):401-410, 1958.
The shortest queue problem. Shlomo Halfin, Journal of Applied Probability. 224Shlomo Halfin. The shortest queue problem. Journal of Applied Probability, 22(4):pp. 865-878, 1985.
Heavy-traffic limits for queues with many exponential servers. Shlomo Halfin, Ward Whitt, Operations Research. 293Shlomo Halfin and Ward Whitt. Heavy-traffic limits for queues with many exponential servers. Operations Research, 29(3):567-588, 1981.
Heavy traffic limit theorems for a queueing system in which customers join the shortest line. Zhang Hanqin, Wang Rongxin, Advances in Applied Probability. 212Zhang Hanqin and Wang Rongxin. Heavy traffic limit theorems for a queueing system in which customers join the shortest line. Advances in Applied Probability, 21(2):pp. 451-469, 1989.
Two similar queues in parallel. J F C Kingman, The Annals of Mathematical Statistics. 3241961J. F. C. Kingman. Two similar queues in parallel. The Annals of Mathematical Statistics, 32(4):1314-1323, 12 1961.
The power of two choices in randomized load balancing. Parallel and Distributed Systems. M Mitzenmacher, IEEE Transactions on. 1210M. Mitzenmacher. The power of two choices in randomized load balancing. Parallel and Distributed Systems, IEEE Transactions on, 12(10):1094-1104, 2001.
Martingale proofs of many-server heavytraffic limits for Markovian queues. Guodong Pang, Rishi Talreja, Ward Whitt, Probability Surveys. 4Guodong Pang, Rishi Talreja, and Ward Whitt. Martingale proofs of many-server heavy- traffic limits for Markovian queues. Probability Surveys, 4:193-267, 2007.
The Principles of Mathematical Analysis. Walter Rudin, International Series in Pure & Applied Mathematics. McGraw-Hill Publishing Company3rd editionWalter Rudin. The Principles of Mathematical Analysis. International Series in Pure & Applied Mathematics. McGraw-Hill Publishing Company, 3rd edition, 2006.
Optimal control of distributed parallel server systems under the halfin and whitt regime. Tolga Tezcan, Mathematics of Operations Research. 331Tolga Tezcan. Optimal control of distributed parallel server systems under the halfin and whitt regime. Mathematics of Operations Research, 33(1):51-90, 2008.
Queueing system with selection of the shortest of two queues: An asymptotic approach. N D Vvedenskaya, R L Dobrushin, F I Karpelevich, Probl. Peredachi Inf. 321N. D. Vvedenskaya, R. L. Dobrushin, and F. I. Karpelevich. Queueing system with selection of the shortest of two queues: An asymptotic approach. Probl. Peredachi Inf., 32(1):20-34, 1996.
On the optimal assignment of customers to parallel servers. Richard R Weber, Journal of Applied Probability. 152Richard R. Weber. On the optimal assignment of customers to parallel servers. Journal of Applied Probability, 15(2):pp. 406-413, 1978.
Stochastic-process Limits: An Introduction To Stochastic-process Limits And Their Application To Queues. Ward Whitt, Springer Series In Operations Research. SpringerWard Whitt. Stochastic-process Limits: An Introduction To Stochastic-process Limits And Their Application To Queues. Springer Series In Operations Research. Springer, New York, 2002.
Optimality of the Shortest Line Discipline. Wayne Winston, Journal of Applied Probability. 141Wayne Winston. Optimality of the Shortest Line Discipline. Journal of Applied Probability, 14(1):181-189, 1977.
Heavy traffic limit theorems for a sequence of shortest queueing systems. Hanqin Zhang, Guang-Hui Hsu, Rongxin Wang, Queueing Systems. 211-2Hanqin Zhang, Guang-Hui Hsu, and Rongxin Wang. Heavy traffic limit theorems for a sequence of shortest queueing systems. Queueing Systems, 21(1-2):217-238, 1995.
| [] |
[
"The Compton/Schwarzschild duality, black hole entropy and quantum information",
"The Compton/Schwarzschild duality, black hole entropy and quantum information"
] | [
"Marcelo Ferreira Da Silva ",
"Carlos Silva "
] | [] | [] | A new kind of duality has been proposed by Carr related to the quantum description of black holes, the so-called Compton/Schwarzschild duality[1]. In this context, a new form for a Generalized Uncertainty Principle has arisen, which must bring us an interesting new route to a quantum description of spacetime. In the present paper, we shall investigate the consequences of the Compton/Schwarzschild duality to black hole entropy. The results found out reinforce an interesting perspective on the relationship between black holes and quantum information theory that has been recently proposed in the literature: that black hole entropy can assume negative values at the final stage of black hole evaporation. Consequently, in the context of the quantum corrections to gravity proposed by the Compton/Schwarschild duality, the final state of a black hole might correspond to a quantum entangled state, in the place of a remnant. | null | [
"https://arxiv.org/pdf/2205.09502v1.pdf"
] | 248,887,536 | 2205.09502 | 6999d1734cb71dbc942287eb54546f746e4c6293 |
The Compton/Schwarzschild duality, black hole entropy and quantum information
May 20, 2022
Marcelo Ferreira Da Silva
Carlos Silva
The Compton/Schwarzschild duality, black hole entropy and quantum information
May 20, 2022
A new kind of duality has been proposed by Carr related to the quantum description of black holes, the so-called Compton/Schwarzschild duality[1]. In this context, a new form for a Generalized Uncertainty Principle has arisen, which must bring us an interesting new route to a quantum description of spacetime. In the present paper, we shall investigate the consequences of the Compton/Schwarzschild duality to black hole entropy. The results found out reinforce an interesting perspective on the relationship between black holes and quantum information theory that has been recently proposed in the literature: that black hole entropy can assume negative values at the final stage of black hole evaporation. Consequently, in the context of the quantum corrections to gravity proposed by the Compton/Schwarschild duality, the final state of a black hole might correspond to a quantum entangled state, in the place of a remnant.
Introduction
The road to a quantum description of the gravitational phenomena has been a tough challenge. In this sense, efforts have been done by different approaches, which have proposed some interesting results that may underly possible routes to quantum gravity. Among such results, we have three interesting hints that have occupied a lot of space in the literature.
The first of such hints is related to dualities. Particularly, dualities consist of an important feature of string theory where we have several examples of them, such as the T -duality and the S duality. Moreover, in string theory context, holographic dualities such as those related to the AdS/CF T correspondence [2] have found a large number of applications [3]. Actually, holographic dualities have been shown to be a unifying principle in investigations related to quantum gravity, being a possible bridge between string theory and Loop Quantum Gravity [4,5].
A second interesting hint to a quantum description of the gravitational phenomena is the socalled Generalized Uncertainty Principle (GUP), which is rooted in which must be an elementary feature of quantum gravity: the inclusion of a minimal length scale in the description of the physical world [6,7,8,9,10]. In this way, a GUP replaces the usual Heisenberg Uncertainty Principle, assuming a special role in telling us how the measurement process in quantum gravity must be different from that we have in quantum mechanics.
The third hint to quantum gravity is traced by the Bekenstein-Hawking entropy-area formula which arises in the context of black hole thermodynamics. Such a formula occupies a central place in the discussions on quantum gravity since it gives us the number of degrees of freedom we have in nature at the most fundamental level. For instance, the Bekenstein-Hawking formula is one of a few equations in physics where the four fundamental constants in nature appear together. Moreover, it establishes a connection between black hole thermodynamics and quantum information theory, giving rise to the concept, by Wheeler, of "it from bit". Sprouting from such discussions, an interesting result has been introduced by Carr [1], the so-called Compton/Schwarzschild duality. By consisting of a kind of T -duality, it says that in three spatial dimensions, the Compton wavelength (R C ∝ M −1 ) and the Schwarzschild radius (R S ∝ M ) are dual under the transformation M → M 2 P /M , where M P is the Planck mass. This suggests that there could be a fundamental link between elementary particles with M < M P and black holes in the M > M P regime. In the context of such a duality, a new proposal for a GUP has been suggested, which makes it possible to address the two first hints to quantum gravity presented in the paragraphs above through a unique formulation.
In this paper, we shall investigate how one could expand the discussions promoted by the Compton/Schwarzschild duality to include the third hint we have pointed out above, i.e., the Bekenstein-Hawking entropy/area formula. In this way, we shall study black hole thermodynamics in the context of the Compton/Schwarzschild duality, by considering quantum corrections coming from the GUP that arise in the context of such a duality. In this way, we obtain a quantum corrected formula to black hole entropy which possesses a distinct behavior when compared with the usual Bekenstein-Hawking formula. The most intriguing feature of the result obtained in the present paper will be the possibility of having negative values for the black hole entropy at the final stages of the black hole evaporation when the black hole enters the subplanckian regime. In this case, the final state of a black hole could correspond to an entangled quantum state where the quantum information related to the black hole's initial state could be stored. It corresponds to a different scenario that has been proposed by other GUP approaches, where the final stage of a black hole corresponds to a remnant [11,12,13,14].
The paper is organized as follows: in section (2), we shall review the Compton/Schwarzschild duality as it has been introduced by Carr [1]. In section (3), we shall derive a quantum corrected Bekenstein-Hawking formula in the context of the Compton Schwarzschild duality. Section (4) is devoted to conclusions and perspectives.
The Compton/Schwarzschild duality and the GUP
By considering a microscopic description of reality, quantum mechanics says that a central role is assumed by the Heisenberg Uncertainty Principle, which gives us that the uncertainties in the position δx and in the momentum δp of a particle must obey the relation δx ≥ /2δp. It implies that, if the momentum of a particle has an upper bound given by mc, then it is not possible to localize the particle in a region smaller than /2mc. It defines a characteristic length related to the particle, the so-called reduced Compton length, given by /mc.
On the other hand, by considering a macroscopic description of the world, black holes assume a key role. Black holes consist of a region of spacetime where the gravitational field is so strong that not even light can escape from there. The characteristic length of such a region is the black hole Schwarzschild radius R S = 2Gm/c 2 , where m is the black hole mass, which defines the black hole boundary, called the event horizon. Such a putative solution of General Relativity has been considered a gateway for quantum gravity, and it has gotten more attention in the last years due to the detection of gravitational waves by LIGO [15], and the capturing of the first black hole image by the Event Horizon Telescope [16].
It has been suggested that the Schwarzschild radius and the Compton length may intercept around the Planck scale. However, the way it can occur is not well understood yet, since both quantum mechanics and general relativity must break in such a regime. In the sense of shedding more light on this issue, it was proposed by Adler and colleagues that at the Planck scale the Heisenberg Uncertainty Principle must be replaced by a GUP written as [11,12,13,14] δx ≥ δp
+ αl 2 P δp ,(1)
where α is an adimensional constant (normally considered positive), and l P is the Planck length.
In the Eq. (1) above, the first term on the right-hand side is given by the quantum mechanical contribution, while the second term corresponds to the gravitational contribution to the GUP. The GUP in the Eq. (1), where the quantum mechanical and gravitational contributions are added linearly, bring us a crucial consequence for black hole thermodynamics, especially for the final stage of black hole evaporation, where quantum gravity effects must become more evident. For example, in the context of such a GUP, a black hole is prevented to evaporate completely, in a similar way the hydrogen atom is avoided from collapsing by the usual Heisenberg uncertainty principle. In this way, a black hole remnant must be found at the end of the evaporation process. [12,13,14]. Such results have launched several interesting ideas, e.g., the one that black holes could correspond to the origin of dark matter [13,14].
The possibility of a connection between the Schwarzschild radius and the Compton length at the Planck scale has also motivated the proposition of an interesting duality by Carr [1], the so-called Compton/Schwarzschild duality. By consisting of a kind of T -duality, it says that in three spatial dimensions, the Compton wavelength (R C ∝ M −1 ) and Schwarzschild radius (R S ∝ M ) are dual under the transformation M → M 2 P /M , where M P is the Planck mass. This suggests that there could be a fundamental link between elementary particles with M < M P and black holes in the M > M P regime.
However, even though it is possible to find some motivation for such a duality from the GUP (1), it has been argued by Carr that, since the contributions from the Compton length and the Schwarzschild radius are independent, it is more natural to assume that they must be added quadratically:
δx ≥ δp 2 + αl 2 P δp 2 .(2)
It introduces a completely different perspective about the role of quantum corrections coming from a GUP to the gravitational phenomena. In the next section, we shall address how such a version of a GUP can be connected to a new insight that has been launched in the literature: that black holes can store negative entropy [17]. It will be shown that differently than we have in the context of the GUP introduced by Adler et al, the final stage of a black hole might not correspond to a remnant, but an entangled quantum state with a negative entropy associated with it.
Quantum corrected Bekenstein-Hawking formula from the Compton/Schwarzschild duality
Let us consider the Schwarzschild black hole, whose metric is
ds 2 = −(1 − 2M r )dt 2 + (1 − 2M r ) −1 dr 2 + r 2 dΩ 2 2 .(3)
For a black hole that absorbs and emits particles of energy dM ≈ cδp, the increase (decrease) in the horizon area can be expressed through
dA = 8πr h dr h = 32πM dM.(4)
Moreover, the black hole radiation is a quantum effect. Thus it must satisfy the Heisenberg uncertainty relation:
δp i δx j ≥ δ ij .(5)
However, in the cases where gravity becomes important, the Heisenberg principle should be replaced by a GUP. More specifically, here we shall consider the GUP proposed by Carr:
δx 2 ≥ δp 2 + αl 2 p δp/ 2 .(6)
From the expression above, we get the momentum uncertainty for the emitted particle:
δp = δx 2 2 ± 2 δx 4 − 4l 4 p α 2 2l 4 p α 2 (7) = ± 2αl 2 p δx 2 − 2 1 − 4α 2 l 4 p δx 4 ,(8)
where we have taken the minus signal inside the square root to recover the Heisenberg principle at the semi-classical limit. The expansion of the Eq. (8) gives us
δp = δx 2αl 2 p 2 − 2 1 − 2α 2 l 4 p (δx) 4 − 2α 4 l 8 p (δx) 8 − . . .(9)= δx 1 + α 2 l 4 p (δx) 4 + . . . .(10)
We have, from 4 and 5, that the change in the area of the black hole horizon can be written as (for c = 1):
dA = 32πM dp = 32πM 1 δx ,(11)
which is related to a change of black hole entropy given by dS = 8πM dp = 8πM 1 δx ,
On the other hand, the results coming from the GUP proposed by Carr give us the quantum corrected expression for the change of the black hole entropy as dS g = 8πM dp = 8πM
1 δx 1 + α 2 l 4 p (δx) 4 + . . . = 1 + α 2 l 4 p (δx) 4 + . . . dS,(13)
where we have used = 1. Now, by taking δx ≈ 2r s = 2 A/4π = S/π, we shall have
dS g = 1 + 16π 2 α 2 l 4 p S 2 + . . . dS .(14)
By neglecting higher-order terms, we can integrate (14) to find
S gup = A 2 + π 2 α 2 l 4 p 4 + παl 2 p 8 ln A 2 + π 2 α 2 l 4 p παl 2 p − 1 − ln A 2 + π 2 α 2 l 4 p παl 2 p + 1 .(15)
The equation above corresponds to the quantum corrected Bekenstein-Hawking entropy-area law in the context of the Compton/Schwarzschild duality. As one can notice, such an expression will correspond to the classical law when one takes α = 0.
Below, we plot the entropy (15) as a function of the black hole mass, with three different values of α, and considering l p = M p = 1, where M p is the Planck mass. The red line corresponding to α = 0.5, the blue line corresponding to α = 0.25 and the yellow line corresponding to α = 0. From Fig. (1), we see that in the sub-Planckian regime, we have a different behavior of the black hole entropy as compared with the classical entropy, with a bigger deviation for bigger values of α. It is very interesting that, in the subplanckian regime, negative values for the BH entropy have been obtained.
Conclusions and Perspectives
What is the final stage of a black hole? If one considers the usual Bekenstein-Hawking scenario for black hole evaporation, the final stage of a black hole must correspond to the that it completely evaporates into thermal radiation. On the other hand, in the scenario proposed by Adler et al, where quantum gravity corrections due to a GUP are taken into account, the final stage of a black hole must correspond to a remnant.
In the present paper, we have obtained a different scenario. By considering the Compton/Schwarzschild duality proposed by Carr [1], we have obtained an intriguing result where black holes can possess negative entropies at the final stages of their evaporation. Such a feature of black hole entropy found in the present paper appears as a surprising and counter-intuitive feature of the quantum description of the black hole evaporation process that is unparalleled in classical information theory. On the other hand, the concept of negative entropy has assumed an important role in quantum information theory, where a negative entropy is related to a quantum entangled state [17,18,19].
In this way, the results found in the present paper might point to the fact that, at the final stages of black hole evaporation, the black hole must enter into a pre-spacetime regime where the information related to the initial state of the black hole must be stored not in a remnant, as occurs in the Adler approach, but into a quantum entangled state.
Figure 1 :
1Black hole entropy in terms of the BH mass for the sub-Planckian regime.
Sub-Planckian black holes and the Generalized Uncertainty Principle. B J Carr, J Mureika, P Nicolini, 10.1007/JHEP07(2015)052arXiv:1504.07637JHEP. 0752gr-qcB. J. Carr, J. Mureika and P. Nicolini, "Sub-Planckian black holes and the General- ized Uncertainty Principle," JHEP 07, 052 (2015) doi:10.1007/JHEP07(2015)052 [arXiv:1504.07637 [gr-qc]].
The Large N limit of superconformal field theories and supergravity. J M Maldacena, 10.1023/A:1026654312961arXiv:hep-th/9711200Int. J. Theor. Phys. 38hep-thJ. M. Maldacena, "The Large N limit of superconformal field theories and super- gravity," Int. J. Theor. Phys. 38, 1113-1133 (1999) doi:10.1023/A:1026654312961 [arXiv:hep-th/9711200 [hep-th]].
The AdS/CFT Correspondence. V E Hubeny, 10.1088/0264-9381/32/12/124010arXiv:1501.00007Class. Quant. Grav. 3212124010gr-qcV. E. Hubeny, "The AdS/CFT Correspondence," Class. Quant. Grav. 32, no.12, 124010 (2015) doi:10.1088/0264-9381/32/12/124010 [arXiv:1501.00007 [gr-qc]].
Spin networks and the big bang singularity avoidance in the AdS/CFT correspondence. C Silva, 10.1103/PhysRevD.102.046001arXiv:2008.07279Phys. Rev. D. 102446001gr-qcC. Silva, "Spin networks and the big bang singularity avoidance in the AdS/CFT correspondence," Phys. Rev. D 102, no.4, 046001 (2020) doi:10.1103/PhysRevD.102.046001 [arXiv:2008.07279 [gr-qc]].
Spacetime from quantum information: spin networks and the cosmological constant in the AdS/CF T correspondence. C Silva, arXiv:2009.07843gr-qcC. Silva, "Spacetime from quantum information: spin networks and the cosmolog- ical constant in the AdS/CF T correspondence," [arXiv:2009.07843 [gr-qc]].
Minimum Physical Length and the Generalized Uncertainty Principle in String Theory. K Konishi, G Paffuti, P Provero, 10.1016/0370-2693(90)91927-4Phys. Lett. B. 234K. Konishi, G. Paffuti and P. Provero, "Minimum Physical Length and the Gener- alized Uncertainty Principle in String Theory," Phys. Lett. B 234, 276-284 (1990) doi:10.1016/0370-2693(90)91927-4
A Generalized uncertainty principle in quantum gravity. M Maggiore, 10.1016/0370-2693(93)91401-8arXiv:hep-th/9301067Phys. Lett. B. 304hep-thM. Maggiore, "A Generalized uncertainty principle in quantum gravity," Phys. Lett. B 304, 65-69 (1993) doi:10.1016/0370-2693(93)91401-8 [arXiv:hep- th/9301067 [hep-th]].
The Algebraic structure of the generalized uncertainty principle. M Maggiore, 10.1016/0370-2693(93)90785-GarXiv:hep-th/9309034Phys. Lett. B. 319hep-thM. Maggiore, "The Algebraic structure of the generalized uncertainty principle," Phys. Lett. B 319, 83-86 (1993) doi:10.1016/0370-2693(93)90785-G [arXiv:hep- th/9309034 [hep-th]].
Generalized uncertainty principle in quantum gravity from micro -black hole Gedanken experiment. F Scardigli, 10.1016/S0370-2693(99)00167-7arXiv:hep-th/9904025Phys. Lett. B. 452hep-thF. Scardigli, "Generalized uncertainty principle in quantum gravity from mi- cro -black hole Gedanken experiment," Phys. Lett. B 452, 39-44 (1999) doi:10.1016/S0370-2693(99)00167-7 [arXiv:hep-th/9904025 [hep-th]].
Universality of Quantum Gravity Corrections. S Das, E C Vagenas, 10.1103/PhysRevLett.101.221301arXiv:0810.5333Phys. Rev. Lett. 101221301hep-thS. Das and E. C. Vagenas, "Universality of Quantum Gravity Correc- tions," Phys. Rev. Lett. 101, 221301 (2008) doi:10.1103/PhysRevLett.101.221301 [arXiv:0810.5333 [hep-th]].
On gravity and the uncertainty principle. R J Adler, D I Santiago, 10.1142/S0217732399001462arXiv:gr-qc/9904026[gr-qc]Mod. Phys. Lett. A. 141371R. J. Adler and D. I. Santiago, "On gravity and the uncertainty principle," Mod. Phys. Lett. A 14, 1371 (1999) doi:10.1142/S0217732399001462 [arXiv:gr- qc/9904026 [gr-qc]].
The Generalized uncertainty principle and black hole remnants. R J Adler, P Chen, D I Santiago, 10.1023/A:1015281430411arXiv:gr-qc/0106080Gen. Rel. Grav. 33gr-qcR. J. Adler, P. Chen and D. I. Santiago, "The Generalized uncertainty principle and black hole remnants," Gen. Rel. Grav. 33, 2101-2108 (2001) doi:10.1023/A:1015281430411 [arXiv:gr-qc/0106080 [gr-qc]].
Black hole remnants and dark matter. P Chen, R J Adler, 10.1016/S0920-5632(03)02088-7arXiv:gr-qc/0205106Nucl. Phys. B Proc. Suppl. 124gr-qcP. Chen and R. J. Adler, "Black hole remnants and dark matter," Nucl. Phys. B Proc. Suppl. 124, 103-106 (2003) doi:10.1016/S0920-5632(03)02088-7 [arXiv:gr- qc/0205106 [gr-qc]].
Might dark matter be actually black?. P Chen, arXiv:astro-ph/0303349astrophP. Chen, "Might dark matter be actually black?," [arXiv:astro-ph/0303349 [astro- ph]].
Observation of Gravitational Waves from a Binary Black Hole Merger. B P Abbott, LIGO Scientific and Virgo10.1103/PhysRevLett.116.061102arXiv:1602.03837Phys. Rev. Lett. 116661102gr-qcB. P. Abbott et al. [LIGO Scientific and Virgo], "Observation of Gravitational Waves from a Binary Black Hole Merger," Phys. Rev. Lett. 116 (2016) no.6, 061102 doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]].
First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. K Akiyama, 10.3847/2041-8213/ab0ec7arXiv:1906.11238Astrophys. J. Lett. 8751Event Horizon Telescope. astro-ph.GAK. Akiyama et al. [Event Horizon Telescope], "First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole," Astrophys. J. Lett. 875 (2019), L1 doi:10.3847/2041-8213/ab0ec7 [arXiv:1906.11238 [astro-ph.GA]].
Negative entropy and black hole information. D Song, 10.1007/s10773-013-1933-4Int J Theor Phys. 53D.. Song, "Negative entropy and black hole information," Int J Theor Phys 53, 1369-1374 (2014). https://doi.org/10.1007/s10773-013-1933-4.
Negative entropy and information in quantum mechanics. N J Cerf, C Adami, 10.1103/PhysRevLett.79.5194arXiv:quant-ph/9512022Phys. Rev. Lett. 795194quant-phN. J. Cerf and C. Adami, "Negative entropy and information in quantum me- chanics," Phys. Rev. Lett. 79 (1997), 5194 doi:10.1103/PhysRevLett.79.5194 [arXiv:quant-ph/9512022 [quant-ph]].
The thermodynamic meaning of negative entropy. L Rio, J Åberg, R Renner, 10.1038/nature10123Nature. 474L. Rio, J.Åberg, R. Renner, et al. "The thermodynamic meaning of negative entropy". Nature 474, 61-63 (2011). https://doi.org/10.1038/nature10123
| [] |
[
"MAXIMUM LIKELIHOOD ESTIMATION FROM A TROPICAL AND A BERNSTEIN-SATO PERSPECTIVE",
"MAXIMUM LIKELIHOOD ESTIMATION FROM A TROPICAL AND A BERNSTEIN-SATO PERSPECTIVE"
] | [
"Anna-Laura Sattelberger ",
"Robin Van Der Veer "
] | [] | [] | In this article, we investigate Maximum Likelihood Estimation with tools from Tropical Geometry and Bernstein-Sato Theory. We investigate the critical points of very affine varieties and study their asymptotic behavior. We relate these asymptotics to particular rays in the tropical variety as well as to Bernstein-Sato ideals and give a connection to Maximum Likelihood Estimation in Statistics. | 10.1093/imrn/rnac016 | [
"https://arxiv.org/pdf/2101.03570v2.pdf"
] | 231,573,311 | 2101.03570 | bfda177dc8f0946bec9aa1003dd3a40e95810b7b |
MAXIMUM LIKELIHOOD ESTIMATION FROM A TROPICAL AND A BERNSTEIN-SATO PERSPECTIVE
3 Mar 2022
Anna-Laura Sattelberger
Robin Van Der Veer
MAXIMUM LIKELIHOOD ESTIMATION FROM A TROPICAL AND A BERNSTEIN-SATO PERSPECTIVE
3 Mar 202210.1093/imrn/rnac016
In this article, we investigate Maximum Likelihood Estimation with tools from Tropical Geometry and Bernstein-Sato Theory. We investigate the critical points of very affine varieties and study their asymptotic behavior. We relate these asymptotics to particular rays in the tropical variety as well as to Bernstein-Sato ideals and give a connection to Maximum Likelihood Estimation in Statistics.
Introduction
Maximum likelihood estimation. Let X be a smooth closed subvariety of the algebraic torus (C * ) p and denote by t 1 , . . . , t p the coordinates on the torus restricted to X. Given α ∈ C p , the maximum likelihood estimation (MLE) problem is to find the zeroes of the logarithmic one-form dlog(t α 1 1 · · · t αp p ). These zeroes are also referred to as critical points. As the terminology suggests, this problem arises from Statistics. We refer to [20] for further details and examples related to this statistical context. In this article, we investigate this problem from an algebro-geometric point of view. The geometric object underlying our analysis is the critical locus C ⊆ X × P p−1 , consisting of all pairs (x, α) for which the logarithmic differential dlog t α vanishes at x. As proven by Franecki-Kapranov in [14] and later by Huh in [18], for a general data vector α, the number of critical points is finite and equal to the signed Euler characteristic (−1) dim X χ(X) of X. We refer to non-general data vectors as special values. The number of critical points for general α is also called the maximum likelihood degree of X, denoted by d ML (X).
Maximum likelihood estimation approaching non-general data vectors. In this article, we study the behavior of the critical points when approaching a special data vector along a curve consisting of general values. More precisely, we study the subset S F ⊆ P p−1 consisting of all data vectors α for which at least one of the d ML (X) many critical points leaves X as we approach α. We refer to the components of S F that are of codimension one as critical slopes of F. In this case, at least one of the coordinates of some critical point approaches either zero or infinity. To obtain more refined information of the asymptotic behavior, we keep track of the direction in which this critical point escapes by recording it in a set Q F,α ⊆ Z p . The points in Q F,α essentially describe which torus orbits in a tropical compactification of X are approached by the critical points as one approaches α. In Section 2, we introduce both of these objects in the general setting of a smooth variety X with a tuple of nowhere vanishing regular functions F = (f 1 , . . . , f p ) on it. It turns out that both S F and Q F,α are closely related to the divisorial valuations corresponding to the irreducible components of the boundary in a smooth compactification of X with simple normal crossings (SNC) boundary. The critical points for special data vectors were, among others, also studied in [9] for the case of hyperplane arrangements and in [27] using probabilistic methods. To study S F and Q F,α , a good understanding of a smooth compactification and its boundary components is essential, which we tackle in Section 3.
Schön varieties and tropical compactifications. A natural class of compactifications of closed subvarieties X of tori is provided by the tropical compactifications of [31]. Tropical compactifications are constructed by taking the closure X Σ of X in the toric variety associated to a fan Σ whose support is the tropical variety of X. In general, these compactifications are not smooth. If the variety X is schön (see [18,Definition 3.6]), it allows for a tropical compactification that is smooth and whose boundary is a simple normal crossings divisor. This class of subvarieties of tori provides a common generalization of hyperplane arrangement complements and of Newton non-degenerate hypersurfaces as shown in [17]. For these compactifications, we prove the following theorem.
Theorem 3.7. Let E i be an irreducible component of the boundary in a smooth tropical compactification of X with simple normal crossings boundary. Assume that χ(
E • i ) = 0, where E • i := E i \ ∪ j =i E j .
Then for general α in the hyperplane H E i = {ord E i (t 1 )s 1 + · · · + ord E i (t p )s p = 0}, the vector (ord E i (t 1 ), . . . , ord E i (t p )) is contained in Q F,α . In particular, the hyperplane H E i is contained in S F . Moreover, these hyperplanes are the only codimension-one components of S F .
The toric variety associated to Σ has codimension-one torus orbits O τ indexed by the rays τ ∈ Σ. The boundary component E • i is an irreducible component of X Σ ∩ O τ for some τ ∈ Σ and thus the condition that χ(E • i ) = 0 is a property of the ray τ in the tropical variety of X. To translate this condition into a tropical condition, we define the notion of rigid rays in Trop(X); these are the rays for which any small perturbation of the ray changes the associated initial ideal. Proposition 3. 13. Let X be a schön very affine variety and Σ a fan supported on the tropical variety of X such that the closure X Σ of X in the toric variety associated to Σ is smooth and X \ X Σ is a SNC divisor. Assume that for all τ ∈ Σ, the intersection X Σ ∩O τ is connected. If τ is a rigid ray, then for general α in the hyperplane orthogonal to τ , the primitive generator of the ray lies in Q F,α . In particular, P(τ ⊥ ) ⊆ S F . This gives a bijection of the rigid rays in Trop(X) and the codimension-one components of S F . This proposition completely determines the critical slopes of F in terms of tropical data associated to X. By the very definition of Q F,α , this proposition gives a description of the asymptotic behavior of the critical points as we approach the special set of data vectors formed by the hyperplane orthogonal to a rigid ray. We refer to Section 3.1 for a more explicit interpretation of this result in terms of maximum likelihood estimation.
Slopes of Bernstein-Sato varieties. In Section 4, we relate these results to Bernstein-Sato ideals. For a tuple F = (f 1 , . . . , f p ) of regular functions on a smooth algebraic variety, the Bernstein-Sato ideal of F is the ideal B F in C[s 1 , . . . , s p ] consisting of polynomials b for which there exists a global algebraic linear partial differential operator P (s 1 , . . . , s p ) such that
P • f s 1 +1 1 · · · f sp+1 p = b · f s 1 1 · · · f sp p .
This definition is a generalization of the Bernstein-Sato polynomial of a single regular function f . It is an intricate invariant of the tuple F which is related to the singularities of the hypersurface V (f 1 · · · f p ), see for instance [3] or [5] for relations to monodromy eigenvalues. Sabbah [28] showed that the Bernstein-Sato ideal is non-zero and that the codimension-one components of the Bernstein-Sato variety V (B F ) are affine hyperplanes with rational coefficients. The set of Bernstein-Sato slopes of F , denoted by BS F ⊆ C p , is the union of these hyperplanes after translating them to the origin. We denote by P(BS F ) the projectivization of this set. Maisonobe [25] gave a geometric description of the Bernstein-Sato slopes. We denote by Y the closure of X inside C p and will assume that Y is smooth. We then study the Bernstein-Sato ideal of the tuple of coordinate functions on C p restricted to Y . In this setup, we deduce the following theorem using Maisonobe's description of the Bernstein-Sato slopes.
Theorem 4.4. Under the assumptions of Proposition 3.13, the irreducible components of S F ∩ P(BS F ) are exactly the hyperplanes P(τ ⊥ ) for τ a rigid ray contained in R p ≥0 .
A study of BS F using the critical locus was also undertaken in [2] under a different technical assumption, namely that the tuple F is sanséclatements en codimension 0. We would like to point out that Lemma 2.6 therein, treating the special case p = 2, is analogous to our Theorem 4.4. In Example 4.5, we demonstrate that, in general, the sets S F and P(BS F ) are incomparable in the sense that either can contain irreducible components not contained in the other one. Theorem 4.4 explains in a rigorous way the observations made in [29, Example 3.1]. We revisit this example in Section 5.
Theorem 4.4 provides information on the slopes of the Bernstein-Sato variety, but not on the affine translation with which these slopes appear. In Subsection 4.2, we formulate a conjecture related to these affine translates. This conjecture is formulated in terms of the log-canonical threshold polytope of the tuple F . We prove this conjecture for the case of indecomposable central hyperplane arrangements, in which case it proves [3,Conjecture 3] for complete factorizations of hyperplane arrangements. It turns out that in this case the object involved has already extensively been studied under a different name: it is the matroid polytope of [13], and Proposition 2.4 in loc. cit. precisely recovers our conjecture. The Bernstein-Sato ideal of hyperplane arrangements has recently also been studied using different methods in [1], [26], and [32].
In summary, our results connect Tropical Geometry, Bernstein-Sato Theory, and Likelihood Geometry. Among others, our article provides new tools for Algebraic Statistics and Particle Physics: in the recent article [30], Sturmfels and Telen outline a link of MLE for discrete statistical models to scattering amplitudes.
Notation and conventions. By a variety we mean an integral, separated scheme of finite type over the complex numbers, unless explicitly stated otherwise. A property of a variety is called general if it holds true on a Zariski dense open subset of the variety. For a smooth variety X with compactification X ֒→ Y s.t. Y is smooth and
E := Y \ X = q i=1 E i is a divisor with irreducible components E i , we denote by E • i := E i \ j =i E j
the complement of E i by the other components. When X is equipped with a tuple F = (f 1 , . . . , f p ) of regular functions, we denote by H E i the hyperplane
H E i := {ord E i (f 1 )s 1 + · · · + ord E i (f p )s p = 0} ⊆ P p−1 .
We denote by ∆ the formal disc Spec C t , by ∆ • := Spec C((t)) its generic point, and by 0 its closed point. Throughout the article, we will always assume that all cones in a fan are strongly convex.
Asymptotic behavior of critical points
In this section, we introduce the objects of study and investigate basic properties. We study smooth varieties X with a p-tuple F = (f 1 , . . . , f p ) of nowhere vanishing regular functions on X. For α = (α 1 , . . . , α p ) ∈ C p ,
dlog f α = p i=1 α i df i f i ∈ H 0 X, Ω 1 X denotes the logarithmic differential of f α := f α 1 1 · · · f αp p .
Definition 2.1. The critical locus of F , encoding the zeros of dlog f α , is defined to be
C F := (x, α) ∈ X × P p−1 | dlog f α (x) = 0 ⊆ X × P p−1 .
Let X ֒→ Y be a compactification of X and denote by E := Y \ X the boundary of this compactification.
Definition 2.2. The asymptotic critical locus of F with respect to Y , denoted C F,Y , is the closure of C F inside Y × P p−1 , i.e., C F,Y := C F Y ×P p−1 .
Denote by π 1 : Y × P p−1 → Y (resp. π 2 : Y × P p−1 → P p−1 ) the projection to the first (resp. second) factor. Definition 2.3. Associated to F we define the variety
S F := π 2 C F,Y ∩ π −1
1 (E) ⊆ P p−1 and refer to its irreducible components of codimension one as critical slopes of F.
The variety S F is the image of a proper variety and as such a closed subvariety of P p−1 .
Example 2.4. Let X ′ ⊆ C 2 be the complement of the arrangement defined by f = xy(x − y)(x − 1), the four factors of which form the tuple F = (f 1 , f 2 , f 3 , f 4 ). As compactification, we take P 2 . With a computer algebra system one confirms that the critical slopes of F are
{s 1 + s 2 + s 3 + s 4 = 0} ∪ {s 2 + s 3 = 0} ∪ {s 1 + s 2 + s 3 = 0} ∪ 4 i=2 {s i = 0} .
More precisely, {s 1 + s 2 + s 3 + s 4 = 0} = π 2 (C F,P 2 ∩ π −1 1 (H ∞ )), {s 2 + s 3 = 0} = π 2 (C F,P 2 ∩ π −1 1 ([0 : 1 : 0])), {s 1 + s 2 + s 3 = 0} = π 2 (C F,P 2 ∩ π −1 1 ([0 : 0 : 1])), and {s i = 0} comes from {f i = 0} for i = 2, 3, 4. △ Lemma 2.5. Let Z be a variety, X ⊆ Z a locally closed subvariety and x ∈ Z a closed point. Then x ∈ X if and only if there exists a morphism γ : ∆ → Z such that γ(∆ • ) ∈ X and γ(0) = x.
Proof. The existence of such γ clearly implies that x ∈ X. For the reverse implication, let x ∈ X. Take a curve C ⊆ Z with x ∈ C and C ∩ X = ∅. LetC denote the normalization of C and choose a pointx ∈C lying over x. SinceC is smooth, by the Cohen structure theorem we can construct the required morphism as
O X,x → O X,x → O C,x → OC ,x ∼ = C t .
As an immediate consequence, we deduce the following corollary.
Corollary 2.6. A point α ∈ P p−1 lies in S F if and only if there exists a morphism γ : ∆ → Y × P p−1 such that γ(∆ • ) ∈ C F and γ(0) ∈ E × {α}.
To keep track of more refined information about the asymptotic behavior of the critical points, we introduce the following subsets of Z p .
Definition 2.7. Let Y be a compactification of X and let α ∈ P p−1 . Let π 1 : Y × P p−1 → Y denote the projection to the first factor. We denote by Q F,α ⊆ Z p the set of vectors that arise in the following manner. A vector is in Q F,α if and only if it is of the form (ord t (γ * (π * 1 f 1 )), . . . , ord t (γ * (π * 1 f p ))) for some morphism γ : ∆ → Y × P p−1 such that γ(∆ • ) ∈ C F and γ(0) ∈ E × {α}.
For brevity, we denote (ord t (γ * (π * 1 f 1 )), . . . , ord t (γ * (π * 1 f p )) by ord t (γ * (π * 1 F )). In general, it is difficult to compute Q F,α explicitly. In Theorem 3.7 and Proposition 3.13, we will recover it partially. The following lemma justifies suppressing the compactification in the notations S F and Q F,α .
Lemma 2.8. The sets S F and Q F,α , α ∈ P p−1 , do not depend on the choice of a compactification.
Proof. Let j 1 : X ֒→ Y 1 and j 2 : X ֒→ Y 2 be two compactifications of X with boundaries E i = Y i \ X and projections π i,2 :
C F,Y i → P p−1 , i = 1, 2. Let α ∈ π 1,2 (C F,Y 1 ∩ π −1
1,1 (E 1 )). By Corollary 2.6 this is equivalent to the existence of a mor-
phism γ : ∆ → Y 1 × P p−1 , such that (1) γ(∆ • ) ∈ C F ⊆ C F,Y 1 (2) π 1,1 (γ(0)) ∈ E 1 , and (3) π 1,2 (γ(0)) = α. By properness of Y 2 × P p−1 , the restricted morphism γ • : ∆ • → C F can be extended to a morphismγ : ∆ → Y 2 × P p−1 . We need to show thatγ(0) ∈ E 2 × {α}. Assume that π 2,1 (γ(0)) ∈ E 2 .
Then π 2,1 (γ(0)) ∈ X. But this is not possible, since then also π 1,1 (γ(0)) ∈ X, sinceγ and γ coincide on ∆ • . Hence π 2,1 (γ(0)) ∈ E 2 . Similarly, we conclude that π 2,2 (γ(0)) = α, so that α ∈ π 2,2 (C F,Y 2 ∩ π −1 2,1 (E 2 )). The statement for Q F,α follows from the same argument, since ord t (γ * (π * 1,1 (f i )) = ord t (γ * (π * 2,1 (f i )) for all i = 1, . . . , p.
The following observation is immediate from the definitions. Lemma 2.9. The closed points of S F are given by those α ∈ P p−1 for which Q F,α is non-empty, i.e., S F = {α ∈ P p−1 | Q F,α = ∅}.
We recall that for a boundary component E j in a compactification of X, we denote by
H E j the hyperplane ord E j (f 1 )s 1 + · · · + ord E j (f p )s p = 0 in P p−1 . Proposition 2.10. Suppose that j : X ֒→ Y is a compactification with Y smooth and Y \ X a simple normal crossings divisor E with irreducible components E 1 , . . . , E q . Suppose that γ : ∆ → Y × P p−1 is such that γ(∆ • ) ∈ C F and γ(0) ∈ j∈J E i × {α} with J ⊆ {1, . . . , q}. Then α is contained in all hyperplanes {H E j } j∈J , i.e., α ∈ j∈J H E j . In particular, S F ⊆ q
j=1 H E j and hence also the critical slopes of F are among the H E j .
Proof. Since E is a SNC divisor, the locally free sheaf Ω 1 X on X extends to the locally free sheaf Ω 1
Y (log E) ⊆ j * Ω 1 X on Y , for which Ω 1 Y (log(E))| X = Ω 1 X .
It can be verified in local coordinates that for all α ∈ C p , the form dlog
f α ∈ H 0 (Y, j * Ω 1 X ) lies in H 0 (Y, Ω 1 Y (log E))
. We then have the following commutative diagram:
C p H 0 Y, Ω 1 Y (log E) H 0 X, Ω 1 Y (log E) = H 0 X, Ω 1 X
The horizontal arrow defines the incidence variety
C F,Y (log E) := {(y, α) | dlog f α (y) = 0} ⊆ Y × P p−1 ,
where dlog f α is regarded as a global section of Ω 1 Y (log E), and hence dlog f α (y) is an element of the fiber of this locally free sheaf over y. We refer to C F,Y (log E) as the logarithmic critical locus of F . This is a closed, possibly reducible, subvariety of Y × P p−1 . By the commutativity of the diagram above,
C F ⊆ C F,Y (log E) and hence also C F ⊆ C F,Y ⊆ C F,Y (log E).
Let y ∈ j∈J E j . Assume for simplicity of notation that J = {1, . . . , r}. Take local coordinates y 1 , . . . , y n on a small open U around y in which E i ∩ U = {y i = 0} for i = 1, . . . , r. Then in these coordinates
df i f i = d(u i y ord E 1 (f i ) 1 · · · y ord Er (f i ) r ) u i y ord E 1 (f i ) 1 · · · y ord Er (f i ) r = du i u i + ord E 1 (f i ) dy 1 y 1 + · · · + ord Er (f i ) dy r y r ,
where the u i are non-vanishing functions on U . Summing up, we conclude that
dlog f α = θ(α) + r j=1 (ord E j (f 1 )α 1 + · · · + ord E j (f p )α p )dy j y j ,(1)
where θ(α) is a regular 1-form around y, which we write in coordinates as n i=1 θ i (α)dy i . We will denote q j (α) = ord E j (f 1 )α 1 +· · ·+ord E j (f p )α p . In the frame dy 1 /y 1 , . . . , dy r /y r , dy r+1 , . . . , dy n for Ω 1 Y (log E) we conclude that dlog f α is given by y 1 θ 1 (α)+q 1 , . . . , y r θ r (α)+q r , θ r+1 (α), . . . , θ n (α), and thus these are a system of defining
equations for C F,Y (log E) ∩ π −1 1 (U ). It follows that C F,Y (log E) ∩ π −1 1 (∩ j∈J E j ∩ U ) = V (y 1 , . . . , y r , q 1 , . . . , q r , θ r+1 , . . . , θ n ) ⊆ (∩ j∈J E j ∩ U ) × ∩ j∈J H E j .(2)
Since y ∈ j∈J E j was arbitrary, we conclude that
C F,Y (log E) ∩ π −1 1 (∩ j∈J E j ) ⊆ (∩ j∈J E j ) × ∩ j∈J H E j and thus also C F,Y ∩ π −1 1 (∩ j∈J E j ) ⊆ (∩ j∈J E j ) × ∩ j∈J H E j .
For a morphism γ : ∆ → Y × P p−1 as in the statement of the proposition, we thus conclude that
γ(0) ∈ C F,Y ∩ (∩ j∈J E j × {α}) = C F,Y ∩ π −1 1 (∩ j∈J E j ) ∩ π −1 2 (α) ⊆ π −1 1 (∩ j∈J E j ) × ∩ j∈J H E j ∩ {α} .
In particular, (∩ j∈J H E j ) ∩ {α} is non-empty so that indeed α ∈ ∩ j∈J H E j . The last claim of the lemma follows from the first statement and Lemma 2.6.
Remark 2.11. In the above proof, the reason we work with
C F,Y (log E) is simply to give a bound for C F,Y . The upshot of C F,Y (log E) is that C F,Y (log E) ∩ π −1 1 (∩ j∈J E j ∩ U )
is easy to compute, as in Equation (2). On the other hand, computing
C F,Y ∩ π −1 1 (∩ j∈J E j ∩ U ), which is our actual goal, is much more difficult. The reason for this difference is essentially that C F,Y is equal to C F,Y (log E) \ π −1 1 (E)
, and hence is defined by a saturation of the ideal defining C F,Y (log E). The ideal defining C F,Y (log E) can be written down fairly explicitly as in the proof above, but we do not have a general method to analyze the saturation.
We illustrate with an example that in general there is a difference between C F,Y and
C F,Y (log E). Consider the variety U = C 2 \V (f 1 f 2 ), with f 1 = x and f 2 = xy−1. U can be compactified in P 2 , with boundary components V (x), V (xy − z 2 )
, and V (z). We can perform blowups with exceptional locus lying over V (z) to make the boundary E SNC. On can compute that
C F,Y ∩ π −1 1 (C 2 ) = {y = 0} × {s 2 = 0}, while C F,Y (log E) ∩ π −1 1 (C 2 ) = {y = 0} × {s 2 = 0} ∪ {x = 0} × {s 2 = 0}.
Maximum likelihood estimation on schön very affine varieties
In this section, we investigate schön very affine varieties. Those varieties allow for a smooth compactification with SNC boundary obtained from combinatorial data, namely from their tropical variety. For background in Tropical Geometry and tropical compactifications, we refer the reader to [24], [31], and [23]. In particular, we refer to [24, Definitions 3.1.1 and 3.2.1] for the definition of the tropical variety and to [24,Theorem 3.2.3] for equivalent characterizations. We analyze the MLE problem in terms of Tropical Geometry. The main results of this section are Theorem 3.7 and Proposition 3.13 which completely determine the critical slopes from tropical data.
3.1. Zeroes of logarithmic differential forms. Let X ⊆ (C * ) p be a smooth closed subvariety of the algebraic p-torus. We denote by t 1 , . . . , t p the coordinate functions on (C * ) p and by f i := t i | X their restrictions to X. We will assume that there exists a fan structure Σ on Trop(X) such that the closure X Σ of X in the associated toric variety T Σ is proper and smooth and X Σ \ X = X Σ ∩ (T Σ \ (C * ) p ) is a reduced simple normal crossings divisor. If X is schön, it admits such a compactification (see proof of [16,Theorem 2.5]). We denote by {τ i } i∈I the rays in Σ and we denote the primitive ray generator of the ray τ by v τ ∈ Z p .
The irreducible components of the boundary E = X Σ \X are partitioned by the rays in Σ. Those corresponding to τ are the irreducible components {E τ,i } of E τ := X Σ ∩O τ , where O τ ⊆ T Σ is the locally closed torus orbit corresponding to τ . Recall that an alternative characterization of schön very affine varieties is as those closed subvarieties of a torus for which there is a fan structure Σ on Trop(X) such that the multiplication map m : X Σ ×(C * ) p → T Σ is smooth. In particular, it follows from this characterization that every E τ is smooth, and hence if E τ is not irreducible, it is a disjoint union of smooth irreducible components.
By construction of T Σ we have that ord Oτ (t j ) = v τ j . Since E τ is reduced by assumption, it follows that also for every component
E τ,i ⊆ E τ , ord E τ,i (f j ) = (v τ ) j . In particular, for each such E τ,i , H E τ,i equals P(τ ⊥ ), the hyperplane orthogonal to τ .
Recall that for each α ∈ C p , the form dlog f α extends to a global section of Ω 1 X Σ (log E), i.e., a differential one-form on X Σ with logarithmic poles along E. As in the proof of Proposition 2.10, we denote by C F,X Σ (log E) the logarithmic critical locus of F, i.e.,
C F,X Σ (log E) = {(x, α) | dlog f α (x) = 0} ⊆ X Σ × P p−1 .
As also stated there, we have have the following containments:
C F ⊆ C F,X Σ = C F,X Σ ⊆ C F,X Σ (log E).
A priori, C F,X Σ (log E) can have irreducible components that are contained in π −1 1 (E). However, the following lemma assures that in this situation this cannot be the case.
Lemma 3.1 ([18]). The logarithmic critical locus C F,X Σ (log E) is smooth and irre- ducible. Thus C F,X Σ (log E) equals C F,X Σ . Proof. The morphism Ω 1 T Σ (log(T Σ \T))| X Σ → Ω 1 X Σ (log E) is surjective. This morphism is identified with the morphism O p X Σ → Ω 1 X Σ (log E) mapping the ith generator to dlog f e i .
It follows that the kernel of this morphism has constant rank equal to p − n and thus is locally free. Since C F,X Σ (log E) is the projectivization of the total space of the vector bundle corresponding to the kernel, it is smooth and irreducible.
Remark 3.2. These properties of the asymptotic critical locus are not intrinsic to X, but depend on the chosen compactification. For example, in [9], the authors investigate the singularities of C F,P n for hyperplane arrangement complements that are not closed in a tropical compactification but are closed in projective space.
Corollary 3.3. Let (x, α) ∈ C F,X Σ (log E) ∩ π −1 1 (∩ j∈J E τ j ) with J maximal.
Then the sum of the primitive ray generators {v τ j } j∈J lies in Q F,α , i.e., j∈J v τ j ∈ Q F,α .
Proof. By Lemma 3.1, C F,X Σ (log E) is equal to C F , and C F,X Σ (log E) is a projective space bundle over X Σ . Assume for the sake of notation that J = {1, . . . , r}. Take local coordinates y 1 , . . . , y n on U ⊆ X Σ centered at x in which E i ∩ U = {y i = 0} for i = 1, . . . , r. Take a trivialization of C F,X Σ (log E) over U so that
Ψ : C F,X Σ (log E) ∩ π −1 1 (U ) ∼ = U × P p−n−1 .
Denote the image of (x, α) under Ψ by (x, q), q ∈ P p−n−1 . Consider then the curve
γ : ∆ → C F,X Σ (log E), t → Ψ −1 ((t, .
. . , t r times , 0, . . . , 0 n−r times , q)).
By construction, γ(0) = (x, α). It follows immediately from the definition of this γ that ord t (γ * (π * 1 (f i ))) = j∈J ord E j (f i ) = j∈J (v τ j ) i , which proves the claim.
Let α ∈ H Eτ . The local expression for the logarithmic differential form dlog f α that we computed in Equation 1 shows that dlog f α has residue 0 along every irreducible component of E τ . Hence we can restrict dlog f α to E τ to obtain
dlog f α | Eτ ∈ H 0 E τ , Ω 1 Eτ (log ((E \ E • τ ) ∩ E τ )) .
We define
C F,X Σ ,Eτ := {(x, α) ∈ E τ × H Eτ | dlog f α | Eτ (x) = 0} ⊆ E τ × H Eτ .
By a local computation, one obtains the following description of C F,X Σ ,Eτ .
Lemma 3.4. Considering C F,X Σ ,Eτ as a subvariety of X Σ × P p−1 via the inclusion E τ × H Eτ ֒→ X Σ × P p−1 , C F,X Σ ,Eτ coincides with C F,X Σ (log E) ∩ π −1 1 (E τ ).
Proof. We work around a point y ∈ k i=1 E i , with E τ = E 1 , and such that y is not contained in any other boundary component. We take coordinates y 1 , . . . , y n on a small open U around y such that E i ∩ U = {y i = 0} for i = 1, . . . , k. The local expression for dlog f α from Equation 1 gives
dlog f α = k i=1 y i θ i (α) + q i y i dy i + n i=k+1 θ i (α)dy i .
On the other hand, y 2 , . . . , y n are coordinates on E τ ∩ U , and the form dlog f α | Eτ has the local expression
dlog f α | Eτ = k i=2 y i θ i (α) + q i y i dy i + n i=k+1 θ i (α)dy i .
This shows that C F,X Σ ,Eτ and C F,X Σ (log E) ∩ π −1 1 (E τ ) are defined by the same equations.
We will denote E • τ := E τ \ τ ′ =τ E τ ′ = X Σ ∩ O τ . Corollary 3.5. Let α ∈ H Eτ . If dlog f α | Eτ has a zero on E • τ , then v τ ∈ Q F,α .
Proof. If dlog f α | E • τ has a zero x ∈ E • τ , then (x, α) ∈ C F,X Σ ,Eτ , and thus by Lemma 3.4, (x, α) ∈ C F,X Σ (log E). The claim then follows from Corollary 3.3.
We will use the following theorem of Huh in order to apply Corollary 3.5. In this theorem, smoothness is a necessary assumption-the failure of the statement for singular X is explained in [7]. Notice that the boundary components E • τ are disjoint unions of smooth very affine varieties, since they are contained in the torus orbit O τ ∼ = (C * ) p−1 . Hence, Huh's theorem also applies to E • τ . Since the torus orbit O τ is naturally a quotient of (C * ) p , its coordinate ring naturally is a subring of the coordinate ring of (C * ) p . Under this identification, when applying Theorem 3.6 to E • τ ⊆ O τ , the vector space C p−1 in which the α live is naturally identified with τ ⊥ . When applying this theorem to X (and E • τ , resp.), we denote the Zariski dense subset that appears in the theorem by V ⊆ C p (and by V τ ⊆ τ ⊥ , resp.).
Theorem 3.7. Assume that χ(E • τ ) = 0. Then for all α ∈ V τ , the primitive ray generator v τ is contained in Q F,α . In particular, H Eτ ⊆ S F . Moreover, these are the only codimension-one components of S F .
Proof. Since E • τ is smooth and very affine, for every α ∈ V τ the form dlog f α | E • τ has exactly (−1) dim Eτ χ(E • τ ) zeroes, all contained in E • τ by Theorem 3.6. The first claim follows from Corollary 3.5. Since V τ is dense inside τ ⊥ and S F is closed, we conclude that
H Eτ = P(τ ⊥ ) ⊆ S F .
We prove the last claim by reasoning the other way round. We recall that by Proposition 2.10, every codimension-one component of S F is of the form P(τ ′⊥ ) for some ray τ ′ ∈ Σ. Let τ ′ be some ray in Σ and suppose that H E τ ′ ⊆ S F . Then for all
α ∈ H E τ ′ , there exists γ : ∆ → C F,X Σ with γ(∆ • ) ∈ C F and γ(0) ∈ E × {α}. Choosing α away from τ ′′ =τ ′ H E τ ′′ , it follows from Proposition 2.10 that γ(0) ∈ E • τ ′ × {α}. Since γ(0) ∈ C F,X Σ ∩π −1 1 (E • τ ′ ) it follows from Lemma 3.4 that γ(0) ∈ C F,X Σ ,E τ ′ ,
and thus dlog f α | E τ ′ (π 1 (γ(0))) = 0. We conclude that for all α ∈ H E τ ′ the form dlog f α | E τ ′ has a zero on E • τ ′ . Then by Theorem 3.6, E • τ ′ has non-zero Euler characteristic. Remark 3.8. Smooth very affine varieties with zero Euler characteristic can still have "resonant" α for which dlog f α has a zero. In this case, strict subvarieties of some H Eτ might show up as an irreducible component of S F . For relations to resonance varieties of hyperplane arrangements, we refer to [10] and the references therein.
We denote by I ⊳ C[t ±1 1 , . . . , t ±1 p ] the defining ideal of X. For w ∈ R n , in w (I) denotes the initial ideal of I as defined in [24, Section 1.6]. The following lemma gives a description of the components E • τ in terms of initial ideals.
Lemma 3.9. Let τ be a ray in Trop(X) with primitive ray generator w. Under the natural isomorphism O τ ∼ = (C * ) p /T w , the intersection X Σ ∩ O τ corresponds to V (in w (I))/T w .
Proof. For a proof, see [24, page 308].
Recall that we started by taking a fan Σ supported on the tropical variety of X. In general, there are many such Σ and there is no coarsest fan structure on Trop(X). Despite the lack of a coarsest fan structure, Theorem 3.7 guarantees that some rays are present in every fan structure, namely those for which E • τ = X Σ ∩ O τ has non-zero Euler characteristic. We now characterize these rays in terms of initial ideals, under a connectedness assumption. In order to decide in practice if a ray is rigid, one has has to compare the initial ideal of the defining ideal of X w.r.t. this ray with the initial ideal w.r.t. a relative interior point of all neighboring cones. Rigid rays can also be characterized as follows. Proof. The ray τ is rigid if and only if in w (I) is not homogeneous with respect to any weight vector other than scalar multiples of w. In other words, τ is rigid if and only if in w (I) is not preserved under any non-trivial subtorus of (C * ) p other than T w := {(t w 1 , . . . , t wp ) | t ∈ C * }. The claim then follows from Lemma 3.9.
Lemma 3.12. In the situation as above, the following two statements hold true.
(i) If χ(E • τ ) = 0, then E • τ is not preserved by any non-trivial subtorus of O τ . (ii) Assume that E • τ is connected. If E • τ is not preserved by any non-trivial subtorus, then χ(E • τ ) = 0.
Proof. Denote by T the non-trivial subtorus of O τ that preserves E • τ . The action of T on E • τ is free and hence so is the action of the group Z/mZ of m-th roots of unity for all m ∈ Z >0 . Hence E
• τ → E • τ /(Z/mZ) is an m-fold cover. Hence χ(E • τ ) = m · χ(E • τ /(Z/mZ)).
In particular, χ(E • τ ) is divisible by every non-negative number m and hence is 0, concluding the proof of the first statement. Since X is schön, E • τ is smooth, and by connectedness it is thus irreducible. Let F be the irreducible perverse sheaf In summary, we obtain the following implications:
ι * (C E • τ [dim E • τ ]), where ι denotes the inclusion E • τ ֒→ O τ .τ is rigid ⇔ E • τ is not invariant under any non-trivial subtorus of O τ ⇐ χ (E • τ ) = 0,
where the last implication is a bi-implication if we assume in addition that E • τ is connected. Combining this with Theorem 3.7, we obtain the following proposition. Proposition 3.13. Assume that for all τ ∈ Σ the intersection E • τ = X Σ ∩ O τ is connected. If τ is a rigid ray, then for all α ∈ V τ , v τ ∈ Q F,α . In particular, P(τ ⊥ ) ⊆ S F . This gives a bijection of the rigid rays in Trop(X) and the codimension-one components of S F . Example 3.14. We pick up the arrangement from Example 2.4, i.e., X ′ ⊆ C 2 is the complement of the arrangement defined by f = xy(x − y)(x − 1), the four factors of which form the tuple
F = (f 1 , f 2 , f 3 , f 4 ). Mapping X ′ into C 4 via the tuple F , its image is X = V (t 1 − t 4 − 1, t 1 − t 2 − t 3 ) .V (f 1 ) \ V (f 2 f 3 f 4 )
is isomorphic to C * and thus has zero Euler characteristic, or using the fact that the ray generated by e 1 is not rigid. We note that a compactification of X ′ is given by P 2 . To make the boundary of this compactification SNC we have to blow up the triple intersection points, namely the origin and a point on the hyperplane at infinity. Every ray τ in the Gröbner fan that we found above is then of the form P(τ ⊥ ) = H E i for some boundary divisor E i in this compactification. Since X ′ is a hyperplane arrangement complement, it makes sense to investigate the relation between the combinatorics of the arrangement and S F . Recall that the combinatorics of an arrangement are encoded in the its intersection lattice L, which is a poset with one vertex for every subspace that can be formed by intersecting some of the hyperplanes in the arrangement. These vertices are referred to as edges of the arrangement, and are ordered using the reverse inclusion. An edge S is a dense edge if the subposet L ≤S is not a non-trivial product of two posets. An edge S is a flacet if neither L ≤S nor L ≥S is a non-trivial product of two posets. The following diagram depicts the Hasse diagram of the arrangement:
{(0, 0)} {(1, 0)} {(1, 1)} {f 1 = 0} {f 2 = 0} {f 3 = 0} {f 4 = 0} C 2 .
We see from this diagram that there are 5 dense edges: {x = 0}, {y = 0}, {x = y}, {x = 1}, and {(0, 0)}. To understand the relation between the codimension-one components of S F and the intersection lattice, we regard X ′ as the complement of the projective hyperplane arrangementf = uv(u − v)(u − w)w inside P 2 and define f 5 := w. Denote by L the intersection lattice of the affine arrangement defined by the same equation. By [13,Theorem 2.7], the rays in the Gröbner fan on Trop(X) ⊆ R 4 are in bijection with the flacets of M . A concrete recipe to compute the rays is as follows: take a flacet F and for i = 1, . . . , 5 define (v F ) i = 1 if F ⊆ {f i = 0} and (v F ) i = 0 otherwise. Then add a multiple of (1, . . . , 1) to v F to make the last coordinate equal to 0, and then forget the last coordinate. In this example, all dense edges except for {f 1 = 0} are flacets, and it can be verified that the procedure outlined here indeed recovers the 6 rays that we found before.
Finally, let E 1 := V (f 1 ). We notice that for α ∈ {s 1 = 0}, the form dlog f α | E 1 equals (α 2 + α 3 )dy/y. It follows that the form dlog f α | E 1 has a zero on E 1 if and only if α ∈ {s 1 = s 2 + s 3 = 0}. By Corollary 3.5, for general such α, e 1 ∈ Q F,α . In conclusion, although E 1 does not contribute a codimension-one component to S F , it does contribute an embedded component, in the sense that π 2 C F,P 2 ∩ π −1 1 (E • 1 ) = {s 1 = s 2 + s 3 = 0} . △
Maximum likelihood estimation interpretation.
Suppose that X has nonzero Euler characteristic. Then by Theorem 3.6, for a general data vector α, the function f α has exactly (−1) dim X χ(X) many critical points. This number is also called the maximum likelihood degree of X and is denoted by d ML (X). We denote the Zariski open subset of P p−1 for which this statement holds true by V . Consider a rigid ray τ , for which we thus know that χ(E • τ ) = 0. Since the E • τ are smooth very affine varieties as well, we denote their set of general data vectors by V τ ⊆ τ ⊥ . By its very definition, the set Q F, α describes the behavior of critical points of f α as α approaches α. Keeping this in mind, Corollary 3.5 states that as α approaches an α ∈ V τ along a curve in V , at least one of the critical points of f α approaches the torus orbit corresponding to τ . In particular, its limit lies in X Σ \ X.
The asymptotic behavior of critical points as described above is particularly explicit in the d ML (X) = 1 case, i.e., when the signed Euler characteristic of X is 1. In this case the maximum likelihood estimate is unique and determined by the rational map ψ : P p−1 X, mapping α to the unique critical point of f α . Notice that ψ is the rational inverse to the birational morphism π 2 : C F,Y → P p−1 . Then ψ i := f i • ψ are rational functions on P p−1 . We now translate Theorem 3.7 into a statement about the ψ i . Note that v τ ∈ Q F,α if and only if there exists a morphism γ :
∆ → C F,X Σ with γ(∆ • ) ∈ C F , γ(0) ∈ (X Σ \ X) × {α}, and ord t (γ * (π * 1 F )) = v τ .
The following diagram gives an overview of the various objects and morphisms that we have defined:
∆ C F,X Σ C p X Σ P p−1 T Σ C γ π 2 ψ F π 1 ψ t i pr i f i ψ i .
By the definition of ψ, the morphisms π 1 • γ and ψ • π 2 • γ coincide. It follows that
ord t ((ψ • π 2 • γ) * F ) = v τ .
In other words, ord t ((ψ i • π 2 • γ) * f i ) = (v τ ) i . Since this is true for all α ∈ V τ , we conclude that ord H Eτ (ψ i ) = (v τ ) i . Moreover, ψ i has no other zeroes or poles besides the H Eτ . To see this, we note that the same argument shows that every additional pole or zero of ψ i induces a component of S F . Namely, suppose there is an additional variety R ⊆ P p−1 such that ord R (ψ i ) = 0. Then approaching a general point p of R along a curve γ, ψ i (γ(t)) will approach 0 or ∞, and in particular ψ(γ(t)) leaves X as t → 0. Hence, p ∈ S F and thus R ⊆ S F . However, by Theorem 3.7 all components of S F are of the form H Eτ for some rigid ray τ ∈ Trop(X). We deduce the following proposition.
Proposition 3.15. Let X be a schön very affine variety with d ML (X) = 1 and Σ a fan whose support is Trop(X) s.t. for all rays τ ∈ Σ, X Σ ∩ O τ is connected. For a rigid ray τ ∈ Σ, let g τ be a defining equation of τ ⊥ . Then there exist complex numbers c i such that
ψ i = c i τ rigid g (vτ ) i τ .
Since ψ i is homogeneous, the primitive generators of the rigid rays sum up to 0, i.e.,
τ rigid v τ = 0.
The fact that the rigid rays sum up to zero is a special property of schön very affine varieties with maximum likelihood degree one, as will be demonstrated in Example 3.19. Related structure results for general very affine variety with maximum likelihood degree one were obtained in [19] and [12].
Example 3.16. We continue Example 3.14. In this case χ(X) = 1. An explicit computation of the morphism ψ using Mathematica gives the four rational functions
ψ 1 = s 1 + s 2 + s 3 s 1 + s 2 + s 3 + s 4 , ψ 2 = s 2 (s 1 + s 2 + s 3 ) (s 2 + s 3 )(s 1 + s 2 + s 3 + s 4 ) , ψ 3 = s 3 (s 1 + s 2 + s 3 ) (s 2 + s 3 )(s 1 + s 2 + s 3 + s 4 ) , ψ 4 = −s 4 s 1 + s 2 + s 3 + s 4 ,
as predicted by Proposition 3.15. Note that the rigid rays sum up to zero. △
In order to formulate an analogous property for arbitrary maximum likelihood degree, we make use of the following lemma.
Lemma 3.17. Let τ be a rigid ray and α ∈ V τ . Assume that E • τ = X Σ ∩O τ is connected and that d ML (E • τ ) > 0. Let S ⊆ P p−1 be an irreducible smooth curve containing α whose generic point lies in V and which intersects H Eτ transversely. Then there exists a morphism γ : ∆ → C F,X Σ with γ(∆ • ) ∈ C F , γ(0) ∈ E • τ × {α}, π 2 (γ(∆)) ⊆ S, and ord t (γ * (π * 1 F )) = v τ .
Proof. Since α ∈ V τ and d ML (E • τ ) > 0, dlog f α | Eτ has a non-empty, zero-dimensional, reduced zero-scheme Z ⊆ E • τ , by Theorem 3.6. Let x ∈ Z. We conclude that (x, α) ∈ π −1 2 (S). Let C be an irreducible component of C F,X Σ ∩ π −1 2 (S) passing through (x, α). A local computation reveals that
C F,X Σ ∩ π −1 2 (S) ∩ π −1 1 (E τ ) = C F,X Σ ∩ π −1 2 ({α}) ∩ π −1 1 (E τ ) = Z.
It follows that C ∩ π −1 1 (E τ ) ⊆ Z is zero-dimensional, reduced, and non-empty since it contains (x, α). Hence any local equation g for π −1 1 (E τ ) around (x, α) gives a generator for the maximal ideal of O C,(x,α) . By Cohen's structure theorem, t → g gives an isomorphism C t ∼ = O C,(x,α) giving rise to the morphism γ : ∆ → C F,X Σ that we are looking for, since by construction ord t (γ * E τ ) = 1 and thus ord t (γ * (π * 1 F )) = v τ .
Remark 3.18. The difference between this construction and the construction in the proof of Corollary 3.3 is the fact that in this lemma we start by specifying a curve in P p−1 along which we approach α and then lift it to C F,X Σ . Note that the number of components of π −1 2 (S) passing through (x, α) is closely related to [20,Conjecture 3.19], which essentially predicts that there is only a single component passing through (x, α).
We deduce the following consequence for maximum likelihood estimation. Let α ∈ V τ and γ : ∆ → P p−1 with γ(∆ • ) ∈ V approaching α. Let S be a system of equations in C[t 1 , . . . , t p , s 1 , . . . , s p ] defining C F . Then substitute the components of γ for the s i -variables. The system S ′ obtained like that consists of equations in C t [t 1 , . . . , t p ]. A solutionγ ∈ C((t)) p of this system is a solution of the equation dlog f γ(t) (γ(t)) = 0. Lemma 3.17 assures that the system S ′ has a solution and that moreover it has a solution for which ord t (γ i ) = (v τ ) i for all i = 1, . . . , p. In practice, in order to approximate such a solution, one does a formal power series substitution x i = ∞ j=(vτ ) i c i,j t j into the system S ′ and iteratively solves for the c i,j .
Example 3.19. Let g ∈ C[x, y] be a generic conic through (0, 0) and write g = l+q as a sum of a linear and a quadratic part. We embed C 2 into C 3 via the tuple F = (x, y, g) and denote, as usual, f = xyg. Denote by X := F (C 2 ) ∩ (C * ) 3 the intersection of its image with (C * ) 3 . X is a hypersurface in (C * ) 3 cut out by the polynomial h := t 3 − g(t 1 , t 2 ). Since g is generic for its Newton polygon, X is schön (see [17,Section 2]). The rigid rays are given by e 1 , e 2 , e 3 , e 1 + e 2 + e 3 , and −e 1 − e 2 − 2e 3 . We remark that d ML (X) = 3 and that the rigid rays do not sum up to zero, demonstrating that the assumption on the maximum likelihood degree in Proposition 3.15 is indeed necessary. We now illustrate Lemma 3.17 for the schön very affine variety X as above. Consider for example the ray τ = −e 1 − e 2 − 2e 3 . The initial ideal in (−1,−1,−2) (h) is generated by t 3 −q(t 1 , t 2 ). By Lemma 3.9, it follows that X Σ ∩O τ is a copy of P 1 minus 4 points, and thus has Euler characteristic −2. By Theorem 3.6 and Corollary 3.5, this means that as one approaches a general α ∈ {s 1 +s 2 +2s 3 = 0}, at least two of the three maximum likelihood estimates of dlog t α will approach the torus orbit O τ . To make this more explicit, consider for instance g = x + y + x 2 + xy + y 2 , which turns out to be generic enough. Take the point (2, 1, −3/2) ∈ τ ⊥ and the curve t → (2 + t, 1 + t, −3/2) approaching it. We now notice that X equipped with the tuple (t 1 , t 2 , t 3 ) is isomorphic, via the map F , to C 2 \V (f ) with the tuple (x, y, g). To make our computations simpler we will work with the latter. On C 2 we compute that the two components of dlog f γ(0) are zero on C 2 \ V (f ) if and only if the following two equations are fulfilled:
x + 2tx − 2x 2 + 2tx 2 + 4y + 2ty + xy + 2txy + 4y 2 + 2ty 2 = 0, 2x + 2tx + 2x 2 + 2tx 2 − y + 2ty − xy + 2txy − 4y 2 + 2ty 2 = 0.
The fact that v τ = (−1, −1, −2) ∈ Q F,α implies that this system has a solution (η 1 , η 2 ) in t −1 C t , for which g(η 1 , η 2 ) = c −2 t −2 + higher order terms. With Mathematica, we compute that we indeed have a solution with
η 1 = −7 + √ 33 (15 − √ 33)t + O (1) , η 2 = −13 + 3 √ 33 (60 − 4 √ 33)t + O (1) .
This solution converges to (1 : 1 8 (−1 − √ 33) : 0) ∈ P 2 , which is one of the three critical points of f α on P 2 . The other ones are p 2 = (1 : 1 8 (−1+ √ 33) : 0) and p 3 = (3 : −3 : 1). For p 2 , we can construct a similar curve to the one found above. The point p 3 is a point in X and thus we get a solution in C t whose first terms are
(3 − 74t + 3508t 2 + O t 3 , −3 + 62t − 2948t 2 + O t 3 ).
△
We remark that in the previous example the following equality holds:
τ rigid χ (E • τ ) v τ = 0,
in analogy to the second equation in Proposition 3.15. It would be interesting to study if this equality holds true in general.
Bernstein-Sato ideals
In this section, we investigate Bernstein-Sato ideals and the codimension-one components of their vanishing sets. We explain how those can partially be recovered in terms of Q F,α and formulate a conjecture relating those codimension-one components to log-canonical threshold polytopes.
Slopes of Bernstein-Sato ideals.
Let Y ⊆ C p be a smooth closed subvariety the affine space. We consider the tuple of regular functions G = (g 1 , . . . , g p ) on Y consisting of the restriction of the coordinates t 1 , . . . , t p on C p to Y . We denote their product by g := p i=1 g i . We denote by X the very affine variety Y ∩ (C * ) p and on it we consider the tuple of nowhere vanishing functions F = (f 1 , . . . , f p ) consisting of the restriction of the coordinates t 1 , . . . , t p on C p to X. For a smooth affine algebraic variety with a tuple of regular functions, the Bernstein-Sato ideal is defined as follows.
Definition 4.1. The Bernstein-Sato ideal of the tuple G on Y , denoted by B G , is the ideal consisting of all polynomials b ∈ C[s 1 , . . . , s p ] for which there exists a differential operator P (s 1 , . . . , s p ) ∈ H 0 (Y, D Y [s 1 , . . . , s p ]) such that
P • g s 1 +1 1 · · · g sp+1 p = b · g s 1 1 · · · g sp p , where D Y [s 1 , . . . , s p ] := D Y ⊗ O Y O Y [s 1 , .
. . , s p ] is the sheaf of algebraic linear partial differential operators on Y with formal variables s 1 , . . . , s p adjoined.
Sabbah [28] showed that every codimension-one component of V (B G ) is an affine hyperplane. The set of Bernstein-Sato slopes of G, denoted by BS G , is defined to be the union of these hyperplanes after translating them to the origin. Since this is a homogeneous variety by definition, we will also consider the projective version of this variety, denoted by P(BS G ) ⊆ P p−1 .
W G := p i=1 α i dg i g i (x), α | x ∈ X, α ∈ C p ⊆ T * X × C p .
Then
BS G = π 2 W G T * Y ×C p ∩ V (π * 1 (π * g)) ,
where π 1 , π 2 are the first and second projection from T * Y × C p , and π : T * Y → Y is the natural map.
Note that the description of BS G in this theorem is very similar to the definition of S F . There are two differences between the objects. In the definition of S F , the second factor is equal to P p−1 rather than C p . The second difference is the fact that in the definition of S F , the set C F is closed inside a compactification of X, whereas in Theorem 4.2, the closure of W G is taken inside the non-compact affine variety Y . Hence there will be contributions to S F from boundary components at infinity that are not relevant from the perspective of the Bernstein-Sato ideal. The following lemma explains how to identify contributions to BS G in terms of Q F,α . Lemma 4.3. Let α ∈ P p−1 and denote by L α ⊆ C p the line through the origin corresponding to α.
If Q F,α ∩ Z p ≥0 = ∅, then L α ⊆ BS G .
Proof. Let i : Y ֒→ Y be a compactification of Y , and thus also of X. By the assumption, there exists γ :
∆ → Y × P p−1 with γ(∆ • ) ∈ C F , γ(0) ∈ Y \ X × {α},
and such that moreover ord t (γ * (π * 1 f i )) ≥ 0 for i = 1, . . . , p. Notice that γ(∆ • ) ∈ C F implies that π 1 (γ(∆ • )) ∈ X ⊂ (C * ) p . Then the fact that ord t (γ * (π * 1 f i )) ≥ 0 implies that π 1 (γ(0)) ∈ Y ∩ C p = Y . This means that π 1 (γ(0)) ∈ Y \ X = V (g). Now let α ∈ L α . By the local triviality of the tautological line bundle, we get a lift η : ∆ → C p of π 2 • γ with η(0) =α. Then we obtain
∆ (π 1 •γ,η) −→ Y × C p −→ T * Y Y × C p ,
where T * Y Y denotes the zero section of T * Y . By construction, for (y, β) in the image of (π 1 • γ, η), dlog g β (y) is in T * Y Y. Hence the image of the morphism (π 1 • γ, η) lies in W G . Since π 1 (γ(0)) ∈ V (g), we conclude that (π 1 (γ(0)), η(0)) ∈ W G ∩ V (g) and thus, by Theorem 4.2,α ∈ BS G .
We denote by P(BS G ) the projectivization of the hyperplanes in BS G , and by X Σ a tropical compactification of X as in Section 3.
Theorem 4.4.
Assume that X is schön and that for all τ ∈ Σ the intersection X Σ ∩O τ is connected. Then the codimension-one irreducible components of S F ∩ P(BS G ) are exactly the hyperplanes P(τ ⊥ ) for τ a rigid ray contained in R p ≥0 .
Proof. By Proposition 3.13, the codimension-one components of S F are exactly the hyperplanes P(τ ⊥ ) for τ a rigid ray. Again by Proposition 3.13, all α ∈ P(τ ⊥ ) satisfy the condition of Lemma 4.3. Thus, L α is contained in BS G and thus α lies in P(BS G ). Hence P(τ ⊥ ) ⊆ P(BS G ). We now show that these indeed recover all components in the intersection. As in Section 3, denote by Σ a fine enough fan structure on Trop(X) such that the closure X Σ of X in the associated toric variety is smooth and the boundary X Σ \ X is a SNC divisor. Starting from an arbitrary fan we can always refine it to obtain a fan satisfying this condition, as in [16, proof of 2.5].
Denote by Σ C p the standard fan for C p . Let Σ ′ be a refinement of Σ of the type mentioned above for which the following is true: if the relative interiors of two cones σ 1 ∈ Σ ′ and σ 2 ∈ Σ C p intersect, then σ 1 ⊆ σ 2 . Denote by X Σ ′ the closure of X in the associated toric variety T Σ ′ . LetẊ be the variety obtained by removing from X Σ ′ all divisors corresponding to rays in Σ ′ \ R p ≥0 . This gives a log-resolution µ :Ẋ → Y of the divisor of g inside Y . Moreover, the irreducible components of the divisor µ −1 (V (g)) are in bijection with the rays in Σ ′ ∩ R p ≥0 . By [6,Lemma 4.4.6], the irreducible components of BS G are among the hyperplanes orthogonal to the rays in Σ ′ ∩ R p ≥0 . Let C be any component of S F ∩ P(BS G ). Since C is in P(BS G ), we find that C = P(τ ⊥ ) for some ray τ ∈ Σ ′ ∩ R p ≥0 . Since C is also a component of S F , this ray must be rigid. It follows that C is indeed of the form P(τ ⊥ ) for some rigid ray in R p ≥0 , concluding the proof.
Example 4.5. We pick up Example 3.16. Using the library dmod.lib [22] in the computer algebra system Singular [11], we compute the Bernstein-Sato slopes to be
BS G = {s 1 + s 2 + s 3 = 0} ∪ 4 i=1 {s i = 0}.
It follows that S F intersected with the projectivized hyperplanes of BS G equals
S F ∩ P (BS G ) = {s 1 + s 2 + s 3 = 0} ∪ 4 i=2 {s i = 0} .
In particular, the components {s 1 + s 2 + s 3 + s 4 = 0} and {s 2 + s 3 = 0} are contained in S F but not in BS G . This is explained by Theorem 4.4 since the rigid rays introducing these components to S F are −e 1 − e 2 − e 3 − e 4 and −e 2 − e 3 , which are not contained in R 4 ≥0 . On the other hand, the component {s 1 = 0} is contained in BS G but not in S F . This shows that in general the sets of components of BS G and S F are incomparable. Finally, we combine Theorem 4.4 with Proposition 3.15. We then find that the linear forms appearing in the numerators and denominators of the MLE as determined in Example 3.16 are among the defining equations for the hyperplanes in BS G . This explains the geometry behind the observation made in [29] suggesting a link between the MLE and the Bernstein-Sato ideal of the parametrization of the algebraic model of the statistical experiment. △
LCT-polytopes.
In the previous subsection, we partially recovered the Bernstein-Sato slopes in terms of the critical slopes of the very affine variety. We now formulate a conjecture related to the translations using log-canonical threshold (LCT) polytopes.
Definition 4.6. Let µ : Y ′ → Y be a log-resolution of V (g) with µ −1 (V (g)) = q i=1 E i . Denote by a ij := ord E i (g j ) and by k i := ord E i (K Y ′ /Y ) + 1, where K Y ′ /Y denotes the relative canonical divisor. The LCT-polytope of G is We call an affine hyperplane H ⊆ C p facet-defining if dim(H ∩ LCT(G)) = p − 1 and H ∩ LCT(G) ⊆ ∂(LCT(G)). The facet-defining hyperplanes are thus all of the form {ord E i (g 1 ) s 1 + · · · + ord E i (g p ) s p = k i } , but in general not all these hyperplanes are facet-defining. The relevance of the facetdefining hyperplanes is highlighted in the following theorem: If {ord E i (f 1 )s 1 + · · · + ord E i (f p )s p = k i } is a facet-defining hyperplane and ord E i (f j ) = 0 for all j = 1, . . . , p, then {ord E i (f 1 )s 1 + · · · + ord E i (f p )s p = −k i } is an irreducible component of V (B G ).
It follows from the proof of Theorem 4.4 that if X is schön we can construct a log resolution µ :Ẋ → Y of the divisor of g as the closure of X inside a toric variety with fan Σ ⊆ R p ≥0 ∩ Trop(X). In this log resolution the components of µ −1 (V (g)) correspond to the rays in Σ, and if τ is a ray in Σ, ord Eτ (g j ) = (v τ ) j . We denote by k τ the integer ord Eτ (K X Σ /X ) + 1. It follows that
LCT(G) = (s 1 , . . . , s p ) ∈ R p ≥0 | p j=1 (v τ i ) j s j ≤ k τ i for i = 1, . . . , q .
By Theorem 4.4, for every rigid ray τ in R p ≥0 , τ ⊥ ∈ BS G , i.e., τ ⊥ is a Bernstein-Sato slope of G. This means precisely that at least one affine translate of τ ⊥ lies in V (B G ). A natural candidate for such a translate is the one corresponding to the log-canonical threshold, leading to the following conjecture. Let Y ⊆ C p be a linear subspace such that the restrictions of the coordinate functions to Y define a central indecomposable hyperplane arrangement. In this situation, the rigid rays in Trop(X) are (1, . . . , 1) and (−1, . . . , −1). These rays are rigid since X is homogeneous and they are the only rigid rays since the arrangement is indecomposable. The matroid polytope of the matroid associated to the arrangement is the intersection of LCT(G) with the hyperplane H := {s 1 + · · · + s p = dim Y }. It is shown in [13] that the matroid polytope has dimension p − 1. Hence H is facetdefining and by Theorem 4.7 H is an irreducible component of V (B G ). This proves [3, Conjecture 3] for complete factorizations of hyperplane arrangements. Recall that a factorization of a hyperplane arrangement is complete if each of its factors is linear. △
Example: Flipping a biased coin
In this section, we pick up and continue the example of flipping a biased coin twice that was studied in [12] and [29]. Consider the smooth curve X in P 2 defined by the homogeneous polynomial f = det p 0 p 1 p 0 + p 1 p 2 = p 0 p 2 − (p 0 + p 1 )p 1 .
Funding. The second author is supported by a PhD Fellowship from FWO (Research Foundation -Flanders).
Theorem 3. 6
6([14, 18]). Let Z be a smooth very affine variety contained in a torus with coordinates t 1 , . . . , t p . Then there exists a Zariski dense open subset of C p , s.t. for all α in this subset, the form p i=1 α i dt i t i has exactly (−1) dim(Z) χ(Z) many zeroes on Z.
Definition 3 .
310. A ray R ≥0 · w ⊆ Trop(X) is called rigid if for all v ∈ R p \ Rw, in w (I) = in w+ǫv (I) for all 1 ≫ ǫ > 0.
Lemma 3 . 11 .
311The ray τ = R ≥0 · w ⊆ Trop(X) is rigid if and only if E • τ = X Σ ∩ O τ is not invariant under any non-trivial subtorus of O τ .
By
Lemma 3.9, all rays of the Gröbner fan on Trop(X) are rigid. They are generated by the vectors e 2 , e 3 , e 4 , −e 2 − e 3 , e 1 + e 2 + e 3 , and −e 1 − e 2 − e 3 − e 4 . It follows from Proposition 3.13 that the codimension-one part of S F equals {s 1 + s 2 + s 3 + s 4 = 0} ∪ {s 2 + s 3 = 0} ∪ {s 1 + s 2 + s 3 = 0} ∪ 4 i=2 {s i = 0} , coinciding with what was computed in Example 2.4. Notice that {s 1 = 0} does not show up in S F . There are two equivalent ways of explaining this: using the fact that
Theorem 4.2 ([25, Résultat 6]). Let
s 1 , . . . , s p ) ∈ R p ≥0 | p j=1 a ij s j ≤ k i for i = 1, . . . , q ⊆ R p ≥0 .
Conjecture 4 . 8 .
48For every rigid ray τ ∈ Trop(X)∩R p ≥0 , {(v τ ) 1 s 1 +· · ·+(v τ ) p s p = k τ } is facet-defining.
By [15, Theorem 0.3], F and hence also E • τ is preserved by a non-trivial subtorus of O τ .
Acknowledgments. We are grateful to Nero Budur, Johannes Nicaise, Yue Ren, and Bernd Sturmfels for insightful discussions. We thank the anonymous referees for carefully reading our article and the valuable feedback.As pointed out in[12,Example 2], this is the implicit representation of the statistical model describing the following experiment: Flip a biased coin. If it shows head, flip it again. Here, the p 0 , p 1 and p 2 are to be thought of as representing the probabilities of three possible outcomes of the experiment. Since these outcomes must sum up to one, we impose the additional condition that p 0 + p 1 + p 2 = 0 and hence consider the variety X \ H, where H is the collection of hyperplanes {p 0 p 1 p 2 (p 0 + p 1 + p 2 ) = 0} in P 2 . We embed X \ H into the 3-torus via the morphismWe denote by X the image of P and by Y the closure of X inside C 3 . The ideal defining X is generated by the two polynomialsThis ideal of C[t 0 , t 1 , t 2 ] is prime and hence Y is defined by the same equations. UsingGfan[21], we compute that the primitive ray generators in the Gröbner fan of Y are given by the three rows w 1 , w 2 , w 3 of the matrix Since the tropical variety of Y is one-dimensional, all three rays are indeed rigid. Hence, by Proposition 3.13, the hyperplanes orthogonal to them recover the codimension-one components of S F , where F = (t 0 | X , t 1 | X , t 2 | X ). In other words, the codimension-one part of S F is equal toThis model has maximum likelihood degree one. We denote the maximum likelihood estimator by ψ : P 2 X. As described in Proposition 3.15, each ψ i = f i • ψ is a rational function on P 2 whose numerator and denominator are products of the linear terms 2s 0 +s 1 , s 1 +s 2 , and 2s 0 +2s 1 +s 2 . More precisely, there exist complex constants c 1 , c 2 , c 3 such that ψ (s 0 , s 1 , s 2 ) = c 1 (2s 0 + s 1 ) 2 (2s 0 + 2s 1 + s 2 ) 2 , c 2 (2s 0 + s 1 ) (s 1 + s 2 ) (2s 0 + 2s 1 + s 2 ) 2 , c 3 (s 1 + s 2 ) (2s 0 + 2s 1 + s 2 ) .For c 1 = c 2 = c 3 = 1, this recovers the MLE computed in[12]. Finally, we compute the Bernstein-Sato ideal of the tuple of coordinate functions on Y . Under the isomorphismUsing Singular, we find that the Bernstein-Sato ideal of this tuple is generated byand thus the Bernstein-Sato slopes of G are BS G = V (2s 0 + s 1 ) ∪ V (s 1 + s 2 ) . Indeed, the components correspond to w 1 and w 2 . The ray w 3 does not contribute to the Bernstein-Sato slopes since it is not contained in R 3 ≥0 .
Combinatorially determined zeroes of Bernstein-Sato ideals for tame and free arrangements. D Bath, J. Singul. 20D. Bath. Combinatorially determined zeroes of Bernstein-Sato ideals for tame and free arrange- ments. J. Singul., 20:165-204, 2020.
Éventails associésà des fonctions analytiques. J Briançon, P Maisonobe, M Merle, Tr. Mat. Inst. Steklova. 238J. Briançon, P. Maisonobe, and M. Merle.Éventails associésà des fonctions analytiques. Tr. Mat. Inst. Steklova, 238:70-80, 2002.
Bernstein-Sato ideals and local systems. N Budur, Ann. Inst. Fourier (Grenoble). 652N. Budur. Bernstein-Sato ideals and local systems. Ann. Inst. Fourier (Grenoble), 65(2):549-603, 2015.
Estimates for zero loci of Bernstein-Sato ideals. N Budur, R Van Der, A Veer, Van Werde, arXiv:2111.03334PreprintN. Budur, R. van der Veer, and A. Van Werde. Estimates for zero loci of Bernstein-Sato ideals. Preprint arXiv:2111.03334, 2021.
Zero loci of Bernstein-Sato ideals. N Budur, R Van Der, L Veer, P Wu, Zhou, Invent. Math. 225N. Budur, R. van der Veer, L. Wu, and P. Zhou. Zero loci of Bernstein-Sato ideals. Invent. Math., 225:45-72, 2021.
Zero loci of Bernstein-Sato ideals-II. N Budur, R Van Der, L Veer, P Wu, Zhou, Selecta Math. 322021N. Budur, R. van der Veer, L. Wu, and P. Zhou. Zero loci of Bernstein-Sato ideals-II. Selecta Math. (N.S.), 27(32), 2021.
Bounding the maximum likelihood degree. N Budur, B Wang, Math. Res. Lett. 226N. Budur and B. Wang. Bounding the maximum likelihood degree. Math. Res. Lett., 22(6), 2014.
Multivariable Hodge theoretical invariants of germs of plane curves. P Cassou-Nogués, A Libgober, J. Knot Theory Ramifications. 2006P. Cassou-Nogués and A. Libgober. Multivariable Hodge theoretical invariants of germs of plane curves. J. Knot Theory Ramifications, 20(06):787-805, 2011.
Critical points and resonance of hyperplane arrangements. D Cohen, G Denham, M Falk, A Varchenko, Canad. J. Math. 635D. Cohen, G. Denham, M. Falk, and A. Varchenko. Critical points and resonance of hyperplane arrangements. Canad. J. Math., 63(5):1038-1057, 2011.
Vanishing products of one-forms and critical points of master functions. D C Cohen, G Denham, M Falk, A Varchenko, Arrangements of Hyperplanes -Sapporo. Tokyo, JapanMathematical Society of JapanD. C. Cohen, G. Denham, M. Falk, and A. Varchenko. Vanishing products of one-forms and critical points of master functions. In Arrangements of Hyperplanes -Sapporo 2009, pages 75- 107, Tokyo, Japan, 2012. Mathematical Society of Japan.
Singular 4-1-3 -A computer algebra system for polynomial computations. W Decker, G.-M Greuel, G Pfister, H Schönemann, W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann. Singular 4-1-3 -A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, 2020.
Discrete Statistical Models with Rational Maximum Likelihood Estimates. E Duarte, O Marigliano, B Sturmfels, Bernoulli. 2712021E. Duarte, O. Marigliano, and B. Sturmfels. Discrete Statistical Models with Rational Maximum Likelihood Estimates. Bernoulli, 27(1):135-154, 02 2021.
Matroid polytopes, nested sets and Bergman fans. E M Feichtner, B Sturmfels, Port. Math. Nova Série. 624E. M. Feichtner and B. Sturmfels. Matroid polytopes, nested sets and Bergman fans. Port. Math. Nova Série, 62(4):437-468, 2005.
The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varietes. J Franecki, M Kapranov, Duke Math. J. 1041J. Franecki and M. Kapranov. The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varietes. Duke Math. J., 104(1):171-180, 2000.
Faisceaux pervers ℓ-adiques sur un tore. O Gabber, F Loeser, Duke Math. J. 833O. Gabber and F. Loeser. Faisceaux pervers ℓ-adiques sur un tore. Duke Math. J., 83(3):501-606, 1996.
The homology of tropical varieties. P Hacking, Collect. Math. 59P. Hacking. The homology of tropical varieties. Collect. Math., 59:263-273, 2007.
Newton polyhedra and toroidal varieties. Funkcional'nyi Analiz i ego Priloženija. A Hovanskiǐ, 11A. Hovanskiǐ. Newton polyhedra and toroidal varieties. Funkcional'nyi Analiz i ego Priloženija, 11:56-64, 1977.
The maximum likelihood degree of a very affine variety. J Huh, Compos. Math. 1498J. Huh. The maximum likelihood degree of a very affine variety. Compos. Math., 149(8):1245-1266, 2013.
Varieties with maximum likelihood degree one. J Huh, J. Algebr. Stat. 5J. Huh. Varieties with maximum likelihood degree one. J. Algebr. Stat., 5:1-17, 2014.
Likelihood geometry. J Huh, B Sturmfels, Combinatorial algebraic geometry. New YorkSpringer2108J. Huh and B. Sturmfels. Likelihood geometry. In Combinatorial algebraic geometry, volume 2108 of Lecture notes in mathematics, pages 63-117. Springer, New York, 2014.
Gfan, a software system for Gröbner fans and tropical varieties. A N Jensen, A. N. Jensen. Gfan, a software system for Gröbner fans and tropical varieties. Available at http://home.imf.au.dk/jensen/software/gfan/gfan.html.
dmod lib: A Singular:Plural library for algorithms for algebraic D-modules. V Levandovskyy, J Martín-Morales, V. Levandovskyy and J. Martín-Morales. dmod lib: A Singular:Plural library for algorithms for algebraic D-modules. https://www.singular.uni-kl.de/Manual/4-2-0/sing_537.
Some results on tropical compactifications. M Luxton, Z Qu, Trans. Amer. Math. Soc. 3639M. Luxton and Z. Qu. Some results on tropical compactifications. Trans. Amer. Math. Soc., 363(9):4853-4876, 2011.
Introduction to tropical geometry, volume 161 of Graduate studies in mathematics. D Maclagan, B Sturmfels, American Mathematical SocietyProvidence, R.I.D. Maclagan and B. Sturmfels. Introduction to tropical geometry, volume 161 of Graduate studies in mathematics. American Mathematical Society, Providence, R.I., 2015.
Filtration relative, l'idéal de Bernstein et ses pentes. P Maisonobe, hal-01285562v2P. Maisonobe. Filtration relative, l'idéal de Bernstein et ses pentes. 2016. hal-01285562v2.
L'idéal de Bernstein d'un arrangement libre d'hyperplans linéaires. P Maisonobe, arXiv:1610.03356P. Maisonobe. L'idéal de Bernstein d'un arrangement libre d'hyperplans linéaires. arXiv:1610.03356, 2016.
Probabilistic algorithm for computing data-discriminants of likelihood equations. J I Rodriguez, X Tang, Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation. the 2015 ACM on International Symposium on Symbolic and Algebraic ComputationJ. I. Rodriguez and X. Tang. Probabilistic algorithm for computing data-discriminants of like- lihood equations. Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, 2015.
Proximitéévanescente. I. La structure polaire d'un D-module. C Sabbah, Compos. Math. 623C. Sabbah. Proximitéévanescente. I. La structure polaire d'un D-module. Compos. Math., 62(3):283-328, 1987.
A.-L Sattelberger, B Sturmfels, arXiv:1910.01395D-modules and holonomic functions. A.-L. Sattelberger and B. Sturmfels. D-modules and holonomic functions. arXiv:1910.01395, 2019.
Likelihood equations and scattering amplitudes. B Sturmfels, S Telen, Algebraic Statistics. 122B. Sturmfels and S. Telen. Likelihood equations and scattering amplitudes. Algebraic Statistics, 12(2):167-186, 2021.
Compactifications of subvarieties of tori. J Tevelev, Amer. J. Math. 1294J. Tevelev. Compactifications of subvarieties of tori. Amer. J. Math., 129(4):1087-1104, 2007.
Bernstein-Sato ideals and hyperplane arrangements. L Wu, J. Pure Appl. Algebra. 226106987L. Wu. Bernstein-Sato ideals and hyperplane arrangements. J. Pure Appl. Algebra, 226:106987, July 2022.
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"Thermalization of boosted charged AdS black holes by an ionic Lattice"
] | [
"Akihiro Ishibashi \nDepartment of Physics\nKinki University\nHigashi-Osaka577-8502Japan\n",
"Kengo Maeda \nFaculty of Engineering\nShibaura Institute of Technology\n330-8570SaitamaJapan\n"
] | [
"Department of Physics\nKinki University\nHigashi-Osaka577-8502Japan",
"Faculty of Engineering\nShibaura Institute of Technology\n330-8570SaitamaJapan"
] | [] | We investigate thermalization process of boosted charged AdS black holes in the Einstein-Maxwell system in the presence of an ionic lattice formed by spatially varying chemical potential. We calculate perturbations of the black holes by the lattice and investigate how the momentum relaxation occurs through umklapp scattering. In the WKB approximation, both of the momentum relaxation rate and entropy production rate are analytically obtained and the first law of black holes is derived in the irreversible process. Interestingly, both the analytical and numerical calculations show that the momentum relaxation rate or the entropy production rate does not approach zero in the zero temperature limit unless the velocity of the black hole is zero. In the dual field theory side, this indicates that persistent current does not exist even in the zero temperature limit, implying that the "ionic lattice" does not behave as a perfect lattice in a strongly coupled dual field theory. | 10.1103/physrevd.88.066009 | [
"https://arxiv.org/pdf/1308.5740v2.pdf"
] | 118,748,547 | 1308.5740 | 9e1f70f1d8676b08b8255f0dbd3863ec7b8d4a29 |
Thermalization of boosted charged AdS black holes by an ionic Lattice
13 Sep 2013 (Dated: May 11, 2014)
Akihiro Ishibashi
Department of Physics
Kinki University
Higashi-Osaka577-8502Japan
Kengo Maeda
Faculty of Engineering
Shibaura Institute of Technology
330-8570SaitamaJapan
Thermalization of boosted charged AdS black holes by an ionic Lattice
13 Sep 2013 (Dated: May 11, 2014)arXiv:1308.5740v2 [hep-th]PACS numbers:
We investigate thermalization process of boosted charged AdS black holes in the Einstein-Maxwell system in the presence of an ionic lattice formed by spatially varying chemical potential. We calculate perturbations of the black holes by the lattice and investigate how the momentum relaxation occurs through umklapp scattering. In the WKB approximation, both of the momentum relaxation rate and entropy production rate are analytically obtained and the first law of black holes is derived in the irreversible process. Interestingly, both the analytical and numerical calculations show that the momentum relaxation rate or the entropy production rate does not approach zero in the zero temperature limit unless the velocity of the black hole is zero. In the dual field theory side, this indicates that persistent current does not exist even in the zero temperature limit, implying that the "ionic lattice" does not behave as a perfect lattice in a strongly coupled dual field theory.
I. INTRODUCTION
The AdS/CFT correspondence [1] is a useful tool to study strongly correlated condensed matter systems using a classical theory of general relativity in asymptotically AdS spacetime. For example, charged AdS black holes dressed with a complex scalar field have been investigated as a holographic gravity model of a superfluid or a superconductor (See Ref. [2] for a nice review of holographic superconductor models). Although typical phenomena of superconductors such as energy gap are explained by the gravity model, there was a serious defect in the model; there is no mechanism of momentum dissipation, as translational invariance is imposed. This implies that even in the non-superconducting state, current is conserved and the DC-conductivity becomes infinite.
In condensed matter systems, momentum conservation is not generally satisfied via umklapp scattering process since translational invariance is broken by the presence of an ionic lattice or impurity. To incorporate the dissipation mechanism in the holographic model, charged AdS black holes with no translational symmetry has been constructed perturbatively or numerically in the non-superconducting state [3][4][5]. In Refs. [4,5], conductivity in the normal phase has been investigated and it was shown that the Drude peak appears instead of delta function at zero frequency. This indicates that the holographic model yields finite DC-conductivity in the normal phase. In the superconducting state, charged hairly AdS black hole solutions with an ionic lattice has been constructed and it has been shown that delta function appears, indicating the existence of a superfluid component [6]. This agrees with an earlier result derived from a toy model of holographic superconductors, showing that the delta function is stable against adding a lattice [7].
The appearance of the superfluid component shown in Ref. [6,7] predicts the existence of "persistent current"in the holographic superconductor model with no translational symmetry. In condensed matter systems, it appears as a result of macroscopic quantum coherence state and the "persistent current" does not decay (at least within the age of our universe). So, it is quite interesting to construct such a black hole solution in AdS with "persistent current" to realize a macroscopic quantum coherence state in the classical theory of general relativity dual to the superconducting state. Even in the normal (non-superconducting) state, there are several results indicating that there should be "persistent current" in the zero temperature limit [4,5,8]. Ref. [8] analytically showed that momentum relaxation rate is given as a function of a positive power of temperature by computing retarded Green's functions at low temperature and frequencies for a static black hole in AdS 2 × R 2 (See also [9] for a different system from the present one, in which there is a static black hole solution that corresponds to an insulating phase in which DC-conductivity decreases as temperature goes to zero.). Refs. [4,5] numerically estimated the relaxation time from the Drude form and found that it becomes infinite as temperature goes to zero in the presence of a non-perturbative lattice. These results suggest that if momentum or current is initially given, it does not decay in the zero temperature limit even in the presence of an ionic lattice. In other words, there should be extremal charged stationary black hole solutions where horizon has nonzero velocity with respect to the lattice, corresponding to the existence of "persistent current" in the dual field theory side.
From the perspective of classical general relativity, the rigidity theorem [10,11] guarantees the existence of an additional symmetry for such a stationary black hole: It states that if a stationary black hole is non-static and the horizon cross-sections are compact, then there must be a one-parameter cyclic isometry group which is spacelike near infinity 1 . Note that in order to prove the theorem, the analyticity is required. In the presence of a lattice, it is clear that such a Killing symmetry which would correspond to the translational symmetry for the planar horizon case does not exist if the analyticity on the AdS black holes is imposed. So, it agrees with the observation in the dual field theory side that initial current or momentum decays according to the finite DC resistivity (or finite momentum relaxation rate) in finite temperature case. Even in the zero temperature case, a version of the rigidity theorem still holds [13] and no counter-example has been found until now. This implies that unless analyticity on the horizon (or other relevant technical conditions) is violated, one would not be able to construct an extremal black hole with "persistent current."
Although there have been several studies [4][5][6]8] on DC resistivity by perturbing static charged AdS black holes, little is known about the nature of charged AdS black holes with momentum or current. Motivated by this, we investigate how initial momentum or current of the AdS black holes decays in the presence of an ionic lattice in the normal state. In the absence of a lattice, charged AdS black hole solutions with "persistent current" can be constructed by boosting planar Reissner-Nordström AdS black hole solution. In this paper, as a first step, we investigate the stability of the black holes against a perturbed ionic lattice and how thermalization occurs by losing the momentum. In addition, we examine if the extremal black hole with "persistent current" exists in the zero temperature limit, avoiding the apparent inconsistency between no translational symmetry due to the lattice structure and the consequence of a Killing symmetry due to the rigidity theorem for stationary black holes. The perturbation is given by adding a small spatially varying chemical potential, corresponding to the spatially varying gauge potential at infinity. At first order in the perturbation, there is no dissipation and we can construct charged AdS black hole solutions with "persistent current" in the presence of the ionic lattice. At second order in the perturbation, however, dissipation occurs and the momentum is lost by the lattice.
We conduct perturbation analysis by exploiting the gauge invariant formalism developed by Kodama-Ishibashi [14]. We first numerically calculate the momentum relaxation rate from the two-decoupled scalar mode equation and investigate the dependence of the temperature and the velocity of the black hole. For a fixed temperature, the momentum relaxation rate is proportional to the velocity of the black hole and it does not decay to zero in the zero temperature limit. We next obtain the analytical expression of the momentum relaxation rate and the entropy production rate in the WKB formalism in the large limit of the wave number of the lattice. We also show that the physical process version of the first law is satisfied in the irreversible process and that thermalization always occurs by losing the momentum, being independent of the temperature.
The organization of the paper is as follows. We first derive two master equations of the scalar mode perturbations and mention to the boundary conditions in Sec. II. In Sec. III, we numerically calculate the momentum relaxation rate for various temperature and wave numbers. In Sec. IV, we obtain an analytical expression of the momentum relaxation rate in the WKB formalism. We also calculate the entropy production rate under the WKB approximation and derive the physical process version of the first law of the black hole in the irreversible process in Sec. V. Sec. VI. is devoted to conclusion and discussions.
II. PERTURBATIONS OF BOOSTED REISSNER-NORTSTRÖM ADS BLACK HOLES
In this section, we consider perturbations of boosted Reissner-Nortström AdS black holes by adding an ionic lattice on the boundary theory. As realized in Ref. [3], it can be constructed by considering a spatially varying chemical potential on the boundary theory. According to the AdS/CFT correspondence [1], the chemical potential µ is given by the boundary value of the temporal component of the gauge potential, A t . So, this corresponds to the scalar-type perturbations in the classification of gauge-invariant perturbations [14].
In the following subsection A, we give the perturbed equations and derive two master variables, following [14]. In the subsection B, the boundary conditions for the perturbations are summarized.
A. perturbed equations
We begin with the following four-dimensional Einstein-Maxwell system with action
S = d 4 x √ −g R + 6 L 2 − 1 2 F 2 , (2.1)
where L is the AdS radius. In the absence of the lattice, we are interested, as our background black hole, the following boosted planar Reissner-Nortström AdS black hole solution given by
ds 2 = − f cosh 2 β − r 2 L 2 sinh 2 β dt 2 * + dr 2 f + 2 f L − r 2 L sinh β cosh βdt * dx * + (r 2 cosh 2 β − f L 2 sinh 2 β)dx * 2 + r 2 dŷ 2 , f (r) = r 2 L 2 − 2M r + Q 2 r 2 , (2.2)
where β is a boost parameter. The null geodesic generator l µ 2 on the horizon and the surface gravity κ * associated with l µ are
l = L∂t * + tanh β ∂x * , κ * = L 2 cosh β f ,r (r + ) = L cosh β 3r + 2L 2 − Q 2 2r 3 + , (2.3)
where r + is the radius of the horizon satisfying f (r + ) = 0. We are primarily interested in considering static perturbations of the following type,
δA t (r,x * ) = ǫa t (r)e −ik * x * ,(2.4)
on the boosted black hole background (2.2), where here and hereafter ǫ denotes an infinitesimally small parameter andk * is a wave number of the lattice. However, we do not attempt to calculate such static perturbations directly on the above boosted black hole background, as they are complicated on the frame (t * ,x * ). Instead, we introduce another inertial frame, (t,x) by the boost
t /L x = cosh β − sinh β − sinh β cosh β t * /L x * ,(2.5)
and consider relevant perturbations in this new frame. By doing so, we can exploit the established pertubation formulas for static black holes [14], as in this new frame (t,x), our background black hole becomes static 3 . Hereafter, we shall call the former frame (t * ,x * ) and the latter frame (t,x) the static lattice frame and the static black hole frame, respectively. The static perturbation (2.4) corresponds to the time-dependent perturbation with e i(kx−ωt) in the the static black hole frame wherek andω are given byω
=k * L sinh β,k = −k * cosh β. (2.6)
In the black hole static frame, it is convenient to use (t, u, x, y) coordinate system and the corresponding frequency ω, wave number k defined by
u = r + r , t = r + L 2t , x = r + Lx , y = r + Lŷ , k = L r +k , ω = L 2 r +ω . (2.7)
Since the perturbation corresponds to the scalar-type perturbation, by choosing the gauge as f a (a = t, r) = H T = 0 in Ref. [14], the perturbed metric in the black hole static frame is given by
ds 2 = L 2 u 2 −g(u){1 + ǫ(X(u) + 2H L (u))e −iωt+ikx }dt 2 + 1 − ǫ(X(u) + 2H L (u))e −iωt+ikx g(u) du 2 − 2ǫL 2 u 2 r 2 + g(u) Z(u)e −iωt+ikx dtdu + (1 + 2ǫH L (u))e −iωt+ikx (dx 2 + dy 2 ) , g(u) = (1 − u)(1 − ξu)(1 + (1 + ξ)u + (1 + ξ + ξ 2 )u 2 ),(2.8)
where the black hole horizon is located at u = 1 and ξ is a non-extremal parameter in the range of 0 ≤ ξ ≤ 1 (ξ = 0 and ξ = 1 correspond to the AdS black hole without charge and the extremal black hole, respectively.). The mass and charge density are the functions of r + and ξ,
M := r 3 + 2L 2 (1 + ξ)(1 + ξ 2 ), Q := r 2 + L ξ(1 + ξ + ξ 2 ) (2.9)
and the only non-zero component of the background Maxwell field F µν is
F ut = √ 2L 2 r 2 + Q. (2.10)
The scalar-type metric perturbation variables, H L , X, Z, are gauge invariant in our present gauge choice (see (5.7a), (5.27) of [14]), while the scalar-type perturbation of the Maxwell field, δF µν is given, in terms of a function A(u) (see (5.17) of [14]), by
δF tu = ǫ ω 2 g A + (gA ′ ) ′ e −iωt+ikx , δF tx = iǫkgA ′ e −iωt+ikx , δF ux = ǫ ωk g Ae −iωt+ikx ,(2.11)
where here and hereafter prime denotes the derivative with respect to u.
The perturbed Einstein equations are then reduced to the following three coupled differential equations
g 2 H ′′ L − ω 2 − k 2 g H L + k 2 2 gX − L 2 ω 2 u r 2 + Z iω = 0, (2.12) 4ugH ′ L − 2(2g + ug ′ )H L − 2gX − L 2 k 2 u 3 r 2 + Z iω = 0, (2.13) Z ′ + i ωr 2 + L 2 u 2 X − 2 √ 2iQω L 2 A = 0, (2.14)
where the last two equations correspond to the Hamiltonian and the momentum constraint equations, respectively when evolution in the u direction is regarded as "time" evolution. The Maxwell equation yields the evolution equation for A as
g 2 A ′′ + gg ′ A ′ + (ω 2 − k 2 g)A − 2 √ 2L 2 Q r 2 + gH L = 0. (2.15)
Following Ref. [14], we shall introduce a master variable Φ as
Φ(u) = 4ωr 2 + H L − 2iL 2 uZ ωr + uh(u) , (2.16)
where h is a function of u defined by
h(u) = r 2 + L 2 k 2 − g ′ u .
(2.17)
Then, using Eqs. (2.12), (2.13), (2.14), and (2.15) we obtain two coupled differential equations for Φ and A,
g(gΦ ′ ) ′ + (ω 2 − V Φ )Φ = 4 √ 2Qg L 2 r + h 2 u {2r 2 + (k 2 + 4ξ(1 + ξ + ξ 2 )u 2 )g + L 2 h(k 2 u 2 + ug ′ − 2g)}A, V Φ = g L 2 r 2 + u 2 h 2 U Φ , U Φ = −2r 4 + (k 2 + ξ(1 + ξ + ξ 2 )u 2 )uh ′ g + h(k 2 + 4ξ(1 + ξ + ξ 2 )u 2 )r 4 + k 2 u 2 + L 2 r 2 + uh(ugh ′′ + (ug ′ − 2g)h ′ ), (2.18) g(gA ′ ) ′ + (ω 2 − V A )A + √ 2Q r + g gΦ ′ + h ′ h g − k 2 2 u Φ = 0, V A = 8Q 2 g 2 r 2 + h + k 2 g. (2.19)
It is easy to check that X, H L , and Z are derived from the master variables Φ and A as
H L (u) = − r + 2L 2 gΦ ′ + g h ′ h − k 2 u 2 Φ + 2 √ 2Qg L 2 h A, (2.20) X(u) = r + 2L 4 h {L 2 uhg ′ + 4r 2 + g(k 2 + 4u 2 ξ(1 + ξ + ξ 2 )) + 2L 2 g(uh ′ − h)}Φ ′ + r + 2L 2 gh 2 {−2g 2 hh ′ + 2ug 2 h ′2 + h 2 (2uω 2 + g(ug ′′ − g ′ ))} + 1 L 4 r + uh 2 (k 2 + 4ξ(1 + ξ + ξ 2 )u 2 ){2r 4 + ugh ′ − hk 2 r 4 + u 2 } Φ + 4 √ 2Qug L 2 h A ′ + 2 √ 2Q L 4 h 2 r 2 + {L 4 u 2 h 2 + 2L 2 r 2 + (g − k 2 u 2 )h − 2r 2 + g(2r 2 + (k 2 + 4ξ(1 + ξ + ξ 2 )u 2 ) + L 2 uh ′ )}A, (2.21) Z(u) = iωr 3 + g L 4 u Φ ′ − 1 2 g ′ g − 2h ′ h Φ − 4 √ 2Q r + h A . (2.22)
Introducing two master variables Φ ± as
Φ ± = a ± (u)Φ + b ± A, a + (u) = r 2 + k 2 Q 2L 2 + 3(1 + k 2 δ)M Q r + u, a − (u) = 6(1 + k 2 δ)M − 4Q 2 r + u, b + = 6 √ 2 (1 + k 2 δ)M, b − = − 8Q √ 2 , δ = −1 + 1 + 16k 2 ξ(1+ξ+ξ 2 ) 9(1+ξ) 2 (1+ξ 2 ) 2 2k 2 ,(2.g(gΦ ′ ± ) ′ + (ω 2 − V ± )Φ ± = 0, V ± = g L 2 r 2 + b ± uh 2 U ± , U ± = 4 √ 2Qg{(2r 3 + a ± (k 2 + 4u 2 ξ(1 + ξ + ξ 2 )) + L 2 h( √ 2Qub ± − 2r + a ± ))} + L 2 uhr + (k 2 b ± hr + + 4 √ 2Qa ± (k 2 u + g ′ )). (2.24)
As mentioned in [14], Φ − and Φ + correspond to the gravitational and electromagnetic modes, respectively. For later convenience, we express Φ and A by the master variables Φ ± as
Φ = b − Φ + − b + Φ − b − a + − b + a − , A = a + Φ − − a + Φ + b − a + − b + a − . (2.25)
B. Boundary conditions
The solutions of the two decoupled second order different equations (2.24) are completely determined by imposing the following four boundary conditions, i. e. , asymptotic boundary conditions and ingoing boundary conditions on the horizon. As the asymptotic boundary conditions, we require that the metric asymptotically approaches AdS and the amplitude of the electric field along x * -direction, E x * is independent of k * .
By Eqs. (2.24), we obtain the asymptotic behaviors of Φ and A as
A(u) ≃ a + O(u), Φ(u) ≃ φ 0 + φ 1 u.(2.H L (u) ≃ 4aL 2ξ(1 + ξ + ξ 2 ) − (φ 1 k 2 + 3φ 0 (1 + ξ)(1 + ξ 2 ))r + 2k 2 L 2 + O(u), (2.27) X(u) + 2H L (u) ≃ O(u), (2.28) Z(u) ≃ − iωr 2 + k 2 L 4 u {4aL 2ξ(1 + ξ + ξ 2 ) − (φ 1 k 2 + 3φ 0 (1 + ξ)(1 + ξ 2 ))r + } + O(1). (2.29)
Thus, we shall impose
φ 1 = 4La 2ξ(1 + ξ + ξ 2 ) − 3φ 0 (1 + ξ + ξ 2 + ξ 3 )r + k 2 r + (2.30) so that O(H L ) = O(X) = O(uZ) = u,
as an asymptotic boundary condition at infinity (u = 0). Another asymptotic boundary condition is concerned with the electric field along x * -direction. By requiring that the amplitude of the electric field E x * (u = 0) = δF t * x * (u = 0) be independent of k * and using the relation (2.6), we obtain the second asymptotic boundary condition as
|δF tx (u = 0)| = |δF t * x * (u = 0)| ∼ ǫk * cosh βA ′ (0) = ǫ C, (2.31)
where C is a constant. With respect to the boundary condition on the horizon, we impose the ingoing boundary condition since we require causal propargation on the perturbation. In terms of the tortoise coordinate, u * , the boundary condition is represented by
Φ ± ∼ e i(kx−ω(t−u * )) ∼ e ik(x+v(t−u * )) , u * = u 0 du g , (2.32)
where u * is in the range of 0 ≤ u * ≤ ∞ and v := tanh β is the velocity of the black hole.
III. NUMERICAL CALCULATION OF MOMENTUM RELAXATION
In this section, we numerically calculate the rate of momentum relaxation G by the ionic lattice and investigate how G depends on the temperature and the velocity v of the boosted black hole. To define the expectation value of the energy-momentum tensor T ab in the dual field theory, it is convenient to use the following coordinate system
ds 2 = N 2 dz 2 + γ ab (dx a + N a dz)(dx b + N b dz) (a, b =t * ,x * ,ỹ), → L 2 z 2 (dz 2 + η ab dx a dx b + O(z 3 )), z → 0,(3.1)
where the spacetime is foliated by z = const. timelike hypersurfaces homeomorphic to the AdS boundary (z = 0) and η ab = diag (−1, 1, 1). Then, T ab in the dual field theory is defind by (See Appendix A in Ref. [3])
T ab = lim z→0 2 L z 5 γ ab K − K ab − 2 L γ ab , (3.2)
where γ ab and K ab are the induced metric and the extrinsic curvature on each timelike hypersurface, respectively. Note that the last term on the r. h. s. of Eq. (3.2) is the holographic counter-term. Since the metric (3.1) is obtained from Eq. (2.2) by coordinate transformation z = L/r,t * =t * /L,x * =x * ,ỹ =ŷ, T ab of the background spacetime (2.2) becomes
Tt * t * = LM (1 + 3 cosh 2β), Tx * x * = LM (3 cosh 2β − 1), Tt * x * = 6LM cosh β sinh β, Tỹỹ = 2LM. (3.
3)
The electric current J a is also defined by
J a = lim z→0 L z 3 √ 2F aµ n µ ,(3.4)
where n µ is a unit normal outward-pointing vector orthogonal to each hypersurface. By using lim z→0 n z = −z/L and F zt * = √ 2Q cosh β, we obtain the background value of Jt * as
Jt * = −2Q cosh β. (3.5)
We find the first law of the thermodynamics for the boosted black holes δ Tt * t * = T δs + vδ Tt * x * ,
where T = κ * /2π is the temperature and s is the entropy density defined by the horizon area Σ = r 2 + cosh β per unit length ∆x * = ∆ỹ * = 1 as
s = 4πΣ = 4πr 2 + cosh β. (3.7)
To derive Eq. (3.6), we have used that δ Jt * = 0 because in our model, the total charge does not change during the evolution. Eq. (3.6) is simply derived from the series of the background stationary boosted black hole solutions. In Sec. V, we derive the first law in the irreversible process where momentum relaxation occurs by an ionic lattice in the WKB approximation. The conservation law of T ab is derived from the constraint equation in the bulk,
0 = −D b (γ ab K − K ab ) + F bµ n µ F a b ,(3.8)
where D a is the covariant derivative with respect to the induced metric γ ab . Substituting Eqs. (3.2) and (3.4) into Eq. (3.8), we obtain
∂ b T ab = √ 2 J b lim z→0 (η ac F cb ). (3.9)
Since we are interested in the rate of spatially averaged momentum loss, G, we shall define a spatial average A of any quantity A(x) satisfying a periodic boundary condition
A(x + L) = A(x) as A = x+L x A(x)dx L . (3.10)
Then, G is derived from Eq. (3.9) by substituting a =x * and taking the spatial average:
G := ∂t * Tx * t * = ∂ b Tx * b . (3.11)
This is in fact a quantity of O(ǫ 2 ) because of the following reasoning. As ηx * x * Fx * t * is already O(ǫ), the r. h. s. of Eq. (3.9) is also O(ǫ) or even higher. Since the zeroth order current of J b is spatially homogeneous and Fx * t * = δFx * t * ∼ ǫe −ik * x * = 0, G at O(ǫ) must be zero, and hence G should be O(ǫ 2 ). In addition to the boundary condition (2.31), we also impose the homogeneous electric field to be zero at all orders, i. e. , Fx * t * | z=0 = 0, as we are not interested in the case where the electric current increases by the homogeneous electric field. This implies that G at leading order includes only the first order of J b and it does not include any second order perturbations of the gauge field and the metric, F (2) , h (2) , defined by
F µν = F µν + ǫF (1) µν + ǫ 2 F (2) µν + · · · , g µν = g (0) µν + ǫh (1) µν + ǫ 2 h (2) µν + · · · . (3.12)
It is also noteworthy that G does not include any first order perturbations of the metric because the metric is asymptotically AdS, as described in Eq. (3.1). So, one obtains
G = 2 Ft * z Fx * t * = 2 (Ft * z + δFt * z )δFx * t * = 2 δFt * z δFx * t * = − 2r 4 + L 4 δF t * u δF t * x * . (3.13)
This indicates that momentum relaxation does not occur at linear order but, it does at the second order, O(ǫ 2 ). To derive the last equation in Eq. (3.13), we have used coordinate transformation z = uL/r + ,t * = Lt * /r + ,x * = Lx/r + , andỹ = Ly/r + . In the static lattice coordinate, δF t * u and δF t * x * are derived from Eq. (2.11) by the boost Eq. (2.5) as
δF ut * = −ǫ cosh β ω 2 A g + (gA ′ ) ′ + sinh β ωk g A e −ik * x * = ǫ(c R + ic I )e −ik * x * , δF t * x * = δF tx = iǫkgA ′ e −ik * x * = ǫ(d R + id I )e −ik * x * ,(3.14)
where c R , c I , d R , and d I are some real functions of u. Then, taking the real value of δF t * u and that of δF t * x * in calculating the spatial average of Eq. (3.13), we obtain
G = − 2r 4 + L 4 Re[δF t * u ]Re[δF t * x * ] = ǫ 2 r 4 + L 4 lim u→0 (c R d R + c I d I ). (3.15)
We numerically solve the two decoupled Eqs. (2.24) under the boundary conditions, (2.30), (2.31), and (2.32) from u = 1 to u = 0 and evaluate Eq. (3.15). To minimize numerical errors, we replace the second derivative A ′′ in Eq. (3.14) by the first ones by Eq. (2.19).
In Fig. 1 we show the velocity dependence v (= tanh β) of L 4 G/r 4 + ǫ 2 normalized by C = 1 in Eq. (2.31) for k = 1/2. As expected in condensed matter systems, G is proportional to β ≃ v for any ξ when β is small. This indicates that in the dual field theory, the equation of motion for momentum dissipation is given by d Tx * t * dt * ≃ −γv (3.16) in the presence of an ionic lattice, where γ is a function of temperature T and the horizon radius r + . When v is small enough compared with the velocity of light, v ≪ 1, the expectation value of the background spacetime becomes Tx * t * ∼ v by Eq. (3.3). Since we consider perturbations around the expectation value, we obtain Tx * t * ≃ Tx * t * ∼ v. Substituting this into Eq. (3.16), we find conventional behavior of dissipation observed in condensed matter systems,
v ∼ v 0 e − t τ ,(3.17)
where v 0 is the initial velocity of the black hole and τ is the relaxation time determined by the amplitude of the lattice ǫ, wave number k * , and temperature and so on. In Fig. 2, we show the normalized η = L 4 G/r 4 + ǫ 2 at smaller wave length, k = 1 for β = 0.1, where the ξ dependence of η is the same as the one in Fig. 1. In either case, we find that the rate of momentum loss does not approach zero in the extremal limit, ξ → 1, independent of the parameters, β and the wave number k. This implies that there is no stationary charged AdS black holes with "persistent current" even in the zero temperature limit in the presence of ionic lattice. In other words, the ionic lattice cannot behave as a perfect lattice with no dissipation in the zero temperature limit. At first glance, this might appear to be a contradiction to the results [4,5,8], which state that the dissipation disappears in the zero temperature limit. Actually, there is no discrepancy between the present analysis and the previous result, since in terms of perturbation, the order of effects considered are essentially different between the two: Although the present analysis includes the lattice effect as a perturbation, we take into account the non-zero current already at zeroth order as an initial state and calculate the momentum relaxation rate at non-linear order. In contrast, in Refs. [4,5], the lattice effect is included non-linearly, but the current is not considered at the background level and the results concern, in essence, linear response induced by a small electric field 4 .
Let us consider the time evolution of the charged black hole with initial momentum in the zero temperature limit. Due to the non-zero momentum relaxation rate at the initial state, G = 0 at ξ = 1, the total momentum would be lost gradually and the entropy of the black hole should increase because it is an irreversible process. In other words, the black hole necessarily heats up even though initially we start from ξ = 1. One might wonder if the perturbed solutions with ξ → 1 is indeed zero temperature solution because it is already perturbed by an ionic lattice. By the perturbation, the temperature would slightly change to O(ǫ), as the amplitude of the perturbation is O(ǫ). However, the effect only appears at higher order corrections for G, i. e. , O(ǫ 3 ) or even higher because G is already O(ǫ 2 ) for any temperature. So, the fact that G = 0 at ξ = 1 is a little bit surprising because the thermal fluctuations go to zero and the umklapp scattering must disappear in the extremal limit, ξ → 1 unless a residual resistance remains. We will discuss the irreversible process in Sec. V.
IV. WKB ANALYSIS OF MOMENTUM RELAXATION
In this section, we support the numerical results in the previous section by deriving the momentum relaxation rate G analytically in the large limit of wave number k by using WKB approximation. The analytic expression of G will explain the reason why it does not approach zero in the zero temperature limit.
Rewriting the master variables Φ ± in Eq. (2.23) as Φ ± = e kS± , we can expand S ± and the potential V ± in Eq. (2.24) as a series in 1/k as
S ± = S 0± + S 1± k + S 2± k 2 + · · · , V ± = k 2 V 0± + V 1± k + V 2± k 2 + · · · . (4.1)
Then, substitution of Eq. (4.1) into Eq. (2.24) yields the following equations
dS 0± du * 2 = V 0± − v 2 , d 2 S 0± du 2 * + 2 dS 0± du * dS 1± du * − V 1± = 0, · · · , (4.2)
where V i± (i = 0, 1) is given by
V 0± = g(u), V 1± = ∓2 ξ(1 + ξ + ξ 2 ) ug(u). (4.3)
As shown in Fig. 3, there is only one turning point u * = u * 0 (v 2 − g(u * 0 ) = 0) for the potential, as g is a monotonously decreasing function of u. So, for u * 0 < u * < ∞, the solution of Eq. (4.2) is given in terms of
Q := v 2 − g as Φ − ≃ D − Q 1 4 exp −ik u * u * 0 √ Qdu * exp i ξ(1 + ξ + ξ 2 ) u * u * 0 ug √ Q du * + D + Q 1 4 exp ik u * u * 0 √ Qdu * exp −i ξ(1 + ξ + ξ 2 ) u * u * 0 ug √ Q du * , Φ + ≃D − Q 1 4 exp −ik u * u * 0 √ Qdu * exp −i ξ(1 + ξ + ξ 2 ) u * u * 0 ug √ Q du * +D + Q 1 4 exp ik u * u * 0 √ Qdu * exp i ξ(1 + ξ + ξ 2 ) u * u * 0 ug √ Q du * ,(4.4)
and for 0 ≤ u * ≤ u * 0 as
Φ − ≃ C − |Q| 1 4 exp k u * 0 u * |Q|du * exp ξ(1 + ξ + ξ 2 ) u * 0 u * ug |Q| du * + C + |Q| 1 4 exp −k u * 0 u * |Q|du * exp − ξ(1 + ξ + ξ 2 ) u * 0 u * ug |Q| du * , Φ + ≃C − |Q| 1 4 exp k u * 0 u * |Q|du * exp − ξ(1 + ξ + ξ 2 ) u * 0 u * ug |Q| du * +C + |Q| 1 4 exp −k u * 0 u * |Q|du * exp ξ(1 + ξ + ξ 2 ) u * 0 u * ug |Q| du * . (4.5)
Since we require the ingoing boundary condition (2.32) at the horizon u * = ∞, the coefficients D + andD + are set to be zero (Note that k < 0). Then, the standard connection formulas around the turning point is given bỹ
C + = −iD − e πi 4 ,C − = 1 2D − e πi 4 , C + = −iD − e πi 4 , C − = 1 2 D − eD − = 4L(2Γ 2 ζ 2 − i) r + k(2Γ 2 − iζ 2 )D − + O 1 k 2 ,(4.7)
where
Γ = exp −k u * 0 0 |Q|du * , ζ = exp ξ(1 + ξ + ξ 2 ) u * 0 0 ug |Q| du * . (4.8)
Φ(0) and A(0) are also expressed byD − as
Φ(0) = − 2e 3πi 4 L 3 Γ(ζ 4 − 1)D − r 4 + (1 − v 2 ) 1/4 (2iΓ 2 ζ + ζ 3 ) ξ(1 + ξ + ξ 2 ) k 2 + O 1 k 4 , A(0) = e 3πi 4 L 2 (1 + 4Γ 2 )ζD − 2 √ 2r 3 + (1 − v 2 ) 1/4 (2Γ 3 − iΓζ 2 ) ξ(1 + ξ + ξ 2 )k + O 1 k 3 . (4.9)
The unknown coefficientD − is determined by the boundary condition (2.31) as
D − = 2 √ 2r 3 + e − iπ 4 Γζ ξ(1 + ξ + ξ 2 ) C L 2 k(1 − v 2 ) 1/4 (1 + 2iΓ 2 ζ 2 ) + O C k 2 .
(4.10)
Thus, we obtain
Φ(0) = O(k −3 ), A(0) = O(k −2 ), Φ ′ (0) = O(k −4 ). (4.11)
In the large k limit, G is calculated as
G = ∂t * Tx * t * = − 2r 4 + L 4 1 2 (δF t * u δF * t * x + δF * t * u δF t * x ) = −ǫ 2 r 4 + C 2 (ζ 2 + ζ −2 ) L 4 Γ 2 . (4.12)
Here, to obtain the real value of G, we replaced δF t * u δF t * x by (δF t * u δF * t * x + δF * t * u δF t * x )/2, where δF * is the complex conjugate of δF and we used the approximation
g(gA ′ ) ′ + ω 2 A u=0 ≃ V A (0)A(0) = k 2 A(0) (4.13)
in the large k limit. As shown in Fig. 3, the potential V 0± = g at leading order is a monotonously decreasing function of u and g ≤ 1. Hence we obtain
ln Γ = −k u0 0 |Q| g du < −k u0 0 |Q| v 2 du < −k 1 0 du v 2 = − k v 2 ,(4.14)
where u 0 is the turning point defined by v 2 = g(u 0 ). This indicates that Γ never diverges as ξ → 1 for any fixed non-zero v, and thus the rate of momentum loss never goes to zero even in the zero temperature limit by Eq. (4.12).
In this sense, the analytical result agrees with the numerical results in the previous section.
It is noteworthy that the energy density does not change during the loss of the momentum because we consider static perturbations in the static lattice frame. By using the fact
δF ux * (0) = ǫ lim u→0 sinh β ω 2 A g + (gA ′ ) ′ + cosh β ωk g A e −ik * x * ≃ ǫ lim u→0
sinh β k 2 A + cosh β ωkA e −ik * x * = 0, (4.15) and substituting a = t * in Eq. (3.9), we can easily check
∂t * Tt * t * ∼ δF t * x * δF * ux * + δF * t * x * δF ux * ≃ 0. (4.16)
Here, we used Eq. (4.13) and ω = −k tanh β in the second line in Eq. (4.15). As the total energy is conserved during the relaxation of the momentum, the kinetic energy associated with the momentum should be converted into thermal energy via the umklapp scattering in the dual field theory. We will address this issue in the next section.
V. THE FIRST LAW OF THE BLACK HOLES IN THE MOMENTUM RELAXATION PROCESS
In the previous sections, we have seen that momentum relaxation occurs for any temperature through the perturbed gauge field δA µ on the boundary. Since this is an irreversible process, entropy should be produced, although the energy density does not change. From the perspective of the bulk theory side, both the gravitational and electromagnetic waves are induced from the ionic lattice boundary condition (2.31) and fall into the horizon. As the area of the black hole increases during the process, the entropy density also does. The process is very similar to the Penrose process (see the text book [15]) in the sense that entropy is produced in the process that the angular momentum of a rotating black hole is extracted. However, one of the key differences from the Penrose process is that energy cannot be extracted from the boosted black hole because we consider static perturbations (in the static lattice frame).
Another key difference is that the process continues until the momentum of the boosted black hole becomes zero.
In this section, we calculate the entropy production rate under the WKB approximation, following the Hawking-Hartle formula [16]. We also check that the first law of the black hole is satisfied in the irreversible process and that thermalization occurs.
Let ρ and σ respectively be the expansion and the shear of the null geodesic congruence l along the horizon. The evolution equations are given by
l µ ∂ µ ρ = κ * ρ − ρ 2 − |σ| 2 − 1 2 T µν l µ l ν , l µ ∂ µ σ = κ * σ + Ψ 0 ,(5.1)
where Ψ 0 is a component of the Weyl tensor in the Newman-Penrose formalism (see Ref. [16]) and κ * is the surface gravity on the horizon given in Eq. (2.3).
Since we consider perturbations around the stationary black hole, ρ is very small and hence ρ, σ, and T µν l µ l ν can be expanded as
ρ = ǫρ (1) + ǫ 2 ρ (2) + · · · , σ = ǫσ (1) + · · · , T µν l µ l ν = ǫT (1) + ǫ 2 T (2) + · · · . (5.2)
Let us define an advanced coordinate v * as dv * = (dt − dr/f ) cosh β/L, where it coincides with the time coordinatẽ t * at infinity introduced in Sec. III. Then, l = ∂ v * and, as shown in Appendix A, we obtain
dΣ dv * ≃ 2ǫ 2 Σ κ * |σ (1) | 2 + 1 2 T (2) ,(5.3)
where Σ is an element of surface of the null congruence on the horizon defined by dΣ dv * = 2ρΣ. (5.4) Since Σ is proportional to the entropy density per unit area ∆x * = ∆ỹ * = 1, Eq. (5.3) describes the entropy production rate per unit time ∆t * = 1 in the static lattice frame.
Near the horizon, the component of the Weyl tensor, Ψ 0 is represented by the master variable Φ (2.16) as
Ψ 0 (r + ) = ǫL 2k2 e i(kx−ωt) (f (2H L + X) − Z) 8r 2 + cosh 2 β + O(ǫ 2 ) = − iǫr + ωk 2 (2iω − g ′ (1)) 8L 2 cosh 2 β e i(kx−ωt) Φ(1) + (ǫ 2 ). (5.5)
To derive the second equality, we used gΦ ′ ≃ iωΦ and the approximation
X ≃ r + 2L 2 g ′ Φ ′ + r + ω 2 L 2 g Φ, Z ≃ iωr 3 + L 4 gΦ ′ − 1 2 g ′ Φ (5.6)
near the horizon. Substituting Eq. (5.5) into Eq. (5.1) and replacing l µ ∂ µ → −iLω/ cosh β, we obtain σ (1) as
σ (1) = ir 2 + ωk 2 4L 3 cosh β e i(kx−ωt) Φ. (5.7)
From Eqs. (2.23) and (2.25), Φ reduces to
Φ ≃ − 8QΦ + + 6k 2 δM Φ − √ 2(b − a + − b + a − ) (5.8)
under the WKB approximation. Then, at the leading order in k, the spatial average of |Φ| 2 becomes
|Φ(1)| 2 = 2L 2 C 2 r 2 + v √ 1 − v 2 Γ 2 k 6 1 ζ 2 + ζ 2 − 2 cos Θ , Θ := 2 ξ(1 + ξ + ξ 2 ) ∞ u * 0 ug √ Q du * .T (2) = T µν (2) l µ (0) l ν (0) = 1 ǫ 2 (δF tx )(δF tx ) * r + L 2 2 g xx = r 2 + L 4 cosh 2 β |ikgA ′ e −i(ωt−kx) | 2 = vr 2 + √ 1 − v 2 C 2 4L 4 Γ 2 ζ 2 + 1 ζ 2 + 2 cos Θ .dΣ dv * ≃ ǫ 2 r 2 + √ 1 − v 2 C 2 2L 4 Γ 2 ζ 2 + 1 ζ 2 v Σ κ * . (5.11)
By Eqs (2.3), (3.7), (4.12), (4.16), and (5.11), it is easy to check that the first law in the dynamical process is satisfied;
0 = dE = T ds + vdL,(5.12)
where E and L are the energy density Tt * t * and the momentum density Tx * t * of the background spacetime, respectively. This equation means that L must decrease to satisfy the second law of the black hole thermodynamics, ds ≥ 0. In other words, the momentum relaxation by the ionic lattice is an irreversible process associated with the entropy production.
The thermalization is guaranteed in the irreversible process as follows. The deviation of the temperature T = κ * /2π in Eq. (2.3) becomes
dT = − tanh β T dβ + L 2π cosh β 3dr + 2L 2 + 3Q 2 dr + 2r 4 + − QdQ r 3 + ≥ L 2π cosh β 3dr + 2L 2 + 3Q 2 dr + 2r 4 + + Q 2 tanh βdβ r 3 + ≥ T r + dr + > 0. (5.13)
Here, we used the facts that T ≥ 0, dβ < 0, and dQ = −Q tanh βdβ, which represents d Jt * = 0, to derive the second inequality. In the third inequality, we used ds = 2π(4r + cosh βdr + + 2r 2 + sinh βdβ) ≥ 0. Eq. (5.13) means that the kinetic energy associated with the initial momentum is converted into thermal energy, and then, the temperature increases during the irreversible process.
VI. CONCLUSION AND DISCUSSIONS
We have investigated adiabatic evolution of charged boosted AdS black holes by perturbation of an ionic lattice. At linear order in the perturbation, we constructed charged stationary AdS black hole solutions with an ionic lattice. At second order, however, the momentum relaxation occurs by the lattice and the rate of momentum loss is proportional to the velocity of the black hole, as shown in Sec. III. In conventional condensed matter systems, the equation of motion for an electron with charge e, mass m, and velocity v is effectively given by
m dv dt = eE −γv, (6.1)
where E is the electric field andγ is a positive constant determined by temperature and so on. As shown in Eq. (3.16), we have verified that the equation is satisfied in the presence of initial velocity when the electric field is zero. The coefficient γ in Eq. (3.16) corresponding to the coefficientγ is a complicated function of temperature, but it never becomes zero in the zero temperature limit. This is supported by the analysis of WKB approximation in Sec. IV. This indicates that "persistent current" cannot exist in a condensed matter system that is dual to the present charged AdS black hole at zero temperature. Even though this result itself is consistent with the rigidity theorem for stationary black holes, there seems to be an apparent discrepancy between our result and the previous results [4,5,8] which state that DC-conductivity becomes infinite at zero temperature. One of the reasons is that we calculate non-linear perturbations beyond a linear response theory, as mentioned in Sec. III. Another reason is that we take into account the effect of non-zero current at zeroth order in the perturbations. In Ref. [8], the rate of momentum loss can be obtained by calculating retarded Green's functions of the perturbed gauge potential δA t ∼ e −i(ωt+kx) in the limit of zero frequency, ω → 0 for a static black hole in AdS 2 × R 2 . In our setting, the perturbation of the gauge potential A t is given by δA t ∼ e −ik * x * in the frame where the black hole has initial momentum or velocity v. If we consider the perturbation in another frame where the velocity of the black hole is zero, the form becomes δA t ∼ e ik(vt+x) (k < 0), implying that the lattice is moving along x-direction with constant velocity −v. Thus, in such a static black hole frame, the perturbations correspond to non-zero frequencies −kv, being different from the perturbation considered in Ref. [8].
In the static black hole frame, energy is always pumped from the boundary into the bulk by the moving lattice and then, the initially static black hole starts moving until the velocity of the black hole reaches the one of the lattice. During the pumping, the energy is always absorbed into the black hole and the entropy is produced during the process. This means that thermalization always occurs even in the zero temperature limit, as shown in Sec. V. In the original frame where the black hole has initial momentum and velocity, the velocity continues to decrease until it becomes zero, keeping the total energy fixed. Since the entropy production is independent of the frame, we have clarified the dissipation mechanism of the momentum loss caused by the "friction" between the lattice and the velocity of the black hole. As a consistency check, we have also derived the first law of black hole in the irreversible process. Although our analysis is limited to the framework of perturbation, we expect that the same result should also be obtained even for fully dynamical, non-perturbative case. It would be interesting to explore non-perturbative dynamics of the present system by using numerical methods.
There remains several open questions. For example, how can our result be interpreted in the dual field theory side? Within the framework of our present analysis, the current always decays even in zero temperature and therefore the lattice cannot be interpreted as a perfect lattice, contrary to the prediction of [4]. One possibility would be that there is a residual resistance in the strongly coupled field theory dual to the black hole. The residual resistance can be caused by impurity or strong interactions between quasiparticles even in the zero temperature. It is interesting to explore our result from the perspective of the dual field theory. Another open question is whether one can construct black hole solutions in the presence of ionic lattice dual to a superconducting state with "persistent current." To reconcile with the symmetry consequence of the rigidity theorem, we need to construct a black hole solution with momentum where the horizon is static with respect to the lattice but some condensation of complexed scalar field outside the horizon moves along the lattice. It would be reported in the near future [17].
FIG. 1 :FIG. 2 :
12(color online) η = L 4 G/(ǫ 2 r 4 + ) is plotted for k = 1/2 for various β. β = 0.1, β = 0.07, and β = 0.05 correspond to a circle, a rhombus, and a square, respectively. (color online) η := L 4 G/(ǫ 2 r 4 + ) is plotted for k = 1 and β = 0.1.
FIG. 3 :
3V0± = g is shown for ξ = 0.99 (solid curve) and ξ = 0.2 (dotted curve). The horizontal line (dashed line) corresponds to v 2 = 1/5.
. (2.25),(4.5), and (4.6), we can expand Eq. (2.30) as a series in 1/k and D − can be expressed byD − as,
WKB approximation, the magnitudes of A and Φ are very small, as A = O(k −2 ), Φ = O(k −3 ) by Eq. (2.25). So, the metric components near the horizon are also small as X ∼ Z ∼ H L = O(k −1 ), by Eqs. (2.12) and (5.6). This implies that the metric fluctuation near the horizon does vanish in the large k limit and the second term in the r. h. s. of Eq. (5.3) is simplified to
23 )
23two coupled Eqs. (2.18) and (2.19) are reduced to the two decoupled equations,
H L , X, and Z are asymptotically expanded as a series in u as26)
Substituting this into Eqs. (2.20), (2.21), and (2.22),
Quite recently, a stationary black hole solution with no such a killing orbit was numerically found. The horizon, however, is noncompact, evading the rigidity theorem[12].
The freedom for multiple scaling is fixed in another coordinate system adopted in Sec. III 3 Strictly speaking, the frame (t,x) is not inertial frame because β is not constant during the momentum relaxation process. However, the process is almost adiabatic as we consider perturbation and hence we treat it as almost constant.
We thank S. A. Hartnoll for discussion.
AcknowledgmentsWe wish to thank Gary T. Horowitz for valuable discussions. It is also a pleasure to acknowledge helpful discussions with S. A. HartnollSubstituting Eq. (6.2) into Eq. (5.4) and integrating it by once, we obtain3)The first teleological term in the r. h. s. necessarily appears because a black hole is defined as a region from which light cannot escape to infinity. So, following the Hawking-Hartle formula in Ref.[16], we shall assume thatThus, by setting Σ = Σ i + δΣ, we havewhere we assumed that v * 1 ≫ κ −1 * . This immediately yields Eq. (5.3).
The Large N limit of superconformal field theories and supergravity. J Maldacena, Adv. Theor. Math. Phys. 2J. Maldacena, "The Large N limit of superconformal field theories and supergravity" Adv. Theor. Math. Phys. 2, 231-252 (1998);
Large N field theories, string theory and gravity. O Aharony, S Gubser, J Maldacena, H Ooguri, Y Oz, Phys. Rep. 323O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, "Large N field theories, string theory and gravity" Phys. Rep. 323, 183-386 (2000).
Theory of Superconductivity. G T Horowitz, Lect. Notes Phys. 828313G. T. Horowitz, "Theory of Superconductivity", Lect. Notes Phys. 828 313 (2011).
Inhomogeneous charged black hole solutions in asymptotically anti-de Sitter spacetime. K Maeda, T Okamura, J Koga, Phys. Rev. D. 8566003K. Maeda, T. Okamura, and J. Koga, "Inhomogeneous charged black hole solutions in asymptotically anti-de Sitter spacetime", Phys. Rev. D 85, 066003 (2012).
Optical Conductivity with Holographic Lattices. G T Horowitz, J E Santos, D Tong, JHEP. 07168G. T. Horowitz, J. E. Santos, and D. Tong, "Optical Conductivity with Holographic Lattices," JHEP 07, 168 (2012).
Further Evidence for Lattice-Induced Scaling. G T Horowitz, J E Santos, D Tong, JHEP. 11102G. T. Horowitz, J. E. Santos, and D. Tong, "Further Evidence for Lattice-Induced Scaling," JHEP 11, 102 (2012).
G T Horowitz, J E Santos, arXiv:1302.6586General Relativity and the Cuprates. hep-thG. T. Horowitz and J. E. Santos, "General Relativity and the Cuprates," [arXiv:1302.6586 [hep-th]].
Towards the Lattice Effects on the Holographic Superconductor. N Iizuka, K Maeda, JHEP. 11117N. Iizuka and K. Maeda, "Towards the Lattice Effects on the Holographic Superconductor," JHEP 11, 117 (2012).
Locally critical umklapp scattering and holography. S A Hartnoll, D M Hofman, Phys. Rev. Lett. 108241601S. A. Hartnoll and D. M. Hofman, "Locally critical umklapp scattering and holography," Phys. Rev. Lett. 108 241601 (2012).
Metal-insulator transition in holography. A Donos, S A Hartnoll, arXiv:1212.2998hep-thA. Donos and S. A. Hartnoll, "Metal-insulator transition in holography", arXiv:1212.2998[hep-th].
S W Hawking, G F R Ellis, The large scale structure of space-time. Cambridge University PressS. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press (1973).
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric. S Hollands, A Ishibashi, R M Wald, Commun. Math. Phys. 271699S. Hollands, A. Ishibashi, and R. M. Wald, "A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmet- ric," Commun. Math. Phys. 271 699 (2007).
Stationary holographic plasma quenches and numerical methods for non-Killing horizons. P Figueras, T Wiseman, Phys. Rev. Lett. 110171602P. Figueras and T. Wiseman, "Stationary holographic plasma quenches and numerical methods for non-Killing horizons," Phys. Rev. Lett. 110 171602 (2013).
On the 'Stationary Implies Axisymmetric' Theorem for Extremal Black Holes in Higher Dimensions. S Hollands, A Ishibashi, Commun. Math. Phys. 291403S. Hollands and A. Ishibashi, "On the 'Stationary Implies Axisymmetric' Theorem for Extremal Black Holes in Higher Dimensions," Commun. Math. Phys. 291 403 (2009).
Master equations for perturbations of generalised static black holes with charge in higher dimensions. H Kodama, A Ishibashi, Prog. Theor. Phys. 11129H. Kodama and A. Ishibashi, "Master equations for perturbations of generalised static black holes with charge in higher dimensions," Prog. Theor. Phys. 111 29 (2004).
General Relativity. R M Wald, The University of Chicago PressR. M. Wald, "General Relativity," The University of Chicago Press (1984).
S Chandrasekhar, The mathematical Theory of Black Holes. Oxford University PressS. Chandrasekhar, "The mathematical Theory of Black Holes," Oxford University Press (1983).
. N Iizuka, A Ishibashi, K Maeda, Work in progressN. Iizuka, A. Ishibashi, and K. Maeda, Work in progress.
| [] |
[
"Variable resolution Poisson-disk sampling for meshing discrete fracture networks",
"Variable resolution Poisson-disk sampling for meshing discrete fracture networks"
] | [
"Johannes Krotz \nDepartment of Mathematics\nUniversity of Tennessee\n37919KnoxvilleTennesseeUSA\n",
"Matthew R Sweeney \nComputational Earth Science (EES-16)\nEarth and Environmental Sciences\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n",
"Carl W Gable \nComputational Earth Science (EES-16)\nEarth and Environmental Sciences\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n",
"Jeffrey D Hyman \nComputational Earth Science (EES-16)\nEarth and Environmental Sciences\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n",
"Juan M Restrepo \nOak Ridge National Laboratory\n37830Oak RidgeTennesseeUSA\n"
] | [
"Department of Mathematics\nUniversity of Tennessee\n37919KnoxvilleTennesseeUSA",
"Computational Earth Science (EES-16)\nEarth and Environmental Sciences\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA",
"Computational Earth Science (EES-16)\nEarth and Environmental Sciences\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA",
"Computational Earth Science (EES-16)\nEarth and Environmental Sciences\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA",
"Oak Ridge National Laboratory\n37830Oak RidgeTennesseeUSA"
] | [] | We present the near-Maximal Algorithm for Poisson-disk Sampling (nMAPS) to generate point distributions for variable resolution Delaunay triangular and tetrahedral meshes in two and three-dimensions, respectively. nMAPS consists of two principal stages. In the first stage, an initial point distribution is produced using a cell-based rejection algorithm. In the second stage, holes in the sample are detected using an efficient background grid and filled in to obtain a nearmaximal covering. Extensive testing shows that nMAPS generates a variable resolution mesh in linear run time with the number of accepted points. We demonstrate nMAPS capabilities by meshing three-dimensional discrete fracture networks (DFN) and the surrounding volume. The discretized boundaries of the fractures, which are represented as planar polygons, are used as the seed of 2D-nMAPS to produce a conforming Delaunay triangulation. The combined mesh of the DFN is used as the seed for 3D-nMAPS, which produces conforming Delaunay tetrahedra surrounding the network. Under a set of conditions that naturally arise in maximal Poisson-disk samples and are satisfied by nMAPS, the two-dimensional Delaunay triangulations are guaranteed to only have wellbehaved triangular faces. While nMAPS does not provide triangulation quality bounds in more than two dimensions, we found that low-quality tetrahedra in 3D are infrequent, can be readily detected and removed, and a high quality balanced mesh is produced. | 10.1016/j.cam.2022.114094 | [
"https://arxiv.org/pdf/2111.13742v1.pdf"
] | 244,714,402 | 2111.13742 | 217c4f40efc67701bcbf4b30034184295891f8a6 |
Variable resolution Poisson-disk sampling for meshing discrete fracture networks
Johannes Krotz
Department of Mathematics
University of Tennessee
37919KnoxvilleTennesseeUSA
Matthew R Sweeney
Computational Earth Science (EES-16)
Earth and Environmental Sciences
Los Alamos National Laboratory
87545Los AlamosNew MexicoUSA
Carl W Gable
Computational Earth Science (EES-16)
Earth and Environmental Sciences
Los Alamos National Laboratory
87545Los AlamosNew MexicoUSA
Jeffrey D Hyman
Computational Earth Science (EES-16)
Earth and Environmental Sciences
Los Alamos National Laboratory
87545Los AlamosNew MexicoUSA
Juan M Restrepo
Oak Ridge National Laboratory
37830Oak RidgeTennesseeUSA
Variable resolution Poisson-disk sampling for meshing discrete fracture networks
Discrete Fracture NetworkMaximal Poisson-disk SamplingMesh GenerationConforming Delauany Triangulation
We present the near-Maximal Algorithm for Poisson-disk Sampling (nMAPS) to generate point distributions for variable resolution Delaunay triangular and tetrahedral meshes in two and three-dimensions, respectively. nMAPS consists of two principal stages. In the first stage, an initial point distribution is produced using a cell-based rejection algorithm. In the second stage, holes in the sample are detected using an efficient background grid and filled in to obtain a nearmaximal covering. Extensive testing shows that nMAPS generates a variable resolution mesh in linear run time with the number of accepted points. We demonstrate nMAPS capabilities by meshing three-dimensional discrete fracture networks (DFN) and the surrounding volume. The discretized boundaries of the fractures, which are represented as planar polygons, are used as the seed of 2D-nMAPS to produce a conforming Delaunay triangulation. The combined mesh of the DFN is used as the seed for 3D-nMAPS, which produces conforming Delaunay tetrahedra surrounding the network. Under a set of conditions that naturally arise in maximal Poisson-disk samples and are satisfied by nMAPS, the two-dimensional Delaunay triangulations are guaranteed to only have wellbehaved triangular faces. While nMAPS does not provide triangulation quality bounds in more than two dimensions, we found that low-quality tetrahedra in 3D are infrequent, can be readily detected and removed, and a high quality balanced mesh is produced.
Introduction
There are a number of methods used to model flow and the associated transport of chemical species in low-permeability fractured rock, such as shale and granite. The most common are continuum models, which use effective medium parameters [21,40,54,55,66,68] and discrete fracture network/matrix (DFN) models, where fractures and the networks they form are explicitly represented [10,42,56]. In the DFN methodology, individual fractures are represented as planar N − 1 dimensional objects embedded within an N dimensional space. Each fracture in the network is resolved with a computational mesh and the governing equations for flow and transport are solved thereon. While the explicit representation of fractures allows DFN models to represent a wider range of transport phenomena and makes them a preferred choice when linking network attributes to flow properties compared to continuum methods [25,31,29], it also leads to unique and complex issues associated with mesh generation. Both conforming methods, where the mesh conforms to intersections [30,50,51], and non-conforming methods, which use more complex discretization schemes so the mesh does not need to be conforming [5,18,57,58], have been developed. If the volume representing the rock matrix surrounding the fracture network also needs to be meshed, then the complications associated with mesh generation are compounded for both techniques [4].
We present the near-Maximal Algorithm for Poisson-disk Sampling (nMAPS) that generates a conforming variable resolution triangulation mesh on and around three-dimensional discrete fracture networks. Although the theory of Poisson disk sampling to produce a point distribution is straightforward, details of the implementation, algorithm, termination criteria, and maximizing efficiency are nuanced and often problem-dependent. nMAPS addresses these issues in the context of mesh generation using a two-stage approach. The first stage is based on the framework presented by Dwork et al., [14] and uses a rejection algorithm to generate an initial Poisson-disk sample point set. The second phase is based on the framework presented by Mitchell et al., [47] and adds additional points to fill gaps in the covering, which maximizes point density without violating the restrictions of a Poisson-disk sampling. nMAPS combines the two methods to achieve nearmaximal coverage with a run time that scales linearly in the number of accepted points. nMAPS efficiency results from a novel rejection technique that uses a quadrilateral or hexahedral background mesh search based on already sampled points. This method significantly shortens the time needed to reach a sufficient, i.e., near-maximal, point density due to a large reduction in computational operations. Once the point set is generated, the conforming Delaunay algorithm presented by Murphy et al., [49] is used to generate a Delaunay triangulation in two or three dimensions. Although nMAPS does not guarantee that the point distribution's density is maximal, i.e., no additional points can be added without violating the restrictions on distances between points, the sample sets are sufficiently maximal to produce high-quality meshes. Comparison with previously implemented meshing techniques for variable resolution conforming triangulations showed that nMAPS produces meshes comparable in quality but requires considerably less computational time. Moreover, the nMAPS framework is presented in a general manner that can be easily extended beyond mesh generation for DFNs.
In section 2, we describe the challenges associated with DFN mesh generation and the general properties of maximal Poisson-disk sampling. In section 3, we provide a detailed explanation of nMAPS, for both 2D fracture networks and 3D volume meshing. In section 4, we assess the quality of the mesh and performance, i.e., run times, of 2D and 3D examples using nMAPS. In section 5, we provide a few concluding remarks.
Background
Discrete Fracture Networks & Mesh Generation
Due to the epistemic uncertainty associated with hydraulic and structural properties of subsurface fractured media, fracture network models are typically constructed stochastically [52,53,55]. In the DFN methodology, individual fractures are placed into the computational domain with locations, sizes, and orientations that are sampled from appropriate distributions based on field site characterizations. The fractures form an interconnected network embedded within the porous medium. Each fracture must be meshed for computation so that the governing equations for flow and transport can be numerically integrated to simulate physical phenomena of interest.
Formally, each fracture in a DFN can be represented as a planar straight-line graph (PSLG) composed of a set of line segments that represent the boundary of the fracture and a set of line segments that represent where other fractures intersect it. In this manner, each fracture can be described by a set of boundary points on the PSLG, denoted {p}, and a set of intersection lines { i,j }, where the subscripts i and j indicate that this line corresponds to the intersection between the ith and jth fractures. Once {p} and { i,j } are obtained for every fracture in the network, a point distribution covering each fracture must be generated. If a conforming numerical scheme is used, then all cells of { i,j } are discretized lines in the mesh that must coincide between intersecting fractures. So long as minimum feature size constraints are met, a conforming triangulation method, such as that presented in Murphy et al., [49], can be implemented to connect the vertices such that all lines of intersection form a set of connected edges in a triangulation.
In general, one wants to properly resolve all relevant flow and transport properties of interest while minimizing the number of nodes in the mesh, and these two goals compete. The first of these conditions depends partially on the mesh quality, which is controlled partially by the second. A starting point for the notion of mesh quality is the minimum angle condition, i.e., that the smallest angle should be bounded away from zero, as poorly shaped elements can affect the condition number of the linearized system of equations [1,63,69]. While a uniform resolution mesh is straightforward to generate and the minimum angle condition easily met, it is computationally more expensive than a variable resolution mesh refined in areas of interest, which can be tailored to reduce the number of nodes in the mesh. A variable resolution mesh can be appropriate for single-phase flow simulations or in particle tracking simulations where the spatially variable resolution does not adversely affect transport properties. Specifically, Eulerian formulations of transport will have spatially variable numerical diffusion on a variable resolution mesh, which means that the mesh needs to have a sufficiently small refinement to accurately capture solute fronts [2]. Variable resolution mesh generation is more complex than its uniform counterpart. One of the principal complications is creating a smooth transition of cell sizes. Jumps in the computed fields of interest and other numerical artifacts can occur if the transition is not sufficiently smooth.
Most methods to generate a conforming DFN mesh use a uniform point distribution and modify connectivity locally to conform to intersections [50,51]. When using a conforming mesh, the numerical methods for resolving flow and transport in the network are typically simpler and have fewer degrees of freedom compared to non-conforming mesh methods [20]. Similarly, almost all nonconforming numerical methods use a uniform resolution, but some create variable resolutions across fractures, still uniform within a single plane, in an attempt to reduce the number of total nodes in the mesh [6]. Using a variable mesh resolution in non-conforming schemes could drastically reduce the number of nodes in the mesh while retaining the the ability to retain a high order of accuracy. However, it is rarely implemented due to the associated meshing complications [8].
The generation of a variable resolution, unstructured conforming mesh is quite rare, even with the advantages noted above. One technique in use is the Features Rejection Algorithm for Meshing (FRAM) that addressed the issues associated with conforming DFN mesh creation by coupling it with network generation [30]. Through this technique, FRAM allows for the creation of a variable resolution mesh that smoothly coarsens away from intersections where pressure gradients in flow simulations are typically the highest. FRAM has been implemented in the computational suite dfnWorks [33], which has been used to probe fundamental aspects of geophysical flows and transport in fractured media [27,28,35,36,44,62] as well as practical applications including hydraulic fracturing operations [26,37,43], inversion of micro-seismicity data for characterization of fracture properties [48], the long term storage of spent civilian nuclear fuel [25], and geo-sequestration of carbon dioxide into depleted reservoirs [32]. However, the current method for mesh generation uses an inefficient iterative refinement technique to produce the point distribution used to generate the mesh. Sweeps of a refinement algorithm are applied to an initially coarse triangulation based on the boundary set {p}. If an edge in the mesh is larger than the current maximum edge length, then a new point is added to the mesh at the midpoint of that edge to split it into two new edges. In practice, the edge splitting is done using Rivara refinement [60,61]. The resulting field is then smoothed using Laplacian smoothing in combination with Lawson flipping [38]. This process is repeated until all edges meet the assigned target edge length, which could be a spatially variable field based on the distance to { i,j }, for example. While the resulting mesh quality is quite good, the process is rather slow and cumbersome.
If these implementation complexities can be addressed, then the superior modeling qualities of variable resolutions can be made practical. To accomplish this, we present nMAPS, where the final vertex distribution is directly created rather than iteratively derived. While the method was initially designed to specifically improve FRAM, we provide the details in a general format such that it can be implemented for mesh generation in general. Specifically, it can be used as the basis for any DFN meshing methodology, including both conforming and non-conforming techniques. Details are given for Delaunay triangulations, which are of importance in many two-point flux finite volume solvers as they are used to generate the Voronoi control volumes on which these solvers compute. In the next section, we recount the properties of maximal Poisson-disk sampling that we used to design and implement nMAPS, including theoretical bounds on mesh gradation that ensure high-quality variable mesh resolutions.
Maximal Poisson-disk Sampling
Meshes produced from a point distribution that is dense yet cluster free have provable high quality bounds under specific conditions [7,12,16]. Maximal Poisson-disk samples fulfill these conditions [22]. Similar quality bounds can be established for sphere-packings, whose radii are Lipschitz continuous with respect to their location [45,46,64]. Traditionally, Poisson-disk sampling is performed using an expensive dart-throwing algorithm [13]. These algorithms struggle to achieve maximality as the probability to select a free spot becomes decreasingly small as the number of points sampled (n) increases. The algorithm in [47], which is based on these dart-throwing algorithms, was the first to guarantee maximality. It achieves maximality with run times of O(n log(n)) by using a regular grid for acceleration and sampling from polygonal regions in its second phase. In practice, performance close to O(n) has been reported [15,17,47]. Another algorithm that is not based on dart-throwing was proposed in [9]. While the method does not guarantee maximality, it did show linear performance in the number of points sampled. This algorithm was extended to variable radii by [14]. Other authors have provided algorithms that produce variable Poissondisk samplings on 3D-surfaces [23,24]. Their triangulation-based algorithm runs in linear time, and while theoretically not guaranteeing maximality, their numerical experiments indicate that near-maximality can certainly be achieved. A summary of recent developments in this area, along with a comparison of different methods, can be found in [67].
Maximal Poisson-disk samplings X on a domain Ω ⊆ R d are random selections of points X = {x i } n i=1 , that fulfill the following properties: 1. empty disk property:
∀i = j ∈ {1, ..., n} : |x i − x j | > r.
We will call r the inhibition radius, 2. maximality:
Ω = n i=1 B R (x i ),
where B ε (x) = {y ∈ Ω : |x − y| < y} is the open ball of radius ε around x. R will be called the coverage radius. [47] Intuitively, the empty disk property says that every sample point is at the center of a d-dimensional ball or disk that does not contain any other points of the sampling. Maximality implies that these balls cover the whole domain, i.e., there is no point y ∈ Ω that is not already contained in one of the balls around a point in the sample.
It is useful to generalize these definitions, such that both the inhibition and the coverage radius depend on the sampling points, i.e., r = r(x i , x j ) and
R = R(x i , x j ) for all x i , x j ∈ X.
We hereafter refer to this construct as a variable radius maximal Poisson-disk sampling, and we refer to a Poisson-disk sampling with constant radii as a fixed-radii maximal Poisson-disk sampling [47].
A common approach is to assign each point x ∈ Ω a positive radius ρ(x) and have r(x i , x j ) be a function of ρ(x i ) and ρ(x j ). Natural choices for r(x i , x j ) are, for example, ρ(x i ) or ρ(x j ) for i < j, thereby determining the inhibition radius depending on the ordering on X. Order independent options include min(ρ(x i ), ρ(x j )), max(ρ(x i ), ρ(x j )) or ρ(x i ) + ρ(x j ). The last of these options corresponds to a sphere packing [47]. The coverage radius can but does not have to be different from ρ.
Delaunay triangulations maximize the smallest angle of all triangulations based on a generating point distribution [41]. Since numerical errors in many applications tend to increase if these angles are small [69], Delaunay triangulations often are a triangulation of choice. Moreover, the dual of the Delaunay triangulation is a Voronoi tessellation, which in a certain sense is optimal for two-point flux finite volume solvers [19] that are commonly used in subsurface flow and transport simulators such as fehm [70], tough2 [59], and pflotran [39]. In the case of maximal Poisson-disk samplings, we can go one step further and give a lower bound on these angles. We provide a brief summary of the proofs found in [47] and highlight the most important results we use. We then proceed with the new bounds. Lemma 1. The smallest angle α in any triangle is greater than arcsin r 2R , where r is the length of the shortest edge and R the radius of the circumcircle or
sin(α) ≥ r 2R (1)
Proof. This is a direct corollary of the central angle theorem.
This Lemma allows us to give explicit bounds for maximal Poisson-disk samplings. While we will focus entirely on inhibition radii given by r(
x i , x j ) = min(ρ(x i ), ρ(x j )), where ρ(x) is some positive function, comparable results can be found for different r(x i , x j ) in a similar fashion. Lemma 2. Let ε ≥ 0 and ρ : R d → R (d ≥ 2)
be a positive Lipschitz continuous function with Lipschitz constant L with Lε < 1. Let X be a variable maximal Poisson-disk sampling on the domain Ω ⊂ R d with inhibition radius r(x, y) = min(ρ(x), ρ(y)) and coverage radius R(x, y) ≤ (1 + ε)r(x, y) with ε > 0. Let the triangle ∆ be an arbitrary element of the Delaunay triangulation of X (d = 2) or an arbitrary 2-dimensional face of a cell of the Delaunay triangulation of X (d > 2). If the circumcenter of ∆ is contained in Ω, then each angle α of ∆ is greater or equal to
arcsin 1−L−εL 2+2ε or sin(α) ≥ 1 − L − εL 2 + 2ε .
Proof. Let α be the smallest angle of ∆ and x, y ∈ X be the vertices of the shortest edge of ∆, i.e., the vertices opposite to α. Without loss of generality, assume
ρ(x) ≤ ρ(y). Since X is a Poisson-disk sampling |x − y| ≥ min (ρ(x), ρ(y)) = ρ(x). Now, let z ∈ Ω be the circumcenter of ∆. Since X is maximal, there exists v ∈ X with |z − v| ≤ R(z, v) ≤ (1 + ε)ρ(z)
. Because ∆ was retrieved from a Delaunay triangulation v cannot be contained in the interior of ∆'s circumcircle. Hence,
|z − x| ≤ |z − v| ≤ (1 + ε)ρ(z) ≤ (1 + ε) (ρ(x) + L|z − x|) .
Rearranging this inequality yields
|z − x| ≤ ρ(x) 1 + ε 1 − L − εL .
The result follows by applying Lemma 1 after noticing that |x − y| is the length of the shortest edge and that |z − x| is the radius of the circumcirle.
Remark. Note that for d > 2 the same result is true, if we assume the circumcenter of the d-simplex, of which ∆ is a face, is contained in Ω instead of the circumcenter of ∆ itself. The proof is identical.
Remark. While this result allows us to control the quality of 2D-triangulations of maximal Poisson-disk samplings, it can also be used to gauge how close a given Poisson-disk sampling is to being maximal.
The previous Lemma only gives us bounds on all triangles, if their circumcenters are contained in Ω. The next two Lemmas will give sufficient conditions to guarantee exactly this as long as Ω is a polytope.
Lemma 3.
Let Ω ⊂ R 2 be a polygonal region and X a maximal Poisson-disk sampling containing all vertices of Ω. Let the inhibition radius r(x, y) be defined like in the previous lemma. Furthermore, let the coverage radius of X ∩ δΩ fulfill
R δ (x, y) < r(x,y) √ 2(1+L) , i.e., |x − y| < √ 2 1+L r(x, y) for all x, y ∈ δΩ ∩ X.
Then the circumcenter of all triangles in the Delaunay triangulation of X are contained inΩ.
Proof. Suppose this claim is wrong. Let ∆ be a triangle in the Delaunay triangulation with circumcenter z /
∈ Ω. For this to be possible the circumcircle needs to be cut into at least two pieces by δΩ, separating z and the vertices of ∆. Since ∆ is part of a Delaunay triangulation and all vertices of Ω are part of the sampling, this is done by (at least) one segment of a straight line, i.e., δΩ contains a secant of the circumcircle.
Next, let b 1 , b 2 ∈ δΩ be the two boundary points closest to the circumcircle on either side of that line segment and let B be the disk bounded by the circumcircle. Note thatB ∩ Ω contains ∆ and is itself entirely contained in the disk of radius 1
2 |b 1 − b 2 | < 1 2 √ 2 1+L r(b 1 , b 2 ) around 1 2 (b 1 + b 2 ). Now, let x / ∈ {b 1 , b 2 } be a vertex of ∆ and let b ∈ {b 1 , b 2 }
be the point of the two, that is closer to x. We already established that x lies within the just mentioned ball around 1
2 (b 1 + b 2 )
. Let x p be the projection of x onto the line segment connecting b 1 and b 2 .
Then |x − x p | < 1 2 √ 2 1+L r(b 1 , b 2 ), because x lies within the circle of that radius, |x p − b| < 1 2 √ 2 1+L r(b 1 , b 2 ), because b is the closer of the two points b 1 , b 2 and therefore |x − b| = |x − x p | 2 + |b − x p | 2 < r(b 1 , b 2 ) 1 + L ≤ ρ(b) 1 + L ≤ ρ(b).(2)
Since |x − b| ≥ min(ρ(x), ρ(b)) this implies |x − b| ≥ ρ(x). However assuming this and applying the Lipschitz condition on (2) gives us
|x − b| < ρ(b) 1 + L ≤ 1 1 + L (ρ(x) + L|x − b|) ≤ 1 + L 1 + L |x − b|,
which is a contradiction.
Remark. Although Lemma 3 generalizes to higher dimensions, it is not very practical because it is difficult to guarantee the bounds on R δ if the boundary is more than 1-dimensional. However, it is still possible to get some bounds on the radii of the circumcircles and then, using Lemma 1, on the angles, if the distance of non-boundary points is greater than some lower bound a > 0. In fact, using notation from the previous proof, let ∆ again be an d-simplex with circumcenter outside of Ω and x / ∈ δΩ on of its points. Since the circumsphere of any simplex in a Delaunay triangulation does not contain any other points the radius of the intersection with δΩ is bounded by R δ . Using simple geometric arguments, one can show that this feature forces the radius of ∆'s circumsphere R to fulfill the following inequality
R 2 ≤ (R − a) 2 + R δ 2 ⇒ R ≤ a 2 + R δ 2 2a ≤ R δ 2 a .(3)
If R δ (x, y) < r(x, y) there is a lower bound on a, continuously depending on R δ , solely due to the fact, that we have a Poisson-disk sampling.
If R δ = R δ (x p , b)
is any bigger, a needs to be bounded artificially. This implies that the angle bounds change continuously, if the conditions for Lemma 3 cannot be met they still can be relatively controlled by the choice of the artificial bound on a.
Under the conditions of the previous Lemmas, the simplices of the twodimensional Delaunay triangulation resulting from the maximal Poisson-disk sampling are guaranteed to only have well-behaved triangular faces. In three or more dimensions, however, this does not imply that the simplices themselves are well-behaved. It is possible for a 3D Delaunay triangulation to contain slivers, which are tetrahedra whose four points are all positioned approximately on the equator of their circumsphere. In [11] slivers are characterized as tetrahedra whose points are all close to a plane and whose orthogonal projection onto that plane is a quadrilateral. In [3] slivers are equivalently classified as tetrahedra with a dihedral angle close to 180 • containing their own circumcenter. All faces of a sliver can be equilateral triangles, yet they can still have arbitrarily small dihedral angles. This latter feature can cause large numerical errors in the computational physics simulations performed on the mesh. Therefore, slivers should be eschewed as much as possible. While slivers cannot be entirely avoided, one can show that if the points x, y, z, w of a maximal Poisson-disk sampling form a sliver, the distance between w and the plane spanned by x, y, z is very small [11]. We use this property to minimize slivers around the DFN and the faces of the surrounding volume by using the 2D sampling of the DFN as the seed of the 3D sampling and enforce a minimum distance between the DFN and subsequent points in the 3D sample. This procedure leads to slivers being scarce in the initial 3D triangulation. Moreover, given any three points in the mesh, the probability a fourth point will produce is a sliver quite small in practice. Therefore, if the points producing slivers are removed and resampled, the resulting new triangulation will most likely have fewer slivers than the previous one. We tested this rejection/resampling algorithm and found it improves the overall quality of the mesh.
nMAPS Workflow overview
The nMAPS workflow contains the following high-level steps:
1. Generation of a DFN using dfnWorks [34]. 2. Deconstruct the DFN into individual fractures/polygons. 3. Generation of a 2D-variable-radii Poisson-disk sampling on each polygon. 4. Construct a conforming Delaunay triangulation of each fracture based on the Poisson-disk sample point set. 5. Merge individual fracture meshes into a single DFN. 6. Generation of a 3D variable radii Poisson-disk sampling of the volume surrounding the meshed DFN. 7. Mesh generation of the resulting 3D sample set, identify low-quality tetrahedra and remove two of their nodes that are not located in the DFN mesh. 8. Repeat steps 6-7 with the remaining nodes as the seed set until all slivers are removed.
A graphical version of the workflow is presented in Figure 1. DFN generation was discussed in section 2.1, so we begin with a description of the 2D sampling method.
Two-Dimensional Sampling Method
We generate a 2D Poisson-disk sampling in a successive manner using a rejection method. First, we provide a general overview of the method. This method can be performed on every fracture in the network independent of the other fractures. In each step, a new point candidate is generated. It is accepted if it does not break the empty disk property with any of the already accepted points. For the sampling in two dimensions, we use a variable inhibition radius that increases linearly with distance from intersections on the fracture. Specifically, we reject a candidate point y, if there is an already excepted point x such that the condition
|x − y| ≥ r(x, y) = min(ρ(x), ρ(y))(4)
is violated. In this equation, ρ(x) as a piecewise linear function given by
ρ(x) = ρ(D(x)) = H 2 for D(x) ≤ F H A(D(x) − F H) + H 2 for F H ≤ D(x) ≤ (R + F )H (AR + 1 2 )H otherwise(5)
Here D(x) is the Euclidean distance between x and the closest intersection. H, A, R, and F are parameters that determine the global minimum distance between two points (H/2), the distance around an intersection where the local inhibition radius remains at its minimum value (F H), the global maximum inhibition radius (ARH + H/2), and the slope at which the inhibition radius grow with D(x) (A). Since ρ(D) is piecewise linear, it is a Lipschitz-function with Lipschitz-constant A.
If the sampling has a coverage radius R(x, y) ≤ (1 + ε)r(x, y) for some ε > 0, then the conditions of (2) hold. To satisfy the conditions of (3) as well, and thereby ensure angle bounds on all triangles in a Delaunay triangulation, we begin by sampling points along the boundary and enforce a maximum distance of r(x,y) √ 2(1+L) between points. Next, we generate new candidates by randomly sampling within an annulus around an already accepted point; this is illustrated in Figure 2. The minimum distance another point could be from the center point and still preserve the empty disk property determines the inner radius. The maximum distance a point be from to the center in a maximal sampling determines the outer radius. For our choice of inhibition radius, and assuming it has the same radius as the coverage radius, these distances are ρ(x)
1+A for the inner radius and 2ρ(x) 1−A for the outer radius. We will now go over the individual steps of the 2D algorithm in detail. These steps are also presented in the pseudocode in Algorithm 1 (Section 3.4) and are illustrated in A-D of Fig. 1. The notation used in the pseudocode is found in the table at the start of the same section. First, a 1D Poisson-disk sampling along the boundary of the polygon is generated as a seed set (line 2). Next, k candidate points at a time (line 12) around each already accepted point are sampled and determined whether they are accepted or not (lines 13 through
21)
. k is a positive integer and a user-defined parameter. We use cell lists to find points around a candidate that could potentially cause a violation of the empty disk. A visual depiction of these cells is provided in Fig. 3(a). The size of these cells is chosen to contain at most one point, which allows us to skip distance calculations with points beyond a certain cutoff (line 16). Unlike the previously mentioned algorithms, we label cells containing points as occupied and those cells that are too close to an accepted point to contain the candidate point. Specifically, if a candidate x lies in a cell C and any other cell D with diam(C ∪ D) ≤ r in (x) is occupied, then x is immediately rejected as it conflicts with the point in D (line 13). On the other hand, if dist(C, D) > ρ(x), then there is no need to calculate the distance between x and any potential element of D because they cannot violate the empty disk property. An example of this property is shown in Fig. 3(b). We use this technique to our advantage in two ways. First, it allows us to reject many candidates without calculating any distances to nearby points, which provides substantial speedup compared to previous methods and especially for large values of k. Second, unmarked cells that contain space for another point are easily identified, which allows us to find under-sampled regions after the initial sweep terminates (line 32). If a point is accepted, then it is added to the end of the sample set (line 22). Note that this newly accepted point will be used as a seed for sampling another k point when its turn comes up in the queue. Thus, the method grows the sample set within the primary while loop. If all k candidates around a point are rejected, then we move on to the next already accepted point (line 29). The algorithm terminates once every accepted point has been used as a sampling center (line 30). Next, we detect unmarked or under-sampled cells and randomly place points within them (line 33). Then, the main algorithm is restarted with the seed set, including these newly added points, and ends once the termination criterion is met once again (line 46). While this process of resampling sweeps in under-sampled cells can be performed multiple times, we found that a single resampling was sufficient to increase the quality of the sampling to acceptable levels, and additional sweeps were unnecessary.
Once the point distribution is created, the conforming Delaunay triangulation method of Murphy et al., [49] is used to generate the mesh. We recount the general idea of the method here for completeness. To create a conforming Delaunay triangulation that preserves the lines of fracture intersections as a set of triangle edges to be created, it is sufficient that the circumscribed circle of each segment of the discretized line of intersection be empty of any other point in the distribution prior to connecting the mesh. To achieve this sufficient condition, any point within the circumscribed circle of each segment of the discretized lines of intersection is removed from the point distribution. Next, a two-dimensional unconstrained Delaunay triangulation algorithm is used to connect the resulting point set. Because of the construction method, i.e., empty regions around the lines of intersection, the line segments that represent lines of fracture intersection naturally emerge in the triangulation. In turn, the Delaunay triangulation will conform to all of the fracture intersection line segments. Once every fracture polygon is triangulated, they are all joined together to form the mesh of the entire DFN.
Three-dimensional Sampling Method
nMAPS uses a similar method to that presented in the two-dimensional section to generate a point distribution in three-dimensional space. The primary difference is that candidates are generated on a spherical shell around accepted nodes instead of an annulus. The 3D variant of ρ(x) given by
ρ(x) = ρ(D(x)) = ρ 2 (x p ) for D(x) ≤ F ρ 2 (x p ) A(D(x) − F r 2D (x p )) + H 2 for F ρ 2D (x p ) ≤ D(x) ≤ ρmax−r2(xp) A ρ max otherwise(6)
where x p the fracture point closest to x, ρ 2 (x p ) is its 2D inhibition radius on the fracture, and D(x) is the distance between x and x p . Just as in the 2D case, equation (6) is a piecewise linear function in D(x). It is constant within a distance of ρ 2 (x p )F of the fracture network and increases linearly with a slope of A until a given maximal inhibition radius of ρ max is reached. In practice these parameters can be chosen to be the same as their 2D counterparts. One difference from the 2D case is that in addition to rejecting all candidates that violate the empty-disk property (4), we also reject a candidate x if it is within a distance of ρ(x)/2 to a boundary or fracture. This latter piece prevents slivers from having three nodes located on a single fracture or the domain's boundary. In turn, this limits the circumradius of tetrahedra with circumcenter outside of the domain; cf. lemma 3 and subsequent remark for additional details. The pseudocode of the 3D-sampling method provided is in Algorithm 2 in Section 3.4. The initial sampling process is identical to the 2D version, therefore, we will focus on the initialization and resampling, which is where the methods differ. The seed set is made up of points sampled along the boundary of the 3D volume and those of the DFN generated by the 2D algorithm (line 2). Neighbor cells can still be used in the same way as in 2D to speed up the rejection of candidates. Unlike in 2D, a maximal Poisson-disk sampling in 3D does not guarantee sliver-free triangulation, which is why we do not use the cell lists to find undersampled cells in 3D. Instead, once the algorithm terminates, the resulting sampling is triangulated (line 10), slivers identified (line 11), and two nodes of every sliver, with a preference for nodes that are neither on a boundary or a fracture, are removed (line 12). While the definition of a sliver given earlier in Section 2.2 allows for a bit of leeway in what is considered a small or large dihedral angle, we successfully replaced tetrahedra with dihedral angles outside of [8 • , 170 • ] and aspect ratios bigger than 0.2. It is worth noting that more traditional ways of sliver-removal like perturbation [65] or exudation [11] can break the empty disk property and are not used in nMAPS. Then the algorithm is restarted with the remaining nodes as the seed set (line 15). This process is repeated until all slivers are removed (line 16). With this approach, we have obtained triangulations with no elements of dihedral angles of less than 8 • , examples provided in the next section. The method for generating the conforming mesh is similar to that for the 2D case, but spheres around triangle cells of the fracture planes are excavated.
Pseudocode for the 2D and 3D sampling algorithms
Notation for Pseudocodes:
Input: •D 3 ⊂ R 3 : Cubical Domain ( * ) •DF N ⊂ D 3 : Generated by dfnWorks ( * ) •F l ⊂ R 3 :
l-th fracture of the DFN •q (1) l,m and q (1) l,m :
Endpoints of intersection between fractures F l and F m User defined parameters: •H/2:
Minimum distance between points •F :
HF distance round intersections with constant density •R:
ARH + H/2 is maximum distance between points •A:
Maximum slope of inhibition radius •k:
Number of concurrently sampled candidates Additional notation:
•G: Square cells covering F l with diam(g) ≤ H/2 for all g ∈ G. •ρ(x):
Piecewise linear function defined in (5)-2D or (6)-3D •r(x, y):
Inhibition radius min(ρ(x), ρ(y)) •R(x, y):
Coverage radius •C(x) ∈ G:
Grid cell containing the point x. along boundary δF l as seed. 3: for x ∈ X do 4:
G occ ← G occ ∪ N − (x)
Initialize occupied cells 5: end for 6: Sampling:
7: i ← 1
Start sampling at first accepted point 8: n ← |X| Will increase as more points are accepted 9: while i ≤ n do if C(p j ) ∈ G occ then 14: reject p j Cell already blocked by existing point's inhibition radius 15: else 16: for y ∈ N + (p j ) do 17: if |p j − y| < r(p j , y) then if p j was not rejected then 23:
X ← X ∪ {p j } Accept p j and add it to the sampling set 24: G occ ← G occ ∪ N − (p j ) Update occupied cells 25: n ← n + 1 Ensures sampling around newly accepted points 26: end if 27: end for 28: until All k of the p j are rejected 29: i ← i + 1 Start sampling around next accepted point 30: end while Terminate here or start resampling
Continuation of Algorithm 1 (2D Resampling) 31: Resampling: (Optional) 32: for C ∈ G \ G occ do 33: p ∈ C Generate a random candidate on each cell 34: for y ∈ N + (p) do 35: if |p − y| < r(p, y) then 36: reject
G occ ← G occ ∪ N − (x)
Initialize occupied cells 5: end for 6: Sampling: 7: The sampling process in 3D is the same as the 2D method (Algorithm 1) except for the following:
A. New candidates are generated on a spherical shell instead of an annulus for T ∈ T (X) do 11: if T is a sliver then 12:
B. A candidate p / ∈ δD 3 is rejected if dist(p,
X ← X \ {x, y}, where x, y ∈ T are two random points not contained in the boundary or the DFN Minimum distance of points to DFN and boundary (7B above) ensures that this is possible. Rerun algorithm from line 6 16: until T (X) contains no more slivers.
Results
In this section, we provide examples of nMAPS and compare its performance with previous methods.
Two-dimensional Examples
We begin with a network composed of four disc-shaped fractures. Figure 4 shows the triangulation produced using a variable radius sampling on a single fracture that contains three intersections. Triangles are colored by their maximum edge length to demonstrate how their size increases with distance from the intersections. Figure 5, shows the triangulation of a constant-radius sampling (uniform resolution mesh) on the fracture but re-assembled into the network. Figure 6 shows a meshed network that contains 25 fractures whose radii are generated from an exponential distribution with a decay exponent of 0.3. The largest number of intersections on a fracture is eight within the network; this is not a constraint of generation nor the sampling technique. The inhibition radius parameters are H = 0.1, A = 0.1, F = 1, and R = 40. The mesh contains
Run Time Analysis
Next, we present an analysis of the run time and quality of the sampling on a DFN for varying sample sizes, variations of the parameter k, and different numbers of resampling attempts. All data is generated using the DFN shown in Fig. 6. All samplings are performed on Fujitsu Laptop with 4 2.5 GHz intel Core i5 processors and 16GB of RAM Different numbers of nodes were achieved by changing the parameter H, the minimum distance between points. All data is from independent samplings. The plot in Figure 8 shows the run time prior to the resampling process against the number of points sampled. The color corresponds to the value of the parameter k, which controls the number of concurrent samples. We see an increase in run time with increasing values of k. The run times for samples with the same k are positioned along straight lines of slope one, indicating a linear dependence of the total run time and the number of points sampled. The red lines in the plot have a slope of one for reference. Figure 9 shows the relation between the parameter k and the run time. Colors correspond to different numbers of points. As already established, the run time increases linearly with the number of points sampled. The run time in terms of k exhibits a slightly sublinear behavior. The linear fit (black) of the data on the log-log plot has a slope of 0.7(9) ± 0.00 (7). While this fitting error of ≈ 9% is not insignificant, comparing the data to the two lines of slope 1 (red) in the plot indicates that the run time does not increase more than linearly with k. Figure 10 shows a comparison of runtime for nMAPS variable-radii sampling with and without direct rejection of candidates using the background grid to find nearby points. Recall that the original method presented by [9] and [14] did not use direct rejection implemented in this manner. Thus, nMAPS without direct rejection of candidates is equivalent to the implementation of those algorithms. Data points generated by nMAPS with direct rejection are represented by a filled circle, and data points generated without direct rejection are empty squares. All data points are colored by their k value. nMAPS with direct rejection has a shorter run time for every pair of data points. The difference between the methods increases with larger values of k because more candidates are rejected with larger values of k, but the direct rejection method does so without a distance calculation. For k = 5, the speed difference between the algorithms is slightly less than a factor of 2, whereas, for k = 160, the advantage has grown to about an order of magnitude. These plots highlight the performance gained by using the additional features found in nMAPS compared to previous methods.
A comparison of nMAPS with the method used in dfnWorks [33] to create a variable resolution point distribution is shown in Table 1. The method implemented in dfnWorks uses an iterative Rivara refinement algorithm to generate the nodes of the mesh as described in Section 2.1. We consider three different DFN to characterize the difference between the methods. The first is the deterministic network of four ellipses shown in Fig. 5. The second is the network with fractures sampled from an exponential distribution containing 25 Note that nMAPS is intrinsically parallelizable, and is implemented to mesh each fracture independently on a separate processor. The original meshing technique in dfnWorks is parallelized in the same manner. The mesh resolution and setup were consistent between the methods so that the number of nodes in the final mesh is roughly the same. In all cases, nMAPS was faster than the iterative method, and the speedup improved with the number of fractures. However, differences in network properties also likely play a role in the speedup, which is a feature that we do not explore in this study.
Quality and resampling
The maximality and density of nMAPS depends on the choice of k and the number of times resampling is performed. Figure 11 shows the total number of points accepted after a different number of resampling sweeps for various values of k. Foremost, notice that the density of points increases with k. This rate of growth is largest for small values of k and is lower at higher values. On the lower end of the k scale, resampling increases the point density significantly, whereas there is barely a difference when k > 100. The first resampling is particularly effective in adding additional nodes, whereas the difference between each additional resampling decreases thereafter. Given that resampling adds negligible run time to the original sampling process due to the efficient background mesh look-up in nMAPS, there is a trade-off between higher k and more resampling sweeps. Adopting the latter method, i.e., a larger number of sweeps, can yield better performance in terms of obtaining a higher density with shorter run times. For example, a run at k = 5 with a few resampling sweeps results in a density comparable to a run with more than 10 times higher k performed without resampling, but the former is significantly faster. Recall that run time increased linear with k. Similar conclusions can be reached when looking at the quality of resulting triangulations rather than just the density of the Poisson-disk sampling. Figure 12 shows the smallest minimum angle in a triangulation using nMAPS for variable k with different numbers of resampling sweeps. We can see for k
80 this angle appears to be at around 25 • regardless of the number of repetitions. The theoretical bound for a maximal Poisson-disk sampling (with r(x, y) = R(x, y)) for the settings used to generate these data points is 27.04 • . Solving the the angle bounds from Lemma 2 for ε shows us that in this sampling R(x, y)
(1 + 0.1)r(x, y). Given that nMAPS is a stochastic method and identical inhibition/coverage radii are not exactly guaranteed, these results can be considered very good. While the quality of triangulations for smaller k values without resampling is significantly lower, it is worth emphasizing that a single repetition resolves this issue and produces triangulations with qualities on par with those produced using significantly higher k values. Thus, nMAPS can be run using a single or low double-digit value of k, perform a single resampling sweep, and will produce a triangulation just as good as a higher value of k would have produced but in a fraction of the time.
Three-Dimensional Example
While nMAPS is primarily designed to optimize 2D sampling, we conclude with an example where these 2D samplings are combined with a 3D sampling of the surrounding volume to showcase how nMAPS can be used to produce high-quality 3D triangulations as well. We use a simple network of seven square fractures for clarity in the visualization. The mesh produced by the 3D algorithm is shown in Fig. 14. The tetrahedra are colored according to their maximum edge length to highlight how the point density depends on the distance from the DFN. Note that both the mesh of the DFN and the volume are variable resolution depending on the distance from the fracture intersections.
Histograms presented in Fig. 13 show the distribution of mesh quality measures of the tetrahedra in the 3D triangulation. Tetrahedra with either a dihedral angle of less than 8 • or an aspect ratio of less than 0.2 are discarded before the sampling algorithm was restarted. Histogram (a) shows the distribution of the minimum dihedral angle of each tetrahedron. As expected, no dihedral angle below 8 • remains, while the vast majority exceeds values of 30 • . Histogram (b) shows that despite not optimizing with respect to the maximum dihedral angle, none of these angles exceed 165 • . Histogram (c) shows a sharp cut-off at 0.2 in the distribution of aspect ratios, indicating that the aspect ratio is likely to have been the driving factor for a majority of the resamplings. This example was run through the sliver-removal resampling process 17 times to obtain its triangulation quality. In each of these steps, only 200 or fewer of ≈ 50000 nodes were removed before the resampling. This low value of removed points indicates both the scarcity of slivers in samples generated through Algorithm 2 and that the vertices of these slivers can successfully be removed and replaced in a way that does not give rise to new slivers.
Conclusions
We have presented the near-Maximal Algorithm for Poisson-disk Sampling (nMAPS) to produce a point distribution designed to generate high-quality variable resolution Delaunay triangulations in two and three dimensions. We provided the theoretical basis on which nMAPS is built as well as details of its implementation. nMAPS uses efficient rejection and resampling techniques to achieve near maximality and linear run time scaling in the number of points accepted. We demonstrated that nMAPS could successfully generate variableradii Poisson-disk samples on polygonal regions, networks of polygons, and the surrounding volume they are embedded in. Meshes generated using the point distributions produced by nMAPS show a quality nearly matching theoretical bounds for maximal Poisson-disk samplings, in which coverage and inhibition radii coincide. It is worth noting that near maximality is reached for a coverage radius just slightly larger than the inhibition radius. We determined that mesh quality produced by nMAPS is comparable to the method previously used in dfnWorks but runs in a significantly shorter time. Thus, nMAPS is significantly faster than the previous conforming variable mesh strategies due to our efficient rejection techniques that omit costly distance calculations. It achieves mesh quality only marginally worse than what is theoretically possible. Moreover, it provides an iterative method where slivers in 3D volume meshes can be removed entirely from the domain within certain bounds.
It is worth mentioning that nMAPS is fast and simple to run in a parallel fashion in the context of mesh generation for DFNs, further improving the overall run time performance. nMAPS is intrinsically parallelizable by working on each fracture independently on a different processor, as was done in the performance comparison in Table 1. Based on the grid structure used to accept and reject candidates, 2D and 3D can also be further parallelized by dividing their domain into several pieces, which could be sampled individually on different processors while needing to communicate only cell information along the boundaries of the split domains. However, once these point distributions are produced, they all must reside on a single processor to connect them into a Delaunay mesh.
As a final comment, it's worth noting that nMAPS is not restricted to variable resolution DFN mesh generation for conforming numerical schemes. As shown in Fig. 5, it can be easily used to generate a uniform resolution mesh if desired. Moreover, nMAPS can be readily used to create meshes of fractures for non-conforming methods. One merely needs to skip the step in the algorithm where any point within the circumscribed circle of each segment of the discretized lines of intersection is removed from the point distribution. Omitting this step and performing a triangulation, not necessarily a Delaunay triangulation, will produce a mesh suitable for most non-conforming numerical schemes. Finally, the general framework of nMAPS can be applied to efficiently produce arbitrary triangulations in two and three dimensions so long as the sampling can be constrained to within the domain.
3 .
3Method: The near-Maximal Algorithm for Poisson-disk Sampling (nMAPS)
Use boundary points as the seed for nMAPS D. When the algorithm terminates, find the undersampled regions and sample new points therein. Repeat from step C.
Figure 1 :
1Overview of the nMAPS workflow from the creation of DFN to the final mesh (left 1-5) and during 2D-sampling (right A-D).
Figure 2 :
2Visualisation of a single sampling step. The current point is in the center, the new candidates in the annulus (k = 4). The inner circle is bounded by the inhibition radius of the current point. The outer circle is bounded by the maximum distance a point can be away from the current point if the Poisson-disk sampling was maximal.
Figure 3 :
3Visualization of how the grid is used to find possibly conflicting points. The new candidate is red, already accepted points are green, and cells containing conflicting points are shaded grey. The red circle shows the inhibition radius of the candidate, and blue circles show the furthest cells a point in the center cell could conflict with.
•N + (x) : {g ∈ G : dist(C(x), g) ≤ ρ(x)}: Cells that can contain points y with |x − y| ≤ r(x, y) •N − (x): {g ∈ G : diam(g ∪ C(x)) ≤ ρ(x) 1+A }: Cells, where for all their points y |x − y| ≤ r(x, y) •G occ x∈X N − (x): Cells on which X is already maximal. •T (X): Delaunay triangulation of X ( * ) Output: •X : Poisson-disk sampling on the l-th fracture ( * ): Only used in 3D sampling Algorithm 1 2D Poisson-disk sampling 1: Initializing: 2: X ⊂ F l Generate a 1D Poisson-disk sampling with R(x, y) ≤ r(
j ∈ {1, ..., k} do 12:p j ∈ F l Generate k new candidate points on the annulus around x i 13:
δD 3 )
3<
Figure 4 :
4Triangulation of variable radii Poisson-disk sampling on fracture with three intersections. Parameters H = 0.01, R = 40, A = 0.1, and F = 1. Triangles are colored according to their maximum edge length. The lines of intersection are shown as spheres.
Figure 5 :
5Triangulation of a uniform resolution Poisson-disk sampling based mesh reassembled into the whole DFN.
Figure 6 :
6Triangulation of a variable radii Poisson-disk sampling reassembled into original DFN. The network contains 25 fractures whose radii are generated from an exponential distribution with a decay exponent of 0.3. Parameters used in the sampled: H = 0.1,R = 40,A = 0.1,F = 1. The mesh contains 23195 nodes and 47367 triangles. The minimum angle is ≥ 25 • , the maximum angle ≤ 120 • , and all aspect ratios are ≥ 0.47. 23,195 nodes and 47,367 triangles. The quality of the triangulation is presented in the histograms shown in Figure 7: distribution of (a) minimum angle, (b) maximum angle, and (c) aspect ratio. With the exception of two elements, all of the triangles have a minimum angle greater than 27 • . The theoretical minimum angle in a maximal Poisson-disk sampling with Lipschitz constant A = 0.1 is 27.04 • . The two exceptions are 25 • and 26 • . In terms of the maximum angle, there are very few triangles with angles larger than 110 • and none larger than 120 • . The largest maximum angle theoretically possible in a maximal Poisson-disk sampling with this Lipschitz-constant is 125.92 • . The vast majority of aspect ratios are greater than 0.8 with only a marginal number of triangles having a value less than 0.6 and none less than 0.47.
Figure 7 :Figure 8 :
78Histograms of selected quality measures of the triangulation of variable radii Poissondisk sampling on a fracture with three intersections. Parameters: H = 0.01, R = 40, A = 0.1, and F = 1. (a): Minimum angle (≥ 25 • ), (b): Maximum angle (≤ 120 • ), Run time of nMAPS plotted as a function of points sampled prior to the resampling process. Data points are generated using the same DFN. Different point densities are generated by changing the minimum inhibition radius H 2 between every pair of points. Data points are colored by the value of k. Other parameters are set to A = 0.1, R = 40, F = 1. Comparison to lines of slope 1 (red) indicates that the run time increases at an approximately linear rate.
Figure 9 :Figure 10 :
910Run time per node of nMAPS plotted as a function of concurrently sampled points k prior to the resampling process. Data points generated using the same DFN. Different point densities generated by changing the minimum inhibition radius H 2 between every pair of points. Data points are colored by on the total number of points sampled. Other parameters are set to A = 0.1, R = 40, F = 1. Linear fit (black) with slope 0.7(9) ± 0.00(7). Comparison to lines of slope 1(red) indicate sublinear behavior. Comparison of run time for an implementation of[9,14] (squares) and nMAPS algorithm (circles) plotted as a function of number of nodes and k value. Data points generated using the same DFN and different point densities using H. Data points are colored depending on the value of k. Other parameters are set to A = 0.1, R = 40, F = 1.
DFN
fractures shown inFig. 6. The final network is composed of a single family of disc-shaped fractures whose fracture lengths are sampled from a truncated power-law with exponent 1.8, minimum length 1 m, maximum length 25 m within a cubic domain with sides of length 100 m. There are 8,417 fractures in this network. In the first two examples, the algorithms were run on a MacBook Pro laptop with 8 2.9 GHz Intel Core i9 processors and 32GB of RAM. The third example was run on a Linux server with 64 AMD Opteron(TM) Processor 6272 (1469.697 MHz) and 252GB of RAM.
Figure 11 :
11The total number of points accepted after resampling is plotted as a function of k and colored by the number of resamplings. Other parameters are set to A = 0.1, R = 40, F = 1.
Figure 12 :
12Smallest minimum angle of the triangulation produced using nMAPS. Points are colored by the number of resamplings. Increasing k or the number of sweeps increases the minimum angle in the mesh. The latter of these requires much less time.
Figure 13 :
13Histograms of quality measures of the 3D mesh produced using nMAPS around a seven fracture DFN. Parameters: H = 0.01, R = 40, A = 0.1, and F = 1. (a): Minimum angle (all values ≥ 8 • ), (b): Maximum angle (all values ≤ 165 • ), (c): Aspect Ratio (all values ≥ 0.2) Figure 14: (Left) 3D Mesh produced by nMAPS of and around a seven fracture DFN. Parameters: H = 0.25, R = 100, A = 0.125, and F = 1. (Right) Close up of the conforming mesh. Tetrahedra are colored according to their maximum edge length.
Rerun algorithm from line 9 (Note: i is not reset)p
Empty disk property violated
37:
end if
38:
end for
39:
if p was not rejected then
40:
X ← X ∪ {p}
Accept p and add it to the sampling set
41:
G occ ← G occ ∪ N − (p)
Update occupied cells
42:
n ← n + 1
43:
end if
44: end for
45:
46: Algorithm 2
3D Poisson-disk sampling + resampling
1: Initializing:
2: X ⊂ D 3 Use Algorithm 1 to generate a Poisson-disk sampling on δD 3 and
the DFN
3: for x ∈ X do
4:
On the angle condition in the finite element method. I Babuska, A K Aziz, SIAM Journal on Numerical Analysis. 132I. Babuska and A. K. Aziz. On the angle condition in the finite element method. SIAM Journal on Numerical Analysis, 13(2):214-226, 1976.
A comparison of Eulerian and Lagrangian transport and non-linear reaction algorithms. A David, Tomás Benson, Diogo Aquino, Nicholas Bolster, Engdahl, V Christopher, Daniel Henri, Fernandez-Garcia, Advances in Water Resources. 99David A Benson, Tomás Aquino, Diogo Bolster, Nicholas Engdahl, Christo- pher V Henri, and Daniel Fernandez-Garcia. A comparison of Eulerian and Lagrangian transport and non-linear reaction algorithms. Advances in Water Resources, 99:15-37, 2017.
Dihedral bounds for mesh generation in high dimensions. Marshall Bern, Paul Chew, David Eppstein, Jim Ruppert, Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '95. the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '95USAMarshall Bern, Paul Chew, David Eppstein, and Jim Ruppert. Dihedral bounds for mesh generation in high dimensions. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '95, pages 189-196, USA, 1995. Society for Industrial and Applied Mathematics.
Flow in fractured porous media: A review of conceptual models and discretization approaches. Transport in Porous Media. Inga Berre, Florian Doster, Eirik Keilegavlen, Inga Berre, Florian Doster, and Eirik Keilegavlen. Flow in fractured porous media: A review of conceptual models and discretization approaches. Trans- port in Porous Media, Oct 2018.
A PDE-constrained optimization formulation for discrete fracture network flows. Stefano Berrone, Sandra Pieraccini, Stefano Scialo, SIAM J. Sci. Comput. 352Stefano Berrone, Sandra Pieraccini, and Stefano Scialo. A PDE-constrained optimization formulation for discrete fracture network flows. SIAM J. Sci. Comput., 35(2):B487-B510, 2013.
Parallel meshing, discretization, and computation of flow in massive discrete fracture networks. Stefano Berrone, Stefano Scialò, Fabio Vicini, SIAM Journal on Scientific Computing. 414Stefano Berrone, Stefano Scialò, and Fabio Vicini. Parallel meshing, dis- cretization, and computation of flow in massive discrete fracture networks. SIAM Journal on Scientific Computing, 41(4):C317-C338, 2019.
Guaranteed-quality triangular mesh generation for domains with curved boundaries. Charles Boivin, Carl Ollivier-Gooch, Charles Boivin and Carl Ollivier-gooch. Guaranteed-quality triangular mesh generation for domains with curved boundaries, 2001.
Comparison of the response to geometrical complexity of methods for unstationary simulations in discrete fracture networks with conforming, polygonal, and non-matching grids. Andrea Borio, Alessio Fumagalli, Stefano Scialò, Computational Geosciences. 251Andrea Borio, Alessio Fumagalli, and Stefano Scialò. Comparison of the response to geometrical complexity of methods for unstationary simulations in discrete fracture networks with conforming, polygonal, and non-matching grids. Computational Geosciences, 25(1):143-162, 2021.
Fast Poisson disk sampling in arbitrary dimensions. Robert Bridson, SIGGRAPH '07. Robert Bridson. Fast Poisson disk sampling in arbitrary dimensions. In SIGGRAPH '07, 2007.
Modeling fracture flow with a stochastic discrete fracture network: Calibration and validation: 2. The transport model. M C Cacas, E Ledoux, G De Marsily, A Barbreau, P Calmels, B Gaillard, R Margritta, Water Resour. Res. 263M. C. Cacas, E. Ledoux, G. De Marsily, A. Barbreau, P. Calmels, B. Gaillard, and R. Margritta. Modeling fracture flow with a stochastic discrete fracture network: Calibration and validation: 2. The transport model. Water Resour. Res., 26(3):491-500, 1990.
Sliver exudation. Siu-Wing Cheng, Tamal K Dey, Herbert Edelsbrunner, Michael A Facello, Shang-Hua Teng, Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SCG '99. the Fifteenth Annual Symposium on Computational Geometry, SCG '99New York, NY, USAAssociation for Computing MachinerySiu-Wing Cheng, Tamal K. Dey, Herbert Edelsbrunner, Michael A. Facello, and Shang-Hua Teng. Sliver exudation. In Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SCG '99, pages 1-13, New York, NY, USA, 1999. Association for Computing Machinery.
Guaranteed-quality Delaunay meshing in 3d (short version). L , Paul Chew, Proceedings of the Thirteenth Annual Symposium on Computational Geometry, SCG '97. the Thirteenth Annual Symposium on Computational Geometry, SCG '97New York, NY, USAAssociation for Computing MachineryL. Paul Chew. Guaranteed-quality Delaunay meshing in 3d (short version). In Proceedings of the Thirteenth Annual Symposium on Computational Geometry, SCG '97, pages 391-393, New York, NY, USA, 1997. Association for Computing Machinery.
Stochastic sampling in computer graphics. Robert L Cook, ACM Trans. Graph. 51Robert L. Cook. Stochastic sampling in computer graphics. ACM Trans. Graph., 5(1):51-72, January 1986.
Fast variable density poisson-disc sample generation with directional variation for compressed sensing in mri. Nicholas Dwork, Corey A Baron, Ethan M I Johnson, O' Daniel, John M Connor, Peder E Z Pauly, Larson, Magnetic Resonance Imaging. 77Nicholas Dwork, Corey A. Baron, Ethan M.I. Johnson, Daniel O'Connor, John M. Pauly, and Peder E.Z. Larson. Fast variable density poisson-disc sample generation with directional variation for compressed sensing in mri. Magnetic Resonance Imaging, 77:186-193, 2021.
Efficient maximal Poisson-disk sampling. Mohamed S Ebeida, Andrew A Davidson, Anjul Patney, Patrick M Knupp, Scott A Mitchell, John D Owens, ACM Trans. Graph. 304Mohamed S. Ebeida, Andrew A. Davidson, Anjul Patney, Patrick M. Knupp, Scott A. Mitchell, and John D. Owens. Efficient maximal Poisson-disk sampling. ACM Trans. Graph., 30(4), July 2011.
Efficient and good Delaunay meshes from random points. Mohamed S Ebeida, Scott A Mitchell, Andrew A Davidson, Anjul Patney, Patrick M Knupp, John D Owens, Computer-Aided Design. 4311Solid and Physical ModelingMohamed S. Ebeida, Scott A. Mitchell, Andrew A. Davidson, Anjul Patney, Patrick M. Knupp, and John D. Owens. Efficient and good Delaunay meshes from random points. Computer-Aided Design, 43(11):1506-1515, 2011. Solid and Physical Modeling 2011.
A simple algorithm for maximal Poisson-disk sampling in high dimensions. Mohamed S Ebeida, Scott A Mitchell, Anjul Patney, Andrew A Davidson, John D Owens, Computer Graphics Forum. Mohamed S. Ebeida, Scott A. Mitchell, Anjul Patney, Andrew A. Davidson, and John D. Owens. A simple algorithm for maximal Poisson-disk sampling in high dimensions. Computer Graphics Forum, 2012.
Flow simulation in three-dimensional discrete fracture networks. J Erhel, J-R De Dreuzy, B Poirriez, SIAM J. Sci. Comput. 314J Erhel, J-R de Dreuzy, and B Poirriez. Flow simulation in three-dimensional discrete fracture networks. SIAM J. Sci. Comput., 31(4):2688-2705, 2009.
Finite volume methods. Robert Eymard, Thierry Gallouët, Raphaèle Herbin, Handbook of numerical analysis. 7Robert Eymard, Thierry Gallouët, and Raphaèle Herbin. Finite volume methods. Handbook of numerical analysis, 7:713-1018, 2000.
Conforming, nonconforming and non-matching discretization couplings in discrete fracture network simulations. Alessio Fumagalli, Eirik Keilegavlen, Stefano Scialò, Journal of Computational Physics. 376Alessio Fumagalli, Eirik Keilegavlen, and Stefano Scialò. Conforming, non- conforming and non-matching discretization couplings in discrete fracture network simulations. Journal of Computational Physics, 376:694-712, 2019.
A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. M Hh Gerke, Th Van Genuchten, Water Rescour. Res. 292HH Gerke and M Th Van Genuchten. A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Rescour. Res., 29(2):305-319, 1993.
Tetrahedral meshing via maximal Poisson-disk sampling. Jianwei Guo, Dong-Ming Yan, Li Chen, Xiaopeng Zhang, Oliver Deussen, Peter Wonka, Computer Aided Geometric Design. 43Geometric Modeling and ProcessingJianwei Guo, Dong-Ming Yan, Li Chen, Xiaopeng Zhang, Oliver Deussen, and Peter Wonka. Tetrahedral meshing via maximal Poisson-disk sampling. Computer Aided Geometric Design, 43:186-199, 2016. Geometric Modeling and Processing 2016.
Efficient maximal poisson-disk sampling and remeshing on surfaces. Jianwei Guo, Dong-Ming Yan, Xiaohong Jia, Xiaopeng Zhang, Computers & Graphics. 46Shape Modeling InternationalJianwei Guo, Dong-Ming Yan, Xiaohong Jia, and Xiaopeng Zhang. Efficient maximal poisson-disk sampling and remeshing on surfaces. Computers & Graphics, 46:72-79, 2015. Shape Modeling International 2014.
Efficient triangulation of poisson-disk sampled point sets. The Visual Computer. Jianwei Guo, Dongming Yan, Guanbo Bao, Weiming Dong, Xiaopeng Zhang, Peter Wonka, KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: This research was partially funded by National Natural Science Foundation of China (Nos. 61372168, 61172104, 61331018, and 61271431), the KAUST Visual Computing Center, and the National Science Foundation. 30Jianwei Guo, Dongming Yan, Guanbo Bao, Weiming Dong, Xiaopeng Zhang, and Peter Wonka. Efficient triangulation of poisson-disk sampled point sets. The Visual Computer, 30(6-8):773-785, may 2014. KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: This research was partially funded by National Natural Science Foundation of China (Nos. 61372168, 61172104, 61331018, and 61271431), the KAUST Visual Computing Center, and the National Science Foundation.
A comparative study of discrete fracture network and equivalent continuum models for simulating flow and transport in the far field of a hypothetical nuclear waste repository in crystalline host rock. Teklu Hadgu, Satish Karra, Elena Kalinina, Nataliia Makedonska, Jeffrey D Hyman, Katherine Klise, Hari S Viswanathan, Yifeng Wang, Journal of Hydrology. 553Teklu Hadgu, Satish Karra, Elena Kalinina, Nataliia Makedonska, Jeffrey D. Hyman, Katherine Klise, Hari S. Viswanathan, and Yifeng Wang. A comparative study of discrete fracture network and equivalent continuum models for simulating flow and transport in the far field of a hypothetical nuclear waste repository in crystalline host rock. Journal of Hydrology, 553:59 -70, 2017.
Discontinuities in effective permeability due to fracture percolation. J D Hyman, S Karra, J W Carey, C W Gable, H S Viswanathan, E Rougier, Z Lei, Mech. Mater. 119J. D. Hyman, , S. Karra, J. W. Carey, C. W. Gable, H. S. Viswanathan, E. Rougier, and Z. Lei. Discontinuities in effective permeability due to fracture percolation. Mech. Mater., 119:25 -33, 2018.
Flow channeling in fracture networks: Characterizing the effect of density on preferential flow path formation. J D Hyman, Water Resources Research. J. D. Hyman. Flow channeling in fracture networks: Characterizing the effect of density on preferential flow path formation. Water Resources Research, 2020.
Emergence of stable laws for first passage times in three-dimensional random fracture networks. J D Hyman, M Dentz, A Hagberg, P Kang, Phys. Rev. Lett. 12324248501J. D. Hyman, M. Dentz, A. Hagberg, and P. Kang. Emergence of stable laws for first passage times in three-dimensional random fracture networks. Phys. Rev. Lett., 123(24):248501, 2019.
Linking structural and transport properties in three-dimensional fracture networks. J D Hyman, M Dentz, A Hagberg, P Kang, J. Geophys. Res. Sol. Ea. J. D. Hyman, M. Dentz, A. Hagberg, and P. Kang. Linking structural and transport properties in three-dimensional fracture networks. J. Geophys. Res. Sol. Ea., 2019.
Conforming Delaunay triangulation of stochastically generated three dimensional discrete fracture networks: A feature rejection algorithm for meshing strategy. J D Hyman, C W Gable, S L Painter, N Makedonska, SIAM J. Sci. Comput. 364J. D. Hyman, C. W. Gable, S. L. Painter, and N. Makedonska. Conforming Delaunay triangulation of stochastically generated three dimensional discrete fracture networks: A feature rejection algorithm for meshing strategy. SIAM J. Sci. Comput., 36(4):A1871-A1894, 2014.
Dispersion and mixing in threedimensional discrete fracture networks: Nonlinear interplay between structural and hydraulic heterogeneity. J D Hyman, J Jiménez-Martínez, Water Resources Research. 545J. D. Hyman and J. Jiménez-Martínez. Dispersion and mixing in three- dimensional discrete fracture networks: Nonlinear interplay between struc- tural and hydraulic heterogeneity. Water Resources Research, 54(5):3243- 3258, 2018.
Characterizing the impact of fractured caprock heterogeneity on supercritical CO 2 injection. J D Hyman, Joaquin Jimenez-Martinez, Carl W Gable, H Philip, Rajesh J Stauffer, Pawar, Transp. Porous Media. 1313J. D. Hyman, Joaquin Jimenez-Martinez, Carl W Gable, Philip H Stauffer, and Rajesh J Pawar. Characterizing the impact of fractured caprock hetero- geneity on supercritical CO 2 injection. Transp. Porous Media, 131(3):935- 955, 2020.
dfnworks: A discrete fracture network framework for modeling subsurface flow and transport. Satish Jeffrey D Hyman, Nataliia Karra, Makedonska, W Carl, Gable, L Scott, Hari S Painter, Viswanathan, Computers & Geosciences. 84Jeffrey D Hyman, Satish Karra, Nataliia Makedonska, Carl W Gable, Scott L Painter, and Hari S Viswanathan. dfnworks: A discrete fracture network framework for modeling subsurface flow and transport. Computers & Geosciences, 84:10-19, 2015.
dfnworks: A discrete fracture network framework for modeling subsurface flow and transport. Satish Jeffrey D Hyman, Nataliia Karra, Makedonska, W Carl, Gable, L Scott, Hari S Painter, Viswanathan, Computers & Geosciences. 84Jeffrey D Hyman, Satish Karra, Nataliia Makedonska, Carl W Gable, Scott L Painter, and Hari S Viswanathan. dfnworks: A discrete fracture network framework for modeling subsurface flow and transport. Computers & Geosciences, 84:10-19, 2015.
Matrix diffusion in fractured media: New insights into power law scaling of breakthrough curves. Jeffrey D Hyman, Harihar Rajaram, Shriram Srinivasan, Nataliia Makedonska, Satish Karra, Hari Viswanathan, Gowri Srinivasan, Geophys. Res. Lett. 4623Jeffrey D. Hyman, Harihar Rajaram, Shriram Srinivasan, Nataliia Makedon- ska, Satish Karra, Hari Viswanathan, and Gowri Srinivasan. Matrix diffusion in fractured media: New insights into power law scaling of breakthrough curves. Geophys. Res. Lett., 46(23):13785-13795, 2019.
Anomalous transport in three-dimensional discrete fracture networks: Interplay between aperture heterogeneity and particle injection modes. P Kang, J D Hyman, W S Han, M Dentz, Water Resour. Res. P. Kang, J. D. Hyman, W. S. Han, and M. Dentz. Anomalous transport in three-dimensional discrete fracture networks: Interplay between aperture heterogeneity and particle injection modes. Water Resour. Res., 2020.
Effect of advective flow in fractures and matrix diffusion on natural gas production. S Karra, Makedonska, Hs Viswanathan, J D Painter, Hyman, Water Resour. Res. 5110S Karra, N Makedonska, HS Viswanathan, SL Painter, and JD Hyman. Effect of advective flow in fractures and matrix diffusion on natural gas production. Water Resour. Res., 51(10):8646-8657, 2015.
Anisotropic smoothing and solution adaption for unstructured grids. Ahmed Khamayseh, Andrew Kuprat, International Journal for Numerical Methods in Engineering. 3918Ahmed Khamayseh and Andrew Kuprat. Anisotropic smoothing and solu- tion adaption for unstructured grids. International Journal for Numerical Methods in Engineering, 39(18):3163-3174, 1996.
PFLOTRAN user manual: A massively parallel reactive flow and transport model for describing surface and subsurface processes. P C Lichtner, G E Hammond, C Lu, S Karra, G Bisht, B Andre, R T Mills, J Kumar, No.: LA-UR-15-20403Los Alamos National LaboratoryReportP.C. Lichtner, G.E. Hammond, C. Lu, S. Karra, G. Bisht, B. Andre, R.T. Mills, and J. Kumar. PFLOTRAN user manual: A massively parallel reactive flow and transport model for describing surface and subsurface processes. Technical report, (Report No.: LA-UR-15-20403) Los Alamos National Laboratory, 2015.
Modeling multiscale-multiphasemulticomponent reactive flows in porous media: Application to co 2 sequestration and enhanced geothermal energy using PFLO-TRAN. Peter Lichtner, Satish Karra, Al-Khoury, R., Bundschuh, J.CRC PressComputational Models for CO 2 Geo-sequestration & Compressed Air Energy StoragePeter Lichtner and Satish Karra. Modeling multiscale-multiphase- multicomponent reactive flows in porous media: Application to co 2 sequestration and enhanced geothermal energy using PFLO- TRAN. In Al-Khoury, R., Bundschuh, J. (eds.) Computational Models for CO 2 Geo-sequestration & Compressed Air Energy Storage (http://www.crcnetbase.com/doi/pdfplus/10), pages 81-136. CRC Press, 2014.
The Delaunay triangulation learner and its ensembles. Yehong Liu, Guosheng Yin, Computational Statistics & Data Analysis. 107030Yehong Liu and Guosheng Yin. The Delaunay triangulation learner and its ensembles. Computational Statistics & Data Analysis, page 107030, 2020.
Porous media equivalents for networks of discontinuous fractures. Jcs Long, Remer, P A Wilson, Witherspoon, Water Resour. Res. 183JCS Long, JS Remer, CR Wilson, and PA Witherspoon. Porous media equivalents for networks of discontinuous fractures. Water Resour. Res, 18(3):645-658, 1982.
. A E Lovell, S Srinivasan, S Karra, D O'malley, N Makedonska, H S , A. E. Lovell, S. Srinivasan, S. Karra, D. O'Malley, N. Makedonska, H. S. .
Extracting hydrocarbon from shale: An investigation of the factors that influence the decline and the tail of the production curve. G Viswanathan, J W Srinivasan, L P Carey, Frash, Water Resour. Res. Viswanathan, G. Srinivasan, J. W. Carey, and L. P. Frash. Extracting hydrocarbon from shale: An investigation of the factors that influence the decline and the tail of the production curve. Water Resour. Res., 2018.
Evaluating the effect of internal aperture variability on transport in kilometer scale discrete fracture networks. N Makedonska, J D Hyman, S Karra, S Painter, C W Gable, H Viswanathan, Adv. Water Resour. 94N. Makedonska, J. D. D Hyman, S. Karra, S. L Painter, C. W. W Gable, and H. S Viswanathan. Evaluating the effect of internal aperture variability on transport in kilometer scale discrete fracture networks. Adv. Water Resour., 94:486-497, 2016.
A Delaunay based numerical method for three dimensions: Generation, formulation, and partition. Gary L Miller, Dafna Talmor, Shang-Hua, Noel Teng, Walkington, Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, STOC '95. the Twenty-Seventh Annual ACM Symposium on Theory of Computing, STOC '95New York, NY, USAAssociation for Computing MachineryGary L. Miller, Dafna Talmor, Shang-Hua Teng, and Noel Walkington. A Delaunay based numerical method for three dimensions: Generation, formulation, and partition. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, STOC '95, pages 683-692, New York, NY, USA, 1995. Association for Computing Machinery.
Control volume meshes using sphere packing: Generation, refinement and coarsening. Gary L Miller, Dafna Talmor, Shang-Hua, Noel Teng, Han Walkington, Wang, Fifth International Meshing Roundtable. Gary L. Miller, Dafna Talmor, Shang-Hua Teng, Noel Walkington, and Han Wang. Control volume meshes using sphere packing: Generation, refinement and coarsening. In In Fifth International Meshing Roundtable, pages 47-61, 1996.
Variable radii Poisson-disk sampling. Scott Mitchell, Alexander Rand, Mohamed Ebeida, Chandrajit Bajaj, Proceedings of the 24th Canadian Conference on Computational Geometry. the 24th Canadian Conference on Computational Geometry012012Scott Mitchell, Alexander Rand, Mohamed Ebeida, and Chandrajit Bajaj. Variable radii Poisson-disk sampling. Proceedings of the 24th Canadian Conference on Computational Geometry, CCCG 2012, 01 2012.
Sequential geophysical and flow inversion to characterize fracture networks in subsurface systems. M K Mudunuru, S Karra, N Makedonska, T Chen, Stat. Anal. Data. Min. 105M. K. Mudunuru, S. Karra, N. Makedonska, and T. Chen. Sequential geophysical and flow inversion to characterize fracture networks in subsurface systems. Stat. Anal. Data. Min., 10(5):326-342, 2017.
A point-placement strategy for conforming Delaunay tetrahedralization. Michael Murphy, M David, Carl W Mount, Gable, International Journal of Computational Geometry & Applications. 1106Michael Murphy, David M Mount, and Carl W Gable. A point-placement strategy for conforming Delaunay tetrahedralization. International Journal of Computational Geometry & Applications, 11(06):669-682, 2001.
A new approach to simulating flow in discrete fracture networks with an optimized mesh. H Mustapha, K Mustapha, SIAM J. Sci. Comput. 291439H. Mustapha and K. Mustapha. A new approach to simulating flow in discrete fracture networks with an optimized mesh. SIAM J. Sci. Comput., 29:1439, 2007.
An efficient method for discretizing 3d fractured media for subsurface flow and transport simulations. Hussein Mustapha, Roussos Dimitrakopoulos, Thomas Graf, Abbas Firoozabadi, International Journal for Numerical Methods in Fluids. 675Hussein Mustapha, Roussos Dimitrakopoulos, Thomas Graf, and Abbas Firoozabadi. An efficient method for discretizing 3d fractured media for sub- surface flow and transport simulations. International Journal for Numerical Methods in Fluids, 67(5):651-670, 2011.
Characterization, modeling, monitoring, and remediation of fractured rock. Medicine, National Academies PressEngineering National Academies of SciencesEngineering National Academies of Sciences, Medicine, et al. Characteri- zation, modeling, monitoring, and remediation of fractured rock. National Academies Press, 2020.
Rock fractures and fluid flow: contemporary understanding and applications. National Academy PressNational Research CouncilNational Research Council. Rock fractures and fluid flow: contemporary understanding and applications. National Academy Press, 1996.
Use of variable-scale pressure test data to estimate the log hydraulic conductivity covariance and dispersivity of fractured granites near oracle, arizona. P Shlomo, Joseph S Neuman, Depner, J. Hydrol. 1021-4Shlomo P Neuman and Joseph S Depner. Use of variable-scale pressure test data to estimate the log hydraulic conductivity covariance and dispersivity of fractured granites near oracle, arizona. J. Hydrol., 102(1-4):475-501, 1988.
Trends, prospects and challenges in quantifying flow and transport through fractured rocks. S P Neuman, Hydrogeol. J. 131S.P. Neuman. Trends, prospects and challenges in quantifying flow and transport through fractured rocks. Hydrogeol. J., 13(1):124-147, 2005.
A variable aperture fracture network model for flow and transport in fractured rocks. A Nordqvist, Y W Tsang, C F Tsang, Björn Dverstorp, Johan Andersson, Water Resources Research. 286A. Wille Nordqvist, Y. W. Tsang, C. F. Tsang, Björn Dverstorp, and Johan Andersson. A variable aperture fracture network model for flow and transport in fractured rocks. Water Resources Research, 28(6):1703-1713, 1992.
A mixed hybrid mortar method for solving flow in discrete fracture networks. G Pichot, J-R De Erhel, Dreuzy, Appl. Anal. 8910G Pichot, J Erhel, and J-R de Dreuzy. A mixed hybrid mortar method for solving flow in discrete fracture networks. Appl. Anal., 89(10):1629-1643, 2010.
A generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks. G Pichot, J-R De Erhel, Dreuzy, SIAM J. Sci. Comput. 341G Pichot, J Erhel, and J-R de Dreuzy. A generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks. SIAM J. Sci. Comput., 34(1):B86-B105, 2012.
Tough2 user's guide version 2. Karsten Pruess, M Curtis, G J Oldenburg, Moridis, Karsten Pruess, Curtis M Oldenburg, and GJ Moridis. Tough2 user's guide version 2. 1999.
Algorithms for refining triangular grids suitable for adaptive and multigrid techniques. Cecilia Rivara, International journal for numerical methods in Engineering. 204M Cecilia Rivara. Algorithms for refining triangular grids suitable for adaptive and multigrid techniques. International journal for numerical methods in Engineering, 20(4):745-756, 1984.
Mesh refinement processes based on the generalized bisection of simplices. Maria-Cecilia Rivara, SIAM J. Numer. Anal. 213Maria-Cecilia Rivara. Mesh refinement processes based on the generalized bisection of simplices. SIAM J. Numer. Anal., 21(3):604-613, 1984.
Characterizing the influence of fracture density on network scale transport. T Sherman, J D Hyman, M Dentz, D Bolster, e2019JB018547,2020.e2019JB01854710.1029/2019JB018547J. Geophys. Res. Sol. Ea. 1251T. Sherman, J. D. Hyman, M. Dentz, and D. Bolster. Characterizing the influence of fracture density on network scale transport. J. Geophys. Res. Sol. Ea., 125(1):e2019JB018547, 2020. e2019JB018547 10.1029/2019JB018547.
An Analysis of the Finite-Element Method. Gilbert Strang, George J Fix, D S Griffin, Journal of Applied Mechanics. 411Gilbert Strang, George J. Fix, and D. S. Griffin. An Analysis of the Finite-Element Method. Journal of Applied Mechanics, 41(1):62-62, 03 1974.
Well-spaced points for numerical methods. Dafna Talmor, Guy Blelloch, Alan M Frieze, Noel J Walkington, Teng Shang Hua, Technical reportDafna Talmor, Guy Blelloch, Alan M. Frieze, Noel J. Walkington, and Shang hua Teng. Well-spaced points for numerical methods. Technical report, 1997.
Interleaving Delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. Jane Tournois, Camille Wormser, Pierre Alliez, Mathieu Desbrun, ACM SIGGRAPH 2009 Papers, SIGGRAPH '09. New York, NY, USAAssociation for Computing MachineryJane Tournois, Camille Wormser, Pierre Alliez, and Mathieu Desbrun. Inter- leaving Delaunay refinement and optimization for practical isotropic tetra- hedron mesh generation. In ACM SIGGRAPH 2009 Papers, SIGGRAPH '09, New York, NY, USA, 2009. Association for Computing Machinery.
Tracer transport in a stochastic continuum model of fractured media. Yw Tsang, Tsang, B Hale, Dverstorp, Water Resour. Res. 3210YW Tsang, CF Tsang, FV Hale, and B Dverstorp. Tracer transport in a stochastic continuum model of fractured media. Water Resour. Res, 32(10):3077-3092, 1996.
A survey of blue-noise sampling and its applications. Jian Dong Ming Yan, Bin Wei Guo, Xiao Peng Wang, Peter Zhang, Wonka, Journal of Computer Science and Technology. 303SpringerDong Ming Yan, Jian Wei Guo, Bin Wang, Xiao Peng Zhang, and Peter Wonka. A survey of blue-noise sampling and its applications. Journal of Computer Science and Technology, 30(3):439-452, may 2015. Publisher Copyright: © 2015, Springer Science+Business Media New York.
A numerical dual-porosity model with semianalytical treatment of fracture/matrix flow. Gang Robert W Zimmerman, Teklu Chen, Gudmundur S Hadgu, Bodvarsson, Water Resour. Res. 297Robert W Zimmerman, Gang Chen, Teklu Hadgu, and Gudmundur S Bodvarsson. A numerical dual-porosity model with semianalytical treatment of fracture/matrix flow. Water Resour. Res, 29(7):2127-2137, 1993.
On the finite element method. M Zlamal, Numerische Mathematik. 12M. ZLAMAL. On the finite element method. Numerische Mathematik, 12:394-409, 1968.
FEHM: A control volume finite element code for simulating subsurface multi-phase multi-fluid heat and mass transfer. G Zyvoloski, UR-07-3359Los Alamos Unclassified Report LA. G Zyvoloski. FEHM: A control volume finite element code for simulating subsurface multi-phase multi-fluid heat and mass transfer. Los Alamos Unclassified Report LA-UR-07-3359, 2007.
| [] |
[
"ALGEBRAIC COMBINATORICS Access to articles published by the journal Algebraic Combinatorics on the website On prime order automorphisms of generalized quadrangles",
"ALGEBRAIC COMBINATORICS Access to articles published by the journal Algebraic Combinatorics on the website On prime order automorphisms of generalized quadrangles"
] | [
"Santana F Afton ",
"Eric Swartz ",
"Santana F Afton ",
"Eric Swartz "
] | [] | [
"Algebraic Combinatorics"
] | In this paper, we study prime order automorphisms of generalized quadrangles. We show that, if Q is a thick generalized quadrangle of order (s, t), where s > t and s + 1 is prime, and Q has an automorphism of order s + 1, thenwith a similar inequality holding in the dual case when t > s, t + 1 is prime, and Q is a thick generalized quadrangle of order (s, t) with an automorphism of order t + 1.In particular, if s + 1 is prime and if there exists a natural number n such thatthen a thick generalized quadrangle Q cannot have an automorphism of order s + 1, and hence the automorphism group of Q cannot be transitive on points. These results apply to numerous potential orders for which it is still unknown whether or not generalized quadrangles exist, showing that any examples would necessarily be somewhat asymmetric. Finally, we are able to use the theory we have built up about prime order automorphisms of generalized quadrangles to show that the automorphism group of a potential generalized quadrangle of order (4, 12) must necessarily be intransitive on both points and lines. | 10.5802/alco.89 | [
"https://alco.centre-mersenne.org/article/ALCO_2020__3_1_143_0.pdf"
] | 73,669,189 | 1809.05569 | 05d13fe856e22c327e6d3bbe4fd70d9c343b0316 |
ALGEBRAIC COMBINATORICS Access to articles published by the journal Algebraic Combinatorics on the website On prime order automorphisms of generalized quadrangles
2020. 2020
Santana F Afton
Eric Swartz
Santana F Afton
Eric Swartz
ALGEBRAIC COMBINATORICS Access to articles published by the journal Algebraic Combinatorics on the website On prime order automorphisms of generalized quadrangles
Algebraic Combinatorics
312020. 202010.5802/alco.89This article is licensed under the CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE. Algebraic Combinatorics is member of the Centre Mersenne for Open Scientific Publishing
In this paper, we study prime order automorphisms of generalized quadrangles. We show that, if Q is a thick generalized quadrangle of order (s, t), where s > t and s + 1 is prime, and Q has an automorphism of order s + 1, thenwith a similar inequality holding in the dual case when t > s, t + 1 is prime, and Q is a thick generalized quadrangle of order (s, t) with an automorphism of order t + 1.In particular, if s + 1 is prime and if there exists a natural number n such thatthen a thick generalized quadrangle Q cannot have an automorphism of order s + 1, and hence the automorphism group of Q cannot be transitive on points. These results apply to numerous potential orders for which it is still unknown whether or not generalized quadrangles exist, showing that any examples would necessarily be somewhat asymmetric. Finally, we are able to use the theory we have built up about prime order automorphisms of generalized quadrangles to show that the automorphism group of a potential generalized quadrangle of order (4, 12) must necessarily be intransitive on both points and lines.
Introduction
Following [26], a finite generalized quadrangle Q is an incidence structure (P, L, I), where P is the set of points, L is the set of lines (which is disjoint from P), and I is a symmetric point-line incidence relation satisfying the following axioms: Point-line incidence: Each point is incident with t + 1 lines and each line is incident with s + 1 points, where s, t ∈ N, and two distinct points (respectively, lines) are mutually incident with at most one line (respectively, point). GQ Axiom: Given a point P and a line not incident with P , there is a unique pair (P , ) ∈ P × L such that P I I P I . A generalized quadrangle with s + 1 points incident with a given line and t + 1 lines incident with a given point is said to have order (s, t), and such a generalized quadrangle is said to be thick if both s > 1 and t > 1. Much like the situation with projective planes, there are some arithmetical restrictions on the possible order (s, t) of a generalized quadrangle; see Lemma 2.1. However, these restrictions leave a large number of cases about which absolutely nothing is known, and, while projective planes are only known to exist when their order is a prime power, there exist generalized quadrangles of orders (q, q), (q, q 2 ), (q 2 , q), (q 2 , q 3 ), (q 3 , q 2 ), (q − 1, q + 1), and (q + 1, q − 1), where q is a prime power [26]. The existence of generalized quadrangles of orders (q − 1, q + 1) and (q + 1, q − 1) especially make it entirely unclear whether generalized quadrangles of other orders will exist.
Generalized quadrangles (and, more generally, generalized n-gons) were invented by Jacques Tits [29] to help better understand certain classical groups by providing natural geometric objects on which the groups act. The automorphism group of a finite generalized quadrangle is the set of permutations of the point set that preserve collinearity. While the definition of a generalized quadrangle is purely combinatorial, the known examples of generalized quadrangles all have nontrivial (and, typically, quite robust) automorphism groups and often arise from algebraic constructions; see [13,23,25,26]. Moreover, many examples of generalized quadrangles of order (q −1, q +1) and (q +1, q −1) have automorphism groups that are point-transitive and line-transitive, respectively. For this reason, given a hypothetical order (s, t) of a generalized quadrangle Q, it is natural to study the possible automorphisms of Q in the same spirit that the possible automorphisms of the "missing" Moore graph have been studied; see [10, pp. 89-91] and [21]. Toward this end, we prove the following result: Theorem 1.1. Let Q be a thick generalized quadrangle of order (s, t), where s > t and s + 1 is prime. If Q has an automorphism of order s + 1, then
s t 2 s + 1 s + 1 t t(s + t).
If Q is a generalized quadrangle of order (s, t), where s, t satisfy both the hypotheses and the inequality of Theorem 1.1, then we cannot say much. The real strength of this result arises in the situation where s, t satisfy all of the numerical constraints of Theorem 1.1 except for the inequality, in which case we can make the following conclusion: The dual of a generalized quadrangle Q with point set P and line set L comes from switching the roles of points and lines to create a new generalized quadrangle Q with point set L and line set P. Viewed through the lens of the dual quadrangle, Corollary 1.2 can be rephrased to obtain results about the line-transitivity of the automorphism group of certain potential generalized quadrangles.
Corollary 1.2. Let Q beCorollary 1.3. Let Q be a thick generalized quadrangle of order (s, t), where t > s and t + 1 is prime. If t s 2 t + 1 t + 1 s > s(s + t),
then Q does not have an automorphism of order t + 1 and the automorphism group of Q cannot be line-transitive.
At first glance, the inequalities listed above seem to be rather weak. However, the following corollaries show the power of the result when one parameter is (relatively speaking) much bigger than the other. For thick generalized quadrangles, when s < t 2 , the current best theoretical upper bound is s t 2 − t (see Lemma 2.1), and, for this reason, hypothetical generalized quadrangles of order (q 2 − q, q) (and, dually, (q, q 2 − q)) have been the subject of recent investigation [1,22], although still very little is known about such (potential) generalized quadrangles. In particular, Corollary 1.5 shows that the automorphism group of a generalized quadrangle of order (q 2 − q, q), where q 2 − q + 1 is prime, is not point-transitive.
We are further able to use the theory that we have built up to study the automorphism group of a potential generalized quadrangle of order (4,12). The best known result thus far comes from [1], which states that if such a generalized quadrangle contains an ovoid, a set of st + 1 pairwise noncollinear points, then its automorphism group cannot be point-transitive. We are able to say considerably more: Theorem 1.6. If Q is a generalized quadrangle of order (4,12), then the automorphism group of Q cannot be transitive on either points or lines.
While there are certainly "regular" combinatorial structures that are asymmetric, Theorem 1.6 makes it much more unlikely that such a generalized quadrangle exists. Moreover, it is likely that the techniques used in the proof of Theorem 1.6 can be used to prove that the automorphism groups of generalized quadrangles of other potential orders cannot be point-or line-transitive.
It is possible that the only generalized quadrangles that have a point-transitive automorphism group either arise from classical groups or have order (q−1, q+1), where q is a prime power; see [26]. On the other hand, unlike the case for projective planes [17,18], other than what is known about specific families of generalized quadrangles, there is a much smaller body of knowledge when it comes to point-transitive generalized quadrangles: the results either involve conditions that are much stronger than mere point-transitivity [2,3,4,5,6,7] or involve groups G that act regularly on the point set, i.e. G is transitive on the point set, but the stabilizer in G of each point is trivial [12,14,16,28,30]. The possible orders ruled out by Corollaries 1.2, 1.4, 1.5 and Theorem 1.6 appear to be among the first combinatorially feasible parameters (in the sense of the results contained in Lemma 2.1) that can a priori be ruled out from admitting generalized quadrangles with point-transitive automorphism groups. This paper is organized as follows. In Section 2, we provide the background material necessary for the rest of the paper. In Section 3, we prove various preliminary results about automorphisms of prime order of generalized quadrangles. Section 4 is dedicated to the proof of Theorem 1.1, and Section 5 is dedicated to the consequences of Theorem 1.1 and, in particular, contains proofs of Corollaries 1.2, 1.4, and 1.5. Section 6 contains results that apply specifically to generalized quadrangles of order (4,12), culminating in the proof of Theorem 1.6. Finally, we include in Appendix A some tables which list all possible orders (s, t) with t 100 to which Corollary 1.2 applies.
Algebraic Combinatorics, Vol. 3 #1 (2020)
Background
Let Q be a generalized quadrangle with point set P and line set L. We say that two points P, P are collinear if there is a line incident with both P and P , in which case we write P ∼ P . Similarly, we say that two lines , are concurrent if there is a point incident with both and , and we write ∼ . Our convention here will be that P ∼ P . Given a set X of points,
X ⊥ := {P ∈ P : P ∼ Q for all Q ∈ X}.
We define Z ⊥ for a set Z of lines analogously. If the role of "point" and of "line" (as well as the values of s and t) are interchanged for Q, then the result is also a generalized quadrangle and is called the dual of Q.
A grid is an incidence structure (P, L, I) such that for some integers s 1 ,
s 2 ∈ N we have P = {P i,j : 0 i s 1 , 0 j s 2 }, L = { 0 , . . . , s1 , 0 , . . . , s2 } with incidence defined by P i,j I k if and only if i = k and P i,j I k if and only if j = k.
A dual grid is the point-line dual of a grid, and, instead of s 1 and s 2 , it is defined in terms of parameters t 1 and t 2 . It is easy to see that a grid is a generalized quadrangle if and only if s 1 = s 2 , and the generalized quadrangles with t = 1 are precisely the grids with s 1 = s 2 (= s). The dual result holds for dual grids.
The following omnibus lemma contains basic results about the parameters s and t.
(i) |P| = (s + 1)(st + 1) and |L| = (t + 1)(st + 1); (ii) s + t divides st(s + 1)(t + 1); (iii) if s, t > 1, then t s 2 and s t 2 ; (iv) if 1 < s < t 2 , then s t 2 − t, and if 1 < t < s 2 , then t s 2 − s.
The following result of Payne is an application of the so-called Higman-Sims technique and is crucial to the proof of Theorem 1.1. A more general result is actually proved in [24], and a proof of just Lemma 2.2 is given in [
|Y | = n, and X ⊆ Y ⊥ . Then (m − 1)(n − 1) s 2 . Dually, if X and Y are disjoint sets of pairwise nonconcurrent lines of a generalized quadrangle Q of order (s, t) with t > 1 such that |X| = m, |Y | = n, and X ⊆ Y ⊥ , then (m − 1)(n − 1) t 2 .
Let x be an automorphism of Q. Given a point P of Q, there are three possibilities:
(i) P x = P , (ii) P x = P but P x ∼ P , (iii) P x ∼ P .
With this in mind, we define P 0 (x) to be the set of points fixed by x, P 1 (x) to be the set of points that are not fixed by x but are sent to collinear points, and P 2 (x) to be the set of points sent to noncollinear points by x. For each i, we define α i (x) := |P i (x)|. For the lines of Q, we define L 0 (x), L 1 (x), and L 2 (x) analogously, and for each i we define
β i (x) := |L i (x)|.
The following result is known as Benson's Lemma and is a fundamental result regarding automorphisms of generalized quadrangles. Although we state the result here only in terms of generalized quadrangles, it should be noted that Benson's Lemma is a special case of a more general idea, which allows the character of an automorphism group of an association scheme to be calculated on one of its eigenspaces. This Algebraic Combinatorics, Vol. 3 #1 (2020) technique, attributed to Graham Higman, is described in [10, pp. 89-91] and has been applied to both distance-regular and strongly regular graphs; see, for instance [11,15,21].
Lemma 2.3 ([8, Lemma 4.3])
. If x is an automorphism of a finite generalized quadrangle of order (s, t), then
(t + 1)α 0 (x) + α 1 (x) ≡ (st + 1) (mod s + t).
The following result relates the total number of points sent to collinear points by x to the total number of lines sent to collinear lines by x.
Lemma 2.4 ([26, 1.9.2])
. If x is an automorphism of a finite generalized quadrangle of order (s, t), then
(1 + t)α 0 (x) + α 1 (x) = (1 + s)β 0 (x) + β 1 (x).
Given an automorphism x of Q, the substructure Q x fixed by x must have one of a few types. The following result lists these possible types. For convenience, our delineation into types is slightly different than what is listed in [26].
(3) The substructure Q x is a grid with s 1 < s 2 . (3 ) The substructure Q x is a dual grid with t 1 < t 2 . (4) The substructure Q x is a generalized subquadrangle of order (s , t ).
Note that we allow in (4) the possibility that Q x is a grid or a dual grid, i.e. we allow either s = 1 or t = 1.
Finally, we introduce some terminology from permutation group theory that will be used in Section 6. The action of a group G on a set Ω is said to be quasiprimitive if every nontrivial normal subgroup of G is transitive on Ω. Quasiprimitive groups are a generalization of primitive permutation groups, since, if G acts on Ω and G contains a normal subgroup N that is intransitive on Ω, then the set of orbits of N on Ω is a G-invariant partition of Ω. For a characterization of quasiprimitive permutation groups, see [27, Section 2].
Automorphisms of prime order
In this section, we collect a number of basic results about prime order automorphisms of generalized quadrangles. Throughout this section, Q will refer to a generalized quadrangle of order (s, t) with point set P and line set L. For a given automorphism x of Q, the type of Q x refers to its designation in Lemma 2.5.
Algebraic Combinatorics, Vol. 3 #1 (2020) 147 Santana F. Afton & Eric Swartz Lemma 3.1. Let x be an automorphism of Q with order p, where p is a prime. For i = 1, 2, we have α i (x), β i (x) ≡ 0 (mod p). Moreover, α 0 (x) ≡ (s + 1)(st + 1) (mod p)
and
β 0 (x) ≡ (t + 1)(st + 1) (mod p).
Proof. The set P 1 (x) can be partioned into orbits of x , and, since none of these points are fixed, each orbit has size p. This implies that α 1 (x) ≡ 0 (mod p). By duality, β 1 (x) ≡ 0 (mod p), and analogous arguments show that Proof.
α 2 (x), β 2 (x) ≡ 0 (mod p). The results for α 0 (x), β 0 (x) follow from (s + 1)(st + 1) = |P| = α 0 (x) + α 1 (x) + α 2 (x) and (t + 1)(st + 1) = |L| = β 0 (x) + β 1 (x) + β 2 (x).
Since Q x has type (0), it follows that α 0 (x) = β 0 (x) = 0, implying that p | (s + 1)(st + 1) and p | (t + 1)(st + 1). If p (st + 1), then p | (s + 1) and p | (t + 1) by Euclid's Lemma. Finally, if p is an odd prime and s + 1 ≡ t + 1 ≡ 0 (mod p), then
st + 1 ≡ (−1)(−1) + 1 ≡ 2 ≡ 0 (mod p).
Lemma 3.3. Let x be an automorphism of Q of order p, where p is a prime. If Q x has type (1), then t + 1 ≡ 0 (mod p). If Q x has type (1 ), then s + 1 ≡ 0 (mod p).
Proof. We will prove the result for Q x of type (1); the analogous result for type (1 ) follows by duality. Since Q x has type (1), there are no fixed lines, but there is at least one fixed point. Let P be any fixed point, and let L(P ) be the lines incident with P . Since x is an automorphism, if ∈ L(P ), then x ∈ L(P ). Since no line is fixed by x, |L(P )| = t + 1, and L(P ) can be partitioned into orbits of x , it follows that t + 1 ≡ 0 (mod p).
Lemma 3.4. Let x be an automorphism of Q of order p, where p is a prime.
(i) If Q x has type (2) and α 0 (x) = 1, then s + 1 ≡ 1 (mod p).
(ii) If Q x has type (2) and α 0 (x) 2, then t + 1 ≡ 1 (mod p).
(iii) If Q x has type (2 ) and β 0 (x) = 1, then t + 1 ≡ 1 (mod p).
(iv) If Q x has type (2 ) and β 0 (x) 2, then s + 1 ≡ 1 (mod p).
Proof. We will prove the results for Q x of type (2) and note that the analogous results for type (2 ) follow by duality. Assume first that α 0 (x) = 1, and let P be this unique fixed point. By assumption, there is at least one fixed line incident with P , and the s remaining points of are partitioned into orbits of size p of x , proving that s + 1 ≡ 1 (mod p). Now assume that α 0 (x) 2. Let P, Q be two distinct fixed points, where we may assume by hypothesis that P ∼ P for every P ∈ P 0 (x). Hence P ∼ Q, and so x also fixes the unique line with which P and Q are mutually incident. Since none of the other t lines incident with Q are fixed by x, these t lines are partitioned into orbits of size p, proving that t + 1 ≡ 1 (mod p).
Algebraic Combinatorics, Vol. 3 #1 (2020) Lemma 3.5. Let x be an automorphism of Q of order p, where p is a prime. If Q x has type (2), then
α 0 (x) ≡ 1 + sβ 0 (x) (mod p). If Q x has type (2 ), then β 0 (x) ≡ 1 + tα 0 (x) (mod p).
Proof. We will prove the result for Q x of type (2); the result when Q x is of type (2 ) follows by duality. Let P be the distinguished point with which all fixed lines of x are incident and all fixed points of x are collinear. For any fixed line , let s 0 ( ) be the number of fixed points incident with other than P and let s 1 ( ) be the number of points incident with not fixed by x. Noting that the s 1 ( ) points of that are not fixed by x are partitioned into orbits of size p of x , we have s 1 ( ) ≡ 0 (mod p), which implies that s 0 ( ) ≡ s (mod p) since s 0 ( ) + s 1 ( ) = s. If the β 0 (x) lines fixed by x are 1 , . . . , β0(x) , then
α 0 (x) = 1 + β0(x) i=1 s 0 ( i ) ≡ 1 + β0(x) i=1 s (mod p),
and so α 0 (x) ≡ 1 + sβ 0 (x) (mod p), as desired.
Lemma 3.6. Let x be an automorphism of Q of order p, where p is a prime. If Q x has type (3), then t + 1 ≡ 2 (mod p) and s 1 ≡ s 2 ≡ s (mod p). If Q x has type (3 ), then s + 1 ≡ 2 (mod p) and t 1 ≡ t 2 ≡ t (mod p). In particular, if Q x has type (3) or type (3 ), then p < max{s, t}, and, if Q is a thick generalized quadrangle and Q x has either type (3) or type (3 ), then p < min{s, t}.
Proof. We will prove the result for Q x of type (3); the result when Q x is of type (3 ) follows by duality. Let P be a fixed point of the grid. Since exactly two lines incident with P are fixed by x, the remaining lines incident with P must be partitioned into x -orbits of size p, and hence t + 1 ≡ 2 (mod p). Now, there are two types of lines in the grid: those containing s 1 + 1 fixed points and those containing s 2 + 1 fixed points. For a line of Q fixed by x containing s 1 + 1 fixed points in Q x , the remaining (s + 1) − (s 1 + 1) points incident with are partioned into x -orbits, and so s 1 ≡ s (mod p). Analogously, we have s 2 ≡ s (mod p). Finally, the prime p divides (t + 1) − 2 = t − 1, so p t − 1 if t > 1, and, since s 1 < s 2 s, p divides s − s 1 > 0 and p s − s 1 . The result follows.
Lemma 3.7. Let x be an automorphism of Q of order p, where p is a prime. If s + 1 ≡ 0, 1, 2 (mod p) and t + 1 ≡ 0, 1, 2 (mod p), then either Q x has type (0) and st + 1 ≡ 0 (mod p), or Q x has type (4).
Proof. We will proceed through the types listed in Lemma 2.5. If Q x has type (0), then since p divides neither s + 1 nor t + 1, it follows that st + 1 ≡ 0 (mod p) by Lemma 3.2. If Q x has type (1) or (1 ), then either s + 1 ≡ 0 (mod p) or t + 1 ≡ 0 (mod p) by Lemma 3.3, contrary to our hypotheses. If Q x has type (2) or (2 ), then either s + 1 ≡ 1 (mod p) or t + 1 ≡ 1 (mod p) by Lemma 3.4, a contradiction to our hypotheses. Finally, if Q x has type (3) or type (3 ), then either s + 1 ≡ 2 (mod p) or t + 1 ≡ 2 (mod p) by Lemma 3.6, again a contradiction.
Lemma 3.8. Let x be an automorphism of Q of order p, where p is a prime. If Q x has type (4) and is a subquadrangle of order (s , t ), then s ≡ s (mod p) and t ≡ t (mod p).
Algebraic Combinatorics, Vol. 3 #1 (2020)
Proof. Let be any line fixed by x. By hypothesis, there are exactly s + 1 points fixed by x on , and hence there are (s + 1) − (s + 1) = (s − s ) points on that are not fixed by x. These remaining (s − s ) points are partitioned into orbits of x of size p, and hence s ≡ s (mod p). By duality, t ≡ t (mod p). Lemma 3.9. Let x be an automorphism of Q of order p, where p is a prime. If Q x has type (4) and is a proper subquadrangle of order (s, t ), then α 0 (x) = (s + 1)(st + 1)
α 1 (x) = 0 α 2 (x) = s(s + 1)(t − t )
and β 0 (x) = (t + 1)(st + 1)
β 1 (x) = (t − t )(s + 1)(st + 1) β 2 (x) = (t + 1)(st + 1) − (t(s + 1) − st + 1)(st + 1).
Proof. First, α 0 (x) = (s + 1)(st + 1), since Q x is a subquadrangle of order (s, t ). Similarly, β 0 (x) = (t + 1)(st + 1). We will now show that α 1 (x) = 0. Let P be a point that is not fixed by x. If is a line fixed by x, then, since Q x is a subquadrangle of order (s, t ), all points incident with are fixed by x. This means that P is not incident with , and so, by the GQ Axiom, there exists a unique point Q on with which P is collinear. Let be the line incident with both P and Q. Since Q is incident with , Q is fixed. The line cannot be fixed, since P is not fixed by x. However, ( ) x is also incident with Q, and, by the GQ Axiom, there are no triangles, which means P ∼ P x and α 1 (x) = 0. The value of β 1 (x) now follows from Lemma 2.4, and the values of α 2 (x) and β 2 (x) follow from |P| = α 0 (x) + α 1 (x) + α 2 (x) and |L| = β 0 (x) + β 1 (x) + β 2 (x), respectively. Proof. By Lemmas 2.3 and 3.9,
(t + 1)(s + 1)(st + 1) ≡ st + 1 (mod s + t).
The result follows after simplification of this expression.
Lemma 3.11. Let p be a prime such that p s, and suppose x is an automorphism of Q of order p such that Q x has type (4). Then s = s, t < t, t ≡ t (mod p), and s + t divides st (st + 1).
Proof. By Lemma 3.8, s ≡ s (mod p), and, since p s s , we have s = s. The result now follows from Lemmas 3.8 and 3.10.
Algebraic Combinatorics, Vol. 3 #1 (2020) Lemma 3.12. Let Q be a generalized quadrangle of order (s, t), where s > t and s + 1 is prime. If x is an automorphism of Q of order s + 1, then Q x has type (1 ).
Proof. Assume first that Q x has type (0). Since s > t, (s + 1) (t + 1), and hence st + 1 ≡ 0 (mod s + 1) by Lemma 3.2. However, this implies that
s(t − 1) ≡ (st + 1) − (s + 1) ≡ 0 (mod s + 1).
By Euclid's Lemma, either (s + 1) | s or (s + 1) | (t − 1), which is impossible since s + 1 > s, t − 1. Hence Q x cannot have type (0).
If Q x has type (1), then t + 1 ≡ 0 (mod s + 1) by Lemma 3.3, which is impossible since s > t. If Q x has either type (2) or type (2 ), then either s + 1 ≡ 1 (mod s + 1) or t + 1 ≡ 1 (mod s + 1) by Lemma 3.4, again a contradiction. If Q x has either type (3) or type (3 ), then either s + 1 ≡ 2 (mod s + 1) or t + 1 ≡ 2 (mod s + 1) by Lemma 3.6, another contradiction. Finally, if Q x has type (4), then by Lemma 3.11 we have s = s and (s + 1) divides (t − t ). However, t − t is both smaller than s + 1 and nonzero, a contradiction. Proof. First, if t = 1, then Q is a grid with automorphism group isomorphic to the wreath product Sym(s + 1) wr 2, and so p s + 1, with an analogous result holding in the dual grid case. Hence we may assume that Q is a thick generalized quadrangle. Assume p > (s + 1) and p > (t + 1). Let x be an automorphism of Q of order p. Since Q is thick, we have s + 1 ≡ 0, 1, 2 (mod p) and t + 1 ≡ 0, 1, 2 (mod p). By Lemma 3.7, if p (st + 1), then Q x has type (4). However, since p > s + 1, by Lemma 3.11 we have that s = s and p | (t − t ), a contradiction since p > s + 1 > t − t > 0. The result follows.
It should be noted that this last lemma yields a definitive list of primes that could be orders of automorphisms of a generalized quadrangle Q without knowing any information about Q other than its order (s, t).
Proof of the inequality
This section is devoted to the proof of Theorem 1.1.
Proof of Theorem 1.1. Let Q be a thick generalized quadrangle of order (s, t), where s > t and s + 1 is prime, and let Q have an automorphism x of order s + 1. By Lemma 3.12, Q x has type (1 ), thus no points are fixed by x and at least one line is fixed by x. Let be a line fixed by x. Since Q x has type (1 ), no points incident with are fixed and all fixed lines are pairwise nonconcurrent, and so the lines concurrent with are divided into t distinct orbits of x of size s + 1. If is any other fixed line, then |{ , } ⊥ | = s + 1, i.e. there is a unique x -orbit of lines concurrent with that is also concurrent with . By the Pigeonhole Principle, one of the t different x -orbits of lines concurrent with , which we name X, is also concurrent with at least (β 0 (x) − 1)/t fixed lines other than . If Y is the set of (β 0 (x) − 1)/t + 1 lines that are all nonconcurrent, fixed by x, and incident with each line in X, then, by Lemma 2.2,
s β 0 (x) − 1 t t 2 .
Algebraic Combinatorics, Vol. 3 #1 (2020)
To finish the proof, we provide a lower bound on β 0 (x). Since s+1 is prime, by Lemma 3.1 we have β 0 (x) ≡ (t + 1)(st + 1) (mod s + 1), which equivalently implies that
β 0 (x) ≡ −(t 2 − 1) (mod s + 1).
Thus, there exists some k ∈ N such that
β 0 (x) = k(s + 1) − (t 2 − 1).
If k < t 2 /(s + 1), then
β 0 (x) = k(s + 1) − (t 2 − 1) < t 2 s + 1 (s + 1) − (t 2 − 1) = 1,
which implies that β 0 (x) < 1, a contradiction, since Q x has type (1 ). Thus k t 2 s+1 , and so
t 2 s + 1 (s + 1) − (t 2 − 1) β 0 (x).
This means
s t 2 s + 1 · s + 1 t − t = s t 2 s+1 (s + 1) − (t 2 − 1) − 1 t s β 0 (x) − 1 t t 2 .
Simplifying, we have
s t 2 s + 1 s + 1 t t(s + t),
as desired.
Consequences of the inequality
In this section, we present some consequences of Theorem 1.1. First, we have immediately Corollary 1.2, which says that a generalized quadrangle of order (s, t), where s > t, s + 1 is prime, and
s t 2 s + 1 s + 1 t > t(s + t),
is not point-transitive.
Proof of Corollary 1.2. Assume that Q has order (s, t), where s > t > 1 and s + 1 is prime. If Q has an automorphism group G that is transitive on points and P is a point of Q, then |G| = |P||G P | = (s + 1)(st + 1)|G P |.
Since the prime (s + 1) divides |G|, G must have an element of order s + 1, which means s and t must satisfy the hypotheses of Theorem 1.1. The result follows.
We can also now prove Corollary 1.3.
Algebraic Combinatorics, Vol. 3 #1 (2020) and, by Theorem 1.1 and Corollary 1.2, such a generalized quadrangle cannot have an automorphism of order s + 1 and cannot be point-transitive.
One particular application of the inequality is Corollary 1.5, which states that, if Q is a generalized quadrangle of order (q 2 − nq, q) and q 2 − nq + 1 is prime with q > 2n, then Q is not point-transitive. These conditions apply to numerous potential generalized quadrangles whose existence is not known, for instance orders (12,4), (30,6), (42, 7), and (72, 9); see Appendix A for many more instances.
Proof of Corollary 1.5. Let Q be a generalized quadrangle of order (q 2 −nq, q), where q > 2n and q 2 − nq + 1 is prime. In this instance,
q 2 q 2 − nq + 1 = 2, and 2 q 2 − nq + 1 q = 2q − 2n + 2 q = 2q − 2n + 1, and so (q 2 − q) q 2 q 2 − q + 1 q 2 − q + 1 q = (q 2 − q)(2q − 2n + 1) = q ((q − 1)(2q − 2n + 1)) > q · q 2 > q (q 2 − nq) + q
when q > 2n, and hence by Theorem 1.1 and Corollary 1.2, such a generalized quadrangle cannot have an automorphism of order q 2 − nq + 1 and cannot be pointtransitive.
It is unknown whether q 2 −nq+1 is prime for infinitely many positive integer values of q for a fixed n. However, the following conjecture from number theory supports this conclusion.
Conjecture 5.1 ([9]). Let f (x) = a d x d + · · · + a 1 x + a 0 be a polynomial with integer coefficients. The set {k ∈ Z : f (k) is prime} is infinite if the following three conditions hold:
(i) a d = 1, (ii) f is irreducible over Z, (iii)
The set of integers f (Z) = {f (n) : n ∈ Z} has greatest common divisor 1.
For f (x) = x 2 − nx + 1, it is plain to see that f satisfies conditions (i) and (ii) when n = 2, and f (n) = 1, showing (iii). The numerical evidence in Appendix A lends evidence that there could indeed be infinitely many such pairs where (s, t) = (q 2 − nq, q) that satisfy s + t | st(st + 1) where s + 1 is prime.
It is an interesting question as to whether generalized quadrangles of such orders actually exist. While the asymmetry of such examples is potential evidence against existence, combinatorial regularity also does not necessitate symmetry. 6. Automorphisms of a generalized quadrangle of order (4,12) This section is dedicated to analyzing the structure of the automorphism group of a generalized quadrangle of order (4,12), if one were to exist. Throughout this section, Q will be a generalized quadrangle of order (4,12) with point set P, line set L, and automorphism group G. As in the previous sections, for x ∈ G, the type of Q x refers to its designation under Lemma 2.5.
Algebraic Combinatorics, Vol. 3 #1 (2020) Lemma 6.1. If p is a prime that divides |G|, then p 7.
Proof. Let p be a prime dividing |G|. By Lemma 3.13, p 13. We know that no automorphism of order 13 exists by Corollary 1.3, and so we assume p = 11 and let x be an element of G of order 11. By Lemma 3.2, Q x cannot be of type (0); by Lemma 3.3, Q x cannot be of type (1) or type (1 ); by Lemma 3.4, Q x cannot be of type (2) or (2 ); by Lemma 3.6, Q x cannot be of type (3) or (3 ); and, by Lemma 3.11, Q x cannot be of type (4). Therefore, if p divides |G|, then p 7.
Lemma 6.2. If x ∈ G is an element of order 7, then α 0 (x) = 0.
Proof. Let x be an element of G of order 7. By Lemma 3.3, Q x cannot be of type (1) or of type (1 ). By Lemma 3.4, Q x cannot be of type (2) or of type (2 ). By Lemma 3.6, Q x cannot be of type (3) or of type (3 ) By Lemma 3.11, Q x cannot be of type (4). Therefore, Q x is of type (0) and α 0 (x) = 0. Proof. Let X be a Sylow 7-subgroup of G, and let y ∈ X. If y is not the identity and y fixes any point of Q, then y |y|/7 is an element of order 7 that fixes a point of Q, a contradiction to Lemma 6.2. This implies that X acts semiregularly on P, and so |X| divides |P| = 245. The result follows.
Lemma 6.4. If h ∈ G is an element of order 5, then α 0 (h) = 0.
Proof. Let h be an element of G of order 5. By Lemma 3.2, Q h cannot be of type (0). By Lemma 3.3, Q h cannot be of type (1). By Lemma 3.4, Q h cannot be of type (2) or of type (2 ). By Lemma 3.6, Q h cannot be of type (3) or of type (3 ). By Lemma 3.11, Q h cannot be of type (4). Therefore, Q h is of type (1 ) and α 0 (h) = 0. Lemma 6.5. A Sylow 5-subgroup of G has order at most 5.
Proof. Let H be a Sylow 5-subgroup of G, and let y ∈ H. If y is not the identity and y fixes any point of Q, then y |y|/5 is an element of order 5 that fixes a point of Q, a contradiction to Lemma 6.4. This implies that H acts semiregularly on P, and so |H| divides |P| = 245. The result follows. Lemma 6.6. If G is transitive on P, then the action of G on P is not quasiprimitive, i.e. G must contain a nontrivial normal subgroup that is intransitive on P.
Proof. Assume that the action of G on P is quasiprimitive. By [27, Theorem 1], since |P| is not a prime power, G must have a nonabelian minimal normal subgroup N = T k , where T is a nonabelian finite simple group and k ∈ N, such that N is transitive on P. Moreover, by Lemma 6.1, no prime greater than 7 divides |N |. Assume k 2. Since the largest power of 5 to divide |G| is 5, in this case 5 |T |. However, since N is transitive on P, 5 divides |N |, and so 5 divides |T |, a contradiction. Hence N = T is a finite nonabelian simple group.
On the other hand, the only primes that can divide |T | are 2, 3, 5, 7. Moreover, 5 and 7 all must divide |T |, since T is transitive on P, and the largest power of 5 dividing |T | is 5 and the largest power of 7 dividing |T | is 49. By [19,Theorem II], there is no such finite simple group. Hence the action of G on P cannot be quasiprimitive. Proof. Assume that G is transitive on P. It suffices to show that either the normalizer of a 5-subgroup contains an element of order 7 or the normalizer of a 7-subgroup contains an element of order 5, since, in either case, the normalizing element is forced to be in the centralizer of the p-subgroup.
Since G is transitive but not quasiprimitive on P by Lemma 6.6, G must contain an intransitive normal subgroup N . Let P be a Sylow p-subgroup of N for some prime p. By the Frattini Argument (see, for instance, [20, Theorem 1.13]), G = N G (P )N . This means that |G| divides |N G (P )| · |N |.
Since N is intransitive on P, there are four possibilities: (i) there are 5 distinct N -orbits of size 49, (ii) there are 7 distinct N -orbits of size 35, (iii) there are 35 distinct N -orbits of size 7, or (iv) there are 49 distinct N -orbits of size 5. Consider first the case when there are 5 distinct N -orbits of size 49. Since N is transitive on a set of size 49, 49 | |N |. Let P be a Sylow 7-subgroup of N , which has size 49. Since G is transitive on the five N -orbits, 5 | |G : N |. Since G is not divisible by 25, this implies that 5 |N |. However, since 5 divides |G|, |G| divides |N G (P )| · |N |, and 5 does not divide |N |, we have that 5 divides |N G (P )|, and so G contains an element of order 5 that normalizes (and hence centralizes) a Sylow 7-subgroup of G.
We proceed similarly in the remaining cases: if there are 7 distinct N -orbits of size 35, then 7 divides |N G (P )|, where P is a Sylow 5-subgroup of N ; if there are 35 distinct N -orbits of size 7, then 5 divides |N G (P )|, where P is a Sylow 7-subgroup of N ; and, finally, if there are 49 distinct N -orbits of size 5, then 7 divides |N G (P )|, where P is a Sylow 5-subgroup of N . In any case, if G is transitive on P, then G must contain an element of order 35, as desired.
We are now ready to prove Theorem 1.6.
Proof of Theorem 1.6. Let Q be a generalized quadrangle of order (4,12), and let G = Aut(Q). By Lemma 6.1, 13 |G|, and so G cannot be transitive on lines.
Assume that G is transitive on points. By Lemma 6.7, G must contain an element h of order 35. Since |h 5 | = 7, by Lemma 6.2, α 0 (h) = 0. Consider P 1 (h), the set of points sent to distinct collinear points by h. The orbits of h have size 5, 7, or 35, and P 1 (h) is made up of these orbits. However, since |h 5 | = 7 and |h 7 | = 5, both h 5 and h 7 are semiregular on P by Lemmas 6.2 and 6.4, and so no orbit of h can have size 5 or 7. Thus α 1 (h) ≡ 0 (mod 35).
On the other hand, by Benson's Lemma (Lemma 2.3), α 1 (h) ≡ 1 (mod 16). By the Chinese Remainder Theorem, this means that α 1 (h) ≡ 385 (mod 560). Since α 1 (h) |P| = 245, we reach a contradiction, and so G cannot be transitive on P, as desired.
Finally, we remark that, while the techniques used in this section relied heavily on the exact values of s and t, the ideas used here should be applicable to other relatively small values of s and t.
Manuscript received 14th September 2018, revised 25th June 2019, accepted 29th June 2019.
.
Let x be an automorphism of a generalized quadrangle Q. The substructure Q x fixed by x is one of the following: (0) The substructure Q x is empty; that is, there are no fixed points and there are no fixed lines. (1) At least one point is fixed, all fixed points are noncollinear, and no lines are fixed. (1 ) At least one line is fixed, all fixed lines are nonconcurrent, and no points are fixed. (2) There exists some fixed point P such that P ∼ P for each fixed point P , there exists at least one fixed line, and every fixed line is incident with P . (2 ) There exists some fixed line such that ∼ for each fixed line , there exists at least one fixed point, and every fixed point is incident with .
Lemma 3 . 10 .
310Let x be an automorphism of Q of order p, where p is a prime. If Q x has type (4) and s = s, then s + t divides st (st + 1).
Lemma 3 . 13 .
313Let p be a prime that divides the order of the automorphism group of a finite generalized quadrangle Q of order (s, t). If p (st+1), then p max{s+1, t+1}.
Lemma 6.3. A Sylow 7-subgroup of G has order at most 49.
Lemma 6 . 7 .
67If G is transitive on P, then G contains an element of order 35.Algebraic Combinatorics, Vol. 3 #1 (2020)
a thick generalized quadrangle of order (s, t), where s > t and s + 1 is prime. If then Q does not have an automorphism of order s + 1 and the automorphism group of Q cannot be point-transitive.s
t 2
s + 1
s + 1
t
> t(s + t),
Algebraic Combinatorics, Vol. 3 #1 (2020) Corollary 1.4. Let Q be a thick generalized quadrangle of order (s, t). If s + 1 is a prime and if there exists a natural number n such thatt 2
n + 1
+ t s + 1 <
t 2
n
,
then Q cannot have an automorphism of order (s + 1) and cannot have a point-
transitive group of automorphisms.
Corollary 1.5. If Q is a generalized quadrangle of order (q 2 − nq, q), where n and
q are positive integers with 2n < q and q 2 − nq + 1 is prime, then Q cannot have an
automorphism of order q 2 − nq + 1, and, moreover, Q does not have a point-transitive
group of automorphisms.
26, 1.4.1]. Lemma 2.2 ([24, Theorem I.2]). Let X and Y be disjoint sets of pairwise noncollinear points of a generalized quadrangle Q of order (s, t) with s > 1 such that |X| = m,
Acknowledgements. The authors would like to thank the anonymous referees for many useful suggestions that helped improve this paper.On prime order automorphisms of generalized quadranglesProof of Corollary 1.3. This follows immediately from point-line duality and Corollary 1.2.At first glance, the inequalitydoes not seem like much of a restriction. However, as we will see, when s is much larger than t, there are often situations when the ceiling functions contained in the inequality make a drastic difference. Corollary 1.4 shows that one implication of Theorem 1.1 is that, if s + 1 is prime and if there exists a natural number n such thatthen a generalized quadrangle of order (s, t) cannot have an automorphism of order s + 1 and cannot be point-transitive.Proof of Corollary 1.4. Assume that s + 1 is prime and thats t. If s = t, then t 2 /(n + 1) 1, which implies thata contradiction. Hence s > t. On the other hand, sincewe have n < t 2 s + 1 < n + 1, and so t 2 s + 1 = n + 1.Moreover, since t 2 /(n + 1) + t s + 1 (n + 1)(s + 1) t 2 + (n + 1)t, and, since t > 1, (n + 1)s t 2 + (n + 1)(t − 1) > t 2 . Thus,Algebraic Combinatorics, Vol. 3 #1 (2020)Appendix A. CalculationsAs t increases, there seems to be a steady increase in the proportion of feasible parameters (s, t) of generalized quadrangles that satisfy the hypotheses of Corollary 1.2 and hence cannot be point-transitive if they exist. Now, we give tables of all possible orders (s, t) of generalized quadrangles with t 100 that satisfy the hypotheses of Corollary 1.2. The tag ( * * * ) denotes that this order (s, t) has the form s = t 2 − nt where s + 1 is prime and 2n < t.Algebraic Combinatorics, Vol. 3 #1 (2020)
280, 30) (1012, 46) (2380, 60) (72, 9) *** (420, 30) (456, 48) (3540, 60) *** (40, 10) (232, 32) (540, 48) (1830, 61) (60, 12) (672, 32) *** (1128, 48) (1860, 62) (66, 12) (330, 33) (1296, 48) *** (2542, 62) *** (78, 13) (442, 34) (1666, 49) *** (3906, 63) *** (156, 13) *** (1122, 34) *** (460, 50) (576, 64) (126, 14) *** (280, 35) (700, 50) (768, 64) (210, 15) *** (490, 35) (970, 50) (976, 64) (112, 16) (700, 35) *** (1200, 50) (1216, 64) (240, 16) *** (396, 36) (1450, 50) *** (1600, 64. *** ; ***, 28) *** (6302550, 51) *** (760, 65) (210, 18) (612, 36) (796, 52) (910, 65) (306, 18) *** (630, 36) (1300, 52). 423626) (946, 44) (826, 59) (22, 6) (442, 26) *** (1276, 44) *** (660, 60) (30, 6) *** (540, 27) *** (1408, 44) *** (672, 60. 1326, 52) (3510, 65) *** (148, 20) (456, 38) (540, 54) (1408, 66) (180, 20) (546, 39) (918, 54) (2112, 66) (190, 20) (1482, 39) *** (936, 54) (2346, 66) (280, 20) *** (616, 40) (1566, 54) *** (4422, 67) *** (316, 20) (760, 40) (2376, 54) *** (1666, 68) (330, 20) (820, 41) (1870, 55) *** (3060, 68) *** (126, 21) (546, 42) (2970, 55) *** (1380, 69) (210, 21) (732, 42) (616, 56) (2346, 69) (420, 21) *** (1162, 42) (742, 56) (910, 70) (462, 22) *** (1722, 42) *** (856, 56) (2380, 70) (136, 24) (430, 43) (1008, 56) (4830, 70) *** (276, 24) (316, 44) (1288, 56) (1096, 72) (336, 24) *** (616, 44) (2296, 56) *** (1656, 72) (456, 24) *** (676, 44) (1596, 57) (2520, 72) (600, 25) *** (682, 44) (3306, 58) *** (2556, 7226) (946, 44) (826, 59) (22, 6) (442, 26) *** (1276, 44) *** (660, 60) (30, 6) *** (540, 27) *** (1408, 44) *** (672, 60) (42, 7) *** (378, 28) (576, 45) (1038, 60) (28, 8) (756, 28) *** (630, 45) (1740, 60) (40, 8) *** (270, 30) (990, 45) (2136, 60) (36, 9) (280, 30) (1012, 46) (2380, 60) (72, 9) *** (420, 30) (456, 48) (3540, 60) *** (40, 10) (232, 32) (540, 48) (1830, 61) (60, 12) (672, 32) *** (1128, 48) (1860, 62) (66, 12) (330, 33) (1296, 48) *** (2542, 62) *** (78, 13) (442, 34) (1666, 49) *** (3906, 63) *** (156, 13) *** (1122, 34) *** (460, 50) (576, 64) (126, 14) *** (280, 35) (700, 50) (768, 64) (210, 15) *** (490, 35) (970, 50) (976, 64) (112, 16) (700, 35) *** (1200, 50) (1216, 64) (240, 16) *** (396, 36) (1450, 50) *** (1600, 64) (136, 17) (408, 36) (612, 51) (2016, 64) (96, 18) (556, 36) (2550, 51) *** (760, 65) (210, 18) (612, 36) (796, 52) (910, 65) (306, 18) *** (630, 36) (1300, 52) (2080, 65) (130, 20) (852, 36) (1326, 52) (3510, 65) *** (148, 20) (456, 38) (540, 54) (1408, 66) (180, 20) (546, 39) (918, 54) (2112, 66) (190, 20) (1482, 39) *** (936, 54) (2346, 66) (280, 20) *** (616, 40) (1566, 54) *** (4422, 67) *** (316, 20) (760, 40) (2376, 54) *** (1666, 68) (330, 20) (820, 41) (1870, 55) *** (3060, 68) *** (126, 21) (546, 42) (2970, 55) *** (1380, 69) (210, 21) (732, 42) (616, 56) (2346, 69) (420, 21) *** (1162, 42) (742, 56) (910, 70) (462, 22) *** (1722, 42) *** (856, 56) (2380, 70) (136, 24) (430, 43) (1008, 56) (4830, 70) *** (276, 24) (316, 44) (1288, 56) (1096, 72) (336, 24) *** (616, 44) (2296, 56) *** (1656, 72) (456, 24) *** (676, 44) (1596, 57) (2520, 72) (600, 25) *** (682, 44) (3306, 58) *** (2556, 72)
2052, 76) (2436, 84) (3796, 91) (2850, 76) (2856, 84) (8190, 91) *** (5700, 76) *** (4200, 84) *** (3082, 92) (4642, 77) (4326, 84) (6256, 92) *** (936, 78) (4956, 84) *** (2790, 93) (1950, 78) (6036, 84) (1692, 94) (2766, 78) (6580, 84) (4512, 96) (6006, 78) *** (990, 85) (4560, 96) (6162, 79) *** (1360. 1050, 75) (1596, 84) (5580, 90) *** (1800, 75) (1876, 84) (6210, 90) *** (4200, 75) *** (2268, 84) (8010, 90) *** (1596, 76) (2296, 84. 3853432, 72) (1092, 84) (3060, 90) (5112, 72) *** (1276, 84) (3186, 90) (1776, 74) (1582, 84. 4656, 97) (1200, 80) (3612, 86) (1288, 98) (1216, 80) (4902, 86) *** (3136, 98) (1360, 80) (2436, 87) (4018, 98) (1720, 80) (1870, 88) (2376, 99) (2080, 80) (3916, 89) (2926, 99) (2620, 80) (1530, 90) (3168, 99) (2800, 80) (1548, 90) (4356, 99) (3120, 80) (1800, 90) (5346, 99) *** (3760, 80) *** (1860, 90) (1900, 100) *** (4240, 80) *** (2010, 90) (4950, 100) (4240, 80) *** (2016, 90) (9900, 100) ** (4720, 80) *** (2178, 90) (6480, 81) *** (2250, 90Algebraic Combinatorics, Vol. 3 #1 (2020) (3432, 72) (1092, 84) (3060, 90) (5112, 72) *** (1276, 84) (3186, 90) (1776, 74) (1582, 84) (3690, 90) (1050, 75) (1596, 84) (5580, 90) *** (1800, 75) (1876, 84) (6210, 90) *** (4200, 75) *** (2268, 84) (8010, 90) *** (1596, 76) (2296, 84) (2002, 91) (2052, 76) (2436, 84) (3796, 91) (2850, 76) (2856, 84) (8190, 91) *** (5700, 76) *** (4200, 84) *** (3082, 92) (4642, 77) (4326, 84) (6256, 92) *** (936, 78) (4956, 84) *** (2790, 93) (1950, 78) (6036, 84) (1692, 94) (2766, 78) (6580, 84) (4512, 96) (6006, 78) *** (990, 85) (4560, 96) (6162, 79) *** (1360, 85) (6112, 96) (880, 80) (3570, 85) (4656, 97) (1200, 80) (3612, 86) (1288, 98) (1216, 80) (4902, 86) *** (3136, 98) (1360, 80) (2436, 87) (4018, 98) (1720, 80) (1870, 88) (2376, 99) (2080, 80) (3916, 89) (2926, 99) (2620, 80) (1530, 90) (3168, 99) (2800, 80) (1548, 90) (4356, 99) (3120, 80) (1800, 90) (5346, 99) *** (3760, 80) *** (1860, 90) (1900, 100) *** (4240, 80) *** (2010, 90) (4950, 100) (4240, 80) *** (2016, 90) (9900, 100) ** (4720, 80) *** (2178, 90) (6480, 81) *** (2250, 90)
. Algebraic Combinatorics. 31Algebraic Combinatorics, Vol. 3 #1 (2020) References
Ovoids of generalized quadrangles of order (q, q 2 − q) and Delsarte cocliques in related strongly regular graphs. Mohammad Adm, Ryan Bergen, Ferdinand Ihringer, Sam Jaques, Karen Meagher, Alison Purdy, Boting Yang, J. Combin. Des. 265Mohammad Adm, Ryan Bergen, Ferdinand Ihringer, Sam Jaques, Karen Meagher, Alison Purdy, and Boting Yang, Ovoids of generalized quadrangles of order (q, q 2 − q) and Delsarte cocliques in related strongly regular graphs, J. Combin. Des. 26 (2018), no. 5, 249-263.
Generalised quadrangles with a group of automorphisms acting primitively on points and lines. John Bamberg, Michael Giudici, Joy Morris, Gordon F Royle, Pablo Spiga, J. Combin. Theory Ser. A. 1197John Bamberg, Michael Giudici, Joy Morris, Gordon F. Royle, and Pablo Spiga, Generalised quadrangles with a group of automorphisms acting primitively on points and lines, J. Combin. Theory Ser. A 119 (2012), no. 7, 1479-1499.
Generalized quadrangles and transitive pseudo-hyperovals. John Bamberg, Stephen Peter Glasby, Tomasz Popiel, Cheryl E Praeger, J. Combin. Des. 244John Bamberg, Stephen Peter Glasby, Tomasz Popiel, and Cheryl E. Praeger, Generalized quad- rangles and transitive pseudo-hyperovals, J. Combin. Des. 24 (2016), no. 4, 151-164.
A classification of finite antiflag-transitive generalized quadrangles. John Bamberg, Eric Cai Heng Li, Swartz, Trans. Amer. Math. Soc. 3703John Bamberg, Cai Heng Li, and Eric Swartz, A classification of finite antiflag-transitive gen- eralized quadrangles, Trans. Amer. Math. Soc. 370 (2018), no. 3, 1551-1601.
A classification of finite locally 2-transitive generalized quadrangles. , A classification of finite locally 2-transitive generalized quadrangles, https://arxiv. org/abs/1903.07442, 2019.
Point-primitive, line-transitive generalised quadrangles of holomorph type. John Bamberg, Tomasz Popiel, Cheryl E Praeger, J. Group Theory. 202John Bamberg, Tomasz Popiel, and Cheryl E. Praeger, Point-primitive, line-transitive gener- alised quadrangles of holomorph type, J. Group Theory 20 (2017), no. 2, 269-287.
Simple groups, product actions, and generalized quadrangles. Nagoya Math. J. 234, Simple groups, product actions, and generalized quadrangles, Nagoya Math. J. 234 (2019), 87-126.
On the structure of generalized quadrangles. Clark T Benson, J. Algebra. 15Clark T. Benson, On the structure of generalized quadrangles, J. Algebra 15 (1970), 443-454.
Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la composition des entiers en facteurs. Victor Bouniakowsky, Sc. Math. Phys. 6Victor Bouniakowsky, Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la composition des entiers en facteurs, Sc. Math. Phys. 6 (1857), 305-329.
Permutation groups. J Peter, Cameron, Cambridge University Press45CambridgePeter J. Cameron, Permutation groups, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, Cambridge, 1999.
Automorphisms of strongly regular graphs with applications to partial difference sets. Ellen Stefaan De Winter, Zeying Kamischke, Wang, Des. Codes Cryptogr. 793Stefaan De Winter, Ellen Kamischke, and Zeying Wang, Automorphisms of strongly regular graphs with applications to partial difference sets, Des. Codes Cryptogr. 79 (2016), no. 3, 471- 485.
Generalized quadrangles with an abelian Singer group. De Stefaan, Koen Winter, Thas, Des. Codes Cryptogr. 391Stefaan De Winter and Koen Thas, Generalized quadrangles with an abelian Singer group, Des. Codes Cryptogr. 39 (2006), no. 1, 81-87.
The automorphism group of Payne derived generalized quadrangles. Adv. Math. 2141, The automorphism group of Payne derived generalized quadrangles, Adv. Math. 214 (2007), no. 1, 146-156.
Generalized quadrangles admitting a sharply transitive Heisenberg group. Des. Codes Cryptogr. 471-3, Generalized quadrangles admitting a sharply transitive Heisenberg group, Des. Codes Cryptogr. 47 (2008), no. 1-3, 237-242.
On automorphisms of a distance-regular graph with intersection array {56, 45, 1; 1, 9. Alexander L Gavrilyuk, Aleksander A Makhnev, Dokl. Akad. Nauk. 565Alexander L. Gavrilyuk and Aleksander A. Makhnev, On automorphisms of a distance-regular graph with intersection array {56, 45, 1; 1, 9, 56}, Dokl. Akad. Nauk 432 (2010), no. 5, 583-587.
Regular groups on generalized quadrangles and nonabelian difference sets with multiplier −1. Dina Ghinelli, Geom. Dedicata. 412Dina Ghinelli, Regular groups on generalized quadrangles and nonabelian difference sets with multiplier −1, Geom. Dedicata 41 (1992), no. 2, 165-174.
Transitive projective planes. Nick Gill, Adv. Geom. 74Nick Gill, Transitive projective planes, Adv. Geom. 7 (2007), no. 4, 475-528.
Transitive projective planes and insoluble groups. Trans. Amer. Math. Soc. 3685, Transitive projective planes and insoluble groups, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3017-3057.
Simple groups of order divisible by at most four primes. Bertram Huppert, Wolfgang Lempken, Proc. F. Scorina Gomel State Univ. F. Scorina Gomel State Univ16Bertram Huppert and Wolfgang Lempken, Simple groups of order divisible by at most four primes, Proc. F. Scorina Gomel State Univ. 16 (2000), no. 3, 64-75.
I , Martin Isaacs, Finite group theory. Providence, RIAmerican Mathematical Society92I. Martin Isaacs, Finite group theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008.
Search for properties of the missing Moore graph. Martin Mačaj, Jozef Širáň, Linear Algebra Appl. 4329Martin Mačaj and Jozef Širáň, Search for properties of the missing Moore graph, Linear Algebra Appl. 432 (2010), no. 9, 2381-2398.
Ovoids and bipartite subgraphs in generalized quadrangles. Alexander A Makhnev, Jr , Alexander A Makhnev, Mat. Zametki. 736Alexander A. Makhnev, Jr. and Alexander A. Makhnev, Ovoids and bipartite subgraphs in generalized quadrangles, Mat. Zametki 73 (2003), no. 6, 878-885.
Automorphism groups of generalized quadrangles via an unusual action of PΓL(2, 2 h ). Christine M O'keefe, Tim Penttila, European J. Combin. 232Christine M. O'Keefe and Tim Penttila, Automorphism groups of generalized quadrangles via an unusual action of PΓL(2, 2 h ), European J. Combin. 23 (2002), no. 2, 213-232.
An inequality for generalized quadrangles. Stanley E Payne, Proc. Amer. Math. Soc. 711Stanley E. Payne, An inequality for generalized quadrangles, Proc. Amer. Math. Soc. 71 (1978), no. 1, 147-152.
The fundamental theorem of q-clan geometry. Des. Codes Cryptogr. 81-2Special issue dedicated to Hanfried Lenz, The fundamental theorem of q-clan geometry, Des. Codes Cryptogr. 8 (1996), no. 1-2, 181-202, Special issue dedicated to Hanfried Lenz.
E Stanley, Joseph A Payne, Thas, Finite generalized quadrangles. ZürichEMSsecond ed.Stanley E. Payne and Joseph A. Thas, Finite generalized quadrangles, second ed., EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2009.
Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs. Cheryl E Praeger, An O , J. London Math. Soc. 2Cheryl E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc. (2) 47 (1993), no. 2, 227-239.
. Algebraic Combinatorics. 31Algebraic Combinatorics, Vol. 3 #1 (2020)
On generalized quadrangles with a point regular group of automorphisms. Eric Swartz, European J. Combin. 79Eric Swartz, On generalized quadrangles with a point regular group of automorphisms, European J. Combin. 79 (2019), 60-74.
Sur la trialité et certains groupes qui s'en déduisent. Jacques Tits, Inst. Hautes Etudes Sci. Publ. Math. 2Jacques Tits, Sur la trialité et certains groupes qui s'en déduisent, Inst. Hautes Etudes Sci. Publ. Math. 2 (1959), 14-60.
A generalized quadrangle with an automorphism group acting regularly on the points. Satoshi Yoshiara, European J. Combin. 282Satoshi Yoshiara, A generalized quadrangle with an automorphism group acting regularly on the points, European J. Combin. 28 (2007), no. 2, 653-664.
. F Santana, Afton, Department Of Mathematics, P O Mary, Box, 8795Williamsburg, VA 23187-8795, USA School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA [email protected] F. Afton, Department of Mathematics, College of William & Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, USA School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332- 0160, USA E-mail : [email protected]
. P.O. Box. 8795Eric Swartz, Department of Mathematics, College of William & [email protected] Swartz, Department of Mathematics, College of William & Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, USA E-mail : [email protected]
. Algebraic Combinatorics. 31Algebraic Combinatorics, Vol. 3 #1 (2020)
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"Matthieu Schaller \nLorentz Institute for Theoretical Physics\nLeiden University\nPO Box 9506NL-2300 RALeidenthe Netherlands\n\nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n",
"Richard G Bower \nInstitute for Computational Cosmology\nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamUK\n",
"Joop Schaye \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n"
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"Institute for Computational Cosmology\nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamUK",
"Department of Physics\nKavli Institute for Astrophysics and Space Research\nMassachusetts Institute of Technology\n02139CambridgeMAUSA",
"Lorentz Institute for Theoretical Physics\nLeiden University\nPO Box 9506NL-2300 RALeidenthe Netherlands",
"Leiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands",
"Institute for Computational Cosmology\nDepartment of Physics\nUniversity of Durham\nSouth RoadDH1 3LEDurhamUK",
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] | [
"MNRAS"
] | Smoothed Particle Hydrodynamics (SPH) is a ubiquitous numerical method for solving the fluid equations, and is prized for its conservation properties, natural adaptivity, and simplicity. We introduce the Sphenix SPH scheme, which was designed with three key goals in mind: to work well with sub-grid physics modules that inject energy, be highly computationally efficient (both in terms of compute and memory), and to be Lagrangian. Sphenix uses a Density-Energy equation of motion, along with variable artificial viscosity and conduction, including limiters designed to work with common sub-grid models of galaxy formation. In particular, we present and test a novel limiter that prevents conduction across shocks, preventing spurious radiative losses in feedback events. Sphenix is shown to solve many difficult test problems for traditional SPH, including fluid mixing and vorticity conservation, and it is shown to produce convergent behaviour in all tests where this is appropriate. Crucially, we use the same parameters within Sphenix for the various switches throughout, to demonstrate the performance of the scheme as it would be used in production simulations. Sphenix is the new default scheme in the Swift cosmological simulation code and is available open-source. | 10.1093/mnras/stab3166 | [
"https://arxiv.org/pdf/2012.03974v2.pdf"
] | 227,745,707 | 2012.03974 | 43d7425b64eb2c636241e052e04b23042e845ace |
Sphenix: Smoothed Particle Hydrodynamics for the next generation of galaxy formation simulations
29 October 2021
Josh Borrow
Institute for Computational Cosmology
Department of Physics
University of Durham
South RoadDH1 3LEDurhamUK
Department of Physics
Kavli Institute for Astrophysics and Space Research
Massachusetts Institute of Technology
02139CambridgeMAUSA
Matthieu Schaller
Lorentz Institute for Theoretical Physics
Leiden University
PO Box 9506NL-2300 RALeidenthe Netherlands
Leiden Observatory
Leiden University
P.O. Box 95132300 RALeidenThe Netherlands
Richard G Bower
Institute for Computational Cosmology
Department of Physics
University of Durham
South RoadDH1 3LEDurhamUK
Joop Schaye
Leiden Observatory
Leiden University
P.O. Box 95132300 RALeidenThe Netherlands
Sphenix: Smoothed Particle Hydrodynamics for the next generation of galaxy formation simulations
MNRAS
000000029 October 2021Preprint 29 October 2021 Compiled using MNRAS L A T E X style file v3.0galaxies: formationgalaxies: evolutionmethods: N-body simulationsmethods: numericalhydrodynamics
Smoothed Particle Hydrodynamics (SPH) is a ubiquitous numerical method for solving the fluid equations, and is prized for its conservation properties, natural adaptivity, and simplicity. We introduce the Sphenix SPH scheme, which was designed with three key goals in mind: to work well with sub-grid physics modules that inject energy, be highly computationally efficient (both in terms of compute and memory), and to be Lagrangian. Sphenix uses a Density-Energy equation of motion, along with variable artificial viscosity and conduction, including limiters designed to work with common sub-grid models of galaxy formation. In particular, we present and test a novel limiter that prevents conduction across shocks, preventing spurious radiative losses in feedback events. Sphenix is shown to solve many difficult test problems for traditional SPH, including fluid mixing and vorticity conservation, and it is shown to produce convergent behaviour in all tests where this is appropriate. Crucially, we use the same parameters within Sphenix for the various switches throughout, to demonstrate the performance of the scheme as it would be used in production simulations. Sphenix is the new default scheme in the Swift cosmological simulation code and is available open-source.
INTRODUCTION
There have been many approaches to solving the equations of motion for a collisional fluid in a cosmological context over the years, from simple first order fixed-grid (Cen 1992) to high-order discontinuous Galerkin schemes solved in an adaptive environment (Guillet et al. 2019). Because Smoothed Particle Hydrodynamics (SPH) strikes the sweet spot between computational cost, stability, and adaptivity, it has been used throughout the astronomical community for nearly five decades. SPH was originally developed by Lucy (1977) and Gingold & Monaghan (1977), and was used initially to model individual stars, as this problem was well suited to Lagrangian schemes. Shortly after, further applications of the method were deployed for the study of the fragmentation of gas clouds (Wood 1981), and for the formation of planets (Benz 1988).
The practical use of SPH in a cosmological context began with Hernquist & Katz (1989), which provided a novel solution to the large dynamic range of time-steps required to evolve a cosmological fluid, and was cemented by the Gadget-2 code (Springel 2005) that was made public and exploited worldwide to model galaxy formation processes within this context for the first time (e.g. Dolag et al. 2004;Ettori et al. 2006;Crain et al. 2007). The base SPH model released in Gadget-2, however, was relatively simple, consisting of a fixed artificial viscosity coefficient and scheme based on Monaghan (1992). Improved models existed, such as those presented in Monaghan (1997), but the key that led to the community rallying around Gadget-2 was both its open source nature and scalability, with Gadget-2 able to run on hundreds or thousands of cores.
The popularity of Gadget-2, and similar codes like GASOLINE (Wadsley et al. 2004), along with its relatively simple hydrodynamics model, led to critical works such as Agertz et al. (2007) and Bauer & Springel (2012) that pointed out flaws in their SPH modelling, relative to mesh-based codes of the time. The SPH community as a whole, however, already had solutions to these problems (see e.g. Price 2008) and many robust solutions were proposed and integrated into cosmological modelling codes. In Heß & Springel (2010), the authors experimented with an extension to Gadget-2 using a Voronoi mesh to reduce errors inherrent in SPH and allow for better results on fluid mixing problems, eventually giving rise to the AREPO moving mesh scheme, allowing for significantly improved accuracy per particle but drastically increasing computational cost (Springel 2010;Weinberger et al. 2020). In this case, the authors have steadily increased their computational cost per particle in an attempt to reduce errors inherrent in their hydrodynamics model as much as practicable.
Other authors took different directions, with the GASOLINE code (Wadsley et al. 2004(Wadsley et al. , 2008(Wadsley et al. , 2017 choosing to explicitly average pressures within the SPH equation of motion to alleviate the problems of artificial surface tension; the PHANTOM developers (Price 2008(Price , 2012Price et al. 2018) advocating for artificial conduction of energy; and further developments on the Gadget-2 and updated Gadget-3 code by Hopkins (2013) and Hu et al. (2014) based on the work by Saitoh & Makino (2013) using an explicit smoothed pres-sure scheme to ensure a consistent pressure field over the contact discontinuities that artificial surface tension arises from.
Simultaneously, there was work to reduce the fundamental numerical errors present in SPH taking place by (Cullen & Dehnen 2010;Dehnen & Aly 2012;Read et al. 2010;Read & Hayfield 2012) through the use of improved choices for the SPH kernel, which up until this point was assumed to have little effect on results from SPH simulations. These improved kernels typically have larger 'wings', encompassing more neighbours and providing more accurate reconstructions for smoothed quantities. These more accurate reconstructions are particularly important for the construction of accurate gradients, which enter into 'switches' that control the strength of the artificial viscosity and conduction terms.
The rise of more complex SPH models occurred alongside a significant jump in the complexity of the corresponding galaxy formation models; such an increase in complexity was required as resolutions increased over time, meaning more physics could be modelled directly. Many astrophysical processes take place on scales smaller than what can be resolved in simulations and are included in these so-called galaxy formation 'sub-grid' models. These processes include radiative cooling, which has progressed from a simple one parameter model to element and even ionisation state dependent rates (see e.g. Wiersma et al. 2009;Ploeckinger & Schaye 2020); star formation (see e.g. Cen & Ostriker 1992;, and references therein); and stellar feedback to model supernovae and element outflows (see e.g. Navarro & White 1993;Springel & Hernquist 2003;Schaye 2008, 2012, andreferences therein). The coupling of these processes to hydrodynamics is complex and often overlooked; careful treatment of conservation laws and quirks of the chosen variables used to represent the fluid can frequently hide errors in plain sight .
The development of the Swift code (Schaller et al. 2016) led to a re-implementation of the sub-grid model used for the EAGLE simulation , and a chance to re-consider the Anarchy SPH scheme that was used in the original (Gadget-3 based) code (see Schaller et al. 2015, for details on the scheme). The findings in Oppenheimer et al. (2018) (their Appendix D) and meant that a switch away from the original Pressure-Entropy scheme to one based on a smoothed density field was preferred, along with the key design goals outlined below. This work describes the Sphenix 1 scheme and demonstrates its performance on many hydrodynamics tests. We note here that Sphenix does not give the best performance-per-particle (in both absolute values of L1 norm (see §5.1 for our definition of the L1 norm) compared to the analytical solution, and in terms of convergence speed) compared to other schemes. The moving mesh AREPO (Springel 2010), finitevolume GIZMO (Hopkins 2015), and corrected scheme presented in Rosswog (2020a) will produce improved results. Sphenix however lies in the very low-cost (memory and computation) per particle sweet-spot that traditional SPH schemes occupy, whilst maximising performance with some novel limiters for artificial conduction and viscosity. This makes it an excellent choice for the default SPH scheme in Swift.
The remainder of this paper is organised as follows: in §2 we describe the Swift cosmological simulation code and the time-stepping algorithms present within it. In §3 we describe Sphenix in its entirety. In §4 we describe the artificial conduction limiter used for energetic 1 Note that, similar to the popular Gizmo schemes, Sphenix is not an acronym. feedback schemes. Finally, in §5 we show how Sphenix performs on various hydrodynamics problems.
THE Swift SIMULATION CODE
The Swift 2 simulation code (Schaller et al. 2016) is a hybrid parallel SPH and gravity code, designed to run across multiple compute nodes using MPI, but to utilise threads on each node (rather than the traditional method of using one MPI rank per core). This, along with its task-based parallelism approach, asynchronous communication scheme, and work-splitting domain decomposition system allow for excellent strong-and weak-scaling characteristics (Borrow et al. 2018).
Swift is also designed to be hugely modular, with hydrodynamics schemes, gravity schemes, and sub-grid models able to be easily swapped out. Swift can be configured to use a replica of the Gadget-2 hydrodynamics scheme (Springel & Hernquist 2002), a simplified version of the base PHANTOM scheme (Price et al. 2018), the MFM and MFV schemes described in Hopkins (2015), Sphenix, or a host of other schemes. It can also be configured to use multiple different galaxy formation sub-grid models, including a very basic scheme (constant Λ cooling, no star formation), the EAGLE sub-grid model , a 'Quick Lyman-α" model, the GEAR sub-grid model (Revaz & Jablonka 2012), and some further evolutions including cooling tables from Ploeckinger & Schaye (2020). The gravity solver is interchangeable but the one used here, and throughout all Swift simulations, uses the Fast Multipole Method (Greengard & Rokhlin 1987) with an adaptive opening angle, similar to Dehnen (2014).
Time integration
Swift uses a velocity-verlet scheme to integrate particles through time. This takes their acceleration ( a) from the equation of motion and time-step (∆t) and integrates their position forward in time through a Kick-Drift-Kick scheme as follows:
v t + ∆t 2 = v(t) + ∆t 2 a(t),(1)r (t + ∆t) = r(t) + v t + ∆t 2 ∆t,(2)v (t + ∆t) = v t + ∆t 2 + ∆t 2 a(t + ∆t),(3)
where the first and last equations, updating the velocity, are referred to as the 'kick', and the central equation is known as the 'drift'. The careful observer will note that the 'drift' can be split into as many pieces as required allowing for accurate interpolation of the particle position in-between kick steps. This is important in cosmological galaxy formation simulations, where the dynamic range is large. In this case, particles are evolved with their own, particle-carried timestep, given by
∆t i = C CFL 2γ K h i v sig,i ,(4)
dependent on the Courant-Friedrichs-Lewy (C CFL Courant et al. 1928) constant, the kernel-dependent relationship between cut-off and smoothing length γ K , particle-carried smoothing length h i , and signal velocity v sig,i (see Equation 30). The discussion of the full time-stepping algorithm is out of the scope of this work, but see Hernquist & Katz (1989) and Borrow et al. (2019) for more information.
Time-step Limiter
As the time-step of the particles is particle-carried, there may be certain parts of the domain that contain interacting particles with vastly different time-steps (this is particularly promoted by particles with varied temperatures within a given kernel). Having these particles interact is problematic for a number of reasons, and as such we include the time-step limiter described in Saitoh & Makino (2009);Durier & Dalla Vecchia (2012) in all problems solved below. Swift chooses to limit neighbouring particles to have a maximal time-step difference of a factor of 4.
Sphenix
The Sphenix scheme was designed to replace the Anarchy scheme used in the original EAGLE simulations for use in the Swift simulation code. This scheme had three major design goals:
• Be a Lagrangian SPH scheme, as this has many advantages and is compatible with the EAGLE subgrid model.
• Work well with the EAGLE subgrid physics, namely instantaneous energy injection and subgrid cooling.
• Be highly computationally and memory efficient.
The last requirement precludes the use of any Riemann solvers in socalled Gizmo-like schemes (although these do not necessarily give improved results for astrophysical problem sets, see Borrow et al. 2019); see Appendix A. The second requirement also means that the use of a pressure-based scheme (such as Anarchy) is not optimal, see for more details. The Sphenix scheme is based on so-called 'Traditional' Density-Energy SPH. This means that it uses the smoothed mass density,
ρ( x) = j m j W(| x − x j |, h( x))(5)
where here j are indices describing particles in the system, h( x) is the smoothing length evaluated at position x, and W(r, h) is the kernel function.
In the examples below, the Quartic Spline (M5) kernel,
w(q) = 5 2 − q 4 − 5 3 2 − q 4 + 10 1 2 − q 4 q < 1 2 5 2 − q 4 − 5 3 2 − q 4 1 2 ≤ q < 3 2 5 2 − q 4 3 2 ≤ q < 5 2 0 q ≥ 5 2(6)
with W(r, h) = κ n D w(r/h)/h n D , n D the number of dimensions, and κ 3 = (7/478π) for three dimensions, is used. The Sphenix scheme has also been tested with other kernels, notably the Cubic and Quintic Spline (M4, M6) and the Wendland (C2, C4, C6) kernels (Wendland 1995). The choice of kernel does not qualitatively affect the results in any of the tests in this work (see Dehnen & Aly 2012, for significantly more information on kernels). Higher order kernels do allow for lower errors on tests that rely on extremely accurate reconstructions to cancel forces (for instance the Gresho-Chan vortex, §5.3), but we find that the Quintic Spline provides an excellent trade-off between computational cost and accuracy in practical situations. Additionally, the Wendland kernels do have the benefit that they are not susceptible to the pairing instability, but they must have an ad-hoc correction applied in practical use (Dehnen & Aly 2012, Section 2.5). We find no occurrences of the pairing instability in both the tests and our realistic simulations. The Sphenix scheme is kernel-invariant, and as such can be used with any reasonable SPH kernel.
The smoothing length h is determined by satisfyinĝ
n( x) = j W | x − x j |, h( x) = η h( x) n D ,(7)
with η setting the resolution scale. The precise choice for η generally does not qualitatively change results; here we choose η = 1.2 due to this value allowing for a very low E 0 error (see Read et al. 2010;Dehnen & Aly 2012) 3 , which is a force error originating from particle disorder within a single kernel. In Swift, these equations are solved numerically to a relative accuracy of 10 −4 . The smoothed mass density, along with a particle-carried internal energy per unit mass u, is used to determine the pressure at a particle position through the equation of state
P( x i ) = P i = (γ − 1)u iρi ,(8)
with γ the ratio of specific heats, taken to be 5/3 throughout unless specified. This pressure enters the first law of thermodynamics,
∂u i ∂ q i A i = − P i m i ∂V i ∂ q i ,(9)
with q i a state vector containing both x i and h i as independent variables, A i the entropy of particle i (i.e. this equation only applies to dissipationless dynamics), and V i = m i /ρ i describing the volume represented by particle i. This constraint, along with the one on the smoothing length, allows for an equation of motion to be extracted from a Lagrangian (see e.g. the derivations in Springel & Hernquist 2002;Hopkins 2013),
d v i dt = − j m j f i j P î ρ 2 i ∇ i W i j + f ji P ĵ ρ 2 j ∇ j W ji ,(10)
where W ab = W(| x b − x a |, h( x a )), ∇ a = ∂/∂ x a , and f ab a dimensionless factor encapsulating the non-uniformity of the smoothing length field
f ab = 1 − 1 m b h a n Dna ∂ρ i ∂h i 1 + h i n Dni ∂n i ∂h i −1(11)
and is generally of order unity. There is also an associated equation of motion for internal energy,
du i dt = − j m j f i j P î ρ 2 i v i j · ∇ i W i j ,(12)
3 This corresponds to ∼58 weighted neighbours for our Quartic Spline in a scenario where all neighbours have uniform smoothing lengths. In practical simulations the 'number of neighbours' that a given particle interacts with can vary by even orders of magnitude but Equation 7 must be satisfied for all particles ensuring an accurate reconstruction of the field. More discussion on this choice of smoothing length can be found in (Springel & Hernquist 2002;Monaghan 2002;Price 2007Price , 2012. We chose η = 1.2 based on Figure 3 in Dehnen & Aly (2012), where this corresponds to a very low reconstruction error in the density.
with v i j = v j − v i . Note that other differences between vector quantities are defined in a similar way, including for the separation of two particles x i j = x j − x i .
Artificial Viscosity
These equations, due to the constraint of constant entropy introduced in the beginning, lead to naturally dissipationless solutions; they cannot capture shocks. Shock capturing in SPH is generally performed using 'artificial viscosity'. The artificial viscosity implemented in Sphenix is a simplified and modified extension to the Cullen & Dehnen (2010) 'inviscid SPH' scheme. This adds the following terms to the equation of motion (see Monaghan 1992, and references within),
d v i dt = − j m j ζ i j f i j ∇ i W i j + f ji ∇ j W ji ,(13)
and to the associated equation of motion for the internal energy,
du i dt = − 1 2 j m j ζ i j v i j · f i j ∇ i W i j + f ji ∇ j W ji ,(14)
where ζ i j controls the strength of the viscous interaction. Note here that the internal energy equation of motion is explicitly symmetrised, which was not the case for the SPH equation of motion for internal energy (Eqn. 12). In this case, that means that there are terms from both the i j and ji interactions in Equation 14, whereas in Equation 12 there is only a term from the i j interaction. This choice was due to the symmetric version of this equation performing significantly better in the test examples below, likely due to multiple time-stepping errors within regions where the viscous interaction is the strongest 4 . There are many choices available for ζ i j , with the case used here being
ζ i j = −α V µ i j v sig,i ĵ ρ i +ρ j ,(15)
where
µ i j = v i j · x i j | x i j | v i j · x i j < 0 0 v i j · x i j ≥ 0(16)
is a basic particle-by-particle converging flow limiter (meaning that the viscosity term vanishes when ∇ · v ≥ 0), and
v sig,i j = c i + c j − β V µ i j ,(17)
is the signal velocity between particles i and j, with β V = 3 a dimensionless constant, and with c i the soundspeed of particle i defined through the equation of state as
c i = P î ρ i = (γ − 1)γu i .(18)
Finally, the dimensionless viscosity coefficient α V (Monaghan & Gingold 1983) is frequently taken to be a constant of order unity. In Sphenix, this becomes an interaction-dependent constant (see Morris & Monaghan 1997;Cullen & Dehnen 2010, for similar schemes), with α V = α V,i j , dependent on two particle-carried α values as follows:
α V,i j = 1 4 (α V,i + α V, j )(B i + B j ),(19)
where
B i = |∇ · v i | |∇ · v i | + |∇ × v i | + 10 −4 c i /h i(20)
is the Balsara (1989) switch for particle i, which allows for the deactivation of viscosity in shear flows, where there is a high value of ∇ · v, but the associated shear viscosity is unnecessary. This, in particular, affects rotating shear flows such as galaxy disks, where the scheme used to determine α V,i described below will return a value close to the maximum. The equation for α V,i is solved independently for each particle over the course of the simulation. Note that α V,i is never drifted, and is only ever updated at the 'kick' steps. The source term in the equation for α V,i , designed to activate the artificial viscosity within shocking regions, is the shock indicator
S i = −h 2 i max ∇ · v i , 0 ∇ · v i ≤ 0 0 ∇ · v i > 0(21)
where here the time differential of the local velocity divergence fielḋ
∇ · v i (t + ∆t) = ∇ · v i (t + ∆t) − ∇ · v i (t) ∆t(22)
with ∇ · v i the local velocity divergence field and ∆t the time-step associated with particle i. The primary variable in the shock indicator S i of∇ · v is high in pre-shock regions, with the secondary condition for the flow being converging (∇ · v ≤ 0) helpful to avoid false detections as the Balsara (1989) switch is used independently from the equation that evolves α V,i (this choice is notably different from most other schemes that use B i directly in the shock indicator S i ). This choice allows for improved shock capturing in shearing flows (e.g. feedback events occurring within a galaxy disk). In these cases, the Balsara (1989) switch (which is instantaneously evaluated) rapidly becomes close to 1.0, and the already high value of α V,i allows for a strong viscous reaction from the fluid. The shock indicator is then transformed into an optimal value for the viscosity coefficient as
α V,loc,i = α V,max S i c 2 i + S i ,(23)
with a maximum value of α V,max = 2.0 for α V,loc . The value of α V,i is then updated as follows:
α V,i = α V,loc,i α V,i < α V,loc,i α V,i +α V,loc,i ∆t τ V,i 1+ ∆t τ V,i α V,i > α V,loc,i(24)
where τ V,i = γ K V h i /c i with γ K the 'kernel gamma' a kernel dependent quantity relating the smoothing length and compact support (γ K = 2.018932 for the quartic spline in 3D, Dehnen & Aly 2012) and V a constant taking a value of 0.05. The final value of α V,i is checked against a minimum, however the default value of this minimum is zero and the evolution strategy used above guarantees that α V,i is strictly positive and that the decay is stable regardless of timestep.
Artificial Conduction
Attempting to resolve sharp discontinuities in non-smoothed variables in SPH leads to errors. This can be seen above, with strong velocity discontinuities (shocks) not being correctly handled and requiring an extra term in the equation of motion (artificial viscosity) to be captured. A similar issue arises when attempting to resolve strong discontinuities in internal energy (temperature). To resolve this, we introduce an artificial energy conduction scheme similar to the one presented by Price (2008). This adds an extra term to the equation of motion for internal energy,
du i dt = j m j v D,i j (u i − u j )r i j · f i j ∇ i W i ĵ ρ i + f ji ∇ j W jî ρ j(25)
withr i j the unit vector between particles i and j, and where
v D,i j = α D,i j 2 | v i j · x i j | | x i j | + 2 |P i − P j | ρ j +ρ j .(26)
This conductivity speed is the average of two commonly used speeds, with the former velocity-dependent term taken from Price et al. (2018) (modified from Wadsley et al. 2008), and the latter pressure-dependent term taken from Price (2008). These are usually used separately for cases that aim to reduce entropy generation in disordered fields and contact discontinuities respectively(where initially there is a strong discontinuity in pressure that is removed by the artificial conduction scheme), but we combine them here as both cases are relevant in galaxy formation simulations and use this same velocity throughout our testing, a notable difference with other works using conduction schemes (e.g. Price et al. 2018). Price et al. (2018) avoided pressure-based terms in simulations with selfgravity, but they use no additional terms (e.g. our α D ) to limit conduction in flows where it is not required. This is additionally somewhat similar to the conduction speed used in Anarchy and Hu et al. (2014), which is a modified version of the signal velocity (Eqn. 17) with our speed replacing the sum of sound speeds with a differenced term. Appendix B contains an investigation of the individual terms in the conduction velocity. The interaction-dependent conductivity coefficient,
α D,i j = P i α D,i + P j α D, j P i + P j ,(27)
is pressure-weighted to enable the higher pressure particle to lead the conduction interaction, a notable departure from other thermal conduction schemes in use today. This is critical when it comes to correctly applying the conduction limiter during feedback events, described below. The individual particle-carried α D,i are ensured to only be active in cases where there is a strong discontinuity in internal energy. This is determined by using the following discontinuity indicator,
K i = β D γ K h i ∇ 2 u i √ u i ,(28)
where β D is a fixed dimensionless constant taking a value of 1. The discontinuity indicator enters the time differential for the individual conduction coefficients as a source term,
dα D,i dt = K i + α D,min − α D,i τ D,i ,(29)
with τ D,i = γ K h i /v sig,i , α D,min = 0 the minimal allowed value of the artificial conduction coefficient, and with the individual particle signal velocity,
v sig,i = max j (v sig,i j ),(30)
controlling the decay rate. ∇ 2 u is used as the indicator for a discontinuity, as opposed to ∇u, as it allows for (physical, well represented within SPH) linear gradients in internal energy to be maintained without activating artificial conduction. This is then integrated during 'kick' steps using The final stage of evolution for the individual conduction coefficients is to limit them based on the local viscosity of the fluid. This is necessary because thermal feedback events explicitly create extreme discontinuities within the internal energy field that lead to shocks (see §4 for the motivation leading to this choice). The limit is applied using the maximal value of viscous alpha among the neighbours of a given particle,
α D,i (t + ∆t) = α D,i (t) + dα D,i dt ∆t.(31α V,max,i = max j (α V, j ),(32)
with the limiter being applied using the maximally allowed value of the conduction coefficient,
α D,max,i = α D,max 1 − α V,max,i α V,max ,(33)
with α D,max = 1 a constant, and
α D,i = α D,i α D,i < α D,max α D,max α D,i > α D,max .(34)
This limiter allows for a more rapid increase in conduction coefficient, and a higher maximum, than would usually be viable in simulations with strong thermal feedback implementations. In Anarchy, another scheme employing artificial conduction, the rate at which the artificial conduction could grow was chosen to be significantly smaller. In Anarchy, β D = 0.01, which is 100 times smaller than the value chosen here (Schaye et al. 2015, Appendix A3). This additional conduction is required to accurately capture contact discontinuities with a Density-Energy SPH equation of motion.
MOTIVATION FOR THE CONDUCTION LIMITER
The conduction limiter first described in §3 is formed of two components; a maximal value for the conduction coefficient in viscous flows (Eqn. 34), and one that ensures that a particle with a higher pressure takes preference within the conduction interaction (Eqn. 27). This limiter is necessary due to interactions of the artificial conduction scheme with the sub-grid physics model. Here the EAGLE sub-grid model is shown as this is what Sphenix was designed for use with, however all schemes employing energetic feedback and unresolved cooling times will suffer from the same problems when using un-limited artificial conduction. In short, when an energetic feedback event takes place, the artificial conduction switch is activated (as this is performed by injecting lots of energy into one particle, leading to an extreme value of ∇ 2 u). This then leads to energy leaking out of the particle ahead of the shock front, which is then radiated away as the neighbouring particles can rapidly cool due to their temperature being lower leading to smaller cooling times.
To show the effect of this problem on a real system, we set up a uniform volume containing 32 3 gas particles at approximately solar metallicity (Z = 0.014) and equilibrium temperature (around 10 4 K), at a density of n H = 0.1 cm −3 . The central particle in the volume has approximately the same amount of energy injected into it as in a single EAGLE-like stellar feedback event (heating it to ∼ 10 7.5 K) at the start of the simulation and the code is ran with full sub-grid cooling (using the tables from Wiersma et al. 2009) enabled. The initial values for the artificial viscosity and conduction coefficients are set to be zero (whereas in practice they are set to be their maximum and minimum in 'real' feedback events; this has little effect on the results as the coefficients rapidly stabilise). Fig. 1 shows the energy in the system (with the thermal energy of the 'background' particles removed to ensure a large dynamic range in thermal energy is visible on this plot) in various components. We see that, at least for the case with the limiter applied, at t = 0 there is the expected large injection of thermal energy that is rapidly partially transformed into kinetic energy as in a classic blastwave problem (like the one shown in Fig. 5; in our idealised, non-radiative, Sedov blasts only 28% of the injected thermal energy is converted to kinetic energy). A significant fraction, around two thirds, of the energy is lost to radiation, but the key here is that there is a transformation of the initial thermal injection to a kinetic wave.
In the same simulation, now with the conduction limiter removed (dashed lines), almost all of the injected energy is immediately lost to radiation (i.e. the feedback is unexpectedly inefficient). The internal energy in the affected particle is rapidly conducted to its neighbours (that are then above, but closer to, the equilibrium temperature) which have a short cooling time and hence the energy is quickly lost.
The direct effect of the conduction limiter is shown in Fig. 2, where the same problem as above is repeated ten times with maximal artificial conduction coefficients α D,max of 0 to 2.5 in steps of 0.1 (note that the value of α D,max used in production simulations is 1). We choose to show these extreme values to demonstrate the efficacy of the limiter even in extreme scenarios. The simulations with and without the limiter show the same result at α D,max = 0 (i.e. with conduction disabled) but those without the limiter show a rapidly increasing fraction of the energy lost to cooling as the maximal conduction coefficient increases. The simulations with the limiter show a stable fraction of energy (at this fixed time of t = 25 Myr) in each component, showing that the limiter is working as expected and is curtailing these numerical radiative losses. This result is qualitatively unchanged for a factor of 100 higher, or lower, density background gas (i.e. gas between n H = 0.001 cm −3 and n H = 10.0 cm −3 ). In both of these cases, the conduction can rapidly cause numerical radiative losses, but with the limiter enabled this is remedied entirely. We also note that the limiter remains effective even for extreme values of the conduction parameter (e.g. with α D,max = 100), returning the same result as the case without artificial conduction for this test.
HYDRODYNAMICS TESTS
In this section the performance of Sphenix is shown on hydrodynamics tests, including the Sod (1978) shock tube, Sedov (1959) blastwave, and the Gresho & Sani (1990) vortex, along with many other problems relevant to galaxy formation. All problems are performed in hydrodynamics-only mode, with no radiative cooling or any other additional physics, and all use a γ = 5/3 equation of state
(P = (2/3)u iρ ).
Crucially, all tests were performed with the same scheme parameters and settings, meaning that all of the switches are consistent (even between self-gravitating and pure hydrodynamical tests) unless otherwise stated. This departs from most studies where parameters are set for each problem independently, in an attempt to demonstrate the maximal performance of the scheme for a given test. The parameters used are as follows:
• The quartic spline kernel.
• CFL condition C CFL = 0.2, with multiple time-stepping enabled (see e.g. Lattanzio et al. 1986).
• Viscosity alpha 0.0 ≤ α V ≤ 2.0 with the initial value being α V = 0.1 (similar to Cullen & Dehnen 2010).
• Viscosity beta β V = 3.0 and length V = 0.05 (similarly to Cullen & Dehnen 2010).
• Conduction alpha 0.0 ≤ α D ≤ 1.0 (a choice consistent with Price 2008) with the viscosity-based conduction limiter enabled and the same functional form for the conduction speed (Eqn. 26) used in all simulations.
• Conduction beta β D = 1.0 with the initial value of α D = 0.0. . 4) with the rarefaction wave, contact discontinuity, and shock, shown from left to right. All panels are shown at the same time t = 0.2, and for the same resolution level, using the 64 3 and 128 3 initial conditions for x < 1 and x > 1 respectively.
These choices were all 'calibrated' to achieve an acceptable result on the Sod shock tube, and then left fixed with the results from the rest of the tests left unseen. We choose to present the tests in this manner in an effort to show a representative overview of the performance of Sphenix in real-world conditions as it is primarily designed for practical use within future galaxy formation simulations. The source code required to produce the initial conditions (or a link to download the initial conditions themselves if this is impractical) are available open source from the Swift repository.
Sod shock tube
The Sod (1978) shock tube is a classic Riemann problem often used to test hydrodynamics codes. The tube is made up of three main sections in the final solution : the rarefaction wave (between 0.7 < x < 1.0), contact discontinuity (at position x ≈ 1.2), and a weak shock (at position x ≈ 1.4) at the time that we show it in Figure 3.
Initial Conditions
The initial conditions for the Sod shock tube uses body centred cubic lattices to ensure maximally symmetric lateral forces in the initial state. Two lattices with equal particle masses, one at a higher density by a factor of 8 (e.g. one with 32 3 particles and one with 64 3 particles) are attached at x = 1.0 5 . This forms a discontinuity, with the higher density lattice being placed on the left with ρ L = 1 and the lower density lattice on the right with ρ R = 1/8. The velocities are initially set to zero for all particles and pressures set to be P L = 1 and P R = 0.1. The three purple bands correspond to three distinct regions within the shock tube. The furthest left is the rarefaction wave, which is an adiabatically expanding fluid. The band covers the turnover point of the wave, as this is where the largest deviation from the analytic solution is present. There is a slight overestimation of the density at this turnover point, primarily due to the symmetric nature of the SPH kernel.
Results
The next band shows the contact discontinuity. No effort is made to suppress this discontinuity in the initial conditions (i.e. they are not relaxed). There is a small pressure blip, of a similar size to that seen with schemes employing Riemann solvers such as GIZMO (Hopkins 2015). There is no large velocity discontinuity associated with this pressure blip as is seen with SPH schemes that do not explicitly treat the contact discontinuity (note that every particle present in the simulation is shown here) with some form of conduction, a smoothed pressure field, or other method. Due to the strong discontinuity in internal energy present in this region, the artificial conduction coefficient α D peaks, allowing for the pressure 'blip' to be reduced to one with a linear gradient. offset this, the lattices are placed so that the particles are aligned along the x-axis wherever possible over the interface, however some spurious forces still result. The final section of the system, the rightmost region, is the shock. This shock is well captured by the scheme. There is a small activation of the conduction coefficient in this region, which is beneficial as it aids in stabilising the shock front (Hu et al. 2014). This shows that the conduction limiter ( §4) does not eliminate this beneficial property of artificial conduction within these frequently present weak (leading to α V 1.0) shocks.
In an ideal case, the scheme would be able to converge at second order L 1 ∝ h 2 away from shocks, and at first order L 1 ∝ h within shocks (Price et al. 2018). Here the L 1 norm of a band is defined as
L 1 (K) = 1 n n |K sim ( x) − K ref ( x)|(35)
with K some property of the system such as pressure, the subscripts sim and ref referring to the simulation data and reference solution respectively, and n the number of particles in the system. Fig. 4 shows the convergence properties of the Sphenix scheme on this problem, using the pressure field in this case as the convergence variable. Compared to a scheme without artificial conduction (dotted lines), the Sphenix scheme shows significantly improved convergence and a lower norm in the contact discontinuity, without sacrificing accuracy in other regions.
Sedov-Taylor Blastwave
The Sedov-Taylor blastwave (Sedov blast ;Taylor 1950;Sedov 1959) follows the evolution of a strong shock front through an initially isotropic medium. This is a highly relevant test for cosmological simulations, as this is similar to the implementations used for subgrid (below the resolution scale) feedback from stars and black holes. In SPH schemes this effectively tests the artificial viscosity 10 2 2 × 10 2 3 × 10 2 4 × 10 2 Mean smoothing length h 10 1 10 0 2 × 10 1 3 × 10 1 4 × 10 1 6 × 10 1 L 1 Norm for given variable
L 1 h Pressure P (L 1 h 0.39 ) Radial Velocity v r (L 1 h 0.51 ) Density (L 1 h 0.29 ) Figure 6
. L 1 Convergence with mean smoothing length for various particle fields in the Sedov-Taylor blastwave test, measured at t = 0.1 against the analytic solution within the purple band of Fig. 5. Each set of points shows a measured value from an individual simulation, with the lines showing a linear fit to the data in logarithmic space. Dotted lines for the simulation without conduction are not shown as they lie exactly on top of the lines shown here. scheme for energy conservation; if the scheme does not conserve energy the shock front will be misplaced.
Initial Conditions
Here, we use a glass file generated by allowing a uniform grid of particles to settle to a state where the kinetic energy has stabilised. The particle properties are then initially set such that they represent a gas with adiabatic index γ = 5/3, a uniform pressure of P 0 = 10 −6 , density ρ 0 = 1, all in a 3D box of side-length 1. Then, the n = 15 particles closest to the centre of the box have energy E 0 = 1/n injected into them. This corresponds, roughly, to a temperature jump of a factor of ∼ 10 5 over the background medium.
Results
Fig. 5 shows the particle properties of the highest resolution initial condition (128 3 ) at t = 0.1 against the analytic solution. The Sphenix scheme closely matches the analytic solution in all particle fields, with the only deviation (aside from the smoothed shock front, an unavoidable consequence of using an SPH scheme) being a slight upturn in pressure in the central region (due to a small amount of conduction in this region). Of particular note is the position of the shock front matching exactly with the analytic solution, showing that the scheme conserves energy in this highly challenging situation thanks to the explicitly symmetric artificial viscosity equation of motion. The Sphenix scheme shows qualitatively similar results to the PHANTOM scheme on this problem (Price et al. 2018).
SPH schemes in general struggle to show good convergence on shock problems due to their inherent discontinuous nature. Ideal convergence for shocks with the artificial viscosity set-up used in Sphenix is only first order (i.e. L 1 ∝ h). Fig. 6 shows the L 1 convergence for various fields in the Sedov-Taylor blastwave as a function of mean smoothing length. Convergence here has a best-case of L 1 (v) ∝ h 1/2 in real terms, much slower than the expected L 1 ∝ h −1 . This is primarily due to the way that the convergence is measured; the shock front is not resolved instantaneously (i.e. there is a rise in density and velocity over some small distance, reaching the maximum value at the true position) at the same position as in the analytic solution. However, all resolution levels show an accurately placed shock front and a shock width that scales linearly with resolution (see Appendix D for more information).
Gresho-Chan Vortex
The Gresho-Chan vortex (Gresho & Chan 1990) is typically used to test for the conservation of vorticity and angular momentum, and is performed here in two dimensions. Generally, it is expected that the performance of SPH on this test is more dependent on the kernel employed (see Dehnen & Aly 2012), as long as a sufficient viscositysuppressing switch is used.
Initial Conditions
The initial conditions use a two dimensional glass file, and treat the gas with an adiabatic index γ = 5/3, constant density ρ 0 = 1, in a square of side-length 1. The particles are given azimuthal velocity with the pressure set so that the system is in equilibrium as
v φ = 5r r < 0.2 2 − 5r 0.2 ≤ r < 0.4 0 r ≥ 0.4(36)P 0 = 5 + 12.5r 2
r < 0.2 9 + 12.5r 2 − 20r + 4 log(5r) 0.2 ≤ r < 0.4 3 + 4 log(2) r ≥ 0.4
where r = x 2 + y 2 is the distance from the box centre. Fig. 7 shows the state of a high resolution (using a glass containing 512 2 particles) result after one full rotation at the peak of the vortex (r = 0.2, t = 1.3). The vortex is well supported, albeit with some scatter, and the peak of the vortex is preserved. There has been some transfer of energy to the centre with a higher density and internal energy than the analytic solution due to the viscosity switch (shown on the bottom right) having a very small, but nonzero, value. This then allows for some of the kinetic energy to be transformed to thermal, which is slowly transported towards the centre as this is the region with the lowest thermal pressure. Fig. 8 shows the convergence properties for the vortex, with the Sphenix scheme providing convergence as good as L 1 ∝ h 0.7 for the azimuthal velocity. As there are no non-linear gradients in internal energy present in the simulation there is very little difference between the simulations performed with and without conduction at each resolution level due to the non-activation of Eqn. 31. This level of convergence is similar to the the rate seen in Dehnen & Aly (2012) implying that the Sphenix scheme, even with its less complex viscosity limiter, manages to recover some of the benefits of the more complex Inviscid scheme thanks to the novel combination of switches employed. Fig. 11, with this figure showing the highest resolution simulation performed, using 512 3 particles. This simulation state is also visualised in Fig. 10.
Results
Noh Problem
The Noh (1987) problem is known to be extremely challenging, particularly for particle-based codes, and generally requires a high particle number to correctly capture due to an unresolved convergence point. It tests a converging flow that results in a strong radial shock. This is an extreme, idealised, version of an accretion shock commonly present within galaxy formation simulations.
Initial Conditions
There are many ways to generate initial conditions, from very simple schemes to schemes that attempt to highly optimise the particle distribution (see e.g. Rosswog 2020a). Here, we use a simple initial condition, employing a body-centred cubic lattice distribution of particles in a periodic box. The velocity of the particles is then set such that there is a convergent flow towards the centre of the box,
v = − C − x | C − x|(38)
with C = 0.5L(1, 1, 1), where L is the box side-length, the coordinate at the centre of the volume. This gives every particle a speed of unity, meaning those in the centre will have extremely high relative velocities. We cap the minimal value of | C − x| to be 10 −10 L to prevent a singularity at small radii. The simulation is performed in a dimensionless co-ordinate system, with a box-size of L = 1.
10
30 50 Density [m l 3 ] Figure 10. A density slice through the centre of the Noh probeem at t = 0.6 corresponding to the particle distribution shown in Fig. 9. The Sphenix scheme yields almost perfect spherical symmetry for the shock, but does not capture the expected high density in the central region, likely due to lower than required artificial conductivity (see Appendix E for more information). Figure 11. L 1 convergence test for various particle properties at t = 0.6 for the Noh problem, corresponding to the particle distribution shown in Fig. 9. The lines without conduction are not shown here as there is little difference between the with and without conduction case, due to the extremely strong shock present in this test (leading to low values of the viscosity alpha, Equation 34).
Results
The state of the simulation is shown at time t = 0.6 in Fig. 9 and visualised in Fig. 10, which shows the radial velocity, which should be zero inside of the shocked region (high density in Fig. 10), and the same as the initial conditions (i.e. -1 everywhere) elsewhere. This behaviour is captured well, with a small amount of scatter, corresponding to the small radial variations in density shown in the image.
The profile of density as a function of radius is however less well captured, with some small waves created by oscillations in the artificial viscosity parameter (see e.g. Rosswog 2020b, for a scheme that corrects for these errors). This can also be seen in the density slice, and is a small effect that also is possibly exacerbated by our non-perfect choice of initial conditions, but is also present in the implementation shown in Rosswog (2020a). The larger, more significant, density error is shown inside the central part of the shocked, high-density, region. This error is ever-present in SPH schemes, and is likely due to both a lack of artificial conduction in this central region (as indicated by Noh 1987, note the excess pressure in the centre caused by 'wall heating') and the unresolved point of flow convergence.
The Noh problem converges well using Sphenix, with better than linear convergence for the radial velocity ( Fig. 11; recall that for shocks SPH is expected to converge with L 1 ∝ h).
This problem does not activate the artificial conduction in the Sphenix implementation because of the presence of Equation 34 reducing conductivity in highly viscous flows, as well as our somewhat conservative choice for artificial conduction coefficients (see Appendix E for more details on this topic). However, as these are necessary for the practical functioning of the Sphenix scheme in galaxy formation simulations, and due to this test being highly artificial, this outcome presents little concern.
Square Test
The square test, first presented in Saitoh & Makino (2013), is a particularly challenging test for schemes like Sphenix that do not use a smoothed pressure in their equation of motion, as they typically lead to an artificial surface tension at contact discontinuities (the same ones that lead to the pressure blip in §5.1). This test is a more challenging variant of the ellipsoid test presented in Heß & Springel (2010), as the square includes sharp corners which are more challenging for conduction schemes to capture.
Initial conditions
The initial conditions are generated using equal mass particles. We set up a grid in 2D space with n × n particles, in a box of size L = 1. The central 0.5 × 0.5 square is set to have a density of ρ C = 4.0, and so is replaced with a grid with 2n×2n particles, with the outer region having ρ O = 1.0. The pressures are set to be equal with P C = P O = 1.0, with this enforced by setting the internal energies of the particles to their appropriate values. All particles are set to be completely stationary in the initial conditions with v = 0. The initial conditions are not allowed to relax in any way. Figure 12. The density field for the square test at t = 4, shown at various resolution levels (different columns, numbers at the top denote the number of particles in the system) and with various modifications to the underlying SPH scheme (different rows). The dashed line shows the initial boundary of the square that would be maintained with a perfect scheme due to the uniform pressure throughout. The white circle at the centre of the square shows a typical smoothing length for this resolution level. Vertically, the scheme with no conduction is shown at the top, with the Sphenix scheme in the middle and a scheme with the conduction coefficient set to the maximum level throughout at the bottom. The schemes with conduction maintain the square shape significantly better than the one without conduction, and the Sphenix limiters manage to provide the appropriate amount of conduction to return to the same result as the maximum conduction case.
Results
Sphenix scheme without any artificial conduction enabled (this is achieved by setting α D,max to zero) and highlights the typical end state for a Density-Energy SPH scheme on this problem. Artificial surface tension leads to the square deforming and rounding to become more circular.
The bottom row shows the Sphenix scheme with the artificial conduction switch removed; here α D,min is set to the same value as α D,max = 1. The artificial conduction scheme significantly reduces the rounding of the edges, with a rapid improvement as resolution increases. The rounding present here only occurs in the first few steps as the energy outside the square is transferred to the boundary region to produce a stable linear gradient in internal energy.
Finally, the central row shows the Sphenix scheme, which gives a result indistinguishable from the maximum conduction scenario. This is despite the initial value for the conduction coefficient α D = 0, meaning it must ramp up rapidly to achieve such a similar result. The Sphenix result here shows that the choices for the conduction coefficients determined from the Sod tube ( §5.1) are not only appropriate for that test, but apply more generally to problems that aim to capture contact discontinuities. Figure 13. Density map of the standard Kelvin-Helmholtz 2D test at various resolutions (different columns, with the number of particles in the volume at the top) and at various times (different rows showing times from t = τ KH to t = 10τ KH ). Despite this being a challenging test for SPH, the instability is captured well at all resolutions, with higher resolution levels capturing finer details.
2D Kelvin-Helmholtz Instability
The two dimensional Kelvin-Helmholtz instability is presented below. This test is a notable variant on the usual Kelvin-Helmholtz test as it includes a density jump at constant pressure (i.e. yet another contact discontinuity). This version of the Kelvin-Helmholtz instability is performed in two dimensions. A recent, significantly more detailed, study of Kelvin-Helmholtz instabilities within SPH is available in Tricco (2019). In this section we focus on qualitative comparisons and how the behaviour of the instability changes with resolution within Sphenix.
Initial conditions
The initial conditions presented here are similar to those in Price (2008), where they discuss the impacts more generally of the inclusion of artificial conduction on fluid mixing instabilities. This is set up in a periodic box of length L = 1, with the central band between 0.25 < y < 0.75 set to ρ C = 2 and v C,x = 0.5, with the outer region having ρ O = 1 and v O,x = −0.5 to set up a shear flow. The pressure P C = P O = 2.5 is enforced by setting the internal energies of the equal mass particles. Particles are initially placed on a grid with equal separations. This is the most challenging version of this test for SPH schemes to capture as it includes a perfectly sharp contact discontinuity; see Agertz et al. (2007) for more information.
We then excite a specific mode of the instability, as in typical SPH simulations un-seeded instabilities are dominated by noise and are both unpredictable and unphysical, preventing comparison between schemes. Fig. 13 shows the simulation after various multiples of the Kelvin-Helmholtz timescale for the excited instability, with τ KH given by
Results
τ KH = (1 + χ)λ v √ χ (39) where χ = ρ C /ρ O = 2 is the density contrast,v = v I,x − v O,x = 1 the
shear velocity, and λ = 0.5 the wavelength of the seed perturbation along the horizontal axis (e.g Hu et al. 2014). The figure shows three initial resolution levels, increasing from left to right. Despite this being the most challenging version of the Kelvin-Helmholtz test (at this density contrast) for a Density-Energy based SPH scheme, the instability is captured well at all resolutions, with higher resolutions allowing for more rolls of the 'swirl' to be captured. In particular, the late-time highly mixed state shows that with the conduction removed after a linear gradient in internal energy has been established, the Sphenix scheme manages to preserve the initial contact discontinuity well. Due to the presence of explicit artificial conduction, Sphenix seems to diffuse more than other schemes on this test (e.g. Hu et al. 2014;Wadsley et al. 2017), leading to the erausre of some smallscale secondary instabilities. The non-linear growth rate of the swirls is resolution dependent within this test, with higher-resolution simulations showing faster growth of the largest-scale modes. This is due to better capturing of the energy initially injected to perturb the volume to produce the main instability, with higher resolutions showing lower viscous losses. Fig. 14 shows a different initial condition where the density contrast χ = 8, four times higher than the one initially presented. Because SPH is fundamentally a finite mass method, and we use equalmass particles throughout, this is a particularly challenging test as the low-density region is resolved by so few particles. Here we also excite an instability with a wavelength λ = 0.125, four times smaller than the one used for the χ = 2 test. This value is chosen for two reasons; it is customary to lower the wavelength of the seeded instability as the density contrast is increased when grid codes perform this test as it allows each instability to be captured with the same number of cells at a given resolution level; and also to ensure that this test is as challenging as is practical for the scheme.
Sphenix struggles to capture the instability at very low resolutions primarily due to the lack of particles in the low-density flow (an issue also encountered by Price 2008). In the boundary region the artificial conduction erases the small-scale instabilities on a timescale shorter than their formation timescale (as the boundary region is so large) and as such they cannot grow efficiently. As the resolution increases, however, Sphenix is able to better capture the linear evolution of the instability, even capturing turn-overs and the beginning of nonlinear evolution for the highest resolution. Figure 14. The same as Fig. 13, but this time using initial conditions with a significantly higher (1:8 instead of 1:2) density contrast. The initial instabilities are captured well at all resolution levels, but at the lowest level they are rapidly mixed by the artificial conduction scheme due to the lack of resolution elements in the low-density region.
Blob Test
The Blob test is a challenging test for SPH schemes (see Klein et al. 1994;Springel 2005) and aims to replicate a scenario where a cold blob of gas falls through the hot IGM/CGM surrounding a galaxy. In this test, a dense sphere of cold gas is placed in a hot, low density, and supersonic wind. Ideally, the blob should break up and dissolve into the wind, but Agertz et al. (2007) showed that the inability of traditional SPH schemes to exchange entropy between particles prevents this from occurring. The correct, specific, rate at which the blob should mix with its surroundings, as well as the structure of the blob whilst it is breaking up, are unknown.
Initial Conditions
There are many methods to set up the initial conditions for the Blob test, including some that excite an instability to ensure that the blob breaks up reliably (such as those used in Hu et al. 2014). Here we excite no such instabilities and simply allow the simulation to proceed from a very basic particle set-up with a perfectly sharp contact discontinuity. The initial conditions are dimensionless in nature, as . Time-evolution of the blob within the supersonic wind at various resolution levels (different columns; the number of particles in the whole volume is noted at the top) and at various times (expressed as a function of the Kelvin-Helmholtz time for the whole blob τ KH ; different rows). The projected density is shown here to enable all layers of the three dimensional structure to be seen. At all resolution levels the blob mixes with the surrounding medium (and importantly mixes phases with the surrounding medium), with higher resolution simulations displaying more thermal instabilities that promote the breaking up of the blob.
the problem is only specified in terms of the Mach number of the background medium and the blob density contrast.
To set up the initial particle distribution, we use two body centred cubic lattices, one packed at a high-density (for the blob, ρ blob = 10) and one at low density (for the background medium, ρ bg = 1). The low-density lattice is tiled four times in the x direction to make a box of size 4 × 1 × 1, and at [0.5, 0.5, 0.5] a sphere of radius 0.1 is removed and filled in with particles from the high-density lattice. The particles in the background region are given a velocity of v bg = 2.7 (with the blob being stationary), and the internal energy of the gas everywhere is scaled such that the background medium has a mach number of M = 2.7 and the system is in pressure equilibrium everywhere.
Results
The blob is shown at a number of resolution levels at various times in Fig. 15. At all resolution levels the blob mixes well with the back-ground medium after a few Kelvin-Helmholtz timescales (see Eqn. 39 for how this is calculated; here we assume that the wavelength of the perturbation is the radius of the blob) 6 . The rate of mixing is consistent amongst all resolution levels, implying that the artificial conduction scheme is accurately capturing unresolved mixing at lower resolutions.
The rate of mixing of the blob is broadly consistent with that of modern SPH schemes and grid codes, however our set of initial conditions appear to mix slightly slower (taking around ∼ 4 − 6τ KH to fully mix) than those used by other contemporary works (Agertz et al. 2007;Read & Hayfield 2012;Hu et al. 2014), possibly due to the lack of perturbation seeding (see Read et al. 2010, Appendix B for more details). When testing these initial conditions with a scheme that involves a Riemann solver or a Pressure-based scheme (see Appendix F) the rate of mixing is qualitatively similar to the one presented here. Sphenix is unable to fully capture the crushing of the blob from the centre outwards seen in grid codes and other SPH formulations using different force expressions (Wadsley et al. 2017), rather preferring to retain a 'plate' of dense gas at the initial point of the blob that takes longer to break up. A potential explanation for this difference is some residual surface tension in Sphenix. In these highly dynamic situations, it may not be possible for the artificial conduction to establish a smooth internal energy profile rapidly enough for small-scale instabilities on the surface to assist in the breakup of the blob.
At low resolutions it is extremely challenging for the method to capture the break-up of the blob as there are very few particles in the background medium to interact with the blob due to the factor of 10 density contrast.
Evrard Collapse
The Evrard collapse (Evrard 1988) test takes a large sphere of selfgravitating gas, at low energy and density, that collapses in on itself, causing an outward moving accretion shock. This test is of particular interest for cosmological and astrophysical applications as it allows for the inspection of the coupling between the gravity and hydrodynamics solver.
Initial Conditions
Gas particles are set up in a sphere with an adiabatic index of γ = 5/3, sphere mass M = 1, sphere radius R = 1, initial density profile ρ(r) = 1/2πr, and in a very cold state with u = 0.05, with the gravitational constant G = 1. These initial conditions are created in a box of size 100, ensuring that effects from the periodic boundary are negligible. Unfortunately, due to the non-uniform density profile, it is considerably more challenging to provide relaxed initial conditions (or use a glass file). Here, positions are simply drawn randomly to produce the required density profile.
The Evrard collapse was performed at four resolution levels, with total particle numbers in the sphere being 10 4 , 10 5 , 10 6 , and 10 7 . The gravitational softening was fixed at 0.001 for the 10 6 resolution level, and this was scaled with m −1/3 with m the particle mass for the other resolution levels. The simulations were performed once with artificial conduction enabled (the full Sphenix scheme), and once with it disabled.
Results
The highest resolution result (10 7 particles) with the full Sphenix scheme is shown in Fig. 16. This is compared against a high resolution grid code 7 simulation performed in 1D, and here Sphenix shows an excellent match to the reference solution. The shock at around r = 10 −1 is sharply resolved in all variables, and the density and velocity profiles show excellent agreement. In the centre of the sphere, there is a slight deviation from the reference solution for the internal energy and density (balanced to accurately capture the pressure in this region) that remains even in the simulation performed without artificial conduction (omitted for brevity, as the simulation without conduction shows similar results to the simulation with conduction, with the exception of the conduction reducing scatter in the internal energy profile). This is believed to be an artefact of the initial conditions, however it was not remedied by performing simulations at higher resolutions.
The convergence properties of the Evrard sphere are shown in Fig. 17. The velocity profile shows a particularly excellent result, with greater than linear convergence demonstrated. The thermodynamic properties show roughly linear convergence. Of particular note is the difference between the convergence properties of the simulations with and without artificial conduction; those with this feature of Sphenix enabled converge at a more rapid rate. This is primarily due to the stabilising effect of the conduction on the internal energy profile. As the particles are initially placed randomly, there is some scatter in the local density field at all radii. This is quickly removed by adiabatic expansion in favour of scatter in the internal energy profile, which can be stabilised by the artificial conduction.
nIFTy Cluster
The nIFTy cluster comparison project, Sembolini et al. (2016), uses a (non-radiative, cosmological) cluster-zoom simulation to evaluate the efficacy of various hydrodynamics and gravity solvers. The original paper compared various types of schemes, from traditional SPH (Gadget, Springel 2005) to a finite volume adaptive mesh refinement scheme (RAMSES, Teyssier 2002). The true answer for this simulation is unknown, but it is a useful case to study the different characteristics of various hydrodynamics solvers.
In Fig. 18 the Sphenix scheme is shown with and without artificial conduction against three reference schemes from Sembolini et al. (2016). Here, the centre the cluster was found using the VELOCIraptor (Elahi et al. 2019) friends-of-friends halo finder, and the particle with the minimum gravitational potential was used as the reference point.
The gas density profile was created using 25 equally log-spaced radial bins, with the density calculated as the sum of the mass within a shell divided by the shell volume. Sphenix scheme shows a similar low-density core as AREPO, with the no conduction scheme resulting in a cored density profile similar to the traditional SPH scheme from Sembolini et al. (2016).
The central panel of Fig. 18 shows the 'entropy' profile of the cluster; this is calculated as T n −2/3 e with n e the electron density (assuming primordial gas, this is n e = 0.875ρ/m H with m H the mass of a hydrogen atom) and T the gas temperature. Each was calculated individually in the same equally log-spaced bins as the density profile, with the temperature calculated as the mass-weighted temperature within that shell. The rightmost panel shows this mass-weighted Figure 17. L 1 convergence for various gas properties for the Evrard collapse sphere at t = 0.8. The region considered for convergence here is the purple band shown in Fig. 16. The Sphenix scheme is shown with the points and linear fits in solid, and the same scheme is shown with artificial conduction turned off as dotted lines. Artificial conduction significantly improves convergence here as it helps stabilise the thermal properties of the initially randomly placed particles. temperature profile, with Sphenix showing slightly higher temperatures in the central region than AREPO, matching G2-anarchy instead. This high-temperature central region, along with a lowdensity centre, lead to the 'cored' (i.e. flat, with high values of entropy, at small radii) entropy profile for Sphenix.
The cored central entropy profile with Sphenix is attained primarily due to the artificial conduction scheme and is not due to the other improvements over the traditional SPH base scheme (including for example the artificial viscosity implementation). We note again that there was no attempt to calibrate the artificial conduction scheme to attain this result on the nIFTy cluster, and any and all parameter choices were made solely based on the Sod shock tube in §5.1.
In Fig. 19, a projected mass-weighted temperature image of the cluster is shown. The image demonstrates how the artificial conduction present in the Sphenix scheme promotes phase mixing, resulting in the cored entropy profile demonstrated in Fig. 18.
The temperature distribution in the SPH simulation without conduction appears noisier, due to particles with drastically different phases being present within the same kernel. This shows how artificial conduction can lead to sharper shock capture as the particle distribution is less susceptible to this noise, enabling a cleaner energy transition between the pre-and post-shock region.
CONCLUSIONS
We have presented the Sphenix SPH scheme and its performance on seven hydrodynamics tests. The scheme has been demonstrated to show convergent (with resolution) behaviour on all these tests. In summary:
• Sphenix is an SPH scheme that uses Density-Energy SPH as a base, with added artificial viscosity for shock capturing and artificial conduction to reduce errors at contact discontinuities and to promote phase mixing.
• A novel artificial conduction limiter allows Sphenix to be used with energy injection feedback schemes (such as those used in EA-GLE) by reducing conduction across shocks and other regions where the artificial viscosity is activated.
• The artificial viscosity and conduction scheme coefficients were determined by ensuring good performance on the Sod Shock tube test, and remain fixed for all other tests.
• The modified Inviscid SPH (Cullen & Dehnen 2010) scheme captures strong shocks well, ensuring energy conservation, as shown by the Sedov-Taylor blastwave test, but the smooth nature of SPH prevents rapid convergence with resolution.
• The use of the Balsara (1989) switch in Sphenix was shown to be adequate to ensure that the Gresho-Chan vortex is stable. Convergence on this test was shown to be faster than in Cullen & Dehnen (2010).
• The artificial conduction scheme was shown to work adequately to capture thermal instabilities in both the Kelvin-Helmholtz and Blob tests, with contact discontinuities well preserved when required.
• Sphenix performed well on both the Evrard collapse and nIFTY cluster problems, showing that it can couple to the FMM gravity solver in Swift and that the artificial conduction scheme can allow for entropy cores in clusters.
• Sphenix is implemented in the Swift code and is available fully open source to the community.
Sphenix hence achieves its design goals; the Lagrangian nature of the scheme allows for excellent coupling with gravity; the artificial conduction limiter allows the injection of energy as in the EAGLE sub-grid physics model; and the low cost-per-particle and lack of matrices carried on a particle-by-particle basis provide for a very limited computational cost (see Borrow et al. 2019, for a comparison of computational costs between a scheme like Sphenix and the GIZMO-like schemes also present in Swift).
ACKNOWLEDGEMENTS
The authors thank Folkert Nobels for providing initial conditions for Appendix C, and the anonymous referee for their comments that improved the paper. JB is supported by STFC studentship ST/R504725/
Software Citations
This paper made use of the following software packages: As the simulations presented in this paper are small, test, simulations that can easily be repeated, the data is not made immediately available.
SPHENIX
No Conduction 10 7 10 8 Temperature [K] Figure 19. Image of the nIFTY cluster, as a projected mass-weighted temperature map, shown for the Sphenix scheme with (top) and without artificial conduction enabled (bottom). The image shows a 5 Mpc wide view, centred on the most bound particle in the halo. a small amount of extra data to store things like the particle-carried artificial conduction and viscosity coefficients. The amount of data required increases for more complex models, such as those making use of the full shear tensor, like Wadsley et al. (2017), or additional corrections, like Rosswog (2020b). SPH models using an ALE (Arbitrary Lagrangian-Eulerian) framework (see Vila 1999) require even more information as the particles carry flux information for use in the Riemann solver.
The amount of data required to store a single element in memory is of upmost importance when considering the speed at which a simulation will run. SPH codes, and Swift in particular, are bound by the memory bandwidth available, rather than the costs associated with direct computation. This means any increase in particle cost corresponds to a linear increase in the required computing time for simulation; this is why keeping the particles lean is a key requirement of the Sphenix model. Additionally, in large simulations performed over many nodes, the bandwidth of the interconnect can further become a limitation and hence keeping the memory cost of particles low is again beneficial.
In Fig. A1 we show the memory cost of four models: Traditional SPH (similar to to the one implemented in Gadget-2;Springel 2005), Sphenix, a model with the full shear matrix, and a SPH-ALE model similar to GIZMO-MFM (Hopkins 2015), all implemented in the Swift framework. We see that Sphenix only represents a 25% increase in memory cost per particle for significant improvement over the traditional model.
APPENDIX B: CONDUCTION SPEED
The conduction speed (Eqn. 26) in Sphenix was primarily selected for numerical reasons. In a density based scheme, it is common to see significant errors around contact discontinuities where there are large changes in density and internal energy simultaneously to produce a uniform pressure field. In Fig. 3 we demonstrated the perfor- Figure B1. The pressure contact discontinuity in the Sod Shock (Fig. 3) at a resolution of 32 3 and 64 3 , but using glass files instead of the BCC lattices (this leads to significantly increased particle disorder, but more evenly distributes particles in the x direction enabling this figure to be clearer). Here we show a zoomed-in representation of all particles (blue points) against the analytical solution (purple dashed line). Each sub-figure shows the simulation at the same time t = 0.2, but with different forms for the conduction velocity (see text for details). mance of the Sphenix scheme on one of these discontinuities, present in the Sod Shock.
In Fig. B1 we zoom in on the contact discontinuity, this time using glass files for the base initial conditions (of resolution 32 3 and 64 3 ), allowing for a more even distribution of particles along the horizontal axis. We use five different models,
• Sphenix, the full Sphenix model using the conduction speed from Eqn. 26.
• Pressure Term, which uses only the pressure-based term from Eqn. 26.
• Velocity Term, which only uses the velocity-based term from Eqn. 26.
• No Conduction, which sets the conduction speed to zero.
• Max Conduction, which sets α D to unity everywhere, and uses the conduction speed from Eqn. 26.
The first thing to note here is that the pressure term provides the vast majority of the conduction speed, highlighting the importance of this form of bulk conduction in Sphenix and other models employing a density based equation of motion. Importantly, the conduction allows for the 'pressure blip' to be reduced to a level where there is no longer a discontinuity in pressure (i.e. there is a smooth gradient with x). Although the velocity term is able to marginally reduce the size of the blip relative to the case without conduction, it is unable to fully stabilise the solution alone. Pressure blips can lead to large pressure differences between individual particles, then leading to the generation of a divergent flow around the point where the contact discontinuity resides. This is the primary motivation for the inclusion of the velocity divergence-based term in the conduction speed. Along with the conduction limiter (see Eqn. 28 for the source term), if there is a large discontinuity in internal energy that is generating a divergent flow (and not one that is expected to do so, such as a shock), the velocity-dependent term can correct for this and smooth out the internal energy until the source of divergence disappears.
B1 Alternative Conduction Speeds
The Sphenix conduction speed (Eqn. 26) contains two components: one based on pressure differences and one based on a velocity component. In Sphenix, as in a number of other models, this velocity component really encodes compression or expansion along the axis between particles.
The motivation for some alternative schemes (e.g. those presented in Wadsley et al. 2008Wadsley et al. , 2017 is shear between particles. To test if we see significant differences in our tests, we formulate a new conduction speed,
v D,i j = α D,i j 2 | v i j × x i j | | x i j | + 2 |P i − P j | ρ j +ρ j .(B1)
that focuses on capturing the shear component of the velocity between two particles. We again test this new formulation on some of our example problems. First, the nIFTy cluster, presented in Fig. B2, shows little difference between the two formulations, with both providing a solution similar to other modern SPH schemes and grid codes.
The Kelvin-Helmholtz test again shows little difference (Fig. B3), although there is a slightly increased growth rate of the perturbation at late times for the shear formulation.
We find no discernible difference between the two formulations on the blob test, as this is mainly limited by the choice of Density-SPH as the base equation of motion to correctly capture the initial break up of the blob from the centre outwards.
APPENDIX C: MAINTENANCE OF HYDROSTATIC BALANCE
The form of the conduction speed used in Sphenix based on pressure differences (Eqn. 26) has been conjectured to not allow for the maintenance of a pressure gradient against some external body force (for example a halo in hydrostatic equilibrium; Sembolini et al. 2016). The main concern here is that the pressure difference form of the conduction speed may allow thermal energy to travel down into the gravitational potential, heating the central regions of the halo. As Sphenix uses an additional limiter (Eqn. 28 for the source term) that only activates conduction in regions where the internal energy gradient cannot be represented by SPH anyway, this may be less of a concern. Additionally, there will be no conduction across accretion shocks due to the limiter in Eqn. 33.
In Fig. C1 we show an idealised simulation of an adiabatic halo with an NFW (Navarro et al. 1996) dark matter density profile, and gas in hydrostatic equilibrium. The halo uses a fixed NFW potential in the background, with a mass of 10 13 M , concentration 7.2, and a stellar bulge fraction of 1%. The halo has a gas mass of 1.7 × 10 12 M , resolved by 1067689 particles with varying mass from 10 5 to 1.7 × 10 12 M with the highest resolution in the centre.
The gas in the halo is set up to be isothermal, following (Stern We see no qualitative differences between the two models, with them both providing an entropy core at a similar level.
Shear Compression
1.0 1.5 2.0 Density [m l 2 ] Figure B3. Kelvin-Helmholtz test with a density contrast of ρ C = 2 as in Fig. 13, shown at t = 2τ KH (top) and t = 4τ KH (bottom). We show on the left the simulation with the shear-based conduction speed, and again the compression-based speed on the right. No significant qualitative differences are seen between the two models.
where v c is the circular velocity of the halo. The condition used to set the initial temperatures is v c = c s , and to get the correct normal-isation for pressure and density the gas fraction at R 500,crit is used following Debackere et al. (2020). Fig. C1 shows that there is little difference between the result with conduction, and without. There is a small offset in the centre where the simulation with conduction has a slightly higher energy and slightly lower density, giving a very small overall offset in pressure. This figure is shown at t = 5 Gyr, much longer than any realistic cluster of a similar mass would go without accretion or some other external force perturbing the pressure profile anyway. Finally, the conduction allows the noisy internal energy distribution (and additionally density distribution) to be normalised over time thanks to the inclusion of the pressure differencing term.
APPENDIX D: SEDOV BLAST
In Fig. 6 we presented the convergence properties of the Sedov blast with the Sphenix scheme. The scheme only demonstrated convergence as L 1 (v) ∝ h ∼0.5 , which is much slower than the expected convergence rate of L 1 ∝ h 1 for shock fronts in SPH (that is demonstrated and exceeded in the Noh problem in Fig. 11). This is, however, simply an artefact of the way that the convergence is measured.
In Fig. D1 we show the actual density profiles of the shock front, by resolution (increasing as the subfigures go to the right). Note here that the width of the shock front (from the particle distribution to the right of the vertical line to the vertical line in the analytical solution) does converge at the expected rate of L 1 ∝ 1/n 1/3 ∝ h with n the number of particles in the volume (in 3D).
The Sedov blast, unlike the Noh problem and Sod tubes, does not aim to reproduce a simple step function in density and velocity, but also a complex, expanding, post-shock region. The L 1 convergence is measured 'vertically' in this figure, but it is clear here that the vertical deviation from the analytical solution is not representative of the 'error' in the properties of a given particle, or in the width of the shock front. Small deviations in the position of the given particle could result in changes of orders of magnitude in the value of the L 1 norm measured for it.
Because of this, and because we have demonstrated in other sections that Sphenix is able to converge on shock problems at faster Figure C1. Profiles of the idealised NFW halo (of mass ≈ 10 13 M , at a gas particle mass resolution of 10 5 M ) at t = 5 Gyr after the initial state. Blue points show every particle in the simulation without artificial conduction enabled, with orange showing the simulation with conduction enabled. Here the conduction can allow for a reduction in the scatter in internal energy without leading to significant conduction into the centre. The offset seen in the centre of about a factor of 1.5x originates from the smoothing of the kink around ≈ 0.7 kpc during the initial settling of the halo, and remained stable from that point at around ≈ t = 0.5 Gyr until the end of the simulation. than first order, we believe the slow convergence on the Sedov problem to be of little importance in practical applications of the scheme.
APPENDIX E: CONDUCTION IN THE NOH PROBLEM
In §5.4 we presented the Noh problem, and showed that the Sphenix scheme (like other SPH schemes in general) struggles to capture the high density in the central region due to so-called 'wall heating'. The Sphenix scheme includes a switch to reduce artificial conduction in viscous flows. This is, as presented in §4, to allow for the capturing of energetic feedback events. It does, however, lead to a minor downside; the stabilising effect of the conduction in these shocks is almost completely removed. Usually, the artificial conduc-tion lowers the dispersion in local internal energy values, and hence pressures, allowing for a more regular particle distribution.
In Fig. E1 we show three re-simulations of the Noh problem (at 256 3 resolution) with three separate schemes. The first, the full Sphenix scheme, is simply a lower resolution version of Fig. 10. The second, 'No Conduction Limiter', is the Sphenix scheme, but with Equation 34 removed; i.e. the particle-carried artificial conduction coefficient depends solely on the local internal energy field (through ∇ 2 u and Eqn. 28), instead of also being mediated by the velocity divergence field. The final case, 'Fixed α D = 1.0', shows the case where we remove all conduction switches and use a fixed value for the conduction α D of 1.0. The former two look broadly similar, suggesting that the post-shock region is not significantly affected by the additional Sphenix conduction limiter. Fig. 10) shown for three different artificial conduction schemes (see text). The colour bar is shared between all, and they all use the same, 256 3 , initial condition, and are also all shown at t = 0.6. The case with the fixed, high, conduction coefficient (right) shows the smallest deviation in density in the centre, as the conduction can treat the wall heating present in this test.
The final panel, however, shows the benefits available to a hypothetical scheme that can remove the artificial conduction switch; the central region is able to hold a significantly higher density thanks to energy being conducted out of this region, allowing the pressure to regularise. In addition to the above, this case shows significantly weaker spurious density features (recall that the post-shock, highdensity, region should have a uniform density) because these have been regularised by the conduction scheme.
We present this both to show the drawbacks of the Sphenix artificial conduction scheme, and to show the importance of demonstrating test problems with the same switches that would be used in a production simulation.
APPENDIX F: BLOB TEST
In Fig. 15 we demonstrated the performance of the Sphenix scheme on an example 'blob' test. Here, we show how the same initial conditions are evolved using two schemes: a 'traditional SPH' scheme with fixed artificial viscosity (α V = 0.8) and no artificial conduction (e.g. Monaghan 1992) 9 , and a SPH-ALE (Vila 1999) scheme similar to GIZMO-MFM 10 (Hopkins 2015) with a diffusive slope limiter. This is in an effort to demonstrate how the initial conditions are evolved with a minimally viable non-diffusive scheme, through to what could be considered the most diffusive viable scheme. Fig. F1 shows the result of the blob test with the traditional SPH scheme. Here, as expected, there is a severe lack of mixing, with the artificial surface tension holding the blob together even at the highest resolutions. The lack of phase mixing also contributes to a lack of overall mixing, with the stripped trails (shown most clearly at t = 3τ KH ) adiabatically expanding but crucially remaining distinct from the hot background medium. Fig. F2 shows the result of the blob test with the SPH-ALE (GIZMO-MFM) scheme. This scheme is known to be highly diffusive (due to the less conservative slope limiter employed in the 9 The minimal scheme in Swift. 10 The gizmo-mfm scheme in Swift with a HLLC Riemann solver. Figure F2. A repeat of Fig. 15 but using an SPH-ALE scheme with a diffusive slope limiter. Note however that this is one step lower in resolution, due to the additional computational cost required to perform a simulation including a Riemann solver.
Swift implementation). This follows closely the results seen in e.g. Agertz et al. (2007) for diffusive grid-based codes. Here, the blob is rapidly shattered, and then dissolves quickly into the surrounding media, especially at the lowest resolutions. The Sphenix results in Fig. 15 showed that the blob mixed with the surrounding media, but at a less rapid rate than in the SPH-ALE case. This is somewhat expected, given the trade-off required in the artificial conduction switches (Eqn. 34). We do note, however, that no analytical solution exists for the blob test, and as such all of these comparisons may only be made qualitatively.
In Fig. F3 we examine the effect of removing the conduction limiter from the Sphenix implementation (i.e. Eqn. 34 is removed, allowing α D to vary irrespective of the values of α V ). We see that the inclusion of the limiter does slightly reduce the rate of initial mixing within the blob, but that the effect of the limiter is not particularly strong within this case. Figure F3. The evolution of a single blob (using the medium resolution, 2116547 particle, initial conditions from Fig. 15), to illustrate the effect of turning off the conduction limiter (Eqn. 34; bottom row) in comparison to the full Sphenix scheme (top row). The limiter suppresses some of the initial mixing during the cloud crushing, but does not cause significant qualitative changes in the mixing of the cloud.
Figure 1 .
1Energy in various components as a function of time for a simulated supernova blast (see text for details of the set-up). Blue shows energy in the kinetic phase, orange shows energy in the thermal phase (neglecting the thermal energy of the background) and green shows energy lost to radiation. The solid lines show the simulation performed with the artificial conduction limiter applied, and the dashed lines show the simulation without any such limiter. Simulations performed without the limiter show huge, rapid, cooling losses.
Figure 2 .
2The set-up from Fig. 1 performed for different values for the maximum artificial conduction coefficient α D,max (i.e. a different horizontal axis as Fig. 1, with the same vertical axis), now showing the components of energy in each phase at a fixed time of t = 25 Myr. Colours and line styles are the same as inFig. 1. As well as demonstrating the issue with un-limited conduction, this figure shows that the conduction limiter prevents the loss of additional energy energy relative to a simulation performed without any artificial conduction.
Figure 3 .
3Individual quantities plotted against the analytic solution (purple dashed line) for the Sod shock tube in 3D. The horizontal axis shows the x position of the particles. All particles are shown in blue, with the purple shading in the background showing the regions considered for the convergence (Fig
Figure 4 .
4Pressure convergence for the three regions inFig 3.The solid lines show fits to the data at various resolution levels (points) for each region, with the dotted lines showing convergence speed when the artificial conduction term is removed. The dashed grey line shows the expected speed of convergence for shocks in SPH simulations, to guide the eye, with a dependence of L 1 ∝ h.
Fig. 3
3shows the shock tube at t = 1, plotted against the analytic solution. This figure shows the result from the 64 3 and 128 3 initial condition. In general the simulation data (blue points) shows very close agreement with the analytic solution (purple dashed line).
Figure 5 .
5Particle properties at t = 0.1 shown against the analytic solution (purple dashed line) for the Sedov-Taylor blastwave. A random sub-set of 1/5th of the particles are shown in blue, with the orange points showing the mean value within equally spaced horizontal bins with one standard deviation of scatter. The background purple band shows the region considered for measuring convergence inFig. 6. This figure shows the results for a 128 3 particle glass file.
Figure 7 .
7Gresho vortex at t = 1.3 after one rotation of the vortex peak with the Sphenix scheme using a background resolution of 512 2 and with a mach number of M = 0.33. Here the blue points show all particles in the volume, the purple band the region used for convergence testing in Fig. 8, and the purple dashed line shows the analytic solution. The viscosity switch panel shows a very low maximal value (0.15) relative to the true maximum allowed by the code (α V B = 2.0), with the mean value (orange points with error bars indicating one standard deviation of scatter) of around 0.02 showing an excellent activation of the viscosity reducing switches throughout the Sphenix scheme.
Figure 8 .
8L 1 Convergence with mean smoothing length for various particle fields in the Gresho vortex test, measured against the analytic solution within the shaded region ofFig. 7. Each set of points shows a measured value from an individual simulation, with the lines showing a linear fit to the data in logarithmic space. The solid lines show results obtained with the full Sphenix scheme, with dotted lines showing the results with the artificial conduction scheme disabled.
Figure 9 .
9Noh problem simulation state at t = 0.6, showing a random sub-set of 1/100th of all of the particles plotted as blue points, the analytical solution as a dashed purple line, and binned quantities as orange points with error bars showing one standard deviation of scatter in that bin. The background shaded band shows the region considered for convergence in
Radial
Velocity v r (L 1 h 1.16 ) Pressure P (L 1 h 0.83 ) Density (L 1 h 0.68 )
Fig
. 12 shows the square test at t = 4 for four different resolution levels and three different variations on the Sphenix scheme. By this time the solutions are generally very stable. The top row
Figure 15
15Figure 15. Time-evolution of the blob within the supersonic wind at various resolution levels (different columns; the number of particles in the whole volume is noted at the top) and at various times (expressed as a function of the Kelvin-Helmholtz time for the whole blob τ KH ; different rows). The projected density is shown here to enable all layers of the three dimensional structure to be seen. At all resolution levels the blob mixes with the surrounding medium (and importantly mixes phases with the surrounding medium), with higher resolution simulations displaying more thermal instabilities that promote the breaking up of the blob.
Figure 16 .
16State of the Evrard sphere at t = 0.8 for a resolution of 10 7 particles. A random sub-set of 1/10th of the particles is shown in blue, with the solution from a high resolution 1D grid code shown as a purple dashed line. The orange points with error bars show the median within a radial equally log-spaced bin with the bar showing one standard deviation of scatter. The shaded band in the background shows the region considered for the convergence test inFig. 17.
Figure 18 .
18Thermodynamics profiles for the nIFTy cluster at z = 0 with five codes and schemes. The solid lines show those simulated with Swift, with the blue line showing the full Sphenix scheme, and the orange line showing Sphenix without artificial conduction . The dashed lines were extracted directly from Sembolini et al. (2016) and show a modern Pressure-Entropy scheme (G2-anarchy; Schaye et al. 2015, appendix A), a moving mesh finite volume scheme (AREPO; Springel 2010), and a traditional SPH scheme (G3-music; Springel 2005).
•
Swift (Schaller et al. 2018) • python (van Rossum & Drake Jr 1995), with the following libraries numpy (Harris et al. 2020) scipy (SciPy 1.0 Contributors et al. 2020) numba (Lam et al. 2015) matplotlib (Hunter 2007) swiftsimio (Borrow & Borrisov 2020) 8 DATA AVAILABILITY All code and initial conditions used to generate the simulations is open source as part of Swift version 0.9.0 (Schaller et al. 2018) 8 .
Figure A1 .
A1APPENDIX A: PARTICLE COSTSDifferent SPH models require different information stored per particle. Compared to a basic, 'Traditional' SPH model, Sphenix requires Traditional SPHENIX Shear Matrix SPH-Cost per particle (in bytes) for four different hydrodynamics models (see text for details). Percentages are give relative to the Traditional (similar to Gadget-2, with no artificial conduction and fixed artificial viscosity coefficients) model.
Figure B2 .
B2Reproduction of Fig. 18 but including the line (red) for the version of Sphenix performed with an explicit shear component in the conduction speed.
Figure D1 .
D1The density profile of the Sedov blasts initially presented inFig. 6. The blue points show the positions of every particle in the volume, the purple dashed line the analytical prediction, and the orange points binned means with error bars showing one standard deviation. The shaded band is the region over which the convergence properties were measured. The text at the top notes the total number of particles in each volume.
Figure E1 .
E1Density slice through the centre of the Noh problem (analogue of
Figure F1 .
F1A repeat ofFig. 15but using a 'traditional' SPH scheme without diffusive switches.
1. MS is supported by the Netherlands Organisation for Scientific Research (NWO) through VENI grant 639.041.749. RGB is supported by the Science and Technology Facilities Council ST/P000541/1. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure.
Borrow et al.
For the interested reader, the implementation of the Sphenix scheme was developed fully in the open and is available in the Swift repository at http://swiftsim.com (Schaller et al. 2018), including all of the tests and examples shown below. We use version 0.9.0 of the Swift code for the tests in this work.MNRAS 000, 000-000(0000)
MNRAS 000, 000-000 (0000)
For these reasons all recent works choose symmetric forms for these equations.
This simplistic particle arrangement does cause a slight problem at the interface at higher (i.e. greater than one) dimensions. In 3D, some particles may have spurious velocities in the y and z directions at the interface, due to asymmetries in the neighbours found on the left and right side of the boundary. To
Note that here the Kelvin-Helmholtz timescale is 1.1 times the cloud crushing timescale(Agertz et al. 2007).
HydroCode1D, see https://github.com/bwvdnbro/HydroCode1D and the Swift repository for more details.MNRAS 000, 000-000(0000)
Available from http://swift.dur.ac.uk, https://gitlab.cosma. dur.ac.uk/swift/swiftsim, and https://github.com/swiftsim/ swiftsim.MNRAS 000, 000-000(0000)
. O Agertz, 10.1111/j.1365-2966.2007.12183.xMonthly Notices of the Royal Astronomical Society. 380963Agertz O., et al., 2007, Monthly Notices of the Royal Astronomical Society, 380, 963
. D S Balsara, 10.1111/j.1365-2966.2012.21058.xMonthly Notices of the Royal Astronomical Society. Bauer A., Springel V.4232558PhD thesisBalsara D. S., 1989, PhD thesis Bauer A., Springel V., 2012, Monthly Notices of the Royal Astronomical Society, 423, 2558
. W Benz, 10.1016/0010-4655(88)90027-6Computer Physics Communications. 4897Benz W., 1988, Computer Physics Communications, 48, 97
. J Borrow, A Borrisov, 10.21105/joss.02430Journal of Open Source Software. 52430Borrow J., Borrisov A., 2020, Journal of Open Source Software, 5, 2430
J Borrow, R G Bower, P W Draper, P Gonnet, M Schaller, Proceedings of the 13th SPHERIC International Workshop. the 13th SPHERIC International WorkshopGalway, IrelandBorrow J., Bower R. G., Draper P. W., Gonnet P., Schaller M., 2018, Pro- ceedings of the 13th SPHERIC International Workshop, Galway, Ireland, June 26-28 2018, pp 44-51
J Borrow, B Vandenbroucke, M Schaller, M Schaller, R G Bower, arXiv:2011.11641Proceedings of the 14th SPHERIC International Workshop. the 14th SPHERIC International WorkshopExeter, United Kingdom; ExeterBorrow J., Vandenbroucke B., Schaller M., 2019, in Proceedings of the 14th SPHERIC International Workshop, Exeter, United Kingdom, June 25-27 2019. Exeter, United Kingdom Borrow J., Schaller M., Bower R. G., 2020, arXiv e-prints, p. arXiv:2011.11641
. R Cen, 10.1086/191630The Astrophysical Journal Supplement Series. 78341Cen R., 1992, The Astrophysical Journal Supplement Series, 78, 341
. R Cen, J P Ostriker, 10.1086/186620\apjl. 399113Cen R., Ostriker J. P., 1992, \apjl, 399, L113
. R Courant, K Friedrichs, H Lewy, 10.1007/BF01448839Mathematische Annalen. 10032Courant R., Friedrichs K., Lewy H., 1928, Mathematische Annalen, 100, 32
. R A Crain, V R Eke, C S Frenk, A Jenkins, I G Mccarthy, J F Navarro, F R Pearce, 10.1111/j.1365-2966.2007.11598.xMonthly Notices of the Royal Astronomical Society. 37741Crain R. A., Eke V. R., Frenk C. S., Jenkins A., McCarthy I. G., Navarro J. F., Pearce F. R., 2007, Monthly Notices of the Royal Astronomical Society, 377, 41
. L Cullen, W Dehnen, 10.1111/j.1365-2966.2010.17158.xMonthly Notices of the Royal Astronomical Society. 408669Cullen L., Dehnen W., 2010, Monthly Notices of the Royal Astronomical Society, 408, 669
. Dalla Vecchia, C Schaye, J , 10.1111/j.1365-2966.2008.13322.xMonthly Notices of the Royal Astronomical Society. 3871431Dalla Vecchia C., Schaye J., 2008, Monthly Notices of the Royal Astronom- ical Society, 387, 1431
. Dalla Vecchia, C Schaye, J , 10.1111/j.1365-2966.2012.21704.xMonthly Notices of the Royal Astronomical Society. 426140Dalla Vecchia C., Schaye J., 2012, Monthly Notices of the Royal Astronom- ical Society, 426, 140
. S N B Debackere, J Schaye, H Hoekstra, 10.1093/mnras/stz3446Monthly Notices of the Royal Astronomical Society. 4922285Debackere S. N. B., Schaye J., Hoekstra H., 2020, Monthly Notices of the Royal Astronomical Society, 492, 2285
. W Dehnen, 10.1186/s40668-014-0001-7Computational Astrophysics and Cosmology. 11Dehnen W., 2014, Computational Astrophysics and Cosmology, 1, 1
. W Dehnen, H Aly, 10.1111/j.1365-2966.2012.21439.xMonthly Notices of the Royal Astronomical Society. 4251068Dehnen W., Aly H., 2012, Monthly Notices of the Royal Astronomical Soci- ety, 425, 1068
. K Dolag, M Jubelgas, V Springel, S Borgani, E Rasia, 10.1086/420966The Astrophysical Journal. 60697Dolag K., Jubelgas M., Springel V., Borgani S., Rasia E., 2004, The Astro- physical Journal, 606, L97
. F Durier, Dalla Vecchia, C , 10.1111/j.1365-2966.2011.19712.xMonthly Notices of the Royal Astronomical Society. 419465Durier F., Dalla Vecchia C., 2012, Monthly Notices of the Royal Astronom- ical Society, 419, 465
. P J Elahi, R Cañas, R J J Poulton, R J Tobar, J S Willis, C D P Lagos, C Power, A S G Robotham, 10.1017/pasa.2019.12Publications of the Astronomical Society of Australia3621Elahi P. J., Cañas R., Poulton R. J. J., Tobar R. J., Willis J. S., Lagos C. d. P., Power C., Robotham A. S. G., 2019, Publications of the Astronomical Society of Australia, 36, e021
. S Ettori, K Dolag, S Borgani, G Murante, 10.1111/j.1365-2966.2005.09788.xMonthly Notices of the Royal Astronomical Society. 3651021Ettori S., Dolag K., Borgani S., Murante G., 2006, Monthly Notices of the Royal Astronomical Society, 365, 1021
. A E Evrard, 10.1093/mnras/235.3.911Monthly Notices of the Royal Astronomical Society. 235911Evrard A. E., 1988, Monthly Notices of the Royal Astronomical Society, 235, 911
. R A Gingold, J J Monaghan, 10.1093/mnras/181.3.375Monthly Notices of the Royal Astronomical Society. 181375Gingold R. A., Monaghan J. J., 1977, Monthly Notices of the Royal Astro- nomical Society, 181, 375
. L Greengard, V Rokhlin, 10.1016/0021-9991(87)90140-9Journal of Computational Physics. 73325Greengard L., Rokhlin V., 1987, Journal of Computational Physics, 73, 325
. P M Gresho, S T Chan, 10.1002/fld.1650110510International Journal for Numerical Methods in Fluids. 11621Gresho P. M., Chan S. T., 1990, International Journal for Numerical Methods in Fluids, 11, 621
. P M Gresho, R L Sani, 10.1002/fld.1650110702International Journal for Numerical Methods in Fluids. 11951Gresho P. M., Sani R. L., 1990, International Journal for Numerical Methods in Fluids, 11, 951
. T Guillet, R Pakmor, V Springel, P Chandrashekar, C Klingenberg, 10.1093/mnras/stz314Monthly Notices of the Royal Astronomical Society. 4854209Guillet T., Pakmor R., Springel V., Chandrashekar P., Klingenberg C., 2019, Monthly Notices of the Royal Astronomical Society, 485, 4209
. C R Harris, arXiv:2006.10256arXiv e-printsHarris C. R., et al., 2020, arXiv e-prints, p. arXiv:2006.10256
. L Hernquist, N Katz, 10.1086/191344The Astrophysical Journal Supplement Series. 70419Hernquist L., Katz N., 1989, The Astrophysical Journal Supplement Series, 70, 419
. S Heß, V Springel, 10.1111/j.1365-2966.2010.16892.xMonthly Notices of the Royal Astronomical Society. 4062289Heß S., Springel V., 2010, Monthly Notices of the Royal Astronomical Soci- ety, 406, 2289
. P F Hopkins, 10.1093/mnras/sts210Monthly Notices of the Royal Astronomical Society. 4282840Hopkins P. F., 2013, Monthly Notices of the Royal Astronomical Society, 428, 2840
. P F Hopkins, 10.1093/mnras/stv195Monthly Notices of the Royal Astronomical Society. 45053Hopkins P. F., 2015, Monthly Notices of the Royal Astronomical Society, 450, 53
. C.-Y Hu, T Naab, S Walch, B P Moster, L Oser, 10.1093/mnras/stu1187Monthly Notices of the Royal Astronomical Society. 4431173Hu C.-Y., Naab T., Walch S., Moster B. P., Oser L., 2014, Monthly Notices of the Royal Astronomical Society, 443, 1173
. J D Hunter, 10.1109/MCSE.2007.55Computing in Science & Engineering. 990Hunter J. D., 2007, Computing in Science & Engineering, 9, 90
. R I Klein, C F Mckee, P Colella, 10.1086/173554The Astrophysical Journal. 420213Klein R. I., McKee C. F., Colella P., 1994, The Astrophysical Journal, 420, 213
S K Lam, A Pitrou, S Seibert, 10.1145/2833157.2833162Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC. LLVM '15. the Second Workshop on the LLVM Compiler Infrastructure in HPC. LLVM '15New York, NY, USAAssociation for Computing MachineryLam S. K., Pitrou A., Seibert S., 2015, in Proceedings of the Sec- ond Workshop on the LLVM Compiler Infrastructure in HPC. LLVM '15. Association for Computing Machinery, New York, NY, USA, doi:10.1145/2833157.2833162
. J C Lattanzio, J J Monaghan, H Pongracic, M P Schwarz, 10.1137/0907039SIAM Journal on Scientific and Statistical Computing. 7591Lattanzio J. C., Monaghan J. J., Pongracic H., Schwarz M. P., 1986, SIAM Journal on Scientific and Statistical Computing, 7, 591
. L B Lucy, 10.1086/112164The Astronomical Journal. 821013Lucy L. B., 1977, The Astronomical Journal, 82, 1013
. J J Monaghan, 10.1146/annurev.aa.30.090192.002551Annual Review of Astronomy and Astrophysics. 30543Monaghan J. J., 1992, Annual Review of Astronomy and Astrophysics, 30, 543
. J Monaghan, 10.1006/jcph.1997.5732Journal of Computational Physics. 136298Monaghan J., 1997, Journal of Computational Physics, 136, 298
. J J Monaghan, 10.1046/j.1365-8711.2002.05678.xMonthly Notices of the Royal Astronomical Society. 335843Monaghan J. J., 2002, Monthly Notices of the Royal Astronomical Society, 335, 843
. J Monaghan, R Gingold, 10.1016/0021-9991(83)90036-0Journal of Computational Physics. 52374Monaghan J., Gingold R., 1983, Journal of Computational Physics, 52, 374
. J Morris, J Monaghan, 10.1006/jcph.1997.5690Journal of Computational Physics. 13641Morris J., Monaghan J., 1997, Journal of Computational Physics, 136, 41
. J F Navarro, S D M White, 10.1093/mnras/265.2.271\mnras. 265271Navarro J. F., White S. D. M., 1993, \mnras, 265, 271
. J F Navarro, C S Frenk, S D M White, 10.1086/177173The Astrophysical Journal. 462563Navarro J. F., Frenk C. S., White S. D. M., 1996, The Astrophysical Journal, 462, 563
. W F Noh, 10.1016/0021-9991(87)90074-XJournal of Computational Physics. 7278Noh W. F., 1987, Journal of Computational Physics, 72, 78
. B D Oppenheimer, J Schaye, R A Crain, J K Werk, A J Richings, 10.1093/mnras/sty2281Monthly Notices of the Royal Astronomical Society. 481835Oppenheimer B. D., Schaye J., Crain R. A., Werk J. K., Richings A. J., 2018, Monthly Notices of the Royal Astronomical Society, 481, 835
. S Ploeckinger, J Schaye, 10.1093/mnras/staa2172Monthly Notices of the Royal Astronomical Society. 4974857Ploeckinger S., Schaye J., 2020, Monthly Notices of the Royal Astronomical Society, 497, 4857
. D J Price, 10.1071/AS07022Publications of the Astronomical Society of Australia. 24159Price D. J., 2007, Publications of the Astronomical Society of Australia, 24, 159
. D J Price, 10.1016/j.jcp.2008.08.011Journal of Computational Physics. 22710040Price D. J., 2008, Journal of Computational Physics, 227, 10040
. D J Price, 10.1016/j.jcp.2010.12.011Journal of Computational Physics. 231759Price D. J., 2012, Journal of Computational Physics, 231, 759
. D J Price, 10.1017/pasa.2018.25Publications of the Astronomical Society of Australia. 3531Price D. J., et al., 2018, Publications of the Astronomical Society of Aus- tralia, 35, e031
. J I Read, T Hayfield, 10.1111/j.1365-2966.2012.20819.xMonthly Notices of the Royal Astronomical Society. 4223037Read J. I., Hayfield T., 2012, Monthly Notices of the Royal Astronomical Society, 422, 3037
. J I Read, T Hayfield, O Agertz, 10.1111/j.1365-2966.2010.16577.xMonthly Notices of the Royal Astronomical Society. Read J. I., Hayfield T., Agertz O., 2010, Monthly Notices of the Royal As- tronomical Society, pp no-no
. Y Revaz, P Jablonka, 10.1051/0004-6361/201117402Astronomy & Astrophysics. 53882Revaz Y., Jablonka P., 2012, Astronomy & Astrophysics, 538, A82
. S Rosswog, 10.1093/mnras/staa2591Monthly Notices of the Royal Astronomical Society. 4984230Rosswog S., 2020a, Monthly Notices of the Royal Astronomical Society, 498, 4230
. S Rosswog, 10.3847/1538-4357/ab9a2eThe Astrophysical Journal. 89860Rosswog S., 2020b, The Astrophysical Journal, 898, 60
. T R Saitoh, J Makino, 10.1088/0004-637X/697/2/L99ApJ. 69799Saitoh T. R., Makino J., 2009, ApJ, 697, L99
. T R Saitoh, J Makino, 10.1088/0004-637X/768/1/44The Astrophysical Journal. 76844Saitoh T. R., Makino J., 2013, The Astrophysical Journal, 768, 44
. M Schaller, Dalla Vecchia, C Schaye, J Bower, R G Theuns, T Crain, R A Furlong, M Mccarthy, I G , 10.1093/mnras/stv2169Monthly Notices of the Royal Astronomical Society. 4542277Schaller M., Dalla Vecchia C., Schaye J., Bower R. G., Theuns T., Crain R. A., Furlong M., McCarthy I. G., 2015, Monthly Notices of the Royal Astronomical Society, 454, 2277
M Schaller, P Gonnet, A B G Chalk, P W Draper, 10.1145/2929908.2929916Proceedings of the Platform for Advanced Scientific Computing Conference on -PASC '16. the Platform for Advanced Scientific Computing Conference on -PASC '16Schaller M., Gonnet P., Chalk A. B. G., Draper P. W., 2016, Proceedings of the Platform for Advanced Scientific Computing Conference on -PASC '16, pp 1-10
. M Schaller, P Gonnet, P W Draper, A B G Chalk, R G Bower, J Willis, L Hausammann, 1805.020Astrophysics Source Code Library. Schaller M., Gonnet P., Draper P. W., Chalk A. B. G., Bower R. G., Willis J., Hausammann L., 2018, Astrophysics Source Code Library, p. ascl:1805.020
. J Schaye, Dalla Vecchia, C , 10.1111/j.1365-2966.2007.12639.x\mnras. 3831210Schaye J., Dalla Vecchia C., 2008, \mnras, 383, 1210
. J Schaye, 10.1093/mnras/stu2058Monthly Notices of the Royal Astronomical Society. 446521Schaye J., et al., 2015, Monthly Notices of the Royal Astronomical Society, 446, 521
. 10.1038/s41592-019-0686-2Nature Methods. 17261SciPy 1.0 ContributorsSciPy 1.0 Contributors et al., 2020, Nature Methods, 17, 261
Similarity and Dimensional Methods in Mechanics -NASA/ADS. Sedov. Sedov 1959, Similarity and Dimensional Methods in Mechanics - NASA/ADS
. F Sembolini, 10.1093/mnras/stw250Monthly Notices of the Royal Astronomical Society. 4574063Sembolini F., et al., 2016, Monthly Notices of the Royal Astronomical Soci- ety, 457, 4063
. G A Sod, 10.1016/0021-9991(78)90023-2Journal of Computational Physics. 271Sod G. A., 1978, Journal of Computational Physics, 27, 1
. V Springel, 10.1111/j.1365-2966.2005.09655.xMonthly Notices of the Royal Astronomical Society. 3641105Springel V., 2005, Monthly Notices of the Royal Astronomical Society, 364, 1105
. V Springel, 10.1111/j.1365-2966.2009.15715.xMonthly Notices of the Royal Astronomical Society. 401791Springel V., 2010, Monthly Notices of the Royal Astronomical Society, 401, 791
. V Springel, L Hernquist, 10.1046/j.1365-8711.2002.05445.xMonthly Notices of the Royal Astronomical Society. 333649Springel V., Hernquist L., 2002, Monthly Notices of the Royal Astronomical Society, 333, 649
. V Springel, L Hernquist, 10.1046/j.1365-8711.2003.06206.xMonthly Notices of the Royal Astronomical Society. 339289Springel V., Hernquist L., 2003, Monthly Notices of the Royal Astronomical Society, 339, 289
. J Stern, D Fielding, C.-A Faucher-Giguère, E Quataert, arXiv:1906.07737astro-phStern J., Fielding D., Faucher-Giguère C.-A., Quataert E., 2019, arXiv:1906.07737 [astro-ph]
. G Taylor, 10.1098/rspa.1950.0049Proceedings of the Royal Society of London Series A. 201159Taylor G., 1950, Proceedings of the Royal Society of London Series A, 201, 159
. R Teyssier, 10.1051/0004-6361:20011817Astronomy & Astrophysics. 385337Teyssier R., 2002, Astronomy & Astrophysics, 385, 337
. T S Tricco, 10.1093/mnras/stz2042\mnras. 4885210Tricco T. S., 2019, \mnras, 488, 5210
Mathematical models and methods in applied sciences. J P Vila, 9161Vila J. P., 1999, Mathematical models and methods in applied sciences, 9, 161
. J Wadsley, J Stadel, T Quinn, 10.1016/j.newast.2003.08.004New Astronomy. 9137Wadsley J., Stadel J., Quinn T., 2004, New Astronomy, 9, 137
. J W Wadsley, G Veeravalli, H M P Couchman, 10.1111/j.1365-2966.2008.13260.xMonthly Notices of the Royal Astronomical Society. 387427Wadsley J. W., Veeravalli G., Couchman H. M. P., 2008, Monthly Notices of the Royal Astronomical Society, 387, 427
. J W Wadsley, B W Keller, T R Quinn, 10.1093/mnras/stx1643Monthly Notices of the Royal Astronomical Society. 4712357Wadsley J. W., Keller B. W., Quinn T. R., 2017, Monthly Notices of the Royal Astronomical Society, 471, 2357
. R Weinberger, V Springel, R Pakmor, 10.3847/1538-4365/ab908cThe Astrophysical Journal Supplement Series. 24832Weinberger R., Springel V., Pakmor R., 2020, The Astrophysical Journal Supplement Series, 248, 32
. H Wendland, 10.1007/BF02123482Advances in Computational Mathematics. 4389Wendland H., 1995, Advances in Computational Mathematics, 4, 389
. R P C Wiersma, J Schaye, T Theuns, Dalla Vecchia, C Tornatore, L , 10.1111/j.1365-2966.2009.15331.xMonthly Notices of the Royal Astronomical Society. 399574Wiersma R. P. C., Schaye J., Theuns T., Dalla Vecchia C., Tornatore L., 2009, Monthly Notices of the Royal Astronomical Society, 399, 574
. D Wood, 10.1093/mnras/194.2.201Monthly Notices of the Royal Astronomical Society. 194201Wood D., 1981, Monthly Notices of the Royal Astronomical Society, 194, 201
. G Van Rossum, F L DrakeJr, Python Tutorial. Centrum voor Wiskunde en Informatica. 620van Rossum G., Drake Jr F. L., 1995, in , Vol. 620, Python Tutorial. Centrum voor Wiskunde en Informatica, Amsterdam
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"Astronomy ALMA study of the HD 100453 AB system and the tidal interaction of the companion with the disk",
"Astronomy ALMA study of the HD 100453 AB system and the tidal interaction of the companion with the disk"
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"G Van Der Plas \nUMR 5274\nCNRS\nUniversité Grenoble Alpes\nIPAG (\n38000GrenobleFrance\n",
"F Ménard \nUMR 5274\nCNRS\nUniversité Grenoble Alpes\nIPAG (\n38000GrenobleFrance\n",
"J.-F Gonzalez \nCentre de Recherche Astrophysique de Lyon UMR5574\nCNRS\nUniversité\nClaude Bernard Lyon 1, Ens de Lyon69230Saint-Genis-LavalFrance\n",
"S Perez \nMillenium Nucleus Protoplanetary Disks in ALMA Early Science\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n\nDepartamento de Astronomía\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n",
"L Rodet \nUMR 5274\nCNRS\nUniversité Grenoble Alpes\nIPAG (\n38000GrenobleFrance\n",
"C Pinte \nUMR 5274\nCNRS\nUniversité Grenoble Alpes\nIPAG (\n38000GrenobleFrance\n\nMonash Centre for Astrophysics (MoCA) and School of Physics and Astronomy\nMonash University\nClayton Vic 3800Australia\n",
"L Cieza \nNúcleo de Astronomía\nFacultad de Ingeniería\nUniversidad Diego Portales\nAv Ejército 441SantiagoChile\n",
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] | [] | Context. The complex system HD 100453 AB with a ring-like circumprimary disk and two spiral arms, one of which is pointing to the secondary, is a good laboratory in which to test spiral formation theories. Aims. We aim to resolve the dust and gas distribution in the disk around HD 100453 A and to quantify the interaction of HD 100453 B with the circumprimary disk. Methods. Using ALMA band 6 dust continuum and CO isotopologue observations we have studied the HD 100453 AB system with a spatial resolution of 0. 09 × 0. 17 at 234 GHz. We used smoothed particle hydrodynamics (SPH) simulations and orbital fitting to investigate the tidal influence of the companion on the disk. Results. We resolve the continuum emission around HD 100453 A into a disk between 0. 22 and 0. 40 with an inclination of 29.5 • and a position angle of 151.0 • , an unresolved inner disk, and excess mm emission cospatial with the northern spiral arm which was previously detected using scattered light observations. We also detect CO emission from 7 au (well within the disk cavity) out to 1. 10, overlapping with HD 100453 B at least in projection. The outer CO disk position angle (PA) and inclination differ by up to 10 • from the values found for the inner CO disk and the dust continuum emission, which we interpret as due to gravitational interaction with HD 100453 B. Both the spatial extent of the CO disk and the detection of mm emission at the same location as the northern spiral arm are in disagreement with the previously proposed near co-planar orbit of HD 100453 B. Conclusions. We conclude that HD 100453 B has an orbit that is significantly misaligned with the circumprimary disk. Because it is unclear whether such an orbit can explain the observed system geometry we highlight an alternative scenario that explains all detected disk features where another, (yet) undetected, low mass close companion within the disk cavity, shepherds a misaligned inner disk whose slowly precessing shadows excite the spiral arms. | 10.1051/0004-6361/201834134 | [
"https://www.aanda.org/articles/aa/pdf/2019/04/aa34134-18.pdf"
] | 102,486,890 | 1902.00720 | 81d280304b13681cc6234e6155a52d8e57b9cc68 |
Astronomy ALMA study of the HD 100453 AB system and the tidal interaction of the companion with the disk
G Van Der Plas
UMR 5274
CNRS
Université Grenoble Alpes
IPAG (
38000GrenobleFrance
F Ménard
UMR 5274
CNRS
Université Grenoble Alpes
IPAG (
38000GrenobleFrance
J.-F Gonzalez
Centre de Recherche Astrophysique de Lyon UMR5574
CNRS
Université
Claude Bernard Lyon 1, Ens de Lyon69230Saint-Genis-LavalFrance
S Perez
Millenium Nucleus Protoplanetary Disks in ALMA Early Science
Universidad de Chile
Casilla 36-DSantiagoChile
Departamento de Astronomía
Universidad de Chile
Casilla 36-DSantiagoChile
L Rodet
UMR 5274
CNRS
Université Grenoble Alpes
IPAG (
38000GrenobleFrance
C Pinte
UMR 5274
CNRS
Université Grenoble Alpes
IPAG (
38000GrenobleFrance
Monash Centre for Astrophysics (MoCA) and School of Physics and Astronomy
Monash University
Clayton Vic 3800Australia
L Cieza
Núcleo de Astronomía
Facultad de Ingeniería
Universidad Diego Portales
Av Ejército 441SantiagoChile
S Casassus
Millenium Nucleus Protoplanetary Disks in ALMA Early Science
Universidad de Chile
Casilla 36-DSantiagoChile
Departamento de Astronomía
Universidad de Chile
Casilla 36-DSantiagoChile
M Benisty
UMR 5274
CNRS
Université Grenoble Alpes
IPAG (
38000GrenobleFrance
Departamento de Astronomía
Universidad de Chile
Casilla 36-DSantiagoChile
Astronomy ALMA study of the HD 100453 AB system and the tidal interaction of the companion with the disk
10.1051/0004-6361/201834134Received 23 August 2018 / Accepted 31 January 2019& Astrophysics A&A 624, A33 (2019)protoplanetary disks -planet-disk interactions -stars: individual: HD 100453 -stars: variables: T Tauri, Herbig Ae/Be - binaries: general
Context. The complex system HD 100453 AB with a ring-like circumprimary disk and two spiral arms, one of which is pointing to the secondary, is a good laboratory in which to test spiral formation theories. Aims. We aim to resolve the dust and gas distribution in the disk around HD 100453 A and to quantify the interaction of HD 100453 B with the circumprimary disk. Methods. Using ALMA band 6 dust continuum and CO isotopologue observations we have studied the HD 100453 AB system with a spatial resolution of 0. 09 × 0. 17 at 234 GHz. We used smoothed particle hydrodynamics (SPH) simulations and orbital fitting to investigate the tidal influence of the companion on the disk. Results. We resolve the continuum emission around HD 100453 A into a disk between 0. 22 and 0. 40 with an inclination of 29.5 • and a position angle of 151.0 • , an unresolved inner disk, and excess mm emission cospatial with the northern spiral arm which was previously detected using scattered light observations. We also detect CO emission from 7 au (well within the disk cavity) out to 1. 10, overlapping with HD 100453 B at least in projection. The outer CO disk position angle (PA) and inclination differ by up to 10 • from the values found for the inner CO disk and the dust continuum emission, which we interpret as due to gravitational interaction with HD 100453 B. Both the spatial extent of the CO disk and the detection of mm emission at the same location as the northern spiral arm are in disagreement with the previously proposed near co-planar orbit of HD 100453 B. Conclusions. We conclude that HD 100453 B has an orbit that is significantly misaligned with the circumprimary disk. Because it is unclear whether such an orbit can explain the observed system geometry we highlight an alternative scenario that explains all detected disk features where another, (yet) undetected, low mass close companion within the disk cavity, shepherds a misaligned inner disk whose slowly precessing shadows excite the spiral arms.
Introduction
Protoplanetary (PP) disks are a natural byproduct of star formation. These disks dissipate with a typical timescale of two to three million years (see e.g., the review by Williams & Cieza 2011, and references therein) and planet formation during the evolution and dissipation of the disk appears to be the rule rather than the exception (e.g., Dressing & Charbonneau 2015). The mechanisms that allow the gas and small dust grains in the disk to coalesce into planetary systems are not clear yet and high angular resolution studies of PP disks are necessary to solve this part of the planet formation puzzle.
Our current best tools to study PP disks at high spatial resolution are (sub-)mm interferometers such as ALMA and
The reduced datacube and continuum images (FITS files) are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/ qcat?J/A+A/624/A33 extreme AO high-contrast imagers such as the Gemini Planet Imager (Gemini/GPI; Macintosh et al. 2014) and the Spectro-Polarimetric High-contrast Exoplanet REsearch (VLT/SPHERE; Beuzit et al. 2008). Each of them now routinely yields spatial resolutions below 0. 1 but each traces different regions of the disks. The scattered light traces the small ≈micron sized dust grains high up in the disk surface, while the longer wavelength observations can trace both the larger, typically mm sized, dust grains in the disk mid plane, as well as the intermediate disk layers through many different molecular gas lines.
As we observe PP disks at increasingly high spatial resolution it becomes clear that substructures in these disks are common, and that understanding these substructures is essential to understand disk evolution and planet formation. The most common structures found so far are (1) opacity cavities ranging between a few to over 100 au that sometimes contain a small misaligned inner disk (i.e., HD 142527, see Marino et al. 2015), where this disk also casts a shadow on the outer disk (Casassus et al. 2012), (2) (multiple) rings and / or cavities (e.g., Andrews et al. 2016;Avenhaus et al. 2018), (3) large spiral arms (such as for example, HD 142527, see Christiaens et al. 2014), or HD 100453, see Wagner et al. (2015), and (4) azimuthal dust concentrations with various contrast often interpreted as dust trapping in vortices (such as for example IRS 48 and HD 34282, see van der Marel et al. 2013;van der Plas et al. 2017a). All of these features can be induced by the gravitational interaction with a forming body (e.g., a planet) but also by other processes that do not require a massive body within the disk such as snow lines (Lecar et al. 2006;Stammler et al. 2017), a pressure gradient at the edge of a dead zone (Lovelace et al. 1999), self-induced dust traps (Lyra & Kuchner 2013;Gonzalez et al. 2017), stellar fly-by (Quillen et al. 2005), and others. Studying these features using different proxies narrows down their possible origins and thus helps building a list of processes that are dominant in disk dispersal and planet formation. The nearby HD 100453 AB system is an ideal candidate for such a study.
HD 100453 A at 103 +3 −4 pc (Gaia Collaboration 2016) is an A9Ve star with an age of 10±2 Myr and a mass of 1.7 M (Collins et al. 2009). It is orbited by a companion (hereafter called HD 100453 B) that was first noticed by Chen et al. (2006) and later confirmed to be comoving by Collins et al. (2009). The spectral type of HD 100453 B was estimated to be between M4V and M4.5V with a mass of 0.2 ± 0.04 M (Collins et al. 2009). Wagner et al. (2018) recently published new astrometric measurements further confirming the bound nature of the companion orbit, and placing it at a projected separation of ≈1. 05 from the primary at a position angle of 131.95 • at the time of the observations we present in this manuscript.
The disk surrounding HD 100453 A is highly structured and complex. There is little material accreting onto the central star with an upper limit to the accretion rate of 1.4 × 10 −9 M yr −1 (Collins et al. 2009). Wagner et al. (2015) resolved a disk cavity and an outer disk between 0. 18 and 0. 25 in radius, as well as two nearly symmetric spiral arms extending out to r = 38 au (distances scaled to a distance of 103 au), and Benisty et al. (2017) saw two symmetric shadows on the outer disk, all in scattered light. Similar features have been detected in other transition disks (objects whose inner disk regions have undergone substantial clearing, see e.g., Espaillat et al. 2014) such as the ones around HD 135344 B (Stolker et al. 2016) and HD 142527 . In these cases the shadow cast by a small, misaligned, inner disk was deemed responsible. NIR infrared interferometric observations have indeed detected such a misaligned inner disk around HD 100453 A with a half light radius of ≈1 au (Menu et al. 2015;Lazareff et al. 2017), and Min et al. (2017) calculate a position angle and inclination for the inner disk (i = 45 • , PA = 82 • ) and for the outer disk (i = −38 • , PA = 142 • ) using the assumption that the shadows are cast by the inner disk. Finally, Meeus et al. (2003) report an unresolved detection of the disk at 1.2 mm with 265 ± 21 mJy, and Wagner et al. (2018) use part of the data we present here to determine a counterclockwise rotation direction for the disk. The grand design spiral arm structure in this system has been connected to the companion by Dong et al. (2016), who used hydrodynamical and radiative transfer simulations to show that a close-to-coplanar orbit of the companion can explain the main disk features detected in scattered light assuming the disk is oriented close to face-on.
In this paper, we present high angular resolution ALMA band 6 observations of the HD 100453 system to measure the dust and gas distribution in the disk (Sects. 2 and 3). We use hydrodynamical SPH and radiative transfer models to investigate whether our observations are consistent with the previously suggested coplanar companion as origin for the spiral arms (Sect. 4), and we discuss our results in Sect. 5. We conclude in Sect. 6 that this is unlikely to be the case and offer an alternative scenario to explain the system geometry where an as of yet undetected companion inside the disk cavity drives a slowly precessing misaligned inner disk whose shadow cast on the outer disk triggers the spiral arms.
Observations and data reduction
ALMA early science cycle 3 observations were conducted in a compact configuration on April 23 2016 with 13.1 minutes of total time on-source and in an extended configuration on September 8 2016 with 26.2 min of total time on-source. The array configuration provided baselines ranging between respectively 15 and 463 m, and between 15 and 2483 m. During the observations the precipitable water vapor had a median value at zenith of respectively 1.64 and 0.56 mm.
Two of the four spectral windows of the ALMA correlator were configured in time division mode (TDM) to maximize the sensitivity for continuum observations (128 channels over 1.875 GHz usable bandwidth). These two TDM spectral windows were centered at 234.16 and 216.98 GHz. The other two spectral windows were configured in frequency division mode (FDM) to target the 12 CO J = 2-1, 13 CO J = 2-1 and C 18 O J = 2-1 lines with a spectral resolution of 61, 122, and 122 kHz respectively. The data were calibrated and combined using the Common Astronomy Software Applications pipeline (CASA, McMullin et al. 2007, version 4.7.2).
Inspection of the calibrated visibilities showed a 16% difference in amplitude between the two observations at short baselines. We assumed that the emission from the midplane was constant in the 4.5 month period spanning the observations and that the difference is due to calibration uncertainties. Inspection of the calibrator archives did not lead us to favor one calibration over the other, and we decided to scale the flux of the compact array configuration to match the extended array configuration data. We estimate the absolute flux calibration to be accurate within ∼20%, details of the observations and calibration are summarized in Table 1.
We imaged the continuum visibilities with the clean task in CASA (Högbom 1974) using Briggs and superuniform weightings, which resulted in a restored beam size of respectively 0. 23 × 0. 15 at PA = 25.1 • (Briggs) and 0. 17 × 0. 09 at PA = 14.1 • (superuniform). The dynamic range of these images was limited by the bright continuum source and we performed two rounds of phase only self-calibration, resulting in a final RMS of 0.05 mJy beam−1 (peak signal-to-noise ratio, S/N, of 159) for the images created using superuniform weighting, and 0.04 mJy beam −1 (peak S/N of 362) for the images created using Briggs weighting. We show the resulting continuum map in Fig. 1.
We applied the self-calibration solutions obtained for the continuum emission to the CO visibilities and subtracted the continuum emission using the CASA task uvcontsub. We imaged the line data with a velocity resolution of 0.2 km s −1 using natural weighting to maximize sensitivity which resulted in a restored beam of 0. 29 × 0. 23. The CO line emission detections were summarized using the integrated intensity (moment 0), intensity-weighted mean velocity (moment 1) and peak intensity (moment 8) maps as well as the integrated spectra. These are shown in Figs. 2-4 for the 12 CO, 13 CO and C 18 O J = 2-1 transitions, respectively.
Results
We detect and resolve the 1.4 mm dust continuum emission and the 12 CO, 13 CO and C 18 O J = 2-1 emission lines. We determine the continuum flux and geometry by fitting several disk components to the visibilities and report the measured fluxes and derived geometry for the both the dust and gas emission in this section. We also report upper limits for the emissions coming from the location of HD 100453 B.
1.4 mm continuum emission
The dust continuum emission of HD 100453 shown in Fig. 1 is concentrated into a ring that peaks at 0. 32 with an azimuthal variation along the ring of ≈30% between the maximum at PA = 331 • and the minimum at PA = 180 • . There is also excess emission present at the stellar position that, when convolved with the beam, connects with the outer disk along the beam major axis (see the inset in Fig. 1).
Disk geometry
We use the fitting library uvmultifit (Martí-Vidal et al. 2014) to quantify the inclination, position angle and spatial distribution of the disk emission. From our first look it is apparent that the emission can be broken up into several components, and we start by fitting the most obvious component (a uniform disk)
to the visibilities after which we progressively add components to the model based on the imaged residuals. We end up using the following components to reach a satisfactory fit (no more recognisable structure in the residual emission): (1) a disk with a uniform surface brightness, (2) a ring, (3) a Gaussian, and (4) a point source. The order in which these components are added does not influence the final fitting results. The components for the disk, ring and central component all share the same offset in RA and Dec from the phase center, the axis ratio, and the position angle, while the flux and semi major axis are left unconstrained. We fit these geometries for each of the continuum windows (at 217 and 234 GHz) separately to allow the detection of possible changes in flux due to the spectral slope of the dust emission α (S ν ∝ ν −α ).
We achieve the best fit with a combination of a disk, ring, Gaussian and a point source component (Fig. 5). The Gaussian component is offset from the center of the cavity with 0. 09 and 0. 20 in RA and DEC respectively, and has a semi major axis of 0. 19, an axis ratio of 0.61, and a PA of 104.7 • . This feature overlaps with the northern spiral arm detected in scattered light and we discuss it further in the next Sect. (3.1.2).
The flux of the unresolved central component at 234 GHz is 1.3 ± 0.1 mJy, the combined flux of all components is 149.2 ± 3.0 mJy. The uniform disk is constrained between 0. 22 and 0. 40 and inclined by 29.5 ± 0.5 • with a position angle of 151.0 ± 0.5 • . An unresolved ring of emission at 0. 48 ± 0. 01 containing ≈13% of the total flux improves the fit to the visibilities further. It is unclear from our data whether this represents a real structure such as a second ring or spiral arms, or that it is an artifact of our use of a uniform disk with a discontinuity in flux at the inner and outer edge (in other words, the unresolved ring takes the place of a tapered or power-law outer disk). The spectral index for the disk component is 2.4 ± 0.1, and between 3.0 and 3.6 for the other components, respectively.
The best fit parameters for the fitted components are summarized in Table 2 and visualized in Fig. 5, where we compare the imaged model and residuals to the HD 100453 disk and show the real part of the visibilities for the data, the model, and their difference.
A mm counterpart to the northern spiral arm or a vortex?
There is significant residual emission at the same location as the northern spiral arm detected in scattered light when only considering axisymmetric components for the disk. These residuals can be fit with a single elliptical Gaussian containing 8.7 mJy of flux at the same position and with a similar positioning as the northern spiral arm as seen in scattered light (Table 2 and the right panel of Fig. 6).
To better compare this emission to the spirals detected in scattered light we subtract the best-fit disk, ring, and central component from the data in visibility space and image the residuals. We show these residuals together with the SPHERE image published in Benisty et al. (2017) in the right panel of Fig. 6. The other two panels in that figure show the two datasets imposed over each other to illustrate their relative spatial extent.
All the scattered light emission including the two spiral arms is contained within the region where mm emission is detected, with the bulk of the scattered light emission originating from within the mm emission disk cavity. Comparison of the cavity outer radius with the scattered light data presented by Benisty et al. (2017), as shown in the left panel of Fig. 6, highlights a striking similarity between the two datasets in their deviation from circular symmetry. Both maps show an almost hexagonal shape of the cavity border suggesting that whatever mechanism is shaping the disk cavity is not acting in an azimuthally symmetric way. Table 2. Best-fit parameters with their respective 1 σ uncertainty in parenthesis, obtained from fitting components to the continuum visibilities: a disk with a radially constant surface brightness, an unresolved ring, a point source, and a Gaussian.
Component ∆RA ∆Dec S ν,234.2 GHz α Semi major axis Inclination PA ( ) ( ) (mJy) ( ) ( • ) ( • ) Disk -a -a 152.5 (0.5) 2.4 (0.1) 0.22 (0.01), 0.40 (0.01) b 29.5 (0.5) 151.0 (0.5) Ring -a -a 18.9 (0.6) 3.0 (0.4) 0.48 (0.01) Fixed Fixed Point -a -a 1.3 (0.1) 3.5 (1.3) - - - Gaussian 0.09 c 0.20 c 8.7 (0.3) 3.6 (0.6) 0.19 (0.01) 52.2 (5.0) 104.7 (3.0) All components 149.2 (3.0) 2.6 (0.1)
Notes. The center, position angle and inclination for the 3 first components have been fixed during the fitting. The spectral slope α (5th col.) is calculated following S ν ∝ ν −α using measurements at 234.2 and 217.0 GHz (1.28 and 1.38 mm). (a) The first three components have been fixed to the center fitted for the disk component. (b) Contains two values for the disk component: the inner and outer radius. (c) Offset relative to the center of the best-fit disk and ring component.
The mm residual emission is unresolved in the radial direction, recovered from our data regardless of the weighting applied during the imaging, and matches both the radial extent and positioning of the northern spiral arm (Fig. 6, right panel). Given the quality of our data, however, it is not clear whether this really is a mm counterpart to that spiral arm. Another viable origin for this emission would be a vortex such as detected in the HD 135344 B disk (van der Marel et al. 2016). That vortex is cospatial with the end of the spiral arm detected in scattered light, and in itself is likely responsible for launching the spiral arm due to its mass, while being induced by an interior body (Cazzoletti et al. 2018).
Future higher resolution observations are needed to disentangle the nature of the excess mm emission. However, both scenarios mentioned above lead to the same conclusion which we will explore in the remainder of this manuscript: that HD 100453 B does not induce the twin spiral arms seen in scattered light. Either the excess mm emission comes from a vortex which in itself induces the northern spiral arm, or it is the mm counterpart of the northern spiral arm. This latter option makes the northern spiral arm the primary arm (containing most mass) which is inconsistent with the position and orbital motion previously derived for of HD 100453 B. We only refer to the spiral arms scenario in the following analysis and discussion sections in order to keep them as concise as possible, and reiterate in our conclusions the two likely scenarios for the excess emission.
Dust mass estimates for the circum-primary and circum-secondary disks
To convert the measured continuum emission into a dust mass we assume that the emission is optically thin and of a single temperature following
log M dust = log S ν + 2log d − log κ ν − log B ν ( T dust ),(1)
where S ν is the flux density, d is the distance, κ ν is the dust opacity, and B ν ( T dust ) is the Planck function evaluated at the average dust temperature (Hildebrand 1983). We adopt a dust opacity of 2.31 cm 2 g −1 at 1.28 mm, calculated using astronomical silicate (Draine & Lee 1984;Laor & Draine 1993;Weingartner & Draine 2000), with a grain size distribution with sizes between 0.1 and 3000 µm distributed following a power law with a slope of −3.5.
Typically the dust temperature is estimated extrapolating from the mass averaged dust temperature in grids of radiative transfer disk models that cover a range of stellar and disk parameters (e.g., Andrews et al. 2013;van der Plas et al. 2016). At the moment these grids only consider "full disks" and thus do not give accurate dust temperatures for disks like the one around HD 100453 A which consists of a relatively narrow ring of dust.
Instead we perform a radial decomposition of the disk intensity using the radiative transfer code MCFOST (Pinte et al. 2006(Pinte et al. , 2009 as was previously done for HL Tau by Pinte et al. (2016), to estimate the dust temperature in the disk. Shortly, we fix the disk (1). We do not detect any signal at the location of HD 100453 B. We measure the continuum RMS in a circular region centered on the companion location with a diameter of 0. 20 (20.6 au) using the Briggs-weighted images for the best compromise between spatial resolution and sensitivity. The rms at the location of the companion is 0.033 mJy, leading to a 3 σ upper limit of 0.099 mJy. To calculate a limit on the amount of dust that can be present around HD 100453 B we estimate the average dust temperature using the stellar luminosity determined from the BHAC2015 evolutionary tracks (Baraffe et al. 2015) for a 0.2 M , 10 Myr old star. The expected average dust temperature in a disk with an outer radius of 10 au around such a star is 22 K following
CO J = 2-1 isotopologue emission lines
We detect spatially and spectrally resolved emission from the 12 CO, 13 CO and C 18 O J = 2-1 emission lines from the HD 100453 disk and show the moment maps and line profiles in Figs. 2-4, respectively.
We estimate the systemic velocity from the 12 CO J = 2-1 emission line at v LSR = 5.25 ± 0.10 km s −1 , based on the center of the line profile and the channel maps. The 12 CO emission line is detectable up to projected velocities of ± 7.0 km s −1 from the systemic velocity, which translates to a distance from the central star of 7.4 au assuming the gas is in Keplerian rotation in a disk inclined by 29.5 • around a 1.70 M star 1 . The outer radius as measured from 12 CO emission above 3σ in the moment maps is 1. 10. We make a first order estimate of the disk inclination using the axis ratio measured from the moment maps, and find that the inclination for the 13 CO (31 ± 5 • ) and C 18 O (35 ± 5 • ) emission is in agreement with the inclination determined from the continuum data, while the 12 CO emission appears more inclined (49 ± 5 • ). The ratio of the line flux from the 12 CO, 13 CO and C 18 O J = 2-1 emission lines is 6.0:2.3:1.0 which is similar to the isotopologue ratio detected from more massive disks around other Herbig Ae/Be stars (e.g., Perez et al. 2015) and indicates that the CO emission is optically thick for at least the 12 CO emission. We make an estimate of the optical depths for each isotopologue from the detected line ratios under the assumption that the emission comes from an isothermal slab (see for details e.g., Sect. 3.3 in Perez et al. 2015). We adopt a 12 CO to 13 CO ratio of 76 (Stahl et al. 2008) and a 12 CO to C 18 CO ratio of 500 (Wilson & Rood 1994) and find optical depths of τ 12 CO ≈ 39, τ 13 CO ≈ 0.5 and τC 18 O ≈ 0.1. The integrated line fluxes, spatial extent and geometry of all CO line emission are summarized in Table 3.
The velocity field of the disk is globally coherent with Keplerian rotation although there are hints of a deviation present in the outer disk where the CO emission at systemic velocity appears to be rotated clockwise by several degrees. To highlight this rotation of the velocity field we show the disk major and minor axis as determined from the dust emission together with a line following approximately the emission at zero projected velocity to guide the eye in the second panel of Figs. 2-4. Furthermore, despite the presence of a small misaligned inner disk the velocity map of the CO emission lacks the typical "s" shaped pattern expected at the location of the warped inner disk as described by for example Rosenfeld et al. (2014) and detected in other Herbig Ae/Be disks such as HD 142527 and HD 97048 (van der Plas et al. 2017b). The fact that the velocity field inside the cavity appears to be consistent with Keplerian rotation despite the presence of a misaligned inner disk in the cavity is possibly due to insufficient spatial resolution or to a lack of sensitivity of our observations. We explore possible deviations from Keplerian rotation in the disk further in Sect. 4.2.
As already remarked upon by Wagner et al. (2018) the disk rotation direction is counter-clockwise if we follow the interpretation by Benisty et al. (2017) that the faint spiral structure seen toward the southwest of the disk in scattered light is actually scattering from a spiral arm on the opposite face of the disk and thus that the southwest part of the disk is the side nearest to us. This means that the spiral arms seen in scattered light are trailing. Table 3. Line fluxes, spectral resolution, spatial extent and inclination for the CO J = 2-1 isotopologue emission.
Line
Line flux Error a Channel width rms b Radius c i c (Jy km s −1 ) (Jy km s −1 ) (m s −1 ) (mJy beam −1 ) ( ) 3.2.1. CO gas mass and the gas to dust ratio Deriving a total gas mass from CO emission is a highly uncertain endeavour given the large uncertainties on, among other things, the local conditions at the emitting surface, the amount of CO in gas phase, and the conversion between CO mass and total gas mass (see e.g., Miotello et al. 2017;Krijt et al. 2018). Large parametric disk grids relating a suit of disk parameters to simulated observable CO line fluxes can somewhat alleviate these uncertainties. We use the grid of disk models calculated by Williams & Best (2014) to estimate a total gas mass based on the isotopologue CO line ratio for the disk around HD 100453 A between 0.001 and 0.003 M depending on the 12 CO/C 18 O ratio assumed (550 or 1650). Using the dust mass of 0.07 M jup derived in Sect. 3.1.3 we arrive at a gas to dust ratio of 15-45. This value is in agreement with a previous upper limit on the disk gas mass by Collins et al. (2009) who suggested that the outer disk is significantly depleted in gas with an estimated gas to dust ratio between a few and a few tens.
Analysis
One of the reasons why the HD 100453 system is of interest is the possible connection between the two spiral arms detected in scattered light and the 0.2 M companion orbiting at a projected distance of 1. 05 (108 au) from the central star. Such a companion, if in a low eccentricity and close to co-planar orbit, would excite two spiral arms similar to those detected in scattered light (Dong et al. 2016;Wagner et al. 2018). This tidal interaction would also truncate the circumprimary (CP) disk at a fraction of between ≈1/2 and 1/3 of the semi-major axis (Artymowicz & Lubow 1994), in agreement with the outer radius of the disk as detected in scattered light and mm continuum emission.
However, our observations bring several discrepancies with this interpretation. The 12 CO gas disk extends to 1. 10 (113 au) and overlaps with the projected position of the secondary. Furthermore, hydro simulations for spiral arms induced by co-planar orbiting planets indicate that the surface density enhancement is expected to be higher in the primary arm (the one pointing to the perturber) than in the secondary arm (Fung & Dong 2015), which means the southern spiral arm is the primary if it were induced by a co-planar companion. Yet, we only detect mm emission from the location of northern spiral arm (cf. Sect. 3.1.2). If the mm continuum excess detected in the northern arm is indicating that this arm is the more massive one then, because it is not pointing to the perturber, it is not clear anymore that the spirals are driven by the companion M star, in particular if it is in a co-planar and prograde orbit with the disk.
Lastly, despite the proximity of the companion to the CP disk we detect no emission from a circumsecondary or circumbinary disk. This is contrary to the idea that a recent flyby, prograde and co-planar, as such an interaction would likely lead to a significant amount of dust and gas being captured by the by the interloper (Cuello et al. 2019).
If any of the three arguments above is correct, it would challenge the proposed co-planarity of the orbit of the companion and its dominant role in the excitation of the spiral arms. We investigate the influence of a co-planar orbit companion using gas+dust SPH simulations in Sect. 4.1 and the possible deviation of the gas kinematics in the outer disk in Sect. 4.2. We also reassess the orbital parameters to further check the viability of a co-planar orbit for the HD 100453 AB system in Sect. 4.3.
SPH simulations
We study the tidal influence of a co-planar companion on the gas and dust content of the circumprimary disk via global 3D simulations with the SPH code PHANTOM (Price et al. 2018a). Gas and dust are treated as separate sets of particles interacting via aerodynamic drag according to the algorithm described in Laibe & Price (2012), using 7.5 × 10 5 SPH particles for the gas and 2.5 × 10 5 for the dust and setting the initial dust-to-gas mass ratio to 1%. The grain size is set to 1 mm. We adopt for our simulations the same parameters for the binary orbit and for the disk as in Dong et al. (2016). The primary and secondary stars, treated as sink particles, have masses M A = 1.7 and M B = 0.3 M 2 and are separated by a = 120 au on a circular orbit, co-planar with the disk. We initially set the inner and outer disk radii to r in = 12 and r out = 96 au (we note that r out is outside the Roche lobe of the primary) and its mass to M d = 0.003 M , with power-law profiles Σ ∝ r −1 for the surface density and T ∝ r −0.5 for the temperature. Contrary to Dong et al. (2016), we do not seek here to reproduce the pitch angle of the spirals and adopt a more conventional disk aspect ratio of H/r = 0.05 at 12 au , with a vertically isothermal profile. We set the SPH artificial viscosity in order to obtain an average Shakura & Sunyaev (1973) viscosity of α SS = 5 × 10 −3 (Lodato & Price 2010). The accretion radius of both stars is set to 12 and 5 au, respectively. We run the simulation for ten orbits of the binary, at which time the disk has reached a quasi-steady state.
To facilitate a more quantitative comparison between our simulations and the observed data we use the radiative transfer code MCFOST to convert the results of our simulation into images Fig. 7. Comparison between the ALMA observations (top row) and the ray-traced SPH simulations (bottom row). Panels from left to right: CO integrated intensity (moment 0) map, CO intensity weighted velocity field (moment 1) map, and the 1.4 mm dust emission map. The purple star represents the location of HD 100453 B in all panels. For the CO moment 1 map (middle panel) we only include emission that is within a certain fraction of the peak emission in the image channels. The maximum observed dynamic range in our observations is 40, and we construct the model moment 1 maps using only emission that is brighter than a fraction of 3/40 of the peak intensity. Top left and right panels: 2, 10 and 30 σ contours using yellow lines. For the bottom panels we use the dynamic range from the observations to approximate these contours as fraction of the maximum emission in the simulated maps.
at relevant frequencies. We convolve the resulting images with a Gaussian of the same FWHM as the beam in the observations and scale the maximum intensity in the convolved image to that of the observed images. We show the simulated 1.4 mm continuum map, the integrated CO intensity, and the CO velocity field, together with the observed maps, in Fig. 7. In our simulations two other disks quickly form from the material that was part of the circumprimary disk and located outside the primary's Roche lobe: a circumsecondary and a circumbinary disk. In the simulated 1.4 mm map most of the continuum emission originates from the circumprimary disk. Compared to this disk the peak flux from the circumsecondary and circumbinary disks are weaker by a factor 80 and 300, respectively. For comparison, the observed ratio between the peak flux from the circumprimary disk and the background rms is ≈400. The dust grains in the circumprimary disk get concentrated in a smaller disk with two faint spiral arms whose primary arm is marginally brighter than the secondary arm. The CO velocity field in the simulations shows a twist in the same manner as seen in the observations, and the CO disk becomes more elongated as it fills the Roche lobe of the primary. We discuss these results further in Sect. 5.1.
Quantifying the disk warp
The velocity field of the disk around HD 100453 A shows deviations from a pure Keplerian rotation, most notably through a twist in the iso-velocity contours at systemic velocity (highlighted in Figs. 2-4 with a purple line). To better quantify these deviations we fit the observed velocity field of the CO gas using the methodology introduced in Perez et al. (2015) which we shortly summarize in the next paragraph. We note that we restrict our analysis to quantifying the velocity field and the warp in the circumprimary disk. We do not optimize on the disk structure other than obtaining a reasonable fit and will explore the intracavity column density and kinematics in an upcoming paper using higher sensitivity and resolution observations.
We fit a parametric model of the 12 CO gas allowing for a warp (a different inclination) and PA of the inner disk w.r.t the outer disk which starts at 38 au. The parametric model follows Casassus et al. (2015) and adopts the surface density parameters fitted by Wagner et al. (2018) to the lower resolution compact array configuration part of the dataset also presented in this manuscript, with exception of the CO scale height (H/r), the power law for the radial surface density (γ), and the characteristic radius (r c ). Following our choice for H/r described in Sect. 4.1 we choose a more conventional value for the disk aspect ratio of 0.05. Because no value for γ is mentioned in Wagner et al. (2018) we use a standard value of 1. Finally, we are unable to reproduce the outer disk extent with a large value for R c of 27 ± 1 au, and instead use a value of 10 au which better reproduces the extent of the outer disk.
Our four free parameters are the inclination angle and PA for the inner and outer disk: {i out , PA out , i in , PA in }. We compare Fig. 8. Best fit model for the intensity-weighted velocity field of the 12 CO emission in the disk (middle panels) for a Keplerian disk (bottom row) and a warped disk (top row). We note that we only compare the velocity field in those regions where the CO emission is above a 5σ threshold in the observed moment 0 map. We show the observed intensity-weighted velocity field in the top left panel, and the residual after subtracting the model from the observations in the right panels.
the model and data via the computation of first moment maps. Optimization is done by minimizing χ 2 = (M1 o − M1 m ), where M1 o and M1 m correspond to observed and model first moment. The comparison is done only in the pixels where the observed signal in the zeroth moment is above 5σ. First, we performed a simple χ 2 search using 0.5 • steps around {i out , PA out } = 25, 140. We fix the outer disk values to those that yield a minimum in χ 2 . Then, we do the same search but for the intra-cavity angles {i in , PA in }. We adopt the best fit values and repeat the exploration for the outer disk parameters. We repeat the same for the intra-cavity angles. These steps are repeated until the variation is <0.5 • (our step). The final best fit parameters are {i out , PA out } = {19.5, 139.5} for the outer disk parameters, and {i in , PA in } = {24.0, 146.0} for the inner disk.
We compare the observed moment 1 map both with a purely Keplerian disk and with our best-fit solution for a disk warp in Fig. 8. The velocity residuals show that the Keplerian model (bottom right panel) cannot account for the velocity field in the inner region, and produces red and blue residuals that have a different PA from the outer disk. A mildly warped disk (represented here by an inner region with different PA and inclination) yields, as expected, a better fit to the data than the purely Keplerian model. The best fit inclination for the CO outer disk is 10.0 • lower than the value derived from the dust continuum disk (i.e closer to face-on), while the inner disk inclination is halfway between these values. Similarly the best fit PA for the CO outer disk is 11.5 • lower compared to the value derived from the continuum, while the inner disk PA is 5.0 • lower compared to the PA of the dust disk. The most significant residual after subtracting the best fit warp model is approximately at the stellar position, where our model overpredicts the beam-averaged velocity by 0.7 km s −1 in a region the same size as our beam, and which thus is likely unresolved.
Orbital fitting of HD 100453 AB
In the previous sections, we show that a co-planar model for the orbit of HD 100453 B may not succeed as well as first thought to match the observations, and in particular the CO data. Wagner et al. (2018) present the most complete set of astrometric data for this system yet and we re-assess the orbit and the assumption of co-planarity starting from the same astrometric data.
We fit the relative orbit of HD 100453 B with respect to the HD 100453 A disk assuming a Keplerian orbit projected on the plane of the sky. In this formalism, the astrometric position of the companion can be written as:
x = ∆Dec = r (cos(ω + θ) cos Ω − sin(ω + θ) cos i sin Ω) ,(2)y = ∆RA = r (cos(ω + θ) sin Ω + sin(ω + θ) cos i cos Ω) ,(3)
where Ω is the longitude of the ascending node (measured counterclockwise from north), ω is the argument of the periastron, i is the inclination, θ is the true anomaly, and r = a(1 − e 2 )/(1 + e cos θ) is the radius, where a stands for the semi-major axis and e for the eccentricity. The orbital fit we performed uses the observed astrometry measurements given in Wagner et al. (2018, Table 2) to derive probability distributions for elements P (period), e, i, Ω, ω, and time for periastron passage t p . Elements a and P can be deduced from one another through Kepler's third law. We used two complementary fitting methods, as described in (LSLM) algorithm to search for the model with the minimal reduced χ 2 , and (ii) a more robust statistical approach using the Markov-chain monte carlo (MCMC) Bayesian analysis technique (Ford 2005(Ford , 2006 to probe the distribution of the orbital elements. Ten chains of orbital solutions were conducted in parallel, and we used the Gelman-Rubin statistics as a convergence criterion (see Ford 2006, for details). We picked randomly a sample of 500 000 orbits into those chains following the convergence. This sample is assumed to be representative of the probability (posterior) distribution of the orbital elements, for the given priors. We chose the priors to be uniform in x = (ln P, e, cos i, Ω + ω, ω − Ω, t p ) following Ford (2006). As explained therein, for any orbital solution, the couples (ω,Ω) and (ω + π,Ω + π) yield the same astrometric data, this is why the algorithm fits Ω + ω and ω − Ω, which are not affected by this degeneracy. The system distance and total mass used for the fitting are 103 pc and 1.9 M . We calculate the relative inclination between the orbit of HD 100453 B and the HD 100453 A + disk system using the longitude of node Ω (equivalent to the PA for disks) which is the angle of the intersection line between the disk and sky plane, measured from the north, and the inclination i, which is the angle between the disk and sky planes. The relative inclination between two planes depends on i 1 and i 2 , but also on the difference Ω 1 − Ω 2 following: cos i r = cos i 1 cos i 2 + sin i 1 sin i 2 cos(Ω 1 − Ω 2 ).
(4)
Despite that the astrometric measurements only cover a small fraction of the orbit, we obtain a consistent fit to the orbit with χ 2 r values between 0.5 and 2. A sample of the best-fit orbits is shown in Fig. 9, the corner plot showing the posteriors for the orbital fitting in Fig. A.1. The previously mentioned inherent ambiguity of direct imaging regarding the couple (Ω,ω) induces a bimodal posterior distribution for these two parameters. Radial velocity data are needed in order to remove the degeneracy. On the other hand, the loosely constrained and probably low eccentricity prevents a robust determination of the argument of periastron and the periastron passage. A longer orbital coverage would be necessary to resolve a clear curvature in the orbit and further constrain all the orbital elements.
We are able to reasonably constrain the semi-major axis of the orbit to be close to the projected value and the eccentricity to be low. Our results are mostly in agreement with the results presented in Wagner et al. (2018) but with two deviations. Firstly, whereas Wagner et al. (2018) conclude that the inclination of the companion's orbit is co-planar with the disk to within a few σ, our calculations indicate that a co-planar orbit is not favored with a most likely relative inclination of 60 • (right panel of Fig. 9). This is most likely because Wagner et al. (2018) did not account for the longitude of node in their determination of the relative inclination. Secondly, the likelihood for the orbital eccentricity of the companion in our calculation peaks at zero eccentricity and it is safe to assume the orbit is bound as the probability distribution of the eccentricity rules out solutions with an eccentricity higher than 0.5 at a 97% probability. This value is lower than that found by Wagner et al. (2018) who find a probability distribution that peaks between values of 0.1 and 0.2.
Discussion
In this section, we tie together our observations with the outcomes of the analysis and discuss the most likely orbit for HD 100453 B, the origin of the detected spiral arms, and the implications thereof on the origin of the disk inner cavity.
The orbit of HD 100453 B
Given the current evidence we deem it unlikely that the double armed spiral pattern in this system is excited by an external companion in a close-to co-planar orbit as previously suggested. Even though the outer edge of the dust disk extends to a radius in agreement with tidal truncation by such a companion, the gas disk is not. This disk, as traced by CO, extends out up to a distance greater than the projected separation of the companion. We also do not detect emission from the circumsecondary disk.
To test for the influence of tidal truncation on the gas in the disk we simulated a system similar to HD 100453. These simulations show that the material that was originally outside the primary's Roche lobe is captured into a circumsecondary disk or ejected onto a circumbinary ring. After a fast initial redistribution of disk material the system continues to evolve on a viscous timescale. This timescale is shorter for the smaller circumsecondary disk, possibly explaining its non-detection. The circumbinary disk is more significant in our simulations and it is possible that it would survive even up to the current age of the system. The properties of this disk heavily depend on the companion orbit and such a circumbinary disk may even not be present for significantly misaligned orbits. This needs to be tested with future simulations.
A misaligned orbit for HD 100453 B would explain the large extent of the observed circumprimary CO disk as for such orbits the tidal torque on the disk reduces with a factor of ≈cos 8 (i) for misalignment angle i (Lubow et al. 2015). A secondary on a misaligned orbit can of course also be comfortably outside the primary's Roche lobe while its projected location is close to or overlapping with the disk edge.
The observed CO disk does show signs of dynamical disturbance through the warped circumprimary disk and through the more elongated spatial distribution of the 12 CO emission compared to the dust disk geometry and the rarer CO isotopologues. In our SPH simulations small amounts of gas fill the Roche lobe of the primary which closely mimics the more stretched out 12 CO disk (cf. the 2 and 30 σ contours of the model CO emission shown in the bottom left panel of Fig. 7). This more elongated structure for the 12 CO emission is in qualitative agreement with the distribution of CO gas in our simulations and we interpret it as a reservoir of lower-density material which is distributed along the major axis by tidal interactions between the gas disk and the companion. Connecting the warped CO disk to HD 100453 B also hints at an inclination for the companion orbit that is closer to face-on than that of the circumprimary disk because the inclinations derived for the inner and outer disk are progressively closer to face-on compared to the inclination calculated from the (midplane) dust emission (cf. Sect. 4.2).
Our orbital fitting shows that while we cannot constrain the relative inclination between the companion and the disk, a co-planar orbit is not favored. Rather, the probability density function for the relative inclination peaks at a misalignment of ≈60 • . Two quantities that we can reasonably constrain are a low eccentricity orbit and a semi-major axis close to the projected separation.
Finally, the southwestern spiral pointing toward the companion is expected to contain more mass if an external companion on a co-planar orbit was to excite the double armed spiral. We find instead a mm counterpart to the northern spiral suggesting that this is the primary spiral arm.
Given the above, we re-evaluate the causal connection between the external companion and the double spiral arms. The CO disk does show signs of tidal disturbance and while the orbit of the companion is of low eccentricity, it most likely is significantly misaligned compared to the plane of the disk. Such a misaligned orbit allows for weaker truncation of the circumprimary disk and explains both the warped outer disk and the CO emission that is seen up to distances similar to the separation of the companion.
Such a companion that orbits in a plane that is misaligned compared to the disk could still excite double spiral arms, but it is as of yet unclear what those spiral arms would look like in terms of opening angle and surface density contrast. We therefore consider alternative scenarios that could also generate the observed spiral pattern in Sect. 5.3.
The disk cavity + misaligned inner disk
We resolve an inner cavity extending up to 23 au from the mm dust continuum emission that contains an unresolved mm counterpart to the small misaligned inner disk previously detected using near-and mid-IR interferometric observations (Menu et al. 2015;Lazareff et al. 2017), and whose presence is corroborated by two shadows cast on the outer disk . The spectral index we determine for the inner disk from the limited 0.1 mm bandwidth available is 3.5 ± 1.3. This is consistent with emission originating from a dusty disk. The NIR emission from the inner disk is best fit with a Gaussian with an inclination of 48 • and a PA of 80 • (Lazareff et al. 2017), significantly misaligned with respect to the values we determine for the circumprimary disk. The size of the cavity detected in scattered light is ≈21 au (Wagner et al. 2015), comparable to the size of the cavity in mm emission. The geometry of the outer cavity wall deviates at both wavelengths from circular symmetry and has a more hexagonal shape. Secular precession resonances in young binary systems with mass ratios on the order of 0.1 can generate large misalignments between the circumstellar disk and a companion (Owen & Lai 2017), and these authors suggest that the misalignment seen in HD 100453 could have been generated by resonance crossing and that such a scenario implies that a low-mass (between ≈0.01 and 0.1 M ) companion is residing inside the cavity with an orbit that is aligned with the outer disk. Such a companion in a circular orbit would need to orbit at ≈13 au to truncate the circumbinary 3 disk at 23 au (Artymowicz & Lubow 1994). More A&A 624, A33 (2019) eccentric orbits would allow for values smaller than 13 au for the companion orbit.
It is interesting to note that the HD 100453 system shares many similarities with the much better studied HD 142527 system, such as a small and misaligned inner disk, a large disk cavity, spiral arms and shadows cast by a misaligned inner disk detected in scattered light, and azimuthally asymmetric mm emission in the outer disk. Recent work by Price et al. (2018b) shows that the interaction of a companion inside the cavity on an inclined and eccentric orbit can reproduce the spirals, shadows, and horseshoe geometry of dust emission detected in that disk, as well as a non-circular geometry of the outer cavity wall.
The presence of such a close-in companion in this system is supported by the lack of detected CO emission from the HD 100453 disk within 7 au. Furthermore, the lower-than expected observed velocities of the CO gas close to the stellar position (≈10-20% lower than the local Keplerian velocity, cf. Fig. 8) is consistent with a velocity signature left by a close-in companion. Pérez et al. (2018) show that the spiral wakes left by these bodies imprint asymmetric velocity patterns, where the maximum deviation from Keplerian rotation occurs at the outer spiral wake launched by a giant planet. Once these kinematic signatures get convolved with our beam they would appear similar to the deviation we detect.
These lines of reasoning all point toward the presence of a companion inside the cavity. Determining the precise properties of such a companion requires more data and investigations and is outside the scope of this work, but the constraints on the orbital parameters and mass are that it should be able to drive the misalignment of the inner disk while not leaving a gravitational fingerprint on the velocity field of the CO gas in the cavity that would have stood out in our observations. Juhász et al. (2015) argue that planet-induced spiral arms are unlikely to be detected with current instruments, and suggest that all as of yet observed spiral arms are instead pressure scaleheight perturbations. Together with the relative brightness of the northern spiral arm and the extent of the CO disk this motivates us to explore alternative origins for the spiral arms.
Possible origins of the spiral arms
Self-gravity can cause parts of the disk to collapse and form spiral arms in the process if the disk is sufficiently massive. Typically a disk needs to hold ≈10% of the mass of the central star for gravitational instabilities to become relevant (see for example the review by Kratter & Lodato 2016), a condition that is far from fulfilled in the disk around HD 100453 A. A gravitationally unstable disk is unlikely to be the cause for the detected spiral arms. Stellar fly-by scenarios can, under certain circumstances, also generate two near-symmetric spiral arms in disks (Pfalzner 2003), but the low eccentricity of the orbit of HD 100453 B indicates the companion is on a bound orbit which makes a recent fly-by an equally unlikely candidate for provoking the spiral arms.
A companion inside the disk cavity could drive a slow precession of the misaligned inner disk. If the direction of this precession is prograde and a region of the outer disk rotates at the same frequency as the shadow cast by this precessing inner disk (≈85 yr for a launching location for the spirals of 0. 22), spiral arms whose pitch angle much resemble those caused by a planet can develop at the location of the shadow (Montesinos et al. 2016;Montesinos & Cuello 2018). Slight asymmetries in the tilted inner disk affect the depth of shadows and thus the relative strength of the spiral arms. A weaker shadow on the western disk then would be able to explain the non-detection of a mm counterpart to the southern spiral arm.
Conclusions
We resolve the disk around HD 100453 A into a disk of dust continuum emission between 23 and 41 au, an unresolved inner disk, and excess mm emission at the location of the northern spiral arm detected using scattered light imaging. Two likely interpretations for this excess emission are (1) that it is a mm counterpart to the spiral arm, or (2) that it is a narrow vortex associated with the spiral arm either through having a common origin or by inducing the spiral arm. We do not detect emission from the location of HD 100453 B and put a 3σ upper limit on the dust content for that disk of 0.03 Earth masses. The CO emission from the circumprimary disk extends out to 1. 10 and shows a velocity pattern that is mostly Keplerian but with a 10 • warp between inner and outer disk. The morphology of the 12 CO disk is more elongated along the major axis when compared to the 13 CO, C 18 O, and mm dust emission, likely as a consequence of tidal disruption of the circumprimary disk by HD 100453 B.
Our fit to the orbit of HD 100453 B suggests a significantly misaligned orbit with respect to the circumprimary disk. Such an orbit is supported by our SPH simulations, which show that a companion on a co-planar orbit cannot reproduce the detected spatial extent of the CO disk nor our detection of mm emission from the northern spiral arm. It is possible that a companion at larger separation and/or on an inclined orbit reproduces the morphology of the detected CO emission better but it is as of yet unclear if a companion on a sufficiently misaligned orbit can qualitatively reproduce the spiral arm morphology of this system. Pending detailed calculations of the impacts by a significantly inclined orbit of the companion on the circumprimary disk we suggest an alternative scenario that could also generate the observed spiral pattern.
Given the relatively low mass of the disk and the low eccentricity of the orbit of HD 100453 B we deem a recent fly-by or a gravitational instability in the disk unlikely to provoke the spiral arms. Instead, we suggest that comoving shadows of a precessing inner disk as possible cause for the detected spiral arms. A small misaligned inner disk has been detected using near infrared interferometry and its shadows are visible on the outer disk at roughly the same location as the launching points of the spiral arms. Such a misaligned inner disk, the non-detection of CO emission from the inner 7 au, and the 23 au large cavity in the dust disk, all can be explained by a companion inside the disk cavity orbiting at a distance between a few and ≈13 au.
All features described in this manuscript are illustrated in The black lines and points depict the best fitting orbit (better χ 2 ), obtained with the LSLM algorithm. The color scale is logarithmic, blue corresponds to 1 orbit and red to 1000.
Fig. 2 .
2Summary of the 12 CO line emission in HD 100453. We show the integrated intensity (moment 0, left panel), intensity-weighted velocity (moment 1, 2nd panel), peak intensity (moment 8, 3rd panel) and the integrated emission line (right panel). The moment 1 + 8 maps were made using a 3 σ cutoff from images reconstructed using natural weighting to maximize sensitivity. Over plotted in the 1 st panel is the 25 σ(1.27 mJy beam −1 ) contour of the continuum emission shown in Fig. 1. The beam is shown in orange in the bottom left of each panel. We show the approximate position of HD 100453 B during our observations (1. 05 at PA = 132 • Wagner et al. 2018) with a purple star in the 2 nd panel, together with two dotted lines that show the major and minor disk axis of a disk with a semi major axis value listed in Table 3 and the inclination and position angle determined from fitting the continuum emission. The purple line highlights the clockwise rotation of the velocity field discussed in Sect. 4.2. The line profile shown in the right panel shows the integration boundaries used to calculate the total line emission (a half line width of 7.0 km s −1 ), the systemic velocity of 5.25 km s −1 , and the level of the continuum emission used to calculate the integrated line flux.
Fig. 3 .
3Same as Fig. 2 but for the 13 CO J = 2-1 line emission.
Fig. 4 .
4Same as Fig. 2 but for the C 18 O J = 2-1 line emission.
Fig. 5 .
5Comparison of ALMA band 6 data (left panel) with the best-fit composite model (second panel). Third panel: imaged residual visibilities. This panel also includes ellipses representing the fitted disk and the central components in yellow thick lines, the outer ring with a yellow thin line, and the Gaussian component with a dark solid line. Units of all intensity scales are in mJy beam −1 . Top right panel: real part of the visibilities as function of the deprojected baseline for the data (black dots) and model (red line). Bottom panel: residuals. The visibilities are binned in sets of 200.
Fig. 6 .
6Left panel: J-band Q φ image reproduced fromBenisty et al. (2017) in arbitrary intensity units with an overlay of the 12 and 25 σ contours of the ALMA data presented inFig. 1. Central panel: inverted counterpart to the images shown in the left panel but with arbitrary contours of the SPHERE data overlayed on the ALMA data. Right panel: ALMA residuals after subtracting the best-fit disk, ring and central point source components summarized inTable 2in visibility space, imaged using superuniform weighting. The same contours as shown for the SPHERE images in the central panel are again overlayed. The Gaussian component of the ALMA continuum emission appears to coincide well with the northern spiral arm seen in scattered light.inclination and PA and match the model radial surface density profile in an iterative procedure to the observed one. See Sect. 3 ofPinte et al. (2016) for a full description. The dust mass in the resulting model is 0.07 M jup and the mass averaged dust temperature in the resulting model is 27 K which translates in a dust disk mass of 0.09 M jup applying Eq.
Fig. 5 of van der Plas et al. (2016). This puts an upper limit of 0.03 M Earth on the amount of dust around HD 100453 B.
Line fluxes have been calculated from the natural-weighted images by integrating the emission around the systemic velocity at 5.25 km s −1 assuming a half line width of 7.0 km s −1 . (a) The error on the integrated line flux was estimated from the rms of the integrated spectrum outside the line boundaries and does not include calibration uncertainties. (b) 1 σ rms per channel. (c)The radius is measured along the semi-major axis of the moment 1 maps shown in the 2nd panel of Figs. 2-4 that were made using CO emission detected above 3σ in the channel maps.
Fig. 9 .
9Chauvin et al. (2012): (i) a least squares Levenberg-Summary of the orbital fitting results. Panel a: plots of a hundred trajectories obtained with the MCMC algorithm for the orbit of HD 100453 B. A cartoon of the dust disk is shown at the center. Panel b: evolution of separation with respect to time. The three shades of gray represent the 1, 2 and 3 σ intervals. Panel c: similar to the 2nd panel, but for the evolution of position angle with respect to time. Panel d: posterior distribution of the relative inclination between the HD 100453 B orbit and the disk plane.
Fig. A. 1 .
1Fig. B.1 together with a list of relevant figures in which they are visible. Distribution and correlations of each of the orbital element fitted by the MCMC algorithm.
Table 1 .
1Details of the observations.Fig. 1. Continuum image of HD 100453 for the ALMA band 6 observations, reconstructed using superuniform weighting resulting in a 0. 09 × 0. 17 beam. Over plotted are contours at 12, 25 and 100× the rms value of 0.05 mJy beam −1 . The beam is shown in orange in the bottom left, and a 0. 3 wide inset of the disk cavity with stretched colors highlights the emission at the stellar position. We note that the color scale is negative.UT date
Number Baseline range pwv
Calibrators:
antennas (m)
(mm) Flux
bandpass
Gain
2016 Apr 23 42
15-463
1.64
J1107-4449 J1107-4449 J1132-5606
2016 Sep 08 36
15-2483
0.56
J1107-4449 J1107-4449 J1132-5606
This is an upper limit as both beam dilution as higher velocity gradients make the CO emission more difficult to detect at higher velocities and at closer distance.
We note that in this section we followDong et al. (2016) in using a companion mass of 0.3 M (fromChen et al. 2006), whereas in the rest of this manuscript we adopted a companion mass of 0.2 M(Collins et al. 2009). The impact of using a lighter companion in our simulations would be to decrease the size of its Roche lobe and the amount of mass the secondary can capture.A33, page 7 of 15 A&A 624, A33(2019)
A33, page 10 of 15 G. van der Plas et al.: HD 100453 with ALMA
We refer to this disk as the circumprimary disk in the rest of this manuscript because, while likely, the presence of a companion inside the cavity has not been confirmed by direct obesrvations.
A33, page 15 of 15
Acknowledgements. This paper makes use of data from ALMA programme 2015.1.00192.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science This project is supported by CNRS, the OSUG@2020 labex and the Programme National de Planetologie (PNP, INSU) and Programme National de Physique Stellaire (PNPS, INSU). The orbital fit presented in this paper was performed using the Froggy platform of the CIMENT infrastructure (https://ciment. ujf-grenoble.fr), which is supported by the Rhone-Alpes region (GRANT CPER07 13 CIRA), the OSUG@2020 labex (reference ANR10 LABX56) and the Equipe@Meso project (reference ANR-10-EQPX-29-01) of the programme Investissements d'Avenir, supervised by the Agence Nationale pour la Recherche. SPH simulations were run at the Common Computing Facility (CCF) of LABEX LIO.
. S M Andrews, K A Rosenfeld, A L Kraus, D J Wilner, http:/linker.aanda.org/10.1051/0004-6361/201834134/1ApJ. 771129Andrews, S. M., Rosenfeld, K. A., Kraus, A. L., & Wilner, D. J. 2013, ApJ, 771, 129
. S M Andrews, D J Wilner, Z Zhu, http:/linker.aanda.org/10.1051/0004-6361/201834134/2ApJ. 82040Andrews, S. M., Wilner, D. J., Zhu, Z., et al. 2016, ApJ, 820, L40
. P Artymowicz, S H Lubow, http:/linker.aanda.org/10.1051/0004-6361/201834134/3ApJ. 421651Artymowicz, P., & Lubow, S. H. 1994, ApJ, 421, 651
. H Avenhaus, S P Quanz, A Garufi, http:/linker.aanda.org/10.1051/0004-6361/201834134/4ApJ. 86344Avenhaus, H., Quanz, S. P., Garufi, A., et al. 2018, ApJ, 863, 44
. I Baraffe, D Homeier, F Allard, G Chabrier, http:/linker.aanda.org/10.1051/0004-6361/201834134/5A&A. 57742Baraffe, I., Homeier, D., Allard, F., & Chabrier, G. 2015, A&A, 577, A42
. M Benisty, T Stolker, A Pohl, http:/linker.aanda.org/10.1051/0004-6361/201834134/6A&A. 59742Benisty, M., Stolker, T., Pohl, A., et al. 2017, A&A, 597, A42
J.-L Beuzit, M Feldt, K Dohlen, http:/linker.aanda.org/10.1051/0004-6361/201834134/7Proc. SPIE. SPIE7014701418Beuzit, J.-L., Feldt, M., Dohlen, K., et al. 2008, Proc. SPIE, 7014, 701418
. S Casassus, M S Perez, A Jordán, http:/linker.aanda.org/10.1051/0004-6361/201834134/8ApJ. 75431Casassus, S., Perez, M. S., Jordán, A., et al. 2012, ApJ, 754, L31
. S Casassus, G Van Der Plas, M S Perez, http:/linker.aanda.org/10.1051/0004-6361/201834134/9Nature. 493191Casassus, S., van der Plas, G., Perez, M. S., et al. 2013, Nature, 493, 191
. S Casassus, S Marino, S Pérez, http:/linker.aanda.org/10.1051/0004-6361/201834134/10ApJ. 81192Casassus, S., Marino, S., Pérez, S., et al. 2015, ApJ, 811, 92
. P Cazzoletti, E F Van Dishoeck, P Pinilla, http:/linker.aanda.org/10.1051/0004-6361/201834134/11A&A. 619161Cazzoletti, P., van Dishoeck, E. F., Pinilla, P., et al. 2018, A&A, 619, A161
. G Chauvin, A.-M Lagrange, H Beust, http:/linker.aanda.org/10.1051/0004-6361/201834134/12A&A. 54241Chauvin, G., Lagrange, A.-M., Beust, H., et al. 2012, A&A, 542, A41
. X P Chen, T Henning, R Van Boekel, C A Grady, http:/linker.aanda.org/10.1051/0004-6361/201834134/13A&A. 445331Chen, X. P., Henning, T., van Boekel, R., & Grady, C. A. 2006, A&A, 445, 331
. V Christiaens, S Casassus, S Perez, G Van Der Plas, F Ménard, http:/linker.aanda.org/10.1051/0004-6361/201834134/14ApJ. 78512Christiaens, V., Casassus, S., Perez, S., van der Plas, G., & Ménard, F. 2014, ApJ, 785, L12
. K A Collins, C A Grady, K Hamaguchi, http:/linker.aanda.org/10.1051/0004-6361/201834134/15ApJ. 697557Collins, K. A., Grady, C. A., Hamaguchi, K., et al. 2009, ApJ, 697, 557
. N Cuello, G Dipierro, D Mentiplay, http:/linker.aanda.org/10.1051/0004-6361/201834134/16MNRAS. 4834114Cuello, N., Dipierro, G., Mentiplay, D., et al. 2019, MNRAS, 483, 4114
. R Dong, Z Zhu, J Fung, http:/linker.aanda.org/10.1051/0004-6361/201834134/17ApJ. 81612Dong, R., Zhu, Z., Fung, J., et al. 2016, ApJ, 816, L12
. B T Draine, H M Lee, http:/linker.aanda.org/10.1051/0004-6361/201834134/18ApJ. 28589Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89
. C D Dressing, D Charbonneau, http:/linker.aanda.org/10.1051/0004-6361/201834134/19ApJ. 80745Dressing, C. D., & Charbonneau, D. 2015, ApJ, 807, 45
C Espaillat, J Muzerolle, J Najita, http:/linker.aanda.org/10.1051/0004-6361/201834134/20Protostars and Planets VI. TucsonUniversity of Arizona Press497Espaillat, C., Muzerolle, J., Najita, J., et al. 2014, Protostars and Planets VI (Tucson: University of Arizona Press), 497
. S Facchini, A Juhász, G Lodato, http:/linker.aanda.org/10.1051/0004-6361/201834134/21MNRAS. 4734459Facchini, S., Juhász, A., & Lodato, G. 2018, MNRAS, 473, 4459
. E B Ford, http:/linker.aanda.org/10.1051/0004-6361/201834134/22AJ. 1291706Ford, E. B. 2005, AJ, 129, 1706
. E B Ford, http:/linker.aanda.org/10.1051/0004-6361/201834134/23ApJ. 642505Ford, E. B. 2006, ApJ, 642, 505
. J Fung, R Dong, http:/linker.aanda.org/10.1051/0004-6361/201834134/24ApJ. 81521Fung, J., & Dong, R. 2015, ApJ, 815, L21
. T Prusti, Gaia Collaborationhttp:/linker.aanda.org/10.1051/0004-6361/201834134/25A&A. 5951Gaia Collaboration (Prusti, T., et al.) 2016, A&A, 595, A1
. Garcia Lopez, R Natta, A Testi, L Habart, E , http:/linker.aanda.org/10.1051/0004-6361/201834134/26A&A. 459837Garcia Lopez, R., Natta, A., Testi, L., & Habart, E. 2006, A&A, 459, 837
. J.-F Gonzalez, G Laibe, S T Maddison, http:/linker.aanda.org/10.1051/0004-6361/201834134/27MNRAS. 467Gonzalez, J.-F., Laibe, G., & Maddison, S. T. 2017, MNRAS, 467, 1984
. R H Hildebrand, http:/linker.aanda.org/10.1051/0004-6361/201834134/28QJRAS. 24267Hildebrand, R. H. 1983, QJRAS, 24, 267
. J A Högbom, http:/linker.aanda.org/10.1051/0004-6361/201834134/29A&AS. 15417Högbom, J. A. 1974, A&AS, 15, 417
. A Juhász, M Benisty, A Pohl, http:/linker.aanda.org/10.1051/0004-6361/201834134/30MNRAS. 4511147Juhász, A., Benisty, M., Pohl, A., et al. 2015, MNRAS, 451, 1147
. S Khalafinejad, K M Maaskant, N Mariñas, A G G M Tielens, http:/linker.aanda.org/10.1051/0004-6361/201834134/31A&A. 58762Khalafinejad, S., Maaskant, K. M., Mariñas, N., & Tielens, A. G. G. M. 2016, A&A, 587, A62
. S Krijt, K R Schwarz, E A Bergin, F Ciesla, http:/linker.aanda.org/10.1051/0004-6361/201834134/32ApJ. 86478Krijt, S., Schwarz, K. R., Bergin, E. A., & Ciesla, F. J. 2018, ApJ, 864, 78
. G Laibe, D J Price, http:/linker.aanda.org/10.1051/0004-6361/201834134/33MNRAS. 4202345Laibe, G., & Price, D. J. 2012, MNRAS, 420, 2345
. A Laor, B T Draine, http:/linker.aanda.org/10.1051/0004-6361/201834134/34ApJ. 402441Laor, A., & Draine, B. T. 1993, ApJ, 402, 441
. B Lazareff, J.-P Berger, J Kluska, http:/linker.aanda.org/10.1051/0004-6361/201834134/35A&A. 59985Lazareff, B., Berger, J.-P., Kluska, J., et al. 2017, A&A, 599, A85
. M Lecar, M Podolak, D Sasselov, E Chiang, http:/linker.aanda.org/10.1051/0004-6361/201834134/36ApJ. 6401115Lecar, M., Podolak, M., Sasselov, D., & Chiang, E. 2006, ApJ, 640, 1115
. R V E Lovelace, H Li, S A Colgate, A F Nelson, http:/linker.aanda.org/10.1051/0004-6361/201834134/37ApJ. 513805Lovelace, R. V. E., Li, H., Colgate, S. A., & Nelson, A. F. 1999, ApJ, 513, 805
. W Lyra, M Kuchner, http:/linker.aanda.org/10.1051/0004-6361/201834134/38Nature. 499184Lyra, W., & Kuchner, M. 2013, Nature, 499, 184
. K Kratter, G Lodato, http:/linker.aanda.org/10.1051/0004-6361/201834134/39ARA&A. 54271Kratter, K., & Lodato, G. 2016, ARA&A, 54, 271
. G Lodato, D J Price, http:/linker.aanda.org/10.1051/0004-6361/201834134/40MNRAS. 4051212Lodato, G., & Price, D. J., 2010, MNRAS, 405, 1212
. S H Lubow, R G Martin, C Nixon, http:/linker.aanda.org/10.1051/0004-6361/201834134/41ApJ. 80096Lubow, S. H., Martin, R. G., & Nixon, C. 2015, ApJ, 800, 96
B Macintosh, J R Graham, P Ingraham, http:/linker.aanda.org/10.1051/0004-6361/201834134/42Proc. Natl. Acad. Sci. Natl. Acad. Sci11112661Macintosh, B., Graham, J. R., Ingraham, P., et al. 2014, Proc. Natl. Acad. Sci., 111, 12661
. S Marino, S Perez, S Casassus, http:/linker.aanda.org/10.1051/0004-6361/201834134/43ApJ. 79844Marino, S., Perez, S., & Casassus, S. 2015, ApJ, 798, L44
. I Martí-Vidal, W H T Vlemmings, S Muller, S Casey, http:/linker.aanda.org/10.1051/0004-6361/201834134/44A&A. 563136Martí-Vidal, I., Vlemmings, W. H. T., Muller, S., & Casey, S. 2014, A&A, 563, A136
J P Mcmullin, B Waters, D Schiebel, W Young, K Golap, http:/linker.aanda.org/10.1051/0004-6361/201834134/45Astronomical Data Analysis Software and Systems XVI. 376127McMullin, J. P., Waters, B., Schiebel, D., Young, W., Golap, K. 2007, Astro- nomical Data Analysis Software and Systems XVI, ASP Conf. Ser., 376, 127
. G Meeus, J Bouwman, C Dominik, L B F M Waters, A De Koter, http:/linker.aanda.org/10.1051/0004-6361/201834134/46A&A. 402767Meeus, G., Bouwman, J., Dominik, C., Waters, L. B. F. M., & de Koter, A. 2003, A&A, 402, 767
. M Min, T Stolker, C Dominik, M Benisty, http:/linker.aanda.org/10.1051/0004-6361/201834134/47A&A. 60410Min, M., Stolker, T., Dominik, C., & Benisty, M. 2017, A&A, 604, L10
. A Miotello, E F Van Dishoeck, J P Williams, http:/linker.aanda.org/10.1051/0004-6361/201834134/48A&A. 599113Miotello, A., van Dishoeck, E. F., Williams, J. P., et al. 2017, A&A, 599, A113
. J Menu, R Van Boekel, T Henning, http:/linker.aanda.org/10.1051/0004-6361/201834134/49A&A. 581107Menu, J., van Boekel, R., Henning, T., et al. 2015, A&A, 581, A107
. M Montesinos, N Cuello, http:/linker.aanda.org/10.1051/0004-6361/201834134/50MNRAS. 47535Montesinos, M., & Cuello, N. 2018, MNRAS, 475, L35
. M Montesinos, S Perez, S Casassus, http:/linker.aanda.org/10.1051/0004-6361/201834134/51ApJ. 8238Montesinos, M., Perez, S., Casassus, S., et al. 2016, ApJ, 823, L8
. J E Owen, D Lai, http:/linker.aanda.org/10.1051/0004-6361/201834134/52MNRAS. 4692834Owen, J. E., & Lai, D. 2017, MNRAS, 469, 2834
. S Perez, S Casassus, F Ménard, http:/linker.aanda.org/10.1051/0004-6361/201834134/53ApJ. 79885Perez, S., Casassus, S., Ménard, F., et al. 2015, ApJ, 798, 85
. S Pérez, S Casassus, P Benítez-Llambay, S Pfalzner, http:/linker.aanda.org/10.1051/0004-6361/201834134/55MNRAS. 480986ApJPérez, S., Casassus, S., & Benítez-Llambay, P. 2018, MNRAS, 480, L12 Pfalzner, S. 2003, ApJ, 592, 986
. C Pinte, F Ménard, G Duchêne, P Bastien, http:/linker.aanda.org/10.1051/0004-6361/201834134/56A&A. 459797Pinte, C., Ménard, F., Duchêne, G., & Bastien, P. 2006, A&A, 459, 797
. C Pinte, T J Harries, M Min, http:/linker.aanda.org/10.1051/0004-6361/201834134/57A&A. 498967Pinte, C., Harries, T. J., Min, M., et al. 2009, A&A, 498, 967
. C Pinte, W R F Dent, F Ménard, http:/linker.aanda.org/10.1051/0004-6361/201834134/58ApJ. 81625Pinte, C., Dent, W. R. F., Ménard, F., et al. 2016, ApJ, 816, 25
. D J Price, J Wurster, T S Tricco, http:/linker.aanda.org/10.1051/0004-6361/201834134/59Proc. Astron. Soc. Aust. 3531Price, D. J., Wurster, J., Tricco, T. S., et al. 2018a, Proc. Astron. Soc. Aust., 35, e031
. D J Price, N Cuello, C Pinte, http:/linker.aanda.org/10.1051/0004-6361/201834134/60MNRAS. 4771270Price, D. J., Cuello, N., Pinte, C., et al. 2018b, MNRAS, 477, 1270
. A C Quillen, P Varnière, I Minchev, A Frank, http:/linker.aanda.org/10.1051/0004-6361/201834134/61AJ. 1292481Quillen, A. C., Varnière, P., Minchev, I., & Frank, A. 2005, AJ, 129, 2481
. K A Rosenfeld, E Chiang, S M Andrews, http:/linker.aanda.org/10.1051/0004-6361/201834134/62ApJ. 78262Rosenfeld, K. A., Chiang, E., & Andrews, S. M. 2014, ApJ, 782, 62
. N I Shakura, R A Sunyaev, http:/linker.aanda.org/10.1051/0004-6361/201834134/63A&A. 24337Shakura, N. I., & Sunyaev, R. A., 1973, A&A, 24, 337
. O Stahl, S Casassus, T Wilson, http:/linker.aanda.org/10.1051/0004-6361/201834134/64A&A. 477865Stahl, O., Casassus, S., & Wilson, T. 2008, A&A, 477, 865
. S M Stammler, T Birnstiel, O Panić, C P Dullemond, C Dominik, http:/linker.aanda.org/10.1051/0004-6361/201834134/65A&A. 600140Stammler, S. M., Birnstiel, T., Panić, O., Dullemond, C. P., & Dominik, C. 2017, A&A, 600, A140
. T Stolker, C Dominik, H Avenhaus, http:/linker.aanda.org/10.1051/0004-6361/201834134/66A&A. 595113Stolker, T., Dominik, C., Avenhaus, H., et al. 2016, A&A, 595, A113
. N Van Der Marel, E F Van Dishoeck, S Bruderer, http:/linker.aanda.org/10.1051/0004-6361/201834134/67Science. 3401199van der Marel, N., van Dishoeck, E. F., Bruderer, S., et al. 2013, Science, 340, 1199
. N Van Der Marel, P Cazzoletti, P Pinilla, A Garufi, http:/linker.aanda.org/10.1051/0004-6361/201834134/68ApJ. 832178van der Marel, N., Cazzoletti, P., Pinilla, P., & Garufi, A. 2016, ApJ, 832, 178
G Van Der Plas, F Ménard, K Ward-Duong, http:/linker.aanda.org/10.1051/0004-6361/201834134/71A55 van der Plas. 81932A&Avan der Plas, G., Ménard, F., Ward-Duong, K., et al. 2016, ApJ, 819, 102 van der Plas, G., Ménard, F., Canovas, H., et al. 2017a, A&A, 607, A55 van der Plas, G., Wright, C. M., Ménard, F., et al. 2017b, A&A, 597, A32
. K Wagner, D Apai, M Kasper, M Robberto, http:/linker.aanda.org/10.1051/0004-6361/201834134/72ApJ. 8132Wagner, K., Apai, D., Kasper, M., & Robberto, M. 2015, ApJ, 813, L2
. K Wagner, R Dong, P Sheehan, http:/linker.aanda.org/10.1051/0004-6361/201834134/73ApJ. 854130Wagner, K., Dong, R., Sheehan, P., et al. 2018, ApJ, 854, 130
. J C Weingartner, B T Draine, http:/linker.aanda.org/10.1051/0004-6361/201834134/74BAAS. 327Weingartner, J. C., & Draine, B. T. 2000, BAAS, 32, 42.07
. J P Williams, W M J Best, http:/linker.aanda.org/10.1051/0004-6361/201834134/75ApJ. 78859Williams, J. P., & Best, W. M. J. 2014, ApJ, 788, 59
. J P Williams, L A Cieza, http:/linker.aanda.org/10.1051/0004-6361/201834134/76ARA&A. 4967Williams, J. P., & Cieza, L. A. 2011, ARA&A, 49, 67
. T L Wilson, R Rood, http:/linker.aanda.org/10.1051/0004-6361/201834134/77ARA&A. 32191Wilson, T. L., & Rood, R. 1994, ARA&A, 32, 191
| [] |
[] | [
"\nDimiter Ostrev\n\n"
] | [
"Dimiter Ostrev\n"
] | [] | This paper proposes and proves security of a QKD protocol which uses two-universal hashing instead of random sampling to estimate the number of bit flip and phase flip errors. This protocol dramatically outperforms previous QKD protocols for small block sizes. More generally, for the two-universal hashing QKD protocol, the difference between asymptotic and finite key rate decreases with the number n of qubits as cn −1 , where c depends on the security parameter. For comparison, the same difference decreases no faster than c ′ n −1/3 for an optimized protocol that uses random sampling and has the same asymptotic rate, where c ′ depends on the security parameter and the error rate.of the protocol output and adversary registers will be able to distinguish between the output and an ideal key. 5. Robustness: the amount and type of noise that the protocol can tolerate without aborting. In particular, the QKD protocol should be able to tolerate at the very least the imperfections of whatever quantum channel and entanglement source is used to implement the protocol.Existing QKD protocols and security proofs exhibit trade-offs between these parameters: improving the security or robustness of the protocol worsens the key rate. These trade-offs are particularly severe when the block size is small. The phenomenon that the key rate of a QKD protocol deteriorates significantly for small block sizes has been called finite size effect [13, Sections II-C and IX].The finite size effect has practical consequences in cases when the quantum phase of the protocol is particularly difficult to implement. As an example, consider the problem of using QKD between users who are far apart on the surface of the earth. The Micius satellite experiment [23] tried to solve this problem by using a satellite to distribute entangled photon pairs to two ground stations that are 1120km apart. However, sending entangled photon pairs from space to earth is difficult. In the Micius experiment, several nights of good weather had to pass until the ground stations accumulated sifted block size 3100. The error rate that the ground stations needed to tolerate was 4.51%. Reference [9] performed a state-of-the-art security analysis on this data, and concluded that security levels better than around 10 −6 lead to no secret key at all, while at security level 10 −6 , only six bits of secret key are extracted. Several nights, 6 bits of secret key, security level 10 −6 : this is not enough to fulfil the promise of QKD for high levels of security, and not enough to justify the complexity and cost of QKD equipment. Something else is needed.Where can further improvement be found? Finite key analysis for QKD protocols of the BBM92 type is already mature. Reference [9] managed to prove a slightly tighter upper bound on the tail probability for random sampling, and thus obtain a small improvement over reference[20]. However, this cannot continue much further: there are also lower bounds on the tail probability for random sampling.Can the equipment for transmitting entangled photon pairs from space to earth be improved by several orders of magnitude? Perhaps, but that appears not so easy to do, and would involve a major technological advance. This paper proposes a different path: to consider QKD protocols where Alice and Bob can apply CNOT gates in addition to single qubit measurements. Later discussion will make clear how this helps, but for now, focus on the widespread belief that protocols with single qubit operations are practical, while other QKD protocols are impractical. It is true that QKD protocols that use the CNOT gate are not easy to implement with present technology. However, this need not remain so in the future. Indeed, what is practical or not practical changes with time. Recall that when the BB84 protocol was published, technology for even single qubit operations was not available. The BB84 protocol started to become practical around two-three decades after its publication. Returning now | null | [
"https://arxiv.org/pdf/2109.06709v2.pdf"
] | 237,502,877 | 2109.06709 | da49ffa4712cb1b5d42c8b372761d36dfc015d37 |
10 Mar 2022
Dimiter Ostrev
10 Mar 2022arXiv:2109.06709v2 [quant-ph] QKD parameter estimation by two-universal hashing leads to faster convergence to the asymptotic rate
This paper proposes and proves security of a QKD protocol which uses two-universal hashing instead of random sampling to estimate the number of bit flip and phase flip errors. This protocol dramatically outperforms previous QKD protocols for small block sizes. More generally, for the two-universal hashing QKD protocol, the difference between asymptotic and finite key rate decreases with the number n of qubits as cn −1 , where c depends on the security parameter. For comparison, the same difference decreases no faster than c ′ n −1/3 for an optimized protocol that uses random sampling and has the same asymptotic rate, where c ′ depends on the security parameter and the error rate.of the protocol output and adversary registers will be able to distinguish between the output and an ideal key. 5. Robustness: the amount and type of noise that the protocol can tolerate without aborting. In particular, the QKD protocol should be able to tolerate at the very least the imperfections of whatever quantum channel and entanglement source is used to implement the protocol.Existing QKD protocols and security proofs exhibit trade-offs between these parameters: improving the security or robustness of the protocol worsens the key rate. These trade-offs are particularly severe when the block size is small. The phenomenon that the key rate of a QKD protocol deteriorates significantly for small block sizes has been called finite size effect [13, Sections II-C and IX].The finite size effect has practical consequences in cases when the quantum phase of the protocol is particularly difficult to implement. As an example, consider the problem of using QKD between users who are far apart on the surface of the earth. The Micius satellite experiment [23] tried to solve this problem by using a satellite to distribute entangled photon pairs to two ground stations that are 1120km apart. However, sending entangled photon pairs from space to earth is difficult. In the Micius experiment, several nights of good weather had to pass until the ground stations accumulated sifted block size 3100. The error rate that the ground stations needed to tolerate was 4.51%. Reference [9] performed a state-of-the-art security analysis on this data, and concluded that security levels better than around 10 −6 lead to no secret key at all, while at security level 10 −6 , only six bits of secret key are extracted. Several nights, 6 bits of secret key, security level 10 −6 : this is not enough to fulfil the promise of QKD for high levels of security, and not enough to justify the complexity and cost of QKD equipment. Something else is needed.Where can further improvement be found? Finite key analysis for QKD protocols of the BBM92 type is already mature. Reference [9] managed to prove a slightly tighter upper bound on the tail probability for random sampling, and thus obtain a small improvement over reference[20]. However, this cannot continue much further: there are also lower bounds on the tail probability for random sampling.Can the equipment for transmitting entangled photon pairs from space to earth be improved by several orders of magnitude? Perhaps, but that appears not so easy to do, and would involve a major technological advance. This paper proposes a different path: to consider QKD protocols where Alice and Bob can apply CNOT gates in addition to single qubit measurements. Later discussion will make clear how this helps, but for now, focus on the widespread belief that protocols with single qubit operations are practical, while other QKD protocols are impractical. It is true that QKD protocols that use the CNOT gate are not easy to implement with present technology. However, this need not remain so in the future. Indeed, what is practical or not practical changes with time. Recall that when the BB84 protocol was published, technology for even single qubit operations was not available. The BB84 protocol started to become practical around two-three decades after its publication. Returning now
Introduction
Quantum Key Distribution allows two users, Alice and Bob, to agree on a shared secret key using an authenticated classical channel and a completely insecure quantum channel. There are information theoretic security proofs for QKD protocols (for example [17,16,8,6,1,21,20] among many others). Quantum key distribution has also been realized experimentally and is commercially available. The rare combination of information theoretic security and practical achievability has attracted considerable attention to QKD.
A QKD protocol has several important parameters:
1. Block size: the number of pairs of qubits that Alice and Bob receive. Following [20,Part 1], this paper considers entanglement based protocols and defines the block size as the number of qubits after sifting.
2. Output size: the number of bits of secret key that the protocol produces.
3. Key rate: the ratio of output size to block size. The higher the key rate is, the more efficiently the protocol converts the available quantum resource to a secret key. 4. Security level: the distance of the output from an ideal secret key. The lower the security level, the better the guarantee that no future evolution to protocols involving the CNOT gate, note that many groups around the world are working on technologies to store and manipulate qubits, striving for better fidelity and more qubits. Thus, there is hope that within the next two-three decades, QKD protocols that use the CNOT gate will also become practical. In summary, the path of QKD protocols that use the CNOT gate also requires a technological advance. However, referring again to the Micius satellite example, it appears easier to make a moderate advance in CNOT gates and a moderate advance in the transmission of entangled photon pairs from space to earth, rather than to put the entire burden on only one of these approaches.
What kind of QKD protocols become possible if the restriction to single qubit operations is lifted, and the CNOT gate is allowed? How do they perform in comparison to protocols with only single qubit operations? This paper presents one QKD protocol that involves the use of CNOT gates: the two-universal hashing QKD protocol, and proves its security. The two-universal hashing QKD protocol is an entanglement based protocol with block size n, that can tolerate any r bit flip errors and any r phase flip errors, and at the end extract n − 2⌈nh(r/n) + 2 log 2 (1/ǫ) + 5⌉ secret key bits, that are ǫ close to an ideal secret key.
For small block sizes, the two-universal hashing QKD protocol dramatically outperforms protocols of the BBM92 type. To illustrate, consider again the security analysis developed in the sequence of papers [21,20,9] applied to the Micius satellite example.
1. Fix the tolerated error rate at 4.51%, the security level at 10 −6 and the output size at 6 bits. The BBM92 type protocol with the security proof developed in [21,20,9] requires block size 3100. The two-universal hashing protocol requires block size 200.
2. Fix the block size at 3100 and fix the error rate at 4.51%. The BBM92 type protocol with the security proof developed in [21,20,9] can extract 6 secret key bits with security level 10 −6 . The two universal hashing protocol can extract 385 secret key bits with security level 10 −80 .
The advantage of the two-universal hashing QKD protocol is particularly noticeable for small block sizes; however, it is not limited to them. For fixed error rate δ = r/n and fixed security parameter ǫ, the asymptotic rate of this protocol is 1 − 2h(δ), and the deviation of finite from asymptotic rate is between (4 log 2 (1/ǫ)+10)/n and (4 log 2 (1/ǫ)+12)/n. By contrast, the deviation of finite from asymptotic key rate for the BBM92 type protocol with the security proof [21,20,9] is of the form cn −1/3 , where c depends on the tolerated error rate and the security level.
What is different about the two-universal hashing protocol and what causes the dramatic improvement in performance? How does the use of CNOT gates help? To start, note that no classical protocol can distinguish between inputs that are suitable for the extraction of a secret key and inputs that are not suitable. The ability to "detect the presence of an eavesdropper" is a uniquely quantum feature and is the central insight that makes QKD possible. This is the task of parameter estimation. The two-universal hashing protocol performs parameter estimation differently from previous protocols. Indeed, it seems natural to expect that in order to perform better the uniquely quantum task of distinguishing suitable from unsuitable inputs, it is advantageous to allow more general quantum operations for Alice and Bob.
The next few paragraphs explain the disadvantages of parameter estimation as performed in QKD protocols with only single qubit operations. For the purpose of this high level discussion, define parameter estimation as a two party LOCC protocol which performs a partial measurement on the input state and outputs a decision: to accept or reject, and outputs a promise on the postmeasurement state in case of acceptance. Parameter estimation protocols can differ in the class of input states on which they accept, the number of ebits they consume, the precision of the promise they provide in case of acceptance, and the probability that parameter estimation accepts but the promise on the post-measurement state does not hold.
Previous QKD protocols perform parameter estimation by random sampling: a random subset of n pe positions is measured and the outcomes are publicly compared. If the error rate on these positions is below some threshold δ, then parameter estimation accepts and outputs the promise that the error rate on the remaining positions is at most δ + ν, where ν is the gap between observed and inferred error rate.
A significant advantage of parameter estimation by random sampling is that it can be implemented with only single qubit operations for Alice and Bob. Unfortunately, this is also the only advantage of random sampling. The promise that the error rate on the remaining positions is at most δ + ν is very weak: n pe ebits have already been sacrificed for parameter estimation, and now further ebits have to be sacrificed for information reconciliation and privacy amplification. The failure probability scales roughly as exp(−4n pe v 2 ), which is also very weak, despite the fact that it involves an exponential function. To see this, suppose that the target failure probability is e −100 and that the target gap is ν = 0.01. Then, n pe has to be chosen to be 250000, clearly orders of magnitude more than can be afforded for block sizes around 1000 or 10000.
By contrast, for the two-universal hashing protocol, 2k ebits are sacrificed for parameter estimation. If the test passes, then Alice and Bob know that the post-parameter-estimation state is a particular Bell state of n − 2k ebits; thus, Alice and Bob do not need to sacrifice any further ebits for information reconciliation and privacy amplification. Moreover, the scaling of the failure probability for parameter estimation with the number of sacrificed ebits does not have the ν 2 coefficient in front of the number of sacrificed ebits.
To obtain such a parameter estimation protocol, the present paper builds on a number of previous ideas. The idea that two universal hashing can be used to estimate the number of errors is partially present in the protocols [21,6], and [6] attributes it to the earlier work [10]. In these protocols, the number of errors in one of the measurement bases is estimated by random sampling, while for the other basis there is a two-universal hash in the information reconciliation phase that is used to ensure correctness. This is also related to the observation [2,Theorem 6], [16,Section 6.3.2] that two-universal hash functions can be used to achieve information reconciliation with minimum leakage.
A combination of several ideas leads to the extension of the use of twouniversal hashing from information reconciliation to a full QKD protocol. Specifically, these ideas are: random matrices over the field with two elements are a two-universal hash family [5], and they are also parity check matrices of classical linear error-correcting codes. Classical linear codes can be used to construct quantum CSS codes [4,18], and CSS codes can be used to design and prove security of QKD protocols [17]. The present paper also uses a number of technical lemmas related to the stabilizer formalism [7,3]. Finally, [1] translates the guarantees of classical random sampling to the quantum case. This served as inspiration for the present paper, which translates the guarantees of classical two-universal hashing to the quantum case.
Another group of related works are those that prove security of classical privacy amplification by arguing that it corresponds to a virtual phase error correction, such as [8,6]. However, note that privacy amplification is a classical protocol for Alice and Bob. As such, privacy amplification requires a promise on the input state to operate, and if an unsuitable input is given, it produces an insecure key. With this in mind, note that the use of virtual phase error correction to prove security of classical privacy amplification is possible, but is not strictly necessary: other proofs of security for privacy amplification exist, for example in [16]. When it comes to the uniquely quantum task of distinguishing suitable from unsuitable inputs, [8,6] resort to the same technique as all other QKD protocols: single qubit operations and random sampling tail bounds. By contrast, the two-universal hashing QKD protocol takes an input state prepared by the adversary, correctly identifies whether the state is suitable or not, and either produces a secure key or aborts. Here, random linear functions and the stabilizer formalism are used to perform the uniquely quantum task: distinguish suitable from unsuitable inputs.
The rest of this paper is structured as follows: Section 2 introduces material that is needed to present and prove the security of the two universal hashing QKD protocol, including the security and robustness criteria for QKD protocols, a number of useful lemmas related to the stabilizer formalism, the use of twouniversal hashing to obtain an optimal information reconciliation protocol, and a number of useful lemmas about random matrices over the field with two elements. Section 3 presents the two-universal hashing QKD protocol and shows that it is secure and robust. Section 4 shows that for fixed security level and tolerated error rate, the finite key rate converges to the asymptotic rate as cn −1 for two-universal hashing and as cn −1/3 for random sampling, where n is the block size. Section 5 concludes and gives some open problems.
Preliminaries
This section presents definitions and results that are used to state and prove the main result on the security and robustness of the two-universal hashing protocol. Subsection 2.1 recalls the standard security criterion for QKD. Then, subsection 2.2 contains a number of lemmas related to the stabilizer formalism; these are used during the security proof. Finally, subsection 2.3 contains lemmas related to two-universal hashing. Subsection 2.3 also discusses an application of two-universal hashing to approximately compute certain functions from partial information about the input; this is used during the security proof.
Security and robustness of quantum key distribution
This section recalls the security and robustness criteria from [16] that ensure that a the key produced by QKD can be used in any application. See [15] for a proof of the equivalence of this security criterion and security in the Abstract Cryptography framework for composable security.
As is common in the QKD literature, this paper assumes that the adversary Eve is active in the quantum phase of the protocol but remains passive during the classical phase, i.e. Eve eavesdrops the classical communication but does not attempt to modify or block it. Under this assumption, an entanglement-based QKD protocol is a completely positive trace preserving map that transforms input states ρ ABE of Alice, Bob and Eve into output statesρ WAWB CE , where W A , W B are registers containing Alice and Bob's output: a secret key or indication ⊥ of protocol abort, and where C is a register containing a transcript of the classical communication between Alice and Bob.
Since registers W A , W B contain classical values, the final stateρ WAWB CE can be decomposed as
ρ WAWB CE = |⊥⊥ ⊥⊥| WAWB ⊗ρ CE (⊥)+ wA,wB |w A w B w A w B | WAWB ⊗ρ CE (w A , w B )
This decomposition is used to formulate the definition of security: Definition 1. A QKD protocol is ǫ secure if for all input states ρ ABE , the output stateρ WAWB CE is ǫ-close in trace distance to the corresponding ideal state
|⊥⊥ ⊥⊥| WAWB ⊗ρ CE (⊥) + w 1 |W | |ww ww| WA WB ⊗ (ρ CE −ρ CE (⊥))
where |W | denotes the size of the secret key space.
Alternatively, ǫ-security can be further subdivided into requirements for secrecy and correctness:
Definition 2. A QKD protocol is ǫ correct if for all input states ρ ABE , the probability P r(W A = W B ) = wA =wB T r(ρ CE (w A , w B ))
that Alice and Bob accept and output different keys is bounded by ǫ.
Definition 3. Alice's key is ǫ secret if for all input states ρ ABE , the reduced output stateρ WACE is ǫ-close in trace distance to the corresponding ideal state
|⊥ ⊥| WA ⊗ρ CE (⊥) + w 1 |W | |w w| WA ⊗ (ρ CE −ρ CE (⊥))
The following lemma establishes the relation between security and correctness plus secrecy: Lemma 1. If a QKD protocol is ǫ secure, then it is ǫ correct and Alice's key is ǫ secret. Conversely, if the protocol is ǫ correct and Alice's key is δ secret, then the protocol is ǫ + δ secure.
Proof. The forward direction follows from monotonicity of the trace distance and its interpretation as distinguishing advantage. The reverse direction follows by considering the hybrid state
|⊥⊥ ⊥⊥| WAWB ⊗ρ CE (⊥) + w |ww ww| WA WB ⊗ w ′ρ CE (w, w ′ )
and the triangle inequality.
Next, note that in the standard definition of QKD security (Definition 1) the ideal state, beyond being ǫ close to the real state, satisfies the following additional conditions:
1. The probabilities of accepting and rejecting are the same for the real and ideal state.
2. The real and ideal state differ only in the accept case.
3. The sub-normalized reduced density matrix of registers C, E in the accept case is equal toρ CE −ρ CE (⊥) for both the real and the ideal state. Now, suppose an ideal state is found that is ǫ close to the real state, but which does not necessarily satisfy these additional conditions. This suffices to demonstrate security:
Lemma 2.
Suppose that for all input states ρ ABE , there exist positive σ accept CE and σ reject CE such that T r(σ accept CE ) + T r(σ reject CE ) = 1 and such that the output stateρ WAWB CE is ǫ-close in trace distance to
|⊥⊥ ⊥⊥| WAWB ⊗ σ reject CE + w 1 |W | |ww ww| WA WB ⊗ σ accept CE
Then, the protocol is 2ǫ secure.
Proof. By assumption,
1 2 ρ CE (⊥) − σ reject CE 1 + wA,wB ρ CE (w A , w B ) − 1 |W | 1(w A = w B )σ accept CE 1 ≤ ǫ
From the triangle inequality it follows that
1 2 ρ CE (⊥) − σ reject CE 1 + ρ CE −ρ CE (⊥) − σ accept CE 1 ≤ ǫ
The lemma then follows by another application of the triangle inequality.
Finally, note that a protocol that always aborts is secure, but not useful. For a useful QKD protocol, the probability of acceptance is bounded below by 1 − δ for some δ ∈ (0, 1) on a suitable class of input states. In the present paper, robustness of the two-universal hashing protocol is shown by giving explicit bounds on the probability of acceptance as a function of the input state.
The Pauli group and the Bell basis
Denote the Pauli matrices by
σ 1 = 0 1 1 0 , σ 2 = 0 −i i 0 , σ 3 = 1 0 0 −1 For a row vector u ∈ F 1×n 2 , denote σ u 1 = σ u1 1 ⊗ . . . σ un 1 , σ u 3 = σ u1 3 ⊗ · · · ⊗ σ un 3
The Pauli group on n qubits is
G n = {ωσ u 1 σ v 3 : ω ∈ {±1, ±i}, u, v ∈ F 1×n 2 }
Matrix multiplication of elements of G n can be performed in terms of u, v, ω:
(ωσ u 1 σ v 3 )(ω ′ σ u ′ 1 σ v ′ 3 ) = ωω ′ (−1) v·u ′ σ u+u ′ 1 σ v+v ′ 3
This also shows that the map F : G n → F 1×2n 2 given by
F (ωσ u 1 σ v 3 ) = u v
is a group homomorphism. Any element of the Pauli group squares to either I or −I; any two elements g, g ′ of the Pauli group satisfy
gg ′ = (−1) F (g)SF (g ′ ) T g ′ g where S ∈ F 2n×2n 2 is the matrix with block form S = 0 I n I n 0
Say that a tuple of elements of the Pauli group
g = g 1 . . . g m is independent if the row vectors F (g i ) ∈ F 1×2n 2
are linearly independent. Given such an independent tuple and given any x ∈ F m 2 , it is possible to find g ∈ G n such that ∀i, gg i = (−1) xi g i g by solving the corresponding linear system of equations over F 2 . A tuple of independent commuting self-adjoint elements of the Pauli group g = (g 1 , . . . g m ) T defines a projective measurement on its joint eigenspaces. The measurement outcomes can be indexed by x ∈ F m 2 and the corresponding projections are given by
P ( g, x) = 2 −m m j=1 (I + (−1) xj g j )
The projections P ( g, x) form a complete set of orthogonal projections. The elements of the Pauli group map these projections to each other under conjugation, as can be seen from Lemma 3 below. Therefore, the projections P ( g, x) all have the same rank 2 n−m .
Lemma 3. For all tuples g = (g 1 . . . g m ) T of independent commuting self- adjoint elements of G n , for all h ∈ G n , for all x ∈ F m 2 , P ( g, x)h = hP ( g, x + F ( g)SF (h) T ) where F ( g) = F (g 1 ) . . . F (g m )
is the matrix with rows F (g 1 ), . . . , F (g m ).
Proof.
P ( g, x)h = 2 −m m j=1 (I + (−1) xj g j ) h = 2 −m h m j=1 (I + (−1) xj+F (gj )SF (h) T g j ) = hP ( g, x + F ( g)SF (h) T )
Now, take a tuple g of m independent commuting self-adjoint elements, take k ≤ m and take a full rank matrix L ∈ F k×m 2 . The matrix L transforms the tuple g to the k-tuple
L g = L g 1 . . . g m = m j=1 g L1j j . . . m j=1 g L kj j
The tuple L g also consists of independent commuting self-adjoint elements. The transformation of g to L g satisfies
M (L g) = (M L) g
for any g, L, M of compatible size. The matrix F (L g) can be expressed in terms of the matrix F ( g):
F (L g) = F ( m j=1 g L1j j ) . . . F ( m j=1 g L kj j ) = m j=1 L 1j F (g j ) . . . m j=1 L kj F (g j ) = LF ( g)
The measurement projections of L g can be expressed in terms of the measurement projections of g.
Lemma 4. For all n ≥ m ≥ k ≥ 1, for all tuples g of m independent commuting self-adjoint elements of G n , for all full rank L ∈ F k×m 2 , for all y ∈ F k 2 ,
P (L g, y) = x∈F m 2 :Lx=y P ( g, x) Proof. Take any i ∈ {1, . . . k}, any x ∈ F m 2 such that Lx = y. Then, m j=1 g Lij j P ( g, x) = m j=1 g Lij j 2 −m m j=1 (I + (−1) xj g j ) = (−1) m j=1 Lij xj P ( g, x) = (−1) yi P ( g, x)
Then, for any x ∈ F m 2 such that Lx = y, P (L g, y)P ( g, x) = P ( g, x) holds. Since {P ( g, x) : Lx = y} is a collection of 2 m−k orthogonal projections of rank 2 n−m and since P (L g, y) has rank 2 n−k , the lemma follows.
The maximally entangled state in C 2 n ⊗ C 2 n is
|ψ = 2 −n/2 z∈F n 2 |zz
The collection
|ψ αβ = I ⊗ σ α T 1 σ β T 3 |ψ , α, β ∈ F n 2 is the Bell basis of C 2 n ⊗ C 2 n .
First, the maximally entangled state has the properties:
Lemma 5. For all matrices M ∈ C 2 n ×2 n , M ⊗ I|ψ = I ⊗ M T |ψ and ψ|I ⊗ M |ψ = 2 −n T r(M ).
Proof. Follows by expanding M in the computational basis.
Pauli group measurements acting on Bell basis states satisfy the following:
Lemma 6. For all tuples g of independent self-adjoint commuting elements of G n such that the associated projections P ( g, x) have only real entries when expressed as matrices in the computational basis, for all α, β ∈ F n 2 , for all x, y ∈ F m 2 ,
(P ( g, x) ⊗ P ( g, y))|ψ αβ = 1 x = y + F ( g)S α β P ( g, x) ⊗ I|ψ αβ
where for an expression that takes the values true or false, 1(expression) takes the corresponding values 1 or 0.
Proof. Follows from Lemma 3 and the relation
M ⊗ I|ψ = I ⊗ M T |ψ
The QKD security proof also uses the following lemma. It gives two equivalent expressions for the projection on the subspace of C 2 n ⊗C 2 n that corresponds to a specific pattern of bit flip errors or a specific pattern of phase flip errors.
Lemma 7. For all n, for all α, β ∈ F n 2 , β ′ ∈F n 2 |ψ αβ ′ ψ αβ ′ | = zA∈F n 2 |z A , z A + α z A , z A + α| α ′ ∈F n 2 |ψ α ′ β ψ α ′ β | = xA∈F n 2 H ⊗2n |x A , x A + β x A , x A + β|H ⊗2n
Proof
|αβ αβ| AB = P σ A 3 σ B 3 , α β ; |ψ αβ ψ αβ | = P σ AB 3 σ AB 1 , α β
The first relation of Lemma 7 now follows from
I I σ A 3 σ B 3 = I 0 σ AB 3 σ AB 1
and Lemma 4. The second relation follows similarly.
Approximately computing certain functions from only a two-universal hash of the input
Let F 2 denote the field with two elements and F n 2 the n-dimensional vector space over this field. Take any subset S ⊂ F n 2 . Consider the function f S : F n 2 → S∪{⊥} given by
f S (α) = α if α ∈ S ⊥ otherwise
If α specifies errors, then f S computes whether α belongs to a set S of acceptable errors, if so computes the entire string α, and otherwise outputs an error message. It is very convenient to have functions of this form when constructing QKD protocols and security proofs. It turns out that it is possible to approximately compute f S (α) given only a two universal hash of the input. Recall [5,22]: Definition 4. A family of functions H from finite set X to finite set Y is two-universal with collision probability at most ǫ if for all x = x ′ ∈ X,
Pr h←H (h(x) = h(x ′ )) ≤ ǫ
where the probability is taken over h chosen uniformly from H. If no explicit value is specified for the collision probability bound, then the default value ǫ = 1/|Y| is taken. Pr h←H (f S (α) = g S (h, h(α))) ≤ ǫ|S|
Proof. The event f S (α) = g S (h, h(α)) implies the event ∃s ∈ S\{α} : h(s) = h(α)
The union bound and Definition 4 give
Pr h←H (f S (α) = g S (h, h(α))) ≤ ǫ|S|
The remainder of this section specializes Theorem 1 to the case that the family H is a family of matrices over F 2 , and the set S is a Hamming Ball.
First, consider the following useful lemmas about random matrices over the field with two elements. Let F n×k 2 to denote the space of n by k matrices over
F 2 .
Recall a property of random linear functions that was observed in [5]:
Lemma 8. Let L be uniformly random in F k×n 2 , and take any fixed x ∈ F n 2 −{0}. Then, Pr L (Lx = 0) = 2 −k .
Proof. Take i such that x i = 1. Then, Lx = L i + L −i x −i , where L i is the i-th column of L and where L −i , x −i are formed from L, x by omitting the i-th column and i-th entry respectively. Now, L i is uniform over F k 2 and independent from L −i , so Lx is also uniform over F k 2 .
Thus, for all y = z ∈ F n 2 , Pr L (Ly = Lz) = 2 −k , so random linear functions are two-universal.
Later on, it will be more convenient to select matrices not from all of F k×n 2 , but from the subset consisting of those matrices of rank k. This subset also satisfies the two-universal condition, as the following two lemmas show.
Pr(LM −1 M x = 0) = k i=1 (2 n−1 − 2 i−1 ) k i=1 (2 n − 2 i−1 ) = 2 n−k − 1 2 n − 1 < 2 −k
completing the proof of Lemma 10.
Interestingly, the collision probability bound ǫ = 2 n−k −1 2 n −1 achieved by the full rank matrices is the lowest possible for a two-universal family F n 2 → F k 2 . This follows from a slight strengthening of [5, Proposition 1]:
Lemma 11. For every family H (not necessarily two-universal) of functions from finite set X to finite set Y, there exist x = x ′ ∈ X such that
Pr h←H (h(x) = h(x ′ )) ≥ |X| |Y| − 1 |X| − 1
Proof. Follow the same proof as [5] until the point they apply the pigeonhole principle. At that point, observe that the number of non-zero terms in the sum is not only less than |X| 2 , as they say there, but is in fact at most |X|(|X| − 1).
In more detail, for h ∈ H, x, x ′ ∈ X, define
δ h (x, x ′ ) = 1 if x = x ′ ∧ h(x) = h(x ′ ) 0 otherwise For every h ∈ H partition X = ∪ y∈Y h −1 (y) then observe that x,x ′ ∈X δ h (x, x ′ ) = y∈Y |h −1 (y)|(|h −1 (y)| − 1) ≥ |X| 2 |Y| − |X|
by the quadratic mean-arithmetic mean inequality. Now, sum over h ∈ H:
h∈H x,x ′ ∈X δ h (x, x ′ ) = x,x ′ ∈X h∈H δ h (x, x ′ ) ≥ |H|( |X| 2 |Y| − |X|) Now, h∈H δ h (x, x ′ ) is non-zero only when x = x ′ . Then, there exist x = x ′ such that h∈H δ h (x, x ′ ) ≥ |H| |X| |Y| − 1 |X| − 1
Later results will also use the fact that a row submatrix of a random invertible matrix has the uniform distribution over full rank matrices: Proof. Pick any fixed full rank Λ ∈ F k×n 2 . Compute Pr(L S = Λ) as the number of ways to choose the remaining rows of L, which is n−k i=1 (2 n − 2 k+i−1 ) divided by the number of invertible matrices in F n×n 2 , which is n i=1 (2 n − 2 i−1 ). Thus,
Pr(L S = Λ) = n−k i=1 (2 n − 2 k+i−1 ) n i=1 (2 n − 2 i−1 ) = 1 k i=1 (2 n − 2 i−1 )
Thus, L S is uniform over the full rank matrices in F k×n 2 .
Applying Theorem 1 when the set S is a Hamming ball requires a bound on the size of Hamming balls. For x, y ∈ F n 2 , let d H (x, y) = |{i : x i = y i }| denote the Hamming distance between them. Let B n (x, r) denote the Hamming ball of radius r around x. Then:
Lemma 13. For all n, r ∈ N such that 2r ≤ n, for all x ∈ F n 2 , |B n (x, r)| < 2 nh(r/n) Proof.
The two-universal hashing QKD protocol and its security
Consider the following family π(n, k, r) of entanglement-based QKD protocols, parameterized by n, k, r ∈ N. The interpretation of the parameters is the following: n is the number of qubits that each of Alice and Bob receive, k is the size of each of their syndrome measurements and n − 2k is the size of their output secret key, and r is the maximum number of bit flip or phase flip errors on which the protocol does not abort. The protocols output a secret key with security guarantees when 2nh(r/n) < 2k < n. It will be clear throughout that the size of the two syndrome measurements can vary independently, and so can the maximum number of tolerated bit flip and phase flip errors, but that would lead to overly complex notation, with five parameters n, k, k ′ , r, r ′ , so it is not pursued explicitly below.
1. Alice and Bob each receive an n qubit state from Eve, and they inform each other that the states have been received. 3. Alice applies the isometry z |z, L 1 z AU ′ A z| A and Bob applies the isometry z |z, L 1 z BU ′ B z| B . This can be done by preparing k ancilla qubits in state 0 and applying a CNOT gate for each entry L 1 (i, j) that equals 1.
Alice and
Alice and Bob measure all qubits in registers
A, B in the |+ , |− basis,
obtaining outcomes x A , x B . Alice and Bob measure all qubits in registers U ′ A , U ′ B in the computational basis, obtaining outcomes u A , u B .
Alice and Bob compute v
A = M 2 x A , v B = M 2 x b , w A = M 3 x A , w B = M 3 x B .
Alice and Bob discard registers
(L 1 , u A + u B ) and t = g Bn(0,r) (M 2 , v A + v B ).
9. If both of these are not ⊥, then Alice takes w A to be the output secret key, and Bob takes w B + M 3 t to be the output secret key.
As is usual in the literature on QKD, the protocol assumes that classical communication takes place over an authenticated channel. Unconditionally secure message authentication with composable security in the Abstract Cryptography framework can be obtained from a short secret key [14], or using an advantage in channel noise [12].
If it is desired that the classical communication is minimized, then the following exchange of messages suffices: Bob confirms to Alice that he has received the qubits, Alice sends to Bob L, u A , v A , Bob informs Alice whether both of s, t are not ⊥. However, the initial formulation above better emphasizes the symmetry of the protocol, and makes clear that it is not important to keep the values u B , v B , s, t secret.
The following theorem establishes the security and robustness of the protocols π(n, k, r). Theorem 2. Take any n, k, r ∈ N such that 2nh(r/n) < 2k < n. Then, the protocol π(n, k, r) is 2 −k/2+nh(r/n)/2+5/2 secure.
Moreover, for any input state ρ AB , the probability that π(n, k, r) accepts on input ρ AB is 2 −k/2+nh(r/n)/2+3/2 close to T r(Π n,r ρ AB Π n,r ), where Π n,r is the projection on the subspace of systems AB spanned by the Bell states with at most r bit flip and at most r phase flip errors.
Proof of Theorem 2
The main idea of the proof of Theorem 2 is that the real values g Bn(0,r) (L 1 , u A + u B ) and g Bn(0,r) (M 2 , v A + v B ) computed during the protocol can be replaced by the corresponding ideal values f Bn(0,r) (α), f Bn(0,r) (β). From now on, use shorthand notation and skip the subscript B n (0, r), thus writing f for f Bn(0,r) and g for g Bn(0,r) .
The steps of the proof of Theorem 2 are the propositions below. Start by writing the action of the protocol as an isometry followed by a partial trace. Proposition 1. Let E real be the completely positive trace preserving transformation applied by the first eight steps of the protocol. Then, for all input states ρ ABE to the protocol, the output state E real (ρ ABE ) of the classical registers L, U A , U B , V A , V B , W A , W B , S, T and the quantum register of Eve equals
T r ABL ′ S ′ T ′ U ′ A U ′ B V ′ A V ′ B W ′ A W ′ B WV real U real (ρ ⊗ |L L|) U † real V † real W † where |L = L √ p L |LL LL ′ is a purification of the choice of random matrix L, where U Real = L,zA,zB |L L| L ⊗ |z A z B z A z B | AB ⊗|L 1 z A , L 1 z A , L 1 z B , L 1 z B , g(L 1 , L 1 (z A +z B )), g(L 1 , L 1 (z A +z B )) UA U ′ A UB U ′ B SS ′
is an isometry that captures the measurement through which Alice and Bob obtain the values u A = L 1 z A and u B = L 1 z B as well as the subsequent computation of the value s = g(L 1 ,
L 1 (z A + z B )), where V Real = L,xA,xB |L L| L ⊗ H ⊗2n |x A x B x A x B |H ⊗2n AB ⊗|M 2 x A , M 2 x A , M 2 x B , M 2 x B , g(M 2 , M 2 (x A +x B )), g(M 2 , M 2 (x A +x B )) VA V ′ A VB V ′ B T T ′|L L| L ⊗ H ⊗2n |x A x B x A x B |H ⊗2n AB ⊗ |M 3 x A , M 3 x A , M 3 x B , M 3 x B WAW ′ A WB W ′ B
is an isometry that captures the measurement through which Alice and Bob ob-
tain the values w A = M 3 x A and w B = M 3 x B .
Proof. Recall the Stinespring dilation theorem [19]. Systematically express each step of the protocol as an isometry followed by a partial trace. The step in which Alice and Bob choose the random matrix L can be expressed as preparing the purification |L LL ′ and then taking T r L ′ .
The steps in which Alice and Bob apply the isometry
zA,zB |z A , z B , L 1 z A , L 1 z B ABU ′ A U ′ B z A , z B | AB
then measure registers U ′ A , U ′ B in the computational basis, discarding the postmeasurement state and keeping only the outcome, then compute the value s can be expressed by the isometry U real followed by T r S ′ U ′
A U ′ B .
The steps in which Alice and Bob measure the qubits in A, B in the |+ , |− basis obtaining x A , x B , then compute v A , v B , w A , w B , t, then discard the postmeasurement state of the qubits in A, B and the outcomes x A , x B can be expressed by the product of isometries WV real followed by T r ABT ′ V ′
A V ′ B W ′ A W ′ B .
Finally, note that all the partial trace operations can be commuted to the end.
Next, note that U real can be approximated by an ideal isometry followed by a simulator isometry.
Proposition 2. Let U ideal = α,β |ψ αβ ψ αβ | AB ⊗ |f (α), f (α) SS ′
This ideal isometry computes whether the number of bit flip errors is acceptable and if so it computes the entire string of bit flip error positions.
Let
U simulator = L,zA,zB |L L| L ⊗|z A z B z A z B | AB ⊗|L 1 z A , L 1 z A , L 1 z B , L 1 z B UAU ′ A UB U ′ B
This isometry captures the measurement through which Alice and Bob obtain the values u A = L 1 z A and u B = L 1 z B . Then:
L|U † ideal U † simulator (U real |L ) ≥ (1 − 2 −k+nh(r/n) )I AB Proof. Simplify: U † simulator U real = L,zA,zB |L L| L ⊗ |z A z B z A z B | AB ⊗ |g(L 1 , L 1 (z A + z B )), g(L 1 , L 1 (z A + z B )) SS ′ Therefore, L|U † ideal U † simulator (U real |L ) = L,zA,zB ,α,β p L |ψ αβ ψ αβ | AB |z A z B z A z B | AB f (α)|g(L 1 , L 1 (z A + z B )) S
Now, apply Lemma 7:
L,zA,zB,α p L β |ψ αβ ψ αβ | AB |z A z B z A z B | AB f (α)|g(L 1 , L 1 (z A +z B )) S = L,zA,zB,α,z ′ A p L |z ′ A , z ′ A +α z ′ A , z ′ A +α| AB |z A z B z A z B | AB f (α)|g(L 1 , L 1 (z A +z B )) S = zA,zB |z A z B z A z B | AB L p L f (z A + z B )|g(L 1 , L 1 (z A + z B )) = zA,zB |z A z B z A z B | AB Pr L (f (z A + z B ) = g(L 1 , L 1 (z A + z B )))
Now, the marginal distribution of L 1 is uniform over the rank k matrices in F k×n 2 because L is selected uniformly among invertible matrices in F n×n 2 (Lemma 12). Complete the proof of Proposition 2 by applying Corollary 1.
Next, perform the same approximation for V real .
Proposition 3. Let V ideal = α,β |ψ αβ ψ αβ | AB ⊗ |f (β), f (β) T T ′|L L| L ⊗ H ⊗2n |x A x B x A x B |H ⊗2n AB ⊗ |M 2 x A , M 2 x A , M 2 x B , M 2 x B VAV ′ A VB V ′ B
This isometry captures the measurement through which Alice and Bob obtain the values v A = M 2 x A and v B = M 2 x B . Then:
L|V † ideal V † simulator (V real |L ) ≥ (1 − 2 −k+nh(r/n) )I AB Proof.
As in the proof of Proposition 2, use Lemma 7 to compute
L|V † ideal V † simulator (V real |L ) = xA,xB H ⊗2n |x A x B x A x B |H ⊗2n AB Pr L (f (x A +x B ) = g(M 2 , M 2 (x A +x B )))
Now, M = (L −1 ) T is uniformly distributed over invertible matrices in F n×n 2 , so Lemma 12 and Corollary 1 complete the proof.
Next, observe that:
Proposition 4. U simulator V real = V real U simulator
Proof. Rewrite:
U simulator = L,zA,zB |L L| L ⊗|z A z B z A z B | AB ⊗|L 1 z A , L 1 z A , L 1 z B , L 1 z B UAU ′ A UB U ′ B = L,uA,uB |L L| L ⊗ |u A , u A , u B , u B UAU ′ A UB U ′ B ⊗ zA:L1zA=uA |z A z A | A ⊗ zB:L1zB =uB |z B z B | B = L,uA,uB |L L| L ⊗|u A , u A , u B , u B UAU ′ A UB U ′ B ⊗P (L 1 ( σ 3 ), u A ) A ⊗P (L 1 ( σ 3 ), u B ) B
where the last step uses Lemma 4 and the notation of Section 2.2 for the tuple σ 3 of single qubit σ 3 operations.
Similarly, rewrite
V Real = L,xA,xB |L L| L ⊗ H ⊗2n |x A x B x A x B |H ⊗2n AB ⊗|M 2 x A , M 2 x A , M 2 x B , M 2 x B , g(M 2 , M 2 (x A +x B )), g(M 2 , M 2 (x A +x B )) VA V ′ A VB V ′ B T T ′ = L,vA,vB |L L| L ⊗ P (M 2 ( σ 1 ), v A ) A ⊗ P (M 2 ( σ 1 ), v B ) B ⊗ |v A , v A , v B , v B , g(M 2 , v A + v B ), g(M 2 , v A + v B VAV ′ A VB V ′ B T T ′
where σ 1 is the tuple of single qubit σ 1 operations. Proposition 4 now follows by observing that the elements of the two tuples L 1 ( σ 3 ) and M 2 ( σ 1 ) commute and therefore for all u, v, the corresponding projections P (L 1 ( σ 3 ), u) and P (M 2 ( σ 1 ), v) also commute.
Next, use propositions 1, 2, 3, 4 to construct an ideal transformation that approximates E real : Proposition 5. Let E ideal be the transformation that prepares |L , then applies isometries U ideal , V ideal , V simulator , U simulator , W, and finally applies
T r ABL ′ S ′ T ′ U ′ A U ′ B V ′ A V ′ B W ′ A W ′ B .
Then, the diamond distance of E real and E ideal is at most 2 −k/2+nh(r/n)/2+3/2 .
Proof. Take any input state ρ ABE and purify it to |φ ABEE ′ . From Proposition 2 deduce that the fidelity of V real U simulator U ideal |φ |L and V real U real |φ |L is at least 1 − 2 −k+nh(r/n) . Using the relation of fidelity and trace distance for pure states [11,Equation 9.99], the trace distance between these two states is Next, from Proposition 4 deduce V real U simulator U ideal |φ |L = U simulator V real U ideal |φ |L Next, from Proposition 3 deduce that the fidelity of U simulator V real U ideal |φ |L and U simulator V simulator V ideal U ideal |φ |L is at least 1−2 −k+nh(r/n) , so the trace distance between them is at most 2 −k/2+nh(r/n)/2+1/2 . Finally, from Proposition 1, the triangle inequality and monotonicity of the trace distance deduce that the trace distance between E real (ρ) and E ideal (ρ) is at most 2 −k/2+nh(r/n)/2+3/2 .
Next, compute the output state of E ideal : Proposition 6. Take any input state ρ ABE and purify it to |φ ABEE ′ . Expand φ in the Bell basis for Alice and Bob:
|φ ABEE ′ = α,p L |L L| L ⊗ |γ αβ γ αβ | EE ′ ⊗ |α, β α, β| ST ⊗ 2 −n |u A , v A , w A u A , v A , w A | UAVAWA ⊗ |u A + L 1 α, v A + M 2 β, w A + M 3 β u A + L 1 α, v A + M 2 β, w A + M 3 β| UB VB WB Proof. Simplify: V ideal U ideal = α,β |ψ αβ ψ αβ | ⊗ |f (α), f (α), f (β), f (β) SS ′ T T ′ Also, WV simulator U simulator = L,uA,uB ,vA,vB ,wA,wB |L L| L ⊗ P L 1 σ 3 M 2 σ 1 M 3 σ 1 , u A v A w A A ⊗ P L 1 σ 3 M 2 σ 1 M 3 σ 1 , u B v B w B B ⊗|u A , u A , u B , u B , v A , v A , v B , v B , w A , w A , w B , w B UAU ′ A UB U ′ B VAV ′ A VB V ′ B WAW ′ A WB W ′ B
using the notation of section 2.2, the observation that the elements of the three tuples L 1 σ 3 , M 2 σ 1 , M 3 σ 1 are independent and commute, and Lemma 4. Next, use Lemma 6 to deduce that
WV simulator U simulator V ideal U ideal |φ |L = L,uA,vA,wA,α,β √ p L |L, L LL ′ ⊗ P L 1 σ 3 M 2 σ 1 M 3 σ 1 , u A v A w A A |ψ αβ AB ⊗ |γ αβ EE ′ ⊗ |u A , u A , u A + L 1 α, u A + L 1 α UAU ′ A UB U ′ B ⊗ |v A , v A , v A + M 2 β, v A + M 2 β VAV ′ A VB V ′ B ⊗ |w A , w A , w A + M 3 β, w A + M 3 β WA W ′ A WB W ′ B ⊗ |f (α), f (α), f (β), f (β) SS ′ T T ′
Next, break this up into a sum of two sub-normalized vectors |τ accept and |τ reject , where |τ accept contains those terms of the sum with α, β ∈ B n (0, r) and |τ reject contains all other terms of the sum. Note that T r S ′ T ′ |τ accept τ reject | = 0 and deduce
E ideal (|φ φ|) = T r ABL ′ S ′ T ′ U ′ A U ′ B V ′ A V ′ B W ′ A W ′ B |τ accept τ accept | + T r ABL ′ S ′ T ′ U ′ A U ′ B V ′ A V ′ B W ′ A W ′ B |τ reject τ reject | Take σ reject LEE ′ ST UAVAWAUB VB WB = T r ABL ′ S ′ T ′ U ′ A U ′ B V ′ A V ′ B W ′ A W ′ B
|τ reject τ reject | Finally, simplify and use Lemma 5 to deduce that
T r ABL ′ S ′ T ′ U ′ A U ′ B V ′ A V ′ B W ′ A W ′ B
|τ accept τ accept | = L,uA,vA,wA,α,β:α,β∈Bn(0,r)
p L |L L| L ⊗ |γ αβ γ αβ | EE ′ ψ αβ |P L 1 σ 3 M 2 σ 1 M 3 σ 1 , u A v A w A A |ψ αβ ⊗ |u A , v A , w A u A , v A , w A | UAVAWA ⊗ |u A + L 1 α, v A + M 2 β, w A + M 3 β u A + L 1 α, v A + M 2 β, w A + M 3 β| UB VB WB ⊗ |α, β α, β| ST = L,p L |L L| L ⊗ |γ αβ γ αβ | EE ′ ⊗ |α, β α, β| ST ⊗ 2 −n |u A , v A , w A u A , v A , w A | UAVAWA ⊗ |u A + L 1 α, v A + M 2 β, w A + M 3 β u A + L 1 α, v A + M 2 β, w A + M 3 β| UB VB WB
which completes the proof.
Finally, note that for any input state ρ ABE , applying the final step of the protocol (the correction of w B ) to E ideal (ρ) produces an ideal state that satisfies the assumptions of Lemma 2 with ǫ = 2 −k/2+nh(r/n)/2+3/2 ; therefore the protocol is 2 −k/2+nh(r/n)/2+5/2 secure. Moreover, for any input ρ ABE = T r E ′ |φ φ| ABEE ′ , the probability that the protocol accepts is within 2 −k/2+nh(r/n)/2+3/2 of α,β∈Bn(0,r) γ αβ |γ αβ = T rΠ n,r ρ AB Π n,r This completes the proof of Theorem 2.
Comparison with previous work
The introduction illustrated the advantage of two-universal hashing over random sampling using specific examples. This section reveals the general pattern behind the examples in the introduction. To study the advantage of the twouniversal hashing protocol for all block sizes, fix values for the tolerated error rate and security level, and consider key rate as a function of block size. How fast does key rate converge to the asymptotic value as block size goes to infinity? Subsection 4.1 gives the rate of convergence for the two-universal hashing protocol. Subsection 4.2 gives a bound on the rate of convergence of the random sampling protocol.
4.1 Key rate of the two-universal hashing protocols π(n, k, r) Given n qubits per side, the target to tolerate δn bit flip and δn phase flip errors, and a target security parameter ǫ, it suffices to choose k = ⌈nh(δ) + 2 log 2 (1/ǫ) + 5⌉. The key rate 1 − 2k/n then satisfies:
1 − 2h(δ) − 4 log 2 (1/ǫ) + 12 n ≤ 1 − 2k n ≤ 1 − 2h(δ) − 4 log 2 (1/ǫ) + 10 n
Therefore, the rate of convergence of the finite to the asymptotic rate is of the form cn −1 .
Key rate of the random sampling protocols
The sequence of works [21,20,9] develops QKD protocols and security proofs optimized for the finite key regime. The current evolution of the entanglementbased protocol can be found in [20,Section 3]; the difference between [9] and [20] is only in the random sampling tail bound that is used. For comparison with the present work we take only the case of perfect measurements in the rectilinear and diagonal basis. A summary of the protocol in this case is as follows:
1. Eve prepares a state of 2n qubits and sends n to Alice and n to Bob.
2. Alice and Bob agree on a uniformly random choice of either the rectilinear or the diagonal basis measurement for each pair of qubits.
3. Alice and Bob select a uniformly random subset of n pe positions to serve for parameter estimation, leaving the remaining n rk = n − n pe to serve as the raw key.
4. Alice and Bob compare their outcomes on the parameter estimation positions. If the error rate on these positions exceeds a threshold δ, Alice and Bob abort.
5. Alice sends a syndrome of her raw key to Bob, and a two-universal hash of her raw key to Bob. Bob uses the syndrome to correct his raw key, and uses the hash to verify that the correction was successful. For simplicity, take the combined length of syndrome and hash to be the theoretical minimum n rk h(δ) − log 2 (ǫ ec ), where ǫ ec is the desired bound on the probability that the hash test passes but Bob's corrected raw key does not match Alice's.
6. Alice and Bob compress their raw keys to shorter output keys of length n out using a two-universal family of hash functions.
The security ǫ qkd of these protocols can be written in the form
ǫ qkd = ǫ ec + inf 0<ν<1/2−δ (ǫ pa (ν) + ǫ pe (ν))
where ǫ ec is the desired bound on the correctness of the protocol, where
ǫ pa (ν) = 1 2 √ ǫ ec 2 (−n rk (1−h(δ+ν)−h(δ))+nout)/2
is a bound on the secrecy of the protocol, and where
ǫ pe (ν) = inf 0<ξ<ν ǫ pe (ν, ξ)
comes from a tail bound for random sampling. The precise form of the function ǫ pe (ν, ξ) is given in [9, Lemma 2] and satisfies the equation
ǫ pe (ν, ξ) 2 2 = exp − 2nn pe ξ 2 n rk + 1 + exp − 2(n + 2)(n 2 rk (ν − ξ) 2 − 1) (n(δ + ξ) + 1)(n(1 − δ − ξ) + 1)
For the purpose of this section, consider the following lower bound on ǫ pe (ν): Lemma 14. Suppose n rk ≥ n/2. Then,
ǫ pe (ν) ≥ 2exp(−2n pe ν 2 )
Proof. Take any ξ ∈ (0, ν). Note that 2nn pe ξ 2 n rk + 1 ≤ 4n pe ν 2 and therefore exp − 2nn pe ξ 2 n rk + 1 ≥ exp −4n pe ν 2
The lemma follows.
The following bound holds on the key rate of the random sampling protocols:
Theorem 3. Fix the block size n, the tolerated error rate δ and the security level ǫ qkd = ǫ ec + inf 0<ν<1/2−δ (ǫ pa (ν) + ǫ pe (ν))
Then, the key rate n out /n is upper bounded by the larger of (1 − 2h(δ))/2 and Proof. Take the optimal ν. In case n rk /n < 1/2, then
n out n ≤ n rk (1 − h(δ + ν) − h(δ)) n ≤ 1 − 2h(δ) 2
Suppose now that n rk /n ≥ 1/2. Simplify the problem by eliminating ǫ ec : note that ǫ ec + ǫ pa (ν) = ǫ ec + 1 2 √ ǫ ec 2 (−n rk (1−h(δ+ν)−h(δ))+nout)/2 ≥ 3 2 4/3 2 (−n rk (1−h(δ+ν)−h(δ))+nout)/3 with equality if and only if ǫ ec = 1 2 4/3 2 (−n rk (1−h(δ+ν)−h(δ))+nout)/3
Use this and Lemma 14 to deduce 3 2 4/3 2 (−n rk (1−h(δ+ν)−h(δ))+nout)/3 + 2exp(−2n pe ν 2 ) ≤ ǫ qkd From this, deduce further: −n rk (1 − h(δ + ν) − h(δ)) + n out ≤ 3 log 2 ǫ qkd + 4 − 3 log 2 (3) −2n pe ν 2 ≤ ln(ǫ qkd ) − ln (2) Rewrite the first inequality as n out ≤ n(1−2h(δ))−n pe (1−2h(δ))−n rk (h(δ+ν)−h(δ))+3 log 2 ǫ qkd +4−3 log 2 (3)
(1) Now, apply the inequality a + b ≥ 3a 1/3 (b/2) 2/3 to the second and third term: n pe (1 − 2h(δ)) + n rk (h(δ + ν) − h(δ)) ≥ 3 2 2/3 n 1/3 pe (1 − 2h(δ)) 1/3 n 2/3 rk (h(δ + ν) − h(δ)) 2/3 ≥ 3 2 2/3 n 1/3 pe (1 − 2h(δ)) 1/3 (n/2) 2/3 (h(δ + ν) − h(δ)) 2/3
Further, use the line through (δ, h(δ)) and (1/2, 1) to obtain h(δ + ν) − h(δ) ≥ ν 1 − h(δ) 1/2 − δ then combine this with n pe ν 2 ≥ 0.5 ln(2/ǫ qkd ) to obtain
n 1/3 pe (h(δ + ν) − h(δ)) 2/3 ≥ 1 − h(δ) 1/2 − δ 2/3 1 2 ln 2 ǫ qkd 1/3
Thus, n pe (1 − 2h(δ)) + n rk (h(δ + ν) − h(δ)) Combining with (1) proves the Theorem.
Conclusion and open problems
The present paper has proposed and proved security of a QKD protocol that uses two-universal hashing instead of random sampling to perform the uniquely quantum task of distinguishing suitable from unsuitable inputs. This protocol dramatically outperforms previous QKD protocols for small block sizes. More generally, the speed convergence to the asymptotic rate for the two-universal hashing protocol is cn −1 , whereas for an optimized random sampling protocol, the speed of convergence is no faster than c ′ n −1/3 .
As discussed already in the introduction, random sampling protocols involve only single qubit preparation and measurement, whereas the two-universal hashing protocol presented here requires Alice and Bob also to be able to store qubits for a short time while they agree on the matrix L, and to apply CNOT gates. It appears that the use of two-qubit gates is necessary for the improved performance, but this has not yet been mathematically proven. Can the speed of convergence cn −1 be achieved using only single qubit operations, or is there some fundamental limit that prevents this? Another line of research related to the distinction between single and two-qubit quantum operations would be to develop quantum hardware capable of performing QKD protocols involving the CNOT gate.
Second, the algorithm given in section 2.3 for computing the function g Bn(0,r) is not efficient. This leads to the following open problem: is there a probability distribution over CSS codes, such that the marginal distributions of the two parity check matrices satisfy a two-universal hashing condition with some good collision probability bound, and such that each of the two parity check matrices has additional structure that allows efficient computation of g Bn(0,r) during the protocol? There is a long history in information theory of approximating the performance of random codes with brute force decoding by more structured codes with efficient decoding, so there is reason to hope that the same can be done in the present case.
Third, the arguments in the present paper are for the case where Alice and Bob can apply perfect quantum operations. It thus remains an open problem to generalize the present security proof to the case of imperfect devices.
Now, let H be a two-universal family from F n 2 to some finite set Y with collision probability bound ǫ.Let S = {s 1 , . . . , s m }. Consider the function g S : H × Y → S ∪ {⊥} given by the deterministic algorithm: 1. On input h, y, 2. For i = 1, . . . , m, if h(s i ) = y, output s i and stop.3. Output ⊥.Then:Theorem 1. For all n ∈ N, for all ǫ, for all two-universal families H : F n 2 → Y with collision probability bound ǫ, for all subsets S ⊂ F n 2 , for all α ∈ F n 2 ,
Lemma 9 .
9For all integers n ≥ k ≥ 1, the number of rank k matrices in F k×n2 is k i=1 (2 n − 2 i−1 )Proof. Given i − 1 linearly independent rows, there are 2 n − 2 i−1 ways to choose the i-th row outside their span.Lemma 10. Take k ≤ n, let L be a uniformly random rank k matrix in F k×n 2 and take anyx ∈ F n 2 − {0}. Then Pr L (Lx = 0) = 2 n−k −1 2 n −1 < 2 −k Proof. Take invertible M ∈ F n×n 2such that M x = (1, 0, . . . , 0) T . Then Pr(Lx = 0) = Pr(LM −1 M x = 0). Now, find the probability that the first column of LM −1 is zero. Note that LM −1 is also uniformly distributed over the rank k matrices in F k×n 2 , so the probability its first column is zero is the number of rank k matrices in F k×(n−1) 2 divided by the number of rank k matrices in F k×n 2 . Lemma 9 implies:
Lemma 12 .
12Take any integers n ≥ k ≥ 1, and any S ⊂ {1, . . . , n} of size k. Let L be uniformly distributed over invertible matrices in F n×n 2 . Let L S denote the matrix formed by rows of L with indices in S. Then, L S is uniformly distributed over full rank matrices in F k×n 2 .
Corollary 1 .
1For all n, k, r ∈ N with 2r ≤ n and k ≤ n, for all α ∈ F n 2 ,Pr L (f Bn(0,r) (α) = g Bn(0,r) (L, Lα)) < 2 −k+nh(r/n)where L is chosen uniformly from the full rank matrices in F k×n 2 .
Bob publicly choose a random invertible L ∈ F n×n 2 .
2Let L 1 , L 2 , L 3 be the matrices formed by the first k rows, the second k rows, and the last n − 2k rows of L. Let M = (L −1 ) T , and let M 1 , M 2 , M 3 be the matrices formed by the first k, second k, and last n − 2k rows of M . L 1 , M 2 are the parity check matrices of a CSS code. L 3 , M 3 contain information about the logical Z and X operators on the codespace.
is an isometry that captures the measurement through which Alice and Bob obtain the values v A = M 2 x A and v B = M 2 x B as well as the subsequent computation of the value t = g(M 2 , M 2 (x A + x B )) and where W = L,xA,xB
This ideal isometry computes whether the number of phase flip errors is acceptable and if so it computes the entire string of phase flip error positions. Let V simulator = L,xA,xB
β∈F n 2
2|ψ αβ AB ⊗ |γ αβ EE ′where |γ αβ are vectors in Eve's space that satisfy α,β∈F n 2 γ αβ |γ αβ = 1Then, there exists σ reject LEE ′ ST UAVAWAUB VB WB that is classical on registers LST U A V A W A U B V B W Band such that at least one of ST contains ⊥ and such that E ideal (|φ φ|) = σ reject LEE ′ ST UAVAWAUB VB WB + L,uA,vA,wA,α,β:α,β∈Bn(0,r)
( 1 −
12h(δ)) − c 1 (ǫ qkd , δ)n −1/3 − c 2 (ǫ qkd )n (ǫ qkd ) = 3 log 2 (1/ǫ qkd ) + 3 log 2 (3) − 4
. Let e 1 , . . . e n denote the standard basis of F 1×n 2 . For i ∈ {1, 3} and R ∈ {A, B}, let σ R 3 denote the tuple σ e1 i , . . . σ eni
acting on register R, and let
σ AB
i
denote the tuple σ e1
i ⊗ σ e1
i , . . . , σ en
i ⊗ σ en
i . Note that for all α, β,
Alice and Bob discard x A , x B , keeping only v A , v B , w A , w B . Thus, in effect, Alice and Bob erase M 1 x A , M 1 x B . Note that the post measurement states in registers A, B, as well as x A , x B have to be discarded in such a way that Eve cannot get them.A, B, U ′
A , U ′
B .
7. 8. Alice and Bob announce u A , u B , v A , v B . Alice and Bob compute s =
g Bn(0,r)
uA,vA,wA,α,β:α,β∈Bn(0,r)
− (1 − 2 −k+nh(r/n) ) 2 ≤ 2 −k/2+nh(r/n)/2+1/2
AcknowledgmentThis work was supported by the Luxembourg National Research Fund, under CORE project Q-CoDe (CORE17/IS/11689058). The author would like to thank Prof. Marco Tomamichel and two anonymous reviewers for helpful comments.
Sampling in a quantum population, and applications. J Niek, Serge Bouman, Fehr, Annual Cryptology Conference. SpringerNiek J Bouman and Serge Fehr. Sampling in a quantum population, and applications. In Annual Cryptology Conference, pages 724-741. Springer, 2010.
Secret-key reconciliation by public discussion. Gilles Brassard, Louis Salvail, Workshop on the Theory and Application of of Cryptographic Techniques. SpringerGilles Brassard and Louis Salvail. Secret-key reconciliation by public dis- cussion. In Workshop on the Theory and Application of of Cryptographic Techniques, pages 410-423. Springer, 1993.
Quantum error correction and orthogonal geometry. A R Calderbank, E M Rains, P W Shor, N J A Sloane, Phys. Rev. Lett. 78A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane. Quantum error correction and orthogonal geometry. Phys. Rev. Lett., 78:405-408, Jan 1997.
Good quantum error-correcting codes exist. Robert Calderbank, W Peter, Shor, Physical Review A. 5421098A Robert Calderbank and Peter W Shor. Good quantum error-correcting codes exist. Physical Review A, 54(2):1098, 1996.
Universal classes of hash functions. Lawrence Carter, Mark N Wegman, Journal of computer and system sciences. 182J Lawrence Carter and Mark N Wegman. Universal classes of hash func- tions. Journal of computer and system sciences, 18(2):143-154, 1979.
Practical issues in quantum-key-distribution postprocessing. Chi-Hang Fred Fung, Xiongfeng Ma, H F Chau, Physical Review A. 811Chi-Hang Fred Fung, Xiongfeng Ma, and H. F. Chau. Practical issues in quantum-key-distribution postprocessing. Physical Review A, 81(1), Jan 2010.
Class of quantum error-correcting codes saturating the quantum hamming bound. Daniel Gottesman, Physical Review A. 5431862Daniel Gottesman. Class of quantum error-correcting codes saturating the quantum hamming bound. Physical Review A, 54(3):1862, 1996.
Simple security proof of quantum key distribution based on complementarity. Masato Koashi, New Journal of Physics. 11445018Masato Koashi. Simple security proof of quantum key distribution based on complementarity. New Journal of Physics, 11(4):045018, 2009.
Security analysis of quantum key distribution with small block length and its application to quantum space communications. Charles Ci-Wen Lim, Feihu Xu, Jian-Wei Pan, Artur Ekert, Physical Review Letters. 12610Charles Ci-Wen Lim, Feihu Xu, Jian-Wei Pan, and Artur Ekert. Secu- rity analysis of quantum key distribution with small block length and its application to quantum space communications. Physical Review Letters, 126(10), Mar 2021.
Estimates for practical quantum cryptography. Norbert Lütkenhaus, Physical Review A. 5953301Norbert Lütkenhaus. Estimates for practical quantum cryptography. Phys- ical Review A, 59(5):3301, 1999.
Quantum computation and quantum information. A Michael, Nielsen, L Isaac, Chuang, Michael A Nielsen and Isaac L Chuang. Quantum computation and quan- tum information. 2010.
Composable, unconditionally secure message authentication without any secret key. Dimiter Ostrev, 2019 IEEE International Symposium on Information Theory (ISIT). IEEEDimiter Ostrev. Composable, unconditionally secure message authentica- tion without any secret key. In 2019 IEEE International Symposium on Information Theory (ISIT), pages 622-626. IEEE, 2019.
Advances in quantum cryptography. S Pirandola, U L Andersen, L Banchi, M Berta, D Bunandar, R Colbeck, D Englund, T Gehring, C Lupo, C Ottaviani, J L Pereira, M Razavi, J Shaari, M Tomamichel, V C Usenko, G Vallone, P Villoresi, P Wallden, Adv. Opt. Photon. 124S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Col- beck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Val- lone, P. Villoresi, and P. Wallden. Advances in quantum cryptography. Adv. Opt. Photon., 12(4):1012-1236, Dec 2020.
Key recycling in authentication. Christopher Portmann, IEEE Transactions on Information Theory. 607Christopher Portmann. Key recycling in authentication. IEEE Transac- tions on Information Theory, 60(7):4383-4396, 2014.
Cryptographic security of quantum key distribution. Christopher Portmann, Renato Renner, Christopher Portmann and Renato Renner. Cryptographic security of quantum key distribution, 2014.
Security of quantum key distribution. Renato Renner, ETH ZurichPhD thesisRenato Renner. Security of quantum key distribution. PhD thesis, ETH Zurich, 2005.
Simple proof of security of the bb84 quantum key distribution protocol. W Peter, John Shor, Preskill, Physical review letters. 852441Peter W Shor and John Preskill. Simple proof of security of the bb84 quantum key distribution protocol. Physical review letters, 85(2):441, 2000.
Multiple-particle interference and quantum error correction. Andrew Steane, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. 452Andrew Steane. Multiple-particle interference and quantum error correc- tion. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 452(1954):2551-2577, 1996.
Positive functions on c*-algebras. Forrest Stinespring, Proceedings of the American Mathematical Society. 62W Forrest Stinespring. Positive functions on c*-algebras. Proceedings of the American Mathematical Society, 6(2):211-216, 1955.
A largely self-contained and complete security proof for quantum key distribution. Marco Tomamichel, Anthony Leverrier, 14Marco Tomamichel and Anthony Leverrier. A largely self-contained and complete security proof for quantum key distribution. Quantum, 1:14, Jul 2017.
Tight finite-key analysis for quantum cryptography. Marco Tomamichel, Charles Ci Wen Lim, Nicolas Gisin, Renato Renner, Nature communications. 31Marco Tomamichel, Charles Ci Wen Lim, Nicolas Gisin, and Renato Ren- ner. Tight finite-key analysis for quantum cryptography. Nature commu- nications, 3(1):1-6, 2012.
New hash functions and their use in authentication and set equality. N Mark, J Lawrence Wegman, Carter, Journal of computer and system sciences. 223Mark N Wegman and J Lawrence Carter. New hash functions and their use in authentication and set equality. Journal of computer and system sciences, 22(3):265-279, 1981.
Entanglement-based secure quantum cryptography over 1,120 kilometres. Juan Yin, Yu-Huai Li, Sheng-Kai Liao, Meng Yang, Yuan Cao, Liang Zhang, Ji-Gang Ren, Wen-Qi Cai, Wei-Yue Liu, Shuang-Lin Li, Nature. 5827813Juan Yin, Yu-Huai Li, Sheng-Kai Liao, Meng Yang, Yuan Cao, Liang Zhang, Ji-Gang Ren, Wen-Qi Cai, Wei-Yue Liu, Shuang-Lin Li, et al. Entanglement-based secure quantum cryptography over 1,120 kilometres. Nature, 582(7813):501-505, 2020.
| [] |
[
"LOW M * -ESTIMATES ON COORDINATE SUBSPACES",
"LOW M * -ESTIMATES ON COORDINATE SUBSPACES"
] | [
"A A Giannopoulos ",
"V D Milman "
] | [] | [] | Let K be a symmetric convex body in R n . It is well-known that for every θ ∈ (0, 1) there exists a subspace F of R n with dim F = [(1 − θ)n] such thatwhere P F denotes the orthogonal projection onto F . Consider a fixed coordinate system in R n . We study the question whether an analogue of ( * ) can be obtained when one is restricted to choose F among the coordinate subspaces R σ , σ ⊆ {1, . . . , n}, with |σ| = [(1 − θ)n]. We prove several "coordinate versions" of ( * ) in terms of the cotype-2 constant, of the volume ratio and other parameters of K.The basic source of our estimates is an exact coordinate analogue of ( * ) in the ellipsoidal case. Applications to the computation of the number of lattice points inside a convex body are considered throughout the paper. | 10.1006/jfan.1996.3054 | [
"https://arxiv.org/pdf/math/9605218v1.pdf"
] | 17,522,940 | math/9605218 | 5db3b36b46656bfa3756b9f33192310ff400dff6 |
LOW M * -ESTIMATES ON COORDINATE SUBSPACES
2 May 1996
A A Giannopoulos
V D Milman
LOW M * -ESTIMATES ON COORDINATE SUBSPACES
2 May 1996
Let K be a symmetric convex body in R n . It is well-known that for every θ ∈ (0, 1) there exists a subspace F of R n with dim F = [(1 − θ)n] such thatwhere P F denotes the orthogonal projection onto F . Consider a fixed coordinate system in R n . We study the question whether an analogue of ( * ) can be obtained when one is restricted to choose F among the coordinate subspaces R σ , σ ⊆ {1, . . . , n}, with |σ| = [(1 − θ)n]. We prove several "coordinate versions" of ( * ) in terms of the cotype-2 constant, of the volume ratio and other parameters of K.The basic source of our estimates is an exact coordinate analogue of ( * ) in the ellipsoidal case. Applications to the computation of the number of lattice points inside a convex body are considered throughout the paper.
Introduction
Notation. Our setting is R n equipped with an inner product ., . and the associated Euclidean norm defined by |x| = x, x 1/2 , x ∈ R n . We denote the Euclidean unit ball and the unit sphere by D n and S n−1 respectively, and we write σ for the rotationally invariant probability measure on S n−1 .
Let K be a symmetric convex body in R n . Then, K induces in a natural way a norm . K on R n . In what follows we shall denote by X K the normed space (R n , . K ). As usual, K o = {y ∈ R n : y, x ≤ 1 for every x ∈ K} is the polar body of K, and X K o = (R n , . K o ) is the dual space of X K .
Finally, we consider the integral parameters
M = M K = S n−1 x 2 K σ(dx) 1/2 , M * = M K o = S n−1 x 2 K o σ(dx) 1/2 ,
which are up to a constant the mean widths of K o and K respectively.
Results. The following inequality of the second named author plays an important role in developing a proportional theory of high-dimensional convex bodies:
Theorem A (Low M * -estimate). There exists a function f : (0, 1) → R + such that for every symmetric convex body K in R n and for every θ ∈ (0, 1) one can find a subspace F of R n with dim F = [(1 − θ)n] satisfying (1.1)
x
K ≥ f (θ) M K o |x| , x ∈ F.
Theorem A was originally proved in [M1] and a second proof using the isoperimetric inequality on S n−1 was given in [M2] where it was shown that (1.1) holds with f (θ) ≥ cθ for some absolute constant c > 0 (and with an estimate f (θ) ≥ θ + o(1 − θ) as θ → 1 − ). This was later improved to f (θ) ≥ c √ θ in [PT], see also [M3] for a different proof with this best possible √ θ-dependence. Finally, it was proved in [Go] that one can have
(1.2) f (θ) ≥ √ θ(1 + O( 1 θn )).
Moreover, if we fix some θ ∈ (0, 1) and consider the Grassmannian manifold G n,k of all k-dimensional subspaces of R n , where k = k(θ, n) = [(1 − θ)n], equipped with the Haar probability measure ν n,k , then (1.1) holds true with f (θ) ≥ c √ θ for all subspaces F in a subset A n,k of G n,k which is of almost full measure ν n,k (A n,k ) as n → ∞.
Interchanging the roles of K and K o , we may equivalently read Theorem A in the following geometric form:
(1.3) P F (K) ⊇ c √ θ M K D n ∩ F,
where P F denotes the orthogonal projection onto F . In this paper we will follow the tradition and continue calling an inclusion of the type (1.3) a "low M * -estimate" (for K o ). Among other applications of (1.3), let us mention the quotient of subspace theorem and the reverse Santaló inequality [M1], [BM].
Let {e 1 , . . . , e n } be an arbitrary but fixed orthonormal basis of R n with respect to ., . . For a subset σ ⊆ {1, . . . , n} we naturally define the coordinate subspace R σ = {x ∈ R n : x, e j = 0 if j / ∈ σ}. We write D σ for D n ∩ R σ and Q σ for the unit cube Q n ∩ R σ = [−1, 1] σ in R σ .
Our purpose is to discuss "low M * -estimates" in the form (1.3) when one is restricted to choose F among the coordinate subspaces of R n of a certain dimension m proportional to n.
In Section 2 we study the case of an ellipsoid E in R n . It turns out that for any orthonormal basis of R n one has results analogous to (1.3) with almost the same √ θ-dependence on the parameter θ:
Theorem B (Coordinate low M * -estimate for ellipsoids). Let E be an ellipsoid in R n and θ ∈ (0, 1). Then, there exists σ ⊆ {1, . . . , n}, |σ| ≥ (1 − θ)n, with
P σ (E) ⊇ c √ θ log 1/2 ( 2 θ )M E D σ ,
where P σ denotes the orthogonal projection onto R σ , and c > 0 is an absolute constant.
It is perhaps surprising that this type of geometric result about ellipsoids is new and non-trivial. Note that our investigation of these questions was started from a simpler fact of the same nature about a special class of ellipsoids, which was discovered in [Gi].
It can be checked that Theorem B is optimal apart from the logarithmic term (see Remark 2.5). A result of the same type can be proved for an ellipsoid E of smaller but sufficiently large dimension living in an arbitrary subspace F of R n (Theorem 2.3). We also consider the corresponding problem for sections (instead of projections) of E with coordinate subspaces (Theorem 2.4).
Simple examples show that one cannot achieve the same strong estimate in full generality: for an arbitrary symmetric convex body K and an arbitrary orthonormal basis in R n . Consider e.g the case of the unit cube Q n and the standard basis of R n : observe that M Qn ≃ log n/n, while the radius of the largest Euclidean ball contained in any coordinate projection of Q n is 1. In Section 3 we give a general low M * -estimate in terms of the cotype-2 constant C K of X K :
Theorem C (M * -estimate in terms of C K ). For an arbitrary symmetric convex body K in R n and for any θ ∈ (0, 1), one can find σ ⊆ {1, . . . , n}, |σ| ≥ (1 − θ)n, satisfying
P σ (K) ⊇ c 1 θ log 2 ( 2 θ )h(C K )M K D σ ,
where h(y) = y log 2y, y ≥ 1, and c 1 > 0 is an absolute constant.
Let us note that one can give a simpler argument, based on the isomorphic Sauer-Shelah lemma of S. J. Szarek and M. Talagrand and a factorization theorem of B. Maurey, which results in a weaker estimate of the same type (we sketch it in Remark 3.6). We also obtain results of the same nature in which M K is replaced by various other "volumic" parameters of K or K o (see Remark 3.7).
In Section 4 we give a general low M * -estimate in terms of the volume ratio vr(K) of K:
Theorem D (M * -estimate in terms of vr(K)). Let K be a symmetric convex body in R n . For every θ ∈ (0, 1), there exists σ ⊆ {1, . . . , n}, |σ| ≥ (1 − θ)n, such that
P σ (K) ⊇ 1 [c 2 vr(K)] c 3 log( 2 θ ) θ M K D σ ,
where c 2 , c 3 > 0 are absolute constants.
In Sections 5 and 6 we give some further applications of the low M * -estimate for ellipsoids. We demonstrate an exact dependence between coordinate sections of an ellipsoid and its polar in the spirit of [M5]. We also apply Theorems 2.2 and 2.4 to questions related to the number of integer or "almost integer" points inside an ellipsoid.
Recall that the cotype-2 constant C K of X K is the smallest constant λ > 0 for which
Ave ε j =±1 m j=1 ε j x j 2 K 1/2 ≥ 1 λ m j=1 x j 2 K 1/2
holds for all choices of m ∈ N and {x j } j≤m in X K . We refer to [MS] and [TJ] for basic facts about type, cotype and p-summing operators which are used below. The letter c will always denote an absolute positive constant, not necessarily the same in all its occurrences. By |.| we denote the cardinality of a finite set, volume of appropriate dimension, and the Euclidean norm (this will cause no confusion).
Ellipsoidal case
In this Section we consider the case of an arbitrary ellipsoid E in R n . There exists a linear isomorphism T : R n → R n such that T (E) = D n . It will be convenient for us to write E in the form
(2.1) E = {x = n j=1
x j e j ∈ R n : | n j=1
x j u j | ≤ 1}, where u j = T (e j ), j = 1, . . . , n. Writing E in this way, we can easily express M E in terms of the u j 's as follows:
(2.2) M E = S n−1 x 2 T −1 (Dn) σ(dx) 1/2 = S n−1 | n j=1 x j u j | 2 σ(dx) 1/2 = 1 n n j=1 |u j | 2 1/2 .
Under the extra assumption that |u j | ≤ 1, j = 1, . . . , n, an estimate for coordinate projections of E was given in [Gi] in connection with the problems of the Banach-Mazur distance to the cube and the proportional Dvoretzky-Rogers factorization. Its proof combines the structure of the ellipsoid with the well-known Sauer-Shelah lemma and factorization arguments analogous to the ones in [BT,Theorem 1.2]:
Lemma 2.1. Let E τ = {x = j∈τ x j e j ∈ R τ : | j∈τ x j u j | ≤ 1}, where u j ∈ R n , j ∈ τ , with |u j | ≤ 1. Then, for every ζ ∈ (0, 1) there exists σ ⊆ τ, |σ| ≥ (1 − ζ)|τ |, such that P σ (E τ ) ⊇ c ζ D σ ,
where c > 0 is an absolute constant.
One more step is needed in order to obtain a low M * -estimate for coordinate subspaces in the ellipsoidal case:
Theorem 2.2. Let E be an ellipsoid in R n . For every θ ∈ (0, 1) there exists a subset σ of {1, . . . , n} with |σ| ≥ (1 − θ)n, such that
P σ (E) ⊇ c √ θ log 1/2 ( 2 θ )M E D σ ,
where c > 0 is an absolute constant.
Proof: We write E in the form (2.1) and assume as we may that M E = 1. If ρ = {j ≤ n : |u j | ≥ 2/θ}, then by (2.2) we have 2|ρ|/θ ≤ j≤n |u j | 2 = n, hence |ρ| ≤ θn/2. Consider the sets of indices:
τ 0 = {j ≤ n : |u j | ≤ 1}, τ k = {j ≤ n : e k−1 < |u j | ≤ e k }, k ≥ 1. If k 0 = [log( 2/θ)] + 1, we have | 0≤k≤k 0 τ k | ≥ n − |ρ| ≥ (1 − θ 2 )n. We define ζ k = θn 2 e k / √ |τ k | k e k √ |τ k |
for all k ≤ k 0 with τ k = ∅, and consider the set I = {k ≤ k 0 : τ k = ∅ and ζ k < 1}. For each k ∈ I we can apply Lemma 2.1 for the
ellipsoid E τ k = E ∩ R τ k to find σ k ⊆ τ k with |σ k | ≥ (1 − ζ k )|τ k | such that (2.3) P σ k (E τ k ) ⊇ c 1 √ ζ k e k D σ k ,
where c 1 is the constant from Lemma 2.1. Finally, we set σ = k∈I σ k . Note that the above choice of ζ k 's implies that
| k 0 k=0 τ k | − |σ| ≤ k 0 k=0 ζ k |τ k | = θn 2 ,
and therefore, |σ| ≥ (1 − θ)n.
Suppose that w ∈ D σ . If we write w = k∈I w k , where w k = P σ k (w), then by (2.3),
(2.4) w ∈ 1 c 1 k∈I |w k | e k √ ζ k P σ k (E ∩ R τ k ) ⊆ 1 c 1 k∈I |w k | e k √ ζ k P σ (E),
and since w ∈ D σ was arbitrary, an application of the Cauchy-Schwartz inequality shows that
(2.5) D σ ⊆ 1 c 1 k∈I e 2k ζ k 1/2 P σ (E).
Inserting our ζ k 's in the sum above, we conclude that
(2.6) D σ ⊆ 1 c 2 √ θn k 0 k=0 e k |τ k | P σ (E).
It remains to give an upper bound for the sum k≤k 0 e k |τ k |: to this end, note that for k = 1, . . . , k 0 , we have |τ k |e 2k−2 ≤ j∈τ k |u j | 2 ≤ n and thus e k |τ k | ≤ e √ n for k = 1, . . . , k 0 which allows a first upper bound of the order of k 0 √ n.
We partition the set of indices {0, 1, . . . , k 0 } setting |ϕ s | ≤ k 0 e s−1 , for all s ≤ s 0 . By the definition of ϕ s and by (2.7), we can now estimate the sum in (2.6) as follows:
ϕ 0 = {k ≤ k 0 : |τ k | ≤ 1 k 0 n e 2k−2 }, ϕ s = {k ≤ k 0 : e s−1 k 0 n e 2k−2 < |τ k | ≤ e s(2.8) k 0 k=0 e k |τ k | = s 0 s=0 k∈ϕs e k |τ k | ≤ s 0 s=0 |ϕ s | e k e s/2 √ n √ k 0 e k−1 ≤ e √ n √ k 0 s 0 s=0 k 0 e s−1 e s/2 ≤ e 2 ( ∞ s=0 e −s/2 ) √ n k 0 ≤ c 3 k 0 √ n.
Therefore, (2.6) becomes
(2.9) D σ ⊆ 1 c 4 √ θ k 0 P σ (E),
which completes the proof, since k 0 ≃ log(2/θ) and we had assumed that M E = 1.
We proceed to prove an extension of Theorem 2.2 concerning the case where E is an ellipsoid of dimension m < n living in an arbitrary m-dimensional subspace F of R n . If m is proportional to n, with m/n sufficiently close to 1, then we still have coordinate projections of E of large dimension containing large Euclidean balls. This result will be useful for our treatment of the general case in Sections 3 and 4:
Theorem 2.3. Let ε ∈ (0, 1) and F be a subspace of R n with dim F = m ≥ (1 − ε)n. Then, for every non-degenerate ellipsoid E in F and for every ζ ∈ [c 1 ε log( 2 ε ), 1) there exists σ ⊆ {1, . . . , n} with |σ| ≥ (1 − ζ)n, such that P σ (E) ⊇ c √ ζ 2 √ 2 log 1/2 ( 2 ζ )M E D σ ,
where c is the constant from Theorem 2.2 and c 1 = max{ 8 c 2 , 1 log 2 }. Proof: Suppose that an ellipsoid E is given in F . We can find an orthonormal basis {w 1 , . . . , w m } of F and λ 1 , . . . , λ m > 0 such that
E = {x ∈ F : m j=1 x, w j 2 λ 2 j ≤ 1}.
We extend to an orthonormal basis {w j } j≤n of R n and consider the ellipsoid
E ′ = {x ∈ R n : m j=1 x, w j 2 λ 2 j + n j=m+1 x, w j 2 b 2 ≤ 1}, where b = √ ε/M E . It is easy to check that (2.10) M 2 E ′ = 1 n m j=1 1 λ 2 j + n − m b 2 = mM 2 E + (n − m)M 2 E /ε n ≤ 2M 2 E .
Let ζ ∈ [c 1 ε log( 2 ε ), 1). Applying Theorem 2.2 for E ′ and taking into account (2.10), we find σ ⊆ {1, . . . , n} with |σ| ≥ (1 − ζ)n for which
(2.11) P σ (E ′ ) ⊇ c √ ζ √ 2 log 1/2 (2/ζ)M E D σ .
Since ζ ≥ c 1 ε log( 2 ε ) and the function ζ/ log( 2 ζ ) is increasing on (0,1), one can easily check that
(2.12) c √ ζ √ 2 log 1/2 ( 2 ζ ) ≥ 2 √ ε.
On the other hand, we clearly have E ′ ⊆ E+bD n and hence P σ (E ′ ) ⊆ P σ (E)+bD σ . Combining this with (2.11) and (2.12) we conclude that
(2.13) c √ ζ √ 2 log 1/2 ( 2 ζ )M E D σ ⊆ P σ (E) + 1 2 c √ ζ √ 2 log 1/2 ( 2 ζ )M E D σ .
Claim: If A and B are convex symmetric bodies in R σ and A ⊆ B + 1 2 A, then A ⊆ 2B.
[One easily checks that A ⊆ (1 + 1 2 + . . . + 1 2 k )B + 1 2 k A and the claim follows by letting k → ∞.]
Our claim and (2.13) imply that
P σ (E) ⊇ c 2 √ 2 √ ζ log 1/2 ( 2 ζ )M E D σ ,
and the proof of the theorem is complete.
Our next result concerns coordinate sections of ellipsoids: again, we are interested in finding large balls contained in them. Using a result of [AM] which was recently improved in [T] (in our case each of them works equally well), we can give an essentially optimal answer to this question when the dimension of the coordinate sections is small (of order roughly not exceeding √ n):
Theorem 2.4. Let E be an ellipsoid in R n . For every m ≤ c √ n we can find a subset σ of {1, . . . , n} of cardinality |σ| = m, such that
E ∩ R σ ⊇ c ′ √ mM E D σ .
In the statement above, c and c ′ are absolute positive constants.
Proof: We write E in the form (2.1). As a consequence of (2.2), observe that for every s ≤ n the following identity holds:
(2.14)
Ave |τ |=s M 2 E∩R τ = n − 1 s − 1 / n s 1 s n j=1 |u j | 2 = M 2 E ,
where the average is over all τ ⊆ {1, . . . , n} with |τ | = s. This means in particular that for every s ≤ n we can find τ with |τ | = s for which
M E∩R τ ≤ M E .
Assume that m ≤ c √ n is given, where c > 0 is an absolute constant to be chosen.
We choose s = [ m 2 c 2 ] and find τ with |τ | = s and
M E∩R τ ≤ M E . Observe that Ave ε j =±1 j∈ϕ ε j e j E ≤ |τ |M E∩R τ ≤ |τ |M E .
Hence, if c is small enough, the results of [AM] or [T] allow us to find ϕ ⊆ τ with |ϕ| = 2m such that
(2.15) j∈ϕ ε j e j E ≤ c 1 |τ |M E , for all (ε j ) j∈ϕ ∈ {−1, 1} ϕ , where c 1 is a positive absolute constant. In other words, the coordinate section of E by R ϕ satisfies (2.16) E ∩ R ϕ ⊇ 1 c 1 |τ |M E Q ϕ .
This means that the identity operator id :
ℓ ϕ ∞ → X E ∩ R ϕ has norm id ≤ c 1 |τ |M E , and this implies that π 2 (id) ≤ c 1 K G |τ |M E where K G is Grothendieck's constant. Applying Pietch's factorization theorem we can find (λ i ) i∈ϕ , i∈ϕ λ 2 i = 1: (2.17) i∈ϕ t i e i E ≤ c 1 K G |τ |M E i∈ϕ λ 2 i t 2 i 1/2
for every choice of reals (t i ) i∈ϕ . By Markov's inequality, we find σ 1 ⊆ ϕ,
|σ 1 | ≥ |ϕ|/2 ≥ m, such that |λ i | ≤ √ 2 √ |ϕ|
for all i ∈ σ 1 . Then, for any (t i ) i∈σ 1 we have
(2.18) i∈σ 1 t i e i E ≤ c 1 K G |τ |M E √ 2 |ϕ| i∈σ 1 t 2 i 1/2 .
The choice of |τ | and |ϕ| shows that
(2.19) E ∩ R σ 1 ⊇ c ′ √ mM E D σ 1 ,
for some absolute constant c ′ > 0, and we conclude the proof by choosing any σ ⊆ σ 1 of cardinality |σ| = m.
Remark 2.5. An iteration of the argument above shows that one can extend the range of m's for which Theorem 2.4 holds to e.g the set {1, . . . , [ √ n]}, with some loss in the constant c ′ . The dependence on m is sharp as it can be seen by the following example:
consider the ellipsoid E = {(t j ) j≤n ∈ R n : | t j u j | n+1 ≤ 1},
where u j = e j +e n+1 , j = 1, . . . , n, and {e j } j≤n+1 is the standard orthonormal basis in R n+1 . Given any σ ⊆ {1, . . . , n} with |σ| = m, we have that ( t √ m , . . . , t √ m ) ∈ E ∩ R σ precisely when (1 + m)t 2 ≤ 1. In particular, we must have |t| ≤ 1 √ m . This means that the largest ball contained in E ∩ R σ cannot have radius larger than 1 √ m . On the other hand, observe that M E = √ 2.
The same example shows that the estimate in Theorem 2.2 is best possible apart from the log 1/2 ( 2 θ ) term. By Lemma 2.1, this logarithmic term can be removed if all the u j 's are of about the same Euclidean norm.
General case: estimate in terms of the cotype-2 constant
In this Section we study the general case, that is K is an arbitrary symmetric convex body in R n , and {e j } j≤n is a fixed orthonormal basis. We shall make use of the maximal volume ellipsoid E of K and of the better information we have for coordinate projections of ellipsoids. For this purpose we will also need an estimate for the parameters
A m (K) = sup{(|K ∩ F |/|E ∩ F |) 1/m : dim F = m}, m = 1, . . . , n.
It was proved in [BM] that the volume ratio vr(
K) = (|K|/|E|) 1/n of K is bounded by f (C K ) = cC K [log C K ] 4
, with the power of log C K improved to 1 in [MiP]. A third proof of the same fact is given in [M4], where it is also shown that vr(K) ≤ ch(C K ), where h(y) = y log 2y, y ≥ 1, and c > 0 is an absolute constant. Our first lemma is a modification of the argument presented in [M4] which provides an estimate for A m (K), m ≤ n, in terms of C K :
Lemma 3.1. Let K be a symmetric convex body in R n , and E be the maximal volume ellipsoid of K. If F is an m-dimensional subspace of R n , then
|K ∩ F | |E ∩ F | 1/m ≤ ch( n/mC K ),
where h(y) = y log 2y, y ≥ 1.
Proof: We may clearly assume that E = D n . The proof will be based on an iteration schema, analogous to the one in [M4].
We set K 0 = K, α 0 = n, β 0 = n, and for j = 1, 2, . . . we define:
(i) α j = log α j−1 = log (j) n, the j-iterated logarithm of n, (ii) β j = α j M (K j−1 ∩F ) o , (iii) K j = K ∩ β j D n .
Note that for every j the maximal volume ellipsoid of K j is D n . Also, C K j ≤ 2C K and d(X K j , ℓ n 2 ) ≤ β j . By Sudakov's inequality [Su], [P1] the covering number of K j−1 ∩ F by β j D n ∩ F can be estimated as follows:
N(K j−1 ∩ F, β j D n ∩ F ) = N ≤ exp(c 1 mM 2 (K j−1 ∩F ) o /β 2 j ) = exp(c 1 m/α 2 j ),
and since, by Brunn's theorem,
|K j−1 ∩ (x + β j D n ∩ F )| ≤ |K j−1 ∩ β j D n ∩ F |, x ∈ F, we have |K j−1 ∩ F | ≤ N|K j ∩ F | and hence (3.1) |K j−1 ∩ F | 1/m ≤ exp( c 1 α 2 j ) |K j ∩ F | 1/m
By well-known results of [DMT], [MP], and [P2] we have the string of inequalities
M (K j ∩F ) o ≤ c 2 n m M K o j ≤ c 3 n m T 2 (X K o j ) ≤ c 4 n m C K j log(2d(X K j , ℓ n 2 ))
and therefore
M (K j ∩F ) o ≤ 2c 4 n m C K log(2β j ).
It follows that the sequence {β j } j≥0 satisfies the relation
(3.2) β j+1 ≤ 2c 4 n m C K α j log(2β j ).
We stop this procedure at the smallest t for which α t < 6c 4 . Induction and (3.2) show that
(3.3) β t ≤ 36c 2 4 n m C K log( n m C K ) + 6c 4 ≤ c ′ h( n/mC K ).
By (3.1) we see that
(3.4) |K ∩ F | 1/m ≤ |K t ∩ F | 1/m exp(c 1 [ 1 α 2 1 + . . . + 1 α 2 t ]) ≤ c ′′ |K t ∩ F | 1/m , since 1 α 2 j
is easily seen to be uniformly bounded. Taking into account (3.3), (3.4) and the Blaschke-Santaló inequality we conclude that
(3.5) |K ∩ F | |D n ∩ F | 1/m ≤ c ′′ |D n ∩ F | |(K t ∩ F ) o | 1/m ≤ c ′′ M (Kt∩F ) o ≤ 2c 4 c ′′ n m C K log(2c ′ h( n/mC K )) ≤ ch( n/mC K ).
Simple examples (see Remark 3.3) show that one cannot compare M K and M E even if C K is small: the only estimate one can give is that M E ≤ √ nM K , which is a direct consequence of the fact that K ⊆ √ nE by John's theorem. However, there exist subspaces F of R n of proportional dimension on which we can compare M K with M E∩F reasonably well:
Lemma 3.2. Let E be the maximal volume ellipsoid of K. For every ε ∈ (0, 1) there exists a subspace F of R n with dim F = m ≥ (1 − ε)n such that
M E∩F ≤ ch(C K ) log( 2 ε ) √ ε M K ,
where h(y) = y log 2y, y ≥ 1, and c > 0 is an absolute constant.
Proof: Let {w 1 , . . . , w n } be an orthonormal basis of R n and λ 1 ≥ . . . ≥ λ n > 0 such that
E = {x ∈ R n : n j=1 x, w j 2 λ 2 j ≤ 1}.
For k = 1, . . . , n, set W k = span{w k , . . . , w n }. By Lemma 3.1 we have
(3.6) |K ∩ W k | |E ∩ W k | 1 n−k+1 ≤ c 1 h( n n − k + 1 C K ).
Note that E ∩ W k ⊆ λ k (D n ∩ W k ), and hence
(3.7) |K ∩ W k | |E ∩ W k | 1 n−k+1 ≥ 1 λ k |K ∩ W k | |D n ∩ W k | 1 n−k+1 ≥ 1 λ k M K∩W k ≥ 1 c 2 λ k n n−k+1 M K .
Combining (3.6), (3.7) we obtain
(3.8) 1 λ k ≤ c 1 c 2 n n − k + 1 h( n n − k + 1 C K )M K , k = 1, . . . , n.
Given ε ∈ (0, 1), let m = [(1 − ε)n] and set F m = span{w 1 , . . . , w m }. By (3.8) we can estimate M E∩Fm as follows:
(3.9) M E∩Fm = 1 m m k=1 1 λ 2 k 1/2 ≤ c 1 c 2 C K M K m k=1 n 2 m(n − k + 1) 2 log 2 (2 n n − k + 1 C K ) 1/2 ≤ c 1 c 2 C K log(2 n n − m C K ) n n − m M K ≤ ch(C K ) log( 2 ε ) √ ε M K .
Remark 3.3. The estimate (3.9) is essentially sharp, even if C K is small: to see this, consider the class of bodies K = K(a, b; s) = {x ∈ R n : j≤s
|x j | a + j>s |x j | b ≤ 1},
where a, b are positive parameters and s ∈ {0, 1, . . . , n − 1}. It is clear that the ellipsoid of maximal volume in K is E = E(a, b; s) = {x ∈ R n : j≤s
|x j | 2 a 2 + j>s |x j | 2 b 2 ≤ 1}.
It is also clear that both the cotype-2 constant and the volume ratio of K are uniformly bounded (independently of n, s, a and b).
Given ε ∈ (0, 1), choose b = a √ ε, s = m = (1 − ε)n. Then, it is easy to check that
M K ≃ √ n √ ε/a, while M E∩Fm ≃ √ n/a.
Also, we can have the ratio M E /M K as close to √ n as we like: choose, for example, s = n − 1 and b = a n−1 . Then,
M K ≃ 1/ √ nb while M E ≃ 1/b.
Combining Theorem 2.3 and Lemma 3.2 we prove our M * -estimate in terms of the cotype-2 constant of X K :
Theorem 3.4. Let K be a symmetric convex body in R n , and X K = (R n , . K ). For every θ ∈ (0, 1) there exists σ ⊆ {1, . . . , n}, |σ| ≥ (1 − θ)n, such that
P σ (K) ⊇ cθ log 2 ( 2 θ )h(C K )M K D σ ,
where h(y) = y log 2y, y ≥ 1, and c > 0 is an absolute constant.
Proof: Let E be the maximal volume ellipsoid of K, and set ε = ε(θ) = θ/c 2 log( 2 θ ), where c 2 > 0 is a constant to be chosen. By Lemma 3.2 we can find a subspace F of R n with dim F ≥ (1 − ε)n such that
(3.10) M E∩F ≤ c 3 h(C K ) log( 2 ε ) √ ε M K .
Observe that if c 2 is large enough, then θ ≥ c 1 ε log( 2 ε ) where c 1 is the constant in Theorem 2.3. Thus, we can apply Theorem 2.3 for E ∩ F to find σ ⊆ {1, . . . , n} with |σ| ≥ (1 − θ)n for which
(3.11) P σ (E ∩ F ) ⊇ c √ θ 2 √ 2 log 1/2 ( 2 θ )M E∩F D σ .
Combining (3.10) with (3.11) we finish the proof.
Remark 3.5. It should be noted that the estimate given by Theorem 3.4 is exact not only when C K is small (like e.g in the ellipsoidal case), but in the whole range [1, √ n]
of possible values of C K i.e even if C K is extremely large. This can be easily seen if one considers the case of B n p , p > 2, the unit ball of ℓ n p , and the standard coordinate system in R n . Fix for example θ = 1 2 . Then, the radius of the largest Euclidean ball inscribed in any [ n 2 ]-dimensional coordinate projection of B n p is 1, and the well-known estimates for C B n p and M B n p show that Theorem 3.4 is sharp apart from logarithmic terms. We do not know if the "almost linear" dependence on θ which our method provides is optimal. However, the ellipsoidal case shows that √ θ dependence is the best one might hope for.
Remark 3.6. One can give a weaker estimate, analogous to the one obtained in Theorem 3.4, using the isomorphic Sauer-Shelah lemma of Szarek-Talagrand [ST] and a factorization result of Maurey [Ma] (see also [TJ]). Starting with the body K and the orthonormal basis {e j } j≤n , we have the inequality
Ave ε j =±1 n j=1 ε j e j K ≤ √ nM K ,
and therefore, by Markov's inequality we can find A ⊆ {−1, 1} n of cardinality |A| ≥ 2 n−1 such that ε j e j K ≤ 2M K √ n whenever ε = (ε 1 , . . . , ε n ) ∈ A. If we view A as a set of points in R n , this means that A ⊆ 2M K √ nK. On the other hand, the
isomorphic Sauer-Shelah lemma shows that for some absolute constant c 1 > 0 and for every θ ∈ (0, 1) there exists σ ⊆ {1, . . . , n}, |σ| ≥ (1 − θ 2 )n, with co(P σ (A) ⊇ c 1 θ 2 Q σ , and hence
P σ (K) ⊇ c 1 θ 4M K √ n Q σ . It follows that if Y = (R σ 1 , . K o ), then id : ℓ σ 1 ∞ → Y * has norm id ≤ 4M K √ n
c 1 θ , and Maurey's theorem shows that
π 2 (id) ≤ c 2 M K √ n θ g(Y * ), where g(Y * ) = C Y * 1 + log(C Y * )
. Then, we can apply Pietch's factorization theorem in the context of [BT,Theorem 1.2
] to find σ ⊆ σ 1 with |σ| ≥ (1− θ 2 )|σ 1 | ≥ (1−θ)n for which i∈σ t 2 i 1/2 ≤ c 3 M K g(Y * ) θ 3/2 i∈σ t i e i K o
is true for all (t i ) i∈σ . Taking polars in R σ and using the fact that C Y * ≤ c 4 C K Rad X K , we conclude that
P σ (K) ⊇ cθ 3/2 f (K)M K D σ ,
where c > 0 is an absolute constant, and f (K) = C K Rad X K 1 + log(C K Rad X K ).
Remark 3.7. One can modify the proof of Theorem 3.6 to give analogous estimates in which M K is replaced by other "volumic" parameters of K or K o . Consider e.g the sequence of volume numbers of K o
(3.12) v s (K o ) = max{(|P F (K o )|/|D n ∩ F |) 1/s : dim F = s},
where s = 1, . . . , n. As a consequence of the Aleksandrov-Fenchel inequalities, one can easily see that {v s (K o )} s≤n is non increasing (see [P1]):
(3.13) v 1 (K o ) ≥ v 2 (K o ) ≥ . . . ≥ v n (K o ) = v.rad(K o ).
Let K be a symmetric convex body in R n and let E be the ellipsoid of maximal volume in K as in Lemma 3.4. Using the inverse Santaló inequality in (3.6), (3.7) we get
(3.14) 1 λ k ≤ |D n ∩ W k | |K ∩ W k | 1 n−k+1 |K ∩ W k | |E ∩ W k | 1 n−k+1 ≤ c |P W k (K o )| |D n ∩ W k | 1 n−k+1 c 1 h( n n − k + 1 C K )
for k = 1, . . . , n. By the definition (3.11) of v n−k+1 (K o ) this means that
(3.15) 1 λ k ≤ c 2 h(C K )v n−k+1 (K o ) n n − k + 1 log(2 n n − k + 1 ).
Inserting this estimate in (3.9) we obtain:
(3.16) M 2 E∩Fm = 1 m m k=1 1 λ 2 k ≤ c 2 2 h 2 (C K ) log 2 ( 2n n−m+1 ) m m k=1 n n − k + 1 v 2 n−k+1 (K o ).
The monotonicity of volume numbers shows that
v n−k+1 (K o ) ≤ v n−m+1 (K o ), k = 1,M E∩Fm ≤ c ′ h(C K )v [θn] (K o ) log 3/2 ( 2 θ ),
and, using Theorem 2.3 exactly as in the proof of Theorem 3.4, we can find σ ⊆ {1, . . . , n} with |σ| ≥ (1 − c 1 θ log( 2 θ ))n for which
(3.18) P σ (K) ⊇ c √ θ log 3/2 ( 2 θ )v [θn] (K o )h(C K ) D σ .
A similar argument shows that for some σ of the same cardinality we have
(3.19) P σ (K) ⊇ c √ θw [θn] (K) log 3/2 ( 2 θ )h(C K ) D σ ,
where w s (K) = min{(|K ∩ F |/|D n ∩ F |) 1/s : dim F = s}, s = 1, . . . , n.
General case: estimate in terms of the volume ratio
In this Section we use the volume ratio vr(K) of K instead of the cotype-2 constant of X K as a parameter for our low M * -estimate. Let E be the maximal volume ellipsoid of K. We start with a lemma which estimates the covering number N(K, E) in terms of the volume ratio vr(K) = (|K|/|E|) 1/n . Our proof is based on Lemma 4.4 from [MS2], actually the argument given there leads to a stronger estimate, but we include a simple proof of what we need here for the sake of completeness. Recall that N(K, L) is the smallest natural number N for which there exist x 1 , . . . , x N ∈ R n with K ⊆ i≤N (x i + L):
Lemma 4.1. Let K and L be two symmetric convex bodies in R n such that L ⊆ K. Then,
N(K, L) ≤ c n |K| |L| ,
where c > 0 is an absolute constant.
Proof: Consider a set N of points in K such that x−x ′ L ≥ 1 for every x, x ′ ∈ N, x = x ′ , which has the maximal possible cardinality. Observe that the sets 2 3 x + L 3 , x ∈ N have disjoint interiors and, since L ⊆ K, they are all contained in K. We easily deduce that (4.1) |N| ≤ 3 n |K| |L| .
Finally, it is clear that K ⊆ x∈N (x + L), which completes the proof.
Suppose that K is any symmetric convex body in R n and E is the ellipsoid of maximal volume in K. The analogue of Lemma 3.2 in the "volume ratio" formulation is the following:
Lemma 4.2. Let E be the maximal volume ellipsoid of K. For every ε ∈ (0, 1) there exists a subspace F of R n with dim F = m ≥ (1 − ε)n, such that
M E∩F ≤ [c vr(K)] 1/ε M K ,
where c > 0 is an absolute constant.
Proof: As in the proof of Lemma 3.2, let
E = {x ∈ R n : n j=1 x, w j 2 λ 2 j ≤ 1},
where {w 1 , . . . , w n } is an orthonormal basis of R n and λ 1 ≥ . . . ≥ λ n > 0. Fix k ∈ {1, . . . , n} and consider the subspace W k = span{w k , . . . , w n }. According to Lemma 4.1, we can find x 1 , . . . , x N ∈ K such that N = N(K, E) ≤ [c 1 vr(K)] n and K ⊆ (x i + E). Project all the (x i + E)'s onto W k . Then,
(4.2) K ∩ W k ⊆ P W k (K) ⊆ j≤N P W k (x j + E) = j≤N (P W k (x i ) + E ∩ W k ),
and hence, N(K ∩ W k , E ∩ W k ) ≤ N(K, E). Thus, we can estimate the ratio of the volumes of K ∩ W k and E ∩ W k using (4.2):
(4.3) |K ∩ W k | |E ∩ W k | 1 n−k+1 ≤ [N(K, E)] 1 n−k+1 ≤ [c 1 vr(K)] n n−k+1 .
Combining with (3.7) we get
(4.4) 1 λ k ≤ c 2 n n − k + 1 [c 1 vr(K)] n n−k+1 M K , k = 1, . . . , n.
We continue as in Lemma 3.2: Given any ε ∈ (0, 1), we consider the first m for which m ≥ (1 − ε)n and set F m = span{w 1 , . . . , w m }. In view of (4.5) we can compare M E∩Fm with M K as follows: Remark 4.3. By well-known results of S.J. Szarek and N. Tomczak-Jaegermann (see [Sz], [STJ]) which were extending previous work of Kashin, if E is the maximal volume ellipsoid of K, then for every k = 1, . . . , n − 1 there exist k-dimensional subspaces F of R n for which E ∩ F ⊆ K ∩ F ⊆ (c vr(K)) n n−k E ∩ F , and this obviously implies that M E∩F ≤ [c vr(K)] n n−k M K∩F . This leads to the same estimate as in Lemma 4.2 above, actually if E = D n this is true for all subspaces F in a subset A of G n,k with almost full measure ν n,k (A) > 1 − 2 −n . The argument provided by Lemmata 4.1 and 4.2 gives a concrete example of a subspace on which the weaker "M E∩F and M K∩F " comparison is true: it can be chosen as the k-dimensional subspace which is coordinate with respect to E and corresponds to the k largest semiaxes of E. If E = D n , then this weak comparison is true for all F ∈ G n,k .
Combining Lemma 4.2 with Theorem 2.3 we prove our volume-ratio result:
Theorem 4.4. Let K be a symmetric convex body in R n . For every θ ∈ (0, 1) there exists σ ⊆ {1, . . . , n}, |σ| ≥ (1 − θ)n, such that
P σ (K) ⊇ 1 [c 1 vr(K)] c 2 log( 2 θ ) θ M K D σ ,
where c 1 , c 2 are absolute positive constants.
Proof: Let E be the maximal volume ellipsoid of K, and set ε = ε(θ) = θ c 2 log( 2 θ ) , where c 2 > 0 is a constant to be chosen. Using Lemma 4.2 we find a subspace F of R n with dim F ≥ (1 − ε)n such that
(4.6) M E∩F ≤ [c 4 vr(K)] 1/ε M K .
If c 2 is large enough, we easily check that θ ≥ c 1 ε log( 2 ε ) where c 1 is the constant in Theorem 2.3. We can therefore apply Theorem 2.3 for E ∩ F to find σ ⊆ {1, . . . , n} with |σ| ≥ (1 − θ)n, such that
(4.7) P σ (E ∩ F ) ⊇ c √ θ 2 √ 2 log 1/2 ( 2 θ )M E∩F D σ .
Combining (4.6) with (4.7) we conclude the proof.
For classes of spaces with uniformly bounded volume ratio, Theorem 4.3 gives an optimal answer as long as, say, θ ≥ 1 2 . The estimate obtained "explodes" if vr(K) is large or if θ is needed to be close to 0.
Linear duality relations for coordinate sections of ellipsoids
Let K be a symmetric convex body in R n . We introduce the integer valued functions t, t c : R + → N defined by t(r) = t(K, r) = max{k ≤ n : there exists a subspace E with dimE = k, such that 1 r |x| ≤ x for every x ∈ E} and t c (r) = t c (K, r) = max{k ≤ n : there exists a coordinate subspace E with dim E = k such that 1 r |x| ≤ x for every x ∈ E}.
It is easy to see that if K is an ellipsoid in R n , then t(K, r) + t(K o , 1 r ) ≥ n. In [M5] it is proved that for every body K, for every r > 0, and for every τ ∈ (0, 1), one has a similar duality relation:
(5.1) t(K, r) + t(K o , 1 τ r ) ≥ (1 − τ )n − C,
where C > 0 is a universal constant. The proof of this fact is based on the strong form (1.2) of the low M * -estimate and on the "distance lemma": if 1 a |x| ≤ x ≤ b|x| for every x ∈ R n and if (M K /b) 2 + (M K o /a) 2 = s > 1, then ab ≤ 1 s−1 . In this Section we establish a coordinate version of (5.1) in the ellipsoidal case. Our estimate depends on how close the ellipsoid is to being in M-position:
Definition: For a symmetric convex body K in R n we denote by λ K its volume radius: λ K = (|K|/|D|) 1/n . We also write N K for N(K, λ K D) and say that K is in M δ -position if δ ≥ 1 n log N K . Our first lemma provides some simple estimates which show that this position is "stable" under the operations of taking intersection or convex hull with a ball:
Lemma 5.1. Let K be a symmetric convex body in R n , and let r, r 1 > 0 be given. Define K r = K ∩ rD and K r 1 = co(K ∪ r 1 D). Then,
(i) N Kr ≤ max{3 n N 2 K , 9 n N K }. (ii) N K r 1 ≤ 5 n N K .
Proof: (i) From the Brunn-Minkowski inequality it easily follows that |K ∩ rD| ≥ |K ∩ (x + rD)|, x ∈ R n . This implies that |K| ≤ N(K, rD)|K ∩ rD| or, equivalently, (5.2) λ n K ≤ N(K, rD)λ n Kr . We distinguish two cases:
(1) If λ K < r, then N(K, rD) ≤ N K and, by (5.2), λ n K ≤ N K λ n Kr . It follows that
N Kr ≤ N(K, λ Kr D) ≤ N K N(D, λ Kr λ K D) ≤ N K N(D, 1 N K D) ≤ 3 n N 2 K .
(2) If λ K > r, then N(K, rD) ≤ N(K, λ K D)N(D, r λ K D) ≤ N K 3 n ( λ K r ) n and hence, by (5.2), ( r λ Kr ) n ≤ 3 n N K . It follows that N Kr ≤ N(rD, λ Kr D) ≤ 3 n ( r λ Kr ) n ≤ 9 n N K .
(ii) We obviously have λ K r 1 ≥ max{λ K , r 1 }. Also, K r 1 ⊆ K + r 1 D, which gives
N K r 1 ≤ N(K r 1 , 2λ K r 1 D)N(D, 1 2 D) ≤ 5 n N(K + r 1 D, (λ K + r 1 )D) ≤ 5 n N K .
For an arbitrary symmetric convex body K, one has in general the information λ K M K ≥ 1 as a consequence of the polar coordinates formula for volume. Our next lemma provides an "inverse" inequality in terms of the parameters N K o and b = sup{ x : x ∈ S n−1 }:
Lemma 5.2. Let K be a symmetric convex body in R n , and assume that x ≤ b|x| for all x ∈ R n . Then,
M K ≤ c λ K N t/n K o
where c > 0 is an absolute constant, and t ≤ C( b M K ) 2 . Proof: Using Theorem 6 from [BLM] (to be more precise, using an argument identical to the one given there and the observation that what is really used is the ratio b/M K ), one can find orthogonal transformations u 1 , . . . , u t ∈ O(n) such that
(5.4) M K 2 D ⊆ T = 1 t t i=1 u i (K o ) ⊆ 2M K D, with t ≤ C( b M K ) 2 , where C > 0 is an absolute constant. On observing that N(T, λ K o D) ≤ [(N(K o , λ K o D)] t = N t K o ,
we can estimate M K by (5.4) as follows:
(5.5) M K ≤ 2( |T | |D| ) 1/n ≤ 2λ K o N t/n K o .
Finally, the Blaschke-Santaló inequality implies that λ K λ K o ≤ 1, and hence the proof of the Lemma is complete.
We can now pass to the proof of the main result of this section:
Theorem 5.3. Let E be an ellipsoid in R n , and assume that both E and E o are in M δ -position. For every r > 0 and every τ ∈ (0, 1) we have
t c (E, r) + t c (E o , u(τ, δ) r ) ≥ (1 − τ )n,
where u(τ, δ) = c log( 2 τ ) √ τ e cδ log 2 ( 2 τ ) τ , and c > 0 is an absolute constant.
Proof: Let r > 0 and τ ∈ (0, 1) be given. Consider the body E r = E ∩ rD. Since E r is √ 2-isomorphic to an ellipsoid, one can easily check that Theorem 2.2 holds for E r : for every θ ∈ (0, 1) we can find σ ⊆ {1, . . . , n} with |σ| ≥ (1 − θ)n such that P σ (E o r ) ⊇ [g(θ)/M(E o r )]D σ , where g(θ) = c √ θ/2 log(2/θ) and c is the same constant as in Theorem 2.2.
We distinguish three cases: (1)).
Case 1: M (E o r ) r ∈ [g(τ ), g
In this case, consider any λ ∈ (τ, 1] with 1 r M(E o r ) < g(λ). We can find σ 1 ⊆ {1, . . . , n} with |σ 1 | ≥ (1 − λ)n such that
P σ 1 (E o r ) ⊇ g(λ) M(E o r ) D σ 1 ,
and it is easy to check that, for every x ∈ R σ 1 , max{ x , 1 r |x|} = x Er > 1 r |x|, which means that 1 r |x| ≤ x , i.e (5.6) t c (E, r) ≥ (1 − λ)n.
Taking the infimum of all λ's for which M (E o r ) r < g(λ), we conclude that (5.6) also holds for the solution in λ of the equation M(E o r ) = rg(λ). Now, choose µ ∈ (0, 1) such that (1 − λ) + (1 − µ) = 1 − τ , and r 1 > 0 satisfying M((E r ) r 1 )r 1 < g(µ) (this is always possible since the left hand side is decreasing in r 1 and tends to zero as r 1 → ∞). Since (E r ) r 1 is 2-isomorphic to an ellipsoid, we can find σ 2 ⊆ {1, . . . , n}, |σ 2 | ≥ (1 − µ)n, with
P σ 2 ((E r ) r 1 ) ⊇ g(µ) M((E r ) r 1 ) D σ 2 , thus max{r 1 |x|, x E o r } = x [(Er) r 1 ] o ≥ g(µ) M ((Er) r 1 ) |x| > r 1 |x|, i.e x E o ≥ x E o r > r 1 |x| on R σ 2 , which means that (5.7) t c (E o , 1 r 1 ) ≥ (1 − µ)n.
Again, we may take r 1 to be the solution of the equation M((E r ) r 1 )r 1 = g(µ) in r 1 .
Combining (5.6) with (5.7) we obtain
(5.8) t c (E, r) + t c (E o , 1 r 1 ) ≥ (1 − τ )n,
and it remains to compare r with r 1 . Let us write W for the body (E r ) r 1 . By the way W has been constructed, it is easily checked that the following are satisfied: and making use of (i)-(iii) and of Lemma 5.2 we arrive at (5.9) r r 1 ≤ c g(λ)g(µ) N
C/ng 2 (µ) E o N C/ng 2 (λ) E .
Note that, at some point, we also used the fact that λ E λ E o ≃ 1. Finally, assuming that both E and E o are in M δ -position, we rewrite (5.9) as follows:
(5.10) r r 1 ≤ c g(λ)g(µ) e Cδ/g 2 (λ)g 2 (µ) .
We have λ + µ = 1 + τ and with this condition we can easily check that 1 g(λ)g(µ) ≤ c log( 2 τ ) √ τ , which completes the proof in this case.
Case 2: M (E o r ) r ≥ g(1).
We choose r 1 > 0 such that M((E r ) r 1 )r 1 = g(τ ) and as above we conclude that t c (E o , 1 r 1 ) ≥ (1 − τ )n. The estimate for r/r 1 is done exactly in the same way, the only difference being that now r/M(E o r ) ≤ 1/g(1). Case 3: M ((Er) o ) r < g(τ ). This is the simplest case since we already have t c (E, r) ≥ (1 − τ )n.
Integer points inside an ellipsoid: some remarks
Consider an arbitrary ellipsoid E in R n . Write E in the form (2.1), so that j≤n |u j | 2 = nM 2 E . Without loss of generality we may assume that the |u j |'s are arranged in the increasing order, therefore a simple application of Markov's inequality shows that (6.1) |u j | ≤ n n − j + 1 M E , j = 1, . . . , n.
Recall that the j-th successive minimum λ j (E) of E is defined by λ j (E) = min{λ > 0 : dim(span(λE ∩ Z n )) ≥ j}. Inequality (6.1) gives an estimate on the successive minima of E in terms of M E :
Fact I: Let E be an ellipsoid in R n . Then, λ j (E) ≤ n n−j+1 M E , j = 1, . . . , n. In particular, if M E ≤ 1 then E contains an integer point different from the origin.
Note that if M E > 1 then E may contain no integer points other than the origin. Consider for example a ball of radius r = 1 M E . Let us concentrate on the case M E < 1. If M E < |D n | 1/n /2, then we obviously have |E| > 2 n and Minkowski's theorem with its relatives start giving estimates on the cardinality of the set of integer points in E. We are interested in the range |D n | 1/n /2 < M E < 1. From Fact I we know that E contains non-trivial integer points, and using M E as a parameter we try to estimate the number of them. Theorem 2.4 can be useful in this direction:
Let D m be the m-dimensional Euclidean unit ball, and define d(t, m) = |tD m ∩ Z m | be the cardinality of the set of integer points in tD m . A simple lower bound for d(t, m) can be given by counting the points with coordinates 0, ±1 in tD m :
(6.2) d(t, m) ≥ [t 2 ] k=0 n k 2 k ≥ n [t 2 ] 2 [t 2 ] .
By Theorem 2.4, for every m ≤ c 1 √ n we can find σ ⊆ {1, . . . , n} with |σ| = m and E ∩ R σ ⊇ c 2 √ mM E D σ , where c 1 , c 2 > 0 are absolute constants. Assuming that M E < c 2 and using (6.2) we have some non-trivial information: It is clear that
(6.3) |E ∩ Z n | ≥ max m {|E ∩ Z σ | : |σ| = m ≤ c 1 √ n}
Thus, we have:
Fact II: Let E be an ellipsoid in R n with M E < c 2 < 1. Then,
|E ∩ Z n | ≥ max
−2 }, s ≥ 1. If s 0 = [log k 0 ] + 2, we have 0≤s≤s 0 ϕ s = {0, 1, . . . , k 0 }, and for every s = 1, . . .
. . . , m, and combining with the fact that C K )v n−m+1 (K o ) log 3/2 ( 2n n − m ).Set m = [(1 − θ)n]. Then, (3.17) can be rewritten as
as ε → 1 − .
(i) M(W )r 1 = g(µ) and M(W o ) ≥ M(E o r ) = rg(λ). (ii) x W ≤ 1 r 1 |x| and x W o ≤ r|x|, x ∈ R n ., where c 1 , c 2 > 0 are absolute constants. This is a simple consequence of Lemma 5.1, since both W and W o are formed from E and E o with two successive operations of taking intersection and convex hull
Embedding of ℓ k ∞ in finite dimensional Banach spaces. N Alon, V D Milman, Israel J. Math. 45N. Alon and V. D. Milman, Embedding of ℓ k ∞ in finite dimensional Banach spaces, Israel J. Math. 45 (1983), 365-380.
New volume ratio properties of convex symmetric bodies. J Bourgain, V D Milman, Inventiones Math. 88J. Bourgain and V. D. Milman, New volume ratio properties of convex symmetric bodies, Inventiones Math. 88 (1987), 319-340.
Invertibility of large submatrices with applications to the geometry of Banach spaces and harmonic analysis. J Bourgain, L Tzafriri, Israel J. Math. 57J. Bourgain and L. Tzafriri, Invertibility of large submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Math. 57 (1987), 137-224.
Minkowski sums and symmetrizations. J Bourgain, J Lindenstrauss, V D Milman, Lecture Notes in Mathematics. 1317J. Bourgain, J. Lindenstrauss and V. D. Milman, Minkowski sums and symmetrizations, Lecture Notes in Mathematics 1317 (1988), 44-66.
The distance between certain n-dimensional spaces. W J Davis, V D Milman, N Tomczak-Jaegermann, Israel J. Math. 39W. J. Davis, V. D. Milman and N. Tomczak-Jaegermann, The distance between certain n-dimensional spaces, Israel J. Math. 39 (1981), 1-15.
A proportional Dvoretzky-Rogers factorization result. A A Giannopoulos, Proc. Amer. Math. Soc. 124A. A. Giannopoulos, A proportional Dvoretzky-Rogers factorization result, Proc. Amer. Math. Soc. 124 (1996), 233-241.
On Milman's inequality and random subspaces which escape through a mesh in R n. Y Gordon, Lecture Notes in Mathematics. 1317Y. Gordon, On Milman's inequality and random subspaces which escape through a mesh in R n , Lecture Notes in Mathematics 1317 (1988), 84-106.
Théorèmes de factorization pour les operateurs linéairesà valeurs dans les espaces L p. B Maurey, Astérisque. 11B. Maurey, Théorèmes de factorization pour les operateurs linéairesà valeurs dans les espaces L p , Astérisque 11 (1974), 1-163.
Geometrical inequalities and mixed volumes in the local theory of Banach spaces. V D Milman, Astérisque. 131V. D. Milman, Geometrical inequalities and mixed volumes in the local theory of Banach spaces, Astérisque 131 (1985), 373-400.
Random subspaces of proportional dimension of finite dimensional normed spaces: approach through the isoperimetric inequality. V D Milman, Lecture Notes in Mathematics. 1166V. D. Milman, Random subspaces of proportional dimension of finite dimensional normed spaces: approach through the isoperimetric inequality, Lecture Notes in Mathematics 1166 (1985), 106-115.
V D Milman, Geometry of Banach Spaces, Proceedings of the Conference. P.F.X. Muller and W. SchachermayerStrobl, AustriaCambridge University Press158V. D. Milman, A note on a low M * estimate, Geometry of Banach Spaces, Proceedings of the Conference held in Strobl, Austria, 1989, edited by P.F.X. Muller and W. Schachermayer, LMS Lecture Note Series 158, Cambridge University Press (1990), 219-229.
Some remarks on Uryshon's inequality and volume ratio of cotype-2 spaces. V D Milman, Lecture Notes in Mathematics. 1267V. D. Milman, Some remarks on Uryshon's inequality and volume ratio of cotype-2 spaces, Lecture Notes in Mathematics 1267 (1987), 75-81.
Spectrum of a position of a convex body and linear duality relations. V D Milman, Proceedings (IMCP) 3, Festschrift in Honor of Professor I. Piatetski-Shapiro (Part II). IMCP) 3, Festschrift in Honor of Professor I. Piatetski-Shapiro (Part IIWeizmann Science Press of IsraelV. D. Milman, Spectrum of a position of a convex body and linear duality relations, Israel Math. Conf. Proceedings (IMCP) 3, Festschrift in Honor of Professor I. Piatetski-Shapiro (Part II), Weizmann Science Press of Israel (1990), 151-162.
Some applications of duality relations. V D Milman, Lecture Notes in Mathematics. 1469V. D. Milman, Some applications of duality relations, Lecture Notes in Mathematics 1469 (1991), 13-40.
Séries de variables aleatoires vectorielles independantes et propriétés geometriques des espaces de Banach. B Maurey, G Pisier, Studia Math. 58B. Maurey and G. Pisier, Séries de variables aleatoires vectorielles independantes et pro- priétés geometriques des espaces de Banach, Studia Math. 58 (1976), 45-90.
Banach spaces with a weak cotype 2 property. V D Milman, G Pisier, Israel J. Math. 54V. D. Milman and G. Pisier, Banach spaces with a weak cotype 2 property, Israel J. Math. 54 (1986), 139-158.
Asymptotic Theory of Finite-Dimensional Normed Spaces. V D Milman, G Schechtman, Lecture Notes in Mathematics. 1200V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, Lecture Notes in Mathematics 1200 (1986).
Global vs. local asymptotic theories of finite dimensional normed spaces. V D Milman, G Schechtman, PreprintV. D. Milman and G. Schechtman, Global vs. local asymptotic theories of finite dimensional normed spaces, Preprint.
The Volume of Convex Bodies and Banach Space Geometry. G Pisier, Cambridge Tracts in Math. 94G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94 (1989).
Un théorème sur les opérateurs linéaires entre espaces de Banach qui se factorisent par un espace de Hilbert. G Pisier, Ann. Sci. Ecole Norm. Sup. 13G. Pisier, Un théorème sur les opérateurs linéaires entre espaces de Banach qui se factorisent par un espace de Hilbert, Ann. Sci. Ecole Norm. Sup. 13 (1980), 23-43.
Subspaces of small codimension of finite dimensional Banach spaces. A Pajor, N Tomczak-Jaegermann, Proc. Amer. Math. Soc. 97A. Pajor and N. Tomczak-Jaegermann, Subspaces of small codimension of finite dimensional Banach spaces, Proc. Amer. Math. Soc. 97 (1986), 637-642.
An isomorphic version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube. S J Szarek, M Talagrand, Lecture Notes in Mathematics. 1376S. J. Szarek and M. Talagrand, An isomorphic version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube, Lecture Notes in Mathematics 1376 (1989), 105-112.
On nearly Euclidean decompositions of some classes of Banach spaces. S J Szarek, N Tomczak-Jaegermann, Compositio Math. 40S. J. Szarek and N. Tomczak-Jaegermann, On nearly Euclidean decompositions of some classes of Banach spaces, Compositio Math. 40 (1980), 367-385.
Gaussian random processes and measures of solid angles in Hilbert spaces. V N Sudakov, Soviet Math. Dokl. 12V. N. Sudakov, Gaussian random processes and measures of solid angles in Hilbert spaces, Soviet Math. Dokl. 12 (1971), 412-415.
On Kashin's almost Euclidean orthogonal decomposition of ℓ n 1. S J Szarek, Bull. Acad. Polon. Sci. 26S. J. Szarek, On Kashin's almost Euclidean orthogonal decomposition of ℓ n 1 , Bull. Acad. Polon. Sci. 26 (1978), 691-694.
Embedding of ℓ n ∞ and a theorem of Alon and Milman. M Talagrand, Operator Theory: Advances and applications. 77M. Talagrand, Embedding of ℓ n ∞ and a theorem of Alon and Milman, Operator Theory: Advances and applications, vol. 77 (1995), 289-293.
N Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals. Pitman Monographs38N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monographs 38 (1989).
. A A Giannopoulos, Iraklion, Crete, Greece E-mailDepartment of Mathematics, University of CreteA. A. Giannopoulos, Department of Mathematics, University of Crete, Iraklion, Crete, Greece E-mail address: [email protected]
| [] |
[
"Charged Particles and the Electro-Magnetic Field in Non-Inertial Frames of Minkowski Spacetime: II. Applications: Rotating Frames, Sagnac Effect, Faraday Rotation, Wrap-up Effect",
"Charged Particles and the Electro-Magnetic Field in Non-Inertial Frames of Minkowski Spacetime: II. Applications: Rotating Frames, Sagnac Effect, Faraday Rotation, Wrap-up Effect"
] | [
"David Alba [email protected] \nINFN di Firenze Polo Scientifico\nvia SansoneSezione\n",
"Luca Lusanna \nSezione INFN di Firenze Polo Scientifico\nVia Sansone 150019, 50019Sesto Fiorentino, Sesto Fiorentino (FI)Italy, Italy\n"
] | [
"INFN di Firenze Polo Scientifico\nvia SansoneSezione",
"Sezione INFN di Firenze Polo Scientifico\nVia Sansone 150019, 50019Sesto Fiorentino, Sesto Fiorentino (FI)Italy, Italy"
] | [] | We apply the theory of non-inertial frames in Minkowski space-time, developed in the previous paper, to various relevant physical systems. We give the 3+1 description without coordinatesingularities of the rotating disk and the Sagnac effect, with added comments on pulsar magnetosphere and on a relativistic extension of the Earth-fixed coordinate system. Then we study properties of Maxwell equations in non-inertial frames like the wrap-up effect and the Faraday rotation in astrophysics. | 10.1142/s0219887810004051 | [
"https://arxiv.org/pdf/0908.0215v1.pdf"
] | 14,352,980 | 0812.3057 | 011ef6e4e29cd7c86811394f63eee53c70e763fe |
Charged Particles and the Electro-Magnetic Field in Non-Inertial Frames of Minkowski Spacetime: II. Applications: Rotating Frames, Sagnac Effect, Faraday Rotation, Wrap-up Effect
3 Aug 2009 August 3, 2009
David Alba [email protected]
INFN di Firenze Polo Scientifico
via SansoneSezione
Luca Lusanna
Sezione INFN di Firenze Polo Scientifico
Via Sansone 150019, 50019Sesto Fiorentino, Sesto Fiorentino (FI)Italy, Italy
Charged Particles and the Electro-Magnetic Field in Non-Inertial Frames of Minkowski Spacetime: II. Applications: Rotating Frames, Sagnac Effect, Faraday Rotation, Wrap-up Effect
3 Aug 2009 August 3, 2009
We apply the theory of non-inertial frames in Minkowski space-time, developed in the previous paper, to various relevant physical systems. We give the 3+1 description without coordinatesingularities of the rotating disk and the Sagnac effect, with added comments on pulsar magnetosphere and on a relativistic extension of the Earth-fixed coordinate system. Then we study properties of Maxwell equations in non-inertial frames like the wrap-up effect and the Faraday rotation in astrophysics.
I. INTRODUCTION
In the first paper [1] (quoted as paper I) we developed the general theory of non-inertial frames in Minkowski space-time, whose starting point are its admissible 3+1 splittings defining the allowed conventions for clock synchronization, namely the allowed notions of instantaneous 3-spaces needed, for instance, for setting a well-posed Cauchy problem for Maxwell equations. In this way the coordinate singularities of the traditional 1+3 approach are avoided by construction. In particular it is shown that rigidly rotating frames are not admissible in special relativity.
Also the formulation of charged particles and of the electro-magnetic in non-inertial frames was given.
In this second paper we reformulate relevant physical system, usually described in the 1+3 framework, in the non-inertial frames based on the admissible 3+1 splittings.
In Section II there is a review of the rotating disk and of the Sagnac effect in the 1+3 point of view followed by their description in the framework of the 3+1 point of view (Subsection A) and by a discussion on the ITRS rotating 3-coordinates fixed on the Earth surface (Subsection B).
In Section III we give the 3+1 point of view in admissible nearly rigidly rotating frames of the Wrap Up effect, of the Sagnac effect and of the inertial Faraday rotation by studying electro-magnetic wave solutions of the non-inertial Maxwell equations.
In the Conclusions we give an overview of the results obtained in these two papers and we identify the still open problems about electro-magnetism in non-inertial frames.
II. THE ROTATING DISK AND THE SAGNAC EFFECT
In this Section we give the description of a rotating disk and of the Sagnac effect starting from an admissible 3+1 splitting of Minkowski space-time of the type of Eqs.(2.14) of I, i.e. whose embedding has the form z µ (τ, σ u ) = x µ (τ )+ǫ µ r R r s (τ, σ) σ s with x µ (τ ) = x µ o +f A (τ ) ǫ µ A describing the world-line of the observer origin of the 3-coordinates on the instantaneous 3-spaces Σ τ . This is the simplest non-inertial frame whose 3-spaces are space-like hyperplanes with admissible differentially rotating 3-coordinates. The rotation matrix R r s (τ, σ) = R r s (α i (τ, σ)) = R r s (F (σ)α i (τ )) (σ = | σ|) is admissible if the function F (σ) satisfies the Møller conditions 0 < F (σ) < 1 A σ and d F (σ) dσ = 0.
An enlarged exposition of the material of this Section with a rich bibliography is given in Section I Subsection D and E and in Section VI Subsections B and C of the first paper in Ref. [2].
While at the non-relativistic level one can speak of a rigid (either geometrical or material) disk put in global rigid rotatory motion, the problem of the relativistic rotating disk is still under debate (see Refs. [3,4]) after one century from the enunciation of the Ehrenfest paradox about the 3-geometry of the rotating disk. The problems arise when one tries to define measurements of length, in particular that of the circumference of the disk. Einstein [5] claims that while the rods along the radius R o are unchanged those along the rim of the disk are Lorentz contracted: as a consequence more of them are needed to measure the circumference, which turns out to be greater than 2π R o (non-Euclidean 3-geometry even if Minkowski space-time is 4-flat) and not smaller. This was his reply to Ehrenfest [6], who had pointed an inconsistency in the accepted special relativistic description of the disk 1 in which it is the circumference to be Lorentz contracted: as a consequence this fact was named the Eherenfest paradox (see the historical paper of Grøn in Ref. [7]).
Since relativistic rigid bodies do not exist, at best we can speak of Born rigid motions [8] and Born reference frames 2 . However Grøn [7] has shown that the acceleration phase of a material disk is not compatible with Born rigid motions and, moreover, we do not have a well formulated and accepted relativistic framework to discuss a relativistic elastic material disk. 1 If R and R o denote the radius of the disk in the rotating and inertial frame respectively, then we have R = R o because the velocity is orthogonal to the radius. But the circumference of the rim of the disk is Lorentz contracted so that 2π R < 2π R o inconsistently with Euclidean geometry. 2 A reference frame or platform is Born-rigid [9] if the expansion Θ and the shear σ µν of the associated congruence of time-like observers vanish, i.e. if the spatial distance between neighboring world-lines remains constant.
As a consequence most of the authors treating the rotating disk (either explicitly or implicitly) consider it as a geometrical entity described by a congruence of time-like worldlines (helices in Ref. [10]) with non-zero vorticity, i.e. non-surface forming and therefore non-synchronizable (see for instance Ref. [11]). This means that there is no notion of instantaneous 3-space where to visualize the disk (see Ref. [3] for the attempts to define rods and clocks associated to this type of congruences): every observer on one of these time-like world-lines can only define the local rest frame and try to define a local accelerated reference frame as said in Section IIB of paper I.
In the 3+1 point of view the disk is considered to be a relativistic isolated system (either a relativistic material body or a relativistic fluid or a relativistic dust as a limit case 3 ) with compact support always contained in a finite time-like world-tube W , which in the Cartesian 4-coordinates of an inertial system is a time-like cylinder of radius R. Each admissible 3+1 splitting of Minkowski space-time, centered on an arbitrary time-like observer and with its two associated congruences of time-like observers (see Section IIB of paper I), gives a visualization of the disk in its instantaneous 3-spaces Σ τ : at each instant τ the points of the disk in W ∩ Σ τ are synchronized and through each one of them pass an Eulerian observer belonging to the surface forming congruence having as 4-velocity the unit normal to the instantaneous 3-spaces Σ τ . Instead the irrotational congruence of the disk is described by the second congruence (whose unit 4-velocity is z µ τ (τ, σ u )/ ǫ g τ τ (τ, σ u ) and whose observers follow generalized helices σ u = σ u o ) associated to the admissible 3+1 splitting: each of the observers of this congruence, whose world-lines are inside W , has no intrinsic notion of synchronization.
As a consequence, each instantaneous 3-space Σ τ of an admissible 3+1 splitting has a well defined (in general Riemannian) notion of 3-geometry and of spatial length: the radius and the circumference of the disk are defined in W ∩ Σ τ , so that the disk 3-geometry is 3+1 splitting dependent. When the material disk can be described by means of a parametrized Minkowski theory, all these 3-geometry are gauge equivalent like the notions of clock synchronization.
The other important phenomenon connected with the rotating disk is the Sagnac effect (see the recent review in Ref. [13] for how many interpretations of it exist), namely the phase difference generated by the difference in the time needed for a round-trip by two light rays, emitted in the same point, one co-rotating and the other counter-rotating with the 3 As an example of a congruence simulating a geometrical rotating disk we can consider the relativistic dust described by generalized Eulerian coordinates of Ref. [12] after the gauge fixing to a family of differentially rotating parallel hyper-planes. disk 4 . This effect, which has been tested (see the bibliography of Refs. [13,17]) for light, X rays and matter waves (Cooper pairs, neutrons, electrons and atoms), has important technological applications and must be taken into account for the relativistic corrections to space navigation, has again an enormous number of theoretical interpretations (both in special and general relativity) like for the solutions of the Ehrenfest paradox. Here the lack of a good notion of simultaneity leads to problems of time discontinuities or desynchronization effects when comparing clocks on the rim of the rotating disk.
Another area which is in a not well established form is electrodynamics in non-inertial systems either in vacuum or in material media (problem of the non-inertial constitutive equations). Its clarification is needed both to derive the Sagnac effect from Maxwell equations without gauge ambiguities [14] and to determine which types of experiments can be explained by using the locality hypothesis (see Section IIB of paper I) to evaluate the electro-magnetic fields in the comoving system (see the Wilson experiment and the associated controversy [18] on the validity of the locality principle) without the need of a more elaborate treatment like for the radiation of accelerated charges. It would also help in the tests of the validity of special relativity (for instance on the possible existence of a preferred frame) based on Michelson-Morley -type experiments [19,20].
Instead (see also Ref. [14]) we remark that the Sagnac effect and the Foucault pendulum are experiments which signal the rotational non-inertiality of the frame. The same is true for neutron interferometry [21], where different settings of the apparatus are used to detect either rotational or translational non-inertiality of the laboratory. As a consequence a null result of these experiments can be used to give a definition of relativistic quasi-inertial system.
Let us remark that the disturbing aspects of rotations are rooted in the fact that there is a deep difference between translations and rotations at every level both in Newtonian 4 For monochromatic light in vacuum with wavelength λ the fringe shift is δz = 4 Ω · A/λ c, where Ω is the Galilean velocity of the rotating disk supporting the interferometer and A is the vector associated to the area | A| enclosed by the light path. The time difference is δt = λ δz/c = 4 Ω · A/c 2 , which agrees, at the lowest order, with the proper time difference δτ = (4 A Ω/c 2 ) (1 − Ω 2 R 2 /c 2 ) −1/2 , A = π R 2 , evaluated in an inertial system with the standard rotating disk coordinates. This proper time difference is twice the time lag due to the synchronization gap predicted for a clock on the rim of the rotating disk with a non-time orthogonal metric. See Refs. [13,14,15] for more details. See also Ref. [16] for the corrections included in the GPS protocol to allow the possibility of making the synchronization of the entire system of ground-based and orbiting atomic clocks in a reference local inertial system. Since usually, also in GPS, the rotating coordinate system has t ′ = t (t is the time of an inertial observer on the axis of the disk) the gap is a consequence of the impossibility to extend Einstein's convention of the inertial system also to the non-inertial one rotating with the disk: after one period two nearby synchronized clocks on the rim are out of synchrony.
mechanics and special relativity: the generators of translations satisfy an Abelian algebra, while the rotational ones a non-Abelian algebra. As shown in Refs. [22], at the Hamiltonian level we have that the translation generators are the three components of the momentum, while the generators of rotations are a pair of canonical variables (L 3 and arctg L 2 L 1 ) and an unpaired variable (| L|). As a consequence we can separate globally the motion of the 3-center of mass of an isolated system from the relative variables, but we cannot separate in a global and unique way three Euler angles describing an overall rotation, because the residual vibrational degrees of freedom are not uniquely defined.
We will now give the 3+1 point of view on these topics (Subsection A), followed by a discussion on the rotating 3-coordinates fixed to the Earth surface (Subsection B).
A. The 3+1 Point of View on the Rotating Disk and the Sagnac Effect.
Let us describe an abstract geometrical disk with an admissible 3+1 splitting of the type (2.14) of I, in which the instantaneous 3-spaces are parallel space-like hyper-planes with normal l µ centered on an inertial observer x µ (τ ) = l µ τ
z µ (τ, σ) = l µ τ + ǫ µ r R r (3) s (τ, σ) σ s . (2.1)
The rotation matrix R (3) describes a differential rotation around the fixed axis "3" (we take a constant ω, but nothing changes with ω(τ ))
R r (3) s (τ, σ) = cos θ(τ, σ) − sin θ(τ, σ) 0 sin θ(τ, σ) cos θ(τ, σ) 0 0 0 1 , θ(τ, σ) = F (σ) ω τ, F (σ) < c ω σ , Ω r s (τ, σ) = R −1 (3) dR (3) dτ r s (τ, σ) = ω F (σ) 0 −1 0 1 0 0 0 0 0 , Ω(τ, σ) = Ω(σ) = ω F (σ). (2.2) A simple choice for the gauge function F (σ) is F (σ) = 1 1+ ω 2 σ 2 c 2
(in the rest of the Section we put c = 1), so that at spatial infinity we get Ω(τ, σ) = ω
1+ ω 2 σ 2 c 2 → σ→∞ 0.
By introducing cylindrical 3-coordinates r, ϕ, h by means of the equations σ 1 = r cos ϕ, σ 2 = r sin ϕ, σ 3 = h, σ = √ r 2 + h 2 , we get the following form of the embedding and of its gradients
z µ (τ, σ) = l µ τ + ǫ µ 1 [cos θ(τ, σ) σ 1 − sin θ(τ, σ) σ 2 ] + + ǫ µ 2 [sin θ(τ, σ) σ 1 + cos θ(τ, σ) σ 2 ] + ǫ µ 3 σ 3 = = l µ τ + ǫ µ 1 r cos [θ(τ, σ) + ϕ] + ǫ µ 2 r sin [θ(τ, σ) + ϕ] + ǫ µ 3 h, ∂z µ (τ, σ) ∂τ = z µ τ (τ, σ) = l µ − ω r F (σ) ǫ µ 1 sin [θ(τ, σ) + ϕ] − ǫ µ 2 cos [θ(τ, σ) + ϕ] , ∂z µ (τ, σ) ∂ϕ = z µ ϕ (τ, σ) = −ǫ µ 1 r sin [θ(τ, σ) + ϕ] + ǫ µ 2 r cos [θ(τ, σ) + ϕ] ∂z µ (τ, σ) ∂r = z µ (r) (τ, σ) = −ǫ µ 1 (cos [θ(τ, σ) + ϕ] − r 2 ωτ √ r 2 + h 2 dF (σ) dσ sin [θ(τ, σ) + ϕ] + + ǫ µ 2 sin [θ(τ, σ) + ϕ] + r 2 ωτ √ r 2 + h 2 cos [θ(τ, σ) + ϕ] ∂z µ (τ, σ) ∂h = z µ h (τ, σ) = ǫ µ 3 − ǫ µ 1 rhωτ √ r 2 + h 2 dF (σ) dσ sin [θ(τ, σ) + ϕ] + + ǫ µ 2 rhωτ √ r 2 + h 2 dF (σ) dσ cos [θ(τ, σ) + ϕ] ,(2.3)
where we have used the notation (r) to avoid confusion with the index r used as 3-vector index (for example in σ r ).
In the cylindrical 4-coordinates τ , r, ϕ and h the 4-metric is
ǫ g τ τ (τ, σ) = 1 − ω 2 r 2 F 2 (σ), ǫ g τ ϕ (τ, σ) = −ω r 2 F (σ), ǫ g ϕϕ (τ, σ) = −r 2 , ǫ g τ (r) (τ, σ) = − ω 2 r 3 τ √ r 2 + h 2 F (σ) dF (σ) dσ , ǫ g τ h (τ, σ) = − ω 2 r 2 h τ √ r 2 + h 2 F (σ) dF (σ) dσ , ǫ g (r)(r) (τ, σ) = −1 − r 4 ω 2 τ 2 r 2 + h 2 dF (σ) dσ 2 , ǫ g hh (τ, σ) = −1 − r 2 h 2 ω 2 τ 2 r 2 + h 2 dF (σ) dσ 2 , ǫ g (r)ϕ (τ, σ) = − ω r 3 τ √ r 2 + h 2 dF (σ) dσ , ǫ g hϕ (τ, σ) = − ω 2 r 2 h τ √ r 2 + h 2 dF (σ) dσ , ǫ g h(r) (τ, σ) = − r 3 h ω 2 τ 2 r 2 + h 2 dF (σ) dσ 2 , with inverse ǫ g τ τ (τ, σ) = 1, ǫ g τ ϕ (τ, σ) = −ω F (σ), ǫ g τ (r) (τ, σ) = ǫ g τ h (τ, σ) = 0, ǫ g (r)(r) (τ, σ) = ǫ g hh (τ, σ) = −1, ǫ g ϕϕ (τ, σ) = − 1 + ω 2 r 2 [τ 2 ( dF (σ) dσ ) 2 − F 2 (σ) r 2 , ǫ g ϕ(r) (τ, σ) = ω r τ √ r 2 + h 2 dF (σ) dσ , ǫ g ϕh (τ, σ) = ω h τ √ r 2 + h 2 dF (σ) dσ . (2.4)
It is easy to observe that the congruence of (non inertial) observers defined by the 4velocity field
z µ τ (τ, σ) ǫ g τ τ (τ, σ) = l µ − ω r F (σ) ǫ µ 1 sin [θ(τ, σ) + ϕ] − ǫ µ 2 cos [θ(τ, σ) + ϕ] 1 − ω 2 r 2 F 2 (σ) , (2.5)
has the observers moving along the world-lines
x µ σo (τ ) = z µ (τ, σ o ) = = l µ τ + r o ǫ µ 1 cos [ω τ F (σ o ) + ϕ o ] + ǫ µ 2 sin [ω τ F (σ o ) + ϕ o ] + ǫ µ 3 h o . (2.6)
The world-lines (2.6) are labeled by their initial value
σ = σ o = (ϕ o , r o , h o ) at τ = 0.
In particular for h o = 0 and r o = R these world-lines are helices on the cylinder in the Minkowski space
ǫ µ 3 z µ = 0, (ǫ µ 1 z µ ) 2 + (ǫ µ 2 z µ ) 2 = R 2 , or r = R, h = 0. (2.7)
These helices are defined the equations ϕ = ϕ o , r = R, h = 0 if expressed in the embedding adapted coordinates ϕ, r, h. Then the congruence of observers (2.5), defined by the foliation (2.1), defines on the cylinder (2.7) the rotating observers usually assigned to the rim of a rotating disk, namely observes running along the helices
x µ σo (τ ) = l µ τ + R ǫ µ 1 cos [Ω(R) τ + ϕ o ] + ǫ µ 2 sin [Ω(R) τ + ϕ o ] after having put Ω(R) ≡ ω F (R).
On the cylinder (2.7) the line element is obtained from the line element ds 2 = g AB dσ A dσ B for the metric (2.4) by putting dh = dr = 0 and r = R, h = 0. Therefore the cylinder line element is
ǫ (ds cyl ) 2 = 1 − ω 2 R 2 F 2 (R) (dτ ) 2 − 2 ω R 2 F (R) dτ dϕ − R 2 (dϕ) 2 .
(2.8)
We can define the light rays on the cylinder, i.e. the null curves on it, by solving the equation
ǫ (ds cyl ) 2 = (1 − R 2 Ω 2 (R)) dτ 2 − 2 R 2 Ω(R) dτ dϕ − R 2 dϕ 2 = 0, (2.9) which implies R 2 dϕ(τ ) dτ 2 + 2 R 2 Ω(R) dϕ(τ ) dτ − (1 − R 2 Ω(R)) = 0. (2.10)
The two solutions
dϕ(τ ) dτ = ± 1 R − Ω(R),(2.11)
define the world-lines on the cylinder for clockwise or anti-clockwise rays of light.
Γ 1 : ϕ(τ ) − ϕ o = + 1 R − Ω(R) τ, Γ 2 : ϕ(τ ) − ϕ o = − 1 R − Ω(R) τ . (2.12)
This is the geometric origin of the Sagnac effect. Since Γ 1 describes the world-line of the ray of light emitted at τ = 0 by the rotating observer ϕ = ϕ o in the increasing sense of ϕ (anti-clockwise), while Γ 2 describes that of the ray of light emitted at τ = 0 by the same observer in the decreasing sense of ϕ (clockwise), then the two rays of light will be re-absorbed by the same observer at different τ -times 5 τ (± 2π) , whose value, determined by the two conditions ϕ(τ (± 2π) ) − ϕ o = ± 2π, is
Γ 1 : τ (+2π) = 2π R 1−Ω(R) R , Γ 2 : τ (−2π) = 2π R 1+Ω(R) R . (2.13)
The time difference between the re-absorption of the two light rays is
∆τ = τ (+2π) − τ (−2π) = 4π R 2 Ω(R) 1 − Ω 2 (R) R 2 = 4π R 2 ω F (R) 1 − ω 2 F 2 (R) R 2 ,
(2.14)
and it corresponds to the phase difference named the Sagnac effect (see footnote 4)
∆Φ = Ω ∆τ, Ω = Ω(R) = ω F (R). (2.15)
We see that we recover the standard result if we take a function F (σ) such that F (R) = 1. In the non-relativistic applications, where F (σ) → 1, the correction implied by admissible relativistic coordinates is totally irrelevant.
With an admissible notion of simultaneity, all the clocks on the rim of the rotating disk lying on a hyper-surface Σ τ are automatically synchronized. Instead for rotating observers of the irrotational congruence there is a desynchronization effect or synchronization gap because they cannot make a global synchronization of their clocks: usually a discontinuity in the synchronization of clocks is accepted and taken into account (see Ref. [16] for the GPS).
To clarify this point and see the emergence of this gap, let us consider a reference observer (ϕ o = const., τ ) and another one (ϕ = const. = ϕ o , τ ). If ϕ > ϕ o we use the notation (ϕ R , τ ),
while for ϕ < ϕ o the notation (ϕ L , τ ) with ϕ R − ϕ o = −(ϕ L − ϕ o ).
Let us consider the two rays of light Γ R − and Γ L − , with world-lines given by Eqs.(2.12), emitted in the right and left directions at the event (ϕ o , τ − ) on the rim of the disk and received at τ at the events (ϕ R , τ ) and (ϕ L , τ ) respectively. Both of them are reflected towards the reference observer, so that we have two rays of light Γ R + and Γ L + which will be absorbed at the event (ϕ o , τ + ). By using Eq.(2.13) for the light propagation, we get
Γ R − : (ϕ − ϕ o ) = 1 − RΩ(R) R (τ − τ − ), Γ R + : (ϕ − ϕ o ) = 1 + RΩ(R) R (τ + − τ ), Γ L − : (ϕ − ϕ o ) = − 1 + RΩ(R) R (τ − τ − ), Γ L + : (ϕ − ϕ o ) = − 1 − RΩ(R) R (τ + − τ ). (2.16)
As shown in Section II, Eqs.(2.17) and (2.18), of the first paper in Ref. [2], in a neighborhood of the observer (ϕ o , τ ) [(ϕ, τ ) is an observer in the neighborhood] we can only define the following local synchronization 6
c ∆ T = 1 − R 2 Ω 2 (R) ∆τ E = 1 − R 2 Ω 2 (R) ∆τ − R 2 Ω 2 (R) 1 − R 2 Ω 2 (R) ∆ϕ. (2.17)
If we try to extend this local synchronization to a global one for two distant observers (ϕ o , τ ) and (ϕ, τ ) in the form of an Einstein convention (the result is the same both for ϕ = ϕ R and ϕ = ϕ L )
τ E = 1 2 (τ + + τ − ) = τ − R 2 Ω(R) 1 − R 2 Ω 2 (R) (ϕ − ϕ o ), (2.18)
we arrive at a contradiction, because the curves defined by τ E = constant are not closed, since they are helices that assign the same time τ E to different events on the world-line of an observer ϕ o = constant. For example (ϕ o , τ ) and ϕ o , τ + 2π R 2 Ω(R) 1−R 2 Ω 2 (R) are on the same helix τ E = constant. As a consequence we get the synchronization gap.
As shown in both papers of Ref. [2], by using the global synchronization on the instantaneous 3-spaces Σ τ we can define a generalization of Einstein's convention for clock synchronization by using the radar time τ . If an accelerated observer A emits a light signal at τ − , which is reflected at a point P of the world-line of a second observer B and then reabsorbed at τ + , then the B clock at P has to be synchronized with the following instant of the A clock
[n = + for ϕ = ϕ R , n = − for ϕ = ϕ L ] τ (τ − , n, τ + ) = 1 2 (τ + + τ − ) − n R Ω(R) 2 (τ + − τ − ) def = τ − + E(τ − , n, τ + ) (τ + − τ − ), with E(τ − , n, τ + ) = 1 − n R Ω(R) 2 , Ω(R) = ω F (R). (2.19)
Finally in the first paper of Ref. [2] [see Eqs.(6.37)-(6.47) of Section VI] there is the evaluation of the radius and the circumference of the rotating disk. If we choose the spatial length of the instantaneous 3-space Σ τ of the admissible embedding (2.1), we get an Euclidean 3-geometry, i.e. a circumference 2π R and a radius R at each instant τ independently from the choice of the gauge function F (σ). With other admissible 3+1 splittings we would get non-Euclidean results: as said they are gauge equivalent when the disk can be described with a parametrized Minkowski theory. Instead the use of a local notion of synchronization from the observers of the irrotational congruence located on the rim of the rotating disk implies a local definition of spatial distance based on the 3-metric 3 γ uv = −ǫ g uv − gτu gτv gττ , i.e. a non-Euclidean 3-geometry. In this case the radius is R, but the circumference is 2π R/ 1 − R 2 Ω 2 (R). However this result holds only in the local rest frame of the observer with the tangent plane orthogonal to the observer 4-velocity (also called the abstract relative space) identified with a 3-space (see Section IIB of paper I).
See Subsection D of Section III for a derivation of the Sagnac effect in nearly rigid rotating frames.
B. The Rotating ITRS 3-Coordinates fixed on the Earth Surface.
The embedding (2.14) of I, describing admissible differential rotations in an Euclidean 3-space, could be used to improve the conventions IERS2003 (International Earth Rotation and Reference System Service) [23] on the non-relativistic transformation from the 4-coordinates of the Geocentric Celestial Reference System (GCRS) to the International Terrestrial Reference System (ITRS), the Earth-fixed geodetic system of the new theory of Earth rotation replacing the old precession-nutation theory. It would be a special relativistic improvement to be considered as an intermediate step till to a future development leading to a post-Newtonian (PN) general relativistic approach unifying the existing non-relativistic theory of the geo-potential below the Earth surface with the GCRS PN description of the geo-potential outside the Earth given by the conventions IAU2000 (International Astronomical Union) [23] for Astrometry, Celestial Mechanics and Metrology in the relativistic framework.
In the IAU 2000 Conventions the Solar System is described in the Barycentric Celestial Reference System (BCRS) as a quasi-inertial frame, centered on the barycenter of the Solar System, with respect to the Galaxy. BCRS is parametrized with harmonic PN 4-coordinates with Cartesian 3-coordinates x i BCRS . This frame is used for space navigation in the Solar System. The geo-center (a fictitious observer at the center of the earth geoid) has a world-line
x µ BCRS = x o BCRS = c t BCRS ; x i BCRS ,y µ BCRS (x o BCRS ) = x o BCRS ; y i BCRS (x o BCRS )
, which is approximately a straight line.
For space navigation near the Earth (for the Space Station and near Earth satellites using NASA coordinates) and for the studies from spaces of the geo-potential one uses the GCRS, which is defined outside the Earth surface as a local reference system centered on the geo-center. Due to the earth rotation of the Earth around the Sun, it deviates from a nearly inertial special relativistic frame on time scales of the order of the revolution time. Its harmonic 4-coordinates Let us now consider the embedding z µ (τ, σ u ) = x µ (τ ) + ǫ µ r R r s (τ, σ) σ s of Eq.(2.14) of I. Let us identify x µ = z µ (τ, σ u ) with the GCRS 4-coordinates x µ GCRS centered on the world-line of the geo-center assumed to move along a straight line. Then, if we identify the space-like vectors ǫ µ r with the GCRS non-rotating spatial axes, we have x µ (τ ) = ǫ µ τ τ = l µ τ , where l µ is orthogonal to the nearly Euclidean 3-spaces t GCRS = const.. for the gauge function (ω can be taken equal to the mean angular velocity for the Earth rotation), will contain three Euler angles determined
x µ GCRS = x o GCRS = c t GCRS ; x i GCRS ,i IT RS = W T (t GCRS ) R T 3 (−θ) C i j x j GCRS , where C = R T 3 (s) R T 3 (E) R T 2 (−d) R T 3 (−E) and W (t GCRS ) = R 3 (−s ′ ) R 2 (x p ) R 1 (y p )by putting R(τ, σ)| F (σ)=1 = C T R 3 (−θ) W (t GCRS ).
In this way a special relativistic version of ITRS could be given as a preliminary step towards a PN general relativistic formulation of the geo-potential inside the Earth to be joined consistently with GCRS outside the Earth. Even if this is irrelevant for geodesy inside the geoid, it could lead to a refined treatment of effects like geodesic precession taking into account a model of geo-potential interpolating smoothly between inside and outside the geoid and the future theory of heights over the reference ellipsoid under development in a formulation of relativistic geodesy based on the use of the new generation of microwave and optical atomic clocks both on the Earth surface and in space.
III. NON-INERTIAL MAXWELL EQUATIONS IN NEARLY RIGID ROTATING FRAMES
In the 3+1 point of view the Maxwell equations (4.17) of I in an arbitrary inertial frame are identical to the Maxwell equations in general relativity, but now the 4-metric is describing only the inertial effects present in the given frame. Therefore we can adapt the techniques used in general relativity to non-inertial frames, for instance the definition of electric and magnetic fields done in Ref. [24] (see Appendix A of paper I) or the geometrical optic approximation to light rays of Ref. [25].
For the 1+3 point of view on this topic see for instance Ref. [26] and its bibliography. In particular, for the treatment of electromagnetic wave in rotating frame by means of Fermi coordinates [27] and for the determination of the helicity-rotation coupling, as a special case of spin-rotation coupling [28,29]. In all these calculations the locality hypothesis (see Section IIB of paper I) is used.
In the case of linear acceleration an analysis of the inertial effects has been done in Ref. [20]. The same non-inertial 4-metric has been used in Ref. [30] to study the optical position meters constituents of the laser interferometers on ground used for the detection of gravitational waves. However the 4-metric used has a bad behavior at spatial infinity, so that the conclusions on the electro-magnetic waves in these frames (even if supposed to hold at distances smaller than those where there are coordinate singularities) are questionable because the Cauchy problem for Maxwell equations is not well posed.
In this Section we study some properties of electro-magnetic waves and of geometrical optic approximation to light rays in the radiation gauge in the admissible rotating noninertial frame defined by the embedding (2.14) of I, ensuring a well-posed Cauchy problem, at small distances from the rotation axis where the O(c −1 ) deviations from rigid rotations is governed by Eqs.(2.15) and (2.16) of I. Even if we will ignore these deviations, doing the calculations in the radiation gauge in locally rigidly rotating frames, they could be taken into account in a more refined version of the subsequent calculations base on the 3+1 point of view, which is free from coordinate singularities. This would also allow to verify the validity of the locality hypothesis. In particular we consider the Phase Wrap Up effect [16,27], the Sagnac effect [14,31] and the Faraday Rotation [32].
A. The 3+1 Point of View on Electro-Magnetic Waves and Light Rays in Nearly
Rigidly Rotating Non-Inertial Frames.
Let us consider a non-inertial frame of the type (2.14) of I with vanishing linear acceleration and τ -independent angular velocity and centered on an inertial observer. In the notation of Eqs.(2.15), (2.16) and (4.47) of I, we have x µ (τ ) = ǫ µ τ τ , i.e. v(τ ) = w(τ ) = 0, and Ω(τ ) = Ω = const. (whose components areΩ r = const.). We will ignore the higher order terms, so that locally we have a rigidly rotating frame, but with more effort small deviations from rigid rotation could be taken into account.
In this case the Hamiltonian (4.35), or (4.51), of I gives the following Hamilton equations for the transverse electro-magnetic field (
A ⊥ = {A ⊥ r =à r ⊥ = A r ⊥ } + O(c −2 )) ∂à r ⊥ (τ, σ) ∂τ = π r ⊥ (τ, σ) − 1 c d 3 σ ′ − Ω · σ ′ × ∂ ′ A ⊥ (τ, σ ′ ) + Ω × A ⊥ (τ, σ ′ ) s P sr ( σ ′ , σ), ∂π r ⊥ (τ, σ) ∂τ = ∆à r ⊥ (τ, σ) − 1 c d 3 σ ′ − Ω · σ ′ × ∂ ′ π ⊥ (τ, σ ′ ) + Ω × π ⊥ (τ, σ ′ ) s P sr ( σ ′ , σ) + + i Q i ˙ η i (τ ) + Ω × η i (τ ) s P sr ( η i , σ). (3.1)
For the study of homogeneous solutions of these equations, i.e. for incoming electromagnetic waves propagating in regions where there are no charged particles, these equations can be replaced with the following ones (we use the vector notation of Section IVC of paper I)
∂ A ⊥ (τ, σ) ∂τ = π ⊥ (τ, σ) − 1 c − Ω · σ × ∂ A ⊥ (τ, σ) + Ω × A ⊥ (τ, σ) , ∂ π ⊥ (τ, σ) ∂τ = ∆ A ⊥ (τ, σ) − 1 c − Ω · σ × ∂ π ⊥ (τ, σ) + Ω × π ⊥ (τ, σ) . (3.2)
As shown in Appendix A of I, this result allows to recover the form given by Schiff in
Appendix A of ref. [24] for the Landau-Lifschitz non-inertial electro-magnetic fields [33].
Let us look at solutions of Eqs.(3.2) in the following two ways.
Going back to an Inertial Frame
Let us look at solution by reverting to an inertial frame.
By introducing the 3-coordinates
X a (τ ) = R a r (τ ) σ r ,(3.3)
at each value of τ by means of a τ -dependent rotation (it would become also point-dependent if we go beyond rigid rotations) we can go from the rigidly rotating non-inertial frame with radar 4-coordinates (τ ; σ u ) to an instantaneously comoving inertial frame, centered on the same inertial observer, with 4-coordinates (τ ; X a ).
Let us assume that the non-inertial transverse electromagnetic potential A ⊥ r (τ, σ u ) can be obtained from the instantaneously comoving inertial transverse potential A (com) ⊥ a (τ, X a (τ )) by using the rotation matrix R(τ )
A ⊥ r (τ, σ u ) = A (com) ⊥ a τ, X a (τ ) = R a s (τ ) σ s R a r (τ ). (3.4) By definition A (com)
⊥ a (τ, X a (τ )) satisfies the inertial Maxwell equations in the radiation gauge (obtainable by putting Eqs. (3.4
) into Eqs.(3.2))
∂ 2 A (com) ⊥ a (τ, X b ) ∂τ 2 − ∆ X A (com) ⊥ a (τ, X b ) = 0, a ∂ ∂X a A (com) ⊥ a (τ, X b ) = 0. (3.5)
This result is in accord with the general covariance of non-inertial Maxwell equations and is also consistent with the locality hypothesis (see Section IIB of paper I) of the the 1+3 approach.
If we consider the following plane wave solution with constant F a andK a and aK a F a = 0 (transversality condition)
A (com) ⊥ a (τ, X b ) = 1 ω F a e i ω c (τ− P aK a X a ) , (3.6)
we get the following expression for the non-inertial solution
A ⊥ r (τ, σ u ) = F a R a r (τ ) e i ω c Φ(τ,σ u ) , Φ(τ, σ u ) = τ −K a R a r (τ ) σ r ≈ | Ω=const. τ 1 + Ω c · σ ×K −K · σ + O(Ω 2 /c 2 ).(3.7)
Eikonal Approximation
Let us now look at solutions by making the following eikonal approximation (without any commitment with the locality hypothesis)
A ⊥ r (τ, σ u ) = 1 ω a r (τ, σ u ) e i ω c Φ(τ,σ u ) + O(1/ω 2 )
. (3.8) and by putting this expression in Eqs. (3.2).
Let us consider the case in which we have ω/c >> 1 e Ω/c << 1, so that Eqs.(3.2) become a power series in ω/c. By neglecting terms in Ω 2 /c 2 and terms in (c/ω) −k for k ≥ 0, the dominant terms are: a) at the order ω/c the equation for the phase Φ, named eikonal equation;
b) at the order (ω/c) o = 1 the equation for the amplitude a r , named first-order transport equation.
These equations have the following form ( a = {a r })
∂Φ ∂τ 2 − 2 Ω c · σ × ∂ Φ − ∂ Φ 2 (τ, σ u ) + O(Ω 2 /c 2 ) = 0 ∂Φ ∂τ ∂ a ∂τ + Ω c × a − Ω c · σ × ∂ a − ∂ a ∂τ Ω c · σ × ∂ Φ − ∂ Φ · ∂ a (τ, σ u ) = = − 1 2 ∂ 2 Φ ∂τ 2 − 2 ( Ω × σ · ∂) ∂Φ ∂τ − △Φ (τ, σ u ) + O(Ω 2 /c 2 )
a · ∂ Φ (τ, σ u ) = 0 (transversality condition). This condition implies that the solution of Eq.(3.8) describes a ray emitted from a source having a characteristic frequency ω when it is at rest in the non-inertial frame. Let us remark that in more general cases this type of boundary conditions are possible only if the 3-metric h rs and the lapse (n) and shift (n r ) functions are stationary in the non-inertial frame.
An expansion in powers of Ω/c of
F (σ u ), namely F (σ u ) = F o (σ u ) + Ω c F 1 (σ u ) + O Ω 2 c 2 ,
gives the following form of the eikonal equation (3.12) implying:
1 − ∂ F o (σ u ) 2 − 2Ω c Ω · σ × ∂F o (σ u ) + ∂ F o (σ u ) · ∂ F 1 (σ u ) + O Ω 2 c 2 = 0,a) the equation 1 − ∂ F o (σ u ) 2
= 0 at the order zero in Ω. Ifk is an arbitrary unit vector (the propagation direction of the plane wave in the inertial limit Ω → 0), its solution is
F o (σ u ) = −k · σ.
(3.13) b) the equationk · ∂ F 1 (σ u ) = −Ω · σ ×k for F 1 (σ u ), after having used Eq. (3.13), at the order one in Ω. Since we have (k · ∂) (Ω · σ ×k) = 0 and (k · ∂) (k · σ) = 1, the solution for
F 1 (σ u ) is F 1 (σ u ) = − Ω · σ ×k (k · σ).
(3.14)
Therefore the solution for Φ is
Φ(τ, σ u ) = τ −k · σ 1 + Ω · σ ×k .Φ ∂ τ = 1 −K a R a r (τ ) ǫ ruvΩ u σ v = 1.
Let us remark that both the solutions (3.7) and (3.15) have the following structurẽ
A r ⊥ (τ, σ u ) ∼ A r (τ, σ u ) e i ϕ(τ,σ u ) , (3.16)
where A r (τ, σ u ) ∼ O(1/ω) is the amplitude and ϕ(τ, σ u ) ∼ O(ω) is the phase. The only difference is that the solution (3.7) holds for every value of ω (also for the small values corresponding to the radio waves of the GPS system), while the solution (3.15) for the phase of the eikonal approximation (3.8) holds only for higher values of ω, corresponding to visible light.
Light Rays
Given the phase of Eq.(3.16), the trajectories of the light rays are defined as the lines orthogonal (with respect to the 4-metric g AB of the 3+1 splitting) to the hyper-surfaces ϕ(τ, σ u ) = const.. Therefore the trajectories σ A (s) (s is n affine parameter) satisfy the equation dσ A (s) ds = g AB (σ(s)) ∂ϕ ∂σ B (σ(s)). (3.17) For instance in the case of our rigidly rotating foliation, for which Eqs.(2.14)-(2.16) of I imply g τ τ = 1, g τ r = −( Ω × σ) r , g rs = −δ rs + O(Ω 2 /c 2 ), Eqs.(3.17) take the form
dτ (s) ds = ω + k · Ω c × σ + O(Ω 2 /c 2 ), dσ r (s) ds = ω Ω c × σ r + k r 1 + Ω c × σ ·k − Ω c ×k r ( k · σ) + O(Ω 2 /c 2 ),(3.18)
whose solution has the form
σ(τ ) − σ(0) =k τ + Ω c ×k τ 2 + O(Ω 2 /c 2 ). (3.19)
This equation shows that in the rotating frame the ray of light appears to deviate from the inertial trajectory σ(τ ) =k τ due to the centrifugal correction c(τ ) = Ω c ×k τ 2 +O(Ω 2 /c 2 ) implyingk · c(τ ) = 0 + O(Ω 2 /c 2 ).
B. Sources and Detectors
To connect the previous solutions to the interpretation of observed data we need a schematic description of sources and detectors.
In many applications sources and detectors are described as point-like objects, which follow a prescribed world-line ζ A (τ ) = (τ, η u (τ )) with unit 4-velocity v A (τ ) =
dζ A (τ ) dτ g CD (ζ(τ )) dζ C (τ ) dτ dζ D (τ ) dτ −1/2 .
This description is enough for studying the influence of the relative motion between source and detector on the frequency emitted from the source and that observed by the detector (it works equally well for the Doppler effect and for the gravitational redshift in presence of gravity). With solutions like Eq.(3.16) the frequency emitted by a source located in ζ s A and moving with 4-velocity v s A and that observed by a detector in ζ r A and moving with
4-velocity v r A are ω s = v s A ∂ A ϕ(ζ s ) and ω r = v r A ∂ A ϕ(ζ r ), respectively.
This justifies the boundary condition (3.11), because sources at rest in the rotating frame with coordinates (τ, σ r ) have 4-velocity v A = (1, 0). However, to measure the electro-magnetic field in assigned (spatial) polarization direction we must assume that the detector is endowed with a tetrad orthonormal with respect to the 4-metric of the 3+1 splitting, such that the time-like 4-vector is the unit 4-velocity of the detector: in 4-coordinates adapted to the 3+1 splitting they are
E A (α) (τ ) = E A (o) (τ ) = v A (τ ); E A (i) (τ ) , g AB (ζ r (τ )) E A (α) (τ ) E B (β) (τ ) = η (α)(β)
(see Section IIB of paper I for the 1+3 point of view). A detector measures the following field strengths along the spatial polariza-
tion directions E A (i) (τ ):Ě (i) = F AB v A E B (i) andB (i) = (1/2) ǫ (i)(j)(k) F AB E A (j) E B (k)
. Let us consider the following two cases.
Detectors at Rest in an Inertial Frame
A detector at rest in the instantaneous inertial frame with coordinates (τ ; X a (τ )) follows the straight world-line ζ µ r,in (τ ) = τ ǫ µ τ + ǫ µ a η a in with η a in = const. and has the 4-velocity u µ = ǫ µ τ . If the reference asymptotic tetrad ǫ µ A of the foliation is related by
ǫ µ A = Λ µ (o) ν e ν (A)
to a tetrad e µ (A) = δ µ A aligned to the axes of the inertial frame in Cartesian coordinates, then a generic time-independent non-rotating tetrad associated with the detector will be G µ
(A) = Λ (A) (B) e µ (B) = Λ (A) (µ) if G µ (τ ) = u µ .
Here the Λ's denote Lorentz transformations. The detector will measure the standard electric and magnetic fieldsĚ
(i) = F µν u µ G ν (i) anď B (i) = (1/2) ǫ (i)(j)(k) F µν G µ (j) G ν (k) .
Sources and Detectors at Rest in Rotating Frames
Lt us now consider sources and detectors at rest in the nearly rigid rotating frame described by the embedding z µ (τ, σ u ) = ǫ µ τ τ + ǫ µ r R r s (τ ) σ s + O(Ω 2 /c 2 ), so that z µ τ (τ, σ u ) = ǫ µ τ + ǫ µ rṘ r s (τ ) σ s + O(Ω 2 /c 2 ) and z µ r (τ, σ u ) = ǫ µ s R s r (τ ) + O(Ω 2 /c 2 ).
The world-line of these objects will have the form ζ µ (τ ) = τ ǫ µ τ +ǫ µ r R r s (τ ) η s o +O(Ω 2 /c 2 ) = ǫ µ A ζ A (τ ) with η r o = const.. We have ζ τ (τ ) = τ and ζ r (τ ) = R r s (τ ) η s o + O(Ω 2 /c 2 ). Therefore these objects coincide with some of the observers belonging at the non-surface forming congruence generated by the evolution vector field as said in Section IIB of paper I. Since the world-lines of the Eulerian observers of the other congruence are not explicitly known, it is not possible to study the behavior of objects coinciding with some of these observers.
Therefore the unit 4-velocity
u µ (τ ) = ǫ µ A v A (τ ) will have the components v A (τ ) propor- tional toζ A (τ ) = 1;Ṙ r s (τ ) η s o + O(Ω 2 /c 2 ) ≈ | Ω=const. 1; R r s (τ ) ( η o × Ω c ) s + O(Ω 2 /c 2 )
, where the definitions after Eq.(2.14) of I have been used.
We can also write u µ (τ ) =ũ A (τ ) z µ A (τ, η u o ) by using the non-orthonormal tetrads z µ A (τ, σ u ). Then we get v τ (τ ) =ũ τ (τ )+O(Ω 2 /c 2 ) and v r (τ ) =ũ τ (τ )Ṙ r s (τ ) η s o +R r s (τ )ũ s (τ )+O(Ω 2 /c 2 ). While the quantities v A (τ ) give the description of the 4-velocity with respect to the asymptotic non-rotating inertial observers, the quantitiesũ A (τ ) explicitly show the effect of the rotation at the position η r o of the object. Therefore it should beũ A (τ ) = (1; 0) at the lowest or-
der: indeed we getũ τ (τ ) = 1+O(Ω 2 /c 2 ) andũ r (τ ) = v s (τ ) R s r (τ )−ũ τ R −1 (τ )Ṙ(τ ) r s η s o = 0 + O(Ω 2 /c 2 ).
For the constant unit normal to the instantaneous 3-spaces we get
l µ = ǫ µ τ =l A (τ, η r o ) z µ (τ, η r o ) withl τ (τ, η r o ) = 1 + O(Ω 2 /c 2 ) andl r (τ, η u o ) = −l τ (τ, η u o ) R −1 (τ )Ṙ(τ ) r s η s o = −( η o × Ω c ) r + O(Ω 2 /c 2 ). Let us introduce an orthonormal tetrad W µ (α) , η µν W µ (α) W ν (β) = η (α)(β) , whose time-like 4- vector is l µ , i.e. We have W µ (o) = l µ = ǫ µ τ = W A (o) ǫ µ A =W A (o) (τ, η u o ) z µ A (τ, η u o ) with .W A (o) = (1; 0) andW A (o) (τ, η u o ) =l A (τ, η u o ) = 1; −( η o × Ω c ) r +O(Ω 2 /c 2 ). The spatial axes W µ (i) = W A (i) ǫ µ A = W A (i) (τ, η u o ) z µ A (τ, η u o ) with l µ W µ (i) = [l A g ABW B ]
(τ, η u o ) = 0 must be non-rotating with respect to the observer with 4-velocity proportional to z µ τ (τ, η u o ). Therefore we must haveW
A (i) = 0;W r (i) withW r (i) = const.. As a consequence we have W A (i) (τ ) = 0; R r s (τ )W s (i) + O(Ω 2 /c 2 ).
The polarization axes of sources and detectors will be defined by a tetrad E µ
(α) (τ, η r o ) = E A (α) (τ, η r o ) ǫ µ A =Ẽ A (α) (τ, η r o ) z µ A (τ, η r o ), η µν E µ (α) E ν (β) = η (α)(β) with the following properties: a) the time-like 4-vector E µ (o) (τ, η r o ) is such that its componentsẼ A (o) (τ, η r o ) coincide with the componentsũ A (τ ) = (1; 0) + O(Ω 2 /c 2 ) of the 4-velocity u µ (τ ) of the object located at ζ µ (τ ) = z µ (τ, η r o ): as a consequence we have E µ (o) (τ, η r o ) = z µ τ (τ, η r o ) + O(Ω 2 /c 2 ) = u µ (τ ); b) the spatial axes E µ (i) (τ, η r o )
, orthogonal to the 4-velocity u µ (τ ), must be at rest in the rotating frame: we have to identify their componentsẼ A (i) (τ, η r o ).
If at the observer position we consider the Lorentz transformation sending l µ to u µ (τ ), i.e. L µ ν (l → u(τ )), its projection Therefore the transformation sending the componentsl A (τ, η u o ) of the unit normal into the componentsũ A (τ ) of the 4-velocity modulo terms of order O(Ω 2 /c 2 ) is
L A B ( β) def = ǫ A µ L µ ν (l → u(τ )) ǫ ν B is aE A (o) (τ, η u o ) =ũ A (τ ) = (1; 0) + O(Ω 2 /c 2 ) = = z A µ ǫ µ C L C D ( β) ǫ D ν z ν Bl B (o) (τ, η u o ) + O(Ω 2 /c 2 ) = = z A µ ǫ µ C L C D ( β) ǫ D ν z ν BW B (o) (τ, η u o ) + O(Ω 2 /c 2 ), E A (i) (τ, η u o ) = z A µ ǫ µ C L C D ( β) ǫ D ν z ν B (τ, η u o )W B (i) . (3.20)
This complete the construction of the non-rotating tetrads E µ (α) (τ, η u o ) for the objects at rest at η r o .
A detector endowed of such a non-rotating tetrad will measure the following projections of the electro-magnetic field strength on its polarization directionŝ
E (i) = F ABũ AẼ B (i) ,B (i) = 1 2 ǫ (i)(j)(k) F ABẼ A (j)Ẽ B (k) . (3.21)
These quantities have to be confronted with the non-inertial electric and magnetic fields E r and B r , whose projections on the non-rotating spatial axesW A (i) = (0;W r (i) ) inside the instantaneous 3-space are
E (i) = E rW r (i) , B (i) = B rW r (i) . (3.22)
Eqs. (3.20) imply the following connection among these quantitieŝ
E (i) = E (i) + O(Ω 2 /c 2 ), B (i) = B (i) − ǫ ijkW r (j) δ rs η o × Ω c s E (k) + O(Ω 2 /c 2 ). (3.23)
For radio wave (like in the case of GPS) the directions G a (i) orẼ r (i) are realized by means of antennas attached to both emitters and receivers. In the optical range the antennas are replaced by components of the macroscopic devices used for the emission and the detection.
C. The Phase Wrap Up Effect
The phase wrap up is a modification of the phase when a receiver in rotational motion analyzes the circularly polarized radiation emitted by a source at rest in an inertial frame.
Till now the effect has been explained by using the 1+3 point of view and the locality hypothesis in Refs. [27], where it shown that it is a particular case of helicity-rotation coupling (the spin-rotation coupling for photons). It has been verified experimentally, in particular in GPS [16], where the receiving antenna on the Earth surface is rotating with Earth.
We will explain the effect by using the non-inertial solution (3.7) and an observer at rest in an inertial frame endowed of the tetrad G µ (A) defined in Subsubsection 1 of Subsection B. We rewrite the spatial axes in the form G a (i) = I a (1) , I a (2) ,K a with the vectors satisfying I (1) · I (2) = 0, I (λ) ·K = 0 (λ = 1, 2), I 2 (λ) = 1. Then we pass to a circular basis by introducing the vectors I (±) = I (1) +i I 2 √ 2 , which satisfyK · I (±) = 0, I 2 (±) = 0 and I (+) · I (−) = 1. In the rotating non-inertial frame a right-circularly polarized wave, emitted in the inertial frame, will have the form (3.7) (K · I (+) = 0 is the transversality condition)
A ⊥r (τ, σ) = F ω I (+)a R a r (τ ) e i ω c Φ(τ, σ) . (3.24)
Let us remark that in the circular basis we have A ⊥ = A nn + A + I (+) + A − I (−) , but the components A n , A ± , coincide with either linearly or circularly polarized states of the electro-magnetic field only forn =k, sinceK = ω ck (K 2 = ω 2 c 2 ) is the wave vector. From Eqs(3.24) we obtain the following non-inertial magnetic and electric fields (2.19) of I
B r = − F c I (+)a R a r (τ ) e i ω c Φ(τ, σ) def = B o I (+)a (K) R a r (τ ) e i ω c Φ(τ, σ) , E r = −i F c I (+)a R a r (τ ) e i ω c Φ(τ, σ) + 1 c ( Ω × σ) × B = def = E o I (+)a R a r (τ ) e i ω c Φ(τ, σ) + E ℓKa R a r (τ ) e i ω c Φ(τ, σ) , B o = − F c , E o = −i F c + 1 c ( Ω × σ) × B · I (−) , E ℓ = 1 c ( Ω × σ) × B ·K. (3.25)
Let us now consider a receiver at rest in the rotating frame. Since its 4-velocity is u A = (1; 0), it can be endowed with the non-rotating tetradW A (α) of Subsubsection 2 of Subsection B. Ifn is the unit vector in the direction of the rotation axis, i.e. if Ω = Ωn, we can choose the spatial axesW r (i) = (ǫ r (1) , ǫ r (2) ,n r ) with ǫ (1) · ǫ (2) = 0, ǫ (λ) ·K = 0, ǫ 2 (λ) = 1. If we introduce the circular basis ǫ (±) = ǫ (1) +i ǫ 2 √ 2 , we haven · ǫ (±) = 0, ǫ 2 (±) = 0, ǫ (+) · ǫ (−) = 1 and R a r (τ ) ǫ (±)
r = ǫ a (±) e [± i Ω c τ ] .
The receiver will measure the following magnetic and electric fields σ)) ,
B n = B rn r = B o I (+)an a exp i ω c Φ , B (±) = B r ǫ r (∓) = B o I (+)a ǫ a (∓) exp i c (∓Ω τ + ω Φ(τ,E n = E rn r = E o I (+)an a + E ℓKan a exp i ω c Φ , E (±) = E r ǫ r (∓) = E o I (+)a ǫ a (∓) + E ℓKa ǫ a (±) exp i c (∓Ω τ + ω Φ(τ, σ)) . (3.26)
In the casen a =K a we find It would be interesting to make the calculation of the deviations of order O(Ω 2 /c 2 ) from rigid rotation, to see whether the result ω → γ (ω ± Ω) (γ is a Lorentz factor), found in Ref. [27] by using the locality hypothesis and supporting the interpretation with the helicityrotation coupling, is confirmed.
B n = B (−) = 0 B (+) = B o e [ i c ((ω−Ω)τ+ K· σ)] , E n = E ℓ e [i ω c (ωτ+ K· σ)] , E (+) = 0 E (+) = E o e [ i c ( (ω−Ω)τ + K· σ) ] .
D. The Sagnac Effect
Following a suggestion of Ref. [14] let us consider the solution (3.8) in the eikonal approximation, which describes the propagation of the radiation along a ray of light whose trajectory is given in Eq. (3.19). This solution allows to get a derivation of the Sagnac effect (described in Section II) along the lines of Ref. [31].
Let us consider two receivers A and B at rest in the rotating frame and characterized by the 3-coordinates η r A and η r B respectively. Let us assume that A and B lie in the same 2-plane containing the origin σ r = 0 and orthogonal to Ω. Therefore we have Ω · η A = Ω · η B = 0.
Let us assume that A and B are both on the trajectory of a ray of light, so that Eq. (3.19) implies the existence of a time τ AB such that we have
η B − η A =k τ AB + Ω c ×k τ 2 AB + O(Ω 2 /c 2 ). (3.28)
The phase difference between A and B at the same instant τ is ∆ϕ AB = [32], were it is induced by the gravitational field (due to the equivalence principle only noninertial frames are allowed in general relativity). Our approach is analogous to the one of Ref. [25] in the case of Post-Newtonian gravity.
Let us consider the amplitude a of the solution (3.8) in the eikonal approximation: it carries the information about the polarization of a ray of light. To study the first-order transport equation for it, the second of Eqs.(3.9), let us make the series expansion
a(τ, σ) = a o (τ, σ) + Ω c a 1 (τ, σ) + O Ω 2 c 2 ,(3.35)
and let us make the ansatz (in an inertial frame it corresponds to a plane wave)
a o (τ, σ) = a o = const., ⇒ ∂ a o ∂τ = 0, ∂ r a o = 0. (3.36)
This ansatz implies the following form of the second and third equation in Eqs. (3.9)
Ω c ∂ a 1 ∂τ +Ω × a o − (k · ∂) a 1 + O Ω 2 c 2 = 0, a o ·k + Ω c a o · k (Ω · σ ×k) − (k · σ) (Ω ×k) + a 1 ·k + O Ω 2 c 2 = 0. (3.37)
To study these equations, let us assume that each rotating receiver is endowed with a tetrad of the type given in Eq. (3.20): the spatial axesW r (i) = (R r 1 (k), R r 2 (k),k r ) with R λ (k) · R λ ′ (k) = δ λλ ′ , R λ (k) ·k = 0.
The second of Eqs.
∂a λ 1 ∂τ = 0, ⇒ a λ 1 (τ ) = Ω × R λ ′ (k) · R λ (k) k · σ a λ ′ o . (3.41)
The final solution for the transverse electro-magnetic potential is
A ⊥ = a o 1 ω R 1 + θ( σ) R 2 (k) −k · σ c ( Ω · R 2 (k))k e ( i ω c Φ) + + a o 2 ω R 2 (k) − θ( σ) R 1 (k) +k · σ c
( Ω · R 1 (k))k e ( i ω c Φ) + O(1/ω 2 ), with θ( σ) = 1 c (k · σ) ( Ω ·k).
(3.42)
The resulting non-inertial magnetic and electric fields are
( B = {B r }, E = {E r }) b( η B ) − b( η A ) = i a o
IV. CONCLUSIONS
The theory of non-inertial frames developed in these two papers is free by construction from the coordinate singularities of all the approaches to accelerated frames based on the 1+3 point of view, in which the instantaneous 3-spaces are identified with the local rest frames of the observer. The pathologies of this approach are either the horizon problem of the rotating disk (rotational velocities higher than c), which is still present in all the calculations of pulsar magnetosphere in the form of the light cylinder, or the intersection of the local rest 3-spaces. The main difference between the 3+1 and 1+3 points of view is that the Møller conditions forbid rigid rotations in relativistic theories.
In this paper we have given the simplest example of 3+1 splitting with differential rotations and we have revisited the rotating disk and the Sagnac effect following the 3+1 point of view. This splitting is also used to give a special relativistic generalization of the nonrelativistic non-inertial International Terrestrial Reference System (ITRS) used to describe fixed coordinates on the surface of the rotating Earth in the conventions IERS2003 [23].
Then we re-examined some properties of the electro-magnetic wave solutions of noninertial Maxwell equations, which till now were described only by means of the 1+3 point of view, in the 3+1 framework, where there is a well-posed Cauchy problem due to the absence of coordinate singularities. By considering admissible nearly rigid rotating frames we recover the results of the 1+3 approach and open the possibility to make these calculations in presence of deviations from rigid rotations.
A still open problem are the constitutive equations for electrodynamics in material media in non-inertial systems. For linear isotropic media see the Wilson-Wilson experiment in Refs. [18] and Refs. [14,34], while for an attempt towards a general theory in arbitrary media (including the premetric extension of electro-magnetism) see Refs. [35] In conclusion we have now a good understanding of particles and electro-magnetism in non-inertial frames in Minkowski space-time, where the 4-metric induced by the admissible 3+1 splitting describes all the inertial effects. Going to canonical gravity, in asymptotically Minkowskian space-times without super-translations and in the York canonical basis of Refs. [36,37], it is possible to see which components remain inertial effects and which become dynamical tidal effects (the physical degrees of freedom of the gravitational field). Moreover the inertial 3-volume element and some inertial components of the extrinsic curvature of the instantaneous 3-spaces become complicated functions of both general relativistic inertial and tidal effects, because they are determined by the solution of the super-Hamiltonian constraint (the Lichnerowicz equation) and of the super-momentum constraints. Finally, in accord with the equivalence principle, the instantaneous 3-spaces are only partially determined by the freedom in choosing the convention for clock synchronization: after such a convention the final instantaneous 3-spaces associated to each solution of Einstein's equations are dynamically determined, because in general relativity the metric structure of space-time is dynamical and not absolute like it happens in special relativity.
where t GCRS is the geocentric coordinate time, are obtained from the BCRS ones by means of a PN coordinate transformation which may be described as a special relativistic pure Lorentz boost without rotations (the parameter is the 3-velocity of the geo-center considered constant on small time scales) plus O(c −4 ) corrections taking into account the gravitational acceleration of the geo-center induced by the Sun and the planets. As a consequence the GCRS spatial axes are kinematically non-rotating in BCRS and the relativistic inertial forces (for instance the Coriolis ones) are hidden in the geodetic precession; the same holds for the aberration effects and the dependence on angular variables. A PN 4-metric, determined modulo O(c −4 ) terms, is given in IAU2000: it also contains the GCRS form of the geo-potential and the inertial and tidal effects of the Sun and of the planets. Again the instantaneous 3-spaces are considered nearly Euclidean (modulo O(c −2 ) deviations) 3-spaces t GCRS = const..
In IAU200 the coordinate times t BCRS and t GCRS are then connected with the time scales used on Earth: SI Atomic Second, TAI (International Atomic Time), TT (Terrestrial Time), T EP H (Ephemerides Time), UT and UT1 and UTC (Universal Times for civil use), GPS (Mastr Time), ST (Station Time).
Finally
we need a 4-coordinate system fixed on the Earth crust. It is the ITRS with 4-coordinates x µ IT RS = x o ItRS def = c t GCRS ; x i IT RS , which uses the same coordinate time as GCRS. It is obtained from GCRS by making a suitable set of non-relativistic time-dependent rigid rotations on the nearly Euclidean 3-spaces t GCRS = const.. The geocentric rectangular 3-coordinates x i IT RS match the reference ellipsoid WGS-84 (basis of the terrestrial coordinates (latitude, longitude, height) obtainable from GPS) used in geodesy. As shown in IERS2003, we have x
are rotation matrices. This convention is based on the new definition of the Earth rotation axis (θ is the angle of rotation about this axis): it is the line through the geo-center in direction of the Celestial Intermediate Pole (CIP) at date t GCRS , whose position in GCRS is n i GCRS = sin d cos E, sin d sin E, cos d . The new non-rotating origin (NLO) of the rotation angle θ on the Earth equator (orthogonal to the rotation axis) is a point named the Celestial Intermediate Origin (CIO), whose position in CGRS requires the angle s, called the CIO locator. Finally in the rotation matrix W T (t GCRS ) (named the polar motion or wobble matrix) the angles x p and y p are the angular coordinates of CIP in ITRS, while the angle s ′ is connected with the re-orientation of the pole from the ITRS z-axis to the CIP plus a motion of the origin of longitude from the ITRS x-axis to the so-called Terrestrial Intermediate Origin (TIO), used as origin of the azimuthal angle.
The proper time τ of the geo-center coincides with c t GCRS modulo O(c −2 ) corrections from the GCRS PN 4-metric.Then a special relativistic definition of ITRS can be done by replacing the rigidly rotating 3-coordinates x i IT RS with the differentially rotating 3-coordinates σ r . The rotation matrix R(τ, σ), with the choice F (σ
look for solutions of the eikonal equation for Φ of the form Φ(τ, σ u ) = τ + F (σ u ),
in the solutions (3.7) and (3.15) of Eqs.(3.2) are different since the solutions have different boundary conditions. The solution (3.7) satisfies also the eikonal equation but not the boundary condition (3.11), since we have ∂
Wigner boost, see Eq.(2.8) of I, with parameter β = {β r = R r s (τ ) η o × Ω c s (so that γ = 1 − β 2 = 1 + O(Ω 2 /c 2 )).
components B (+) , E (+) have the frequency modified to ω → ω − Ω: this is the phase wrap up effect. These are same results as in Ref.[27] at the lowest order in Ω/c. The only new fact is the presence of the component E n = 0.
(3.37) for the unknown a o , a 1 is the transversality condition and itimplies order 0 in Ω → a o ·k = 0 ⇒ a o = a λ o R λ (k), order 1 in Ω a 1 ·k = − a o · k (Ω · σ ×k) + (k · σ) (Ω ×k) = = −a λ o R λ (k) · (Ω ×k) (k · σ). (3.38)Due to the ansatz (3.36) the first of Eqs.(3.37) is of order 1 in Ω and gives the following condition on a 1 project this equation on the directionsk, R λ (k), we get∂ ∂τ ( a 1 ·k) −Ω × a o ·k + (k · ∂) ( a 1 ·k) × R λ ′ (k) · R λ (k) a λ ′ o + (k · ∂) a λ 1 = 0.(3.40) While the first of Eqs.(3.40) is automatically satisfied, the second one is an equation for the components a λ 1 . The simplest solutions are obtained with the following ansatz
where t BCRS is the barycentric coordinate time and the mutually orthogonal spatial axes are kinematically non-rotating with respect to fixed radio sources. This a nearly Cartesian 4-coordinate system in a PN Einstein space-time and there is an assigned 4-metric, determined modulo O(c −4 ) terms and containing the gravitational potentials of the Sun and of the planets, PN solution of Einstein equations in harmonic gauges: in practice it is considered as a special relativistic inertial frame with nearly Euclidean instantaneous 3-spaces t BCRS = const. (modulo O(c −2 ) deviations) and
Sometimes the proper time of the rotating observer is used: dT o = dτ 1 − Ω 2 (R) R 2 .
See Ref.[15] for a derivation of the Sagnac effect in an inertial system by using Einstein's synchronization in the locally comoving inertial frames on the rim of the disk and by asking for the equality of the one-way velocities in opposite directions.
.(3.29)Eq.(3.28) impliesso that we getIf A BAO is the area of the triangle BAO in the 2-plane orthogonal to Ω, we have Ωthe choice of ± depends on the direction of motion of the ray). As a consequence, the phase difference is the sum of the following two termsis present also in the inertial frames, the second termis the extra phase variation due to the rotation of the frame. This is the Sagnac effect.E. The Inertial Faraday RotationLet us give the derivation of the rotation of the polarization of an electro-magnetic wave in a rotating frame, named inertial Faraday rotation, which is important in astrophysicsAs in the case of the Sagnac effect let us consider two receivers A and B at the endpoints of the same light ray described by Eqs,(3.19)and(3.28). The magnetic field observed by A, B(τ, η A ), differs from the one observed by B, B(τ, η B ). Since the phase changes have been already analyzed for the Sagnac effect, let us concetrate on the amplitudes b( η A ) and b( η B ). Since Eq.(3.28) gives η B − η A =k τ AB + O(Ω/c), we find
D Alba, L Lusanna, Charged Particles and the Electro-Magnetic Field in Non-Inertial Frames of Minkowski Spacetime: I. Admissible 3+1 Splittings of Minkowski Spacetime and the Non-Inertial Rest Frames. D.Alba and L.Lusanna, Charged Particles and the Electro-Magnetic Field in Non-Inertial Frames of Minkowski Spacetime: I. Admissible 3+1 Splittings of Minkowski Spacetime and the Non-Inertial Rest Frames.
Simultaneity, Radar 4-Coordinates and the 3+1 Point of View about Accelerated Observers in Special Relativity. D Alba, L Lusanna, gr-qc/0311058D. Alba and L.Lusanna, Simultaneity, Radar 4-Coordinates and the 3+1 Point of View about Accelerated Observers in Special Relativity (2003) (gr-qc/0311058);
Generalized Radar 4-Coordinates and Equal-Time Cauchy Surfaces for Arbitrary Accelerated Observers. gr-qc/0501090)Int.J.Mod.Phys. 161149Generalized Radar 4-Coordinates and Equal-Time Cauchy Surfaces for Arbitrary Accelerated Observers (2005), Int.J.Mod.Phys. D16, 1149 (2007) (gr-qc/0501090).
G Rizzi, M L Ruggiero, gr-qc/0207104)Space Geometry of Rotating Platforms: an Operational Approach. G.Rizzi and M.L.Ruggiero, Space Geometry of Rotating Platforms: an Operational Approach, (gr-qc/0207104).
Lengths on Rotating Platforms. A Tartaglia, Found.Phys.Lett. 1217A.Tartaglia, Lengths on Rotating Platforms, Found.Phys.Lett. 12, 17 (1999).
On the Spatial Geometry of a Non-Inertial Observer in Special Relativity. P Dombrowski, J Kuhlmann, U Proff, Global Riemannian Geometry. T.J.Willmore and N.J.HitchinNew YorkHorwood, WileyP.Dombrowski, J.Kuhlmann and U.Proff, On the Spatial Geometry of a Non-Inertial Observer in Special Relativity, in Global Riemannian Geometry, eds. T.J.Willmore and N.J.Hitchin (Horwood, Wiley, New York, 1984).
Relativity in Rotating Frames. Relativistic Physics in Rotating Reference Frames. G.Rizzi and M.L.DordrechtKluwerG.Rizzi and M.L.Ruggiero eds., Relativity in Rotating Frames. Relativistic Physics in Rotating Reference Frames. (Kluwer, Dordrecht, 2003).
Zum Ehrenfestschen Paradoxon. A Einstein, Phys.Z. 12509A.Einstein, Zum Ehrenfestschen Paradoxon, Phys.Z. 12, 509 (1911);
Die Grunlage der Allgemeinen Relativitatstheorie. Annalen der Physik. 49769Die Grunlage der All- gemeinen Relativitatstheorie, Annalen der Physik 49, 769 (1916);
The Meaning of Relativity. PrincetonPrinceton Univ.PressThe Meaning of Relativity (Princeton Univ.Press, Princeton, 1950).
Gleichförmige Rotation starrer Körper und Relativitätheorie. P Ehrenfest, Phys.Z. 10918P.Ehrenfest, Gleichförmige Rotation starrer Körper und Relativitätheorie, Phys.Z. 10, 918 (1909).
Relativistic Description of a Rotating Disk. Ø Grøn, Int.J.Theor.Phys. 1628Am.J.Phys.Ø. Grøn, Rotating Frames in Special Relativity, Int.J.Theor.Phys. 16, 603 (1977). Relativistic Description of a Rotating Disk, Am.J.Phys. 43, 869 (1975). Covariant Formulation of Hooke's Law, Am.J.Phys. 49, 28 (1981).
Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzipe. M Born, Ann.Phys.(Leipzig). 301M.Born, Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzipe, Ann.Phys.(Leipzig) 30, 1 (1909).
M H Soffel, Relativity in Astrometry, Celestial Mechanics and Geodesy. BerlinSpringerM.H.Soffel, Relativity in Astrometry, Celestial Mechanics and Geodesy (Springer, Berlin, 1989).
Time-like Helices in Flat Space-Time. J L Synge, Proc. Royal Irish Acad. A. Royal Irish Acad. A6527J.L.Synge, Time-like Helices in Flat Space-Time, Proc. Royal Irish Acad. A 65, 27 (1967).
Motion of Charged Particles in Homogeneous Electro-magnetic Fields. E Honig, E L Schuking, C V Vishveshwara, J.Math.Phys. 15774E.Honig, E.L.Schuking and C.V.Vishveshwara, Motion of Charged Particles in Homogeneous Electro-magnetic Fields, J.Math.Phys. 15, 774 (1974).
The Frenet-Serret Formalism and Black Holes in Higher Dimensions. B R Iyer, C V Vishveshwara, Class.Quantum Grav. 5961B.R.Iyer and C.V.Vishveshwara, The Frenet-Serret Formalism and Black Holes in Higher Dimensions, Class.Quantum Grav. 5, 961 (1988);
The Frenet-Serret Description of Gyroscopic Precession. Phys.Rev. 485706The Frenet-Serret Description of Gyroscopic Precession, Phys.Rev. D48, 5706 (1993).
Rotating Frames in SRT: the Sagnac Effect and Related Issues. W A Rodrigues Jr, M Sharif, Found.Phys. 311767W.A.Rodrigues jr and M.Sharif, Rotating Frames in SRT: the Sagnac Effect and Related Issues, Found.Phys. 31, 1767 (2001);
Equivalence Principle and the Principle of Local Lorentz Invariance. Found.Phys. 311785erratum Found.Phys.Equivalence Principle and the Principle of Local Lorentz Invariance, Found.Phys. 31, 1785 (2001) [erratum Found.Phys. 32, 811 (2002)].
Generalized Eulerian Coordinates for Relativistic Fluids: Hamiltonian Rest-Frame Instant Form, Relative Variables, Rotational Kinematics, to appear in Int. D Alba, L Lusanna, hep-th/0209032J.Mod.Phys. D.Alba and L.Lusanna, Generalized Eulerian Coordinates for Relativistic Fluids: Hamil- tonian Rest-Frame Instant Form, Relative Variables, Rotational Kinematics, to appear in Int.J.Mod.Phys. (hep-th/0209032).
The Rest-Frame Instant Form of Relativistic Perfect Fluids with Equation of State ρ = ρ(n, s) and of Non-Dissipative Elastic Materials. L Lusanna, D Nowak-Szczepaniak, hep-th/0003095Int. J. Mod. Phys. 154943L.Lusanna and D.Nowak-Szczepaniak, The Rest-Frame Instant Form of Relativistic Perfect Fluids with Equation of State ρ = ρ(n, s) and of Non-Dissipative Elastic Materials, Int. J. Mod. Phys. A15, 4943 (2000) (hep-th/0003095).
Ring-Laser Tests of Fundamental Physics and Geophysics. G E Stedman, Rep.Prog.Phys. 60615G.E.Stedman, Ring-Laser Tests of Fundamental Physics and Geophysics, Rep.Prog.Phys. 60, 615 (1997).
Sagnac Effect. E J Post, Rev.Mod.Phys. 39475E.J.Post, Sagnac Effect, Rev.Mod.Phys. 39, 475 (1967).
The Relativistic Sagnac Effect: two Derivations (gr-qc/0305084). G Rizzi, M L Ruggiero, G.Rizzi and M.L.Ruggiero, The Relativistic Sagnac Effect: two Derivations (gr-qc/0305084).
Relativity in the Global Positioning System. N Ashby, Living Reviews in Relativity. N.Ashby, Relativity in the Global Positioning System, Living Reviews in Relativity (http://www.livingreviews.org).
Introduction to Relativistic Effects on the Global Positioning System. N Ashby, J J Spilker, Global Positioning System: Theory and Applications. B.W.Parkinson and J.J.SpilkerAmerican Institute of Aeronautics and Astronautics1N.Ashby and J.J.Spilker, Introduction to Relativistic Effects on the Global Positioning Sys- tem, in Global Positioning System: Theory and Applications, Vol.1, eds. B.W.Parkinson and J.J.Spilker (American Institute of Aeronautics and Astronautics, 1995).
General Relativistic Corrections to the Sagnac Effect. A Tartaglia, Phys.Rev. 5864009A.Tartaglia, General Relativistic Corrections to the Sagnac Effect, Phys.Rev. D58, 064009 (1998).
On the Electric Effect of Rotating a Magnetic Insulator in a Magnetic Field. M Wilson, H A Wilson, Proc.R.Soc. London. 8999M.Wilson and H.A.Wilson, On the Electric Effect of Rotating a Magnetic Insulator in a Magnetic Field, Proc.R.Soc. London A89, 99 (1913).
Maxwell's Equations in a Rotating Medium. G N Pellegrini, A R Swift, Am.J.Phys. 63694Is there a Problem?G.N.Pellegrini and A.R.Swift, Maxwell's Equations in a Rotating Medium" Is there a Problem?, Am.J.Phys. 63, 694 (1995).
Measurements on a Rotating Frame in Relativity and the Wilson and Wilson Experiment. T A Weber, Am.J.Phys. 65946T.A.Weber, Measurements on a Rotating Frame in Relativity and the Wilson and Wilson Experiment, Am.J.Phys. 65, 946 (1997).
Applying Covariant versus Contravariant Electromagnetic Tensors to Rotating Media. C T Ridgely, Am.J.Phys. 66414Am.J.Phys.C.T.Ridgely, Applying Relativistic Electrodynamics to a Rotating Material Medium, Am.J.Phys. 66, 114 (1998). Applying Covariant versus Contravariant Electromagnetic Tensors to Rotating Media, Am.J.Phys. 67, 414 (1998).
The Wilson-Wilson Experiment. K T Mcdonald, K.T.McDonald, The Wilson-Wilson Experiment, 2008 (http://cosmology.princeton.edu/∼mcdonald/examples
Improved Laser Test of the Isotropy os Space. A Brillet, J L Hall, Phys.Rev.Lett. 42549A.Brillet and J.L.Hall, Improved Laser Test of the Isotropy os Space, Phys.Rev.Lett. 42, 549 (1979).
Tests of Relativity Using a Cryogenic Optical Resonator. C Braxmaier, H Müller, O Pradl, J Mlynek, A Peters, Phys.Rev.Lett. 8810401C.Braxmaier, H.Müller, O.Pradl, J.Mlynek and A.Peters, Tests of Relativity Using a Cryo- genic Optical Resonator, Phys.Rev.Lett. 88, 010401 (2002).
New Limit on Signals of Lorentz Violation in Electrodynamics. J A Lipa, J A Nissen, S Wang, D A Stricker, D Avaloff, Phys.Rev.Lett. 9060403J.A.Lipa, J.A.Nissen, S.Wang, D.A.Stricker and D.Avaloff, New Limit on Signals of Lorentz Violation in Electrodynamics, Phys.Rev.Lett. 90, 060403 (2003).
P Wolf, S Bize, A Clairon, A N Luiten, G Santarelli, M E Tobar, gr-qc/0210049)Tests of Relativity using a Microwave Resonator. P.Wolf, S.Bize, A.Clairon, A.N.Luiten, G.Santarelli and M.E.Tobar, Tests of Relativity using a Microwave Resonator, (gr-qc/0210049).
Detecting the Effects of Linear Acceleration on the Optical Response of Matter. R Nutze, G E Stedman, Phys.Rev. A58. 82R.Nutze and G.E.Stedman, Detecting the Effects of Linear Acceleration on the Optical Re- sponse of Matter, Phys.Rev. A58, 82 (1997).
H Rauch, S A Werner, Neutron Interferometry: Lessons in Experimental Quantum Mechanics. OxfordClarendon PressH.Rauch and S.A.Werner, Neutron Interferometry: Lessons in Experimental Quantum Me- chanics (Clarendon Press, Oxford, 2000).
Dynamical Body Frames, Orientation-Shape Variables and Canonical Spin Bases for the Nonrelativistic N-Body Problem. D Alba, L Lusanna, M Pauri, hep-th/0011014J. Math. Phys. 43D. Alba, L. Lusanna and M. Pauri, Dynamical Body Frames, Orientation-Shape Variables and Canonical Spin Bases for the Nonrelativistic N-Body Problem , J. Math. Phys. 43, 373 (2002) (hep-th/0011014).
. Iers Conventions, IERS TN. D.D.McCarthy and G.Petit32Verlag des BKGIERS Conventions (2003), eds. D.D.McCarthy and G.Petit, IERS TN 32 (2004), Verlag des BKG.
The IAU 2000 Resolutions for Astrometry, Celestial Mechanics and Metrology in the Relativistic Framework: Explanatory Supplement. M Soffel, S A Klioner, G Petit, P Wolf, S M Kopeikin, P Bretagnon, V A Brumberg, N Capitaine, T Damour, T Fukushima, B Guinot, T Huang, L Lindegren, C Ma, K Nordtvedt, J Ries, P K Seidelmann, D Vokroulicky, ' , C Will, Ch, Xu, astro-ph/0303376Astron.J. 126M.Soffel, S.A.Klioner, G.Petit, P.Wolf, S.M.Kopeikin, P.Bretagnon, V.A.Brumberg, N.Capitaine, T.Damour, T.Fukushima, B.Guinot, T.Huang, L.Lindegren, C.Ma, K.Nordtvedt, J.Ries, P.K.Seidelmann, D.Vokroulicky', C.Will and Ch.Xu, The IAU 2000 Resolutions for Astrometry, Celestial Mechanics and Metrology in the Relativistic Framework: Explanatory Supplement Astron.J., 126, pp.2687-2706, (2003) (astro-ph/0303376).
The IAU Resolutions on Astronomical Reference Systems, Time Scales and Earth Rotation Models. G H Kaplan, astro-ph/0602086Naval Observatory circular No. 179G.H.Kaplan, The IAU Resolutions on Astronomical Reference Systems, Time Scales and Earth Rotation Models, U.S.Naval Observatory circular No. 179 (2005) (astro-ph/0602086).
A Question in General Relativity. L I Schiff, Proc.Nat.Acad.Sci. .Nat.Acad.Sci25391L.I.Schiff, A Question in General Relativity, Proc.Nat.Acad.Sci. 25, 391 (1939).
Electromagnetic Waves in Gravitational Fields. J Plebanski, Phys.Rev. 1111396J.Plebanski, Electromagnetic Waves in Gravitational Fields, Phys.Rev. 111, 1396 (1960).
Electromagnetic Fields in Curved Spacetimes. C G Tsagas, gr-qc/0407080)Class.Quantum Grav. 22C.G.Tsagas, Electromagnetic Fields in Curved Spacetimes, Class.Quantum Grav. 22, 393 (2005) (gr-qc/0407080).
. J D Barrow, C G Tsagas, gr-qc/9704015)Class.Quantum Grav. 142539J.D.Barrow and C.G.Tsagas, , Class.Quantum Grav. 14, 2539 (1997)(gr-qc/9704015).
Electromagnetic Waves in a Rotating Frame of Reference. J C Hauck, B Mashhoon, gr-qc/0304069)Ann.Phys. (Leipzig). 12J.C.Hauck and B.Mashhoon, Electromagnetic Waves in a Rotating Frame of Reference, Ann.Phys. (Leipzig) 12, 275 (2003) (gr-qc/0304069).
Observable Frequency Shifts via Spin-Rotation Couplings. B Mashhoon, R Neutze, M Hannam, G E Stedman, gr-qc/9808077)Phys.Lett. 249161B.Mashhoon, R.Neutze, M.Hannam and G.E.Stedman, Observable Frequency Shifts via Spin- Rotation Couplings, Phys.Lett. A249, 161 (1998) (gr-qc/9808077).
Spin-Gravity Coupling. B Mashhoon, arXiv:0801.2134Acta Phys,Polon. Suppl. 1B.Mashhoon, Spin-Gravity Coupling, Acta Phys,Polon. Suppl. 1, 113 (2008) (arXiv: 0801.2134).
Modification of the Doppler Effect due to the Helicity-Rotation Coupling. B Mashhoon, Phys.Lett. 36652105Nonlocal Electrodynamics of Accelerated SystemsPhys.Rev.B.Mashhoon, Nonlocal Electrodynamics of Accelerated Systems, Phys.Lett. A366, 545 (2007). Modification of the Doppler Effect due to the Helicity-Rotation Coupling, Phys.Lett. A306, 66 (2002). Nonlocal Electrodynamics of Linearly Accelerated Systems, Phys.Rev. A70, 062103 (2004). Nonlocal Electrodynamics of Rotating Systems, Phys.Rev. A72, 052105 (2005).
Pioneer Anomaly and the Helicity-Rotation Coupling. J D Anderson, B Mashhoon, Phys.Lett. 315199J.D.Anderson and B.Mashhoon, Pioneer Anomaly and the Helicity-Rotation Coupling, Phys.Lett. A315, 199 (2003).
Acceleration-Induced Nonlocal Electrodynamics in Minkowski Spacetime. U Muench, F W Hehl, B Mashhoon, Phys.Lett. 271U.Muench, F.W.Hehl and B.Mashhoon, Acceleration-Induced Nonlocal Electrodynamics in Minkowski Spacetime, Phys.Lett. A271, 8 (2000).
Can Einstein's Theory of Gravitation be tested beyond the Geometric Optics Limit?. B Mashhoon, gr-qc/0003022)Class.Quant.Grav. 11681Gen.Rel.Grav.B.Mashhoon, Influence of Gravitation on the Propagation of Electromagnetic Radiation, Phys.Rev. D11, 2679 (1975). Can Einstein's Theory of Gravitation be tested beyond the Ge- ometric Optics Limit?, Nature 250, 316 (1974). Gravitational Coupling of Intrinsic Spin, Class.Quant.Grav. 17, 2399 (2000)(gr-qc/0003022). On the Spin-Rotation Gravity Coupling , Gen.Rel.Grav. 31, 681 (1999).
. S P Tarabrin, A A Seleznyov, arXiv:0804.4292Optical Position Meters analyzed in the Non-Inertial References Frames. S.P.Tarabrin and A.A.Seleznyov, Optical Position Meters analyzed in the Non-Inertial Refer- ences Frames, (arXiv: 0804.4292).
Observable Frequency Shifts via Spin-Rotation Coupling. B Mashhoon, R Neutze, M Hannam, G E Stedman, Phys.Lett. 249161B.Mashhoon, R.Neutze, M.Hannam and G.E. Stedman, Observable Frequency Shifts via Spin- Rotation Coupling, Phys.Lett. A249, 161 (1998).
Gravitational Faraday Rotation in a Weak Gravitational Field. M Sereno, Phys.Rev. 6987501M.Sereno, Gravitational Faraday Rotation in a Weak Gravitational Field, Phys.Rev. D69, 087501 (2004).
M Giovannini, K E Kunze, arXiv 0804.3380Faraday Rotation, Stochastic Magnetic Fields and CMB Maps. M.Giovannini and K.E.Kunze, Faraday Rotation, Stochastic Magnetic Fields and CMB Maps, arXiv 0804.3380
The Rotation of Polarization by Gravitational Waves. V Faraoni, arXiv 0709.0386New Astronomy. 13V.Faraoni, The Rotation of Polarization by Gravitational Waves, New Astronomy 13, 178 (2008) (arXiv 0709.0386).
Search for Gravitational Waves through the Electromagnetic Faraday Rotation. M Halilsoy, O Gurtug, Phys.Rev. 75124021M.Halilsoy and O.Gurtug, Search for Gravitational Waves through the Electromagnetic Fara- day Rotation, Phys.Rev. D75, 124021 (2007).
V Perlick, W Hasse, Gravitational Faraday Effect in Conformally Stationary Spacetimes. 10147V.Perlick and W.Hasse, Gravitational Faraday Effect in Conformally Stationary Spacetimes, Clas.Q.Grav. 10, 147 (1993).
Can the Rotation of the Dark Matter Halo of our Galaxy be detected through its Effect on the Cosmic Microwave Background Polarization?. P Nag, S Bharadwaj, S Kar, arXiv astro- ph/0506009P.Nag, S.Bharadwaj and S.Kar, Can the Rotation of the Dark Matter Halo of our Galaxy be detected through its Effect on the Cosmic Microwave Background Polarization?, arXiv astro- ph/0506009
L Landau, E Lifschitz, The Classical Theory of Fields. CambridgeAddison-WesleyL.Landau and E.Lifschitz, The Classical Theory of Fields (Addison-Wesley, Cambridge, 1951).
Note on the Electrodynamics of Accelerated Systems. E J Post, D D Bahulikar, J.Math.Phys. 121098E.J.Post and D.D.Bahulikar, Note on the Electrodynamics of Accelerated Systems, J.Math.Phys. 12, 1098 (1971).
Kottler-Cartan-van Dantzig (KCD) and Noninertial Systems. E J Post, Found.Phys. 9619E.J.Post, Kottler-Cartan-van Dantzig (KCD) and Noninertial Systems, Found.Phys. 9, 619 (1979).
B M Bolotovskii, S N Stolyarov, Current Status of the Electrodynamics of Moving Media (Infinite Media). 17875B.M.Bolotovskii and S.N.Stolyarov, Current Status of the Electrodynamics of Moving Media (Infinite Media), Sov.Phys.Usp. 17, 875 (1975).
Maxwell's Equations in Minkowski's World: their Premetric Generalization and the Electromagnetic Energy-Momentum Tensor. F W Hehl, arXiv:0807.4249F.W.Hehl, Maxwell's Equations in Minkowski's World: their Premetric Generalization and the Electromagnetic Energy-Momentum Tensor, 2008 (arXiv: 0807.4249).
Forces and Momenta caused by Electromagnetic Waves in Magnetoelectric Media. F W Hehl, Y Obukov, arXiv:0710.2219Foundations of Classical Electrodynamics: Charges, Flux and Metric. BostonBirkhauser37111Electrodynamics of Moving Magnetoelectric Media: Variational ApproachF.W.Hehl and Y.Obukov, Electrodynamics of Moving Magnetoelectric Media: Variational Ap- proach, Phys.Lett. A371, 11 (2007); Forces and Momenta caused by Electromagnetic Waves in Magnetoelectric Media 2007 (arXiv: 0710.2219); Foundations of Classical Electrodynamics: Charges, Flux and Metric (Birkhauser, Boston, 2003).
. V A De Lorenci, G P Goulart, arXiv:0806.4685Magnetoelectric Birifrangence Revisited. V.A.De Lorenci and G.P.Goulart, Magnetoelectric Birifrangence Revisited, (arXiv: 0806.4685).
The York Map as a Shanmugadhasan Canonical Transformationn in Tetrad Gravity and the Role of Non-Inertial Frames in the Geometrical View of the Gravitational Field. D Alba, L Lusanna, gr-qc/0604086, v2Gen.Rel.Grav. 392149see v1 for an expanded versionD.Alba and L.Lusanna, The York Map as a Shanmugadhasan Canonical Transformationn in Tetrad Gravity and the Role of Non-Inertial Frames in the Geometrical View of the Grav- itational Field, Gen.Rel.Grav. 39, 2149 (2007) (gr-qc/0604086, v2; see v1 for an expanded version).
The Einstein-Maxwell-Particle System in the York Canonical Basis of ADM Tetrad Gravity: I) The Equations of Motion in Arbitrary Schwinger Time Gauges. D Alba, L Lusanna, 4087907D.Alba and L.Lusanna, The Einstein-Maxwell-Particle System in the York Canonical Basis of ADM Tetrad Gravity: I) The Equations of Motion in Arbitrary Schwinger Time Gauges., 2009 (arXiv 0907.4087).
| [] |
[
"Multivariate Fractional Components Analysis",
"Multivariate Fractional Components Analysis"
] | [
"Tobias Hartl \nUniversity of Regensburg\n93053RegensburgGermany\n\nInstitute for Employment Research (IAB)\n90478NurembergGermany\n",
"Roland Weigand \nAOK Bayern\n93055RegensburgGermany\n"
] | [
"University of Regensburg\n93053RegensburgGermany",
"Institute for Employment Research (IAB)\n90478NurembergGermany",
"AOK Bayern\n93055RegensburgGermany"
] | [] | We propose a setup for fractionally cointegrated time series which is formulated in terms of latent integrated and short-memory components. It accommodates nonstationary processes with different fractional orders and cointegration of different strengths and is applicable in high-dimensional settings. In an application to realized covariance matrices, we find that orthogonal short-and long-memory components provide a reasonable fit and competitive out-of-sample performance compared to several competing methods. | 10.1093/jjfinec/nbab022 | [
"https://arxiv.org/pdf/1812.09149v2.pdf"
] | 85,531,584 | 1812.09149 | 1345f1acbedd0fd1b0c161d074f358db97baa9b0 |
Multivariate Fractional Components Analysis
January 2019 29 Jan 2019
Tobias Hartl
University of Regensburg
93053RegensburgGermany
Institute for Employment Research (IAB)
90478NurembergGermany
Roland Weigand
AOK Bayern
93055RegensburgGermany
Multivariate Fractional Components Analysis
January 2019 29 Jan 2019Long memoryfractional cointegrationstate spaceunobserved componentsfactor modelrealized covariance matrix JEL-Classification C32C51C53C58
We propose a setup for fractionally cointegrated time series which is formulated in terms of latent integrated and short-memory components. It accommodates nonstationary processes with different fractional orders and cointegration of different strengths and is applicable in high-dimensional settings. In an application to realized covariance matrices, we find that orthogonal short-and long-memory components provide a reasonable fit and competitive out-of-sample performance compared to several competing methods.
Introduction
Multivariate fractional integration and cointegration models have proven valuable in a wide range of empirical applications from macroeconomics and finance. They generalize the standard concept of cointegration by allowing for non-integer orders of integration both for the observations and for equilibrium errors; see Gil-Alana and Hualde (2008) for a literature review. In the field of macroeconomics, such models have turned out to be relevant in analyses of purchasing power parity beginning with Cheung and Lai (1993), of the relation between unemployment and input prices (Caporale and Gil-Alana; and of broader models for economic fluctuations . The empirical finance literature has considered fractional cointegration, e.g., for analysing international bond returns (Dueker and Startz;, for modeling co-movements of stock return volatilities (Beltratti and Morana;, for assessing the link between realized and implied volatility and for quantifying risk in strategic asset allocation problems (Schotman et al.;2008). From a methodological point of view, semiparametric techniques for inference on the cointegration rank, the cointegration space and memory parameters have been very popular among empirical researchers, although the development of optimal parametric inferential methods for models with triangular or fractional vector error correction representations has recently made considerable progress (see, e.g., Robinson and Hualde;2003;Avarucci and Velasco;2009;Lasak;Johansen and Nielsen;.
Despite their flexibility and their computationally simple treatment, semiparametric models are limited in scope since they aim to describe low-frequency properties only and are hence not appropriate for impulse response analysis and forecasting. While semiparametric techniques have been developed to cope with multivariate processes of different integration orders and multiple fractional cointegration relations of different strenghts (Chen and Hurvich;Hualde and Robinson;Hualde;2009), there seems to be a lack of parametric models of such generality. Furthermore, the usual error correction and triangular models with their abundant parametrization are not deemed appropriate for time series of dimension, say, larger than five.
In this paper, we propose new models for multivariate fractionally integrated and cointegrated time series which are formulated in terms of latent purely fractional and additive short-memory components. With a "type II" definition of fractional integration (Robinson;, this approach allows for a flexible modeling of possibly nonstationary time series of different fractional integration orders. It permits cointegration relations of different strengths as well as polynomial cointegration (multicointegration in the terminology of Granger and Lee;1989), i.e., cointegration between the levels of some time series and their (fractional) differences, and guarantees a clear representation of the long-run characteris-tics. Consequently, our model is among the most general setups regarding its integration and cointegration properties, compared to popular existing models for cointegrated processes. The unobserved components formulation benefits the modeling of relatively highdimensional time series. For this situation we propose a parsimonious parametrization based on dimension reduction and dynamic orthogonal components in the spirit of Pan and Yao (2008) and Matteson and Tsay (2011).
In contrast to our parametric approach, latent fractional components have mostly been studied in semiparametric frameworks. Ray and Tsay (2000) use semiparametric memory estimators and canonical correlations to infer the existence of common fractional components, Morana (2004) proposes a frequency domain principal components estimator, Morana (2007) estimates components of a single fractional integration order by univariate permanent-transitory (or persistent-transitory) decompositions followed by a principal component analysis of the permanent (or persistent) components and Luciani and Veredas (2015) estimate their fractional factor model by fitting long-memory models to the principal components of a large panel of time series. In a setup closest to ours, Chen and Hurvich (2006) suggest a semiparametric frequency domain methodology to identify and estimate cointegration subspaces which annihilate fractional components of different memory.
Recent parametric frameworks competing to ours have either been much more restrictive, or have a different focus, e.g., on numerical simulation-based estimation methods, or on panel data analysis. Using a Bayesian approach, Hsu et al. (1998) discuss a bivariate process sharing one stationary long-memory component, while Mesters et al. (2016) consider simulated maximum likelihood estimation of models with one or more latent stationary ARFIMA components. On the other hand, Ergemen and Velasco (2017) as well as Ergemen (2017) focus on the elimination of common fractional components and are motivated as alternatives to prior unit root testing. Contrary to our approach, they eliminate the common factor structure, which they treat as nuisance.
As the second main contribution of this paper, our model is applied to forecasting daily realized covariance matrices. In this setup, the strengths of our approach become apparent. In realized covariance modelling, typically high-dimensional processes with strong persistence and a pronounced co-movement in the low-frequency dynamics are considered. In time series of log variances and z-transformed correlations for six US stocks, we find that common orthogonal short-and long-memory components with two different fractional integration orders provide a reasonable fit. Since the dimension of the dataset is reduced to a smaller number of latent processes, our model becomes a factor model. A pseudo out-ofsample study shows that the fractional components model provides a superior forecasting accuracy compared to several competitor methods. In addition to the favorable forecast properties, our methods can be applied to study the cointegration properties of stock mar-ket volatilities. These are of particular importance for longer-term portfolio hedging and the analysis of systematic risk.
The paper is organized as follows. Section 2 introduces the general setup and clarifies its integration and cointegration properties. In section 3, its relation to existing models for multivariate integrated time series is discussed. In section 4, a specific model appropriate for relatively high-dimensional processes is considered. The empirical application to realized covariance matrices and a pseudo out-of-sample assessment are contained in section 5 before section 6 concludes.
The general setup
We consider a linear model for a p-dimensional observed time series y t , which we label a fractional components (FC) setup,
y t = Λx t + u t , t = 1, . . . , n.(1)
The model is formulated in terms of the latent processes x t and u t where Λ will always be assumed to have full column rank and the components of the s-dimensional x t are fractionally integrated noise according to
∆ d j x jt = ξ jt , j = 1, . . . , s.(2)
In principle, s > p is possible, but we only consider cases where s ≤ p here. For a generic scalar d, the fractional difference operator is defined by
∆ d = (1 − L) d = ∞ j=0 π j (d)L j , π 0 (d) = 1, π j (d) = j − 1 − d j π j−1 (d), j ≥ 1,(3)
where L denotes the lag or backshift operator, Lx t = x t−1 . We adapt a nonstationary type II solution of these processes (Robinson; and hence treat d j ≥ 0.5 alongside the asymptotically stationary case d j < 0.5 in a continuous setup, by setting starting values to zero, x jt = 0 for t ≤ 0. Nonzero initial values have been considered for observed fractional processes by Johansen and Nielsen (2012), but are not straightforwardly handled for our unobserved processes. The solution is based on the truncated operator ∆ −d j + (Johansen;2008) and given by
x jt = ∆ −d j + ξ jt = t−1 i=0 π i (−d j )ξ j,t−i , j = 1, . . . , s.
Without loss of generality let the components be arranged such that d 1 ≥ . . . ≥ d s . We assume d j > 0 for all j in what follows, so that x t governs the long-term characteristics of the observations y t . These are complemented by additive short-run dynamics which we describe by stationary vector ARMA specifications for u t in the general case. This ARMA process is given by
Φ(L)u t = Θ(L)e t , t = 1, . . . , n,(4)
where Φ(L) and Θ(L) are a stable vector autoregressive polynomial and an invertible moving average polynomial, respectively. The disturbances ξ t and e t jointly follow a Gaussian white noise (NID) sequence such that
ξ t ∼ NID(0, Σ ξ ), e t ∼ NID(0, Σ e ) and E(ξ t e t ) = Σ ξe ,(5)
where at this stage, before turning to identified and empirically relevant model specifications below, we do not consider restrictions on the joint covariance matrix, but only require Σ ξ to have strictly positive entries on the main diagonal. Some remarks regarding the general FC setup are in order. The model as given in (1) is not identified without further restrictions on the loading matrix Λ, on the vector ARMA coefficients and on the noise covariance matrix. While restrictions on Σ ξ and Λ may be based on results in dynamic factor analysis as will be seen below, choosing specific parametrizations for u t will depend on characteristics of the data and on the purpose of the empirical analysis. Identified vector ARMA structures like the echelon form (see 2005, chapter 12) can be used for a rich parametrization, while a multivariate structural time series approach as described in Harvey (1991) integrates nicely with the unobserved components framework considered in this paper and allows for more restricted parameterization, e.g., by individual or common stochastic cycle components. Below, we introduce a parsimonious model well-suited to relatively high dimensions which is conceptually based on dimension reduction and orthogonal components.
For a characterization of the integration and cointegration properties of our model, we adapt the definitions of these concepts from Hualde and Robinson (2010), which prove useful here. Hence, a generic scalar process ρ t is called integrated of order δ or I(δ) if it can be written as
ρ t = l i=1 ∆ −δ i + ν it , where δ = max i=1,.
..,l {δ i } and ν t = (ν 1t , . . . , ν lt ) is a finite-dimensional covariance stationary process with spectral density matrix which is continuous and nonsingular at all frequencies. A vector process τ t is called I(δ) if δ is the maximum integration order of its components. We call the process τ t cointegrated if there exists a nonzero vector β such that β τ t is I(γ) where δ − γ > 0 will be referred to as the strength of the cointegration relation. The number of linearly independent cointegration relations with possibly differing γ is called cointegration rank of τ t .
By these definitions, x jt is clearly I(d j ) while both x t and y t are integrated of order d 1 . We observe at least two different integration orders in the individual series of y t whenever Λ i1 = 0 for some i and d 1 > d 2 . More generally,
y it ∼ I(d j ), if Λ i1 = . . . = Λ i,j−1 = 0 but Λ ij = 0.
To state the cointegration properties of the FC setup (1), we assume that s ≤ p, so that all fractional components are reflected by the integration and cointegration structure of y t and that Σ ξ is nonsingular. It is useful to identify all q groups of x jt with identical integration orders and denote their respective sizes by s 1 , . . . , s q , such that d s 1 +...+s j−1 +1 = . . . = d s 1 +...+s j and s = q j=1 s j . Of course, if q = s, then s 1 = . . . = s q = 1 and all components of x t have mutually different integration orders, while for q = 1 it holds that s = s 1 and we observe d 1 = . . . = d s .
To keep notation simple, for a generic matrix A for which a specific grouping of rows and columns is clear from the context, we denote by A (i,j) the block from intersecting the i-th group of rows with the j-th group of columns. A stacking of several groups of rows i, . . . , j and columns k, . . . , l is indicated by A (i:j,k:l) . For a grouping in only one dimension we write A (i) or A (i:j) , where it shall be clear from the context whether a grouping of rows or columns is considered. Furthermore, we denote the column space of a generic k × l matrix A by sp(A) ⊆ R k and its orthogonal complement by sp ⊥ (A). Further, for k > l, the k × (k − l) orthogonal complement of A will be denoted by A ⊥ , which spans the (k − l)-dimensional space sp ⊥ (A).
According to the grouping of equal individual integration orders in x t , we may therefore rewrite the FC process (1) as
y t = Λ (1) x (1) t + . . . + Λ (q) x (q) t + u t .
Here, Λ (j) is a p × s j submatrix of Λ consisting of columns Λ ·i for which s 1 + . . . + s j−1 < i ≤ s 1 + . . . + s j , and x (j) t is a s j -dimensional subprocess of x t corresponding to components with memory parameter d (j) := d s 1 +...+s j−1 +1 = . . . = d s 1 +...+s j . Whenever s 1 < p, there exist p − s 1 linearly independent linear combinations β i y t ∼ I(γ i ) and γ i < d 1 , so that fractional cointegration occurs. Due to our definition of cointegration, this may be a trivial case where a single component y it with integration order smaller than d 1 is selected. Since
Λ (1) ⊥ y t = Λ (1) ⊥ Λ (2) x (2) t + . . . + Λ (1) ⊥ Λ (q) x (q) t + Λ (1) ⊥ u t is integrated of order d (2) , the columns of Λ (1)
⊥ qualify as cointegration vectors and S (1) := sp ⊥ (Λ (1) ) is the (p − s 1 )-dimensional cointegration space of y t .
Whenever s 1 + s 2 < p, there are subspaces of S (1) forcing a stronger reduction in integration orders. More generally, it holds that Λ (1:j) ⊥ y t ∼ I(d (j+1) ) whenever j i=1 s i < p and where we set d (j+1) = 0 for j > s. Analogously to Hualde and Robinson (2010), for s = p and j = 1, . . . , q − 1, we call S (j) := sp ⊥ (Λ (1:j) ) the j-th cointegration subspace of y t , for which S (q−1) ⊂ . . . ⊂ S (1) . For p > s, S (q) ⊂ S (q−1) is a further such subspace. Cointegration vectors in S (q) cancel all fractional components and hence reduce the integration order from d 1 to zero, the strongest reduction possible in our setup.
Besides this general pattern of cointegration relations, our model features an interesting special case with so-called polynomial cointegration, that is, cointegration relations where lagged observations nontrivially enter a cointegration relation. To see this possibility, consider a bivariate example similar to Granger and Lee (1989), where q = p = 2 and ξ 1t = ξ 2t , so that Σ ξ is singular and x 2t = ∆ d 1 −d 2 x 1t . Augmenting the variables by a fractional difference asỹ t := (y 1t , y 2t , ∆ d 1 −d 2 y 2t ) , we obtain a three-dimensional system where levels of y t enter a nontrivial cointegration relation with a fractional difference to achieve a reduction in integration order from d 1 to max{2d 2 − d 1 , 0} < d 2 . Hence, our setup complements the model of Johansen (2008, section 4), which was the first to handle polynomial cointegration in a fractional setup, and the results of Carlini and Santucci de Magistris (2018), who derive a Granger representation for the fractional VECM of Granger (1986) under polynomial cointegration.
Relations to other cointegration models
In this section, we clarify the relation of the fractional components model (1) to popular existing representations for cointegrated processes and show how our model can be represented in alternative ways brought forward in the literature. While our model is among the most general setups with respect to its integration and cointegration properties, the additive modeling of short-run dynamics is new to the literature and gives rise to distinct parametrizations not possible within other representations in a similarly convenient way.
Error correction models. The most popular representation of cointegrated systems in the I(1) setting is the vector error correction form. Since an early mention by Granger (1986), in the fractionally integrated case, e.g., Avarucci andVelasco (2009), Lasak (2010) and Johansen and Nielsen (2012) have recently considered such models. In terms of the integration and cointegration properties, the fractional error correction setups are typically restricted to the special case with q = 2 and s = p, such that the observed variables are integrated of order d (1) and there exist p − s 1 cointegration relations with errors of order
d (2) .
Defining the fractional lag operator L b := 1 − ∆ b (Johansen; 2008), we are able to derive the error correction representation for this special case of our model; see appendix A. It is given by
∆ d (1) y t = αβ L d (1) −d (2) ∆ d (2) y t + κ t ,(6)
where we find αβ = −Λ (2) (Λ (1)
⊥ Λ (2) ) −1 Λ(1)
⊥ to precede the error correction term, while
κ t := M (Λ (1) ξ (1) t + ∆ d (1) u t ) − αβ (Λ (2) ξ (2) t + ∆ d (2) u t )
is integrated of order zero and M is defined in (13). The model differs both from the models of Avarucci and Velasco (2009) and from the representation of Johansen (2008) in the way short-run dynamics are modeled. The literature has considered (fractional) lags of differenced variables and possibly of error correction terms in the VECM representation. Our setup, in contrast, generates autocorrelated κ t by filtering the latent u t with fractional difference operators. Hence, adding lags of ∆ d (1) y t in the model (6) is only an approximate solution and achieving a desired approximation quality may require estimating a large number of parameters.
As we have discussed above, Johansen (2008) proposes a polynomially cointegrated generalization of his VAR d,b model which allows terms integrated of orders d, d − b and d − 2b in the Granger representation (Johansen;2008, theorem 9). Even compared to that specification, our model allows for more general patterns of integration orders and cointegration strengths, since we only assume d j > 0 for all j. More in line with the generality envisaged in this paper, Tschernig et al. (2013, equation 14) present a model with error correction term and different integration orders, while Lasak and Velasco (2014) sequentially fit error correction models to test for cointegration relations of possibly different strengths.
Vector ARFIMA. An interesting special case of (1) occurs for s = p and Λ = I, where each series in y it is driven by a single fractional component and y it ∼ I(d i ). This resembles standard vector ARFIMA models with possibly different integration orders; see, e.g., Lobato (1997) who labels the popularly termed vector ARFIMA class considered here as "model A". A frequently used submodel is the fractionally integrated vector autoregressive model discussed by Nielsen (2004). The main difference to these approaches is our additive modeling of short-run dynamics, whereas in the vector ARFIMA setup weakly dependent vector ARMA instead of white noise processes are passed through the fractional integration filters.
Our model belongs to the class of vector ARFIMA processes for integer d j ∈ {1, 2, . . .}, but not for general fractional integration orders. For the case of integer d j , note that (x t , u t ) is a finite-order vector ARMA process, and hence y t as a linear combination is itself in the ARMA class; see Lütkepohl (1984). For general vector ARFIMA processes, a similar conclusion does not hold. To see this, consider a stylized univariate case of our model with
p = s = 1, where ∆ d x t = ξ t and (1 − φL)u t = e t . First note that (∆ d x t , u t )
has an ARMA structure, and hence (x t , u t ) is a vector ARFIMA process. Expanding
(1 − φL)∆ d x t = (1 − φL)ξ t and (1 − φL)∆ d u t = ∆ d e t ,
we can write the sum, belonging to the fractional components model class, as
(1 − φL)∆ d y t = (1 − φL)ξ t + ∆ d e t .(7)
The right hand side of this expression is not a finite-order MA process in general, as it has nonzero autocorrelations for all lags, and hence, the process does not belong to the ARFIMA class for non-integer d.
Triangular representations. The models discussed so far have restricted integration or cointegration properties as compared to our model. Even in the most general setup of Johansen (2008), the integration orders are restricted to be d, d − b, d − 2b for polynomial cointegration. In contrast, Hualde (2009) and Hualde and Robinson (2010) have proposed a very flexible model which adapts the triangular form of Phillips (1991) and its generalization to processes with multiple unit roots (Stock and Watson; to the fractional cointegration setup.
To derive the triangular representation for our model, we assume that the variables in y t are ordered in a way that Λ (1:j,1:j) is nonsingular for j = 1, . . . , q and restrict attention to the case s = p for notational convenience. The variables are partitioned according to the groups of different integration orders in x t as y (j) t := (y s 1 +...+s j−1 +1 , . . . , y s 1 +...+s j ) , j = 1, . . . , q. The first block in the triangular system is (8) where ω (1) t is integrated of order zero. The general expression for the j-th block of the triangular system is derived in appendix A for j = 2, . . . , q, and given by
∆ d 1 y (1) t = Λ (1,1) ξ (1) t + Λ (1,2) ∆ d 1 −d 2 ξ (2) t + . . . + Λ (1,q) ∆ d 1 −dq ξ (q) t + ∆ d 1 u t (:= ω (1) t ),∆ d (j) y (j) t = Λ (j,1:(j−1)) (Λ (1:(j−1),1:(j−1)) ) −1 ∆ d (j) y (1:(j−1)) t + ω (j) t (9) = −B (j,1) ∆ d (j) y (1) t − . . . − B (j,j−1) ∆ d (j) y (j−1) t + ω (j) t ,
where also ω (j) t is integrated of order zero for j = 2, . . . , q. By inverting the fractional difference operators we obtain
By t = (∆ −d 1 + ω (1) t , . . . , ∆ −dq + ω (q) t ) ,(10)
where B has a block triangular structure such that B (i,i) = I and B (i,j) = 0 for i < j. A re-ordering of the variables in y t yields the representation of Hualde and Robinson (2010). This representation allows for a semiparametric cointegration analysis of our model using the methods of Hualde (2009) and Hualde and Robinson (2010). However, our model differs significantly from straightforward parametrizations of the triangular system, e.g., from assuming a vector ARMA process for ω t , since in our setup ω t as stated in (16) generally contains fractional differences that cannot be represented within the ARMA framework.
State space approaches. Bauer and Wagner (2012) have presented a state space canonical form for multiple frequency unit root processes of different (integer-valued) integration orders. Their discussion is based on unit root vector ARMA models which are separated in pure unit root structures and short-term dynamics. Although the analogy to our model is striking, there are notable differences between their unit root and our fractional setup. Firstly, as discussed in the paragraph on vector ARFIMA models (see (7)), the fractional components setup (1) is not nested within a general class comparable to the vector ARMA models, which form the basis of the discussion in Bauer and Wagner (2012). Secondly, in their setting, the introduction of different integration orders is achieved by repeated summation of lower order integrated processes which themselves enter the observations to achieve polynomial cointegration. This is in contrast to the continuous treatment of integration orders in our (type II) fractional setup.
However, fractional components models could be constructed to straightforwardly extend the setup of Bauer and Wagner (2012). Using the fractional lag operator L b = 1 − ∆ b instead of L in the short-run dynamic specification (4), a stable vector ARMA b process can be defined byΦ(L b )ũ t =Θ(L b )e t under suitable stability conditions (Johansen;2008, corollary 6). Then, replacing u t byũ t in the model setup (1)
with d j restricted to some mul- tiple of b (d j = i j b, i j ∈ {1, 2, . . .})
, the process y t is in the class of vector ARMA b models itself, while unit roots in the vector autoregressive polynomial generate the fractional I(d j ) processes. Such a framework could be treated analogously to Bauer and Wagner (2012), but the restriction that all integration orders are multiples of b makes such a framework somewhat less flexible than ours. 4 A dimension-reduced orthogonal components specification So far, we have considered a general modeling setup and discussed its integration and cointegration properties as well as its relation to existing approaches in the literature. We now turn to the discussion of a specific model from this class which bears potential for parsimonious modeling of long-and short-run dynamics in relatively high-dimensional applications. Besides its general interest, this will be the workhorse specification for the empirical application to realized covariance modeling in section 5.
To introduce the model and emphasize its restrictions as compared to (1), we decompose the short-term dependent process u t into an autocorrelated component, Γ z t , where z t is a vector of s 0 mutually uncorrelated components with s + s 0 ≤ p, and a Gaussian white noise component ε t , respectively. We label the result the dynamic orthogonal fractional components (DOFC) model,
y t = Λ (1) x (1) t + . . . + Λ (q) x (q) t + Γ z t + ε t ,(11)
where x t is generated by purely fractional processes (2) as above, while
(1 − φ j1 L − . . . − φ jk L k )z jt = ζ jt , j = 1, . . . , s 0 ,
are s 0 univariate stationary autoregressive processes of order k. Regarding the noise processes ξ t , ζ t and ε t , we assume mutual independence over leads and lags,
ξ t ∼ NID(0, I), ζ t ∼ NID(0, I) and ε t ∼ NID(0, H),
where H is diagonal with entries h i > 0, i = 1, . . . , p. Note that for s + s 0 < p the DOFC model is a factor model as it allows for dimension reduction. The model as specified in (11) and below is not identifiable without further information. Consideringỹ t := ∆ d (1) y t instead of y t to meet the assumptions of Heaton and Solo (2004), their theorem 4 suggests that groups of common components
∆ d (1) x (1) t , . . . , ∆ d (1) x (q) t , ∆ d (1) z t can be disentangled (up to rotations within these groups) through their different shapes in spectral densities whenever d (1) > . . . > d (q) > 0. Still, there exist observationally equivalent structures withΛ (j) = Λ (j) M −1 andx (j) t = M x (j) t
which satisfy the model restrictions for orthonormal M . Hence, we impose further restrictions on the loading matrices. As is standard practice in dynamic factor analysis, we set the upper triangular elements to zero such that Λ (j) rl = 0 for r < l, j = 1, . . . , q, and Γ rl = 0 for r < l. Certain observables are thus assumed not to be influenced by certain factors.
The model (11) is very parsimonious considering that it includes both a rich fractional structure as well as short-run dynamics with co-dependence. This is possible by comprising three components of parsimony which have been brought forward in the statistical time series literature. Firstly, there are p − s − s 0 ≥ 0 white noise linear combinations of y t . A strict inequality implies a reduced dimension in the dynamics of y t which is characteristic for so-called statistical factor models; see Pan and Yao (2008), Lam et al. (2011) and Lam and Yao (2012). In contrast, the model (1) does not belong to this class in general, since it allows for s ≥ p and general forms of autocorrelation in u t . Secondly, all crosssectional correlation stems from the common components which is a familiar feature from classical factor analysis (Anderson and Rubin;1956). Thirdly, both the fractional and the nonfractional components are mutually orthogonal for all leads and lags.
Combined with semiparametric techniques of fractional integration and cointegration analysis, existing methods for statistical factor and dynamic orthogonal components analysis (Matteson and Tsay; can be used to justify the model assumptions and may be useful in the course of model specification. For final model inference, maximum likelihood estimation based on a state space representation is the preferred method. Both steps will be illustrated in the empirical application of the next section.
An application to realized covariance modeling
We apply the fractional components approach to the modeling and forecasting of multivariate realized stock market volatility which has recently received considerable interest in the financial econometrics literature.
Data and recent approaches
We use the dataset of Chiriac and Voev (2011) which comprises realized variances and covariances from six US stocks, namely (1) American Express Inc., (2) Citigroup, (3) General Electric, (4) Home Depot Inc., (5) International Business Machines and (6) Different transformations of the realized covariance matrices have been applied to fit dynamic models to data of this kind. Weigand (2014) discusses these transforms and considers a general framework nesting several previously applied approaches. His results suggest that applying linear models to a multivariate time series of log realized variances along with z-transformed realized correlations is a reasonable choice in practice. We follow this approach and base our empirical study on the 21-dimensional time series y t = (log(X 11,t ), . . . , log(X 66,t ), Z 21,t , Z 31,t , . . . , Z 65,t ) ,
where X t is the 6 × 6 realized covariance matrix at period t, and the z-transforms are
Z ij,t = 0.5[log(1 + R ij,t ) − log(1 − R ij,t )], R ij,t = X ij,t X ii,t X jj,t .
All time series (grey) of log variances and their maxima and minima for a given day t (black) are depicted in figure 1, while z-transformed correlations are shown in figure 2. Recent approaches to modeling realized covariance matrices have successfully used long-memory specifications (Chiriac and Voev;, or found co-movements between the processes well-represented by dynamic factor structures; see Bauer and Vorkink (2011) and Gribisch (2013). In the related problem of forecasting univariate realized variances, factor models with long-memory dynamics have already been proposed. While Beltratti and Morana (2006) use frequency-domain principal components techniques to assess the low-frequency co-movements, Luciani and Veredas (2015) apply time-domain principal components to their high-dimensional series and apply fractional integration techniques to both estimated factors and idiosyncratic components. Recently, Asai and McAleer (2015) have considered long-memory factor dynamics also for the modeling of realized covariance matrices, where again a semiparametric factor approach precedes a long-memory analysis in their two-step approach.
Our fractional components model DOFC (11), applied to the time series (12), offers various advantages to researchers and practitioners in the field. (a) Our methods offer new insights in the integration and cointegration properties of stock market volatilities, for which fractional components structures of different integration orders have not been investigated so far. (b) Fractional cointegration between variances and correlations is of particular interest for the understanding of longer-term portfolio hedging and systemic risk assessment, but has not found attention in the existing literature. (c) Our state space approach for variances and correlations also features other relevant aspects of volatility modeling. It offers a separation into short-term and long-term components in the spirit of Engle and Lee (1999), directly accounts for measurement noise, and is applicable in datasets of higher dimensions. The parameter-driven state space approach our specification enables yields (d) practicability in case of missing values, while it (e) straightforwardly carries over to stochastic volatility frameworks for daily return data in the spirit of Harvey et al. (1994).
Preliminary analysis and model specification
We investigate whether the constraints imposed in the DOFC model (11) are reasonable for the dataset under investigation. Semiparametric methods are used to assess these restrictions and to obtain reasonable starting values for the parametric estimation of our model.
The model (11) implies that there are s + s 0 components which govern the dynamics of y t , and hence, for p > s + s 0 , there is a dimension reduction in terms of the autocorrelation characteristics. Pan and Yao (2008) study time series with such properties and propose a sequential test to infer the dynamic dimension of the process, allowing for nonstationarity of the autocorrelated components. The algorithm sequentially finds the least serially correlated linear combinations of y t , subsequently testing the null of no autocorrelation of these linear combinations. We apply 3 lags when detecting autocorrelations in what follows.
Applying this approach to our dataset, we do not reject the null for eight linear combinations which can hence be treated as white noise. For the ninth such combination, the p-value for the multivariate Ljung-Box test drops from 0.1935 to 0.0002, so that the white noise hypothesis is rejected for reasonable significance levels. We conclude that there are s + s 0 = 21 − 8 = 13 components which account for the dynamic properties of the process. Pan and Yao (2008) also propose an estimator for the space of dynamic components (x t , z t ) . We call these estimates (rotated by principal components) the factors in what follows.
Our model implies that (x t , z t ) and hence a suitable rotation of the factors can be modelled as s + s 0 univariate time series which are mutually orthogonal at all leads and lags. This corresponds to the notion of dynamic orthogonal components as introduced by Matteson and Tsay (2011) who provide methods to test for the presence of such a structure and to estimate the appropriate rotation. Using first differences of the factors to achieve stationarity as required by Matteson and Tsay (2011) for suitable values of d j , we find highly significant cross-correlations of the raw factors (the test statistic takes the value 4198.94 for a level 0.01 critical value of 625.80) while a dynamic orthogonal structure is not rejected for the rotated series, with a test statistic of 445.55 and a corresponding p-value close to one. The test result also holds if the test is conducted in levels. In what follows, the dynamic orthogonal components are computed from the factors in levels which slightly outperforms the difference-approach in simulations with fractional processes. 1 Due to their dynamic orthogonality, the rotation of Matteson and Tsay (2011) identifies the single processes in (x t , z t ) up to scale, sign and order. A preliminary analysis of the integration orders of x t can hence be undergone by a univariate treatment of these series. We investigate these integration orders by the exact local Whittle estimator allowing for an unknown mean (Shimotsu;.
A possible grouping of components with equal integration orders is assessed by the methods proposed by Robinson and Yajima (2002), with the modifications for possibly nonstationary integration orders by Nielsen and Shimotsu (2007). The specific-to-general approach of Robinson and Yajima (2002) sequentially tests for existence of j = 1, 2, . . . groups of equal integration orders. The sequence is terminated if for some j * there is a grouping for which within-group equality is not rejected, and for j * > 1 the grouping with highest p-value is selected. In our application, we restrict attention to possible groupings where, ford i 1 >d i 2 >d i 3 , there is no group including both i 1 and i 3 but not i 2 . For the tests of equal integration orders within the sequential approach, we consider the Wald test proposed by Nielsen and Shimotsu (2007), jointly testing all hypothesized equalities for a given grouping. We choose m = n 0.5 = 46 as bandwidth and set the trimming parameter h to zero, since the dynamic orthogonal components structure does not permit fractional cointegration.
The estimated integration orders for the 13 dynamic orthogonal components range from 0.0087 to 0.7328 and indicate that some of the components may have short memory while others behave like stationary or nonstationary fractionally integrated processes. We clearly reject equality of all integration orders, while also each of the groupings in two groups can be rejected on a 0.01 significance level. For three groups, we do not reject the hypothesis of equal integration orders within groups. The sequential test for groups with equal memory yields j * = 3 with a p-value of 0.3181, where groups of three (d (1) = 0.6717), seven (d (2) = 0.3448) and three (d (3) = 0.0523) components are identified, respectively. The hypothesis that d (3) = 0 is not rejected. We may therefore treat the members of the third group as short-range dependent and belonging to z t . Thus, s 1 = 3, s 2 = 7 and s 0 = 3 appear as a reasonable specification for model (11) due to the preliminary analysis.
We obtain starting values for the parametric estimator from this procedure. Firstly, d and φ are estimated from the dynamic orthogonal components. Secondly, from regressing observed data on standardized estimated orthogonal components with unit innovation variance, we obtain starting values for h, Λ and Γ , while certain columns of the latter matrices are rotated to satisfy the zero restrictions.
In very high-dimensional cases, the approach of Pan and Yao (2008) is not applicable, but Lam et al. (2011) and Lam and Yao (2012) provide feasible methods for stationary settings and comment on possible extensions to nonstationarity. In cases where the dynamic orthogonal components specification (11) is not appropriate, but the general setup (1) is, a specification search and preliminary estimates for the integration and cointegration parameters of the more general model could be based on the algorithm of Hualde (2009) which is capable of identifying and estimating cointegration subspaces by semiparametric methods.
A parametric fractional components analysis
We proceed with maximum likelihood estimation of the fractional components model using the EM algorithm of the state space representation. Although the exact state space respresentation is easily obtained using the current type II definition of fractional integration, the state dimension grows linearly with n and becomes computationally infeasible. Instead, the latent fractionally integrated components are mapped to approximating ARMA(3,3) dynamics as described and justified by Hartl and Weigand (2018). There, we show by simulation that low-order ARMA approximations (with parameters depending both on d j and on n) provide an excellent approximation performance and outperform truncated moving average and autoregressive representations by large amounts.
We note that an asymptotic theory for maximum likelihood estimation in the fractionally cointegrated state space setup is not available. Certain functions of the parameter estimates are expected to exert nonstandard asymptotic behavior, especially in the nonstationary case d j > 0.5 for some j. However, normal and mixed normal asymptotics have been established and conventional tests and confidence intervals have been justified in different parametric fractional cointegration settings as well as in state space models with common unit root components 2009. We thus use standard parameter tests in what follows, bearing the preceding caveats in mind.
Constant terms are included by a further column c in the observation matrix and estimated along with the free elements of Λ and Γ . Setting the autoregressive order of z t to one and using starting values as described above, we estimate models with q ∈ {1, 2, 3} groups of equal integration orders d (j) > 0 and additional autoregressive components. The Bayesian information criterion (BIC) is used to select sizes s 0 , . . . , s q and the value of q with appropriate in-sample fit. 2 We apply the BIC even if consistency is not established in this fractional setting. We expect that existing results hold for specification choices not involving the fractional components, while it is not clear to what extent the results of 2 Instead of estimating all reasonable combinations of s 0 , . . . , s q for each q, we begin by the optimal grouping for a given q obtained from the semiparametric methods of the previous section. From this specification, denoted as s holds for all j = 0, . . . , q. As a result, also the number of white noise combinations may differ from 8, the result of the semiparametric analysis in the previous section. Chang et al. (2012) carry over to the fractional setup. There, consistency of the BIC is shown for the number of stochastic trends in a unit root state space model.
We complement the semiparametric results of the previous section by a parametric specification search. After diagnostic checking of the selected model, we will take a closer look at its parameter estimates and implied long-run characteristics. The best models for each q are shown in table 1, where estimated integration orders are given along with the log-likelihood (log-lik) and the BIC. Regarding the integration orders, we find that for q > 1 estimates of d (1) are always above 0.5 suggesting nonstationarity of at least s 1 series in y t . Overall, the models with q = 2 are superior, in particular the grouping in s 1 = 2 and s 2 = 9 fractional and s 0 = 2 nonfractional components. This specification is similar to the one selected by the semiparametric approach and also suggests a dynamic dimension of s + s 0 = 13. Interestingly, the same specification with full noise covariance matrix H is inferior (BIC = −16.626) as is the model with a full vector autoregressive matrix Φ (BIC = −17.150). Furthermore, considering more lags in z t does not sufficiently improve the fit (BIC = −17.155 for k = 2, BIC = −17.046 for k = 3 and BIC = −17.139 for k = 4).
We conduct several diagnostic tests on standardized model residuals
e it = v it / F ii,t ,
where v t and F t are filtered residuals and forecast error covariance matrices, respectively. The residuals corresponding to log variances and z-transformed correlations for the first three assets are plotted in figure 3, while residual autocorrelations are depicted in figure 4, autocorrelations of squared residuals in figure 5 and histograms of the residuals along with the normal density in figure 6. The visual inspection shows some but no overwhelming evidence against the model assumptions. Autocorrelation both of residuals and squared residuals are generally below 0.1 in absolute value and mostly within the ± 2 standard error bands which are shown as horizontal lines. Some deviations from normality are visible, but not the sort of skewness and fat tails observed for models of untransformed residual variances and covariances. Table 2 presents the diagnostic tests on standardized residuals. The p-values are shown for the Ljung-Box test (LM) and the ARCH-LM test for conditional heteroscedasticity (CH) for different lag length 5, 10 and 22. Additionally, the Jarque-Bera test result (JB) is shown in the last column. The null of no autocorrelation is not rejected at the 0.01 level for all but two or three residuals, depending on lag length. Clear evidence of conditional heteroskedasticity is found for the residuals of the log variance series, that is e 2t ,e 3t , e 5t , and e 6t , where also the normality assumption is clearly rejected, but also for a few correlation series such as e 15,t or e 19,t . A more flexible data transformation like the matrix Box-Cox approach of Weigand (2014) would typically ameliorate these findings, but we do not follow this approach further here.
Estimates of several of the model parameters are shown in table 3. Along with the maximum likelihood estimates, we also show the mean of the estimators from a modelbased bootstrap resampling exercise with 1000 iterations and generally find a low bias for the corresponding estimates. We also show standard errors, obtained in three ways, namely by the bootstrap (SE.boot), using the information matrix (Harvey;1991, section 3.4.5), denoted by SE.info, and by the sandwich form White (1982), labelled SE.sand in the table. The different methods of computing standard errors give similar results, except for the variance parameters h i , where the sandwich estimates are large compared to the others. Overall, including the parameters not shown in the table, the median ratio between bootstrap and sandwich standard errors is 1.31, while a typical sandwich estimate is 1.20 times larger than the corresponding estimate from the information matrix. We hence use the bootstrap methods in order to avoid a possible underestimation of the variances and spurious inference.
The estimated memory parameters d 1 and d 2 exert a marked difference in the integration orders of fractional components. The two series in the first group are the cause of significant nonstationarity in our dataset. The second group of nine series introduces stationary long-memory persistence. In contrast, the nonfractional components in z t are only mildly autocorrelated, with small but significant autoregression parameters. Figure 7 gives a visual impression of the factor dynamics, showing full sample (smoothed) estimates of the two nonstationary components (above), of the first two stationary long-memory components (middle) and of the short-memory components (below). The ± 2 standard error confidence intervals suggest a relatively precise estimation of the components. The different persistence of the three groups is clearly visible.
We turn to a discussion of the cointegration properties of the estimated system. In our preferred specification with a cointegration rank of p−s 1 = 19, and an 11-dimensional cointegration subspace, the loadings of fractional components provide an easier interpretation than the corresponding cointegration vectors, although the latter can be easily obtained and suitably normalized.
With the abovementioned caveat that asymptotic properties are not available for this fractional cointegration setting, we show t-ratios for constants, for fractional loadings and for nonfractional loadings in table 4, where the bootstrap standard errors are used. The t-ratios for Λ (1) suggest that each of the series in y t is influenced by the nonstationary components, and hence all components of y t are nonstationary themselves. The first component loads very significantly on all variances with the same sign and can hence be interpreted as the main common risk factor. The second component represents joint common nonstationarity of the correlations, which is negatively associated with the IBM return variances. Except those corresponding to the first, the second and the forth stationary components with their equal signs, the columns of Λ (2) have a rather mixed pattern. Like the nonstationary factors, also the I(d (2) ) components affect variance and correlation dynamics at the same time and therefore induce fractional cointegration between log variances and z-transformed correlations.
The finding of nonstationary fractional components affecting variances and correlations at the same time is new to the literature and may have remarkable consequences on portfolio selection and hedging opportunities, even at longer horizons. These effects should also be relevant to systemic risk measures as considered by central banks and regulators worldwide. To shed further light on the practical value of our approach, we turn to an evaluation of the forecasting precision in a real-world scenario in the following section.
An out-of-sample comparison
We assess the forecasting performance of our model by means of an out-of-sample comparison. To avoid reference of the forecasts on the out-of-sample periods, we conduct a semiparametric specification search along the lines of section 5.2 for the first estimation sample only, i.e. for y t , t = 1, . . . , 1508, while t = 1509, . . . , 2156 is reserved for prediction and therefore not used for selecting the specification. In this way, the model for the forecasting comparison includes s 1 = 2, s 2 = 7 and s 0 = 3 components of different integration orders. Rather than conducting comprehensive comparisons of a wide range of available methods which is beyond the scope of this paper, we select straightforward and simple benchmark models which have performed well in previous studies.
We choose the same out-of-sample setup as in Weigand (2014). Thus, for each T ∈ [1508; 2156−h], various competing models are estimated for a rolling sample with n = 1508 observations, y T −1507 , . . . , y T . From these estimates, forecasts of y T +h , h = 1, 5, 10, 20, are computed. Also in line with Weigand (2014), we compute bias-corrected forecasts of the realized covariance matricesX T +h|T by the simulation-based technique discussed there. We evaluate the forecasting accuracy using the ex-post available data of the respective period.
The forecasting precision is assessed using different loss functions defined in appendix B. We consider the Frobenius norm LF T ,h (17), the Stein norm LS T ,h (18) and the asymmetric loss L3 T ,h (19); see Laurent et al. (2011) and Laurent et al. (2013). Additionally, the ex-ante minimum variance portfolio is computed from the forecast and its realized variance LM V T ,h (20) used as a loss with obvious economic relevance. Furthermore, we assess density forecasts f r of the daily returns using covariance matrices, which are evaluated at the daily returns r T +h in a logarithmic scoring rule LD T ,h (21).
As benchmarks, we consider two linear models for the log variance and z-transformed correlation series y t , namely a diagonal vector ARMA(2,1) and a diagonal vector ARFI-MA(1,d,1) model, which have been found to perform well by Weigand (2014). Additionally, the diagonal vector ARFIMA(1,d,1) model is applied to the Cholesky factors of the covariance matrices (Chiriac and Voev;. Furthermore, we consider models with a conditional Wishart distribution, namely the conditional autoregressive Wishart (CAW) model of Golosnoy et al. (2012), a dynamic correlation specification (CAW-DCC) of Bauwens et al. (2012), and additive and multiplicative components Wishart models as proposed by Jin and Maheu (2013). For further details on the comparison models consult appendix B. For each loss function and horizon h, we compute the average losses (risks) for all models and obtain model confidence sets of Hansen et al. (2011), bootstrapping the maxt statistic with a block lengths of max{5, h}. In tables 5, 6, 7 and 8, we present the risks for h = 1, 5, 10, 20. The best performing model ( * * * ) as well as members of the 80% model confidence set ( * * ) and models contained in the 90% but not in the 80% set ( * ) are indicated.
The fractional components model is among the best competitors for all horizons and loss functions. It has lowest risks for almost all setups. Exceptions occur for h ≥ 10 where the ARFIMA model for log variances and z-correlations performs best in some cases. Overall, the ARFIMA model on y t appears as second best in terms of forecasting precision.
The DOFC model is always contained in the 80% model confidence set whereas all other models are rejected at least in some cases. For the Stein loss and the minimum-variance loss, the DOFC model is significantly superior than most competitors for small horizons, while with the Frobenius and asymmetric loss, rejections of other models are achieved for h = 10 and h = 20.
The performance of the fractional components model in terms of density forecasting is noteworthy. In each case there, our model is either the single member or one of two models in the confidence set and hence significantly outperforms most of the competitors. Since the behaviour of future daily returns is usually more important than the realized measures themselves, this finding is particularly strong from a practitioner's perspective.
Overall, we find a very good forecast performance of the model proposed in this paper. Although for some criteria and horizons statistical significance is lacking, the model yields very precise forecasts in relation to different competitors for all considered horizons and for several ways to measure this precision.
Conclusion
We have suggested a general setup and a parsimonious model with very general fractional integration and cointegration properties. We discussed the usefulness of our approach for multivariate realized volatility modeling. In our application it was shown to provide a reasonable in-sample fit and competitive out-of-sample forecasting accuracy.
Several questions remain for further research. From an empirical point of view, we have shown the relevance of a very restricted specification in financial econometrics, but the general setup we introduced has a broader scope. Fractional components models with rich short-run dynamics may be considered for models of smaller dimension. In several empirical setups, fractional integration and cointegration has been found relevant, so that dynamic modeling, forecasting, identification of structural shocks and impulse response analyses in an according framework is a fruitful direction of ongoing research.
A Details on alternative representations
In this appendix we provide more details on the derivation of the alternative representations of the fractional components model (1) which we discuss in section 3.
To derive the error correction representation (6), we start from the FC setup with q = 2 and s = p,
y t = Λ (1) x (1) t + Λ (2) x (2) t + u t ,
from which we note that
Λ (1) ⊥ ∆ d (2) y t = Λ (1) ⊥ Λ (2) ξ (2) t + Λ (1) ⊥ ∆ d (2) u t and Λ (2) ⊥ ∆ d (1) y t = Λ (2) ⊥ Λ (1) ξ (1) t + Λ (2) ⊥ ∆ d (1) u t .
We define
N := Λ (2) (Λ (1) ⊥ Λ (2) ) −1 Λ (1) ⊥ and M := Λ (1) (Λ (2) ⊥ Λ (1) ) −1 Λ (2) ⊥ ,(13)
and make use of I = N + M (Johansen;2008), to obtain
∆ d (1) y t = M (Λ (1) ξ (1) t + ∆ d (1) u t ) + ∆ d (1) −d (2) ∆ d (2) N y t .(14)
Adding and substracting ∆ d (2) N y t on the right side of (14) and the decomposition N = −αβ yields (6). Next, we consider the triangular representation; see (8) and (9). The first block, (8), is easily obtained. Since Λ (1,1) is nonsingular and we also assumed a nonsingular covariance matrix of the white noise sequence ξ t , we find that the first term on the right is I(0) with positive definite spectral density while the other terms have integration orders lower than zero, leading to ω withω j t ∼ I(0) which we can solve for
∆ d (j) x
(1:(j−1)) t = (Λ (1:(j−1),1:(j−1)) ) −1 ∆ d (j) y
(1:(j−1)) t − (Λ (1:(j−1),1:(j−1)) ) −1ωj t .
Substituting this expression into (15) yields the general expression (9) for the j-th block of the triangular system for j = 2, . . . , q, where ω (j) t =ω j t − Λ (j,1:(j−1)) (Λ (1:(j−1),1:(j−1)) ) −1ωj t ,
which can be stated in greater detail as
ω (j) t = −Λ (j,1:(j−1)) (Λ (1:(j−1),1:(j−1)) ) −1 Λ (1:(j−1),j:q) ∆ d (j) x (j:q) t − Λ (j,1:(j−1)) (Λ (1:(j−1),1:(j−1)) ) −1 ∆ d (j) u (1:(j−1)) t + Λ (j,j:q) ∆ d (j) x (j:q) t + ∆ d (j) u (j) t . (16)
This process is the sum of several additive negatively integrated plus a white noise process
Λ (j,j) − Λ (j,1:(j−1)) (Λ (1:(j−1),1:(j−1)) ) −1 Λ (1:(j−1),j) ξ (j) t ,
so that we conclude that ω (j) t is I(0) with positive definite spectral density at zero frequency. We arrive at the representation (10) where B is partitioned into blocks according to
B = I 0 . . . 0 B (1,1) I 0 . . . . . . . . . . . . B (q,1) . . . B (q,q−1) I .
In case p > s, we have y (q+1) t = Λ (q+1,1:q) (Λ (1:q,1:q) ) −1 y
(1:q) t + u (q+1) t − Λ (q+1,1:q) (Λ (1:j,1:j) ) −1 u (1:q) t = B (q+1,1) y (1) t + . . . + B (q+1,q) y (j−1) t + ω (q+1) t ,
and the representation (10) is changed to
By t = (∆ −d 1 + ω (1) t , . . . , ∆ −dq + ω (q) t , ω (q+1) t )
where B is extended by the p − s rows (B (q+1,1) , . . . , B (q+1,q) , I).
B Details on the out-of-sample comparison
In this section we give further details on the out-of-sample evaluation of section 5.4. We state the loss functions to evaluate the forecasts as well as the specifications of the benchmark models and their estimation. For given forecasted realized covariance matrices X T +h|T and realizations X T +h , the loss functions considered in this paper are the Frobenius norm (LF T ,h ), the Stein distance (LS T ,h ), the asymmetric loss (L3 T ,h ), the realized variance of the ex-ante minimum variance portfolio (LM V T ,h ), and the negative log-score of density forecasts f r (LD T ,h ), given by
LF T ,h = k i=1 k j=1 (X ij,T +h − X ij,T +h|T ) 2 ,(17)LS T ,h = tr X −1 T +h|T X T +h − log X −1 T +h|T X T +h − k,(18)L3 T ,h = 1 6 tr X 3 T +h|T − X 3 T +h − 1 2 tr X 2 T +h|T (X T +h − X T +h|T ) ,(19)LM V T ,h = w X T +h w, w = (ι X T +h|T ι) −1 X T +h|T ι, ι = (1, . . . , 1) ,(20)LD T ,h = − log f r (r T +h ).(21)
As comparison models we consider three linear models in transformed covariance matrices, namely the diagonal vector ARMA(2,1) model
(1 − φ i1 L − φ i2 L 2 )(y it − c i ) = (1 + θ i1 L)v it , i = 1, . . . , 21,(22)
for the log variance and z-correlation series y t , a diagonal vector ARFIMA(1,d,1) model
(1 − φ i1 L)(1 − L) d i (y it − c i ) = (1 + θ i1 L)v it , i = 1, . . . , 21,(23)
for y t and the same model (23) applied to Cholesky factors. The same model orders have been used by Chiriac and Voev (2011) and Weigand (2014) and were found to compete favorably with other choices. The dynamic parameters of these models are estimated by Gaussian quasi maximum likelihood equation by equation, with no cross-equation restrictions such as equality of memory parameters. A full covariance matrix of the error terms is estimated from the residuals. The other four benchmark models are based on a conditional Wishart distribution,
X t |I t−1 ∼ W n (ν, S t /ν),(24)
where I t is the information set consisting of X s , s ≤ t, W n denotes the central Wishart density, ν is the scalar degrees of freedom parameter and S t /ν is a (6 × 6) positive definite scale matrix, which is related to the conditional mean of X t by E[X t |I t−1 ] = S t . The baseline CAW(p,q) model of Golosnoy et al. (2012) specifies the conditional mean as
S t = CC + p j=1 B j S t−j B j + q j=1 A j X t−j A j ,(25)
C, B j and A j denoting (6×6) parameter matrices, while the CAW-DCC model of Bauwens et al. (2012) employs a decomposition S t = H t P t H t where H t is diagonal and P t is a welldefined correlation matrix. As a sparse and simple DCC benchmark we apply univariate realized GARCH(p v ,q v ) specifications for the realized variances
H 2 ii,t = c i + pv j=1 b v i,j H 2 ii,t−j + qv j=1 a v i,j X ii,t−j ,(26)
along with the 'scalar Re-DCC' model (Bauwens et al.; for the realized correlation matrix R t ,
P t =P + pc j=1 b c j P t−j + qc j=1 a c j R t−j .(27)
The diagonal CAW(p,q) and the CAW-DCC(p,q) specification with p = p v = p c = 2 and q = q v = q c = 1 are selected since they provide a reasonable in-sample fit among various order choices. They are estimated by maximum likelihood using variance and correlation targeting. Table 1: Estimation results for different specifications of the models estimated in section 5.3. We show the combinations of s j , j = 0, . . . , q with best values of the BIC for q = 1 (above), q = 2 (middle) and q = 3 (below). y 10,t 2.7 4.9 2.5 -9.0 -2.9 2.0 -3.3 -0.9 2.5 3.6 4.2 6.1 -4.0 y 11,t 4.0 5.1 3.5 -9.9 0.6 2.7 -2.8 -1.7 7.1 -1.8 -2.5 3.2 1.7 y 12,t 2.9 3.8 9.7 -9.8 -1.0 2.5 -1.3 0.2 0.5 3.5 3.5 -2.5 3.0 y 13,t 3.9 4.3 5.5 -7.0 -0.9 2.2 -5.7 -2.7 -3.1 0.6 -1.0 0.2 2.7 y 14,t 2.0 4.4 5.1 -9.8 -2.6 2.6 -2.3 -0.8 -1.8 -0.8 6.1 2.6 -3.4 y 15,t 3.4 6.0 6.2 -12.5 0.5 2.5 -1.6 0.6 1.4 -7.2 -0.4 -0.3 3.5 y 16,t 4.4 3.1 5.6 -10.6 -1.1 2.3 -1.7 -0.4 -2.9 5.4 -2.2 1.2 2.2 y 17,t 3.2 4.4 5.7 -12.5 -1.5 2.8 0.5 1.7 -0.8 3.9 4.2 3.8 -0.9 y 18,t 2.5 3.1 6.6 -12.6 0.0 2.8 0.4 0.3 2.3 2.2 1.2 1.5 10.2 y 19,t 3.7 5.2 3.6 -8.9 -1.9 1.7 -2.6 -0.9 -5.1 2.1 -0.7 6.2 -1.9 y 20,t 3.7 4.0 3.4 -8.1 0.1 2.6 -4.7 -3.2 -2.4 0.2 -2.2 3.9 6.4 y 21,t 1.6 4.0 3.1 -9.9 -1.4 3.0 -1.2 -0.5 -0.3 -2.1 4.5 8.0 1.4 Table 4: Bootstrap t-ratios for fractional components loadings (Λ (1) and Λ (2) ) and nonfractional loadings (Γ ) from the DOFC model (11) Figure 6: Histogram of residuals corresponding to log variances and z-transformed correlations for the first three assets for the fractional components model estimated in section 5 and normal density.
s 1 s 2 s 3 s 0 log-lik d (1) d (2) d(
JP-Morgan Chase & Co for the period from 2000-01-01 to 2008-07-30 (n = 2156). The data are available from http://qed.econ.queensu.ca/jae/2011-v26.6/chiriac-voev.
{0} j
{0}, j = 0, . . . , q, we estimate all models characterized by s j . . . , q, given that they satisfy s + s 0 − 1 ≥ s{0} j ≥ 1. The model with the least value of the BIC is selected and its indices denoted as s {1} j , and again models with indices close to s
Figure 3 :
3Residuals corresponding to log variances and z-transformed correlations for the first three assets for the fractional components model estimated in section 5.
Figure 4 :
4Residual autocorrelations for the fractional components model estimated in section 5.
Figure 5 :
5Autocorrelations of squared residuals for the fractional components model estimated in section 5.
Table 2 :
2P-values of diagnostic tests for the residuals from the DOFC model (11) estimated in section 5.3. We conducted Ljung-Box tests for residual correlation (LB), ARCH-LM tests for conditional heteroskedasticity (CH), each with different lags, and Jarque-Bera tests (JB) for deviations from normality.Estimate Mean SE.boot SE.sand SE.infod 1
0.6308 0.6361
0.0190
0.0217 0.0178
d 2
0.3382 0.3334
0.0094
0.0116 0.0086
φ 1
0.2468 0.2360
0.0345
0.0417 0.0348
φ 2
0.0768 0.0636
0.0370
0.0419 0.0402
h 1
0.2028 0.1844
0.1122
0.1106 0.0759
h 2
0.3858 0.3727
0.0522
0.0551 0.0321
h 3
0.3289 0.3309
0.0930
0.0957 0.0714
h 4
0.1758 0.1638
0.1222
0.1371 0.0861
h 5
0.7649 0.7618
0.0558
0.0676 0.0482
h 6
0.2459 0.2413
0.0772
0.0810 0.0588
h 7
0.0615 0.0611
0.0037
0.0079 0.0027
h 8
0.0746 0.0739
0.0032
0.0063 0.0026
h 9
0.0799 0.0793
0.0033
0.0060 0.0027
h 10
0.0778 0.0771
0.0034
0.0060 0.0028
h 11
0.0725 0.0718
0.0036
0.0072 0.0030
h 12
0.0563 0.0557
0.0036
0.0062 0.0025
h 13
0.0545 0.0543
0.0032
0.0063 0.0026
h 14
0.0509 0.0505
0.0031
0.0060 0.0025
h 15
0.0570 0.0564
0.0056
0.0077 0.0045
h 16
0.0739 0.0733
0.0033
0.0059 0.0029
h 17
0.0889 0.0880
0.0036
0.0053 0.0032
h 18
0.0441 0.0438
0.0040
0.0082 0.0030
h 19
0.0919 0.0910
0.0038
0.0059 0.0035
h 20
0.0621 0.0615
0.0037
0.0064 0.0031
h 21
0.0601 0.0595
0.0035
0.0060 0.0032
Table 3 :
3Estimated parameters along with bootstrap mean and standard errors from bootstrap (SE.boot), sandwich (SE.sand) and information matrix (SE.info) as described inHartl and Weigand (2018) for the DOFC model (11) estimated in section 5.3.Λ (1)
Λ (2)
Γ
y 1,t 17.4
11.9
3.6
y 2,t 22.7
1.6
1.8 -11.9
0.4 -0.7
y 3,t 13.1 -1.6
3.4 -6.1 -15.6
1.1 1.3
y 4,t 11.9 -2.9
2.1 -3.3 -5.2 16.2
0.6 -1.6
y 5,t 12.5 -11.1
4.4 -10.7 -1.1 1.1 -3.4
1.6 -1.7
y 6,t 18.9
1.2
2.0 -6.0 -2.3 3.1 -3.1 6.8
1.0 1.7
y 7,t
4.1
6.0
5.6 -8.0 -1.1 2.0 -4.7 -1.5 4.9
-0.5 -2.7
y 8,t
3.5
3.2
5.4 -10.2 -1.2 2.4 -1.6 0.5 4.4 7.4
0.9 0.2
y 9,t
4.7
4.2
2.8 -7.4 -1.0 2.2 -5.8 -2.2 -0.1 3.4 -3.0 3.9 -0.5
estimated in section 5.3. FC 84.28 * * * 0.9660 * * * 1807 * * * 0.7905 * * * 8.1319 * * * FC 170.07 * * 1.7033 * * 2837 * * 0.8102 * * 8.3118 * * * ARMA 172.03 * * 1.6985 * * 2890 * * 0.8088 * 8.3519 ARFIMA 168.55 * * * 1.6716 * * * 2837 * * * 0.8076 * * * 8.3372 ARFIMA.chol 173.43 * * 1.8455 2900 * * 0.8103 * * 8.3893 CAW.diag 176.50 * * 1.7399 * * 2980 * * 0.8110 * * 8.4105 CAW.dcc 178.11 * * 1.7120 * * 2986 * * 0.8101 * * 8.4973 CAW.acomp 177.37 * * 1.7366 * * 2947 * * 0.8093 * * 8.4146 CAW.mcomp 181.04 * * 1.7265 * * 3009 * * 0.8096 * * 8.4248h = 1
LF
LS
L3
LMV
LD
h = 10
LF
LS
L3
LMV
LD
Table 7 :
7Out of sample risks for h = 10 as described in section 5.4. For details on the abbreviations see table 5. FC 199.42 * * * 2.0461 * * 3144 * * * 0.8225 * * 8.3778 * * * ARMA 208.07 * * 2.0980 * * 3231 0.8224 * * 8.4314 ARFIMA 200.33 * * 2.0305 * * * 3162 * * 0.8209 * * * 8.4049 ARFIMA.chol 203.22 * * 2.1910 * * 3201 * * 0.8219 * * 8.4738 CAW.mcomp 209.78 * * 2.0858 * * 3282 * * 0.8225 * * 8.5158h = 20
LF
LS
L3
LMV
LD
CAW.diag
214.60 *
2.1580 * * 3326 *
0.8241 * * 8.5034
CAW.dcc
217.52
2.1698 * * 3331
0.8231 * * 8.6158
CAW.acomp
211.33
2.1165 * * 3289 *
0.8214 * * 8.5028
Table 8 :
8Out of sample risks for h = 20 as described in section 5.4. For details on the abbreviations see table 5.
Series v[, i]^2Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^2 Series v[, i]^25
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Results are available from the authors upon request.
(1) t ∼ I(0). To arrive at the j-th block of the system, consider the expression for y(j) t , ∆ d (j) y (j) t = Λ (j,1) ∆ d (j) x (1) t + . . . + Λ (j,q) ∆ d (j) x (q) t + ∆ d (j) u (j) t . Since ∆ d (j) x (i)t is integrated of order zero or lower for i ≥ j, we can write∆ d (j) y (j) t = Λ (j,1) ∆ d (j) x (1) t + . . . + Λ (j,j−1) ∆ d (j) x (j−1) t +ω j t = Λ (j,1:(j−1)) ∆ d (j) x (1:(j−1)) t +ω j t ,(15)whereω j t ∼ I(0). To substitute for the latent variables in this expression, consider∆ d (j) y(1:(j−1)) t = Λ (1:(j−1),1:(j−1)) ∆ d (j) x(1:(j−1)) t +ω j t ,
AcknowledgementsThe research of Roland Weigand has mostly been done at the Institute of Economics and Econometrics of the University of Regensburg and at the Institute for Labour Market Research (IAB) in Nuremberg. Very valuable comments by Rolf Tschernig, by Enzo Weber and by participants of the Interdisciplinary Workshop on Multivariate Time Series Modeling 2011 in Louvain La Neuve, at the Statistische Woche 2011 in Leipzig, and of research seminars at the Universities of Regensburg, Augsburg and Bielefeld are gratefully acknowledged. The authors are also thankful to Niels Aka for providing R codes to estimate model confidence sets. Tobias Hartl gratefully acknowledges support through the projects TS283/1-1 and WE4847/4-1 financed by the German Research Foundation (DFG).
Statistical inference in factor analysis. T W Anderson, H Rubin, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. J. Neymanthe Third Berkeley Symposium on Mathematical Statistics and ProbabilityUniversity of California PressVAnderson, T. W. and Rubin, H. (1956). Statistical inference in factor analysis, in J. Ney- man (ed.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. V, University of California Press.
Forecasting co-volatilities via factor models with asymmetry and long memory in realized covariance. M Asai, M Mcaleer, Journal of Econometrics. 1892Asai, M. and McAleer, M. (2015). Forecasting co-volatilities via factor models with asym- metry and long memory in realized covariance, Journal of Econometrics 189(2): 251-262.
A Wald test for the cointegration rank in nonstationary fractional systems. M Avarucci, C Velasco, Journal of Econometrics. 1512Avarucci, M. and Velasco, C. (2009). A Wald test for the cointegration rank in nonsta- tionary fractional systems, Journal of Econometrics 151(2): 178 -189.
A state space canonical form for unit root processes. D Bauer, M Wagner, Econometric Theory. 286Bauer, D. and Wagner, M. (2012). A state space canonical form for unit root processes, Econometric Theory 28(6): 1313-1349.
Forecasting multivariate realized stock market volatility. G H Bauer, K Vorkink, Journal of Econometrics. 1601Bauer, G. H. and Vorkink, K. (2011). Forecasting multivariate realized stock market volatility, Journal of Econometrics 160(1): 93-101.
Dynamic conditional correlation models for realized covariance matrices, CORE Discussion Paper 2012-60. L Bauwens, G Storti, F Violante, Bauwens, L., Storti, G. and Violante, F. (2012). Dynamic conditional correlation models for realized covariance matrices, CORE Discussion Paper 2012-60. URL: https://EconPapers.repec.org/RePEc:cor:louvco:2012060
Breaks and persistency: macroeconomic causes of stock market volatility. A Beltratti, C Morana, Journal of Econometrics. 1311-2Beltratti, A. and Morana, C. (2006). Breaks and persistency: macroeconomic causes of stock market volatility, Journal of Econometrics 131(1-2): 151-177.
Unemployment and input prices: A fractional cointegration approach. G M Caporale, L A Gil-Alana, Applied Economics Letters. 96Caporale, G. M. and Gil-Alana, L. A. (2002). Unemployment and input prices: A fractional cointegration approach, Applied Economics Letters 9(6): 347-351.
Resuscitating the co-fractional model of granger. F Carlini, P Santucci De Magistris, Technical reportCarlini, F. and Santucci de Magistris, P. (2018). Resuscitating the co-fractional model of granger (1986), Technical report. URL: https://ssrn.com/abstract=3218361
Using Kalman filter to extract and test for common stochastic trends. Unpublished manuscript. Y Chang, B Jiang, J Y Park, Chang, Y., Jiang, B. and Park, J. Y. (2012). Using Kalman filter to extract and test for common stochastic trends. Unpublished manuscript. URL: https://economics.indiana.edu/home/people/yoosoon
Extracting a common stochastic trend: Theory with some applications. Y Chang, J I Miller, J Y Park, Journal of Econometrics. 1502Recent Development in Financial EconometricsChang, Y., Miller, J. I. and Park, J. Y. (2009). Extracting a common stochastic trend: Theory with some applications, Journal of Econometrics 150(2): 231 -247. Recent Development in Financial Econometrics.
Semiparametric estimation of fractional cointegrating subspaces. W W Chen, C M Hurvich, The Annals of Statistics. 346Chen, W. W. and Hurvich, C. M. (2006). Semiparametric estimation of fractional cointe- grating subspaces, The Annals of Statistics 34(6): 2939-2979.
A fractional cointegration analysis of purchasing power parity. Y.-W Cheung, K S Lai, Journal of Business & Economic Statistics. 111Cheung, Y.-W. and Lai, K. S. (1993). A fractional cointegration analysis of purchasing power parity, Journal of Business & Economic Statistics 11(1): 103-112.
Modelling and forecasting multivariate realized volatility. R Chiriac, V Voev, Journal of Applied Econometrics. 266Chiriac, R. and Voev, V. (2011). Modelling and forecasting multivariate realized volatility, Journal of Applied Econometrics 26(6): 922-947.
Maximum-likelihood estimation of fractional cointegration with an application to U.S. and Canadian bond rates. M Dueker, R Startz, The Review of Economics and Statistics. 803Dueker, M. and Startz, R. (1998). Maximum-likelihood estimation of fractional cointegra- tion with an application to U.S. and Canadian bond rates, The Review of Economics and Statistics 80(3): 420-426.
A permanent and transitory component model of stock return volatility. R Engle, G Lee, Cointegration, Causality, and Forecasting: A Festschrift in Honor of Clive W.J. Granger. R. Engle and H. WhiteOxford University PressEngle, R. and Lee, G. (1999). A permanent and transitory component model of stock return volatility, in R. Engle and H. White (eds), Cointegration, Causality, and Forecasting: A Festschrift in Honor of Clive W.J. Granger, Oxford University Press.
System estimation of panel data models under long-range dependence. Y E Ergemen, 10.1080/07350015.2016.1255217Journal of Business & Economic Statistics. Ergemen, Y. E. (2017). System estimation of panel data models under long-range depen- dence, Journal of Business & Economic Statistics . URL: https://doi.org/10.1080/07350015.2016.1255217
Estimation of fractionally integrated panels with fixed effects and cross-section dependence. Y E Ergemen, C Velasco, Journal of Econometrics. 1962Ergemen, Y. E. and Velasco, C. (2017). Estimation of fractionally integrated panels with fixed effects and cross-section dependence, Journal of Econometrics 196(2): 248-258.
Fractional integration and cointegration: An overview and an empirical application. L Gil-Alana, J Hualde, Palgrave Handbook of Econometrics. K. Patterson and T. MillsMacMillanIIGil-Alana, L. and Hualde, J. (2008). Fractional integration and cointegration: An overview and an empirical application, in K. Patterson and T. Mills (eds), Palgrave Handbook of Econometrics, Vol. II, Palgrave, MacMillan, pp. 434-469.
The conditional autoregressive Wishart model for multivariate stock market volatility. V Golosnoy, B Gribisch, R Liesenfeld, Journal of Econometrics. 1671Golosnoy, V., Gribisch, B. and Liesenfeld, R. (2012). The conditional autoregres- sive Wishart model for multivariate stock market volatility, Journal of Econometrics 167(1): 211-223.
Developments in the study of cointegrated economic variables. C W J Granger, Oxford Bulletin of Economics and Statistics. 483Granger, C. W. J. (1986). Developments in the study of cointegrated economic variables, Oxford Bulletin of Economics and Statistics 48(3): 213-228.
Multicointegration. C W J Granger, T H Lee, Advances in Econometrics: Cointegration, Spurious Regressions, and Unit Roots. G. F. Rhodes and T. B. FombyJAI PressGranger, C. W. J. and Lee, T. H. (1989). Multicointegration, in G. F. Rhodes and T. B. Fomby (eds), Advances in Econometrics: Cointegration, Spurious Regressions, and Unit Roots, JAI Press.
A latent dynamic factor approach to forecasting multivariate stock market volatility. B Gribisch, Beiträge zur Jahrestagung des Vereins für Socialpolitik 2013: Wettbewerbspolitik und Regulierung in einer globalen Wirtschaftsordnung -Session: Volatility, F01-V2. Gribisch, B. (2013). A latent dynamic factor approach to forecasting multivariate stock market volatility, Beiträge zur Jahrestagung des Vereins für Socialpolitik 2013: Wettbe- werbspolitik und Regulierung in einer globalen Wirtschaftsordnung -Session: Volatility, F01-V2. URL: http://hdl.handle.net/10419/79823
The model confidence set. P R Hansen, A Lunde, J Nason, Econometrica. 792Hansen, P. R., Lunde, A. and Nason, J. (2011). The model confidence set, Econometrica 79(2): 453-497.
Approximate State Space Modelling of Unobserved Fractional Components. T Hartl, R Weigand, arXiv.orgHartl, T. and Weigand, R. (2018). Approximate State Space Modelling of Unobserved Fractional Components, Papers 1812.09142, arXiv.org. URL: https://ideas.repec.org/p/arx/papers/1812.09142.html
Forecasting. A C Harvey, Structural Time Series Models and the Kalman Filter, Cambridge Books. Cambridge University PressHarvey, A. C. (1991). Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge Books, Cambridge University Press.
Multivariate stochastic variance models. A Harvey, E Ruiz, N Shephard, The Review of Economic Studies. 612Harvey, A., Ruiz, E. and Shephard, N. (1994). Multivariate stochastic variance models, The Review of Economic Studies 61(2): 247-264.
Identification of causal factor models of stationary time series. C Heaton, V Solo, Econometrics Journal. 72Heaton, C. and Solo, V. (2004). Identification of causal factor models of stationary time series, Econometrics Journal 7(2): 618-627.
Bayesian estimation of common long-range dependent models. N.-J Hsu, B K Ray, F J Breidt, Probability Theory and Mathematical Statistics: Proceedings of the Seventh Vilnius Conference. B. Grigelionis, J. Kubilius, V. Paulauskas, V. Statulevicius and H. PragarauskasVSPHsu, N.-J., Ray, B. K. and Breidt, F. J. (1998). Bayesian estimation of common long-range dependent models, in B. Grigelionis, J. Kubilius, V. Paulauskas, V. Statulevicius and H. Pragarauskas (eds), Probability Theory and Mathematical Statistics: Proceedings of the Seventh Vilnius Conference, VSP.
Consistent estimation of cointegrating subspaces. J Hualde, Universidad Pública de Navarra. PreprintHualde, J. (2009). Consistent estimation of cointegrating subspaces. Universidad Pública de Navarra. Preprint.
Semiparametric inference in multivariate fractionally cointegrated systems. J Hualde, P Robinson, Journal of Econometrics. 1572Hualde, J. and Robinson, P. (2010). Semiparametric inference in multivariate fractionally cointegrated systems, Journal of Econometrics 157(2): 492-511.
Modelling realized covariances and returns. X Jin, J M Maheu, Journal of Financial Econometrics. 112Jin, X. and Maheu, J. M. (2013). Modelling realized covariances and returns, Journal of Financial Econometrics 11(2): 335-369.
A representation theory for a class of vector autoregressive models for fractional processes. S Johansen, Econometric Theory. 243Johansen, S. (2008). A representation theory for a class of vector autoregressive models for fractional processes, Econometric Theory 24(3): 651-676.
Likelihood inference for a fractionally cointegrated vector autoregressive model. S Johansen, M Ø Nielsen, Econometrica. 806Johansen, S. and Nielsen, M. Ø. (2012). Likelihood inference for a fractionally cointegrated vector autoregressive model, Econometrica 80(6): 2667-2732.
Factor modeling for high-dimensional time series: Inference for the number of factors. C Lam, Q Yao, The Annals of Statistics. 402Lam, C. and Yao, Q. (2012). Factor modeling for high-dimensional time series: Inference for the number of factors, The Annals of Statistics 40(2): 694-726.
Estimation of latent factors for high-dimensional time series. C Lam, Q Yao, N Bathia, Biometrika. 984Lam, C., Yao, Q. and Bathia, N. (2011). Estimation of latent factors for high-dimensional time series, Biometrika 98(4): 901-918.
Likelihood based testing for no fractional cointegration. K Lasak, Journal of Econometrics. 1581Lasak, K. (2010). Likelihood based testing for no fractional cointegration, Journal of Econometrics 158(1): 67-77.
Fractional cointegration rank estimation. K Lasak, C Velasco, Tinbergen Institute Discussion Paper 2014-021Lasak, K. and Velasco, C. (2014). Fractional cointegration rank estimation, Tinbergen Institute Discussion Paper 2014-021.
On the forecasting accuracy of multivariate GARCH models. S Laurent, J V K Rombouts, F Violante, Journal of Applied Econometrics. 276Laurent, S., Rombouts, J. V. K. and Violante, F. (2011). On the forecasting accuracy of multivariate GARCH models, Journal of Applied Econometrics 27(6): 934-955.
On loss functions and ranking forecasting performances of multivariate volatility models. S Laurent, J V K Rombouts, F Violante, Journal of Econometrics. 1731Laurent, S., Rombouts, J. V. K. and Violante, F. (2013). On loss functions and rank- ing forecasting performances of multivariate volatility models, Journal of Econometrics 173(1): 1-10.
Consistency of the averaged cross-periodogram in long memory time series. I N Lobato, Journal of Time Series Analysis. 182Lobato, I. N. (1997). Consistency of the averaged cross-periodogram in long memory time series, Journal of Time Series Analysis 18(2): 137-155.
Estimating and forecasting large panels of volatilities with approximate dynamic factor models. M Luciani, D Veredas, Journal of Forecasting. 34Luciani, M. and Veredas, D. (2015). Estimating and forecasting large panels of volatilities with approximate dynamic factor models, Journal of Forecasting 34: 163-176.
Linear transformations of vector ARMA processes. H Lütkepohl, Journal of Econometrics. 263Lütkepohl, H. (1984). Linear transformations of vector ARMA processes, Journal of Econo- metrics 26(3): 283 -293.
H Lütkepohl, New Introduction to Multiple Time Series Analysis. SpringerLütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis, Springer.
Dynamic orthogonal components for multivariate time series. D S Matteson, R S Tsay, Journal of the American Statistical Association. 106496Matteson, D. S. and Tsay, R. S. (2011). Dynamic orthogonal components for multivariate time series, Journal of the American Statistical Association 106(496): 1450-1463.
. G Mesters, S J Koopman, M Ooms, Mesters, G., Koopman, S. J. and Ooms, M. (2016).
Monte Carlo maximum likelihood estimation for generalized long-memory time series models. Econometric Reviews. 354Monte Carlo maximum likeli- hood estimation for generalized long-memory time series models, Econometric Reviews 35(4): 659-687.
Frequency domain principal components estimation of fractionally cointegrated processes. C Morana, Applied Economics Letters. 1113Morana, C. (2004). Frequency domain principal components estimation of fractionally cointegrated processes, Applied Economics Letters 11(13): 837-842.
A small scale macroeconometric model for the Euro-12 area. C Morana, Economic Modelling. 233Morana, C. (2006). A small scale macroeconometric model for the Euro-12 area, Economic Modelling 23(3): 391-426.
Multivariate modelling of long memory processes with common components. C Morana, Computational Statistics & Data Analysis. 522Morana, C. (2007). Multivariate modelling of long memory processes with common com- ponents, Computational Statistics & Data Analysis 52(2): 919 -934.
Efficient inference in multivariate fractionally integrated time series models. M Ø Nielsen, Econometrics Journal. 71Nielsen, M. Ø. (2004). Efficient inference in multivariate fractionally integrated time series models, Econometrics Journal 7(1): 63-97.
Local Whittle analysis of stationary fractional cointegration and the implied-realized volatility relation. M Ø Nielsen, Journal of Business & Economic Statistics. 254Nielsen, M. Ø. (2007). Local Whittle analysis of stationary fractional cointegration and the implied-realized volatility relation, Journal of Business & Economic Statistics 25(4): 427-446.
Determining the cointegrating rank in nonstationary fractional systems by the exact local Whittle approach. M Ø Nielsen, K Shimotsu, Journal of Econometrics. 1412Nielsen, M. Ø. and Shimotsu, K. (2007). Determining the cointegrating rank in nonsta- tionary fractional systems by the exact local Whittle approach, Journal of Econometrics 141(2): 574 -596.
Modelling multiple time series via common factors. J Pan, Q Yao, Biometrika. 952Pan, J. and Yao, Q. (2008). Modelling multiple time series via common factors, Biometrika 95(2): 365-379.
Estimation and inference in large heterogenous panels with a multifactor error structure. H Pesaran, Econometrica. 744Pesaran, H. (2006). Estimation and inference in large heterogenous panels with a multi- factor error structure, Econometrica 74(4): 967-1012.
Optimal inference in cointegrated systems. P C B Phillips, Econometrica. 592Phillips, P. C. B. (1991). Optimal inference in cointegrated systems, Econometrica 59(2): 283-306.
Long-range dependence in daily stock volatilities. B K Ray, R S Tsay, Journal of Business & Economic Statistics. 182Ray, B. K. and Tsay, R. S. (2000). Long-range dependence in daily stock volatilities, Journal of Business & Economic Statistics 18(2): 254-262.
The distance between rival nonstationary fractional processes. P Robinson, Journal of Econometrics. 1282Robinson, P. (2005). The distance between rival nonstationary fractional processes, Journal of Econometrics 128(2): 283-300.
Cointegration in fractional systems with unknown integration orders. P M Robinson, J Hualde, Econometrica. 716Robinson, P. M. and Hualde, J. (2003). Cointegration in fractional systems with unknown integration orders, Econometrica 71(6): 1727-1766.
Determination of cointegrating rank in fractional systems. P M Robinson, Y Yajima, Journal of Econometrics. 1062Robinson, P. M. and Yajima, Y. (2002). Determination of cointegrating rank in fractional systems, Journal of Econometrics 106(2): 217 -241.
Long memory and the term structure of risk. P C Schotman, R Tschernig, J Budek, Journal of Financial Econometrics. 21Schotman, P. C., Tschernig, R. and Budek, J. (2008). Long memory and the term structure of risk, Journal of Financial Econometrics 2(1): 1-37.
Exact local Whittle estimation of fractional integration with unknown mean and time trend. K Shimotsu, Econometric Theory. 262Shimotsu, K. (2010). Exact local Whittle estimation of fractional integration with unknown mean and time trend, Econometric Theory 26(2): 501-540.
A simple estimator of cointegrating vectors in higher order integrated systems. J H Stock, M W Watson, Econometrica. 614Stock, J. H. and Watson, M. W. (1993). A simple estimator of cointegrating vectors in higher order integrated systems, Econometrica 61(4): 783-820.
Long-run identification in a fractionally integrated system. R Tschernig, E Weber, R Weigand, Journal of Business and Economic Statistics. Tschernig, R., Weber, E. and Weigand, R. (2013). Long-run identification in a fractionally integrated system, Journal of Business and Economic Statistics .
Matrix Box-Cox models for multivariate realized volatility, University of Regensburg Working Papers in Business, Economics and Management Information System 478. R Weigand, Weigand, R. (2014). Matrix Box-Cox models for multivariate realized volatility, University of Regensburg Working Papers in Business, Economics and Management Information System 478. URL: https://epub.uni-regensburg.de/29687/
Maximum likelihood estimation of misspecified models. H White, Econometrica. 501White, H. (1982). Maximum likelihood estimation of misspecified models, Econometrica 50(1): 1-26.
In different rows, we consider the fractional components (FC) and several benchmark models, namely a diagonal vector ARMA(2,1) and a diagonal vector ARFIMA(1,d,1) model, the conditional autoregressive Wishart (CAW) model of Golosnoy et al. (2012), a dynamic correlation specification (CAW-DCC) of Bauwens et al. (2012), and additive and multiplicative components Wishart models as proposed by. Jin and MaheuAsterisks denote the best performing model ( * * * ), models in the 80% model confidence set ( * * ) and additional models in the 90% model confidence set ( * ). As loss functions, we consider the Frobenius norm (LF), the Stein norm (LS), the predictive densities (LD), the minimum-variance portfolio variance (LMV) and the L3-Loss (L3)Table 5: Out of sample risks for h = 1 as described in section 5.4. In different rows, we consider the fractional components (FC) and several benchmark models, namely a di- agonal vector ARMA(2,1) and a diagonal vector ARFIMA(1,d,1) model, the conditional autoregressive Wishart (CAW) model of Golosnoy et al. (2012), a dynamic correlation specification (CAW-DCC) of Bauwens et al. (2012), and additive and multiplicative com- ponents Wishart models as proposed by Jin and Maheu (2013). Asterisks denote the best performing model ( * * * ), models in the 80% model confidence set ( * * ) and additional models in the 90% model confidence set ( * ). As loss functions, we consider the Frobenius norm (LF), the Stein norm (LS), the predictive densities (LD), the minimum-variance portfolio variance (LMV) and the L3-Loss (L3).
h = 5 LF LS L3 LMV LD FC 135.28 * * * 1.3766 * * * 2463 * * * 0.8011 * * * 8.2490 * * * ARMA 134.43 * *. 14046h = 5 LF LS L3 LMV LD FC 135.28 * * * 1.3766 * * * 2463 * * * 0.8011 * * * 8.2490 * * * ARMA 134.43 * * 1.4046
Figure 1: Time series plots of realized variances for the dataset described in section 5 (grey) together with maximum and minimum for all periods (black). Figure 1: Time series plots of realized variances for the dataset described in section 5 (grey) together with maximum and minimum for all periods (black). 2000 2002 2004 2006 2008
Time series plots of z-transformed realized correlations for the dataset described in section 5 (grey) together with maximum and minimum for all periods (black). Figure. 2Figure 2: Time series plots of z-transformed realized correlations for the dataset described in section 5 (grey) together with maximum and minimum for all periods (black). 2000 2002 2004 2006
Selected smoothed fractional and nonfractional components (solid) ± 2 standard deviations (dashed) for the fractional components model estimated in section 5. Both nonstationary components (above), the first two stationary long-memory components (middle) and the short-memory components. 7below) are givenFigure 7: Selected smoothed fractional and nonfractional components (solid) ± 2 standard deviations (dashed) for the fractional components model estimated in section 5. Both non- stationary components (above), the first two stationary long-memory components (middle) and the short-memory components (below) are given.
| [] |
[
"Attitude determination for nano-satellites -II. Dead reckoning with a multiplicative extended Kalman filter",
"Attitude determination for nano-satellites -II. Dead reckoning with a multiplicative extended Kalman filter"
] | [
"János Takátsy [email protected] ",
"Tamás Bozóki ",
"Gergely Dálya ",
"Kornél Kapás ",
"László Mészáros ",
"András Pál ",
"J Takátsy ",
"T Bozóki ",
"G Dálya ",
"K Kapás ",
"L Mészáros ",
"A Pál ",
"J Takátsy ",
"G Dálya ",
"K Kapás ",
"L Mészáros ",
"A Pál ",
"J Takátsy ",
"T Bozóki ",
"T Bozóki ",
"\nKonkoly Observatory of the Research Centre for Astronomy and Earth Sciences\nBudapestHungary\n",
"\nInstitute for Particle and Nuclear Physics, Wigner Research Centre for Physics\nEötvös Loránd University\nPázmány Péter stny. 1/A, Budapest H-1117, Konkoly-Thege Miklósút 29-33H-1121BudapestHungary, Hungary\n",
"\nDoctoral School of Environmental Sciences\nInstitute of Earth Physics and Space Science (ELKH EPSS)\nUniversity of Szeged\nCsatkai Endre utca 6-8, Aradi vértanúk tere 1, Szeged H-6720H-9400SopronHungary, Hungary\n"
] | [
"Konkoly Observatory of the Research Centre for Astronomy and Earth Sciences\nBudapestHungary",
"Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics\nEötvös Loránd University\nPázmány Péter stny. 1/A, Budapest H-1117, Konkoly-Thege Miklósút 29-33H-1121BudapestHungary, Hungary",
"Doctoral School of Environmental Sciences\nInstitute of Earth Physics and Space Science (ELKH EPSS)\nUniversity of Szeged\nCsatkai Endre utca 6-8, Aradi vértanúk tere 1, Szeged H-6720H-9400SopronHungary, Hungary"
] | [] | This paper is the second part of a series of studies discussing a novel attitude determination method for nano-satellites. Our approach is based on the utilization of thermal imaging sensors to determine the direction of the Sun and the nadir with respect to the satellite with sub-degree accuracy. The proposed method is planned to be applied during the Cubesats Applied for MEasuring and LOcalising Transients (CAMELOT) mission aimed at detecting and localizing gamma-ray bursts with an efficiency and accuracy comparable to large gamma-ray space observatories. In our previous work we determined the spherical projection function of the MLX90640 infrasensors planned to be used for this purpose. We showed that with the known projection function the direction of the Sun can be located with an overall accuracy of ∼ 40 .In this paper we introduce a simulation model aimed at testing the applicability of our attitude determination approach. Its first part simulates the orbit and rotation of a satellite with arbitrary initial conditions while its second part applies our attitude determination algorithm which is based on a multiplicative extended Kalman filter. The simulated satellite is assumed to be equipped with a GPS system, MEMS gyroscopes and the infrasensors. These 2 instruments provide the required data input for the Kalman filter. We demonstrate the applicability of our attitude determination algorithm by simulating the motion of a nano-satellite on Low Earth Orbit. Our results show that the attitude determination may have a 1σ error of ∼ 30 even with a large gyroscope drift during the orbital periods when the infrasensors provide both the direction of the Sun and the Earth (the nadir). This accuracy is an improvement on the point source detection accuracy of the infrasensors. However, the attitude determination error can get as high as 25 • during periods when the Sun is occulted by the Earth. We show that following an occultation period the attitude information is immediately recovered by the Kalman filter once the Sun is observed again. | 10.1007/s10686-021-09818-5 | [
"https://arxiv.org/pdf/2111.13193v1.pdf"
] | 244,709,535 | 2111.13193 | 05d3003dc0f6e6d5f30af5ba6a9833852d718946 |
Attitude determination for nano-satellites -II. Dead reckoning with a multiplicative extended Kalman filter
25 Nov 2021
János Takátsy [email protected]
Tamás Bozóki
Gergely Dálya
Kornél Kapás
László Mészáros
András Pál
J Takátsy
T Bozóki
G Dálya
K Kapás
L Mészáros
A Pál
J Takátsy
G Dálya
K Kapás
L Mészáros
A Pál
J Takátsy
T Bozóki
T Bozóki
Konkoly Observatory of the Research Centre for Astronomy and Earth Sciences
BudapestHungary
Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics
Eötvös Loránd University
Pázmány Péter stny. 1/A, Budapest H-1117, Konkoly-Thege Miklósút 29-33H-1121BudapestHungary, Hungary
Doctoral School of Environmental Sciences
Institute of Earth Physics and Space Science (ELKH EPSS)
University of Szeged
Csatkai Endre utca 6-8, Aradi vértanúk tere 1, Szeged H-6720H-9400SopronHungary, Hungary
Attitude determination for nano-satellites -II. Dead reckoning with a multiplicative extended Kalman filter
25 Nov 2021Exp Astron manuscript No. (will be inserted by the editor) the date of receipt and acceptance should be inserted laterSpace vehicle instruments (1548)Stellar tracking devices (1633)Pointing accuracy (1271)Astrometry (80)
This paper is the second part of a series of studies discussing a novel attitude determination method for nano-satellites. Our approach is based on the utilization of thermal imaging sensors to determine the direction of the Sun and the nadir with respect to the satellite with sub-degree accuracy. The proposed method is planned to be applied during the Cubesats Applied for MEasuring and LOcalising Transients (CAMELOT) mission aimed at detecting and localizing gamma-ray bursts with an efficiency and accuracy comparable to large gamma-ray space observatories. In our previous work we determined the spherical projection function of the MLX90640 infrasensors planned to be used for this purpose. We showed that with the known projection function the direction of the Sun can be located with an overall accuracy of ∼ 40 .In this paper we introduce a simulation model aimed at testing the applicability of our attitude determination approach. Its first part simulates the orbit and rotation of a satellite with arbitrary initial conditions while its second part applies our attitude determination algorithm which is based on a multiplicative extended Kalman filter. The simulated satellite is assumed to be equipped with a GPS system, MEMS gyroscopes and the infrasensors. These 2 instruments provide the required data input for the Kalman filter. We demonstrate the applicability of our attitude determination algorithm by simulating the motion of a nano-satellite on Low Earth Orbit. Our results show that the attitude determination may have a 1σ error of ∼ 30 even with a large gyroscope drift during the orbital periods when the infrasensors provide both the direction of the Sun and the Earth (the nadir). This accuracy is an improvement on the point source detection accuracy of the infrasensors. However, the attitude determination error can get as high as 25 • during periods when the Sun is occulted by the Earth. We show that following an occultation period the attitude information is immediately recovered by the Kalman filter once the Sun is observed again.
Introduction
Owing to the enormous funding requirements, satellite missions were only conducted by the economically most powerful countries of the world in the first few decades of the space era. However, as a consequence of the explosive technological development, small satellite missions with substantially lower funding requirement -for instance, nano-satellites including as CubeSatsbecame a viable alternative for traditional large-size and high-cost satellites, which made space an achievable goal also for countries/organizations with less financial resources. The last few decades brought along a lot of such missions with more and more scientific aims being targeted by them.
One of the technological difficulties that needs to be handled in connection with small satellite missions is the accurate determination of the satellite's actual orientation, i.e. its attitude. While on large-size satellites this information is usually provided by costly, large-size star trackers, which determine the attitude based on the angular distribution of bright stars in their field of view, these systems do not fit the very restricted size and power budget criteria of nano-satellites. In our recent paper (Kapás et al., 2021) we proposed a new, cost-efficient approach to this problem which is based on the utilization of thermal imaging infrasensors. For this purpose we chose the MLX90640 infrasensor of Melexis (2018), which is a small-size, low-cost sensor having 32×24 pixels and a relatively large, 110×75 degree field of view. This coverage by a single sensor implies that six of these sensors, placed on the six sides of a cube, could cover the full sphere, see Figure 4 of Kapás et al. (2021). This technology might be suitable to even smaller satellites in similar missions like GRBAlpha (Pal et al., 2020). As the spherical projection function of MLX90640 infrasensors (to be used for this purpose) is now known with an overall accuracy of ∼ 40 (Kapás et al., 2021) we now turn to the next step and introduce a simulation model for testing the applicability of our attitude determination approach. As we outline in our recent paper (Dálya et al., 2020), our method is based on a multiplicative extended Kalman filter that uses the information provided by the infrasensors (direction of the Sun and the nadir in the satellite's coordinate frame), the GPS system (the location of the satellite in the Earth centered coordinate frame) and the MEMS gyroscopes (angular velocity of the satellite) carried by the satellite.
The layout of the paper is the following. In Section 2 we describe the simulation of the satellite dynamics and introduce the multiplicative extended Kalman filter method. We demonstrate our results in Section 3, and we summarize our conclusions in Section 4. A detailed description of the equations used for the Kalman filter can be found in Appendix A.
2 Simulation model for testing the on-board attitude determination algorithm
The attitude determination algorithm we developed is aimed to run on-board and therefore it needs to be tested for the different situations possible during a space mission. For this purpose we built a simulation model where all parts of the attitude determination process can be tested independently and as a whole as well. The first part of the code simulates the dynamics of the Sun-Earth-satellite system while its second part determines the attitude of the satellite by applying a multiplicative extended Kalman filter to the simulated data provided by the first part of the code. The goal of this process is to see how the recovered attitudes compare to the 'real' ones.
Simulation of the satellite dynamics
This part of the code calculates the position and the attitude of the satellite in the Earth centered J2000 reference system (where the X and Z axes point towards the former positions of the vernal equinox and the Earth's rotation axis in January 1, 2000 at 12:00 TT, respectively) as well as the position of the Sun and the Earth in the satellite's coordinate system (where the origin of the system is fixed to the center of mass of the cubesat and the axes are parallel to its edges, see Fig. 1). In the code time is expressed in Julian dates and the GPS to J2000 coordinate transformations are implemented as well. The code also determines the 'night' part of the orbit, i.e. where the Sun is occulted by the Earth.
The orbit of a satellite can be characterized by five orbital elements (Ω, longitude of ascending node; i, inclination; ω, argument of periapsis; e, eccentricity; h, altitude of satellite orbit at perigeum; see Fig 1.) which remain constant when assuming a spherical Earth. The actual position of the satellite on this elliptical orbit is given via the true anomaly (ν) which is obtained from the Kepler equation. However, the oblateness of the Earth introduces perturbations, from which the J 2 perturbation has the largest magnitude, and therefore it has to be taken into account while simulating the satellite dynamics. The main effect of the J 2 perturbation is on Ω and ω. The time derivatives Fig. 1 The two main coordinate systems used in the simulations. Index i denotes the J2000 system with the Earth in the origin. The Z i axis coincides with the rotation axis of the Earth in January 1, 2000 at 12:00 TT while the X i axis points to the vernal equinox at the same epoch. The axes of the satellite's coordinate frame are denoted by index s. The orbital parameters displayed in the figure are Ω (longitude of ascending node), i (inclination), ω (argument of periapsis) and ν (true anomaly).
Z i Y i X i Ω ν Z s X s i ω Y s
of these orbital elements are the following (Kozai, 1959):
ω = 3 4 nJ 2 R e p 2 (5 cos 2 i − 1) (1) Ω = − 3 2 nJ 2 R e p 2 cos i (2)
where R e is the radius of the Earth, p = a(1 − e 2 ), n = GM e /a 3 and J 2 1.082629 · 10 −3 .
Next we discuss the description of the rotation of the satellite. As the variation of the gravitational field is negligible in the range of the satellite's dimensions and we did not apply any kind of attitude control in our model, the net angular momentum transfer due to external torques is negligible during one orbit. We also assumed that the satellite frame coincides with the principal frame, however, in case it does not, an additional constant rotation needs to be taken into account between the two frames. The time evolution of the attitude then can be determined using Euler's rotation equation, which describes the evolution of the angular velocity of a rotating rigid body represented in its principal frame (see e.g. Coutsias and Romero, 2004):
d dt ω 1 ω 2 ω 3 = I 2 − I 3 I 1 ω 2 ω 3 I 3 − I 1 I 2 ω 3 ω 1 I 1 − I 2 I 3 ω 1 ω 2 .(3)
where ω is the angular velocity vector of the satellite represented in the principal frame (which coincides with the satellite frame), and I 1 , I 2 and I 3 are the moments of inertia corresponding to the x, y and z axes, respectively. Note that Eq.
(3) could imply complex motions, such as tumbling 1 , which we can readily reproduce within our model (see supplementary material). The attitude can then be calculated by an additional integration over the angular velocities. In this work we use unit quaternions to represent the attitude of the satellite as q s = [n sin(γ/2), cos(γ/2)] T , where n is the axis of the rotation that transforms the J2000 coordinate frame to the satellite frame, γ is the rotation angle and the superscript T denotes transposition (quantites represented in the satellite frame are denoted by the superscript 's'). The quaternion kinematics is given by the following equation (see e.g., Crassidis and Junkins, 2012;Baroni, 2018):
q s = 1 2 Ω (ω s ) q s ,(4)
with the 4 × 4 matrix
Ω (ω s ) = −S(ω s ) ω s −(ω s ) T 0 ,(5)
where S(ω s ) is the matrix representation of the cross product (ω s ×):
S(ω) = 0 −ω 3 ω 2 ω 3 0 −ω 1 −ω 2 ω 1 0 .(6)
Attitude determination with Kalman filter
The Kalman filter is an algorithm that provides an efficient way to estimate the state of a dynamic system by a series of measurements with inaccuracies over time (Kalman, 1960). The estimates produced by the algorithm are more accurate than those based on a single measurement alone since the joint probability distribution of the variables is estimated for each discrete time-step of the process. This leads to the minimization of the mean of the squared error of the estimates. The Kalman filter works in a two-step process with a prediction step (time update) and a measurement step. In the prediction step the filter propagates the estimates of state and uncertainties from current to the next time step. In the measurement step the state of the system is measured with some error and the estimate is updated by the weighted average of the estimate and the measurement, where the weights are determined by the respective uncertainties.
In our specific objective the Kalman filter serves to combine the infrasensor data with the angular velocity information provided by the MEMS gyroscopes. Although the 3 axis MEMS gyroscopes yield an accurate attitude information on a short time period, due to the error accumulation effect known as gyroscope drift an absolute attitude information is required as well, which is provided by the thermal imaging infrasensors in our case. In earlier works this kind of absolute attitude information was usually gathered by a 3-axis magnetometer and an optical Sun sensor (see e.g., Ni et al. 2011;Springmann et al. 2012;Baroni 2018;Gaber et al. 2020). The use of infrasensors is more convenient in the sense that as opposed to magnetometers they can be built-in parts of the satellite and do not need an external boom to be mounted on. The infrasensors determine the direction of the Sun and the nadir in the satellite's coordinate frame, which vectors are known in the Earth centered reference frame as well owing to the location information provided by the onboard GPS. The rotation that transforms the reference frame to the satellite frame (which is indeed equivalent to the attitude of the satellite) is unequivocally determined by the pair of these two vectors in the two coordinate frames. Hence we get a prediction of the system's state from the gyroscope which can be corrected by the absolute attitude information provided by the infrasensors.
Since the quaternion kinematics, described by Eq. (4), is nonlinear in the variables ω s and q s the utilization of an extended Kalman filter is necessary. We use the Multiplicative Extended Kalman Filter (MEKF) method (Lefferts et al., 1982), where a multiplicative error state δq is introduced, which represents a small rotation from the predicted attitude -which contains measurement errors -to the actual attitude (from now on we omit superscripts, since everything is understood to be represented in the satellite frame, unless noted otherwise) 2 : δq = q ⊗q −1 ,
where the circumflex ' ∧ ' denotes the expected (or predicted) value of a quantity. With this multiplicative approach the conservation of the unit quaternion length is guaranteed and the problem of singular covariance matrices, encountered in the additive approach, is avoided as well.
To describe gyroscope measurements we use the model of Lefferts et al. (1982), where in addition to a zero mean Gaussian error η ω a time dependent bias vector β is also introduced, the motion of which is determined by a random walk. Hence, the measured angular velocity ω m is given by
ω m = ω + β + η ω ,(8)
2 Note that the quaternion product ⊗ is conventionally defined here such that ijk = 1 instead of the more commonly used ijk = −1
β = η β ,(9)
where the Gaussian processes η ω and η β have covariance matrices σ 2 ω I 3×3 and σ 2 β I 3×3 , respectively. Therefore the estimated value of the angular momentum isω = ω m −β.
The state space model is then given by the equations:
q = 1 2 Ω (ω) q,(11)β = 0.(12)
The predicted values of the angular momentum and bias vectors are updated through the Kalman filter using the position of the Sun and the nadir both as seen by the satellite and as calculated in the inertial frame. A vector in the inertial frame is transformed to the satellite frame by
r s = A(q s )r i .(13)
It can then be shown that a small multiplicative quaternion error δq creates a small δr s deviation detemined by the following equation (Crassidis and Junkins, 2012):
δr s = 2S A (q s ) r i δq 3 ,(14)
where δq 3 is a three component vector containing the imaginary part of the multiplicative error state δq. This determines the so called sensitivity matrix, which is then used by the Kalman filter to calculate the multiplicative error state from the δr s quantities. Since measurements are made at discrete time-steps, to implement these equations one must first discretize the kinematical equations. The Kalman filter then can be applied after each time-step to refine the attitude information predicted by the kinematical equations. The detailed discretized equations can be found in Appendix A.
Simulation results
The applicability of our attitude determination approach is demonstrated by simulating a satellite on Low Earth Orbit (LEO) described by the following parameters:
-Ω = 0 • , i = 60 • , ω = 0 • , e = 0.01, h = 650 km, -I 1 = 2.75 × 10 −4 kg m 2 , I 2 = 2.75 × 10 −4 kg m 2 , I 2 = 5.5 × 10 −5 kg m 2 , -L 0 = [−4.4 × 10 −6 , 1.925 × 10 −6 , −6.05 × 10 −7 ] kg m 2 /s, Fig. 2 The orbit and rotation of the simulated satellite. Its position is described by its altitude, latitude and longitude (left) while its attitude is described by its right ascension (α), declination (δ) and roll (ρ) (right). For the different parameters that characterize the satellite's motion we refer to the main text. Shaded areas represent those parts of the orbit where the Sun is occulted by the Earth.
where L 0 denotes the initial angular momentum of the satellite in the satellite frame. The chosen I 1 , I 2 and I 3 values correspond to a 3U CubeSat with a size of 10x10x30 cm and total mass of 3.3 kg. The initial attitude was selected randomly. Figure 2 shows how the position and the attitude of the satellite changes during 6 hours on such an orbit. The attitude is represented in the form of right ascension (α), declination (δ) and roll (ρ). The conversion rule between these angles and the quaternion representation of the attitude is given by the following formulas: α = arg(q 1 q 3 + q 2 q 4 , q 2 q 3 − q 1 q 4 ), δ = arg(q 2 4 + q 2 3 − q 2 2 − q 2 1 , 2 (q 2 1 + q 2 2 )(q 2 3 + q 2 4 )), ρ = arg(q 1 q 3 − q 2 q 4 , −q 2 q 3 − q 1 q 4 ),
where arg(x, y) gives the ϕ phase factor of the complex number x+iy = r ·e iϕ . Note also that the domain of δ is [0 • , 90 • ], while it is [0 • , 360 • ) for α and ρ. In the figure shaded areas represent those parts of the orbit where the Sun is occulted by the Earth, i.e. where the number of measured vectors for the MEKF is reduced to one. Since MEMS gyroscopes are available with various precision we investigated three different cases for attitude determination characterized by three different values for gyroscope drifts. For the largest error case we used σ ω = 4.89 × 10 −3 rad/s 1/2 and σ β = 3.14 × 10 −4 rad/s 3/2 as proposed by Baroni (2018), while for our standard and low-error case we used errors 0.1 and 0.3 times those of the high-error case, respectively. The initial parameters of the MEKF were chosen as follows: Fig. 3 The real (orange) and the MEKF recovered (green) attitude for the simulation shown in Fig. 2 for our standard choice of gyroscope error (left), and the error of the attitude determination, i.e. the difference between the real and the recovered attitude elements for the same orbital configuration (right). Shaded areas represent parts of the orbit where the Sun is occulted by the Earth. ([0.25, 0.25, 0.25, 0.01, 0.01, 0.01]). q 0 was selected randomly and the standard deviation of the measured vectors had been set to 0.012 rad (∼ 40 , in accordance with our previous result on the pointing accuracy of MLX90640 infrasensors) and had been added to the input vectors. Sensor data were sampled at 1 Hz.
-β 0 = [0, 0, 0], -P 0 = diag
The recovery of the attitude on the orbit shown in Fig. 2 for our standard choice of gyroscope error is presented in Fig. 3. The left panels of Fig. 3 show that the attitude elements are well recovered when the MEKF works with two input vectors ('day'), i.e. when the infrasensors provide both the direction of the Sun and the Earth (the nadir), while the accuracy breaks down significantly when there is only one input vector (only the nadir direction) available for the MEKF ('night'). This behavior is not surprising, since we are lacking the minimum of two linearly independent vectors necessary to gain information about the absolute attitude of the satellite, and since the bias instability of MEMS gyroscopes is relatively high. The right panels of Fig. 3 show that the difference between the real and the recovered attitude elements may reach 25 • during the 'night' phase in our standard case.
Even though the accuracy of recovering the independent attitude parameters breaks down during 'night', the errors are correlated even in this case due to the information gained from observing the horizon. This is shown in the left panel of Fig. 4, where we plot the y and z components of the quaternion error states (δq y and δq z ). As the information about the horizon determines the orientation of the satellite with respect to the orbital plane, the error of Fig. 4 The y and z components of the quaternion error states, as well as the component corresponding to the rotation in the satellite's orbital plane (left), and the errors of the attitude elements around a 'night' to 'day' transition (right). the quaternion component that describes the rotation within this plane (δq O ) does not increase during the 'night' periods. δq O can be produced as a linear combination of δq y and δq z in our example. The right panel of Fig. 4 shows a short time period around a 'night' to 'day' transition and how the attitude information is immediately recovered once the Sun is visible again.
We also investigated the statistical behavior of the measurement errors. To do so we initialized our simulation with the same parameters except for the direction of angular momentum vector, which we picked randomly. By starting the simulation from several different initial conditions we collected statistical data about the first 'day' and 'night' phases. Figure 5 shows the distribution of the right ascension's measurement error for the 'day' case with different gyroscope precisions (the other two attitude parameters have similar error distributions). The results show that the recov- ery has a 1σ error of ∼ 22 in our standard case, while this error is ∼ 18 and ∼ 32 in the low-and high-error cases, respectively. This is an improvement on the ∼ 40 error of the MLX infrasensor's point source detection accuracy (Kapás et al., 2021), which shows the power of the MEKF method.
In Figure 6 we show the same errors for different parts of the 'night' phase. We divided the 'night' period to four equal-length segments to investigate the evolution of the errors and to avoid creating statistics from time periods with qualitatively different behavior. We see that the distribution of errors gets smeared as time progresses, and also with larger gyroscope errors. In the last segment of the high-error case the distribution is completely smeared so that not much information is retained about the real attitude. This is in accordance with the results of Baroni (2018).
Summary
In the present paper we described a simulation model for testing our new concept aimed at determining the attitude of nano-satellites. The attitude was represented by unit quaternions and a MEKF approach was applied to estimate the most probable state (attitude) of the system. In our model the prediction step of the Kalman filter utilizes gyroscope measurements while its measurement step is based on infrasensor measurements and GPS location information which provide the direction of the Sun and the nadir in the satellite and in the J2000 reference frames, respectively.
The results of our simulations show that an attitude accuracy of 22 is achievable using combined measurements of the infrasensors and MEMS gy-roscopes having a conservative drift. This is an improvement on the accuracy of point source detection with the MLX infrasensors (∼ 40 , see Kapás et al., 2021). This accuracy is gradually lost when the Sun is occulted by the Earth whereupon it can reach values of ∼ 15 − 25 • . The attitude information, however, is recovered within a short time once the Sun is observed again.
During the actual mission the satellites will not have infrasensors on all of their six sides and hence not being able to observe the Sun will be more regular. However, these time periods will be relatively short and the gyroscope drift is expected to be manageable during these intervals.
In this work we simulated a satellite on LEO with an inclination of 60 • . This is a reasonable choice for a particle detector experiment like a GRB detector because on this orbit the satellite evades high latitudes with increased noise contamination from the polar regions but is able to cover a large area of the sky. However, on such an orbit the illumination conditions may change substantially on the timescales of a few months due to the motion of the Earth around the Sun, as well as due to the orbital precession caused by J 2 . However, we consider our simulations to represent the average conditions on such an orbit sufficiently well.
The attitude determination method described in this paper is planned to be used in the CAMELOT mission where the attitude data will also serve as additional information for localizing gamma-ray bursts besides triangulation. An in-orbit demonstration of our experiment is planned to be scheduled for the end of 2022.
β − k+1 =β + k ,(17)
whereω + k = ωm −β + k and
Θ(ω + k ) = cos( 1 2 ||ω + k ||∆t)I 3×3 − S(ψ + k )ψ + k − ψ + k T cos( 1 2 ||ω + k ||∆t) ,(18)ψ + k = sin( 1 2 ||ω + k ||∆t)ω + k ||ω + k || .(19)
The covariance matrices are propagated using
P − k+1 = Φ k P + k Φ T k + G k Q k G T k ,(20)
with Φ being the state transition matrix:
Φ k = Φ 11 Φ 12 Φ 21 Φ 22 ,(21)
where
Φ 11 = I 3×3 − S(ω + k ) sin(||ω + k ||∆t) ||ω + k || + S(ω + k ) 2 [1 − cos(||ω + k ||∆t)] ||ω + k || 2 , Φ 12 = S(ω + k ) [1 − cos(||ω + k ||∆t)] ||ω + k || 2 − I 3×3 ∆t − S(ω + k ) 2 [||ω + k ||∆t − sin(||ω + k ||∆t)] ||ω + k || 3 , Φ 21 = 0 3×3 , Φ 22 = I 3×3 .(22)
The Q and G matrices determining the process noise matrix are given by
Q k = σ 2 ω ∆t + 1/3 σ 2 β ∆t 3 I 3×3 − 1/2 σ 2 β ∆t 2 I 3×3 − 1/2 σ 2 β ∆t 2 I 3×3 σ 2 β ∆t I 3×3 ,(23)G k = −I 3×3 0 3×3 0 3×3 I 3×3 .(24)
A.2 Measurement update
In the measurement update step the MEKF first estimates the quaternion error state δq using the sensitivity matrix determined by Eq. (14) and then updates the attitude utilizing Eq. (7). Supposing there are n vectors measured by the satellite, the quaternion error state and the bias vector error can be obtained using the following formula:
δq 3 k δβ k = K k ,(25)
where r s i denotes a vector measured by the satellite, whiler s i = A(q − )r i i is the predicted value of that vector. K k is the Kalman gain defined the usual way:
K k = P − k H T k H k P − k H T k + R k −1 ,(26)
with H k being the sensitivity matrix:
H k = 2S(r s 1,k ) 0 3×3 2S(r s 2,k ) 0 3×3 . . . . . . 2S(r s n,k ) 0 3×3 ,(27)
and R k the measurement covariance matrix:
R k = diag[σ 2 r 1 I 3×3 , σ 2 r 2 I 3×3 , . . . , σ 2 rn I 3×3 ].
The quaternion state, the bias vector and the covariance matrix are then updated bŷ
q + k = δq k ⊗q − k ,(29)β + k = β − k + δβ k ,(30)P + k = (I − K k H k ) P − k ,(31)
where the quaternion error state is obtained from its imaginary part using the normalization constraint:
δq = δq 3 k T , 1 − δq 3 k T · δq 3 k T .(32)
In our setup the number of measured vectors is n = 2 when the Sun and the horizon is visible at the same time, while it is n = 1 when the Sun is occulted by the Earth.
Fig. 5
5Probability distributions of the right ascension's measurement error, i.e. the difference between the real and the recovered values, during 'day'. The different panels show the cases with low (left), standard (middle), and the high (right) gyroscope error. The red curves represent Gaussian fits.
Fig. 6
6Probability distributions of the right ascension's measurement error during 'night'. The left, middle and right columns correspond to the cases with low, standard and high gyroscope error, respectively, while the different rows represent different equal-length time segments with the top row being the first and the bottom row the last quarter of the 'night' phase.
see https://youtu.be/1n-HMSCDYtM
Acknowledgements The authors would like to thank the support of the Hungarian Academy of Sciences via the grant KEP-7/2018, providing the financial background of our experiments. This research has been supported by the European Union, co-financed by the European Social Fund (Research and development activities at the Eötvös Loránd University's Campus in Szombathely, EFOP-3.6.1-16-2016-00023). We also thank the support of the GINOP-2.3.2-15-2016-00033 project which is funded by the Hungarian National Research, Development and Innovation Fund together with the European Union.Conflict of interestThe authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.Appendix A Time and measurement update with the Kalman filterHere we describe the discretized time update and measurement update steps of the Kalman filter. From now on the superscript '-' denotes propagated states before the measurement update, while the superscript '+' denotes states after the measurement update.A.1 Time updateThe estimated values of the attitude quaternion and the bias vector after a ∆t time-step can be calculated using the following equations(Crassidis and Junkins, 2012):
Kalman filter for attitude determination of a CubeSat using low-cost sensors. L Baroni, Comp. Appl. Math. 37Baroni, L.: Kalman filter for attitude determination of a CubeSat using low-cost sensors. Comp. Appl. Math. 37, 72-83. (2018)
The quaternions with an application to rigid body dynamics. E A Coutsias, L Romero, Sandia TechnicalReportCoutsias, E. A. and Romero, L., The quaternions with an application to rigid body dynamics, Sandia TechnicalReport, SAND2004-0153 (2004)
Optimal Estimation of Dynamic Systems. J L Crassidis, J L Junkins, Chapman & Hall/CRCBoca RatonCrassidis, J.L., Junkins, J.L.: Optimal Estimation of Dynamic Systems. Chapman & Hall/CRC, Boca Raton (2012)
. G Dálya, J Takátsy, T Bozóki, K Kapás, L Mészáros, A Pál, Spie, 10.1117/12.2562114114514Dálya G., Takátsy J., Bozóki T., Kapás K., Mészáros L., Pál A., SPIE, 11451, 114514K. doi:10.1117/12.2562114 (2020)
A Hardware Implementation of Flexible Attitude Determination and Control System for Two-Axis-Stabilized CubeSat. K Gaber, M B El Mashade, G A Aziz, Journal of Electrical Engineering & Technology. 15Gaber, K., El Mashade, M.B., Abdel Aziz, G.A.: A Hardware Implementation of Flexible Attitude Determination and Control System for Two-Axis-Stabilized CubeSat. Journal of Electrical Engineering & Technology. 15, 869-882 (2020)
A New Approach to Linear Filtering and Prediction Problems. R E Kalman, Journal of Basic Engineering. 82Kalman, R. E.: A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering. 82. 35-45. (1960)
Attitude determination for nano-satellites -I. Spherical projections for large field of view infrasensors. K Kapás, T Bozóki, G Dálya, 10.1007/s10686-021-09730-yExp Astron. 51Kapás, K., Bozóki, T., Dálya, G. et al. Attitude determination for nano-satellites -I. Spherical projections for large field of view infrasensors. Exp Astron 51, 515-527 doi: 10.1007/s10686-021-09730-y (2021)
Kalman Filtering for Spacecraft Attitude Estimation. Y Kozai, E Lefferts, L Markley, M Shuster, 5.10.2514/3.56190Journal of Guidance, Control, and Dynamics. 64367Astronomical JournalKozai, Y.: The motion of a close earth satellite. Astronomical Journal, 64, 367. (1959) Lefferts, E., Markley, L., Shuster, M.: Kalman Filtering for Spacecraft Attitude Estimation. Journal of Guidance, Control, and Dynamics. 5. 10.2514/3.56190. (1982)
Melexis MLX90640 32x24 IR array datasheet. Melexis MLX90640 32x24 IR array datasheet (https://www.melexis.com/-/media/files/documents/datasheets/ mlx90640-datasheet-melexis.pdf)
Attitude Determination of Nano Satellite Based on Gyroscope. S Ni, C Zhang, Sun Sensor and Magnetometer. Procedia Engineering. 15Ni, S., Zhang, C.: Attitude Determination of Nano Satellite Based on Gyroscope, Sun Sensor and Magnetometer. Procedia Engineering. 15. 959-963. (2011)
. A Pál, M Ohno, L Mészáros, N Werner, J Ripa, M Frajt, 10.1117/12.2561351SPIE. 11444Pál, A., Ohno, M., Mészáros, L., Werner, N., Ripa, J., Frajt, M. et al. SPIE 11444, 114444V doi: 10.1117/12.2561351 (2020)
Satellite relative motion propagation and control in the presence of J2 perturbations. P Sengupta, Texas A&M UniversityPhD dissertationSengupta, P.: Satellite relative motion propagation and control in the presence of J2 per- turbations. PhD dissertation. Texas A&M University. 2004.
The attitude determination system of the RAX satellite. J Springmann, A Sloboda, A Klesh, M Bennett, J Cutler, Acta Astronautica. 75Springmann, J., Sloboda, A., Klesh, A., Bennett, M., Cutler, J.: The attitude determination system of the RAX satellite. Acta Astronautica. 75. 120-135. (2012)
. N Werner, J Řípa, A Pál, M Ohno, N Tarcai, K Torigoe, K Tanaka, 10699. 106992P. Werner N.,Řípa J., Pál A., Ohno M., Tarcai N., Torigoe K., Tanaka K., et al. SPIE. 10699. 106992P. (2018)
| [] |
[
"Dual Prediction-Correction Methods for Linearly Constrained Time-Varying Convex Programs",
"Dual Prediction-Correction Methods for Linearly Constrained Time-Varying Convex Programs"
] | [
"Andrea Simonetto "
] | [] | [] | Devising efficient algorithms to solve continuouslyvarying strongly convex optimization programs is key in many applications, from control systems to signal processing and machine learning. In this context, solving means to find and track the optimizer trajectory of the continuously-varying convex optimization program. Recently, a novel prediction-correction methodology has been put forward to set up iterative algorithms that sample the continuously-varying optimization program at discrete time steps and perform a limited amount of computations to correct their approximate optimizer with the new sampled problem and predict how the optimizer will change at the next time step. Prediction-correction algorithms have been shown to outperform more classical strategies, i.e., correction-only methods. Typically, prediction-correction methods have asymptotical tracking errors of the order of h 2 , where h is the sampling period, whereas classical strategies have order of h. Up to now, Prediction-correction algorithms have been developed in the primal space, both for unconstrained and simply constrained convex programs. In this paper, we show how to tackle linearly constrained continuously-varying problem by predictioncorrection in the dual space and we prove similar asymptotical error bounds as their primal versions. | 10.1109/tac.2018.2877682 | [
"https://arxiv.org/pdf/1709.05850v2.pdf"
] | 55,410,289 | 1709.05850 | 78ffab908b9d91fe79658acf01e61f4660de54bc |
Dual Prediction-Correction Methods for Linearly Constrained Time-Varying Convex Programs
Andrea Simonetto
Dual Prediction-Correction Methods for Linearly Constrained Time-Varying Convex Programs
1Index Terms-Time-varying convex optimizationprediction- correction methodsparametric programmingdual ascent
Devising efficient algorithms to solve continuouslyvarying strongly convex optimization programs is key in many applications, from control systems to signal processing and machine learning. In this context, solving means to find and track the optimizer trajectory of the continuously-varying convex optimization program. Recently, a novel prediction-correction methodology has been put forward to set up iterative algorithms that sample the continuously-varying optimization program at discrete time steps and perform a limited amount of computations to correct their approximate optimizer with the new sampled problem and predict how the optimizer will change at the next time step. Prediction-correction algorithms have been shown to outperform more classical strategies, i.e., correction-only methods. Typically, prediction-correction methods have asymptotical tracking errors of the order of h 2 , where h is the sampling period, whereas classical strategies have order of h. Up to now, Prediction-correction algorithms have been developed in the primal space, both for unconstrained and simply constrained convex programs. In this paper, we show how to tackle linearly constrained continuously-varying problem by predictioncorrection in the dual space and we prove similar asymptotical error bounds as their primal versions.
I. INTRODUCTION
Continuously varying optimization programs have appeared as a natural extension of time-invariant ones when the cost function, the constraints, or both, depend on a time parameter and change continuously in time. This setting captures relevant control, signal processing, and machine learning problems (see e.g., [1] for a broad overview).
We focus here on linearly constrained time-varying convex programs of the form x˚ptq :" argmin xPR n f px; tq, subject to:
Ax " b,(1)
where t P R`is non-negative, continuous, and it is used to index time; f : R nˆR`Ñ R is a smooth strongly convex function uniformly in time; A P R pˆn and b P R m are a real-valued matrix and vector that represent the linear equality constraints. The goal is to find (and track) the solution x˚ptq of (1) for each time t -hereafter referred to as the optimal solution trajectory. Problem (1) might be solved in a centralized setting based on a continuous time platform [2]; however, here we focus on a discrete time setting. The reason for this choice is motivated A preliminary version of this work with limited results and no proofs has been submitted to the American Control Conference 2018 as [1]. Andrea Simonetto is with the Optimization and Control Group of IBM Research Ireland, Dublin, Ireland. Email: [email protected] . by the widespread use of digital computing units, such as control units (actuators) and digital sensors. In this context, we envision that our optimization problem will change in response to measurements taken at discrete time steps and its solution could provide control actions to be implemented on digital control units, similarly to [3]. In addition, we also envision our methods to be implemented on networks of communicating and computing nodes. In this latter scenario, at each time step the nodes have to send messages among each other before and during the computations. Then, continuous-time settings may be less appropriate, especially when high communication latencies may be expected.
Therefore, we use sampling arguments to reinterpret (1) as a sequence of time-invariant problems. In particular, upon sampling the objective functions f px; tq at time instants t k , k " 0, 1, 2, . . . , where the sampling period h :" t k´tk´1 can be chosen arbitrarily small, one can solve the sequence of time-invariant problems x˚pt k q :" argmin xPR n f px; t k q, subject to: Ax " b.
(
By decreasing h, an arbitrary accuracy may be achieved when approximating problem (1) with (2). However, solving (2) for each sampling time t k may not be computationally affordable in many application domains, even for moderatesize problems. Focusing on unconstrained or simply constrained optimization problems, a series of works among which [4], [5] developed prediction-correction methods to find and track the solution trajectory x˚ptq up to a bounded asymptotical error, in the primal space. This methodology arises from non-stationary optimization [6], parametric programming [3], [7]- [9], and continuation methods in numerical mathematics [10].
This paper extends the current state-of-the-art methods [2], [3], [5] by offering the following contributions.
First, we develop prediction-correction methods to track the solutions of the time-varying linearly constrained problems (1) by leveraging a dual ascent technique. To the author's knowledge, this is the first work that proposes predictioncorrection methods in the dual space. In [11], [12], the authors have developed dual ascent methods for similar problems, but they are correction-only methods and -as we prove herethey have worse tracking capabilities than dual predictioncorrection methods.
Second, our algorithm can handle a rank deficient matrix A, which is a situation ubiquitous in distributed optimization problems. This therefore opens the way to distributed algorithms based on dual decomposition, which have many applications. This was not considered in previous efforts, i.e., in the continuous-time platform of [2] 1 .
In this paper, we derive methods that are proved to track the solution trajectory x˚ptq up to an asymptotical error upper bound, which depends on the sampling period, on the properties of the cost function, and on the number of prediction and correction steps we use, and on the spectral properties of A. With the aid of numerical simulations, we are able to showcase further the performance of the proposed methods and their comparison with the correction-only strategies. In particular, the proposed algorithms outperform the correction-only ones in asymptotic error bounds and they appear also better when computational considerations are taken into account, in most cases.
Organization. In Section II, we introduce the basic assumptions for the linear system Ax " b. Section III covers the required background on time-invariant and correctiononly methods in the dual domain. We present our algorithm in Section IV, while convergence analysis is discussed in Section V. The main result of this paper is presented in Theorem 4. Section VI studies distributed optimization problems. Numerical simulations are presented in Section VII, and we conclude in Section VIII.
Notation. Vectors are written as x P R n and matrices as A P R pˆn . We use }¨} to denote the Euclidean norm in the vector space, and the respective induced norms for matrices and tensors. The image (i.e., the column space) and the nullspace of matrix A are indicated as impAq and nullpAq respectively. The gradient of the function f px; tq with respect to x at the point px, tq is denoted as ∇ x f px; tq P R n , the partial derivative of the same function with respect to (w.r.t.) t at px, tq is written as ∇ t f px; tq P R. Similarly, the notation ∇ xx f px; tq P R nˆn denotes the Hessian of f px; tq w.r.t. x at px, tq, whereas ∇ tx f px; tq P R n denotes the partial derivative of the gradient of f px; tq w.r.t. the time t at px, tq, i.e. the mixed first-order partial derivative vector of the objective. The tensor ∇ xxx f px; tq P R nˆnˆn indicates the third derivative of f px; tq w.r.t. x at px, tq, the matrix ∇ xtx f px; tq " ∇ txx f px; tq P R nˆn indicates the time derivative of the Hessian of f px; tq w.r.t. the time t at px, tq, the vector ∇ ttx f px; tq P R n indicates the second derivative in time of the gradient of f px; tq w.r.t. the time t at px, tq.
II. ASSUMPTIONS FOR Ax " b
We assume that b P impAq, so that the optimization problems (1) (or equivalently (2)) has a solution for each time t. We do not assume that the matrix A P R pˆn is full row rank, so it can be rank deficient.
The singular values of A are ordered as σ max :" σ p ě σ p´1 ě σ min ą σ j "¨¨¨" σ 1 " 0, where σ min is the minimum positive singular value. We call κ A " σ max {σ min .
Since b P impAq, one could eliminate the redundant rows in A (if rank deficient) and construct a full row rank matrix. However, in some cases it is desirable to keep a rank deficient A, since it encodes more linear constraints. This is the 1 The work in [2] differs from the work here, not only because they work in continuous time and they do not consider a rank deficient A. It differs also from the algorithmic perspective: they propose a continuous-time primal-dual algorithm, while here we focus on discrete-time dual ones. case, e.g., in distributed optimization where A describes the communication links that are present. The more links means (in general) faster convergence to the desired solution.
III. TIME-INVARIANT AND CORRECTION-ONLY DUAL
ASCENT
We start by two (known) properties of the primal and dual variables at optimality, for the time-invariant problem (2).
Proposition 1 Let function f p¨; t k q : R n Ñ R be strongly convex with constant m and strongly smooth with constant L. Then the primal optimizer of (2), i.e., x˚pt k q, is unique.
If A is full row rank, the dual optimizer of (2), i.e., λ˚pt k q, is also unique and λ˚pt k q P impAq.
If A is rank deficient, there exists a unique dual optimizer of (2) λ˚pt k q for which λ˚pt k q P impAq.
Proof: Given in Appendix A. Proposition 1 sets the frame for the results of this paper. If A is full row rank, the primal-dual optimizers are unique and the dual optimizer lies in the image of A. If A is rank deficient, the primal optimizer is unique, while the dual is not unique but we will be interested in finding the unique dual optimizer that lies in the image of A. By restricting the search space to the image of A, we will be able to overcome the rank deficiency of A in the proofs, without losing optimality.
Consider now the following iterative algorithm to solve (2), known as dual ascent.
1) Pick px 0 , λ 0 q; Set i " 0; Pick a stepsize α ą 0.
2) Iterate:
x i`1 " argmin xPR n tf px; t k q`λ T i Axu,(3a)λ i`1 " λ i`α pAx i`1´b q.(3b)
We have the following result.
Theorem 1 (Time-invariant dual ascent convergence) Fix the time t k . Let function f p¨; t k q : R n Ñ R be strongly convex with constant m and strongly smooth with constant L. Select x 0 arbitrarily, but λ 0 P impAq. Let the stepsize α be chosen as α ă 2m{σ 2 max . Then, the sequence tpx i , λ i qu iPN generated by recursively applying (3) converges to the unique primal-dual optimizer of (2) px˚pt k q, λ˚pt k q P impAqq. In particular, tλ i u iPN converges Q-linearly to λ˚pt k q P impAq as }λ i`1´λ˚p t k q} ď }λ i´λ˚p t k q} ď i`1 }λ 0´λ˚p t k q}, (4) while tx i u iPN converges R-linearly as
}x i`1´x˚p t k q} ď σ max m }λ i´λ˚p t k q},(5)
where the contraction factor ă 1 is defined as " maxt|1ά σ 2 max {m|, |1´ασ 2 min {L|u.
Proof: Given in Appendix B. Theorem 1 says that the time-invariant iteration (3) converges to the primal-dual optimizer of the time-invariant optimization problem (2). Furthermore, the rate is linear.
In [11], [12], the authors extend the previous results to a running version (or with the nomenclature here, a correctiononly version) of dual ascent. By running we mean an algorithm that adjust the problem on-line while the algorithm is running. In this context, consider the time-varying problem (1) and the running version of the iterations (3) defined by sampling problem (1) at discrete sampling times, as follows:
1) Pick px 0 , λ 0 q; Set k " 0; Pick a stepsize α ą 0.
2) Iterate:
x k`1 " argmin xPR n tf px; t k q`λ T k Axu,(6a)λ k`1 " λ k`α pAx k`1´b q.(6b)
As one can see, this running version of (3) considers functions that change at the same time as the updates are computed (i.e., there is only one time variable k).
Let the following assumptions hold. Assumption 2 Let the distance between optimizers of Problem (1) at two subsequent sampling time t k and t k´1 , i.e., px˚pt k q, λ˚pt k q P impAqq and px˚pt k´1 q, λ˚pt k´1 q P impAqq, be upper bounded for each k ą 0 as,
maxt}x˚pt k q´x˚pt k´1 q}, }λpt k q˚´λ˚pt k´1 q}u ď K. (7)
Then the following result is in place.
Theorem 2 (Running dual ascent convergence) Under Assumptions 1-2, consider the running iterations (6). Select x 0 arbitrarily, but λ 0 P impAq. Let the stepsize α be chosen as α ă 2m{σ 2 max . Then, the sequence tpx k , λ k qu kPN generated by recursively applying (6) converges to the unique primaldual trajectory of (1), px˚pt k q, λ˚pt k q P impAqq, up to a constant error bound linearly as }λ k`1´λ˚p t k`1 q} ď p}λ k´λ˚p t k q}`Kq
ď k`1 }λ 0´λ˚p t 0 q}` K 1´ ,(8)}x k`1´x˚p t k`1 q} ď σ max m p}λ k´λ˚p t k q}`Kq,(9)
where the contraction factor ă 1 is defined as " maxt|1ά σ 2 max {m|, |1´ασ 2 min {L|u.
Proof: The proof is given for example in [12], and it is based on the results of Theorem 1 and the triangle inequality. In particular, for each time t k`1 one can write }λ k`1´λ˚p t k`1 q} ď p}λ k´λ˚p t k`1 q}q ď ď p}λ k´λ˚p t k q}`Kq, (10) which is yield by directly applying the time-invariant results and the triangle inequality, and by leveraging Assumption 2 on the variability of the optimizers.
Theorem 2 is a generalization of Theorem 1 for cases in which the cost function changes continuously in time. The convergence result is similar to those of Theorem 1 but is achieved up to a constant error bound, which is due to the drifting of the primal-dual optimal pair. In the limit, one obtains the bounds,
lim sup kÑ8 }λ k´λ˚p t k q} " K 1´ , (11a) lim sup kÑ8 }x k´x˚p t k q} " σ max mˆ K 1´ `K˙,(11b)
and when K " 0, i.e., we are back in the time-invariant scenario, one re-obtains exact convergence.
Remark 1
The result presented in Theorem 2 can be readily related back to those in [11]. There, Assumption 2 is substituted with an assumption on the primal optimizers variation and their gradients. Given the optimal conditions for (1), one can always translate the latter in the former as
}∇ x f px˚pt k q; t k q´∇ x f px˚pt k´1 q; t k´1 q} ď ď σ max }λpt k q˚´λ˚pt k´1 q} ď σ max K. (12)
IV. PREDICTION-CORRECTION METHODOLOGY
The running dual ascent (6) is agnostic of variations of the cost function, in the sense that it only reacts to variations of the cost but it does not attempt at predicting how the function changes depending either on past data, or on the knowledge of the time derivatives of the function. Recently, e.g., [4], [5], a prediction-correction methodology in discrete time has been put forward to increase the accuracy of running (i.e., correction-only) methods by predicting how the cost function changes in time. The aforementioned works stay in the primal space, while here we will extend them to the dual space.
A. Prediction step
To develop the prediction step, we consider the optimality conditions for (6) at time t k`1 ,
∇ x f px˚pt k`1 q; t k`1 q`A T λ˚pt k`1 q " 0, Ax˚pt k`1 q " b.
(13) At time t k , one cannot solve (13) to determine the next primaldual pair. What one can do is to approximate (13) with the knowledge one has at t k via a backward Taylor expansion as,
∇ x f px˚pt k`1 q; t k`1 q`A T λ˚pt k`1 q « ∇ x f px˚pt k q; t k q∇ xx f px˚pt k q; t k qδx`∇ tx f px˚pt k q; t k qhÀ T pλ˚pt k q`δλq " 0 (14a) Ax˚pt k`1 q " Apx˚pt k q`δxq " b.(14b)
If then, one is provided with the primal-dual optimizers at time t k , one can approximate (or predict) the next primal-dual optimal pair by solving (14) for δx and δλ. That is, one has to solve the following quadratic program min δxPR n
1 2 δx T ∇ xx f px˚pt k q; t k qδx`h ∇ tx f px˚pt k q; t k q T δx,(15a)subject to Aδx " 0. (15b)
We use this reasoning to develop our prediction step. Let px k , λ k q be an approximate primal-dual optimal pair available at time t k . In the prediction step we solve the quadratic problem:
min δxPR n 1 2 δx T ∇ xx f px k ; t k qδx`h ∇ tx f px k ; t k q T δx, (16a) subject to Aδx " 0,(16b)
and we set the predicted pair to
x k`1|k " x k`δ x, λ k`1|k " λ k`δ λ.
To solve (16), various techniques can be applied. If one has access to the full instance, one can find the unique px k , λ k P impAqq by solving (16) at optimality 2 .
Since we would like to devise algorithms that can be implemented in a distributed way, we follow another approach, which is to set up a dual gradient method with the iterations: 1) Pick pδx 0 , δλ 0 " 0q; Set p " 0; Pick a stepsize β ą 0 and a maximum number of iterations P . 2) Iterate till p " P´1:
δx p`1 " argmin δxPR n ! 1 2 δx T ∇ xx f px k ; t k qδxh ∇ tx f px k ; t k q T δx`δλ T p Aδx ) , (17a) δλ p`1 " δλ p`β Aδx p`1 . (17b) 3) Set p x k`1|k " x k`δ x P , p λ k`1|k " λ k`δ λ P This converges to the exact x k`1|k and λ k`1|k as P Ñ 8, due to Theorem 1.
This last option (which determines the solution of (16) only approximately if P stays finite -and that is why we indicate the predicted variable with an hat) is to be preferred in distributed settings (as we will see in Section VI). Of course, to make this last option viable, the maximum number of iterations P needs to be small enough, which will induce an extra error in computing the prediction step.
For the sake of uniformity, from now on, we will indicate with p
x k`1|k and p λ k`1|k both the exact and approximate prediction: in fact, the exact prediction couple is equivalent to the approximate one when P Ñ 8.
B. Correction step
At time t k`1 , when one is allowed to sample the new cost function f p¨; t k`1 q, then a correction step can be performed starting from the (approximate or exact) predicted pair previously computed. The correction step is nothing else than one (or possible multiple) round(s) of the dual ascent iteration as 1) Pick pv 0 " p x k`1|k , ξ 0 " p λ k`1|k q; Set c " 0; Pick a stepsize α ą 0 and a maximum number of iterations C.
2) Iterate till c " C´1: v c`1 " argmin vPR n tf pv; t k`1 q`ξ T c Avu,(18a)ξ c`1 " ξ c`α pAv c`1´b q. (18b) 3) Set x k`1 " v C , λ k`1 " ξ C .
Algorithm 1 Approx. Dual Prediction-Correction (ADuPC)
Require: Initial guess px 0 , λ 0 P impAqq; stepsizes α, β ą 0; number of prediction and correction steps P, C 1: for k " 0, 1, 2, . . . do 2:
// time t k 3:
Prediction step: Compute δx and δλ by approximately solving the quadratic program (16) by using the iterations (17) with pδx 0 " 0, δλ 0 " 0q, stepsize β, and number of iterations P 4:
Set p x k`1|k " x k`δ x, p λ k`1|k " λ k`δ λ 5: // time t k`1 6:
Acquire the updated function f p¨; t k`1 q 7:
Correction step: Compute x k`1 and λ k`1 by using the iterations (18) with pv 0 " p x k`1|k , ξ 0 " p λ k`1|k q, stepsize α, and number of iterations C 8: end for
C. Complete algorithms
In Algorithm 1, we summarize the prediction-correction methodology for the approximate prediction. The algorithm is parametrized over the number of prediction and correction steps that it employs.
In the next section, we study the convergence of Algorithm 1 to a ball around the optimal primal-dual trajectory. The size of the error ball will depends on the sampling period and the number of prediction and correction steps, among other parameters.
V. CONVERGENCE ANALYSIS
To derive our convergence results, we need the following additional assumptions.
Assumption 3
The time derivative of the gradient of the cost function is uniformly upper bounded for all x P R n as }∇ tx f px; tq} ď C 0 , @x P R n , t.
Assumption 4
The cost function has bounded third order derivatives with respect to x and t as
}∇ xxx f px; tq} ď C 1 , }∇ xtx f px; tq} ď C 2 , }∇ ttx f px; tq} ď C 3 , @x P R n , t.
Assumptions 3-4 are common in the time-varying optimization domain when dealing with prediction-correction methods, see [2], [5], [8].
Central to our analysis is the following novel implicit function theorem.
Theorem 3 (Implicit function theorem for Problem (1)) Consider the time-varying problem (1). Let Assumptions 1 and 3 hold. The primal-dual optimal trajectory tx˚pt k q, λ˚pt k q P impAqu is locally Lipschitz in time (i.e., for small enough sampling periods), and in particular,
}x˚pt k q´x˚pt k´1 q} ď κ f κ 2 A`1 m C 0 h " Ophq, (19a) }λ˚pt k q´λ˚pt k´1 q} ď κ f κ A σ min C 0 h " Ophq. (19b)
In addition, if the bounds C 1 , C 2 , C 3 are all identically 0, then the inequalities (19) are valid globally (i.e., the trajectory is globally Lipschitz in time, i.e., (19) are valid for all sampling periods).
Proof: Given in Appendix C. Theorem 3 characterizes how the optimal primal-dual pair changes over time due to functional changes. In particular, Theorem 3 implies that optimizers changes are Lipschitz continuous in time, for sufficiently small sampling periods. As we see, Theorem 3 does not need Assumption 2, which is substituted by the stronger Assumption 3. In particular, one can see that Assumption 2 is automatically enforced, as follows.
Corollary 1 Let Assumption 1 and 3 hold. Then Assumption 2 is automatically satisfied with
K " max " κ f κ 2 A`1 m , κ f κ A σ min * C 0 h.(20)
In addition, the asymptotical error for the running dual ascent (6) is Ophq.
Corollary says that the error bound of the running version of dual ascent is proportional to the sampling period h whenever Assumptions 1 and 3 hold.
We are now ready to prove convergence of the approximate dual prediction-correction algorithm.
Theorem 4 (Convergence of Algorithm 1) Consider the time-varying problem (1). Let Assumptions 1 and 3 hold. Consider P prediction steps and C correction steps, while let the stepsizes for prediction β and correction α be chosen such that β ă 2m{σ 2 max , α ă 2m{σ 2 max . Define the contraction factors for prediction and correction as,
P :" maxt|1´βσ 2 max {m|, |1´βσ 2 min {L|u,(21a)C :" maxt|1´ασ 2 max {m|, |1´ασ 2 min {L|u. (21b)
Select the prediction and correction steps P, C to verify the contraction property
γ 1 :" C C p2 P P`1 q ă 1.(22)
There exists a constant γ 2 ą 0, dependent on the problem parameters, such that if one chooses the sampling period h as
h ă p1´γ 1 q{γ 2 ,(23)
(so that τ phq :" γ 1`γ2 h ă 1), then the sequence of approximate primal-dual optimizers tpx k , λ k qu kPN generated by (1) converges linearly to an error ball around the optimal trajectory. In particular, the convergence rate is τ phq, while the asymptotical error is
lim sup kÑ8 }λ k´λ˚p t k q} " Oˆ C C P P h 1´τ phq˙`Oˆ C C h 2 1´τ phq( 24a) lim sup kÑ8 }x k´x˚p t k q} " O˜ C´1 C P P h 1´τ phq¸`O˜ C´1 C h 2 1´τ phq¸. (24b)
Proof: Given in Appendix D, where the constant γ 2 is characterized as
γ 2 :" κ f κ 2 A mˆκ f κ 2 A`1 m C 1 C 0`C2˙ C´1 C p P P`1 q. (25)
And the asymptotical error is duly spelled out in terms of the problem parameters.
Corollary 2 (Convergence in case of exact prediction)
The results of Theorem 4 are valid for the case of exact prediction, by letting P Ñ 8. In particular, condition (22) is verified for any C ě 1.
Theorem 4 and Corollary 2 dictate how the sequences generated by Algorithm 1 converge to a ball around the optimal primal-dual trajectory. For small enough sampling periods and τ phq different enough than 1, such that the term 1´τ phq is practically a constant for all the considered h, then the error ball is in the order of
Op C C P P hq`Op C C h 2 q,(26)
which becomes a Oph 2 q error bound, every time P is sufficiently large, and goes to zero if the correction step is exact (C Ñ 8), that is every time that we solve the sampled timeinvariant problems at optimality. The error bound Oph 2 q, which is an improvement over a purely running scheme, for which we obtain a Ophq error bound (see Corollary 2), is induced by the newly developed prediction step and it comes at the price of more restrictive conditions on C and the sampling period h, i.e., conditions (22) and (23).
The parameters P and C need to be selected so that condition (22) is satisfied. This can be achieved by computing or estimating P and C via the knowledge (or estimates) of the problem properties (m, L, σ max , σ min ). Assuming that α and β are chosen equal, and, e.g., P " C " 0.8, then the condition can be satisfied, e.g., with P " 1, C ě 5, or P " 5, C ě 2, which is not as costly as it may seem.
As can be seen from the expression of γ 2 and the condition (23), the constraint on the sampling period h becomes tighter when γ 2 is large, that is when the matrix A is illconditioned (κ A is large), the condition number of the problem is large (κ f is large), and when the time variations are important (C 0 and C 2 are large).
VI. DISTRIBUTED OPTIMIZATION PROBLEMS
In this section, we consider specifically distributed optimization problems. We are interested in problems of the form:
min xPR n N ÿ i"1 f i px; tq,(27)
where the time-varying cost functions f i : R nˆR`Ñ R verify Assumption 1. In many settings, one would like to exploit the separable structure of such a cost function to decompose the optimization problem over a network of computing and communicating nodes (e.g., sensors, mobile robots). Let each node be associated with the cost function f i , inducing a oneto-one mapping between nodes and local cost functions. The nodes, i " 1, . . . , N can communicate via links. If two nodes i, j share a link, we say that there is an edge connecting them. This defines a undirected graph G " pV, Eq, with vertex set V " t1, . . . , N u and edge set E. The goal is now to solve (27) by allowing the nodes to communicate through their links only. In this case, an often employed procedure is to give each node a copy of the optimization variable, say y i P R n and to constrain the local variable of node i to be the same as the ones of all the nodes it can communicate with. This leads to the lifted problem
y˚ptq :" argmin y1PR n ,...,y N PR n N ÿ i"1 f i py i ; tq, subject to Ay " 0,(28)
where y " py T 1 , . . . , y T N q T P R nN is the stacked version of all the local decision variables and A P R pˆnN is a constraint matrix (i.e., the incidence matrix), whose blocks specify the fact that 3 y i " y j for all communicating couples pi, jq. When the underlying communication graph is connected, then the lifted problem (27) is equivalent to the original problem (28) in the sense that each of the local optimization variable y i at optimality is x˚.
The lifted problem (28) is an instance of (1), for which the matrix A is in general rank deficient. One could reduce A to be full rank by finding a tree in the communication graph (i.e., by eliminating any linear dependent constraint), but in general one would not like to do that, since in practice convergence rates of distributed algorithms are dictated by how many links the communication graph has. The more undirected links translates in general to faster convergence.
The fact that A is rank deficient is not a problem for the proposed prediction-correction methods. However, an interesting question is whether we can perform any of the two algorithms for prediction-correction is a distributed fashion, i.e., by allowing each node i to communicate only through its 1-hop communication links.
A. Distributed implementation
To obtain a distributed implementation, we require the additional assumption that: 3 We use the convention that y i´yj " 0, for i ď j.
Assumption 5 Communication among the nodes is synchronized; moreover, the algorithmic switching between correction and prediction is also synchronized among the nodes.
Under Assumption 5, we claim that Algorithm 1 can be implemented on a network of communicating nodes as follows.
VII. NUMERICAL EXAMPLES
In this section, we implement our algorithm for a simple numerical example in order to assess its performance in practice. Inspired by [13], we consider the following timevarying optimization problem:
min xPR N ÿ i"1 " 1 2 }x´A cospωt`ϕ i q} 2 2`l ogp1`exppx´a i qq loooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooon ":fipx;tq ,(29)
with ta i u N i"1 and tϕ i u N i"1 drawn from uniform probability distribution of support r´10, 10s and r0, 2πq, respectively, while A " 2.5, ω " π{80. The cost function is strongly convex and strongly smooth and m " 1 and L " 1.25, respectively.
From a control perspective, Problem (29) could represent a rendezvous problem of a group of robots that would like to stay close to their moving target A cospωt`ϕ i q, and to their fixed base station located in a i . Or it could represent a consensus problem, where a group of agents try to reach a compromise on their opinions on a certain matter, trading of a short-term dynamics (represented by A cospωt`ϕ i q, e.g., weekly fluctuations caused by the latest news) and a long-term one (represented by a i ), e.g., long-standing beliefs.
We focus our analysis on a network of computing and communicating nodes and we fix the total number of nodes to N " 250, while their communication graph is randomly generated. The nodes have their local cost function f i px; tq and they have to cooperate to determine the common decision variable x. By leveraging a dual decomposition approach to the described distributed optimization problem, one arrives at the problem min y1PR,...,y N PR
N ÿ i"1 f i py i ; tq, subject to Ay " 0,(30)
where y " py 1 , . . . , y N q T P R N is the stacked version of all the local decision variables and A is the constraint matrix constructed as expressed in Section VI, and in our simulations
A. Analysis correction-only vs. prediction-correction
In the first numerical assessment, we study the proposed algorithm by varying the sampling period h and for different choices of number of prediction and correction steps. In Figure 1, we report the results in terms of asymptotical tracking error, here computed as max kě5000 t}y k´y˚p t k q} 2 u,
whereas the final time of the simulations is k " 10000. We see how a correction-only methodology (i.e., the running dual ascent discussed in Section III) is performing the worst, while the prediction-correction scheme with P " 27 (practically equivalent to the an exact prediction algorithm) is performing the best. Using a large number of prediction steps requires more computational/communication effort and therefore there is a natural trade-off between the number of prediction steps one can run and the tracking error (here captured by varying P ). We note that, even a small number of prediction steps are beneficial in terms of asymptotical error. Figure 1 depicts how the tracking error depends on the sampling period h by the help of the two dashed lines, indicating a Ophq and Oph 2 q dependence as expected by our theoretical analysis (Note that τ phq varies less than 1% for the considered sampling periods). In particular, a purely running scheme would have an asymptotical error of Ophq [Cf. Corollary 1], while a prediction-correction one would have an error that approaches Oph 2 q when P is chosen bigger and bigger [Cf. Equation (26)].
We note that all the problem parameters tC i u 3 i"0 can be determined and all the conditions of Theorem 4 are verified.
B. Analysis at fixed run time
Our second assessment regards performance of the algorithms keeping the run time per sampling period fixed, which is extremely relevant in real situations.
Every time a new function is available, a number of correction steps are performed. The number depends on how fast we need the corrected variable to be available and the computational/communication time necessary to compute it. We fix at r 1 h, with r 1 ă 1 the time allocated for the correction steps, while t C is the time to perform one correction step. For the above considerations, we can afford to run
C " tr 1 h{t C u,(32)
correction steps. After the corrected variable is available, one can use it for the decision making process (which may require extra time to be performed). For the time-varying algorithm perspective, one can use the variable to either run P gradient prediction, or C 1 extra correction steps (to improve the corrected variable for having a better starting point when a new function becomes available). Fix at r 2 h, with r 2 ă 1 the time allocated for the prediction (or extra correction) steps. The affordable number of prediction steps can be determined considering that P prediction steps require a time equal tō t`P t P , wheret is the time required to evaluate once the Hessian, and time derivative of the gradient in (17a), while t P is the time to perform one prediction calculation (including communication latencies). Thus, P " tp r 2 h´t q{t P u.
The affordable extra correction steps C 1 can be computed as in (32), substituting r 1 with r 2 .
In the simulation example, we choose r 1 " r 2 " 0.5, while by running the experiments on a 2.7 GHz Intel Core i5 and by mapping the results on simple computational nodes, we empirically fix t C " 21 ms,t " 8 ms, t P " 3 ms. Note that we have include a communication latency of 1 ms in both t C and t P , which simulates the need for communication to agree on the common decision variable, as expressed in Section VI. Note that the correction step takes longer that a prediction step for at least two reasons. First, in the correction step one has to solve iteratively the optimization problem associated with the Lagrangian (18a) (here we use a Newton method), while for the prediction step such optimization problem is quadratic and unconstrained (cf. (17a)), so analytically solvable. Second, the aforementioned optimization problems depends on parameters (Hessian, gradient, time derivative of the gradient) that in the correction step changes for all c P r0, C´1s, while they are the same for the prediction step for all p P r0, P´1s and they can be computed once.
In Figure 2, we report the results in terms of asymptotical tracking error (31) for the sampling period range h P r0.04, 5.12s s. In the simulations, the prediction and correction steps are determined by using (32) and (33), so that when h " 0.08 s, then P " 10 and C " C 1 " 1, while for h " 5.12 s, P " 850 and C " C 1 " 121 (note that when h " 0.04 s, the prediction-correction algorithm does not satisfy the convergence assumption of Theorem 4).
We also consider the situation in which one can use the whole sampling period to do correction, that is r 1 " 1, while r 2 " 0, and we call this case total correction. In this case when h " 0.08 s, then the correction steps are C 2 " 3, while for h " 5.12 s, C 2 " 243. This total correction situation is particularly interesting when one has to make a choice whether to stop the correction steps to perform prediction, or to continue to do correction steps till a new function evaluation becomes available. Note that the correction+extra correction strategy is different from the total correction one, since the error is computed with the corrected variable (which is used for the decision making process), that is after r 1 h. The numerical results suggest that a prediction-correction strategy achieves a lower asymptotical error than performing both correction+extra correction and total correction up to a certain sampling period. This is reasonable to expect, since as P and C grow, the error of the prediction-correction strategy goes as Oph 2 q, while the ones of the correction only schemes go as Ophq. This can be formalized as follows: the correction+extra correction strategy has an asymptotical primal error bound of
Err C`EC " σ max m C´1 C˜ C`C 1 C K 1´ C C`Ķ Ý ÝÝÝÝÝ Ñ C,C 1 Ñ8 Op C´1 C hq;
(34) the total correction strategy has an asymptotical primal error bound of
Err TC " σ max m C 2´1 C˜ C 2 C K 1´ C 2 C`K¸Ý ÝÝÝÑ C 2 Ñ8 Op C 2´1 C hq;
(35) while the prediction-correction has an asymptotical primal error bound of
Err PC " O˜ C´1 C P P h 1´τ phq¸`O˜ C´1 C h 2 1´τ phq¸Ý ÝÝÝÝ Ñ P,CÑ8 Op C´1 C h 2 q;
(36) where (36) is due to (24), while (34) and (35) are generalizations of (9) for multiple correction steps [see Appendix F]. As we see, (34) does not depend on C 1 (the extra correction terms), which make these calculations superfluous, while C 2 ě 2C (in our case), which makes (35) ă (34). Finally, (36) is better than (35) and (34) for small h.
The simulations indicate that, when the sampling period is small, performing prediction-correction is better than the presented alternatives, even taking into account computational and communication requirements. In particular, (i) w.r.t. cor-rection+extra correction: if one has time available after the decision variable needs to be delivered and before the new cost function becomes available, doing prediction rather than extra correction appears to be the best choice; (ii) w.r.t. total correction: it may be wise to stop the correction steps (even if one has still time before delivering the decision variable) and start the prediction ones.
C. Further numerical studies
We report here further numerical studies which are qualitatively very similar to the ones just presented. In particular, we report that both (i) changing the condition number of the function κ f from 1.25 to 3.25 [ Figure 3] and (ii) changing the condition number of the incidence matrix κ A from 2.39 to 4.28 [ Figure 4], require more prediction and correction steps to achieve the same asymptotical error bounds; whereas (iii) increasing the number of nodes from N " 250 to N " 500 (while having κ A " 2.54) [ Figure 5], has very limited effect in the number of prediction and correction steps required.
VIII. CONCLUSIONS
We have developed dual prediction-correction methods to track the solution trajectory of time-varying linearly constrained convex programs. The proposed methods have a better theoretical and numerical performance with respect to more classical strategies. We have characterized the convergence properties and asymptotical tracking error of all the methods and shown how the error depends on the problem instance parameters and sampling period. Proof: Call f k :" f p¨; t k q. The primal optimizer of (2) x˚pt k q is unique since f k is strongly convex. Examine the optimality condition,
∇ x f k px˚pt k qq`A T λ˚pt k q " 0.(37)
By strong smoothness ∇ x f k px˚pt k qq is unique. In fact, if there were two distinct ∇ x f k px 1 q and ∇ x f k px 2 q, for the same x 1 " x 2 , then one could derive a contradiction by using the strong smooth inequality
}∇ x f k px 1 q´∇ x f k px 2 q} ď L}x 1´x2 }.(38)
Thus, A T λ˚pt k q is unique. By the fundamental theorem of linear algebra (or alternatively, Fredholm alternative theorem) [14], λ˚pt k q can be decomposed in two parts as λ˚pt k q " λ0 pt k q`v k , for which λ0 pt k q P impAq and v k P nullpA T q.
In the full rank case, the nullspace of A T is void and λ˚pt k q P impAq. In the rank deficient case, we only concentrate on λ0 pt k q P impAq. Uniqueness of λ0 pt k q is proven by contradiction: assume that λ0 pt k q is not unique and one has two variables λ 1 ‰ λ 2 for which A T λ 1 " A T λ 2 . Since both variables lie in the image of A (and not in the nullspace of A T ), it has to be A T λ 1 ‰ 0, A T λ 2 ‰ 0 as well as for any of their linear combinations. Therefore, it has to be A T pλ 1´λ2 q ‰ 0 for all λ 1 ‰ λ 2 , from which a contradiction arises. Therefore λ0 pt k q must be unique.
APPENDIX B PROOF OF THEOREM 1
Proof: The proof is reported here for completeness, it can be found e.g. in [12], [15]. Call f k :" f p¨; t k q. The proof relies on the properties of the conjugate function of f k , defined as f ‹ k pyq :" sup xPR n ty T x´f k pxqu, and on the properties of the differential operator Bg of a convex function g. In particular, if f k is strongly convex for all x P R n with parameter m, then f ‹ k is strongly smooth with parameter 1{m for all y P R n , while if f k is strongly smooth with parameter L for all x P R n , then f ‹ k is strongly convex with parameter 1{L for all y P R n . Furthermore, for the differential operators of f k and f ‹ k , one has Bf´1 k " Bf ‹ k . Consider now (3a), which can be written in terms of optimality condition as
Bf k px i`1 q`A T λ i " 0 ðñ x i`1 " Bf ‹ k p´A T λ i q,(39)
where we have used the identity Bf k " ∇f k , since f k is differentiable. The dual function q k pλq :" min xPR n tf k pxqλ T i pAx´bqu has gradient,
Bq k pλq " Ax i`1´b " ABf ‹ k p´A T λ i q´b.(40)
Full row rank A. Due to (40), the negative of the dual function´q k is strongly smooth with constant σ 2 max {m and strongly convex with constant σ 2 min {L. The dual ascent (3) is a dual gradient iteration on´q k and it converges for all α ă 2m{σ 2 max , with linear convergence rate " maxt|1ά L{σ 2 min |, |1´αm{σ 2 max |u. Therefore,
}λ i`1´λ˚p t k q} ď }λ i´λ˚p t k q},(41)
which is claim (4). By using the optimality condition for (3a)
,
∇ x f k px i`1 q`A T λ i " ∇ x f k px˚pt k qq`A T λ˚pt k q " 0.
(42) By algebraic manipulations and by using strong convexity,
m}x i`1´x˚p t k q} ď }∇ x f k px i`1 q´∇ x f k px˚pt k qq} " }A T pλ i´λ˚p t k qq} ď σ max }λ i´λ˚p t k q},(43)
from which claim (5).
Rank deficient A. To prove the contraction property in this case, we only need to re-work the strong convexity property of´q k , since now σ 1 " 0. To do that, we need to show that the functions´q k have a strong convex-like property for all λ P impAq and that the iterations (3) generates λ i P impAq (i.e., keeps the dual variable feasible). The second claim is easy to show since λ 0 P impAq, b P impAq and
λ i`1 " λ i`α pAx i`1´b q P impAq.(44)
To show the first claim, we recall that Bq k pλq " ABf ‹ k p´A T λq´b (as proved in the proof of full rank A). Therefore, for all λ, µ P impAq:
pBq k pλq´Bq k pµqq T pµ´λq " " pBf ‹ k p´A T λq´Bf ‹ k p´A T µqq T A T pµ´λq ě ě σ 2 min {L}λ´µ} 2 ,(45)
where the last inequality comes from the fact that λ, µ P impAq and by the fact that A T pµ´λq " 0 iff µ " λ, for the fundamental theorem of linear algebra [14]. Result (45) implies strong monotonicity of´Bq k pλq for all λ P impAq, and therefore strong convexity of´q k pλq for all λ P impAq. Then the contraction property follows from the fact that´q k is both strongly smooth with constant σ 2 max {m k as easy to show, and strongly convex (over the restricted domain). The rest follows as in the proof of the full row rank case.
APPENDIX C PROOF OF THEOREM 3
In order to prove Theorem 3, we need a general result on quadratic programs of a special form. (47)
Proof: The optimality condition for (46): Qδx˚`cÀ T δλ˚" 0 yields, δx˚"´Q´1pc`A T δλ˚q.
(48)
The dual problem of (46) reads,
min δλ 1 2 δλ T AQ´1A T δλ`c T Q´1A T δλ,(49)
whose optimality condition reads,
AQ´1A T δλ˚"´AQ´1c.(50)
If A is full row rank, then δλ˚is unique and δλ˚P impAq, otherwise there exists a unique δλ˚P impAq (see Proposition 1). We focus on the unique δλ˚P impAq. In this case A T δλ˚‰ 0 if δλ˚‰ 0 and therefore we can multiply both sides of (50) by δλ˚, T , obtaining,
δλ˚, T AQ´1A T δλ˚"´δλ˚, T AQ´1c.(51)
Bounding,
σ 2 min L }δλ˚} 2 ď }δλ˚, T AQ´1A T δλ˚} " }δλ˚, T AQ´1c} ď }δλ˚}}A}}Q´1}}c} ď }δλ˚} σ max m }c},(52)
We are now ready for the proof of Theorem 3. Proof: To determine the bounds in (19), we use a Taylor expansion. In particular, call Q " ∇ xx f px˚pt k q; t k q and c " h∇ tx f px˚pt k q; t k q. Then, x˚pt k`1 q can be computed as the solution of
∇ x f px; t k`1 q`A T λ " Qpx´x˚pt k qq`c`h.o.t.À T pλ´λ˚pt k qq " 0,(54)
where h.o.t. stands for the higher order terms of the expansion. The results provided in (19) will be valid when the higher order terms are negligible with respect to the leading terms (i.e., locally), or when C 1 " C 2 " C 3 " 0, i.e., when the higher order terms are identically zero. Problem (54) can be put in the form of (46) by neglecting the h.o.t., and in particular, its solution is }δx˚} " }x˚pt k`1 qx˚p t k q} and }δλ˚} " }λ˚pt k`1 q´λ˚pt k q}. By the bounds on Q " ∇ xx f px˚pt k q; t k q, the upper bound on }c} " }h∇ tx f px˚pt k q; t k q} ď C 0 h, and by using Proposition 2, ones derives the claims (19a) and (19b).
APPENDIX D PROOF OF THEOREM 4 A. Preliminaries and definitions
We begin the convergence analysis by deriving an upper bound on the norm of the approximation error incurred by the Taylor expansion in (14). In particular, given the optimal primal-dual solutions px˚pt k q, λ˚pt k q P impAqq and px˚pt k`1 q, λ˚pt k`1 q P impAqq at t k and t k`1 , respectively, compute the optimal prediction step via the Taylor approximation (14) and indicate the optimal prediction as pxk`1 |k " x˚pt k q`δx, λk`1 |k " λ˚pt k q`δλq. The objective is to bound the error:
e k :" maxt}xk`1 |k´x˚p t k`1 q}, }λk`1 |k´λ˚p t k`1 q}u,(55)
which is committed when the optimal couple px˚pt k`1 q, λ˚pt k`1 qq is replaced by the predicted one pxk`1 |k , λk`1 |k q.
To ease notation, we define the following problem specific quantities:
∆ 1 :" κ f κ 2 A`1 m , ∆ 2 :" κ f κ A σ min ,(56a)∆ 3 :" C 1 C 2 0 ∆ 2 1 {2`∆ 1 C 2 C 0`C3 {2, (56b) ∆ 4 :" ∆ 1 C 1 C 0`C2 . (56c)
Proposition 3 Let Assumptions 1, 3, and 4 hold. The following holds:
}xk`1 |k´x˚p t k`1 q} ď ∆ 1 ∆ 3 h 2 (57a) }λk`1 |k´λ˚p t k`1 q} ď ∆ 2 ∆ 3 h 2 ,(57b)
where ∆ 1 , ∆ 2 , and ∆ 3 are defined in (56). Thus, the error e k is upper bounded as
e k ď maxt∆ 1 , ∆ 2 u ∆ 3 h 2 " Oph 2 q.(58)
Proof: Let us start by simplifying the notation.
Define
∇ x f i " ∇ x f px˚pt k`i q; t k`i q, Q i " ∇ xx f px˚pt k`i q; t k`i q (59a) c i " ∇ tx f px˚pt k`i q; t k`i q, x i " x˚pt k`i q, x " xk`1 |k , (59b) λ i " λ˚pt k`i q, λ " λk`1 |k .
(59c)
With this notation in place, e k " maxt}x´x 1 }, }λ´λ 1 }u [Cf. (55)]. In addition, px, λq is computed by the optimal conditions [Cf. (14)]
∇ x f 0`Q0 px´x 0 q`h c 0`A T λ " 0, Ax " b. (60) while x 1 is the solution of ∇ x f 1`A T λ 1 " 0, Ax 1 " b.(61)
Consider the solution mapping:
sppq :" ! y, µ P impAqˇ∇ x f 0`Q0 py´x 0 q`h c 0`A T µ`p " 0 Ay " b )
.
The mapping sppq is every-where single-valued (due to Proposition 1), while for any two values of the parameter p, say p 1 and p 2 , then, Q 0 pypp 1 q´ypp 2 qq`A T pµpp 1 q´µpp 2 qq`p 1´p2 " 0, (63a)
Apypp 1 q´ypp 2 qq " 0. (63b)
By using Proposition 2 on (63) with c " p 1´p2 ,
}ypp 1 q´ypp 2 q} ďˆL m σ 2 max σ 2 min`1˙1 m }p 1´p2 } " " ∆ 1 }p 1´p2 }, (64a) }µpp 1 q´µpp 2 q} ď L m σ max σ 2 min }p 1´p2 } " ∆ 2 }p 1´p2 }.(64b)
Let p 1 " 0 and p 2 " ∇ x f 1´p ∇ x f 0`Q0 px 1´x0 q`h c 0 q, one obtains ypp 1 q " x, µpp 1 q " λ, and ypp 2 q " x 1 , and µpp 2 q " λ 1 , which means,
e k " maxt}x´x 1 }, }λ´λ 1 }u ď max t∆ 1 , ∆ 2 u} ∇ x f 0´∇x f 1`Q0 px 1´x0 q`h c 0 }. (65)
Consider now the right-hand-side of (65): it is nothing else but the error of the truncated Taylor expansion of ∇ x f 1 :
∇ x f 1´∇x f 0 " Q 0 px 1´x0 q`hc 0` ,(66)
where the error can be bounded as
} } ď 1 2´} ∇ xxx f }}x 1´x0 } 2`h }∇ txx f }}x 1´x0 }h }∇ xtx f }}x 1´x0 }`h 2 }∇ ttx f }¯,(67)
and by using the upper bounds in Assumption 4
,
}∇ x f 0´∇x f 1`Q0 px 1´x0 q`h c 0 } ď ď 1 2 C 1 }x 1´x0 } 2`h C 2 }x 1´x0 }`1 2 h 2 C 3 . (68)
By using the bound (19) on the variability of the optimizers x 1 and x 0 , then
}p 1´p2 } ďˆ1 2 h 2 C 1 ∆ 2 1 C 2 0`h 2 C 2 ∆ 1 C 0`1 2 h 2 C 3˙" ∆ 3 h 2 ,(69)
which, by substituting into (64), proves Proposition 3.
We then look at the optimal prediction error, i.e., the distance between the exact predicted pair px k`1|k , λ k`1|k q and the primal-dual optimizer at time step t k`1 , px˚pt k`1 q, λ˚pt k`1 q P impAqq can be bounded as the following proposition.
Proposition 4 Under the same assumptions and notation of Theorem 4, let px k`1|k , λ k`1|k q be the exact predicted step obtaining by solving (16) at optimality. Let ∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 be defined as in (56). We have that }x k`1|k´x˚p t k`1 q} ď p1`h ∆ 1 ∆ 4 q}x k´x˚p t k q}∆
1 ∆ 3 h 2 ,(70a)}λ k`1|k´λ˚p t k`1 q} ď }λ k´λ˚p t k q}h ∆ 2 ∆ 4 }x k´x˚p t k q}`∆ 2 ∆ 3 h 2 . (70b)
Proof: We proceed as in the proof of Proposition 3. We use similar simplifications of (59), as
∇ x f k " ∇ x f px k ; t k q, Q k " ∇ xx f px k ; t k q (71a) c k " ∇ tx f px k ; t k q, x " x k`1|k , λ " λ k`1|k . (71b)
while ∇ x f 1 , x 1 , and λ 1 , Q 0 , c 0 , ∇ x f 0 , x 0 , and λ 0 are defined just as in (59). The error }x k`1|k´x˚p t k`1 q} is now }x´x 1 }, while }λ k`1|k´λ˚p t k`1 q} is now }λ´λ 1 }.
The vectors x, λ are computed by x " x k`δ x and λ " λ k`δ λ, where the increments are computed via the optimality conditions of (16),
Q k δx`h c k`A T δλ " 0, Aδx " 0.(72)
In addition, define the exact prediction computed starting from px˚pt k q, λ˚pt k qq as pxk`1 |k , λk`1 |k q and the increments δx˚" xk`1 |k´x˚p t k q " xk`1 |k´x 0 and δλ˚" λk`1 |kλ˚p t k q " λk`1 |k´λ 0 , which are computed by [Cf. (60) or equivalently (15)]
Q 0 δx˚`h c 0`A T δλ˚" 0, Aδx˚" 0.(73)
The error }x´x 1 } can be upper bounded as
}x´x 1 } ď }δx`x k´p δx˚`x 0 q}`}xk`1 |k´x 1 } ď }x k´x0 }`}δx´δx˚}`∆ 1 ∆ 3 h 2 ,(74)
and similarly the error }λ´λ 1 } can be upper bounded as
}λ´λ 1 } ď }λ k´λ0 }`}δλ´δλ˚}`∆ 2 ∆ 3 h 2 . (75)
Consider the solution mapping:
sppq :" " y, µ P impAqˇˇˇˇQ k y`h c k`A T µ`p " 0 Ay " 0 * .
(76) The mapping sppq is every-where single-valued (due to Proposition 1). In addition, by looking at two different parameters p 1 and p 2 as done similarly in (63) and by using Proposition 2, then we derive that everywhere (i.e., for all p 1 , p 2 ), }ypp 1 qý pp 2 q} ď ∆ 1 }p 1´p2 } and }µpp 1 q´µpp 2 q} ď ∆ 2 }p 1´p2 }. Set p 1 " 0 and p 2 " pQ 0´Qk qδx˚`h pc 0´ck q, so that ypp 1 q " δx, µpp 1 q " δλ, and ypp 2 q " δx˚, and µpp 2 q " δλ˚. Then,
}p 1´p2 } " }pQ k´Q0 qδx˚`h pc k´c0 q}.(77)
We proceed now to bound }pQ k´Q0 qδx˚`h pc k´c0 q}, by using Assumption 4 }pQ k´Q0 qδx˚`h pc k´c0 q} ď
ď C 1 }x k´x0 }}δx˚}`hC 2 }x k´x0 }. (78)
The next step is to upper bound }δx˚}. For this purpose, we use Proposition 2 on Problem (73). In particular, by (53), one has }δx˚} ď ∆ 1 C 0 h, and therefore,
}p 1´p2 } ď hp∆ 1 C 1 C 0`C2 q}x k´x0 }.(79)
By putting together (79) with the fact that }ypp 1 q´ypp 2 q} ď ∆ 1 }p 1´p2 } and }µpp 1 q´µpp 2 q} ď ∆ 2 }p 1´p2 } and with (74)-(75), the bounds (70) are proven.
B. Main algorithm's convergence
We divide the proof in different steps.
Step 1: we bound the prediction error by using Proposition 4; Step 2: we bound the correction error; Step 3: we put the previous steps together and derive the convergence requirements and results.
Prediction error. The distance between the approximate prediction pp x k`1|k , p λ k`1|k q and the exact prediction px k`1|k , λ k`1|k q can be bounded by using Theorem 1. First, notice that, for Theorem 1 applied to iterations (6), one has }δλ P´δ λ} ď P P }δλ 0´δ λ}, (80a)
}δx P´δ x} ď σ max m P´1 P }δλ 0´δ λ},(80b)
or equivalently, by putting δλ 0 " 0,
} p λ k`1|k´λk`1|k } ď P P }λ k´λk`1|k }, (81a) }p x k`1|k´xk`1|k } ď σ max m P´1 P }λ k´λk`1|k }.(81b)
By putting together Proposition 4, (81), and (19), we obtain for the total error after prediction for the dual variable as } p λ k`1|k´λ˚p t k`1 q} ď } p λ k`1|k´λk`1|k }} λ k`1|k´λ˚p t k`1 q} ď ď P P }λ k´λk`1|k }`}λ k`1|k´λ˚p t k`1 q} ď P P p}λ k´λ˚p t k q}`}λ˚pt k q´λ˚pt k`1 q}} λ˚pt k`1 q´λ k`1|k }q`}λ k`1|k´λ˚p t k`1 q} ď P P }λ k´λ˚p t k q}`p P P`1 q}λ k`1|k´λ˚p t k`1 q}` P P }λ˚pt k q´λ˚pt k`1 q} ď α 2 }λ k´λ˚p t k q}`hα 1 }x k´x˚p t k q}`hα 0 , (82)
where we have set α 2 " 2 P P`1 , α 1 " ∆ 2 ∆ 4 p P P`1 q, and α 0 " P P p∆ 2 ∆ 3 h`∆ 2 C 0 q`∆ 2 ∆ 3 h.
Correction error. We look now at the correction step, which by using Theorem 1, one can derive }λ k`1´λ˚p t k`1 q} ď C C } p λ k`1|k´λ˚p t k`1 q}, (83a)
}x k`1´x˚p t k`1 q} ď σ max m C´1 C } p λ k`1|k´λ˚p t k`1 q}.(83b)
with C " maxt|1´αm|, |1´αL|u. And by putting together the result (82) with (83), we obtain the error bounds, }λ k`1´λ˚p t k`1 q} ď C C pα 2 }λ k´λ˚p t k q}h α 1 }x k´x˚p t k q}`hα 0 q, (84a) }x k`1´x˚p t k`1 q} ď σ max m C´1 C pα 2 }λ k´λ˚p t k q}h α 1 }x k´x˚p t k q}`hα 0 q. (84b)
Global error and convergence. Call a 1 :" C C α 2 , a 2 :" h C C α 1 , u 1 :" h C C α 0 , and γ :" σ max {pm C q. Define z λ,k :" }λ k´λ˚p t k q} and z x,k :" }x k´x˚p t k q}. Then the error dynamics (84) can be written -in the worst case -as
" z λ,k`1 z x,k`1 " " a 1 a 2 γa 1 γa 2 " z λ,k z x,k `" u 1 γu 1 .(85)
Asymptotic stability of the linear system (85) is achieved iff the eigenvalues of the state transition matrix are inside the unit circle, i.e., iff a 1`γ a 2 ă 1 i.e., C C p2 P P`1 qσ max {pm C qph C C ∆ 2 ∆ 4 p P P`1 qq " τ phq ă 1, (86) that is
h ă m σ max " 1´ C C p2 P P`1 q ‰ " C´1 C ∆ 2 ∆ 4 p P P`1 q ‰´1 ,(87)
which is condition (23) when defining
γ 1 " C C p2 P P`1 q,(88)γ 2 " σ max m " C´1 C ∆ 2 ∆ 4 p P P`1 q ‰ " C´1 C κ f κ 2 A mˆκ f κ 2 A`1 m C 1 C 0`C2˙p P P`1 q. (89)
A positive (and therefore implementable) sampling period h exists iff 1´ C C p2 P P`1 q ą 0,
which is condition (22), and in this case,
" z λ,k z x,k " " a 1 a 2 γa 1 γa 2 k " z λ,0 z x,0 `k´1 ÿ τ "0 " pa 1`γ a 2 q τ γpa 1`γ a 2 q τ u 1 ă 8. (91)
The asymptotical error is achieved exponentially fast and it is lim sup kÑ8 }λ k´λ˚p t k q} " u 1 1´pa 1`γ a 2 q " " C C r P P p∆ 2 ∆ 3 h`∆ 2 C 0 q`∆ 2 ∆ 3 hs h 1´τ phq , (92a) lim sup kÑ8 }x k´x˚p t k q} " γu 1 1´pa 1`γ a 2 q "
σ max C´1 C r P P p∆ 2 ∆ 3 h`∆ 2 C 0 q`∆ 2 ∆ 3 hs h r1´τ phqsm .
(92b)
Which concludes the proof.
APPENDIX E PROOF OF CLAIM 1
Proof: To justify the claim, we analyze all the steps of Algorithm 1. First, the prediction step is based on the iterations (6). Let y i,k and δy i,k be the local variables y i and δy i at iteration k; let δλ i,j,k be the dual variable associated with link pi, jq P E at iteration k, then (6) can be rewritten as 1) For all i P V do δy i,p`1 " argmin δyiPR n ! 1 2 δy T i ∇ yiyi f i py i,k ; t k qδy i∇ tyi f py i,k ; t k q T δy iÿ pi,jqPE,iďj δλ T i,j,p δy i´ÿ pi,jqPE,iąj
δλ T i,j,p δy i ) ,(93a)
2) Communicate δy i,p`1 with neighbors; 3) For all i P V do δλ i,j,p`1 " δλ i,j,p`β I iďj pδy i,p`1´δ y j,p`1 q,
where I iďj is 1 if i ď j, and´1 otherwise.
This justifies the fact that the prediction step can be implemented in a distributed fashion with synchronous communication (Assumption 5). Each node maintains local copies δy i,k , δλ i,j,k which converge to the primal-dual optimizers of the prediction step. Second, we analyze the correction step, which is based on the iterations (18). It is easy to see that also (18) can be written in a similar fashion as (93), thereby allowing distributed computation of the correction direction with synchronous communication (Assumption 5).
Provided now that the switching between prediction and correction step is synchronized (Assumption 5) then Algorithm 1 can be implemented in a distributed fashion.
APPENDIX F ASYMPTOTICAL ERROR BOUNDS
We prove here both (34) and (35). For (34), by similar arguments as the one of the proof of Theorem 2, we can write, }λ k`1´λ˚p t k`1 q} ď C C p} Ă λ k´λ˚p t k q}`Kq ď C C p C 1 C }λ k´λ˚p t k q}`Kq }x k`1´x˚p t k`1 q} ď σ max m C´1 C p C 1 C }λ k´λ˚p t k q}`Kq,
and therefore,
Err C`EC " lim sup kÑ8 }x k´x˚p t k q} " σ max m C´1 C˜ C`C 1 C K 1´ C C`K¸,(95)
from which (34).
As for (35), }λ k`1´λ˚p t k`1 q} ď C 2 C p}λ k´λ˚p t k q}`Kq }x k`1´x˚p t k`1 q} ď σ max m C 2´1 C p}λ k´λ˚p t k q}`Kq,
and therefore,
Err TC " lim sup kÑ8 }x k´x˚p t k q} "
σ max m C 2´1 C˜ C 2 C K 1´ C 2 C`K¸,(97)
from which (35).
Assumption 1
1Let time-varying function f : R nˆR`Ñ R be strongly convex with constant m and strongly smooth with constant L, uniformly in time (i.e., for each time t ě 0). Define the condition number of f as κ f :" L{m, uniformly in time.
Claim 1
1Consider the time-varying problem (27), the communication graph G " pV, Eq, and the matrix A induced by the communication graph. Under Assumption 5, the predictioncorrection Algorithm 1 can be implemented in a distributed fashion, by allowing communication only via the edge set E. Proof: Given in Appendix E. The total communication budget per time step per node (intended as the number of scalar variable transmitted) is pP`CqN i n, where N i the number of neighbors of node i, while P and C are the number of prediction and correction iteration respectively.
Fig. 1 :
1Asymptotical tracking error as a function of sampling period for different choices number of prediction steps P .κ A " 2.39. Problem (30) is specific version of problem (28), which we have analyzed in Section VI.
Fig. 2 :
2Asymptotical tracking error as a function of sampling period for different algorithms, keeping the run time constant.
Fig. 3 :
3Asymptotical tracking error performance for k f " 3.25 and other parameters left the same.
Fig. 4 :
4Asymptotical tracking error performance for k A " 4.28 and other parameters left the same.
Fig. 5 :
5Asymptotical tracking error performance for N " 500, κ A " 2.54 and other parameters left the same.
T Qδx`c T δx, subject to Aδx " 0,(46)with unique primal-dual optimizers pδx˚, δλ˚P impAqq. Let the eigenvalues of Q be lower bounded by m and upper bounded by L, while let the matrix A have the singular values ordered as in Section II. Then,
and finally, by dividing by the non-zero, finite }δλ˚}, one derives the claim (47) on the dual variable. For the claim (47) on the primal variable, one uses (48), which can be upper bounded as }δx˚} ď }Q´1}p}c}`}A T
This can be done e.g., by off-the-shelf solvers, or by a custom-made Newton's method which can also employ Krylov-subspace based solvers for the resulting linear system. Note that some of the computations could be made off-line since A is time-invariant.
Prediction-Correction Dual Ascent for Time-Varying Convex Programs. A Simonetto, Proceedings of the American Control Conference. the American Control Conferenceto appear)A. Simonetto, "Prediction-Correction Dual Ascent for Time-Varying Convex Programs," in Proceedings of the American Control Conference (to appear), 2018.
Prediction-Correction Interior-Point Method for Time-Varying Convex Optimization. M Fazlyab, S Paternain, V Preciado, A Ribeiro, IEEE Transactions on Automatic Control. to appearM. Fazlyab, S. Paternain, V. Preciado, and A. Ribeiro, "Prediction- Correction Interior-Point Method for Time-Varying Convex Optimiza- tion," IEEE Transactions on Automatic Control (to appear), 2017.
Real-Time Nonlinear Optimization as a Generalized Equation. V M Zavala, M Anitescu, SIAM Journal of Control and Optimization. 488V. M. Zavala and M. Anitescu, "Real-Time Nonlinear Optimization as a Generalized Equation," SIAM Journal of Control and Optimization, vol. 48, no. 8, pp. 5444 -5467, 2010.
Decentralized Prediction-Correction Methods for Networked Time-Varying Convex Optimization. A Simonetto, A Koppel, A Mokhtari, G Leus, A Ribeiro, IEEE Transactions on Automatic Control. 6211A. Simonetto, A. Koppel, A. Mokhtari, G. Leus, and A. Ribeiro, "Decentralized Prediction-Correction Methods for Networked Time- Varying Convex Optimization," IEEE Transactions on Automatic Con- trol, vol. 62, no. 11, pp. 5724 -5738, 2017.
Prediction-Correction Algorithms for Time-Varying Constrained Optimization. A Simonetto, E Dall'anese, IEEE Transactions on Signal Processing. 6520A. Simonetto and E. Dall'Anese, "Prediction-Correction Algorithms for Time-Varying Constrained Optimization," IEEE Transactions on Signal Processing, vol. 65, no. 20, pp. 5481 -5494, 2017.
Introduction to Optimization. B T Polyak, Optimization Software, IncB. T. Polyak, Introduction to Optimization. Optimization Software, Inc., 1987.
Strongly Regular Generalized Equations. S M Robinson, Mathematics of Operations Research. 51S. M. Robinson, "Strongly Regular Generalized Equations," Mathemat- ics of Operations Research, vol. 5, no. 1, pp. 43 -62, 1980.
An Euler-Newton Continuation method for Tracking Solution Trajectories of Parametric Variational Inequalities. A L Dontchev, M I Krastanov, R T Rockafellar, V M Veliov, SIAM Journal of Control and Optimization. 5151A. L. Dontchev, M. I. Krastanov, R. T. Rockafellar, and V. M. Veliov, "An Euler-Newton Continuation method for Tracking Solution Trajec- tories of Parametric Variational Inequalities," SIAM Journal of Control and Optimization, vol. 51, no. 51, pp. 1823 -1840, 2013.
A Prediction-Correction Path-Following Algorithm for Dual-Degenerate Parametric Optimization Problems. V Kungurtsev, J Jäschke, SIAM Journal on Optimization. 271V. Kungurtsev and J. Jäschke, "A Prediction-Correction Path-Following Algorithm for Dual-Degenerate Parametric Optimization Problems," SIAM Journal on Optimization, vol. 27, no. 1, pp. 538 -564, 2017.
Numerical Continuation Methods: An Introduction. E L Allgower, K Georg, Springer-VerlagE. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction. Springer-Verlag, 1990.
D-MAP: Distributed Maximum a Posteriori Probability Estimation of Dynamic Systems. F Y Jakubiec, A Ribeiro, IEEE Transactions on Signal Processing. 612F. Y. Jakubiec and A. Ribeiro, "D-MAP: Distributed Maximum a Pos- teriori Probability Estimation of Dynamic Systems," IEEE Transactions on Signal Processing, vol. 61, no. 2, pp. 450 -466, 2013.
Time-Varying Convex Optimization via Time-Varying Averaged Operators. A Simonetto, arXiv:1704.07338v1A. Simonetto, "Time-Varying Convex Optimization via Time-Varying Averaged Operators ," arXiv: 1704.07338v1, 2017.
Optimal Scaling of a Gradient Method for Distributed Resource Allocation. L Xiao, S Boyd, Journal of Optimization Theory and Applications. 1293L. Xiao and S. Boyd, "Optimal Scaling of a Gradient Method for Distributed Resource Allocation," Journal of Optimization Theory and Applications, vol. 129, no. 3, pp. 469 -488, 2006.
Introduction to Linear Algebra. G Strang, Wellesley-Cambridge Press and SIAMfifth ed.G. Strang, Introduction to Linear Algebra. Wellesley-Cambridge Press and SIAM, fifth ed., 2016.
Primer on Monotone Operator Methods. E K Ryu, S Boyd, Applied Computational Mathematics. 151E. K. Ryu and S. Boyd, "Primer on Monotone Operator Methods," Applied Computational Mathematics, vol. 15, no. 1, pp. 3 -43, 2016.
| [] |
[
"An introduction to varieties in weighted projective space",
"An introduction to varieties in weighted projective space"
] | [
"Timothy Hosgood "
] | [] | [] | Weighted projective space arises when we consider the usual geometric definition for projective space and allow for non-trivial weights. On its own, this extra freedom gives rise to more than enough interesting phenomena, but it is the fact that weighted projective space arises naturally in the context of classical algebraic geometry that can be surprising. Using the Riemann-Roch theorem to calculate ℓ(E, nD) where E is a nonsingular cubic curve inside P 2 and D = p ∈ E is a point we obtain a non-negatively graded ring R(E) = ⊕ n 0 L(E, nD). This gives rise to an embedding of E inside the weighted projective space P(1, 2, 3).To understand a space it is always a good idea to look at the things inside it. The main content of this paper is the introduction and explanation of many basic concepts of weighted projective space and its varieties. There are already many brilliant texts on the topic ([14, 10], to name but a few) but none of them are aimed at an audience with only an undergraduate's knowledge of mathematics. This paper hopes to partially fill this gap whilst maintaining a good balance between 'interesting' and 'simple'.1The main result of this paper is a reasonably simple degree-genus formula for nonsingular 'sufficiently general' plane curves, proved using not much more than the Riemann-Hurwitz formula. | null | [
"https://arxiv.org/pdf/1604.02441v3.pdf"
] | 10,613,754 | 1604.02441 | 97f7cced8b4ed08249ad32313ad7b475a5cd6a11 |
An introduction to varieties in weighted projective space
17 Apr 2016 April 19, 2016
Timothy Hosgood
An introduction to varieties in weighted projective space
17 Apr 2016 April 19, 2016
Weighted projective space arises when we consider the usual geometric definition for projective space and allow for non-trivial weights. On its own, this extra freedom gives rise to more than enough interesting phenomena, but it is the fact that weighted projective space arises naturally in the context of classical algebraic geometry that can be surprising. Using the Riemann-Roch theorem to calculate ℓ(E, nD) where E is a nonsingular cubic curve inside P 2 and D = p ∈ E is a point we obtain a non-negatively graded ring R(E) = ⊕ n 0 L(E, nD). This gives rise to an embedding of E inside the weighted projective space P(1, 2, 3).To understand a space it is always a good idea to look at the things inside it. The main content of this paper is the introduction and explanation of many basic concepts of weighted projective space and its varieties. There are already many brilliant texts on the topic ([14, 10], to name but a few) but none of them are aimed at an audience with only an undergraduate's knowledge of mathematics. This paper hopes to partially fill this gap whilst maintaining a good balance between 'interesting' and 'simple'.1The main result of this paper is a reasonably simple degree-genus formula for nonsingular 'sufficiently general' plane curves, proved using not much more than the Riemann-Hurwitz formula.
Introduction
My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.
Herman Weyl
Notation, conventions, assumptions, and citations
Effort has been made to avoid using too much presupposed knowledge on the behalf of the reader, though a working knowledge of group theory, ring theory, topology, Riemann surfaces, and suchlike is needed, and familiarity with some level of commutative algebra or algebraic geometry, as well as projective geometry, would not go amiss. Most 'big' theorems or ideas are at least stated before use or mentioned in passing. For the sake of not getting too off course, some other sources will be referenced in the main text, usually in lieu of proofs.
However, as we progress we require more and more tools, and it would be pure folly to try to keep this text entirely self-contained when so many brilliant books and notes have already been written about so many of the subjects which we touch upon in our journey. So at the beginning of a section we might state a few results, or a topic, that we assume the reader already has knowledge of, along with a reference for reading up on it if appropriate. In general, this paper is aimed at readers of the same level as the author: very early graduate students.
All the assumed knowledge of algebraic geometry can be found in [15] (available online), which is also just a very useful introduction to algebraic geometry as a whole. We quite often quote results of affine algebraic geometry (though always try to remember to cite some sort of reference), but usually avoid quoting results of projective algebraic geometry, since this should really be a special case of the things that we're proving here. 2 Another very enlightening book by the same author is [16] (also available online) which deals with the commutative algebra side of algebraic geometry.
There are a few other useful texts to have at hand as a reference (or simply as a better written exposition) for most general algebraic geometry and commutative algebra, as well as the underlying category and scheme theory (if that floats your boat). One is [1]. To quote from the CRing Project website: 3 The CRing project is an open source textbook on commutative algebra, aiming to comprehensively cover the foundations needed for algebraic geometry at the level of EGA or SGA. It is a work in progress.
The other is [18], whose purpose is eloquently summarised in the text itself:
This book is intended to give a serious and reasonably complete introduction to algebraic geometry, not just for (future) experts in the field. [ For this reason, the book is written as a first introduction, but a challenging one. This book seeks to do a very few things, but to try to do them well. Our goals and premises are as follows.
These are not 'classic' references in the way that [8] and [6] are, but they are freely available online, which means that even without access to a library they are easy (and free) to get hold of.
The main three references, and inspiration, for this paper were [14,10,2].
List of notation and assumptions
The following is a list of any notational quirks or assumed conventions used throughout this paper, unless otherwise stated:
• A ⊂ B means that A is a proper subset of B, and A ⊆ B means that A is a notnecessarily proper subset of B;
• if A and B are disjoint sets then we (may) write A ⊔ B to mean the union A ∪ B;
• rings are commutative and unital, with 0 = 1;
• k refers to an algebraically closed field; 4
• N = {1, 2, . . .}, so 0 ∈ N;
• when we say 'graded ring' we mean specifically a Z 0 -graded ring;
• for a, b ∈ Z we write a | b to mean that a divides b, and a ∤ b to mean that a does not divide b;
• G m = k × is the multiplicative group of a field;
• G a = k is the field considered just as an additive group;
• µ n is the cyclic group of order n, usually realised as n-th roots of unity in k;
• ω n is a primitive n-th root of unity (so µ n = ω n );
• R G denotes the subring of R fixed by G, where R is a ring and G is a group that acts on R;
• k[x 0 , . . . , x n ] is the polynomial ring in n + 1 independent indeterminates, i.e. the x i are assumed to be algebraically independent, and the same goes for anything similar, such as k [u, v, w], unless otherwise explicitly stated;
• f : A ։ B means that f is a surjection;
• f : A ֒→ B means that f is an injection (and so we usually think of it as an embedding of A into B);
• (x 1 , . . . , x i , . . . , x n ) = (x 1 , . . . , x i−1 , x i+1 , . . . , x n ) -that is, a hat above an element of a set or ordered n-tuple means ommision of that element;
• the phrase 'projective space', without either of the words 'straight' or 'weighted' in front of it, is always taken to refer to both straight and weighted projective space 5 .
The main aims
This paper is meant to serve as a beginner's guide to weighted projective space, its varieties, and some basic algebra that follows on from this. In some places the explanations might be quite slow or dense, and the notation seemingly clumsy and complicated, but this is because nearly all of the currently existing literature on these topics is relatively advanced (at least, to an undergraduate or early graduate), so this paper aims to fill a gap in the market, as it were. Whenever there has been a decision to make in terms of either leaving some detail out or including it, the latter has almost always been chosen, even if this might not have been the best editorial choice. This is because this paper is not intended to be a textbook, where making the reader puzzle out details themselves helps enormously with their eventual understanding of the topic, but instead really as an exercise for the author -to never leave a proof or explanation as 'an exercise for the reader'. This was how many of the proofs were originally passed over in the rough notes that came before this paper. But after having written the first draft of this paper, I then had to play the role of the reader, which meant having to complete all the 'exercises' anyway.
At the end of the day though, this paper was written by an undergraduate student as a summer project. Because of this, the author doesn't have a complete understanding of each and every topic contained within, let alone of the surrounding area of mathematics and the inner workings of the machinery. Sometimes questions are left unanswered -we apologise for this.
An overview of the journey
We start off in Section 2 with the preliminary definitions of weighted projective space, giving an example or two, and introduce the idea of affine patches to mirror that of straight projective space. Section 3 introduces the idea of a weighted projective variety and the topology to which it gives rise. Again, this is very similar to the usual notion of projective varieties and the Zariski topology on straight projective space, and so it seems natural to consider the idea of some sort of weighted projective Nullstellensatz, which we do in Section 3.2. Finally in this section we define the coordinate ring of a weighted projective variety, which has the expected definition, bar the weighting of the algebraic indeterminates. 5 Obviously both of these will be defined later on, don't worry.
The Nullstellensatz and the coordinate ring let us now do what algebraic geometry always does: study the link between algebra and geometry. This is the main motivation of Section 4. We use the Nullstellensatz to explain the Proj construction and gain some intuition with this algebraic definition of weighted projective space, noting the similarities to that of the Spec construction in affine algebraic geometry. Using the idea of truncation of a graded ring we show that we can often reduce a weighted projective space to a simpler one, where any (n − 1) of the weights are coprime. We give these 'fully simplified' weighted projective spaces the nice name of 'well-formed'. To ensure that we don't get lost in the heady realms of abstract algebra, this section ends with a few worked examples and applications.
Sticking with the theme of following a standard algebraic geometry course we look next, in Section 5, at plane curves, which are varieties in weighted projective 2-space given by the vanishing of a single weighted-homogeneous polynomial: X = V(f ) ⊂ P(a 0 , a 1 , a 2 ). We show that sufficiently nice plane curves are also Riemann surfaces, and then use some of the associated machinery (such as the Riemann-Hurwitz formula) to see what else we can find out. This section culminates with a version of the degree-genus formula for weighted projective plane curves.
Finally, Section 6 is a brief introduction to a few further ideas that follow on from the theory that we have developed up until this point. We cover (speedily and not overly rigorously) some ideas about the Hilbert polynomial of a variety and the Hilbert Syzygy Theorem. It is probably the most interesting of all sections, but is purely expository, with references to various sources that deal with the subject matter in more detail and depth.
Acknowledgements
Many thanks go to Balázs Szendrői for suggesting the subject of, and then supervising, this project. His patience in answering my multitude of questions and explaining so many things was the main reason that the author was able to write this paper.
Thanks must also be given to Edmund de Unger Academic Purposes fund from Hertford College, Oxford, without which the author would not have been able to live in Oxford throughout the composition of this paper.
Corrections
It is absurdly unlikely that there are no mistakes in this paper. If you find any, or have any recommendations for rewording or reordering of sections, please do email the author at [email protected] including the version date (as found on the title page) in your message. Source files for this document are at github.com/thosgood/introduction-to-wps and are available for use and modification.
Apologies are made in advance for the often overly-florid language that is most likely not-at-all suitable for rigorous mathematics.
Weighted projective space
At a first glance, it's very possible that the idea of weighted projective space seems like a needless generalization of the usual projective space (herein referred to as straight projective space) to which we have all grown to know and love, especially when we show that we can simply embed weighted projective space into a large enough straight projective space. But it turns out that, quite often, using a weighted projective space instead of its embedding results in much simpler equations and a generally more manageable beast. So some projective varieties can be more easily described as weighted projective spaces (similar to how some seemingly complicated varieties can turn out to be just a Veronese embedding of some smaller straight projective space), but there is also the fact that weighted projective space itself encompasses the idea of straight projective space and allows us to study interesting ideas in more generality. We cover some particularly natural uses and applications of weighted projective space in Section 6.1.
Like in a lot of algebraic geometry, there are two main ways of approaching a topic: geometrically and algebraically. The geometric way below mirrors the method usually used in approaching projective space and is a very 'hands-on' construction. The algebraic way uses some language from very basic scheme theory, dealing with the Proj construction, and is covered in Section 4. Let a = (a 0 , . . . , a n ) with a i ∈ N, and define the corresponding action of G m (which we write as G (a) m to avoid confusion) on A n+1 \ {0} by λ · (x 0 , . . . , x n ) = (λ a0 x 0 , . . . , λ an x n ).
Geometric construction of weighted projective space
(2.1.2)
We call a 1 , . . . , a n the weights.
We use this action to define weighted projective space as follows.
Definition 2.1.3 [Weighted projective space]
Let a = (a 0 , . . . , a n ) be a weight. Define a-weighted projective space as the quotient P(a 0 , . . . , a n ) = (
A n+1 \ {0})/G (a) m .
We write points in P(a 0 , . . . , a n ) as |x 0 : . . . : x n | a , which represents the equivalence class of the point (x 0 , . . . , x n ) ∈ A n+1 \ {0}, omitting the subscript a if it is clear that we are working in P(a) = P(a 0 , . . . , a n ).
It is easier to deal with weighted projective space once we have some technical lemmas and another perspective under our belt, and so the majority of big examples will come at the end of a later section, but here are a few right now to help build some kind of intuition.
Example 2.1.4 [Recovering straight projective space]
If we set all a i to be 1 then the above definition coincides with that of straight projective space:
P(1, . . . , 1) = (A n+1 \ {0})/G m = P n
where |x 0 : . . . : x n | ∼ λ|x 0 : . . . : x n | = |λx 0 : . . . : λx n |, and we more commonly write [x 0 : . . . :
x n ] for the coordinates.
Example 2.1.5 [Something that looks a bit like a cone]
We cover this example in more detail later, since it turns out to be an interesting one, so here we look at it quite briefly and not particularly rigorously. Consider P(1, 1, 2). Just like in straight projective space, any point is invariant under scaling, but here with respect to the weighting. For example, |1 : 0 : 2| = 3 · |1 : 0 : 2| = |3 : 0 : 18|. Now we try to get a bit more of a grasp on the space as a whole. Define a map ϕ : |x 0 :
x 1 : x 2 | → [x 2 0 : x 0 x 1 : x 2 1 : x 2 ].
We claim that this map has its image in P 3 . Firstly, since at least one of the x i is non-zero (by the definition of weighted projective space), at least one of the monomials will also be non-zero. But then all that we need to check is that the image is invariant under scaling, i.e.
that λ · [x 2 0 : x 0 x 1 : x 2 1 : x 2 ] = [λx 2 0 : λx 0 x 1 : λx 2 1 : λx 2 ] = [x 2 0 : x 0 x 1 : x 2 1 : x 2 ]
. Simply using definitions we get that
[x 2 0 : x 0 x 1 : x 2 1 : x 2 ] = ϕ(|x 0 : x 1 : x 2 |) = ϕ λ 1 2 · |x 0 : x 1 : x 2 | = ϕ |λ 1 2 x 0 : λ 1 2 x 1 : λx 2 | = [λx 2 0 : λx 0 x 1 : λx 2 1 : λx 2 ].
Even though we haven't really defined what an isomorphism should be for weighted projective spaces, it makes sense to think that, if we can find an inverse map that is also given by polynomials in each coordinate, then we can think of P(1, 1, 2) and its image under ϕ in P 3 as isomorphic. That is, we can think of ϕ as an embedding of P(1, 1, 2) in P 3 .
To construct our inverse map we take some point [y 0 , y 1 , y 2 , y 3 ] in the image. Sadly, even though k is algebrically closed, we can't take |y 1/2 0 : y 1/2 2 : y 3 | as our inverse map, since this is not polynomial in each coordinate. But we do know that
y 0 = x 2 0 , y 1 = x 0 x 1 , y 2 = x 2 1 , y 3 = x 2
for some |x 0 : x 1 : x 2 | ∈ P(1, 1, 2), and so
|x 0 : x 1 : x 2 | = x 0 · |x 0 : x 1 : x 2 | = |x 2 0 : x 0 x 1 : x 2 0 x 2 | = |y 0 : y 1 : y 0 y 3 |; x 1 · |x 0 : x 1 : x 2 | = |x 0 x 1 : x 2 1 : x 2 1 x 2 | = |y 1 : y 2 : y 2 y 3 |,
where we choose whichever option gives us a point in P(1, 1, 2), i.e. depending on whether or not all of y 0 , y 1 , y 3 are zero.
So we have two mutually inverse polynomial maps, which we (for the moment) are content with calling an isomorphism:
ϕ : P(1, 1, 2) → X ⊂ P 3 |x 0 : x 1 : x 2 | → [x 2 0 : x 0 x 1 : x 2 1 : x 2 ] ϕ −1 : X → P(1, 1, 2)
|y 0 : y 1 : y 2 : y 3 | → |y 0 : y 1 : y 0 y 3 | if y 0 , y 1 , y 3 = 0; |y 1 : y 2 : y 2 y 3 | otherwise.
Then understanding P(1, 1, 2) becomes a matter understanding the set X ⊂ P 3 . We don't know yet what properties X has exactly, but we will find out later on. One thing to notice though is that, on the patch {x 2 = 0} of X ⊂ P 3 , we have something that looks a lot like the Veronese embedding of P 1 into P 2 : the rational normal curve of degree 2, also known as the flat conic.
The way that we came up with the idea of the map in Example 2.1.5 was to construct all the degree two monomials, but where we consider x 2 as being a degree two element already. Generally, though, we see that it looks like we should be able to embed weighted projective space into some straight projective space of high enough dimension, using something that looks remarkably like a Veronese embedding.
It's not entirely clear why exactly this should work at this point, but it turns out to be a much easier idea to justify once we have introduced the notion of weighted projective spaceusing the Proj construction, and truncation, later on.
Coordinate patches
On straight projective space, picking some 0 i n, we have the standard decomposition
P n = {[x 0 : . . . : x n ] | x i = 0} ⊔ {[x 0 : . . . x n ] | x i = 0} ∼ = P n−1 ⊔ A n
where the sets H i = {x i = 0} are closed, and so the U i = {x i = 0} are open. 6 It makes sense to consider the same sets H i and U i in weighted projective space, and even though we have yet to really define a Zariski-style topology 7 it turns out that these sets are closed and open (respectively) as before, in both the Zariski and quotient topologies.
In this decomposition the U i , often called coordinate patches or affine patches, turn out to be very useful, since they cover the whole of projective space and are isomorphic to affine space, which is much easier to visualise geometrically in most cases. So given some projective variety, we can see how it intersects with the affine patches U i and study these using all our familiarity with affine space.
But in weighted projective space we have a slight issue, namely that the U i are not isomorphic to A n , but instead some quotient of A n by a finite group. They still deserve the name 'affine patches' then, since the quotient of an affine variety by a finite group is again an affine variety 8 but a little bit more work needs to be done to see what exactly they look like.
Definition 2.2.1 [Quotient of affine space by a cyclic group]
Define an action of µ ai on A n , called the action of type 1 ai (a 0 , . . . , a i , . . . , a n ), by
ω ai · (x 0 , . . . , x i , . . . , x n ) = (ω a0 ai x 0 , . . . , ω ai ai x i , . . . , ω an ai x n ). (2.2.2)
This induces an action of µ ai on k[x 0 , . . . , x i , . . . , x n ] given by ω ai · x j = ω aj ai x j and thus 9 gives rise to the affine quotient variety
A n /µ ai = mSpec k[x 0 , . . . , x i , . . . , x n ] µ a i
as well as the quotient map π i = (ι i ) # : A n → A n /µ ai corresponding to the inclusion
ι i : k[x 0 , . . . , x i , . . . , x n ] µ a i ֒→ k[x 0 , . . . , x i , . . . , x n ].
Lemma 2.2.3 [Affine patches in wps]
With U i = {|x 0 : . . . : x n | ∈ P(a 0 , . . . , a n ) : x i = 0} and the quotient A n /µ ai defined as in
Definition 2.2.1 we have U i ∼ = A n /µ ai
where we mean isomorphic in the usual sense: there exists an algebraic morphism given by a polynomial map with polynomial inverse. We often write
A i = A n /µ ai .
Proof. We can write every point in U i as |x 0 : . . . : 1 : . . . : x n | where the 1 is in the i-th place. Write a point in A i as [(y 1 , . . . , y n )], that is, the equivalence class of y = (y 1 , . . . , y n ) ∈ A n consisting of the points in the µ ai -orbit of y. Define the morphism ϕ i : U i → A i by ϕ i : |x 0 : . . . : 1 : . . . : x n | → (x 0 , . . . , 1, . . . , x n ) which we need to show is a well-defined map. Our concern is that this map might rely on our choice of x i . By definition, where ω i = ω ai is a primitive a i -th root of unity, |x 0 : . . . : 1 : . . . : x n | = |ω a0 i x 0 : . . . : ω ai i : . . . : ω an i x n | = |ω a0 i x 0 : . . . : 1 : . . . : ω an i x n |.
Now
ϕ i |ω a0 i x 0 : . . . : 1 : . . . : ω an i x n | = (ω a0 i x 0 , . . . , 1, . . . , ω an i x n ) = ω i · (x 0 , . . . , 1, . . . , x n ) = ϕ i |x 0 : . . . : 1 : . . . : x n | , and so our map is well defined. Define its inverse morphism ϕ −1 i : A i → U i by ϕ −1 i : (y 1 , . . . , y n ) → |y 1 : . . . : y i−1 : 1 : y i : . . . : y n | and note that it is indeed an inverse to ϕ. We can show that it is well defined in exactly the same way that we did for ϕ i , and clearly both ϕ i and ϕ −1 i are polynomial in each coordinate.
Although the U i give us a nice affine space to work with, it is a quotient space and so can be quite tricky to realise at times. Much easier is the idea of looking at the covering space of the affine patches, since it turns out that if the covering space of the affine patch has certain nice properties then so too does the affine patch. We now define a few terms to avoid confusion after all this talk of 'affine patches'.
Definition 2.2.4 [Quotient and covering affine patches]
Given some subset X ⊆ P(a 0 , . . . , a n ) we define the quotient affine patches X i ⊆ A i and the covering affine patches X i ⊆ A n of X as
X i = X ∩ U i ⊆ A i = A n /µ ai X i = π −1 i (X ∩ U i ) ⊆ A n
where we use the isomorphism U i ∼ = A i and the quotient map π i : A n → A i .
So we have two different types of affine patches to think of: those that we glue together to get the ambient weighted projective space, which are in some way 'folded up' (the quotient affine patches); and those that come from 'unfolding' the aforementioned ones (the covering affine patches). The reason for considering both is that they can be equally useful, but in different ways. Really, the covering affine patches come into their own in Section 5, since we already know many useful facts about varieties in A n .
Weighted projective varieties
Although we are dealing with so-called weighted-homogeneous ideals here, we are really just looking at homogeneous ideals (i.e. graded submodules where we consider the ring as a module over itself) of a graded ring. Thus most standard proofs can be used for the vast majority of the lemmas in this section, and at times we simply refer to them instead of providing our own. A good reference is [1, Chapter 6, Section 1].
Before we define the notion of weighted projective varieties we need to cover a few formalities. The main one is 'how do we define evaluating a polynomial at a point in weighted projective space, and is this well defined?'. It turns out that, just as in straight projective space, evaluating a polynomial at a point is not well defined, but seeing whether or not a point is a zero of a polynomial is, as long as our polynomial is homogeneous, but in a slightly different sense. We start off this section in quite a dull manner, with quite a few definitions in a row, and most of them quite expected or natural, but then play around with them to see what we can get. That is, we think of x i as a degree a i monomial and thus, for example,
deg n i=0 x ci i = n i=0 a i c i .
If we omit the subscript a and simply write k[x 0 , . . . , x n ] then we mean the polynomial ring in n + 1 variables with the usual grading, i.e. a i = 1 for all i.
We sometimes use the phrase weighted degree to be clear that we are including the weighting in our calculation of the degree. Note that deg λ = 0 for any λ ∈ k. For a general polynomial f ∈ k a [x 0 , . . . , x n ] we define the degree deg f as the maximum of all the degrees of the monomials in f . 10 Note 3.0. 6 An important thing to note is that this weighting changes the grading of the ring, but it doesn't change the underlying k-algebra structure. So k a [x 0 , . . . , x n ] is Noetherian. Further, it means that until Section 4, when we start talking about the graded ring structure, our choice of notation k a [x 0 , . . . , x n ] vs. k[x 0 , . . . , x n ] is reasonably arbitrary, and doesn't particularly matter.
Definition 3.0.7 [Weighted-homogeneous polynomial] Let f ∈ k[x 0 , . . . , x n ]
where wt x i = a i for some weight a = (a 0 , . . . , a n ). We say that f is a-weighted-homogeneous of degree d 11 if each monomial in f is of weighted degree d, i.e. there exist c i ∈ k and some m ∈ N such that
f = m i=1 c i n j=0 x d (i) j j
and, for all 0 i n, n j=0 a j d
(i) j = d.
We write k a [x 0 , . . . , x n ] d ⊂ k a [x 0 , . . . , x n ] to mean the additive group of all weightedhomogeneous polynomials of d.
Note that if f is a-weighted-homogeneous of degree d then for any λ ∈ G m then, by Definition 3.0.7, f (λ a0 x 0 , . . . , λ an x n ) = λ d f (x 0 , . . . , x n ). So let p = |p 0 : . . . : p n | ∈ P(a 0 , . . . , a n ) and f ∈ k a [x 0 , . . . , x n ]. By definition we also have that p = |λ a0 p 0 : . . . : λ an p n | for any λ ∈ G m . In particular we can assume that λ = 1. But then, using Eq. Thus the idea of evaluating an a-weighted-homogeneous polynomial f at a point p ∈ P(a) doesn't make sense in general, but looking at the points p ∈ P(a) at which f vanishes does make sense. That is, it is well defined to write that f (p) = 0 for some f ∈ k a [x 0 , . . . , x n ] and p ∈ P(a). We will come back to this point shortly, in Section 3.1.
Definition 3.0.9 [Weighted-homogeneous ideal]
We say that an ideal I ⊳ k a [x 0 , . . . , x n ] is a-weighted-homogeneous 12 if it is generated by a-weighted-homogeneous elements (of not necessarily the same degree). Example 3.0.10 Let a = (1, 3, 3, 4) and I, J ⊳ k a [w, x, y, z] be given by I = (w 2 , wx + z, x 3 + w 2 yz), J = (w 2 , w 4 + y).
Then I is weighted-homogeneous, but J isn't, since deg w 4 = 4 = 3 = deg y.
f = deg f i=0 f i for unique f i ∈ k a [x 0 , . . . , x n ] i ∩ I.
Proof. This proof was originally going to be a simple reference to some pre-existing proof, but apparently everybody else has had the same idea -the author couldn't find a reference which stated this fact and didn't leave its proof as an exercise to the reader. If this paper is good for nothing else, at least it might provide a reference for people looking for an easy-to-find proof of this fact.
Note that, for any f ∈ I, writing 13
f i = f ∩ k a [x 0 , . . . , x n ] i we have f = deg f i=0 f i
with the f i uniquely determined by f . So the above condition is equivalent to requiring that f ∩ k a [x 0 , . . . , x n ] i ∈ I for all i (since this intersection is empty, and thus trivially in I, for i > d). Yet another way of phrasing this is that I must satisfy I = i (I ∩ k a [x 0 , . . . , x n ] i ).
If I satisfies this above condition then for each i we find g
(i) j ∈ (I ∩ k a [x 0 , . . . , x n ] i ) such that (g (i) 1 , . . . , g (i) ni ) = I ∩ k a [x 0 , . . . , x n ] i .
By definition, each g
(i)
j must be weighted-homogeneous of degree i. But then
I = i (I ∩ k a [x 0 , . . . , x n ] i ) = i (g (i) 1 , . . . , g (i) ni ) = (∪ i {g (i) 1 , . . . , g (i) ni })
is a way of writing I as being generated by weighted-homogeneous elements. So I is weighted-homogeneous. 12 From now on we will stop pointing out that we might sometimes omit the weight a if it is clear which weight we are working with, but this is still the case. The convention that the author has tried to stick to is to simply say 'weighted-homogeneous' if we have already said that I ⊳ ka[x 0 , . . . , xn], since then the weight a is clear. 13 When we say f ∩ ka[x 0 , . . . , xn] i we mean write f = f j , where each f j is a sum of monomials of degree j, and then define f ∩ ka[x 0 , . . . , xn
] i = {f j } ∩ ka[x 0 , . . . , xn] i = f i .
For the other direction, we assume that I is weighted-homogeneous, so I = (g 1 , . . . , g n ) for some weighted-homogeneous g i . Note that
{g i } ⊆ i (I ∩ k a [x 0 , . . . , x n ] i ). Thus i (I ∩ k a [x 0 , . . . , x n ] i ) ⊆ I = (g 1 , . . . , g n ) ⊆ i (I ∩ k a [x 0 , . . . , x n ] i ),
and so these ideals must in fact be equal. The idea of saying that some polynomial in a weighted-homogeneous ideal vanishes at some point is still well defined, since every polynomial in the ideal is a sum of multiples of weighted-homogeneous polynomials.
Weighted projective varieties
Motivated by our previous discussion (defining what it means for a weighted-homogeneous polynomial to vanish at a point in weighted projective space) we can now define the idea of a weighted projective variety, much in the same way as one would for straight projective space. Conversely, let V ⊆ P(a 0 , . . . , a n ) be a subset of weighted projective space. Define the ideal associated to V by
I(V ) = {f ∈ k a [x 0 , . . . , x n ] | f (p) = 0 for all p ∈ V and f is a-weighted-homogeneous}.
We say that a subset V ⊆ P(a) is a weighted projective variety if it is of the form V(I) for some weighted-homogeneous ideal I ⊳ k a [x 0 , . . . , x n ].
If we have two varieties V ⊆ W then we say that V is a subvariety of W . A weighted projective variety is said to be irreducible if it has no non-trivial decomposition into subvarieties,
i.e. V = V 1 ∪ V 2 with V 1 , V 2 = ∅, V .
It's important to point out that, although we call I(V ) the ideal associated to V , we have yet to prove that it actually is an ideal. We do this in Lemma 3.2.1.
Note 3.1.2
As a slight notational quirk we write V I to mean the composition V • I, since this looks a lot less messy. So instead of writing, for example, V(I(V(I))), we write V I V(I).
Example 3.1.3
Here are some simple examples of V and I using just the definitions that we have so far. We will develop and uncover some more advanced machinery and techniques later on in this section and the next.
• V(x i ) = {|x 0 : . . . : x n | ∈ P(a 0 , . . . , a n ) | x i = 0}, and so U i = P(a 0 , . . . , a n ) \ V(x i );
• Let X = V(x a2 1 − x a1 2 ) ⊆ P(a 0 , a 1 , a 2 ). Then if x = |x 0 : x 1 : x 2 | ∈ X we must have that x a2 1 = x a1 2 .
Splitting this into two cases (x 1 , x 2 = 0 and x 1 , x 2 = 0) gives us
X = {|x 0 : 1 : 1|} ∪ {|1 : 0 : 0|}
where we use that fact that |x 0 : x 1 :
x 2 | = (1/x 1 ) 1/a1 · |x 0 : x 1 : x 2 | for the first case; • What is I V(x 2 i )? Well x 2 i = 0 if and only if x i = 0, since k is a field, so V(x 2 i ) = {|x 0 : . . . : x n | ∈ P(a 0 , . . . , a n ) | x i = 0} = V(x i ).
But the x j (for j = i) can take any value in k (as long as they aren't all simultaneously zero), so the only polynomials f ∈ k a [x 0 , . . . ,
x n ] that satisfy f (x) = 0 for all x ∈ V(x 2 i ) = V(x i ) are those of the form x i g(x) for some polynomial g ∈ k a [x 0 , . . . , x n ]. That is, I V(x 2 i ) = I V(x i ) = (x i ) ⊳ k a [x 0 , . . . , x n ].
We will later see that weighted projective varieties are in fact also projective varieties in the usual sense, and so they really are deserving of the name 'varieties'. But then it doesn't seem too unreasonable to hope that we could define some sort of Zariski topology on weighted projective space with our weighted projective varieties, so we state and prove a lemma that lets us do so. Conversly, f g ∈ IJ for all f ∈ I and g ∈ J. Since k a [x 0 , . . . , x n ] is Noetherian,
(i) V(I) ∪ V(J) = V(IJ) (ii) V(I) ∩ V(J) = V(I + J) (iii) ∅ = V(k a [x 0 , . . . , x n ]);I = (f 1 , .
. . , f k ) and J = (g 1 , . . . , g l ).
So if x vanishes on all of IJ then in particular it vanishes on
{f i g j | 1 i k, 1 j l} = k i=1 {f i g j | 1 j l}.
If f i (x) = 0 for some i then we must have that g j (x) = 0 for all x and so x ∈ J, and if not then
x ∈ I. Thus V(IJ) ⊆ V(I) ∪ V(J). (ii) If x ∈ V(I) ∩ V(J) then f (x) = g(x) = 0 for all f ∈ I and g ∈ J. Thus (f + g)(x) = 0 for all (f + g) ∈ I + J (which is all of I + J by definition).
Conversely, since 0 ∈ I, J, for all f ∈ I and g ∈ J we know that f = f + 0, g = 0 + g ∈ I + J. So if x ∈ V(I + J) then it vanishes in particular on all of I and all of J.
(iii) The only 'point' which vanishes on x i for all i is 0, but 0 ∈ P(a 0 , . . . , a n ), so
V(k a [x 0 , . . . , x n ]) ⊆ V(x 0 , . . . , x n ) = ∅.
(iv) Every point in P(a 0 , . . . , a n ) vanishes on the zero polynomial.
Lemma 3.1.5
An arbitrary sum
I = α∈A I α = β∈B f β f β ∈ I β and B ⊂ A is finite of weighted-homogeneous ideals is a weighted-homogeneous ideal.
Proof. We first claim that an abitrary sum of ideals is an ideal. It is clear that 0 ∈ I, as well as rf ∈ I for any f ∈ I and r ∈ k a [x 0 , . . . , x n ], so it remains only to show that I is closed under finite sums. Let f = β∈B f β and g = β∈C g β be elements of I. We note that
f = β∈D f ′ β , where D ⊇ B is finite and f ′ β = f β if β ∈ B; 0 if β ∈ D \ B. So let D = B ∪ C. Then f + g = β∈D (f ′ β + g ′ β ) ∈ I, since (f ′ β + g ′ β ) ∈ I β for each β ∈ D.
To show further that this sum is a weighted-homogeneous ideal we use Lemma 3.0.11. By definition, every element f in the arbitrary sum J = I∈I I is a finite sum of elements in the summands. That is, there exist I 1 , . . . , I k ∈ I and f i ∈ I i such that
f = k i=1 f i .
But each I i is weighted-homogeneous, and so each f i can be written as a sum of weightedhomogeneous elements g
(i) j where deg g (i) j = j. Thus f = l j=1 k i=1 g (i) j
is an expression for f as a sum of weighted-homogeneous elements, and so J is a weightedhomogeneous ideal.
Corollary 3.1.6 An arbitrary intersection of weighted projective varieties is a weighted projective variety:
I∈I V(I) = V I∈I I = V(J) where I∈I I = J ⊳ k a [x 0 , . . . , x n ] is a weighted-homogeneous ideal.
With Lemma 3.1.4 and Corollary 3.1.6 in hand, we can now make the definition that we would like to.
Definition 3.1.7 [Zariski topology]
The Zariski topology on P(a 0 , . . . , a n ) is given by defining the closed sets of P(a 0 , . . . , a n ) to be those of the form V(I) for some weighted-homogeneous ideal I ⊳ k a [x 0 , . . . , x n ], that is, the weighted projective varieties.
One final thing to note before moving onto the Nullstellensatz is how we can use the construction of weighted projective space to understand these weighted projective varieties. The way that we define f (p) = 0 for some a-weighted-homogeneous f and point p ∈ P(a) is really by requiring that f (p) = 0, wherep ∈ A n+1 \ {0} is a representative of p. We use the requirement of f being a-weighted-homogeneous to ensure that this definition is well-defined under a change of representatives.
So we can think of V(I) as a quotient of the affine 'cone' 14 :
V(I) = V aff (I) \ {0} G m ⊆ A n+1 \ {0} G m (3.1.8) where V aff (I) = {x ∈ A n+1 | f (x)
= 0 for all f ∈ I} and we consider I ⊳ k[x 0 , . . . , x n ] as an ideal in the usual polynomial ring (i.e. with all weights equal to 1, though really this doesn't matter, since the weights affect only the graded structure of the ring).
Definition 3.1.9 [Affine cone]
Given X = V(I) for some weighted-homogeneous ideal I ⊳ k a [x 0 , . . . , x n ] we writeX to mean V aff (I), so that Eq. (3.1.8) can be written as
X =X ∩ (A n+1 \ {0}) G m .
Note that we don't simply writeX \ {0} in Definition 3.1.9 since we don't know a priori that 0 ∈X, but we clear this up in Lemma 3.2.9.
The weighted projective Nullstellensatz
Lemma 3.2.1 [Five useful facts]
Let I, J ⊳ k a [x 0 , . . . , x n ] be weighted-homogeneous ideals and let V, W ⊆ P(a 0 , . . . , a n ). Then with V and I be defined as in Definition 3.1.1 we have that
(i) I(V ) ⊆ k a [x 0 , . . . , x n ] is a radical weighted-homogeneous ideal; (ii) If I ⊆ J then V(I) ⊇ V(J); (iii) If V ⊆ W then I(V ) ⊇ I(W ); (iv) I ⊆ I V(I); (v) V(I) = V I V(I).
Proof. (i) The fact that I(V ) is an ideal is reasonably straightforward: if f and g vanish at all points of V then so too does f + g, as does f h for any other polynomial h.
Since k a [x 0 , . . . , x n ] is Noetherian we know then that
I(V ) = (f 1 , . . . , f m ) for some f i ∈ I(V ).
But by definition, f ∈ I(V ) means that f must be a-weighted-homogeneous.
Thus I(V ) is generated by weighted-homogeneous elements, and so is an a-weightedhomogeneous ideal.
Finally, say that
f k ∈ I(V ). Then 0 = (f k )(p) = f (p) k and so f (p) = 0 since k a [x 0 , . . . , x n ] is an integral domain. So f ∈ I(V ) and hence I(V ) is radical. (ii) If p ∈ V(J) then f (p) = 0 for all f ∈ J,
and thus for all f ∈ I, so p ∈ V(I).
(iii) If some polynomial vanishes at all points in W then it vanishes in particular at all points in V ⊆ W .
(iv) Let f ∈ I, so that by definition f (x) = 0 for all x ∈ V(I) and thus f ∈ I V(I).
(v) Let x ∈ V(I), then f (x) = 0 for all f ∈ I V(I), and so x ∈ V I V(I). Conversely, use (ii) with (iv) to get that V I V(I) ⊆ V(I).
Definition 3.2.2 [Relevant ideals]
An ideal I ⊳ k a [x 0 , . . . , x n ] is said to be relevant if it satisfies the following two conditions:
(i) it is strictly contained inside the irrelevant ideal 15 (x 0 , . . . , x n );
(ii) V(I) = ∅.
Note 3.2.3
If I is weighted-homogeneous then the first condition in Definition 3.2.2 is actually redundant: it is enough to simply ask that V(I) = ∅, since this implies that I is strictly contained inside the irrelevant ideal. 16 To see this, we can use the affine Nullstellensatz and Equation (3.1.8) to get that
(x 0 , . . . , x n ) ⊆ I =⇒ V aff (I) ⊆ {0} ⇐⇒ V(I) = ∅.
So V(I) = ∅ implies that (x 0 , . . . , x n ) ⊆ I. In particular, (x 0 , . . . , x n ) = I. But if I is weighted-homogeneous then I ⊆ (x 0 , . . . , x n ), since all of its generators must be of the form a i x i . Thus if V(I) = ∅ then I is strictly contained inside the irrelevant ideal.
Just as there are multiple equivalent definitions for an ideal to be homogeneous, there are multiple equivalent definitions for an ideal to be relevant. Sometimes one is more useful than the other, so we list four equivalent conditions here, and when we use 'the' definition of a relevant ideal in a proof or suchlike we mean any one of the following conditions.
Lemma 3.2.4
Let I ⊳ k a [x 0 , . . . , x n ] be a weighted-homogeneous ideal. Then the following are equivalent:
(i) I is relevant; (ii) I is strictly contained inside k a [x 0 , . . . , x n ] and is not equal to the irrelevant ideal; (iii) (x 0 , . . . , x n ) ⊆ rad(I).
Proof. (i) ⇐⇒ (ii): If I is strictly contained inside the irrelevant ideal then it is clearly strictly contained inside the whole ring.
Conversely, if I is strictly contained inside the whole ring then it cannot have any constant generators, since (θ) = k a [x 0 , . . . , x n ] for any θ ∈ k. Thus I is strictly contained inside the irrelevant ideal (since by assumption it is not equal to it).
(i) ⇐⇒ (iii): By the the affine Nullstellensatz we know that I aff V aff (I) = rad(I), and so using 17 Lemma 3.2.1 (iii) gives
V(I) = ∅ ⇐⇒ V aff (I) ⊆ {0} ⇐⇒ I aff ({0}) ⊆ I aff V aff (I) ⇐⇒ (x 0 , . . . , x n ) ⊆ rad(I).
Definition 3.2.5 [Maximal weighted-homogeneous ideals]
An ideal I ⊳ k a [x 0 , . . . , x n ] is said to be a maximal weighted-homogeneous ideal if it is relevant and maximal amongst relevant weighted-homogeneous ideals.
That is, if J ⊳ k a [x 0 , . . . , x n ] is a weighted-homogeneous ideal such that I J, then J is irrelevant (so J = (x 0 , . . . , x n ) or J = k a [x 0 , . . . , x n ]).
With all of these definitions out of the way, we now start our journey towards the weighted projective Nullstellensatz. We do this by proving a few technical lemmas, and then the Nullstellensatz drops out quite easily and naturally from them. Really then, the way to understand this train of thought is to read Theorem 3.2.10 first and then come back to Lemmas 3.2.6 and 3.2.9 and Corollary 3.2.7, otherwise it might seem like the lemmas are pulled from thin air.
Lemma 3.2.6
Let I ⊳ k a [x 0 , . . . , x n ] be a weighted-homogeneous ideal. Then rad(I) ⊳ k a [x 0 , . . . , x n ] is also a weighted-homogeneous ideal.
Proof. Let f ∈ rad(I), so that f k ∈ I for some k ∈ N. Write d = deg f and let
f i = f ∩ k a [x 0 , . . . , x n ] i for 0 i d
(since for i > d this intersection will be empty). Then by Lemma 3.0.11 it is enough to show that f i ∈ rad(I) for all 0 i d, since the f i are uniquely determined by f .
We
look first at f d . Because f k d = f k ∩ k a [x 0 , . .
. , x n ] d (since it is the only term of high enough degree) and I is weighted-homogeneous, we must have that f k d ∈ I, and so f d ∈ rad(I). But then f − f d ∈ rad(I) is a polynomial of strictly smaller degree with homogeneous components f 0 , . . . ,
f d−1 , thus (f − f d ) k ′ ∈ I for some k ′ ∈ N, so we repeat the above process with f d−1 to show that f k ′ d−1 ∈ I, and thus f d−1 ∈ rad(I).
After repeating this finitely many times (since the total degree strictly decreases each time) we have that f i ∈ rad(I) for all 0 i d.
Corollary 3.2.7
Let I ⊳ k a [x 0 , . . . , x n ] be a weighted-homogeneous ideal. Then I aff V aff (I) is also a weightedhomogeneous ideal.
Proof. The affine Nullstellensatz tells us that I aff V aff (I) = rad(I), so by Lemma 3.2.6 we are done.
Lemma 3.2.8
Let I ⊳ k a [x 0 , . . . , x n ] be a maximal weighted-homogeneous ideal. Then I is radical.
Proof. By Lemma 3.2.4 we know that (x 0 , . . . , x n ) ⊆ rad(I). Thus V(rad(I)) ⊆ ∅, and V(rad(I)) = ∅. So rad(I) is relevant. Further, rad(I) is a weighted-homogeneous ideal, by Lemma 3.2.6. We also know that I ⊆ rad(I), but if I is a proper subset of rad(I) then this gives us our contradiction, since I is maximal amongst radical weighted-homogeneous ideals. Hence I = rad(I).
The next lemma (Lemma 3.2.9) is really the key to the weighted projective Nullstellensatz, but it is largely just technicalities and abstract faff, so we state and prove it as a separate lemma just to make the statement and proof of Theorem 3.2.10 a bit more slick.
Lemma 3.2.9
Let I ⊳ k a [x 0 , . . . , x n ] be a weighted-homogeneous relevant ideal and X = V(I). Then
I(X) = I aff (X).
Proof. Let f be a generator of I aff (X) (of which there are finitely many, since k[x 0 , . . . , x n ] is Noetherian). By Corollary 3.2.7 we know that f is a-weighted-homogeneous. Also f (x) = 0 for allx ∈X, and so in particular f (x) = 0 for allx ∈X \ {0}. Combining these two facts we see that f ∈ I(X). Then, since all the generators of I aff (X) are in I(X),
I aff (X) ⊆ I(X).
For the other inclusion we use the fact that I is relevant, and so X = V(I) = ∅. We also know that X = V I(X), thus V I(X) = ∅. Hence I(X) is also relevant. So there are no constant polynomials in I(X), since otherwise I(X) would be the whole of k a [x 0 , . . . , x n ], contradicting the fact that it is relevant. Hence if f ∈ I(X) then f (0) = 0.
Also Proof. Write X = V(I). Then Lemma 3.2.9 tells us that I V(I) = I(X) = I aff (X) = rad(I).
, if f ∈ I(X) then f (x) = 0 for all x ∈ X, i.e. f (x) = 0 for all representativeŝ x ∈X \ {0}. So f ∈ I aff (X \ {0}). But since f (0) = 0 as well, f ∈ I aff (X), hence I(X) ⊆ I aff (X).
Corollary 3.2.11 [Applied weighted projective Nullstellensatz]
The maps V and I give us an inclusion reversing bijection between weighted projective varieties and radical weighted-homogeneous relevant ideals:
radical w.h. relevant ideals I ⊳ k a [x 0 , . . . , x n ] I V acts as the identity here V −→ I ←− weighted projective varieties ∅ = X = V(I) ⊆ P(a 0 , . . . , a n ) V I acts as the identity here I ⊆ J =⇒ V(J) ⊆ V(I) I(Y ) ⊆ I(X) ⇐= X ⊆ Y.
Further, under this bijection, prime weighted-homogeneous ideals correspond to irreducible varieties, and maximal weighted-homogeneous ideals 18 to points.
Proof. The first part of the statement follows directly from Theorem 3.2.10.
Next we consider the statement about prime ideals and irreducible varieties. Assume first that I ⊳ k a [x 0 , . . . , x n ] is a prime weighted-homogeneous relevant ideal. Let
X = V(I) = V(I 1 ) ∪ V(I 2 ) = X 1 ∪ X 2
be a decomposition of X, and assume that I i is radical (since V(I i ) = V(rad(I i )) by taking V of both sides of Theorem 3.2.10). We want to show that
X 1 = ∅ or X 1 = X. Since X i ⊆ X we know that V(X) ⊆ V(X i ), i.e. I ⊆ I i . We also know that V(I 1 ) ∪ V(I 2 ) = V(I 1 I 2 )
, and thus I = I 1 I 2 (since the product of radical ideals is again radical). But I is prime, and so, since (trivially by the above equality) I 1 I 2 ⊆ I, we must have I i ⊆ I (without loss of generality assume that I 1 ⊆ I). So I 1 ⊆ I ⊆ I 1 , hence I 1 = I and X 1 = X.
For the other direction, assume that X = V(I) is an irreducible weighted projective variety and let f g ∈ I be such that f, g ∈ I. Then X ⊆ V(f ) ∪ V(g), and so 18 Recall Definition 3.2.5. is a non-trivial decomposition, because f, g ∈ I = I(X), and thus X ∩ V(f ) X (and similarly for g).
X = (X ∩ V(f )) ∪ (X ∩ V(g))
Finally, the statement concerning maximal weighted-homogeneous ideals and points. Let I ⊳ k a [x 0 , . . . , x n ] be a maximal weighted-homogeneous ideal in the sense of Definition 3.2.5. In particular then, I is relevant, and thus V(I) = ∅, so let x ∈ V(I). Then I V(I) ⊆ I({x}), and using Lemma 3.2.8 we see that I V(I) = rad(I) = I. Since J is maximal weighted-homogeneous ideal, we know that either
I({x}) is irrelevant, or I({x}) = I. Say that I({x}) is irrelevant, then {x} = V I({x}) = ∅ is a contradiction. So we must have that I({x}) = I. Thus V(I) = V I({x}) = {x} is a point.
Now let p = |p 0 : . . . : p n | ∈ P(a 0 , . . . , a n ) be a point and define the ideal J ⊳k a [x 0 , . . . , x n ] by
J = n k=0 k =i (p a k i x ai k − p ai k x a k i ) so that J is weighted-homogeneous with J ⊆ I({p}). Thus I({p}) = ∅, and hence I({p}) is relevant. Let a ⊳ k a [x 0 , . . . , x n ] be
Note 3.2.12
Since the bijection in Corollary 3.2.11 is only between non-empty weighted projective varieties and radical weighted-homogeneous relevant ideals, most of the theorems that we cover from now on concerning a weighted projective variety X will include the hypothesis that X is non-empty. Usually these theorems will be trivially true if X = ∅, but it is important to note that the proofs we give assume (if stated) that X = V(I) is non-empty and thus I is relevant.
Coordinate rings
We define the idea of the coordinate ring of a weighted projective variety in this section, but it won't be until Section 4.1 that we have a proper understand of it, or even much of a use.
Definition 3.3.1 [Weighted-homogeneous coordinate rings]
Let X = V(I) be a non-empty weighted projective variety. Then define the weightedhomogeneous coordinate ring of X to be
S(X) = k a [x 0 , . . . , x n ] I(X) .
If we write A(Y ) to mean the coordinate ring of an affine variety Y then
S(X) = k a [x 0 , . . . , x n ] I(X) = k[x 0 , . . . , x n ] I aff (X) = A(X).
That is, the weighted-homogeneous coordinate ring of a non-empty weighted projective variety is simply the coordinate ring of its affine cone, but with the a-weighted grading. It's important to realise that we can't really think of the elements of S(X) as polynomial functions on X, since 'evaluating a function at a point in weighted projective space' is not a well-defined concept in general. They can be thought of as polynomial functions on the affine cone though, but this isn't always much use, since a morphism of affine cones doesn't necessarily descend everywhere to a morphism of weighted projective varieties (this will be covered more in Section 4.2).
Equally important is the fact that isomorphic weighted projective varieties might have non-isomorphic weighted-homogeneous coordinate rings (again, covered in Section 4.2).
An algebraic approach 4.1 Explaining Proj with the Nullstellensatz
For this section we take for granted a knowledge of graded rings (see [1,Chapter 6]). Many of the theorems in this section could be stated and proved for general graded rings R, but we tend to consider only the case when R is also a finitely-generated k-algebra, to avoid straying too far into the world of schemes.
Recall Section 1.1.1 -when we say that a ring is graded we mean specifically Z 0 -graded.
The Proj construction is defined for any graded ring R, but here we only really define it for specific types of graded rings 19 , namely quotients of k a [x 0 , . . . , x n ] by radical a-weightedhomogeneous relevant ideals I ⊳ k a [x 0 , . . . , x n ], that is, coordinate rings S(V(I)). Note that requiring I to be radical simply means that k R, so that R is in a sense non-trivial.
Definition 4.1.1 [Proj of a finitely-generated k-algebra]
Let R be a finitely-generated k-algebra of the form
R = k[y 0 , . . . , y n ] = k a [x 0 , . . . , x n ] I (4.1.2)
where y i is the image of x i in the quotient by I (so wt y i = a i = wt x i ), and I is a radical a-weighted-homogeneous relevant ideal. We define the set 20 Proj R by
Proj R = {p ⊳ R | p is an a-weighted-homogeneous prime ideal with R + ⊆ p}
where R + = (y 0 , . . . , y n ) denotes the irrelevant ideal.
Note 4.1.3
The more general definition of Proj R, for any graded ring R with irrelevant ideal R + , is where we define the set
Proj R = {p ⊳ R | p is a homogeneous prime ideal with R + ⊆ p}. Example 4.1.4 [Proj k a [x 0 , . . . , x n ]]
Let's start off by looking at what seems like it might be the simplest example: when I = {0}. By Corollary 3.2.11 we know that weighted-homogeneous prime ideals p ⊳ k a [x 0 , . . . , x n ] correspond to irreducible varieties in P(a 0 , . . . , a n ), and that maximal weighted-homogeneous ideals correspond to points. So if we split up Proj k a [x 0 , . . . , x n ] into maximal and primebut-not-maximal weighted-homogeneous ideals and use this bijection coming from V and I then we can consider P(a 0 , . . . , a n ) as a set of points in weighted projective space and irreducible weighted projective varieties:
Projk a [x 0 , . . . , x n ] = {p ∈ P(a 0 , . . . , a n )} V(m) where m is maximal weighted-homogeneous ⊔ {X ⊆ P(a 0 , . . . , a n )}. V(p) where p is prime-but-not-maximal weighted-homogeneous
So if we just consider maximal ideals then Proj k a [x 0 , . . . , x n ] is simply P(a 0 , . . . , a n ), but when we throw in the other prime ideals as well we enrich the structure slightly: it is a set containing all the points of P(a 0 , . . . , a n ) as well as all the irreducible weighted projective varieties in P(a 0 , . . . , a n ). Note 4.1. 5 We could use this as an alternative definition of P(a 0 , . . . , a n ): P(a 0 , . . . , a n ) = Proj k a [x 0 , . . . , x n ].
In fact, we might as well do so from now on, but when we speak of 'points in P(a 0 , . . . , a n )' we still mean points in the old sense, and we still refer to the points of Proj k a [x 0 , . . . , x n ] that correspond to varieties as 'varieties'. This is an almost identical situation to when we define A n = Spec k[x 0 , . . . , x n ]. For more on this, see [16,Sections 1.3,1.6] or [18,Section 3.2] So it seems like just considering maximal ideals will give us pretty much the whole picture of what Proj R looks like, and the prime ideals will tell us what the varieties inside Proj R look like. We now make this rigorous.
Let R be a finitely-generated algebra as in Eq. (4.1.2) and p⊳R an a-weighted-homogeneous prime ideal not containing R + = (y 0 , . . . , y n ). Then p corresponds uniquely to the weightedhomogeneous prime idealp ⊳ k a [x 0 , . . . , x n ] with I ⊆p such that p =p/I. 21 Further, since (y 0 , . . . , y n ) ⊆ p we know that (x 0 , . . . , x n ) ⊆p 22 , i.e.p is relevant (sincep is prime and thus radical). The situation unfolds in the same way when m ⊳ R is a maximal weightedhomogeneous ideal, giving us a unique maximal weighted-homogeneous relevant ideal m ⊳ k a [x 0 , . . . , x n ] such that I ⊆m and m =m/I. 21 See [20, Chapter III, Section 8, Theorem 11]. 22 Proof by contradiction, using the fact that if we have ideals i ⊆ a ⊆ b then a/i ⊆ b/i By Corollary 3.2.11,m corresponds to a point pm = V(m) ∈ P(a 0 , . . . , a n ), but I ⊆m means that pm ⊆ V(I). Conversely, given any point in V(I) we see that it corresponds to a maximal weighted-homogeneous relevant ideal of k a [x 0 , . . . , x n ] containing I, and thus to a maximal weighted-homogeneous ideal of R not containing R + . Similarly, the inclusionreversing bijection tells us thatp corresponds to an irreducible weighted projective variety contained inside V(I), i.e. a subvariety of V(I), and vice versa.
So we have the bijective correspondence
{p ∈ Proj R} {Xp ⊆ V(I) | Xp is an irreducible subvariety} (4.1.6)
and, in particular, where we have the same enriched structure as above.
{m ∈ Proj R | m ⊳ R is maximal weighted-homogeneous} {pm ∈ V(I) | pm is a point}.
Proof. This is just using Eqs.
Morphisms between varieties
The idea of morphisms is more complicated than it might sound at first. What we cover here is but a brief part of the whole story, as we take only what we need. Due to time, we (apologetically) might skim over some details, and this section is intended to be more of a motivation 23 The way of making this precise is to use category theory, which turns out to have very exciting applications to algebraic geometry as a whole. There is no quick introduction to this (at least, not that the author can find), but any good book on scheme theory should cover it, but [18] is a particularly good text treating algebraic geometry after covering a reasonable chunk of category theory. Failing that, it seems highly unlikely that [8] wouldn't cover it (and the author apologises for this infuriatingly vague reference).
to read other sources than a complete guide. For more (and better) information, see [8,Chapter II,Section 2], where the real treatment is given using the language of schemes.
In summary: treat this section as a brief vacation from the usual rigour of mathematics to the land of allegory.
We now have some notion of varieties, but as of yet have no rigorous idea of how we should define morphisms between them. Taking a cue from the affine and straight projective cases we think that a preliminary definition could be a map given by a polynomial in each coordinate. But before we make this definition, a thought occurs to us: we've just formalised the underlying algebraic structure of these geometric objects, and in the affine case a morphism of the coordinate rings induces a morphism of the varieties. So let's take this as a definition for the moment and see where we can get with it.
i ) = f i ∈ k a [x 0 , . . . , x n ].
Then we define the morphism F of weighted projective varieties as
F # : X → Y x → |f 0 (x) : . . . : f n (x)|.
We might (and should) worry about whether or not Definition 4.2.1 is well defined. Is it always true that the f i never all vanish simultaneously? Is this map invariant under a different choice of representatives for the point x ∈ P(a 0 , . . . , a n )? Both of these questions can be answered satisfactorily, but we don't do so here, for reasons explained at the end of this section. Sweeping any and all problems of this sort under the proverbial rug, we march onwards.
Definition 4.2.2 [Isomorphism of weighted projective varieties]
Let F : X → Y be a morphism of weighted projective varieties. Then F is an isomorphism if there exists another morphism of weighted projective varieties G : Y → X such that
G • F = id X and F • G = id Y .
But here we hit what seems like a problem: different embeddings of the same variety should definitely be isomorphic by any sensible definition, but we will see that different embeddings might not necessarily have isomorphic coordinate rings. For example, we will see in Section 4.3 that
P 2 = Proj k[x, y] ∼ = Proj k[x, y] (2) = V(v 2 − uw) ⊂ P 3
but the former has coordinate ring k[x, y], and the latter has coordinate ring k[u, v, w]/(v 2 − uw). Since the number of generators is different in each, the two definitely can't be isomorphic as graded rings.
The way to solve this problem is to point out the following: not every morphism of varieties comes form a morphism of coordinate rings. So Definition 4.2.1 sounds great as a partial definition, i.e. that all maps of this form are indeed what we should call a morphism of varieties, but there are other maps that we should also call varieties. It turns out that listening to our original idea of polynomial maps would be sensible.
Definition 4.2.3 [Morphism of weighted projective varieties, attempt 2]
Let F : X → Y be a map of weighted projective varieties, where X ⊂ P(a 0 , . . . , a n ) and Y ⊂ P(b 0 , . . . , b m ). Then F is a morphism if it is a weighted-homogeneous polynomial in each coordinate. That is,
F : X → Y |x 0 : . . . : x n | → |F 0 (x 0 , . . . , x n ) : . . . : F m (x 0 , . . . , x n )| where F i ∈ k a [x 0 , . . . , x n ] is weighted-homogeneous.
Now we think about Definition 4.2.2. It still makes sense as a definition, but there is another way that we could maybe define an isomorphism, using the algebraic structure again. If we had some bijection between prime (and maximal) ideals of the two coordinate rings of our varieties that preserved enough information, such as inclusion, then the varieties should be isomorphic, since the Proj of their coordinate rings will be 'the same', in a sense. This happens as an example in Theorem 4.3.3.
It may seem like we are skirting around the issue of settling on a specific definition, and that's because we are. Really, the only times we will talk about morphisms in this paper is when we are talking about isomorphisms, and then we will come across just two cases:
(i) the underlying graded rings are isomorphic;
(ii) one of the underlying graded rings is a truncation of the other.
In case (i) it is very fair that, if two rings are isomorphic, then their Proj should be isomorphic. In case (ii), Theorem 4.3.3 will apply.
Unfortunately, since not much more theory than this is needed in this paper, not much more theory than this is covered in this paper. The real story is one of morphisms of schemes, and once again the author recommends the invaluable resource that is [8, Chapter II].
Truncation of graded rings
We know that a variety, in the sense we've been describing them so far, depends on its ambient space, and from our experiences with affine varieties we expect that we might be able to embed the same variety in different weighted projective spaces. A classical example is that of a Veronese embedding of a variety from P n → P m for some specific m n. In fact, this is the example that we study in Example 4.3.2.
So since Theorem 4.1.8 gave us a way of converting between algebra and geometry, it seems like we should be able to find some process that we can apply to our coordinate rings that corresponds to an embedding of the associated varieties.
R (d) = i 0 R di . So R (d)
is also a graded ring, with grading given by i. That is, an element that has degree di in R has degree i in R (d) . 24
Example 4.3.2
Let R = k[x, y] with the usual grading (i.e. wt x, y = 1). Then
R (2) = i 0 R 2i = i 0 {f ∈ k[x, y] | deg f = 2i}.
We note then that all polynomials in R of even degree (which are exactly those that are in R (d) ) are generated by x 2 , xy, y 2 . Thus
k[x, y] (2) = k[x 2 , xy, y 2 ].
Now we know that Proj k[x, y] = P(1, 1) = P 1 , but what is Proj k[x, y] (2) ? First, let's write the latter in a form that we know how to deal with:
k[x, y] (2) = k[x 2 , xy, y 2 ] ∼ = k[u, v, w] (uw − v 2 ) .
By Definition 4.3.1 we have that deg x 2 , xy, y 2 = 1, and so taking wt u, v, w = 1 gives us an isomorphism of graded rings. In particular, it's important to note that R ∼ = R (2) here, and in general R ∼ = R (d) . But now we can use Theorem 4.1.8:
Proj k[u, v, w] (uw − v 2 ) = V(uw − v 2 ) ⊆ P(1, 1, 1) = P 2 .
This is exactly the degree-2 Veronese embedding of P 1 ֒→ P 2 . So It turns out that the above example is more than just a lucky coincidence. We have two claims:
(i) Proj R ∼ = Proj R (d) for any graded ring R and d ∈ N;
(ii) Proj k[x 0 , . . . , x n ] (d) (so with wt x i = 1) corresponds to the degree-d Veronese embed- ding P n ֒→ P ( n+d d )−1 .
The second claim is slightly off-topic in a sense, since it is a fact concerning only straight projective space, but we can formulate it to deal with straight projective varieties too. It is a very nice example, but we unfortunately don't have the time to delve into it here any further.
The first claim is one that we can apply to our studies of weighted projective varieties, and so we study it now.
The first claim Theorem 4.3.3
Let R be a graded ring and d ∈ N. Then
Proj R ∼ = Proj R (d) .
Proof. All that we really use in this paper is that Proj R = Proj R (d) as sets. Since really they are endowed with so much more structure than we are covering here (namely, their structure as schemes) the actual proof of this statement uses some ideas and definitions that we haven't mentioned, and don't intend to, for the sake of time. One important thing that we don't show is that
R (d) [f −d ] = R[f −1 ] (d)
, and thus that in particular the degree-0 graded parts are equal. Rather than picking apart a standard proof and discarding the bits that we won't use, and hence potentially taking things out of context and missing the bigger picture, we provide here a sketch proof with commentary, lifted largely from [4,Exercise 9.5]. All of the statements and theorems concerning integral extensions can be found in [4,Chapter 4.4].
For a full proof of this statement see [17,Proposition 5.5.2], for example. And, of course, this (and so much more) is covered in [7,Proposition (2.4.7)].
First of all we note that there is an injective graded ring homomorphism R (d) ֒→ R corresponding to inclusion, but this is not usually an isomorphism. So rather than looking at the underlying rings themselves, we look instead at the structure of their prime ideals.
The above ring extension R (d) ֒→ R is in fact integral, and thus every prime ideal of R (d) is given by the restriction of some prime ideal in R to R (d) , but this is in general not be a bijective correspondence. However, it can be shown 25 that when we consider only weighted-homogeneous prime ideals, we do in fact end up with a bijection p → p ∩ R (d) from weighted-homogeneous prime ideals of R to those in R (d) .
We can use Theorem 4.3.3 to simplify certain weighted projective spaces or varieties in two different ways:
(a) reduce P(a 0 , . . . , a n ) to a well-formed weighted projective space P(a ′ 0 , . . . , a ′ n );
(b) embed P(a 0 , . . . , a n ) ֒→ P N for some large enough N ('straighten out' P(a 0 , . . . , a n )).
Both of these terms will be defined and explained next, before we approach an explicit example and work through it as best we can in Section 4.4. We start with well-formed weighted projective spaces in Section 4.3.2 and then deal with 'straightening out' P(a 0 , . . . , a n ) in Section 4.3.3. Really, the second is sort of a specific case of the first, since straight projective space is a well-formed weighted projective space, but in another sense it is different entirely, since we want to end up in a specific weighted projective space: straight projective space.
Well-formed weighted projective spaces Definition 4.3.4 [Well-formed weights]
We say that a weight a = (a 0 , . . . , a n ) is well-formed if any n − 1 of the a i are coprime. That is, gcd(a 0 , . . . , a i , . . . , a n ) = 1 for all 0 i n.
Definition 4.3.5 [Well-formed weighted projective space]
The weighted projective space P(a 0 , . . . , a n ) is said to be well-formed if the weight (a 0 , . . . , a n ) is well-formed.
Theorem 4.3.6
Given some weight a there is a well-formed weight a ′ such that P(a) ∼ = P(a ′ ). That is, any weighted projective space P(a 0 , . . . , a n ) is isomorphic to a well-formed weighted projective space P(a ′ 0 , . . . , a ′ n ).
Proof. Let R = k a [x 0 , . . . , x n ], so that Proj R = P(a 0 , . . . , a n ). Since we might as well assume that a is not already well-formed (otherwise the proof is trivial) we have two possible cases:
(i) there exists some common factor d of all the a i ;
(ii) a 0 , . . . , a n have no common factor, but there is some j such that a 0 , . . . , a j , . . . , a n have common factor d which is coprime to a j .
We use Theorem 4.3.3 for both cases.
In case (i) we have, for all 0 i n, that d | a i , and thus x i ∈ R dk for all k. Hence
R (d) = k a/d [x 0 , . . . , x n ]. Thus P(a 0 , . . . , a n ) = Proj k a [x 0 , . . . , x n ] ∼ = Proj k a/d [x 0 , . . . , x n ] = P a 0 d , .
. . , a n d .
In case (ii) we see that, since d is coprime to a j , the only a j term that will appear in R dk is a d j and its powers. So R (d) = k a/d [x 0 , . . . , x d j , . . . , x n ], and thus P(a 0 , . . . , a n ) = Proj k a [x 0 , . . . , x n ] ∼ = Proj k a/d [x 0 , . . . , x d j , . . . , x n ] = P a 0 d , . . . , a j , . . . , a n d .
So, given some weight a = (a 0 , . . . , a n ), if they all share some common factor d then we can use case (i) to divide all the a i through by d. We can repeat this until the gcd(a 0 , . . . , a n ) = 1. Then, if any n − 1 of the a i have some common factor d ′ , we can use case (ii) to divide a 0 , . . . , a j , . . . , a n through by d ′ until gcd(a 0 , . . . , a j , . . . , a n ) = 1.
So in light of Theorem 4.3.6 there is usually no loss in generality in assuming that a weighted projective space is well-formed, unless we care about the specific embedding. We will not always assume that P(a 0 , . . . , a n ) is always well-formed, and if we ever do then we will explicitly say so.
Example 4.3.7
Here are two particularly nice cases, the second of which will crop up again in Section 4.4:
• P(a, b) ∼ = P(1, b) ∼ = P(1, 1) = P 1 for any a, b ∈ N;
• P(ab, bc, ca) ∼ = P(b, bc, c) ∼ = P(1, c, c) ∼ = P(1, 1, 1) = P 2 for any a, b, c ∈ N (assumed to be coprime, without loss of generality).
4.3.3
Embedding P(a 0 , . . . , a n ) into P N A good source that also covers the same material as in this section, and where the author first read most of this, is [17,Section 5.5].
Now we take a look at using Theorem 4.3.3 to embed any weighted projective space (or variety) into P N for some large enough N . As we've already mentioned, really this is a specific case of Section 4.3.2, since the weight (1, . . . , 1) is well-formed, but in a sense it is also very different, since we are aiming for a specific well-formed weighted projective space. We start this section with a technical lemma. It turns out that Lemma 4.3.8 is the only new thing that we need to show that embedding into straight projective space is always possible.
Theorem 4.3.9 [Straightening out weighted projective space]
Let X = V(I) ⊆ P(a 0 , . . . , a n ) be an irreducible 27 non-empty weighted projective variety. Then there exists some N large enough, and some projective variety Y ⊆ P N , such that
X ∼ = Y ⊆ P N .
Proof. Let R = k a [x 0 , . . . , x n ]/I. Amongst other things, Theorem 4.3.3 tells us that, for any graded ring S and any d ∈ N, prime weighted-homogeneous ideals in S (d) not containing S (d) + are in exact correspondence with prime weighted-homogeneous ideals in S not containing S + . Thus, since I ⊳ k a [x 0 , . . . , x n ] is a prime weighted-homogeneous relevant ideal (X is non-empty) we know that it corresponds exactly to a prime weighted-homogeneous ideal J ⊳ k a [x 0 , . . . , x n ] (d) not containing k a [x 0 , . . . , x n ]
(d) + . That is, R = k a [x 0 , . . . , x n ] I =⇒ R (d) = k a [x 0 , . . . , x n ] (d) J 26
This theorem is actually true for a general graded ring R, but we state it here in the more specific case, since it is the only one that we use. 27 We include the hypothesis that X is irreducible here just to make the proof easier. In general, given some general weighted projective variety V , we can simply look at its irreducible components V i separately, which all embed into P N i for some N i . Then, since there is a natural embedding P N ֒→ P M for any N M , we can embed V into P N where N = max i {N i } by simply embedding P N i ֒→ P N for each i.
This argument does lack rigour at the end, when we simply 'put all the pieces back together', but we do not have time to cover it here unfortunately. The author does not know of a suitable reference for this proof, but is sure that one must exist somewhere, for what that's worth.
where J is prime weighted-homogeneous and such that V(J) ⊆ Proj k a [x 0 , . . . , x n ] (d) is non-empty (since J doesn't contain k a [x 0 , . . . , x n ] (d) + ). 28 By Lemma 4.3.8 we can find some d ∈ N such that R (d) is generated by R d . In R (d) the elements of R d have degree 1 by definition, and so
R (d) = k[y 0 , . . . , y N ] J
where wt y i = 1 for all i, and J is the image of I in R (d) , which is homogeneous (in the straight sense since a = (1, . . . , 1)) by the above argument. So Theorem 4.3.3 and Theorem 4.1.8 tell us that
X = Proj R ∼ = Proj R (d) = V(J) ⊆ P(1, . . . , 1) = P N .
We showed in Section 4.2 that our definition of isomorphisms agrees with the usual definition of isomorphisms between straight projective varieties. Thus we get the following corollary, which is really just Theorem 4.3.9 phrased in a different way.
Corollary 4.3.10
Let X ⊆ P(a 0 , . . . , a n ) be a non-empty weighted projective variety. Then X can also be thought of as a straight projective variety inside some P N .
A worked example
To check that we have a working understanding of Section 4.3 we now look at an explicit example. This also gives us a chance to see how ideals transform under truncation, and to maybe help clarify the proof of Theorem 4.3.9.
The example that we choose is from [17, Exercise 5, Section 6.5], and is a variation on [14, Example 3.7].
Once again, we point out that we are not really covering the whole picture here -we are dealing simply with the underlying topological spaces. As noted in Theorem 4.3.3, the best source for the gory details is probably [7, Proposition (2.4.7)].
Example 4.4.1
Let f = x 5 + y 3 + z 2 ∈ C (12,20,30) [x, y, z], so that f is weighted-homogeneous of degree 60. Compute
Proj
C (12,20,30)
[x, y, z] (f ) .
Solution. Since f is weighted-homogeneous, (f ) is a weighted-homogeneous relevant ideal.
Further, since f is irreducible and C (12,20,30) [x, y, z] is a UFD 29 the ideal (f ) is prime. By The isomorphism Proj R ∼ = Proj R (d) is induced by the inclusion R (d) ֒→ R (using the ideas from Section 4.2). How canonical this isomorphism is, however, depends on whether we are looking at case (i) or case (ii) from Theorem 4.3.6. That is, if d | a i for all i then R (d) is simply R with all the gradings divided through by d. So our weighted-homogeneous ideal I ⊳ k a [x 0 , . . . , x n ] If, however, we have that d | a i for all i = j, and gcd(d, a j ) = 1, then the isomorphism is a tad less simple. It comes from the correspondence of prime ideals mentioned in our proof of Theorem 4.3.3: weighted-homogeneous prime ideals p ⊳ R (d) correspond uniquely to weighted-homogeneous prime ideals p ′ ⊳ R (d) in such a way that, if f ∈ p, then f d ∈ p ′ . So our first isomorphism is a very natural one, since 12, 20, 30 all divide by 2:
Proj C (12,20,30)
[x, y, z] (f ) ∼ = Proj C (12,20,30) [x, y, z] (f )(2)
= Proj C (6,10,15)
[x, y, z] (f ) .
But from now on, every isomorphism falls into case (ii) -there is no common factor of all of the a i , only for pairs a i , a j . Let's look at the first such one.
By definition, R (ab) = (R (a) ) (b) , and so we are interested first in Proj C (6,10,15)
[x, y, z] (f ) call this S(5)
.
We first look at the 'numerator' of this quotient. Since y, z ∈ R 5k for all k ∈ N, and gcd(5, 6) = 1, we see that C (6,10,15) [x, y, z] (5) = C (6,2,3) [x 5 , y, z].
So let's turn now to the 'denominator'. The ideal in S (5) corresponding to (f ) should be (f 5 ), thus
S (5) = C (6,2,3) [x 5 , y, z] (f 5 ) ,
where f is weighted-homogeneous of degree 30 and so f 5 is weighted-homogeneous of degree 150. But V(f ) = V(f 5 ) ⊆ P(6, 10, 15) by the more general fact that V(g) = V(g k ) whenever 30 g is irreducible and k ∈ N. So Theorem 4. , 30 We might not need such a strong condition on g, but we lose nothing here by erring on the side of caution.
which does makes sense, as f is weighted-homogeneous of degree 30, and thus f ∈ C (6,2,3) [x 5 , y, z] is still weighted-homogeneous, now of degree 6. Finally for this first isomorphism, we simplify things a bit by using the isomorphism of graded rings
C (6,2,3) [x 5 , y, z] (x 5 + y 3 + z 2 ) ∼ = C (6,2,3) [r, s, t] (r + s 3 + t 2 ) .
Putting this all together gives us this composition of maps of rings that all have isomorphic Proj:
C (6,10,15)
[x, y, z] (x 5 + y 3 + z 2 ) S → C (6,10,15) [x, y, z] (x 5 + y 3 + z 2 ) (5) = C (6,2,3) [x 5 , y, z] (x 5 + y 3 + z 2 ) 5 S (5) → C (6,2,3) [x 5 , y, z] (x 5 + y 3 + z 2 ) ∼ = C (6,2,3) [r, s, t] (r + s 3 + t 2 )
S (5) .
Using the same process and notation as above, if we let T = S (5) then we can repeat this with T → T (2) → T (2) , where
T (2) = C (3,1,3) [u, v, w] (u + v 3 + w) . Finally, with U = T (2) we have U → U (3) → U (3) , where U (3) = C (1,1,1) [X, Y, Z] (X + Y + Z) .
So, at long last, we see that
Proj
C (12,20,30)
[x, y, z] (x 5 + y 3 + z 2 ) ∼ = Proj C (1,1,1) [X, Y, Z] (X + Y + Z) = V(X + Y + Z) sitting inside P(1,1,1)=P 2 ∼ = P 1 .
Now, having done this example, let's talk through what it means. First of all, although we have shown that our original weighted projective variety X = V(x 5 +y 3 +z 2 ) ⊂ P (12,20,30) is isomorphic to P 1 , it doesn't mean that it's exactly the same. The affine cone over P 1 is simply the plane A 2 , whereas the affine coneX over X is a degree-5 hypersurface inside A 3 . It turns out in fact that X is a rather special singularity, see [14,Example 3.7] for more information, since it isn't too relevant here (but it is very interesting). 31 Another thing to note is that we travelled down the algebraic path in our solution of Example 4.4.1, but as we might have expected, we could have instead followed a more geometric one. Eq. (3.1.8) tells us that X could be thought of as the quotient variety
X = V aff (x 5 + y 3 + z 2 ) G m ⊆ A 3 \ {0} G m ,
and so we might have tried to construct this quotient explicitly in an attempt to understand the structure of X. We chose not to, because here the algebraic approach gives us a nice way of using all the things that we've found out so far, and because it is arguably much slicker.
But we could also think of weighted projective space as a quotient of straight projective space, and construct X as a quotient of a straight projective variety. What do we mean by this? Well, we claim that we have quotient maps
P n A n+1 \ {0} P(a 0 , . . . , a n ) n i=0 µ a i G (a) m G (1,...,1) m (4.4.2)
where the labels on the arrows are the groups that we quotient by to obtain the surjection. We study at a slightly more specific case of this in Section 5 (looking only at plane curves) but the general idea stays the same. Up until now we have been studying the quotient along the bottom of Eq. (4.4.2), and the upper-left quotient is a specific case of it, giving the usual well-understood case of projective space. The quotient on the right hasn't really been mentioned at all yet, but it comes in useful if we want to use what facts we know about straight projective space (which are sometimes 'nicer' than those about affine space) to gain some know-how about weighted projective spaces.
Plane curves in weighted projective space
In this section we assume familiarity with some of the fundamentals of Riemann surfaces and maps between them (see [12,Chapters 1,2]). We also assume knowledge of orbit spaces, covering spaces, and some other concepts from topology, though these are usually explained when used. Finally, we assume some facts about algebraic curves, though these are stated before being used.
Note 5.0.3
In this section (Section 5) only we write P = P(a 0 , a 1 ,
a 2 ), S = k a [x 0 , x 1 , x 2 ], and f (x) = f (x 0 , x 1 , x 2 )
. But the weight a = (a 0 , a 1 , a 2 ) and the indeterminates x 0 , x 1 , x 2 are still lurking about in the background, and when we say 'weighted-homogeneous' we still mean 'aweighted-homogeneous'. Note that when we write P k we mean projective k-space, as per usual, and we will write P 1 for the projective line, so there should be no ambiguity in our use of P to denote P(a 0 , a 1 , a 2 ). Now seems like a good time to take a look at a specific family of weighted projective varieties, namely plane curves, so that we don't lose our geometric intuition, and so that we have some (comparatively) concrete examples to hold on to and examine. As mentioned at the end of Section 4, we will half-break our promise made in Section 1.1: we assume some prior knowledge of straight projective algebraic geometry, and so don't deal with it as a special case of weighted projective algebraic geometry. Instead, we now shift our viewpoint slightly to think of weighted projective space as a quotient of straight projective space. By doing so, we also get a change to take a break from the algebra side of things and to work primarily with the geometry of these objects that we're studying.
An important thing to be aware of is that the path we follow here is almost certainly much much longer than it needs to be, but (in the view of the author) we uncovers plenty of nice facts along the way, and also develop multiple ways of viewing these objects.
Definition 5.0.4 [Plane curves in weighted projective space]
Let f = f (x 0 , x 1 , x 2 ) ∈ S be a weighted-homogeneous degree d polynomial with no repeated factors. 32 Then a 1 , a 2 ). We say that a plane curve C is irreducible if f has no non-constant factors apart from scalar multiples of itself, since then C cannot be written as a non-trivial union of other plane curves (using Lemma 3.1.4).
C f = V(f ) ⊆ P is a degree-d plane curve in P(a 0 ,
Definition 5.0.5 [Singular points]
Let f ∈ k a [x 0 , x 1 , x 2 ] be a degree-d weighted-homogeneous polynomial. Then we say that
p = (p 0 , p 1 , p 2 ) ∈ A 3 \ {0} is a singular point of f if ∂f ∂x 0 p = ∂f ∂x 1 p = ∂f ∂x 2 p = 0.
We say that f is non-singular if it has no singular points. Similarly, we say that a plane curve C = C f is non-singular if its defining polynomial 33 f is either non-singular, or singular only at points outside of C, i.e. only at points p such that f (p) = 0.
When a = (1, 1, 1), i.e. in the straight case, we see that our definition of plane curves is exactly the same as the usual definition for projective plane curves. We now state a fundamental fact about plane curves in straight projective space.
Some facts about different notions of quotients
In this subsection we state, and cite proofs for, a lot of technical lemmas that we will use in Section 5.2. We can split these into three types by looking at what notion of 'quotient' they 32 The reason that we say this is really just to simplify things without losing generality. We already know that V(f k ) = V(f ), and in a similar way we see that V(f k g) = V(f g). So we might as well assume that our polynomial has no repeated factors, but it isn't entirely necessary for our purposes. 33 So if we dropped the requirement that polynomials have no repeated factors then we'd need to specify which of the infinitely-many defining polynomials we mean -namely the one with no repeated factors. concern: topological spaces, Riemann surfaces, and projective GIT quotients.
We have a few notational notes before we start that seemed rather pointless to put in Section 1.1.1 since they are only really used here. Given a group action G on some space X we write G p for a point p ∈ X to mean the stabiliser subgroup {g ∈ G | g · p = p}. We write σ n to mean a general n-th root of unity (so σ n = ω k n for some 0 k < n). Finally, with a holomorphic map f : X → Y between Riemann surfaces we write mult p (f ) to mean the ramification index, as in [12, Chapter II, Definition 4.2].
Topological quotients (orbit spaces) Lemma [Quotients preserve compactness]
Let X be a compact topological space and G some finite group acting on X. Then the orbit space X/G is a compact topological space.
Proof. This follows from the standard fact that a continuous image of a compact space is compact, and that the quotient map is continuous (by definition of the quotient topology on the quotient space).
Quotients of Riemann surfaces by group actions Definition [Group actions on a Riemann surface]
Let G be a group acting on a Riemann surface X. Then we say that the action of G is
• holomorphic if the bijection ϕ g : X → X given by x → g · x is holomorphic for all g ∈ G;
• effective if the kernel K = {g ∈ G | g · x = x for all x ∈ X} is trivial.
Theorem 5.1.3 [Quotient of a Riemann surface by a group action]
Let G be a finite group acting holomorphically and effectively on a Riemann surface X. Then we can endow the orbit space X/G with the structure of a Riemann surface. Moreover, the quotient map ϑ : X → X/G is holomorphic of degree |G| and mult x (ϑ) = |G x | for any point x ∈ X.
Proof. See [12, Chapter III, Theorem 3.4].
Projective GIT quotients
All of the following is taken from [9], but has been phrased here slightly differently, and in a different order, just to avoid getting too carried away with the vast subject of geometric invariant theory (GIT). We also oversimplify the background machinery wildly, so do check [9,Chapter 4] for the whole story.
Definition 5.1.4 [Linear action of reductive groups on projective varieties]
Let a reductive 34 group G act on a projective variety X ⊆ P n . Then its action is said to be linear if G acts via a homomorphism G → GL(n + 1).
Definition 5.1.5 [Projective GIT quotient ([9, Definition 4.6])]
Let G be a reductive group with a linear action on a projective variety X ⊂ P n . We define the projective GIT quotient variety X/ /G to be the projective variety given by Proj S(X) G , where S(X) is the homogeneous coordinate ring of X.
φ : X ։ X/ /G is said to be geometric if the preimage φ −1 ([x]) of each point [x] ∈ X/ /G is a single orbit in X.
So if φ : X ։ X/ /G is a geometric projective GIT quotient then X/ /G is simply the topological quotient (i.e. the orbit space) X/G. In particular then, X/ /G naturally has the quotient topology coming from the quotient map X → X/G.
Weighted projective plane curves as Riemann surfaces
We now make the assumption that our weighted projective space P is well-formed so that, in particular, the a i are pairwise coprime. Note that we don't lose much generality by doing this, since Theorem 4.3.6 tells us that any weighted projective space is isomorphic to a wellformed one. The only information that we lose by passing to this isomorphic copy is the specific embedding of our original variety, but here we are much less interested in the embedding of plane curves and much more so in their intrinsic nature.
Lemma 5.2.1
Define 36 the homomorphism of graded rings
π # : k a [x 0 , x 1 , x 2 ] → k[y a0 0 , y a1 1 , y a2 2 ] x i → y ai i . Let f ∈ k a [x 0 , x 1 , x 2 ] be weighted-homogeneous of degree d. Then π # (f ) is homogeneous of degree d.
Proof. This follows from the definition of the degree of a weighted-homogeneous polynomial (Definition 3.0.7).
Note 5.2.2
We try to be consistent with the convention that the a-weighted polynomial ring has indeterminates x i and the usual polynomial ring has indeterminates y i , but sometimes we slip up, and sometimes for a good reason (to avoid unnecessary complication with notation at times). It is just a choice of notation, so doesn't affect that maths at all, but can be confusing. Just be aware. 35 Really the map is only defined on X ss ⊂ X, where X ss is a certain subset of X. Again though, this is all much beyond what is needed here. 36 The reason for this choice of notation is that we will eventually show that π # is the pushforward of a quotient map π.
Definition 5.2.3 [Straight cover]
Given some plane curve C = C f ∈ P we define its straight cover C as the straight projective variety
C = C f = V(π # (f )) ⊂ P 2 .
Note that this map is well defined by Lemma 5.2.1.
We need to explain why we give this variety C the name 'straight cover', since it is a very suggestive name. The rest of this section is sort of dedicated to showing why we use this name.
Example 5.2.4
Let f (x, y, z) = x 4 + y 4 + z 2 + xyz ∈ k (1,1,2) [x, y, z] be weighted-homogeneous of degree-4, giving us the plane curve 1, 2).
C = C f = V(x 4 + y 4 + z 2 + xyz) ⊂ P(1,
Then f = π # (f ) = x 4 + y 4 + z 4 + xyz 2 ∈ k[x, y, z] is also degree-4 homogeneous, and
C = C f = V(x 4 + y 4 + z 4 + xyz 2 ) ⊂ P 2 .
It turns out that, for a nice enough plane curve C f , the straight cover C f is non-singular -a fact which we now state and prove, as well as saying what exactly we mean by 'nice enough'.
Lemma 5.2.5
Let f ∈ k a [x 0 , x 1 , x 2 ] be weighted-homogeneous and non-singular. Then f = π # (f ) is homogeneous and non-singular. Equivalently, if we have some non-singular plane curve C f ⊂ P then its straight cover C f ⊂ P 2 is also a non-singular plane curve.
Proof. Define m i (x i ) = x ai i , so that f (x 0 , x 1 , x 2 ) = f (m 0 (x 0 ), m 1 (x 1 ), m 2 (x 2 )
). Then the chain rule tells us that ∂f ∂x i xi=pi = ∂f ∂m i xi=pi ∂m i ∂x i xi=pi for any p = [p 0 : p 1 : p 2 ] ∈ P 2 . So if we can show that neither of the ∂f ∂mi or the ∂mi ∂xi all simultaneously vanish at any point in C f then we know that C f is non-singular.
If a i = 1 for any i then ∂mi ∂xi = 1, and so we cannot ever have all ∂mi ∂xi vanishing simultaneously at any point in P 2 . If a i > 1 for all i then ∂mi ∂xi = a i x ai−1 i , and this is zero if and only if x i = 0, so the only 'point' in P 2 at which all ∂mi ∂xi vanish is [0 : 0 : 0], but this is not a point in P 2 . Thus the ∂mi ∂xi cannot all simultaneously vanish anywhere on P 2 , or thus anywhere in C f . Now, say for a contradiction that p = [p 0 : p 1 : p 2 ] ∈ C f is such that all of the ∂f ∂mi simultaneously vanish at x i = p i . Then p ′ = |p a0 0 : p a1 1 : p a2 2 | ∈ P is such that p ′ ∈ C f (since f (p ′ ) = f (p) = 0) and all of the ∂f ∂xi vanish at x i = p i .
Lemma 5.2.6
Let C = C f ⊂ P be a non-singular plane curve and C its straight cover. Let G = µ a0 ×µ a1 ×µ a2 and define an action of G on C by g · y = (σ a0 , σ a1 , σ a2 ) · [y 0 : y 1 : y 2 ] = [σ a0 y 0 : σ a1 y 1 : σ a2 y 2 ].
Then (i) the orbit space C/G is a compact Riemann surface;
(ii) the quotient map ϑ : C ։ C/G is holomorphic of degree a 0 a 1 a 2 ;
(iii) mult y (ϑ) = |G y | for any point y ∈ C.
Proof. We have already done most of the hard work for this proof, so we just need to fit all the pieces together. Lemmas 5.2.5 and 5.0.6 say that C is a compact Riemann surface, so Lemma 5.1.1 tells us that Y /G is a compact topological space. Theorem 5.1.3 gives us the rest of the claims assuming that we can show that G acts holomorphically and effectively (since it is finite of order a 0 a 1 a 2 ).
The kernel of the action of G on C is K = {g ∈ G | g · y = y for all y ∈ C} but since we have assumed that a is well-formed, and thus that the a i are all pairwise coprime, we know that µ ai ∩ µ aj = {1} for i = j. By definition, g · y = y for all y ∈ C if and only if g = (λ, λ, λ) for some λ ∈ k \ {0}. So the above comment tells us that no g ∈ G is of this form apart from (1, 1, 1), which is the identity in G. Thus the action is effective, since the kernel is trivial. As for the map ϕ g being holomorphic, this follows straight away from the fact that it is an algebraic map. More specifically, it is simply a polynomial map with constant coefficients.
Lemma 5.2.7
Let C = C f ⊂ P be a non-singular plane curve and C ⊂ P 2 be its straight cover, which is also a non-singular plane curve by Lemma 5.2.5. Let G = µ a0 × µ a1 × µ a2 act on C as in Lemma 5.2.6. Then this induces an action of G on the homogeneous coordinate ring S(C), and we have a well-defined GIT quotient
ϕ : C = Proj S(C) ։ Proj S(C) G = C/ /G.
Proof. By Definition 5.1.5, we need to show that G is reductive and has linear action on the homogeneous coordinate ring S(C). Now G is reductive by definition, since it is finite. Further, the action is linear, since it acts diagonally. That is, (σ a0 , σ a1 , σ a2 ) · [y 0 : y 1 :
y 2 ] = σ a0 0 0 0 σ a1 0 0 0 σ a2 y 0 y 1 y 2 .
So we have the well-defined GIT quotient C/ /G = Proj S(C) G , but what does this look like?
The induced action of G on k[y 0 , y 1 , y 2 ] is given by (σ a0 , σ a1 , σ a2 ) · f (y 0 , y 1 , y 2 ) = f (σ a0 y 0 , σ a1 y 1 , σ a2 y 2 ).
Recall that we have assumed that a is well formed, and hence that the a i are all pairwise coprime, so µ ai ∩ µ aj = {1} for i = j. Thus
S(C) G = k[y 0 , y 1 , y 2 ] (f ) G = k[y a0 0 , y a1 1 , y a2 2 ] (f )
since f is already a polynomial in y ai i by definition, and
C/ /G = Proj k[y a0 0 , y a1 1 , y a2 2 ] (f ) .
This section so far has been not much more than a wall of text consisting solely of definitions, lemmas, and proofs, so let's now take a break and have a look at what we've actually discovered and defined.
Given some non-singular plane curve C = C f ⊂ P, by using its straight cover C = C f and the finite group G = µ a0 × µ a1 × µ a2 we can build two more objects, giving us three in total, including C. We later claim (Corollary 5.2.10) that all three actually give us the same thing, and so we have three ways of looking at non-singular plane curves. The three objects we have are (i) the non-singular plane curve C = C f ⊂ P;
(ii) a quotient map ϑ : C ։ C/G of compact Riemann surfaces (Lemma 5.2.6));
(iii) a GIT quotient map ϕ : C ։ C/ /G of projective varieties (Lemma 5.2.7).
Theorem 5.2.8
The objects (i) and (iii) are equivalent. That is, C ∼ = C/ /G as weighted projective varieties.
Proof. It turns out that, not only are these varieties isomorphic, they are 'nicely' isomorphic. That is, the isomorphism of varieties arises from an isomorphism of graded rings, namely
S(C) = k a [x 0 , x 1 , x 2 ] (f ) → k[y a0 0 , y a1 1 , y a2 2 ] (f ) = S(C/ /G) x i → y ai i .
This induces the desired isomorphism of weighted projective varieties C ∼ = C/ /G. In this section we denote this isomorphism by ψ :
C/ /G → C, where ψ : orbit G ([p 0 : p 1 : p 2 ]) → [p a0 0 : p a1 1 : p a2 2 ].
Lemma 5.2.9
The objects (iii) and (ii) are equivalent. That is, the GIT quotient C/ /G is exactly the orbit space C/G, which has all the structure of a compact Riemann surface.
Proof. All that this lemma is really saying is that the quotient C/ /G is geometric, as defined in Definition 5.1.6. So we need to show that the preimage of each point in C/ /G is a single orbit in C.
Let p ∈ C/ /G be a point. By Theorem 5.2.8 we know that we can think of C/ /G as the plane curve C ⊂ P using the isomorphism ψ : C/ /G → C, and so we can think of p as the point ψ(p) = |p 0 : p 1 : p 2 | ∈ P. Then We can now state and prove the main result of Section 5.2, which happens to be no more than a corollary of all the technical heavy lifting we've done already.
ϕ −1 (p) = (ϕ −1 • ψ −1 )(|p 0 : p 1 : p 2 |) = {[σ 0 p 1/a0 0 : σ 1 p 1/a1 1 : σ 2 p 1/a2 2 ] | σ i ∈ µ ai } = {g · [p 1/a0 0 : p 1/a1 1 : p 1/a2 2 ] | g ∈ G}
Corollary 5.2.10
Let C = C f ⊂ P be a non-singular plane curve and C ⊂ P 2 its straight cover. Then the map π : C → C given by π : [y 0 : y 1 : y 2 ] → |y a0 0 : y a1 1 : y a2 2 | is a surjective map of compact Riemann surfaces. Further, π is holomorphic of degree a 0 a 1 a 2 and such that mult y (π) = |G y | for any y ∈ C.
Proof. Although there is a lot of notation here, due to all these isomorphisms and quotient maps, the idea behind this is very simple, and has been pretty much explained by what we have done so far. This is just putting all the pieces together and chasing notation around.
First we look at the map π # : k a [x 0 , . . . , x n ] → k[y a0 0 , y a1 1 , y a2 2 ] from Lemma 5.2.1. We can represent π # by polynomials Π i ∈ k[y a0 0 , y a1 1 , y a2 2 ] where Π i = π(x i ) = y ai i . This induces a map π : Proj k[y a0 0 , y a1 1 , y a2 2 ] → Proj k a [x 0 , . . . , x n ] given by π : p = |p 0 : p 1 : p 2 | → |Π 0 (p) : Π 1 (p) : Π 2 (p)| = |p a0 0 : p a1 1 : p a2 2 |. Lemma 5.2.9 says that C/ /G = C/G, and so ϕ = ϑ, since they both map a point in C to its G-orbit. Then we use the isomorphism ψ : C/G = C/ /G ։ C to define the composition ϑ • ψ which has all the properties of ϑ from Lemma 5.2.6. But ϑ • ψ : [y 0 : y 1 : y 2 ] → orbit G ([y 0 : y 1 : y 2 ]) → |y a0 0 : y a1 1 : y a2 2 |. Thus ϑ • ψ = π, and using Lemma 5.2.6, π is a holomorphic map of degree a 0 a 1 a 2 between compact Riemann surfaces such that mult y (π) = |G y | for any y ∈ C. From all of the above we also get a bonus corollary for free. Even though Corollary 4.3.10 can give us the same result, we mention it here anyway, just to show that this could be an alternative path of getting to it.
Corollary 5.2.12
Let C ⊂ P be a non-singular plane curve. Then C is (isomorphic to) a projective variety.
Proof. We know that C ∼ = C/ /G by Theorem 5.2.8, and projective GIT quotients are, in particular, projective varieties (Definition 5.1.5).
What happens to the affine patches?
We briefly discuss the story of the affine patches now, though we don't dedicate too much time to it, since it is slightly irrelevant compared to the results we move on to state and prove for the rest of the paper. But it is still interesting enough to be worth a mention.
It is natural to think that there should be some way of 'ungluing' a plane curve C ⊂ P into its three affine patches, and then gluing them together in some standard way (homogenising the polynomials in such a way that they agree) to obtain the straight cover C. This is slightly complicated, though, by the fact that the covering affine patches map onto the quotient affine patches by π i (that is, the quotient map for a µ ai action), but the straight cover maps onto the plane curve by π = π ijk (that is, the quotient map for a µ a0 × µ a1 × µ a2 action). In a sense, we have to split C up into its quotient affine patches, map back up to the covering affine patches, and then factor through some sort of patches, also in the affine plane, before finally gluing them back together.
Writing D i to mean the quotient affine patches, D i to mean the covering affine patches, D i to mean the covers of these affine patches we can draw a diagram of the situation: see Figure 5.2.13. We hope that the diagram is at least reasonably helpful and understandable, though we do stress its irrelevance to what is to follow.
A 2 V i P 2 D i C i C A 2 D i A i U i P • D i C i C • π jk ∼ κ π π jk ∼ κ π πi πi ∼ ι ∼ ι inclusion restriction (5.2.13)
The degree-genus formula for weighted projective varieties
In this subsection we often talk of whether or not a polynomial has an x i term, or an x i x k j monomial, or something similar. When we say this, we implicitly mean a non-zero term, whether or not we mention it explicitly. So if, for example, we say that f has a x i term, then we mean that f = λx i + g where λ ∈ k \ {0} and g is some other polynomial in the x i .
What does sufficiently general mean?
In the usual study of plane curves one tends to ignore the edge cases where a certain class of curve is poorly behaved. For example, all conics in P 2 are equivalent via a projective transformation to x 2 + y 2 + z 2 , apart from the singular cases that are equivalent to one of x 2 or x 2 + y 2 . In a sense, it's natural to think that these two examples will be difficult, because we are trying to study degree-2 polynomials in k[x, y, z], but x 2 + y 2 doesn't even have a non-zero z term. It looks like it might be more at home as being classed as a degree-2 polynomial in k[x, y], and x 2 is an even more extreme case. None of this is particularly rigorous, but it gives us a good idea of the restrictions we might want to place upon our polynomials to ensure that they define sufficiently nice plane curve. That is, we want our polynomials to be sufficiently general in some sense.
Definition 5.3.1 [Sufficiently general]
A degree-d weighted-homogeneous polynomial f ∈ k a [x 0 , x 1 , x 2 ]
is sufficiently general if f satisfies the following for each i: We also place some restrictions on the values of d and all the a i :
• d 2; 37 • d a i ; 38
• if a i ∤ d then there exists some j = i such that a i | (d − a j ). 39
A plane curve C = C f ⊂ P is said to be sufficiently general if its defining polynomial f is sufficiently general. 37 Linear polynomials can only involve monomials x i where a i = 1, and so we just end up studying linear polynomials in straight projective space. Hence we don't really lose too much generality in excluding these cases. 38 So in the cases where a j = 1 for some j (that is, all of the not-straight cases) this subsumes the above requirement, since d a j 2 automatically. The reason for this requirement is simply that, if a j > d for some j, then we won't have any x j terms in f , and so we don't really have a 'proper' polynomial in all the x j . 39 This ensures that f can satisfy the second condition: if a i ∤ d then it contains an x j x d−a j i term. In fact, this is arguably the most important condition -we've already required that f has no non-trivial non-constant factors (Definition 5.0.4), and when we combine that with this requirement we see that we satisfy the hypotheses of [10,8.4 Corollary]. That is, we are asking that our plane curves be quasismooth.
The main reason for Definition 5.3.1 is the quasismoothness that it guarantees (see the footnotes), and that explains most of the inner workings of what follows from here on in.
For a more thorough treatment, see [10,Section 8].
One of the reasons that we settle upon Definition 5.3.1 is because it gives us Lemma 5.3.2, which will come in use later. But we also want to make sure that we haven't restricted ourselves so much that we end up studying a tiny subset of all possible plane curves. Another way of looking at Definition 5.3.1 is that, if we say a polynomial is of degree-d then we want it to at least have an x d/ai i terms for all i. Since this might not always be possible, depending on the weighting, we sort of say that if this can't happen then we want the next best thing. (ii) if a i ∤ d then every monomial containing a non-trivial power of x i also contains some non-trivial power of x j for j = i, and so every term of f vanishes at p i , thus p i ∈ C.
Lemma 5.3.2 Let f ∈ k a [x 0 , x 1 , x 2 ] be
Really though, condition (ii) in Definition 5.3.1 is superfluous if we assume that f is also non-singular. The reason that we have that condition is to ensure that if f is sufficiently general and if any of the p i are roots of f then they are not singular points. So if p i ∈ C then C is not singular at p i . But why do we make this restriction?
The chain rule can tell us that f is non-singular if f (1, x, y), f (x, 1, y), and f (x, y, 1) are non-singular for all (x, y) ∈ A 2 \ {(0, 0)} and f is non-singular at each of the p i . So, by the above (which we state and prove in Lemma 5.3.3), it would suffice to check that f is non-singular on the U i to show that f is non-singular on the whole of P. This isn't a fact to which we appeal at all, and so we don't give all the gory details of using the chain rule, but it helps to reassure us slightly that our choice of definition might not be too bad -if we have some non-singular affine curves that glue together to make a weighted projective curve then the resulting curve will also be non-singular. Proof. Assume that p i ∈ C, so that a i ∤ d. By our definition then, f contains an x j x m i term for some j = i, where m = (d − a j )/a i . So ∂f ∂xj contains an x m i term. Further, every other monomial in ∂f ∂xj either has x l j for some l 1 or is x l k for some l 1. Either way, every other monomial vanishes at p i and the only remaining term is x m i , which evaluates to some non-zero scalar. Thus ∂f ∂xj pi = 0, and so p i is not a singular point of f . Finally, we state and prove one more technical lemma here. This one is seemingly unrelated to anything we have mentioned so far, but ends up being a key part in the proof of Theorem 5.3.7, so we get it out of the way now. The proof is messy but, in essence, simple.
Lemma 5.3.4
Let f ∈ k a [x 0 , x 1 , x 2 ] be a sufficiently-general degree-d non-singular weighted-homogeneous polynomial. Then f = π # (f ) (as defined in Lemma 5.2.1) is such that all of
f (0, 1, λ), f (λ, 0, 1), f (1, λ, 0)
have no repeated roots when considered as polynomials in λ.
Proof. We prove that f (0, 1, λ) has no repeated roots, and the other two claims follow in exactly the same manner, mutatis mutandis.
Say for a contradiction that f (0, 1, λ) has a multiple root λ = c, so that (λ−c) 2 | f (0, 1, λ).
Also, since f (0, 1, c) = 0 we know that [0 : 1 : c] ∈ C. We aim to show that [0 : 1 : c] is a singular point of f , contradicting the fact that f is non-singular. For easier reading we split this proof up into three parts -one for each partial derivative of f .
(i) Now, if y = 0 then y d f (0, 1, z/y) = f (0, y, z), and so
f (x, y, z) = f (0, y, z) + d i=1 c i x i y j z k = y d f (0, 1, λ/y) + d i=1 c i x i y j z k = y d z y − c 2 i(x, y, z) + xh(x, y, z) = (z − cy) 2 g(x, y, z) + xh(x, y, z).
where c i ∈ k \ {0} and g, h, i are polynomials in x, y, z. Thus (ii) Next we examine two separate cases:
if c = 0 then (z − cy) 2 = c 2 (y − z/c) 2 , and so (y − z/c) | ∂f ∂y ; if c = 0 then f = z 2 g + xh, and so ∂f ∂y = z 2 ∂g ∂y + x ∂h ∂y .
In both cases we see that ∂f ∂y evaluates to 0 at [x : y : z] = [0 : 1 : c].
(iii) Finally,
∂f ∂x = (z − cy) 2 ∂g ∂x + x ∂h ∂x + h.
Riemann-Hurwitz and the usual degree-genus formula
Here is where all the hard work in Section 5.2 pays of. With our idea of constructing a Riemann surface C ⊂ P 2 for a plane curve C ⊂ P, along with a surjective map of Riemann surfaces π : C ։ C with particularly nice properties (mainly the fact that it is holomorphic, but all of Corollary 5.2.10 comes in useful), we can now look at using one of the particularly powerful theorems from the study of Riemann surfaces: the Riemann-Hurwitz formula.
Theorem 5.3.5 [Riemann-Hurwitz formula]
Let R, S be compact Riemann surfaces and f : R → S be a non-constant holomorphic map. Then
2g R − 2 = deg f (2g S − 2) − b(f ) where b(f ) is the branching index b(f ) = s∈S deg f − |f −1 (s)| = s∈S r∈f −1 (s) (v f (r) − 1)
and g X = 1 2 (χ(X) + 2) is the genus of the Riemann surface X.
Proof. See almost any text on Riemann surfaces, e.g. [12,Chapter II,Theorem 4.16].
Theorem 5.3.6 [Degree-genus formula for straight projective plane curves]
Let C ⊂ P 2 be a non-singular degree-d plane curve. Then C is a Riemann surface with genus g C given by
g C = (d − 1)(d − 2) 2 .
Proof. The fact that C is a Riemann surface has already been proved in Lemma 5.0.6, and [11,Corollary 4.19] gives us the degree-genus formula.
So we take our non-singular plane curve C ⊂ P given by some sufficiently-general degree-d weighted-homogeneous polynomial f ∈ k a [x 0 , x 1 , x 2 ], construct its straight cover C ⊂ P 2 along with a quotient map π : C ։ C. Now C is also a non-singular plane curve defined by a homogeneous polynomial of degree-d, so we have our usual degree-genus formula for C. But then the Riemann-Hurwitz formula tells us the genus of C in terms of its degree d. If we write this all out properly then we expect to get some sort of degree-genus formula for non-singular sufficiently-general plane curves in P, and that is exactly what we get.
Theorem 5.3.7 [Degree-genus formula]
Compare and contrast with [10,Theorem 12.2]. Let C = C f ⊂ P(a 0 , a 1 , a 2 ) be a nonsingular plane curve where f is weighted-homogeneous of degree d and sufficiently general, in the sense of Definition 5.3.1. Then, using the map π as defined in Corollary 5.2.10,
g C = 1 a 0 a 1 a 2 (d − 1)(d − 2) 2 − b(π) 2 + 1 − a 0 a 1 a 2
where the branching index b(π) is given by
b(π) = (d − 1) 3 i=1 (a i − 1) + 3 i=1 a i − 1 if a i | d; a 0 a 1 a 2 − 1 if a i ∤ d.
Proof. First we appeal to Corollary 5.2.10. This, along with the Riemann-Hurwitz formula and the degree-genus formula for straight projective plane curves (Theorems 5.3.5 and 5.3.6), tells us that
2 (d − 1)(d − 2) 2 − 2 = a 0 a 1 a 2 (2g C − 2) + b(πb(π) = p∈C (|G p | − 1) = k i=1 a 0 a 1 a 2 |π −1 (y i )| − 1
where y 1 , . . . , y k are all the branch points of π. So all that remains to do is find and classify all of the branch points of π.
Where would we expect to find branch points? Well, after a little bit of thinking, we see that the only branch points are those who have some zero coordinate, since that is the only way that the size of the preimage can drop. That is,
[σ 0 x 1/a0 0 : σ 1 x 1/a1 1 : σ 2 x 1/a2 2 ] | σ i ∈ µ ai
π −1 (|x0:x1:x2|) < a 0 a 1 a 2 deg π ⇐⇒ x i = 0 for some i where we once again use the assumption that a is well-formed. 40 The points with some zero coordinates (i.e. the branch points y i ) split into two disjoint types: those with just one zero coordinate, and those with two. We introduce some temporary notation: 41
G i = {|x 0 : x 1 : x 2 | ∈ C : x i = 0, x i+1 , x i+2 = 0}.
So with p i as defined in Lemma 5.3.2 we can partition all of the branch points y i into four disjoint sets
{y 1 , . . . , y k } = G 0 ⊔ G 1 ⊔ G 2 ⊔ {p 0 , p 1 , p 2 }.
We look first at the G i : we can see that if x ∈ G i then |π −1 (x)| = a i+1 a i+2 = a 0 a 1 a 2 /a i . So we know how each of the points in G i contribute to b(π) and we are only left wondering how many points there are in each G i . Let's consider the example of G 0 . Without loss of generality we can write points in G 0 as |0 : 1 : λ|. Then asking how many points there are in G 0 is equivalent to asking how many non-zero roots the polynomial g 0 (λ) = f (0, 1, λ) has. The fundamental theorem of algebra tells us that it has d roots overall, but counting multiplicity. However, Lemma 5.3.4 tells us that all of the roots are distinct, and thus g 0 has d distinct roots. By definition, λ = 0 is a root if and only if p i ∈ C if and only if a i ∤ d, and so we see that
|G i | = d if a i | d; d − 1 if a i ∤ d.
Then we look at the p i : we don't need to worry about counting how many p i there are, since Lemma 5.3.2 tells us that p i ∈ C if and only if a i ∤ d, and we can see that 42 |π −1 (p i )| = 1.
Putting this all together we see that
b(π) = (d − 1) 3 i=1 a 0 a 1 a 2 a i+1 a i+2 − 1 + 3 i=1 a0a1a2 ai+1ai+2 − 1 if a i | d; a0a1a2 1 − 1 if a i ∤ d. = (d − 1) 3 i=1 (a i − 1) + 3 i=1 a i − 1 if a i | d; a 0 a 1 a 2 − 1 if a i ∤ d.
as claimed. All that then remains to prove the formula is to rearrange Eq. (5.3.8) into the given form.
An interesting side-effect of Theorem 5.3.7 is that the complicated looking formula must always give an integer whenever d and the a i satisfy the hypotheses, because we know that the genus of a Riemann surface is going to be an integer. This is similar to how the easiest way to show that n! (n−k)!k! is an integer is to note that n k is a way of counting things, and so must be an integer. 6 The view from where we've ended up This final section is disappointingly 43 brief and also not entirely rigorous in the sense that, quite often we simplify things and hope that the reader will consult any of the given references before believing too much in anything read here. Also, some terminology or notation might not be explained. If this is ever the case then it is because it is considered standard (or as standard as mathematical notation can ever be) in the general literature of the subject, so any of the references provided, or other 'classics', should clear up what it means.
As my supervisor once said to me, 'no piece of work is ever complete', and that is particularly true here. We have only scraped the surface of weighted projective space and its varieties, and we have done so in often simple language, which might not be the most natural way of explaining things (it often isn't). But there are a few things that came up during the writing of this text that the author found exciting and hopes that you might too. We present them to you in this section in an attempt to entice and lure people in to this exciting field of mathematics that comes under the vast umbrella that is 'algebraic geometry'.
During this text we have climbed a hill which, although minuscule in comparison to the towering peaks and ranges of mathematics as a whole, is not a hill to be sniffed at. From 42 Since, for example, [σ : 0 : 0] = [ρ : 0 : 0] for any σ, ρ ∈ µ a 0 . 43 Unless you haven't been enjoying this paper, in which case this might be a welcome fact. the top of this hill we can see a little bit more of the surrounding maths than before. It's still looming above us, but is ever so slightly more in focus and seems just that little bit more tangible and achievable. Let's take a look at the view from up here.
Elliptic curves and friends
Let D be an ample divisor 44 on some plane curve C ⊂ P. Then we can define a graded ring
R(C, D) = n 0 L(C, nD)
where L(C, nD) (written as just L(nD) when it is clear that we are working on C) is the Riemann-Roch space of meromorphic functions on C with poles no worse than nD. Using reasonably standard notation we define L(C, nD) by:
L(C, nD) = {f : C → C | f is meromorphic and (f ) + D 0}.
In a loose sense, what it means for a divisor to be ample is that we can reconstruct the curve C from this graded ring R(C, D). To be slightly more precise,
Proj R(C, D) ∼ = C, but the embedding of C given by this Proj construction might not naturally sit inside P. This is why it becomes very useful to look at this idea, since we get different ways of representing the same curve in different weighted projective spaces.
Example 6.1.1 [Elliptic curves]
There is an exercise sheet [13] by Miles Reid on graded rings which covers all of the below, and far far more besides. It should be reasonably easy to find online.
Let C ⊂ P 2 be given by a non-singular homogeneous cubic f ∈ k[x 0 , x 1 , x 2 ], and define the divisor D = p for some point p ∈ C. We can now use the Riemann-Roch theorem which tells us the dimension ℓ(nD) of L(nD) with a correction term involving a canonical divisor κ of C:
ℓ(nD) − ℓ(κ − nD) = deg D + 1 − g.
We know that deg nD = n, since D is just the point p, and we also know the genus g = 1. It is a useful fact that deg κ = 2g − 2 = 0, so here if n 1 then deg(κ − nD) < 0, and thus the correction term ℓ(κ − nD) = 0 disappears. Finally, we know that the only holomorphic functions on C are the constant ones, and thus L(0) = C. So Riemann-Roch tells us that ℓ(np) = 1 n = 0 n n 1.
(6.1.2)
Using this fact, we can try to construct R(C, np) a little bit more explicitly.
We've already said that L(0) = C, so let's look at when n 1, using Eq. (6.1.2) to tell us how many elements we need for a basis 45 : n = 1: Let x ∈ L(p) be such that x = L(p). Since ℓ(p) = 1 we know that L(p) ∼ = C and so we can take x to be the image of 1 ∈ C under this isomorphism. Thus x is an identity map; n = 2: By definition, x 2 ∈ L(2p), since
(x 2 ) + 2p = (x) + (x) + p + p = (x) + p + (x) + p 0.
But x is the identity, and so x 2 is really just a copy of x that naturally lives inside L(2p), and hence still the identity. So let y be such that x, y = L(2p); n = 3: As above, x 3 , xy ∈ L(3p), but we need one more basis element, so let z be such that x, y, z = L(3p); n = 4: Now x 4 , x 2 y, y 2 , xz ∈ L(4p), and no other combinations of x, y, z are, so we have exactly the right amount of elements for a basis; n = 5: Here x 5 , x 3 y, xy 2 , x 2 z, yz ∈ L(5p) gives us exactly the right amount of elements again; n = 6: Things start to change in this case: x 6 , x 4 y, x 2 y 2 , y 3 , x 3 z, xyz, z 2 ∈ L(6p), which is one element too many, so we must have some linear dependence between them all. Since x k is an identity map, we see that the only 'new' elements that we've got in the n = 6 case are y 3 and z 2 (the only ones without a non-trivial power of x), and so the linear dependence must involve them. By doing some linear change of coordinates 46 we can assume that this equation is of the form
z 2 = y 3 + ax 4 y + bx 6 . (6.1.3)
n 7: By doing some dimension counting we can show that x, y, z generate R(C, p) with only one relation between them, namely Eq. (6.1.3). This sort of crosses over to using the Hilbert polynomial and some theory of generating functions, which is covered more in the next section, but the general idea is that
{x α y β z γ ∈ L(np)} = {(α, β, γ) ∈ (N ∪ {0}) 3 : α + 2β + 3γ = n}
which is exactly 47 the t n coefficient of the series expansion of
1 (1 − t)(1 − t 2 )(1 − t 3 )
.
So this tells us that
R(C, D) ∼ = k (1,2,3) [x, y, z] (g 6 )
where g 6 = z 2 − y 3 + ax 4 y + bx 6 (from Eq. (6.1.3)). Thus
C ∼ = Proj R(C, D) ∼ = V(g 6 ) ⊂ P(1, 2, 3)
46 Full details in [8, Chapter IV, Proposition 4.6], or use z → z + α 3 (x, y) and y → y + β 2 (x) with suitable constants, where the subscript represents the degree of the polynomial. 47 See [19,Section 3.15] for this specific problem, and the book as a whole for a great introduction to the theory and applications of generating functions.
is an embedding of C as a degree-6 plane curve in P (1, 2, 3).
We can repeat this story but with D = 2p, as this is still a very-ample divisor. But, up to a constant multiple of the grading (or not even that, depending on our grading convention for truncations), this is the same as looking at the 2-nd truncation:
R(C, 2p) = n 0 L(2np) = m 0 2|m L(mp) ∼ = m 0 L(mp)(2)
= R(C, p) (2) . This then gives us a different embedding, namely
R(C, p) (2) ∼ = k (1,1,2) [x 0 , x 1 , y] (g 4 )
which induces an embedding of C as a degree-4 plane curve in P(1, 1, 2). 48 But why stop at 2p? It seems like we might as well look at kp for k ∈ N. Table 1 summarises what we get when D = kp for different values of k. 49 As a reassuring fact, we see that Theorem 5.3.7 tells us that the genus is 1 for all of these curves, as it should.
k degree of curve(s) ambient space comments 1 6 P(1, 2, 3) the classical Weierstrass equation 2 4 P(1, 1, 2) double cover of P 1 with 4 branch points 3 3 P(1, 1, 1) plane cubic with an inflexion at infinity 4 2, 2 P(1, 1, 1, 1) intersection of two quadrics 5 5 P(1, 1, 1, 1, 1) P 4 section of Grass(2, 5) ⊂ P 9 Table 1: Different embeddings of an elliptic curve C coming from Veronese truncations of R(C, p).
We know 50 that for k 3 the divisor D k is very ample, and we thus 51 get an embedding of C into P k−1 , but as k increases the associated description of C gets more and more complicated. But we also know that a smooth 52 C 7 ⊂ P(1, 2, 3) has genus 1, and is thus also an elliptic curve. 53 Yet it doesn't appear in Table 1 at all, and we've just said that for k 3 all of the embeddings will be in P k−1 . So C 7 never arises from the above method.
It is an interesting question to ask why C 7 never appears from the process in Example 6.1.1. Unfortunately, it is at this point that the author must once more throw up their hands in confession and admit uncertainty. A believe answer is that this specific embedding is not projectively normal, but the details are beyond this text.
Syzygies and some homological algebra
This section raises far more questions than it answers. Whether this is due to a lack of time or a lack of knowledge on the behalf of the author is left intentionally ambiguous.
In Example 6.1.1 we constructed a surjective graded ring homomorphism k a [x 0 , x 1 , x 2 ] → R(C, D) with kernel (g 6 ). Let's consider a more general situation where we have a surjective graded ring homomorphism ϑ : S a = k a [x 0 , . . . , x n ] → R(D) with kernel (g 1 , . . . , g k ), where deg g i = d i . Then we can write the sequence S a R(D) 0 ϑ which is exact at R(D), but not at S a . To make this sequence exact we need a graded ring S ′ along with a graded ring homomorphism ϑ ′ : S ′ → S a such that im ϑ ′ = ker ϑ = (g 1 , . . . , g k ).
Consider the map ϑ ′ : S ′ = k i=1 S a → S a given by ϑ ′ : (f 1 , . . . , f k ) → f 1 g 1 + . . . + f k g k which we write in matrix-like notation as f → (g 1 , . . . , g k )f . This is not a graded ring homomorphism, but if we give S ′ a a different grading then we claim that we can make it one. Write S a [−d i ] to mean the graded ring S a but with a shift of grading by −d i . That is, if f ∈ S a is such that deg f = d then when we consider f as an element of S a [d i ] it has degree
d − d i . Let S ′ = k i=1 S a [−d i ], so that we have the sequence k i=1 S a [−d i ] S a R(D) 0 ϑ ′ ϑ
which is exact at S a and R(D), but not at
k i=1 S a [−d i ].
The question is, if we carry on finding S ′′ , S ′′′ , . . . and ϑ ′′ , ϑ ′′′ , . . . in a similar way, will we ever end up with an exact sequence of finite length? The answer is, maybe surprisingly, yes.
Theorem 6.2.1 [Hilbert syzygy theorem ([4, Theorem 1.13])]
Let R be a finitely-generated k algebra. Then every finitely-generated R-module has a finite graded free resolution of length no more than r, by finitely-generated free modules.
The theorem is a bit stronger than what we've been asking for in our simplified language, but in essence it tells us that we can always find a finite free resolution (i.e. turn our chain into an exact chain of finite length). Let's look at some simple examples.
Example 6.2.2 [k = 1]
If we take k = 1 then we can stop where we stopped above. That is, we have ϑ : S a → R(D) with ker(ϑ) = (g) where deg g = d. Then we have the (short) exact sequence
0 S a [−d] S a R(D) 0 (g) ϑ
Example 6.2.3 [k = 2]
When k = 2 we get a slightly more interesting case. Using the same matrix-like notation as before, we obtain the exact sequence
0 S a [−(d 1 + d 2 )] S a [−d 1 ] ⊕ S a [−d 2 ] S a R(D) 0 g 2 −g 1 (g1, g2) "ϑ" where g2 −g1 : f → (g 2 f, −g 1 f ).
This is all very interesting, but doesn't yet seem to relevant to what we were doing in the last section. But we now provide a few examples and some explanation as to why this is actually all very interesting. Compare and contrast the following with [14, Proposition 4.3, Example 4.4].
Example 6.2.4
We see that a variety which falls under the category of that in Example 6.2.2 (so given by V(g) ⊂ P(a 0 , . . . , a n )) has the Hilbert series
1 − t d (1 − t ai )
and one in the same family as in Example 6.2.3 has Hilbert series
1 − t d1 − t d2 + t d1+d2 (1 − t ai ) = (1 − t d1 )(1 − t d2 ) (1 − t ai ) .
So if we can calculate the Hilbert series of a variety then, by writing it as a fraction and changing the denominator, we can get a rough idea of what embeddings we can get from it. We can see what the weights a i the denominator gives us, and we can say what sort of relations and syzygies its defining polynomials will satisfy by looking at the numerator. Why is this?
Well, let X ⊂ P n be a projective variety with homogeneous coordinate ring S(X) = k[x 0 , . . . , x n ]/ I(X). Then we can construct the Hilbert series P X (t) of X by considering some ample divisor D on X as we did above: Using Riemann-Roch we can usually find ℓ(mD) explicitly, often by choosing D = p to be a specifically nice point in X. Then, ignoring any issues of convergence (by assuming that |t| is small enough), we can try to rewrite this series as a single fraction. This is probably best explained here as an example, since we are being nowhere near rigorous enough to try to explain ourselves in proper mathematical language.
Example 6.2.5 [Elliptic curves, continued]
This example is a continuation Example 6.1.1, and so, in particular, all notation remains the same.
We've already calculated ℓ(np) for p ∈ C, and so we know that the Hilbert polynomial of our elliptic curve is P C,p (t) = m 0 ℓ(mp)t m = 1 + t + 2t 2 + 3t 3 + . . . .
Now, indeed
1 − t 6 (1 − t)(1 − t 2 )(1 − t 3 ) = 1 + t + 2t 2 + . . . = P C,p (t) and by our previous comments this looks like an embedding of C into P(1, 2, 3) as the vanishing of a single degree-6 curve. Similarly,
P C,2p = 1 + ℓ(2p)t + ℓ(4p)t 2 + . . . = 1 − t 4 (1 − t)(1 − t)(1 − t 2 ) P C,3p = 1 + ℓ(3p)t + ℓ(6p)t 2 + . . . = 1 − t 3 (1 − t)(1 − t)(1 − t)
which both agree with the embeddings that we already know. namely C 4 ⊂ P(1, 1, 2) and C 3 ⊂ P 2 .
The idea of generating functions and the combinatorics behind all this can really help to give us some intuition as to why we can read off such data, and why the numerator tells us about the relations. If R(X, D) has generators x 0 , . . . , x n such that x i ∈ L(X, a i D) then to see how many combinations (i.e. products) of these lie in L(kD) we simply look at c k , defined as the k-th coefficient in the series expansion of But if there are some relations between the x i then we won't be getting the full amount of distinct combinations all the time. That is, we won't have c k distinct combinations of the x i in L(X, c k D) when k becomes too large, since there will be some cancellation, and so we need to adjust our fraction to account for this -we need to add in some higher order negative terms to reduce the c j for j k. But then there might be some relations between the relations of the x i , and in that case we have reduced the c j too much and we will actually have more combinations of the x i , and so we will need to put in some positive terms of higher order to make the c j larger, and so on.
We end this paper with one final example, which is again more of an unfinished problem.
Example 6.2.6
Let's try to repeat the method that we applied in Example 6.1.1 to study the genus 3 Riemann surface given by a smooth quartic curve C = C 4 ⊂ P 2 , with the divisor D = p ∈ C. We know that a canonical divisor κ on C has degree 2g − 2 = 4 and is thus linearly equivalent to a hyperplane divisor H = H L for any line L ⊂ P 2 . So by Riemann-Roch, we know that ℓ(np) = n − 2 for n 5, since then deg(κ − nD) < 0 thus ℓ(κ − np) = 0. By definition, we can see that when n = 0, 1 we get ℓ(np) = 1, The question then is: how do we calculate ℓ(np) for n = 2, 3, 4? We know that ℓ(np) is non-decreasing, and if we can further show that ℓ(np), ℓ(κ − np) > 0 then we can use Clifford's theorem to obtain some bounds, but this still gives us a few possible options.
It turns out that, actually, there is no one answer -it depends on the point p that we choose. We always have ℓ(2p) = 1 (by non-hyperellipticity), and we can actually choose p to get any of the possible values of ℓ(3p) and ℓ(4p) that we like. 54 So we have the following possibilities for the Hilbert series coefficients: For the sake of concreteness, let's just examine the first one here: assume that P C (t) = 1 + t + t 2 + t 3 + 2t 4 + 3t 5 + . . . = 1 + t + t 2 + m 4 (m − 2)t m .
Following our naive approach from before, then, we find that we can pick elements v ∈ L(p), w ∈ L(4p), x ∈ L(5p), y ∈ L(6p), z ∈ L(7p) that generate L(np) without any relations until we hit n = 10, and then we have too many elements.
The n-th coefficient of the series expansion of λ(t) = 1/(1−t)(1−t 4 )(1−t 5 )(1−t 6 )(1−t 7 ) tell us how many elements of L(np) we can generate with v, w, x, y, z, as in Example 6.1.1. Since we have too many elements for n 10 we expect our Hilbert polynomial to have the same denominator as λ, but with a negative term of degree 10 in the numerator to lower the coefficients for t n in λ when n 10. This is where combinatoric intuition can help us understand a problem about projective plane curves.
If we do some algebra, we see that P C (t) = 1 + t + t 2 + t 3 + 2t 4 + 3t 5 + . . . = 1 1 − t + t 4 (1 + 2t + 3t 2 + . . .) = 1 1 − t + t 4 d dt 1 + t + t 2 + . . .
= 1 1 − t + t 4 d dt 1 1 − z = 1 1 − t + t 4 (1 − t) 2 = 1 − t + t 4 (1 − t) 2 = . . . = 1 − t 10 − t 11 − 2t 12 − t 13 − t 14 + t 16 + 2t 17 + 2t 18 + 2t 19 + t 20 − t 23 − t 24 − t 25 (1 − t)(1 − t 4 )(1 − t 5 )(1 − t 6 )(1 − t 7 )
.
From this we can see that our discovery that there must be a relation of degree 10 seems correct, and we speculate (and only speculate, not claim with any amount of certainty) by looking at how the signs change that there are also relations of degree 11, 12, 13, 14 with syzygies of degree 16,17,18,19,20 and higher syzygies between them of degree 23, 24, 25. 54 Have a look at a question asked by the author on math.stackexhange.com: [5].
2.1.1 [Weights and their induced action]
Definition 3.0.5 [Weighted polynomial ring] Define the polynomial ring in n + 1 variables with weighting a = (a 0 , . . . , a n ) as k a [x 0 , . . . , x n ] with wt x i = a i .
f
(λ a0 p 0 , . . . , λ an p n ) = f (p 0 , . . . , p n ) if and only if f (p 0 , . . . , p n ) = 0.
Lemma 3.0.11 [Equivalent definition of weighted-homogeneous ideals] An ideal I ⊳ k a [x 0 , . . . , x n ] is weighted-homogeneous if and only if every element f ∈ I can be written as
Lemma 3.0.12 A weighted-homogeneous ideal I ⊳ k a [x 0 , . . . , x n ] is prime if and only if, whenever f g ∈ I for f, g ∈ k a [x 0 , . . . , x n ] with f, g both homogeneous, either f ∈ I or g ∈ I. That is, when considering primality of the ideal, it is enough to check the usual definition on only the homogeneous elements of the ideal. Proof. We can simply appeal to the proof of [1, Chapter 6, Section 1, Lemma 1.2].
Definition 3.1.1 [Weighted projective varieties and their ideals] Let I⊳k a [x 0 , . . . , x n ] be a weighted-homogeneous ideal. Define the weighted projective variety (associated to I) by V(I) = {p ∈ P(a 0 , . . . , a n ) | f (p) = 0 for all f ∈ I}.
Let I, J ⊳ k a [x 0 , . . . , x n ] be weighted-homogeneous ideals. Then
(
iv) P(a 0 , . . . , a n ) = V({0}) Proof. (i) An element of IJ is by definition of the form k f i g i for f i ∈ I and g i ∈ J, which tells us that V(I) ∪ V(J) ⊆ V(IJ), since if x vanishes on either all of I or all of J (or maybe some of both) then it definitely vanishes on all elements of IJ.
Theorem 3.2.10 [Weighted projective Nullstellensatz] Let I ⊳ k a [x 0 , . . . , x n ] be a weighted-homogeneous relevant ideal. Then I V(I) = rad(I).
a relevant weighted-homogeneous ideal with I({p}) ⊆ a. Then ∅ V(a) ⊆ V I({p}) = {p}, and so V(a) = {p}. This then tells us that I({p}) = rad(a). But a ⊆ rad(a) = I({p}) ⊆ a. Hence a = I({p}), and I({p}) is a maximal weightedhomogeneous ideal.
Let I ⊳ k a [x 0 , . . . , x n ] be a radical weighted-homogeneous relevant ideal. Then Proj k a [x 0 , . . . , x n ] I = V(I) ⊆ P(a 0 , . . . , a n ), but where Proj R has this enriched structure of also containing points corresponding to all the irreducible subvarieties of V(I). Equivalently, let X = V(I) ⊆ k a [x 0 , . . . , x n ]. Then Proj S(X) = Proj k a [x 0 , . . . , x n ] I(X) = X ⊆ P(a 0 , . . . , a n )
(4.1.6) and (4.1.7) in the same way as in Example 4.1.4.So in some sense 23 Proj and S(−) are mutual inverses, i.e. Proj S(X)) = X and S(Proj R)) = R.
Definition 4.2.1 [Morphism of weighted projective varieties, attempt 1] Let X ⊂ P(a 0 , . . . , a n ) and Y ⊂ P(b 0 , . . . , b m ) be weighted projective varieties, and f : S(Y ) → S(X) a graded ring homomorphism. Write S(Y ) = k (b0,...,bm) [y 0 , . . . , y m ]/J and S(X) = k a [x 0 , . . . , x n ]/I, where J = I(Y ) and I = I(X), so that f (y
Definition 4.3.1 [Truncation of a graded ring]Let R = ⊕ i 0 R i be a graded ring. Define R (d) , the d-th truncation of R, by
P 1 =
1Proj k[x, y] ∼ = Proj k[x, y] (2) = ν 2 (P 1 ) ⊆ P 2 and hence Proj R ∼ = Proj R (2) when R = k[x, y], and in fact this truncation corresponds exactly to the degree-2 Veronese embedding.
Let 26 R = k a [x 0 , . . . , x n ] I for some radical weighted-homogeneous relevant ideal I ⊳k a [x 0 , . . . , x n ]. Then there exists some d ∈ N such that R (d) is generated by R d . Proof. We can simply apply the more general proof of [17, Lemma 5.5.3].
Theorem 4.1.8 we know that this is the variety V(x 5 + y 3 + z 2 ) ⊆ P(12, 20, 30), but Theorem 4.3.9 makes us wonder what this looks like as a straight projective variety.Using the ideas in the proof of Theorem 4straightening of P(12, 20, 30) = Proj C(12,20,30) [x, y, z]. So now we have to think how the ideal (f ) ⊳ R transforms under these truncations.
Let C ⊂ P 2 be a non-singular plane curve. Then C is a compact Riemann surface. Proof. See [12, Chapter I, Proposition 3.6].
Figure 5 .
52.11 is intended to be an understandable summary of all the confusing notation being thrown around, and the proof of Corollary 5.2.10 essentially aims to prove that the diagram commutes.
(i) if a i | d then f contains an x ) if a i ∤ d then f contains an x j x m i term, where j = i and m = (d − a j )/a i .
a sufficiently-general degree-d weighted-homogeneous polynomial. Let p 0 = |1 : 0 : 0|, p 1 = |0 : 1 : 0|, and p 2 = |0 : 0 : 1| ∈ P. Then p i ∈ C f if and only if a i ∤ d. Proof. By our definition of sufficiently general we have two cases: (i) if a i | d then there is an x d/ai i term in f , and so f (p i ) = 0, and p i ∈ C f ;
Let f ∈ k a [x 0 , x 1 , x 2 ] be a sufficiently-general degree-d weighted-homogeneous polynomial. If p i ∈ C (which happens if and only if a i ∤ d) then p i is not a singular point of C.
∂f ∂z = (z − cy)g + (z − cy) 2 ∂g ∂z + x ∂h ∂z and this evaluates to 0 at [x : y : z] = [0 : 1 : c].
h(0, 1, c) = 0. Thus ∂f ∂x evaluates to 0 at [x : y : z] = [0 : 1 : c].
. . . ] Our intent is to cover a canon completely and rigorously, with enough examples and calculations to help develop intuition for the machinery. [ . . . ] We do not live in an ideal world.
We do not use the word 'simple' here in a patronising way, but rather as a way of saying that we will aim to not drift too far into the colossal world of modern abstract algebraic geometry, and to instead merely nod and wave at it as it passes us by.
This does change from Section 5 onwards though, since it is much easier to use some basic facts about projective plane curves than develop everything from scratch.3 https://math.berkeley.edu/~amathew/cr.html
We do also sometimes use k as a summation index, or a general integer. Hopefully though, it will always be perfectly clear from context which exactly we mean.
Here we mean in the Zariski topology, but it is also true in the quotient topology coming from the definition as P n = (A n \ {0})/Gm.7 This does have the expected definition: simply define closed sets to be those that can be written as V(I) for some ideal I in a suitable polynomial ring. But we do have to worry about which polynomial ring, and what it means to evaluate a polynomial at a point in wps. This is all covered in Section 3.1, in particular in Definition 3.1.7.
See [9, Definition 3.6, Theorem 3.8].9 Again, see [9, Definition 3.6, Theorem 3.8].
Just as we do for polynomials normally. So, for example, f = x 2 + y 2 ∈ k(1,2) [x, y] has deg f = 4.11 Or simply weighted-homogeneous of degree d if it is clear that we are working with the weight a.
It is only a cone though for a 0 , . . . , an = 1, whence we recover straight projective space. The use of the word here is definitely meant to be understood in the sense of a loose meaning, but we will from now one speak of the affine cone anyway, to avoid having to constantly write cone in quotation marks.
For more on the irrelevant ideal (and graded rings in general) see[1, Chapter 6].16 Thanks to Christopher R. Miller for pointing this out and correcting previous versions of this definition, as well as providing some interesting examples.
Or if we are happy with the fact that the irrelevant ideal is radical then we could just take the rad of both sides of (x 0 , . . . , xn) ⊆ I.
Though for the sake of (a poor effort towards) completeness we will mention the more general definition afterwards.20 We make this distinction here because the natural definition of Proj R is as a scheme, i.e. with a topology and a structure sheaf. This doesn't matter too much here, since we try to stick to 'hands-on' algebraic geometry as much as possible, but it is well worth bearing in mind. See[18, Chapter 4.5] for more.
The choice of which grading to use in the truncated ring (either the one that we use here, or simply keeping the grading the same) varies from author to author. As long as you pick one and stick with it doesn't (at least, not as far as this author knows) matter.
This is not the author saying that this proof is left to the reader; this is the author saying that the details are best left to a text on commutative algebra, and not an expository text on weighted projective spaces written by yours truly.
This bit of the argument is admittedly a bit too hand-wavey for the author's liking. Hopefully looking at the example in Section 4.4 will convince the reader that this could indeed be made more rigorous.29 If R is a UFD then R[t] is also a UFD.
See also the answer to one of the author's questions (which also displays their original misunderstandings) for a discussion about the links to the Poincaré Homology Sphere:[3].
We are only ever interested in finite groups here though, and all finite groups are reductive.
Or we could prove this in the other direction: since our field is algebraically closed and all the a i are pairwise coprime we know that |π −1 (p)| = a 0 a 1 a 2 for a point p = |p 0 : p 1 : p 2 | with p 0 , p 1 , p 2 = 0.41 Here we use the convention that x 3 = x 0 .
For our purposes we will only really look at point divisors, that is, D = kp for some specific point p ∈ C and k ∈ N, and these are all ample, and in fact very ample for k 2g C + 1 by [8, Chapter IV, Corollary 3.2(b)].45 Here we skim over issues of linear independence, in keeping with our theme this section of being quick and not very rigorous.
For details. 10, Section 12.6] provides a thorough explanationFor details, [10, Section 12.6] provides a thorough explanation.
. Compare with the list in. 10, Section 12.4Compare with the list in [10, Section 12.4].
The CRing Project. A Collaborative, Open Source Textbook on Commutative Algebra. Shishir Agrawal, Eva Belmont, Zev Chonoles, Rankeya Datta, Anton Geraschenko, Sherry Gong, François Greer, Darij Grinberg, Aise Johan de Jong, Adeel Ahmad Khan, Holden Lee, Geoffrey Lee, Akhil Mathew, Ryan Reich, William Wright, and Moor Xumath.berkleley.eduShishir Agrawal, Eva Belmont, Zev Chonoles, Rankeya Datta, Anton Geraschenko, Sherry Gong, François Greer, Darij Grinberg, Aise Johan de Jong, Adeel Ahmad Khan, Holden Lee, Geoffrey Lee, Akhil Mathew, Ryan Reich, William Wright, and Moor Xu. The CRing Project. A Collaborative, Open Source Textbook on Commutative Algebra. math.berkleley.edu. Oct. 2011.
Weighted projective varieties. Igor Dolgachev, Group Actions and Vector Fields. Berlin HeidelbergSpringerIgor Dolgachev. "Weighted projective varieties". In: Group Actions and Vector Fields. Springer Berlin Heidelberg, 1982, pp. 34-71.
Weighted projective space and Proj. Dorebell, Dorebell. Weighted projective space and Proj. URL: http://math.stackexchange.com/q/1426420.
Introduction to commutative algebra with a view towards algebraic geometry. David Eisenbud, Graduate Texts in Mathematics. David Eisenbud. Introduction to commutative algebra with a view towards algebraic geometry. Graduate Texts in Mathematics, 1995.
Riemann-Roch analysis of point divisor ring on smooth genus 3 Riemann surface. Georges Elencwajg, Georges Elencwajg. Riemann-Roch analysis of point divisor ring on smooth genus 3 Rie- mann surface. URL: http://math.stackexchange.com/questions/1410556.
Éléments de géométrie algébrique: I. Le langage des schémas. Alexander Grothendieck, Publications mathématiques de l'I.H.É.S. 4Alexander Grothendieck. "Éléments de géométrie algébrique: I. Le langage des schémas". In: Publications mathématiques de l'I.H.É.S. 4 (1960), pp. 5-228.
Éléments de géométrie algébrique: II.Étude globalé elémentaire de quelques classes de morphismes. Alexander Grothendieck, Publications mathématiques de l'I. 8Alexander Grothendieck. "Éléments de géométrie algébrique: II.Étude globalé elémentaire de quelques classes de morphismes". In: Publications mathématiques de l'I.H.É.S. 8 (1961), pp. 5-222.
Algebraic Geometry. Robin Hartshorne, SpringerRobin Hartshorne. Algebraic Geometry. Springer, 1977.
Quotients in algebraic and symplectic geometry. Victoria Hoskins, Victoria Hoskins. Quotients in algebraic and symplectic geometry. Dec. 2012. URL: https://www.math.uzh.ch/index.php?file&key1=22003.
Working with weighted complete intersections. Anthony Iano, - Fletcher, Anthony Iano-Fletcher. Working with weighted complete intersections. 2000.
Complex Algebraic Curves. Frances Kirwan, Cambridge University PressFrances Kirwan. Complex Algebraic Curves. Cambridge University Press, 1992.
Algebraic Curves and Riemann Surfaces. Rick Miranda, American Mathematical SocietyRick Miranda. Algebraic Curves and Riemann Surfaces. American Mathematical Society, 1995.
Exercises on graded rings. Miles Reid, Miles Reid. Exercises on graded rings. URL: http://homepages.warwick.ac.uk/~masda/Homework/graded_hom
Graded rings and varieties in weighted projective space. Miles Reid, Miles Reid. Graded rings and varieties in weighted projective space. 2002. URL: http://homepages.warwick.ac.uk/~masda/surf/more/grad.pdf.
Undergraduate Algebraic Geometry. Miles Reid, Cambridge University PressMiles Reid. Undergraduate Algebraic Geometry. Cambridge University Press, 1988.
Undergraduate Commutative Algebra. Miles Reid, Cambridge University PressMiles Reid. Undergraduate Commutative Algebra. Cambridge University Press, 1995.
Jenia Tevelev. Moduli Spaces and Invariant Theory. Jenia Tevelev. Moduli Spaces and Invariant Theory. URL: http://people.math.umass.edu/~tevelev/moduli797.pdf.
The Rising Sea. Ravi Vakil, Ravi Vakil. The Rising Sea. Apr. 2015. URL: http://math.stanford.edu/~vakil/216blog/FOAGapr2915public
. S Herbert, Wilf, Generatingfunctionology, Taylor and FrancisThird editionHerbert S. Wilf. generatingfunctionology. Third edition. Taylor and Francis, 2005.
. Oscar Zariski, Pierre Samuel, Commutative Algebra. 1Van Nostrand Company, IncOscar Zariski and Pierre Samuel. Commutative Algebra: Volume 1. D. Van Nostrand Company, Inc., 1958.
| [] |
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"ASSESSMENT OF GAPSB/SI TANDEM MATERIAL ASSOCIATION PROPERTIES FOR PHOTOELECTROCHEMICAL CELLS Assessment of GaPSb/Si tandem material association properties for photoelectrochemical cells"
] | [
"Lipin Chen \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Mahdi Alqahtani \nDepartment of Electronic and Electrical Engineering\nUniversity College London\nWC1E 7JELondonUnited Kingdom\n\nKing Abdulaziz City for Science and Technology\n\n",
"Christophe Levallois \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Antoine Létoublon \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Julie Stervinou \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Rozenn Piron \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Soline Boyer-Richard \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Jean-Marc Jancu \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Tony Rohel \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Rozenn Bernard \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Yoan Léger \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Nicolas Bertru \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n",
"Jiang Wu \nDepartment of Electronic and Electrical Engineering\nUniversity College London\nWC1E 7JELondonUnited Kingdom\n",
"Ivan P Parkin \nDepartment of Chemistry\nUniversity College London\nWC1H 0AJLondonUnited Kingdom\n",
"Charles Cornet ⋆e-mail:[email protected] \nUniv Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance\n"
] | [
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
"Department of Electronic and Electrical Engineering\nUniversity College London\nWC1E 7JELondonUnited Kingdom",
"King Abdulaziz City for Science and Technology\n",
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
"Univ Rennes\nINSA Rennes\nCNRS\nFOTON -UMR 6082, F-35000RennesInstitutFrance",
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"Department of Chemistry\nUniversity College London\nWC1H 0AJLondonUnited Kingdom",
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] | [] | Here, the structural, electronic and optical properties of the GaP 1-x Sb x /Si tandem materials association are determined in view of its use for solar water splitting applications. The GaPSb crystalline layer is grown on Si by Molecular Beam Epitaxy with different Sb contents. The bandgap value and bandgap type of GaPSb alloy are determined on the whole Sb range, by combining experimental absorption measurements with tight binding (TB) theoretical calculations. The indirect (X-band) to direct (Γ-band) cross-over is found to occur at 30% Sb content. Especially, at a Sb content of 32%, the GaP 1-x Sb x alloy reaches the desired 1.7eV direct bandgap, enabling efficient sunlight absorption, that can be ideally combined with the Si 1.1 eV bandgap. Moreover, the band alignment of GaP 1-x Sb x alloys and Si with respect to water redox potential levels has been analyzed, which shows the GaPSb/Si association is an interesting combination both for the hydrogen evolution and oxygen evolution reactions. These results open new routes for the development of III-V/Si low-cost high-efficiency photoelectrochemical cells. | 10.1016/j.solmat.2020.110888 | [
"https://arxiv.org/pdf/2002.02774v1.pdf"
] | 211,066,209 | 2002.02774 | 1ea00885224ef824a2b0194658d2077caf44584f |
ASSESSMENT OF GAPSB/SI TANDEM MATERIAL ASSOCIATION PROPERTIES FOR PHOTOELECTROCHEMICAL CELLS Assessment of GaPSb/Si tandem material association properties for photoelectrochemical cells
Lipin Chen
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Mahdi Alqahtani
Department of Electronic and Electrical Engineering
University College London
WC1E 7JELondonUnited Kingdom
King Abdulaziz City for Science and Technology
Christophe Levallois
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Antoine Létoublon
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Julie Stervinou
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Rozenn Piron
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Soline Boyer-Richard
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Jean-Marc Jancu
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Tony Rohel
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Rozenn Bernard
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Yoan Léger
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Nicolas Bertru
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
Jiang Wu
Department of Electronic and Electrical Engineering
University College London
WC1E 7JELondonUnited Kingdom
Ivan P Parkin
Department of Chemistry
University College London
WC1H 0AJLondonUnited Kingdom
Charles Cornet ⋆e-mail:[email protected]
Univ Rennes
INSA Rennes
CNRS
FOTON -UMR 6082, F-35000RennesInstitutFrance
ASSESSMENT OF GAPSB/SI TANDEM MATERIAL ASSOCIATION PROPERTIES FOR PHOTOELECTROCHEMICAL CELLS Assessment of GaPSb/Si tandem material association properties for photoelectrochemical cells
1
Here, the structural, electronic and optical properties of the GaP 1-x Sb x /Si tandem materials association are determined in view of its use for solar water splitting applications. The GaPSb crystalline layer is grown on Si by Molecular Beam Epitaxy with different Sb contents. The bandgap value and bandgap type of GaPSb alloy are determined on the whole Sb range, by combining experimental absorption measurements with tight binding (TB) theoretical calculations. The indirect (X-band) to direct (Γ-band) cross-over is found to occur at 30% Sb content. Especially, at a Sb content of 32%, the GaP 1-x Sb x alloy reaches the desired 1.7eV direct bandgap, enabling efficient sunlight absorption, that can be ideally combined with the Si 1.1 eV bandgap. Moreover, the band alignment of GaP 1-x Sb x alloys and Si with respect to water redox potential levels has been analyzed, which shows the GaPSb/Si association is an interesting combination both for the hydrogen evolution and oxygen evolution reactions. These results open new routes for the development of III-V/Si low-cost high-efficiency photoelectrochemical cells.
Introduction
The conversion of solar energy into green hydrogen fuel is one significant milestone on the road to a sustainable energy future [1,2]. Especially under the current background of global energy and environmental crisis, the development of the photoelectrochemical (PEC) water splitting technology, where the sunlight turns the liquid water into gaseous storable hydrogen that can be reused on demand for heat or electricity production, has driven many researches in the past years [3].
However, numbers of challenges remain for the development of this technology, in terms of efficiency, profitability and sustainability. The heart of the PEC conversion process is the choice of an appropriate photoelectrode material with both good bandgap and good band alignment to harvest the largest portion of the solar spectrum and provide sufficient voltage to accomplish the water splitting reactions [4]. Due to the broadness of the solar spectrum, from infrared to ultra-violet light, the needs for combining monolithically different materials with different bandgaps and thus absorbing different wavelengths, is considered today as the main pathway to reach high solar-to-hydrogen (STH) conversion efficiency [5,6]. In the tandem association of two materials, the optimum combination of bandgaps has been widely discussed for different device configurations [7]. Especially, the combination of a 1.7 eV top absorber bandgap with a 1.1 eV bottom absorber is recognized as one of the best tandem materials configuration and gives rise to a theoretical maximum STH efficiency η STH larger than 26% [8]. Different device demonstrations were proposed with a III-V/III-V materials design, mostly based on the Ga(In)As/GaInP association, reaching η STH larger than 15% [8][9][10][11]. In these works, high quality materials are achieved, thanks to the lattice-matching of theses alloys on the expensive GaAs substrate. A way to reduce the cost of tandem materials association is to monolithically integrate a III-V top absorber (1.7 eV bandgap) on the silicon substrate (because of its approximate 1.1eV band gap, earth abundance, low cost and prevalence in the electronics and PV industries) [7]. But by now there are only very few reports on Si-based tandem systems for water splitting, mainly due to the difficulty in growing high quality III-V epilayers on Si. The recent progress in the understanding of III-V/Si epitaxial processes and devices developments gives new hopes for the development of high efficiency III-V/Si PEC devices on the low-cost Si substrate [12,13]. Following this approach, pioneering works were performed on the development of bipolar configured 1.6eV AlGaAs on 1.1eV Si tandem association. Water splitting at a 18.3% conversion efficiency was reported [14]. Very recently, a single InGaN absorber photoanode monolithically integrated on silicon (where the Si (111) substrate was used as a back contact) was proposed with an applied-bias photo-to-current efficiency of 4.1% [15]. GaP-based materials were also proposed as an ideal tandem association with Si for water splitting [16]. The optimum design of GaP/Si tunnel junctions was even considered [17]. TiO 2 -, CoO x -or Ni-based passivation strategies were found to be efficient for photoelectrode stability [18,19]. But this approach still suffers from the indirect and large bandgap (2.26 eV) nature of GaP. Therefore, alloying GaP with other group III-V atoms is needed to achieve a direct bandgap at 1.7 eV. Doscher H. et. al theoretically proposed GaPN/Si and GaPNAs/Si tandem lattice-matched materials for water splitting based on the experiments of GaP/Si photoelectrochemistry and GaPN/Si epitaxial growth [20]. But these N-including lattice-matched materials face the issue of excitons localization effects, which hamper easily charge transport in the developed devices. On the other hand, metamorphic III-V integration on Si is known to generate dislocations that may propagate in the volume and are detrimental for solar devices. Recently, the synthesis of GaPSb was proposed to reduce the bandgap of GaP, with promising properties for PEC operation [21]. The epitaxial growth of Sb-based III-V compounds on Si substrates is particularly interesting in this regard, as it allows relaxation of the crystal stress through a near-perfect misfit dislocation network localized at the III-V/Si interface [12] leading to efficient and stable photonic device demonstrations [22]. Recent work demonstrated efficient operation of a GaPSb/Si photoanode for water splitting, but a complete assessment of the alloy with different compositions for water splitting was not yet given [23].
In this study, we evaluate the potential of GaP 1-x Sb x /Si tandem materials association for the development of efficient III-V/Si photoelectrodes. GaP 1-x Sb x alloys with different compositions were directly grown on Si substrate by molecular beam epitaxy (MBE). The bandgaps and band alignments of the GaP 1-x Sb x alloys were carefully and comprehensively studied over the whole Sb range, by combining the experimental data with tight binding (TB) theoretical calculations. We finally discuss the promises offered by such alloy for its use in photoelectrodes.
Material and device design for solar water splitting
Three GaP 1-x Sb x /Si samples (named as GaPSb-1, GaPSb-2 and GaPSb-3, with increasing Sb amounts) were grown by Molecular Beam Epitaxy (MBE) on HF-chemically prepared n-doped (10 17 cm -3 ) Si(001) substrates, with a 6° miscut toward the [110] direction [24]. The substrates were heated at 800°C for 10 minutes to remove hydrogen at the surface. 1µm-thick GaP 1-x Sb x layers were then grown at 500°C in a conventional continuous MBE growth mode, and at a growth rate of 0.24 ML/s, with a Beam Equivalent Pressure V/III ratio of 5.
The schematic diagrams of the proposed GaP 1-x Sb x /Si tandem device for water splitting and the light absorption of the GaP 1-x Sb x /Si tandem system with around 1.7/1.1eV bandgap combination are shown in Fig.1a and Fig.1b. The sun light first enters the top cell (1.7eV-bandgap targeted) GaP 1-x Sb x layer, in which high-energy photons are absorbed and low-energy photons are transmitted and harvested by the Si substrate or bottom cell, leading to an overall very large light absorption (as shown in Fig.1b). Then the photo-induced charges (electrons and holes) are generated in both layers, and, depending on the doping, one kind of charges flows toward the illuminated surface (the top surface of GaPSb layer) to generate H 2 or O 2 and the other one flow toward the back contact which is connected to the Si substrate (as shown in Fig.1a) and is further extracted to feed the counter-electrode.
Material characterizations and theoretical calculations
The detailed description of the methods for material characterizations and theoretical calculations are given in the supplemental materials. Figure 2 shows the X-ray diffraction (XRD) patterns for the three samples. The miscut of around 6° is observed toward the [110] direction for the three samples based on the positions of Si Bragg peaks, which are in agreement with the substrate specifications. The ω/2θ scans exhibit well-defined GaP 1-x Sb x Bragg peaks for the three samples. Reciprocal space maps (RSM) carried out on either (004) (the insets of Fig.2a, Fig.2b, Fig.2c) or (115) (Fig.S1 in supplemental materials) reflections show a full plastic relaxation of the GaP 1-x Sb x layers for the three samples. GaP 1-x Sb x lattice parameters were extracted from both RSM and ω/2θ scans, leading to very similar values for each sample, confirming the full plastic relaxation rates and giving mean lattice parameters of 0.5665 nm, 0.5835 nm, 0.6093 nm, for the sample GaPSb-1, GaPSb-2, GaPSb-3, respectively. The Sb contents of 0.33, 0.60, and >0.99 are then inferred [25]. Sample GaPSb-3 is almost pure GaSb. These RSM images also exhibit an important Bragg peak broadening due to a relatively large crystal defect density, in low Sb content samples. Furthermore, the XRD analysis does not give any evidence of a phase separation that could occur between GaP and GaSb in the GaP 1-x Sb x alloys in theses growth conditions. Figure 3 shows the scanning electron microscopy (SEM) images of the three GaP 0.67 Sb 0.33 /Si, GaP 0.40 Sb 0.60 /Si, GaSb/Si samples on plane-view (Fig.3 a, c, e) and crosssection view (Fig.3 b, d, f). From these images, it can be observed that the samples GaP 0.40 Sb 0.60 /Si and GaSb/Si exhibit relatively smooth surfaces as compared with the other sample, the GaP 0.67 Sb 0.33 /Si. The roughness observed is attributed to both emergence of some crystal defects such as residual dislocations and residual stress in the sample. The corresponding atomic force microscopy (AFM) images are given in the supplemental materials and the RMS (rootmean-square) roughnesses of the surfaces were found to be 22.80nm for GaP 0.67 Sb 0.33 , 9.32nm for GaP 0.40 Sb 0.60 and 7.91nm for GaSb. It suggests an improvement of the crystal quality with increasing Sb content. This is also confirmed by a very significant Bragg peak sharpening observed on RSM (Fig.2). Nevertheless, the crystal quality of the GaPSb alloy grown on Si has been significantly improved compared with the one obtained in previous work [21]. Ellipsometry measurements were made for GaPSb bandgap determination, independently of the Si substrate. Chemical mechanical polishing (CMP) was performed to obtain smoother surfaces and avoid distortion on ellipsometry measurements. The SEM plane-view images of the three CMP-polished samples are shown in Fig.S2 (supplemental materials), which shows that the surface roughnesses of the three samples are much lower than the as-grown samples. The surface RMS roughnesses of the three CMP samples extracted from AFM measurements are 0.33nm (GaP 0.67 Sb 0.33 ), 0.28nm (GaP 0.40 Sb 0.60 ) and 0.56nm (GaSb), indicating the surfaces of the three samples become very smooth after CMP processes. Then the optical constants of the three GaP 0.67 Sb 0.33 /Si, GaP 0.40 Sb 0.60 /Si, and GaSb/Si samples were measured by variable angle spectroscopic ellipsometry (VASE) at room temperature in the 0.58-5 eV photon energy region. The angles of incidence were set to 60˚ and 70˚. A Tauc-Lorentz model with two oscillators was used to fit the ellipsometry data of the three samples (Fig.S5 in the supplemental materials). From this model, the refractive index (n), extinction coefficient (k) and absorption spectrum of the GaPSb layers of the three samples were extracted independently of the Si substrate, respectively, as shown in Fig.S6 (supplemental materials) and Fig.4 (the red curves). Besides, in order to further integrate and verify the experimental data, ellipsometry measurement was also performed on a GaP/Si sample and the corresponding optical constants extracted based on the Tauc-Lorentz model are shown in Fig.S7 (supplemental materials). The deduced absorption curves of the GaP and GaSb based on ellipsometry measurements show good agreements with both the experimental and theoretical data presented in ref. [26] (see Fig.S8 in the supplemental materials). Based on Tauc plot method, the band gap ranges of the four samples were obtained: Eg=2.25±0.04 eV for GaP; Eg=1.70±0.06 eV for GaP 0.67 Sb 0.33 ; Eg=1.04±0.08 eV for GaP 0.40 Sb 0.60 , and Eg=0.68±0.09 eV for GaSb (the details see the supplemental materials), which are consistent with bandgaps reported for the metamorphic growth of GaPSb on InP substrate [27].
The band structures of the unstrained GaP 1-x Sb x alloys in the whole Sb compositional range have then been calculated by tight-binding calculation, using an extended basis sp3d5s* tight binding Hamiltonian [28]. From the tight binding parameters of GaP and GaSb binary compounds [28], a virtual crystal approximation is performed to obtain the band structures of the different GaP 1-x Sb x alloys at 0K as a function of the Sb content. Then the energies were shifted to take into account the influence of the temperature, in order to get room temperature band structures [29]. Fig.4, which also shows good consistency with the experimental data. From this analysis, we deduce that the Sb content at which a 1.7 eV direct bandgap needed for tandem materials association obtained is 32%. Fig. 5. Room temperature bandgaps of GaP 1-x Sb x alloys with different Sb contents. The green, red and purple solid lines are theoretical curves computed by tight-binding calculations, corresponding to Γ, L and X valleys, respectively. The separated error bar lines correspond to the bandgaps determined experimentally. The black dots correspond to the bandgaps of GaP and GaSb given in ref. [30,31].
For efficient PEC water splitting, another important aspect is to estimate the band edge location with respect to the redox potentials of water. The band alignment (no external electric field applied) of the GaP 1-x Sb x /Si tandem architecture for water splitting is presented in Fig.6 as a function of the Sb content. The VBM (valence band maximum) energy (Ev) of the GaP 1-x Sb x alloy over the whole Sb content range was obtained based on the absolute bandlineup between VBM energy of GaP and GaSb [32], which follows the linear formula Ev(GaP 1-x Sb x )= xEv(GaSb)+ (1-x)Ev(GaP) [33]. In a first approximation, the evolution of the surface acidity with Sb content was neglected. The CBM (conduction band minimum) energy of the GaP 1-x Sb x alloy was obtained by using the results from TB calculations presented in Fig. 5. For this alloy, the minimum of the conduction band is located in the X-valley between 0% and 11%. It then moves to the L valley for Sb contents between 11% and 30%. It finally reaches a direct bandgap configuration (minimum of the CB in the Γ valley) beyond 30% of Sb incorporation. While the precise value predicted by the calculations for the X to L valleys crossover seems hard to confirm experimentally in this work, calculations clearly demonstrate that in any cases, an indirect to direct cross-over is expected for this alloy at around 30% Sb, contrary to previously calculated band structures using density functional theory [21]. The VBM energy of bulk GaP is very close to the oxidation potential of water, which has been verified by many reports [32,[34][35][36]. While, in the different papers, the energy differences (between the water redox levels and GaP VBM energy) have a little difference. So, here we show the redox levels of water based on the VBM position of GaP [32] and the energy differences [32,[34][35][36] with error bars to analyze potential water splitting reactions more accurately and comprehensively (Fig.6). Fig.5 and Fig.6, it can be observed first that, in order to benefit from a direct GaPSb bandgap larger than the one of the silicon (to absorb higher energy solar radiations), the Sb content should lie between 30% (indirect to direct cross-over) and 54% (bandgap equal to the Si one). In this Sb content range, it is also noticed that the bandlineup is of type I, promoting the charge carrier extraction in the silicon under zero bias conditions. Now looking at the bandlineups between GaPSb and the water redox potentials in the 30%-54% Sb content range, it is seen that GaPSb has a strong reduction ability due to its relatively higher CBM and it is around 0.9 eV higher than H+/ H 2 potential at 32% Sb content. On the other hand, the VBM is higher than the O 2 /H 2 O potential in the 30%-54% Sb content range, and due to the linear increase of VBM with the Sb content, the GaPSb with low Sb content (around 30%) is thus more suitable for oxygen evolution reactions (about 0.43 and 0.46 eV higher than O 2 /H 2 O potential at 30% and 32% Sb content). Overall, addition of Sb in GaP will therefore ease photocathode operation, although large currents densities were obtained with a GaPSb/Si photoanode, due to a strong direct bandgap absorption in this alloy [23]. The band lineups calculated in this work thus give a basis for further devices analysis, where the precise determination of Nernstian shifts and band bending will be needed, but it is beyond the scope of this article.
Conclusions
In summary, the structural, electronic and optical properties of the III-V/Si-based tandem materials association GaP 1-x Sb x /Si for PEC water splitting was studied comprehensively for the whole range of Sb composition, and its potential for photoelectrochemical cell evaluated. The GaP 1-x Sb x alloys were directly grown on the Si substrate with different Sb contents. The bandgap values and bandgap types of GaPSb alloys were determined on the whole Sb range, by combining the experimental data with tight binding (TB) theoretical calculations. The indirect (Xband) to direct (Γ-band) cross-over was found to occur at 30% Sb content. Especially, at a Sb content of 32%, the GaP 1-x Sb x alloy reached the ideal 1.7eV direct bandgap, complementary to the Si 1.1eV one. Furthermore, the analysis of the band alignment of GaP 1-x Sb x alloys and Si with respect to water potential levels shows that the GaPSb/Si association is an interesting combination both for the hydrogen evolution and oxygen evolution reactions, suggesting the GaP 1-x Sb x /Si tandem material holds great promise for high-efficiency solar water splitting on the low cost silicon substrate.
⋆E-mail: [email protected]
X-Ray Diffraction (XRD)
The synthesized GaP 1-x Sb x /Si samples were characterized by X-ray diffraction (XRD) on a 4 circles Brucker D8 Diffractometer. A Bartels asymmetric Ge (220) monochromator was used for both line scan and reciprocal space maps (RSM). The detection is ensured by a Lynxeye TM , 1 dimensional position sensitive detector (PSD) allowing a collection angle of 2.6° over 2θ.
The Reciprocal space maps were carried out on either (004) (Fig.1 insets) and (115) (Fig.S1) reflections, which show a full plastic relaxation of the GaP 1-x Sb x layer for the three samples.
Chemical Mechanical Polishing (CMP)
The samples were polished by chemical mechanical polishing method with 1% H 3 PO 4 etching solution at a rate 1 round/s for 30mins.
Scanning Electron Microscopy (SEM)
Scanning electron microscopy images were obtained by using a JEOL JSM-7100 scanning electron microscope. Fig.S2 shows the plane-view SEM images of the GaP 0.67 Sb 0.33 /Si, GaP 0.40 Sb 0.60 /Si, GaSb/Si samples, from which we can find the surface quality of the three samples are much improved.
Atomic Force Microscopy (AFM)
Atomic force microscopy (AFM) measurements were performed based on a Veeco Innova AFM microscope with a highresolution scanning probe. Tapping mode was used with the cantilever tuned around 293KHz.
The atomic force microscopy (AFM) measurements were made to study the surface roughness of the three samples before and after CMP quantitatively. Figure S3 shows the AFM images of the three as-grown samples. The RMS (root-meansquare) roughness of the surfaces were calculated at 22.80 nm for GaP 0.67 Sb 0.33 , 9.32nm for GaP 0.40 Sb 0.60 and 7.91nm for GaSb. Figure S4 shows the AFM images of the three samples after CMP and the surface RMS roughness of the three CMP samples are 0.33nm (GaP 0.67 Sb 0.33 ), 0.28nm (GaP 0.40 Sb 0.60 ) and 0.56nm (GaSb), indicating the surface of the three samples become very smooth after CMP processes.
Ellipsometry Measurement
The optical constants of the GaP 0.67 Sb 0.33 , GaP 0.40 Sb 0.60 , GaSb samples were measured by variable angle spectroscopic ellipsometry (VASE) at room temperature in the 0.58-5 eV photon energy region. The angles of incidence were set to 60˚ and 70˚. A Tauc-Lorentz model with 2 oscillators was used to fit the ellipsometry data and extract the absorption coefficient value. Fig.S5 show the fitting results for the GaP 0.67 Sb 0.33 , GaP 0.40 Sb 0.60 , GaSb samples, respectively. The red and blue lines correspond to experimental spectra where Is and Ic parameters are represented. Is and Ic are related to the well-known ellipsometry variables (amplitude component) and (phase difference) through the following relations: I s = sin(2).sin(), I c = sin(2).cos(). The black lines correspond to the theoretical curves after adjusting the parameters of the Tauc-Lorentz model. From this model, the refractive index (n), extinction coefficient (k) were extracted, as shown in Fig.S6. In order to further integrate and verify the experimental data, ellipsometry measurement (with incidence angle 70˚) was taken on one GaP/Si sample and the corresponding optical constants extracted based on the Tauc-Lorentz model are shown in Fig.S7. The deduced absorption curves of GaP and GaSb based on ellipsometry measurements were compared by the experimental and theoretical data in the reference [1] (Fig.S8), which show good compatibilities.
The absorption spectra were employed to plot the Tauc's curve ((αhν) k vs hν) for the four GaPSb samples ( Figure S9) based on the Tauc's law:
(αhν) k =C(hν-Eg) where: k=1/2 for indirect bandgap GaP and k=2 for direct bandgap GaP 0.67 Sb 0.33 ,GaP 0.40 Sb 0.60 ,GaSb which are verified by the tight binding calculation in the following part, α the absorption coefficient, h the Planck constant, ν the photon frequency and C a constant. Absorption coefficient range 3000-10000cm -1 were used for bandgap evaluation of the direct bandgap GaP 0.67 Sb 0.33 , GaP 0.40 Sb 0.60 , GaSb samples. While for GaP with indirect bandgap, the absorption coefficient corresponding to the bandgap is smaller and the absorption coefficient range was chosen at 100-5000cm -1 . The optical band gap can be obtained by linear extrapolation of the straight-line portion to α=0. Finally, the band gap ranges of the four samples were obtained: Eg=2.25±0.04 eV for GaP; Eg=1.70±0.06 eV for GaP 0.67 Sb 0.33 ; Eg=1.04±0.08 eV for GaP 0.40 Sb 0.60 , and Eg=0.68±0.09 eV for GaSb (as shown in the Fig.S9), which are consistent with bandgaps reported for the metamorphic growth of GaPSb on InP substrate [2].
Tight Binding Calculation
The band diagram of GaP 1-x Sb x alloys were calculated using an extended basis sp3d5s* tight binding Hamiltonian [3]. This method was proved to provide a band structure description with a sub-millielectronvolt precision throughout the Brillouin zone of binary cubic III-V and II-VI [4] semiconductors including quantum heterostructures [5] and surfaces [6]. From the tight binding parameters of GaP and GaSb binary compounds [3], a virtual crystal approximation is performed to obtain band structure of GaP 1-x Sb x alloy at different Sb content. The tight binding parameters of the virtual crystal are an arithmetic mean of the constituent materials weighted to their concentration [7] except for the diagonal matrix elements related to the atomic energies of anion "s-type" and 'p-type" states. For these states, bowing parameters are introduced to model the strong bowing of Γ bandgap of 2.7 eV [7]. With a bowing parameter equal to 9.0 eV for s-state and equal to 2.8 eV for p-state, the experimental bandgaps and absorption curves for different GaP 1-x Sb x alloys are nicely reproduced.
The room temperature band structure of GaP, GaP 0.67 Sb 0.33 , GaP 0.40 Sb 0.60 , GaSb semiconductors as representatives are shown in the Fig.S10, which has been put temperature effect of the energy shift [8] into consideration. We can see the GaP 0.67 Sb 0.33 , GaP 0.40 Sb 0.60 , GaSb semiconductors are direct band transitions and the bandgap energies are 1.673eV, 1.064eV, 0.726eV, respectively, and GaP is indirect band transitions and the bandgap energies are 2.268eV, which are in good agreement with the ellipsometry results.
Fig. 1 .
1(a) Schematic of the proposed GaP 1-x Sb x /Si tandem device for PEC water splitting. (b) Sunlight absorption illustration of GaP 1-x Sb x /Si tandem system with around 1.7/1.1eV bandgap combination for water splitting which can realize high light absorption.
Fig. 2 .
2X-Ray Diffraction patterns of three MBE-grown GaP 1-x Sb x /Si samples with different Sb contents: GaPSb-1 (a), GaPSb-2 (b) and GaPSb-3 (c). The insets show the reciprocal space maps around (004) for the three samples, correspondingly (Sx and Sz are the projected coordinates in the right handed Cartesian, with z axis parallel to the surface normal)[25].
Fig. 3 .
3Scanning electron microscopy (SEM) images of the GaP 0.67 Sb 0.33 , GaP 0.40 Sb 0.60 , GaSb three samples on plane view (a),(c),(e) and cross-section view(b),(d),(f).
Fig. 4 .
4Optical absorption spectra of the GaP 0.67 Sb 0.33 (a), GaP 0.40 Sb 0.60 (b), GaSb (c) semiconductors. The red lines were deduced from ellipsometry measurement and the black lines were obtained based on TB calculation.
The calculated band structures of GaP, GaP 0.67 Sb 0.33 , GaP 0.40 Sb 0.60 , and GaSb semiconductors are given in the Fig.S10]. The bandgap evolutions obtained in the Γ, L and X valleys of the GaP 1-x Sb x alloy for the whole Sb content range are presented in Fig.5, from which we can find both the band gap value and the band gap type (direct or indirect). For the GaP, GaP 0.67 Sb 0.33 , GaP 0.40 Sb 0.60 , and GaSb semiconductors, TB calculations are in good agreement with the experimental absorption measurements. The theoretical absorption curves of GaP 0.67 Sb 0.33 , GaP 0.40 Sb 0.60 , GaSb semiconductors (corresponding to the three MBE grown samples) determined with TB calculations are shown as black lines in
Fig. 6 .
6Band alignment of Γ, L and X valleys of GaP 1-x Sb x and indirect bandgap of Si with respect to water oxidation and reduction potentials. The relative position of redox potentials of water are shown as green bars at pH=0.From the
Fig. S1 .
S1X-Ray diffraction reciprocal space mappings around the (115) Bragg positions for the GaP 0.67 Sb 0.33 (a), GaP 0.40 Sb 0.60 (b), GaSb (c) samples. The red line represents the full relaxation line.
Fig. S2 .
S2Scanning electron microscopy (SEM) plane-view images of the GaP 0.67 Sb 0.33 (a), GaP 0.40 Sb 0.60 (b), GaSb (c) samples after CMP.
Fig. S3 .
S3Atomic force microscopy (AFM) images of the GaP 0.67 Sb 0.33 (a), GaP 0.40 Sb 0.60 (b), GaSb (c) samples before CMP.
Fig. S4 .
S4Atomic force microscopy (AFM) images of the GaP 0.67 Sb 0.33 (a), GaP 0.40 Sb 0.60 (b), GaSb (c) samples after CMP.
Fig. S5 .
S5Experimental ellipsometry spectra of Is and Ic two incidence angles (red and blue lines) and comparison with theoretical curves by using a 2-oscillators Tauc-Lorentz model (black lines) for GaP 0.67 Sb 0.33 (a), GaP 0.40 Sb 0.60 (b), GaSb (c) three samples, respectively.
Fig
Fig. S6. n (optical index real part) and k (optical index imaginary part) optical constants of GaP 0.67 Sb 0.33 (a), GaP 0.40 Sb 0.60 (b), GaSb (c) extracted from the fitting of Fig.S3 (a),(b),(c), respectively.
Fig. S7 .
S7(a) Experimental ellipsometry spectra of Is and Ic two incidence angles (red and blue lines) and comparison with theoretical curves by Tauc-Lorentz model (black lines) for GaP. (b) n (optical index real part) and k (optical index imaginary part) optical constants of GaP extracted from the fitting ofFig.S7 (a).
Fig. S8 .
S8Optical absorption spectra of GaP (a) and GaSb (b). The red lines were deduced from ellipsometry measurements based on our samples. The green experimental data dots and the black theoretical lines are obtained from the literature[1].Fig. S9. Tauc plot of (αhν) k versus photon energy (hν) for GaP (a), GaP 0.67 Sb 0.33 (b), GaP 0.40 Sb 0.60 (c), GaSb (d) samples.
Fig. S10 .
S10Band structures of the bulk unstrained GaP (a), GaP 0.67 Sb 0.33 (b), GaP 0.40 Sb 0.60 (c), GaSb (d) obtained by TB calculation at room temperature.
for Science and Technology (KACST), Riyadh, Saudi Arabia. Lipin Chen acknowledges the China Scholarship Council (CSC) for her Ph.D financial support (No. 2017-6254). Mahdi Alqahtani acknowledges the support and scholarship from King Abdulaziz City for Science and Technology, Riyadh, Saudi Arabia. The authors acknowledge RENATECH (French Network of Major Technology Centers) within Nanorennes for technological support.
AcknowledgementThis research was supported by the French National Research Agency ANTIPODE Project (Grant No. 14-CE26-0014-01), Région Bretagne and King Abdulaziz City
Artificial photosynthesis: solar splitting of water to hydrogen and oxygen. A J Bard, M A Fox, Acc. Chem. Res. 28A.J. Bard, M.A. Fox, Artificial photosynthesis: solar splitting of water to hydrogen and oxygen, Acc. Chem. Res. 28 (1995) 141- 145.
Powering the planet: Chemical challenges in solar energy utilization. N S Lewis, D G Nocera, Proc. Natl. Acad. Sci. 103N.S. Lewis, D.G. Nocera, Powering the planet: Chemical challenges in solar energy utilization, Proc. Natl. Acad. Sci. 103 (2006) 15729-15735.
. D Kang, T W Kim, S R Kubota, A C Cardiel, H G Cha, K.-S , D. Kang, T.W. Kim, S.R. Kubota, A.C. Cardiel, H.G. Cha, K.-S.
Electrochemical Synthesis of Photoelectrodes and Catalysts for Use in Solar Water Splitting. Choi, 10.1021/acs.chemrev.5b00498Chem. Rev. 115Choi, Electrochemical Synthesis of Photoelectrodes and Catalysts for Use in Solar Water Splitting, Chem. Rev. 115 (2015) 12839- 12887. doi:10.1021/acs.chemrev.5b00498.
Photoelectrochemical Hydrogen Production. R. van de Krol, M. GrätzelSpringer US31R. van de Krol, M. Grätzel, eds., Photoelectrochemical Hydrogen Production, Springer US, 2012. https://www.springer.com/gp/book/9781461413790 (accessed January 31, 2019).
Monolithic cells for solar fuels. J Rongé, T Bosserez, D Martel, C Nervi, L Boarino, F Taulelle, G Decher, S Bordiga, J A Martens, 10.1039/C3CS60424AChem. Soc. Rev. 43J. Rongé, T. Bosserez, D. Martel, C. Nervi, L. Boarino, F. Taulelle, G. Decher, S. Bordiga, J. A. Martens, Monolithic cells for solar fuels, Chem. Soc. Rev. 43 (2014) 7963-7981. doi:10.1039/C3CS60424A.
Strategies for stable water splitting via protected photoelectrodes. D Bae, B Seger, P C Vesborg, O Hansen, I Chorkendorff, Chem. Soc. Rev. 46D. Bae, B. Seger, P.C. Vesborg, O. Hansen, I. Chorkendorff, Strategies for stable water splitting via protected photoelectrodes, Chem. Soc. Rev. 46 (2017) 1933-1954.
An analysis of the optimal band gaps of light absorbers in integrated tandem photoelectrochemical water-splitting systems. S Hu, C Xiang, S Haussener, A D Berger, N S Lewis, 10.1039/C3EE40453FEnergy Environ. Sci. 6S. Hu, C. Xiang, S. Haussener, A.D. Berger, N.S. Lewis, An analysis of the optimal band gaps of light absorbers in integrated tandem photoelectrochemical water-splitting systems, Energy Environ. Sci. 6 (2013) 2984-2993. doi:10.1039/C3EE40453F.
Direct solar-to-hydrogen conversion via inverted metamorphic multi-junction semiconductor architectures. J L Young, M A Steiner, H Döscher, R M France, J A Turner, T G Deutsch, 10.1038/nenergy.2017.28Nat. Energy. 217028J.L. Young, M.A. Steiner, H. Döscher, R.M. France, J.A. Turner, T.G. Deutsch, Direct solar-to-hydrogen conversion via inverted metamorphic multi-junction semiconductor architectures, Nat. Energy. 2 (2017) 17028. doi:10.1038/nenergy.2017.28.
A Monolithic Photovoltaic-Photoelectrochemical Device for Hydrogen Production via Water Splitting. O Khaselev, J A Turner, 10.1126/science.280.5362.425Science. 280O. Khaselev, J.A. Turner, A Monolithic Photovoltaic- Photoelectrochemical Device for Hydrogen Production via Water Splitting, Science. 280 (1998) 425-427. doi:10.1126/science.280.5362.425.
Efficient direct solar-to-hydrogen conversion by in situ interface transformation of a tandem structure. M M May, H.-J Lewerenz, D Lackner, F Dimroth, T Hannappel, 10.1038/ncomms9286Nat. Commun. 68286M.M. May, H.-J. Lewerenz, D. Lackner, F. Dimroth, T. Hannappel, Efficient direct solar-to-hydrogen conversion by in situ interface transformation of a tandem structure, Nat. Commun. 6 (2015) 8286. doi:10.1038/ncomms9286.
Monolithic Photoelectrochemical Device for Direct Water Splitting with 19% Efficiency. W.-H Cheng, M H Richter, M M May, J Ohlmann, D Lackner, F Dimroth, T Hannappel, H A Atwater, H.-J Lewerenz, 10.1021/acsenergylett.8b00920ACS Energy Lett. 3W.-H. Cheng, M.H. Richter, M.M. May, J. Ohlmann, D. Lackner, F. Dimroth, T. Hannappel, H.A. Atwater, H.-J. Lewerenz, Monolithic Photoelectrochemical Device for Direct Water Splitting with 19% Efficiency, ACS Energy Lett. 3 (2018) 1795- 1800. doi:10.1021/acsenergylett.8b00920.
Universal description of III-V/Si epitaxial growth processes. I Lucci, S Charbonnier, L Pedesseau, M Vallet, L Cerutti, J.-B Rodriguez, E Tournié, R Bernard, A Létoublon, N Bertru, A Le Corre, S Rennesson, F Semond, G Patriarche, L Largeau, P Turban, A Ponchet, C Cornet, 10.1103/PhysRevMaterials.2.060401Phys. Rev. Mater. 260401I. Lucci, S. Charbonnier, L. Pedesseau, M. Vallet, L. Cerutti, J.-B. Rodriguez, E. Tournié, R. Bernard, A. Létoublon, N. Bertru, A. Le Corre, S. Rennesson, F. Semond, G. Patriarche, L. Largeau, P. Turban, A. Ponchet, C. Cornet, Universal description of III-V/Si epitaxial growth processes, Phys. Rev. Mater. 2 (2018) 060401. doi:10.1103/PhysRevMaterials.2.060401.
Electrically pumped continuous-wave III-V quantum dot lasers on silicon. S Chen, W Li, J Wu, Q Jiang, M Tang, S Shutts, S N Elliott, A Sobiesierski, A J Seeds, I Ross, P M Smowton, H Liu, 10.1038/nphoton.2016.21Nat. Photonics. 10S. Chen, W. Li, J. Wu, Q. Jiang, M. Tang, S. Shutts, S.N. Elliott, A. Sobiesierski, A.J. Seeds, I. Ross, P.M. Smowton, H. Liu, Electrically pumped continuous-wave III-V quantum dot lasers on silicon, Nat. Photonics. 10 (2016) 307-311. doi:10.1038/nphoton.2016.21.
Efficient solar water splitting. S Licht, B Wang, S Mukerji, T Soga, M Umeno, H Tributsch, 2S. Licht, B. Wang, S. Mukerji, T. Soga, M. Umeno, H. Tributsch, Efficient solar water splitting, exemplified by RuO2-catalyzed
. Algaas/Si Photoelectrolysis, J. Phys. Chem. B. 104AlGaAs/Si photoelectrolysis, J. Phys. Chem. B. 104 (2000) 8920- 8924.
Quantum dot activated indium gallium nitride on silicon as photoanode for solar hydrogen generation. P Kumar, P Devi, R Jain, S M Shivaprasad, R K Sinha, G Zhou, R Nötzel, 10.1038/s42004-018-0105-0Commun. Chem. 24P. Kumar, P. Devi, R. Jain, S.M. Shivaprasad, R.K. Sinha, G. Zhou, R. Nötzel, Quantum dot activated indium gallium nitride on silicon as photoanode for solar hydrogen generation, Commun. Chem. 2 (2019) 4. doi:10.1038/s42004-018-0105-0.
A Stress-Free and Textured GaP Template on Silicon for Solar Water Splitting. I Lucci, S Charbonnier, M Vallet, P Turban, Y Léger, T Rohel, N Bertru, A Létoublon, J.-B Rodriguez, L Cerutti, E Tournié, A Ponchet, G Patriarche, L Pedesseau, C Cornet, 10.1002/adfm.201801585Adv. Funct. Mater. 281801585I. Lucci, S. Charbonnier, M. Vallet, P. Turban, Y. Léger, T. Rohel, N. Bertru, A. Létoublon, J.-B. Rodriguez, L. Cerutti, E. Tournié, A. Ponchet, G. Patriarche, L. Pedesseau, C. Cornet, A Stress-Free and Textured GaP Template on Silicon for Solar Water Splitting, Adv. Funct. Mater. 28 (2018) 1801585. doi:10.1002/adfm.201801585.
Design of a latticematched III-V-N/Si photovoltaic tandem cell monolithically integrated on silicon substrate. A Rolland, L Pedesseau, J Even, S Almosni, C Robert, C Cornet, J M Jancu, J Benhlal, O Durand, A L Corre, P Rale, L Lombez, J.-F Guillemoles, E Tea, S Laribi, 10.1007/s11082-014-9909-zOpt. Quantum Electron. 46A. Rolland, L. Pedesseau, J. Even, S. Almosni, C. Robert, C. Cornet, J.M. Jancu, J. Benhlal, O. Durand, A.L. Corre, P. Rale, L. Lombez, J.-F. Guillemoles, E. Tea, S. Laribi, Design of a lattice- matched III-V-N/Si photovoltaic tandem cell monolithically integrated on silicon substrate, Opt. Quantum Electron. 46 (2014) 1397-1403. doi:10.1007/s11082-014-9909-z.
Gallium Phosphide photoanode coated with TiO2 and CoOx for stable photoelectrochemical water oxidation. M Alqahtani, S Ben-Jabar, M Ebaid, S Sathasivam, P Jurczak, X Xia, A Alromaeh, C Blackman, Y Qin, B Zhang, B S Ooi, H Liu, I P Parkin, J Wu, 10.1364/OE.27.00A364Opt. Express. 27M. Alqahtani, S. Ben-Jabar, M. Ebaid, S. Sathasivam, P. Jurczak, X. Xia, A. Alromaeh, C. Blackman, Y. Qin, B. Zhang, B.S. Ooi, H. Liu, I.P. Parkin, J. Wu, Gallium Phosphide photoanode coated with TiO2 and CoOx for stable photoelectrochemical water oxidation, Opt. Express. 27 (2019) A364-A371. doi:10.1364/OE.27.00A364.
S Hu, M R Shaner, J A Beardslee, M Lichterman, B S Brunschwig, N S Lewis, Amorphous TiO2 coatings stabilize Si, GaAs, and GaP photoanodes for efficient water oxidation. 344S. Hu, M.R. Shaner, J.A. Beardslee, M. Lichterman, B.S. Brunschwig, N.S. Lewis, Amorphous TiO2 coatings stabilize Si, GaAs, and GaP photoanodes for efficient water oxidation, Science. 344 (2014) 1005-1009.
H Döscher, O Supplie, M M May, P Sippel, C Heine, A G Muñoz, R Eichberger, H.-J Lewerenz, T Hannappel, Epitaxial III-V films and surfaces for photoelectrocatalysis. H. Döscher, O. Supplie, M.M. May, P. Sippel, C. Heine, A.G. Muñoz, R. Eichberger, H.-J. Lewerenz, T. Hannappel, Epitaxial III-V films and surfaces for photoelectrocatalysis, ChemPhysChem. 13 (2012) 2899-2909.
. A Martinez-Garcia, H B Russell, W Paxton, S Ravipati, S Calero-Barney, M Menon, E Richter, J Young, T Deutsch, M K Sunkara, Unassisted Water Splitting Using a GaSbxP (1-xA. Martinez-Garcia, H.B. Russell, W. Paxton, S. Ravipati, S. Calero-Barney, M. Menon, E. Richter, J. Young, T. Deutsch, M.K. Sunkara, Unassisted Water Splitting Using a GaSbxP (1-x)
. Photoanode, Adv. Energy Mater. 1703247Photoanode, Adv. Energy Mater. (2018) 1703247.
. H Nguyen-Van, A N Baranov, Z Loghmari, L Cerutti, J.-B , H. Nguyen-Van, A.N. Baranov, Z. Loghmari, L. Cerutti, J.-B.
Quantum cascade lasers grown on silicon. J Rodriguez, G Tournet, G Narcy, G Boissier, M Patriarche, E Bahriz, R Tournié, Teissier, 10.1038/s41598-018-24723-2Sci. Rep. 87206Rodriguez, J. Tournet, G. Narcy, G. Boissier, G. Patriarche, M. Bahriz, E. Tournié, R. Teissier, Quantum cascade lasers grown on silicon, Sci. Rep. 8 (2018) 7206. doi:10.1038/s41598-018-24723- 2.
Photoelectrochemical water oxidation of GaP1−xSbx with a direct band gap of 1.65 eV for full spectrum solar energy harvesting. M Alqahtani, S Sathasivam, L Chen, P Jurczak, R Piron, C Levallois, A Létoublon, Y Léger, S Boyer-Richard, N Bertru, J.-M Jancu, C Cornet, J Wu, I P Parkin, 10.1039/C9SE00113ASustain. Energy Fuels. 3M. Alqahtani, S. Sathasivam, L. Chen, P. Jurczak, R. Piron, C. Levallois, A. Létoublon, Y. Léger, S. Boyer-Richard, N. Bertru, J.- M. Jancu, C. Cornet, J. Wu, I.P. Parkin, Photoelectrochemical water oxidation of GaP1−xSbx with a direct band gap of 1.65 eV for full spectrum solar energy harvesting, Sustain. Energy Fuels. 3 (2019) 1720-1729. doi:10.1039/C9SE00113A.
Defects limitation in epitaxial GaP on bistepped Si surface using UHVCVD-MBE growth cluster. T Quinci, J Kuyyalil, T N Thanh, Y P Wang, S Almosni, A Létoublon, T Rohel, K Tavernier, N Chevalier, O Dehaese, J. Cryst. Growth. 380T. Quinci, J. Kuyyalil, T.N. Thanh, Y.P. Wang, S. Almosni, A. Létoublon, T. Rohel, K. Tavernier, N. Chevalier, O. Dehaese, Defects limitation in epitaxial GaP on bistepped Si surface using UHVCVD-MBE growth cluster, J. Cryst. Growth. 380 (2013) 157-162.
Structural and optical analyses of GaP/Si and (GaAsPN/GaPN)/GaP/Si nanolayers for integrated photonics on silicon. T Nguyen Thanh, C Robert, W Guo, A Létoublon, C Cornet, G Elias, A Ponchet, T Rohel, N Bertru, A Balocchi, J. Appl. Phys. 11253521T. Nguyen Thanh, C. Robert, W. Guo, A. Létoublon, C. Cornet, G. Elias, A. Ponchet, T. Rohel, N. Bertru, A. Balocchi, Structural and optical analyses of GaP/Si and (GaAsPN/GaPN)/GaP/Si nanolayers for integrated photonics on silicon, J. Appl. Phys. 112 (2012) 053521.
S Adachi, 10.1063/1.343580Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y. 66S. Adachi, Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y, J. Appl. Phys. 66 (1989) 6030-6040. doi:10.1063/1.343580.
GaPSb: A new ternary material for Schottky diode fabrication on InP. S Loualiche, A Le Corre, S Salaun, J Caulet, B Lambert, M Gauneau, D Lecrosnier, B Deveaud, Appl. Phys. Lett. 59S. Loualiche, A. Le Corre, S. Salaun, J. Caulet, B. Lambert, M. Gauneau, D. Lecrosnier, B. Deveaud, GaPSb: A new ternary material for Schottky diode fabrication on InP, Appl. Phys. Lett. 59 (1991) 423-424.
Empirical spds* tight-binding calculation for cubic semiconductors: General method and material parameters. J.-M Jancu, R Scholz, F Beltram, F Bassani, Phys. Rev. B. 576493J.-M. Jancu, R. Scholz, F. Beltram, F. Bassani, Empirical spds* tight-binding calculation for cubic semiconductors: General method and material parameters, Phys. Rev. B. 57 (1998) 6493.
Temperature dependence of the energy gap in semiconductors. Y P Varshni, Physica. 34Y.P. Varshni, Temperature dependence of the energy gap in semiconductors, Physica. 34 (1967) 149-154.
Temperature Dependence of the Energy Gap in GaAs and GaP. M B Panish, H C Casey, 10.1063/1.1657024J. Appl. Phys. 40M.B. Panish, H.C. Casey, Temperature Dependence of the Energy Gap in GaAs and GaP, J. Appl. Phys. 40 (1969) 163-167. doi:10.1063/1.1657024.
Photoluminescence of high-quality GaSb grown from Ga-and Sb-rich solutions by liquid-phase epitaxy. M Wu, C Chen, 10.1063/1.352216J. Appl. Phys. 72M. Wu, C. Chen, Photoluminescence of high-quality GaSb grown from Ga-and Sb-rich solutions by liquid-phase epitaxy, J. Appl. Phys. 72 (1992) 4275-4280. doi:10.1063/1.352216.
Universal alignment of hydrogen levels in semiconductors, insulators and solutions. C G Van De Walle, J Neugebauer, Nature. 423626C.G. Van de Walle, J. Neugebauer, Universal alignment of hydrogen levels in semiconductors, insulators and solutions, Nature. 423 (2003) 626.
Synthesis and properties of antimonide nanowires. B M Borg, L.-E Wernersson, Nanotechnology. 24202001B.M. Borg, L.-E. Wernersson, Synthesis and properties of antimonide nanowires, Nanotechnology. 24 (2013) 202001.
Efficient solar hydrogen production by photocatalytic water splitting: from fundamental study to pilot demonstration. D Jing, L Guo, L Zhao, X Zhang, H Liu, M Li, S Shen, G Liu, X Hu, X Zhang, Int. J. Hydrog. Energy. 35D. Jing, L. Guo, L. Zhao, X. Zhang, H. Liu, M. Li, S. Shen, G. Liu, X. Hu, X. Zhang, Efficient solar hydrogen production by photocatalytic water splitting: from fundamental study to pilot demonstration, Int. J. Hydrog. Energy. 35 (2010) 7087-7097.
S Chu, W Li, Y Yan, T Hamann, I Shih, D Wang, Z Mi, Roadmap on solar water splitting: current status and future prospects. 122001S. Chu, W. Li, Y. Yan, T. Hamann, I. Shih, D. Wang, Z. Mi, Roadmap on solar water splitting: current status and future prospects, Nano Futur. 1 (2017) 022001.
Heterogeneous photocatalyst materials for water splitting. A Kudo, Y Miseki, Chem. Soc. Rev. 38A. Kudo, Y. Miseki, Heterogeneous photocatalyst materials for water splitting, Chem. Soc. Rev. 38 (2009) 253-278.
S Adachi, 10.1063/1.343580Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y. 66S. Adachi, Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y, J. Appl. Phys. 66 (1989) 6030-6040. doi:10.1063/1.343580.
GaPSb: A new ternary material for Schottky diode fabrication on InP. S Loualiche, A Le Corre, S Salaun, J Caulet, B Lambert, M Gauneau, D Lecrosnier, B Deveaud, Appl. Phys. Lett. 59S. Loualiche, A. Le Corre, S. Salaun, J. Caulet, B. Lambert, M. Gauneau, D. Lecrosnier, B. Deveaud, GaPSb: A new ternary material for Schottky diode fabrication on InP, Appl. Phys. Lett. 59 (1991) 423-424.
Empirical spds* tight-binding calculation for cubic semiconductors: General method and material parameters. J.-M Jancu, R Scholz, F Beltram, F Bassani, Phys. Rev. B. 576493J.-M. Jancu, R. Scholz, F. Beltram, F. Bassani, Empirical spds* tight-binding calculation for cubic semiconductors: General method and material parameters, Phys. Rev. B. 57 (1998) 6493.
Atomistic simulations of the optical absorption of type-II CdSe/ZnTe superlattices. S Boyer-Richard, C Robert, L Gérard, J.-P Richters, R André, J Bleuse, H Mariette, J Even, J.-M Jancu, Nanoscale Res. Lett. 7543S. Boyer-Richard, C. Robert, L. Gérard, J.-P. Richters, R. André, J. Bleuse, H. Mariette, J. Even, J.-M. Jancu, Atomistic simulations of the optical absorption of type-II CdSe/ZnTe superlattices, Nanoscale Res. Lett. 7 (2012) 543.
Calculation of Electronic States in Semiconductor Heterostructures with an Empirical spds* Tight-Binding Model. R Scholz, J.-M Jancu, F Beltram, F Bassani, Phys. Status Solidi B. 217R. Scholz, J.-M. Jancu, F. Beltram, F. Bassani, Calculation of Electronic States in Semiconductor Heterostructures with an Empirical spds* Tight-Binding Model, Phys. Status Solidi B. 217 (2000) 449-460.
Full band approach to tunneling in MOS structures. F Sacconi, A Di Carlo, P Lugli, M Stadele, J.-M Jancu, IEEE Trans. Electron Devices. 51F. Sacconi, A. Di Carlo, P. Lugli, M. Stadele, J.-M. Jancu, Full band approach to tunneling in MOS structures, IEEE Trans. Electron Devices. 51 (2004) 741-748.
Optical properties of potential-inserted quantum wells in the near infrared and Terahertz ranges. F Raouafi, R Samti, R Benchamekh, R Heyd, S Boyer-Richard, P Voisin, J.-M Jancu, Solid State Commun. 236F. Raouafi, R. Samti, R. Benchamekh, R. Heyd, S. Boyer-Richard, P. Voisin, J.-M. Jancu, Optical properties of potential-inserted quantum wells in the near infrared and Terahertz ranges, Solid State Commun. 236 (2016) 7-11.
Temperature dependence of the energy gap in semiconductors. Y P Varshni, Physica. 34Y.P. Varshni, Temperature dependence of the energy gap in semiconductors, Physica. 34 (1967) 149-154.
| [] |
[
"Parton energy loss in a hard-soft factorized approach",
"Parton energy loss in a hard-soft factorized approach"
] | [
"Tianyu Dai \nDepartment of Physics\nDuke University\n27708-0305DurhamNorth CarolinaUSA\n",
"Jean-François Paquet \nDepartment of Physics\nDuke University\n27708-0305DurhamNorth CarolinaUSA\n",
"Derek Teaney \nDepartment of Physics and Astronomy\nStony Brook University\n11794-3800New YorkUSA\n",
"Steffen A Bass \nDepartment of Physics\nDuke University\n27708-0305DurhamNorth CarolinaUSA\n"
] | [
"Department of Physics\nDuke University\n27708-0305DurhamNorth CarolinaUSA",
"Department of Physics\nDuke University\n27708-0305DurhamNorth CarolinaUSA",
"Department of Physics and Astronomy\nStony Brook University\n11794-3800New YorkUSA",
"Department of Physics\nDuke University\n27708-0305DurhamNorth CarolinaUSA"
] | [] | An energetic parton travelling through a quark-gluon plasma loses energy via occasional hard scatterings and frequent softer interactions. Whether or not these interactions admit a perturbative description, the effect of the soft interactions can be factorized and encoded in a small number of transport coefficients. In this work, we present the numerical implementation of a hard-soft factorized parton energy loss model which combines a stochastic description of soft interactions and rate-based modelling of hard scatterings. We introduce a scale to estimate the regime of validity of the stochastic description, allowing for a better understanding of the model's applicability at small and large coupling. We study the energy and fermion-number cascade of energetic partons as an application of the model. * [email protected] 1 The role of hadronic energy loss is still under investigation. SeeRef.[16] for example. 2 We use a parameter "pcut" to define which partons we consider as "energetic". We only track the propagation of these energetic partons with p > pcut. We use pcut = 2 GeV throughout this work since we focus on the energy loss of light partons. | 10.1103/physrevc.105.034905 | [
"https://arxiv.org/pdf/2012.03441v3.pdf"
] | 227,343,940 | 2012.03441 | ff53511243b6a5d44ed6254237b2104f3b6ef4a7 |
Parton energy loss in a hard-soft factorized approach
Tianyu Dai
Department of Physics
Duke University
27708-0305DurhamNorth CarolinaUSA
Jean-François Paquet
Department of Physics
Duke University
27708-0305DurhamNorth CarolinaUSA
Derek Teaney
Department of Physics and Astronomy
Stony Brook University
11794-3800New YorkUSA
Steffen A Bass
Department of Physics
Duke University
27708-0305DurhamNorth CarolinaUSA
Parton energy loss in a hard-soft factorized approach
(Dated: April 25, 2022)
An energetic parton travelling through a quark-gluon plasma loses energy via occasional hard scatterings and frequent softer interactions. Whether or not these interactions admit a perturbative description, the effect of the soft interactions can be factorized and encoded in a small number of transport coefficients. In this work, we present the numerical implementation of a hard-soft factorized parton energy loss model which combines a stochastic description of soft interactions and rate-based modelling of hard scatterings. We introduce a scale to estimate the regime of validity of the stochastic description, allowing for a better understanding of the model's applicability at small and large coupling. We study the energy and fermion-number cascade of energetic partons as an application of the model. * [email protected] 1 The role of hadronic energy loss is still under investigation. SeeRef.[16] for example. 2 We use a parameter "pcut" to define which partons we consider as "energetic". We only track the propagation of these energetic partons with p > pcut. We use pcut = 2 GeV throughout this work since we focus on the energy loss of light partons.
I. INTRODUCTION
The production of energetic hadrons and jets in heavy ion collisions is markedly different from the production of energetic electroweak bosons. The latter clearly exhibit "binary scaling": weak bosons and high-energy photons are produced as if nucleons from each nucleus were independently undergoing inelastic binary collisions [1][2][3][4][5][6][7] (see also Refs. [8,9] and references therein). Hadron and jet measurements, on the other hand, display evident deviations from binary scaling. These deviations are understood to be a consequence of the formation of a quark-gluon plasma in relativistic nuclear collisions: energetic parton production does follow "binary scaling"; it is their subsequent interactions with the plasma that lead to parton energy loss, and consequently to an apparent deviation from binary scaling for hadronic observables.
This characteristic phenomena of jet and hadron "energy loss" in heavy ion collisions has been observed at both the Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC) [10][11][12][13][14][15]. Energetic partons are produced at the earliest stage of heavy ion collisions, and they propagate through all the different phases of the collisions. As a consequence of their interactions with the quark-gluon plasma, the momentum distribution of these energetic partons changes distinctly compared to the baseline observed in proton-proton collisions. 1 This makes them important probes of the deconfined nuclear plasma produced in heavy ion collisions.
A number of different formalisms have been used to model the interaction of energetic light partons 2 with the constituents of the plasma [17][18][19][20][21][22][23][24][25][26][27] (see also Refs. [10,[28][29][30] and references therein). Fundamentally, most parton energy loss formalisms have a well-understood common core, yet applications to heavy ion collisions tend to require approximations and practical considerations that lead to non-negligible differences between parton energy loss models [28][29][30]. One difference between the models is the treatment of the underlying plasma, which is often assumed to be made of a large number of quarks and gluons with energies of 1 GeV in near local thermal equilibrium. Whether these quarks and gluons are treated as dynamical entities or as static scattering centers is one of many differences in the energy loss formalisms [28][29][30]. The above assumption is important, given that the quark-gluon plasma produced in heavy ion collisions is understood to be strongly coupled [31], and a quasiparticle description may not be justified. 3 A different phrasing of the above challenge is that the energy loss of even very energetic partons can be affected by non-perturbative effects from the stronglycoupled plasma. Hard interactions -those with large momentum transfer between the energetic parton and the plasma -are expected to have smaller non-perturbative effects, or even be accessible perturbatively, as a consequence of the running of the QCD coupling. On the other hand, "soft" parton-plasma interactions with small momentum transfer are expected to suffer the largest nonperturbative effects. 4 We note that "hard interactions" and "soft interactions" have various meanings in the literature, but for the purpose of this work, the temperature of the plasma can be considered as the scale separating hard (larger than T ) and soft (smaller than T ) interactions.
A stochastic treatment of these soft interactions of energetic partons provides an alternative approach to account for non-perturbative effects -an approach that is agnostic to the strongly-or weakly-coupled nature of the underlying deconfined plasma. The dynamical details of the large number of soft interactions are encoded in a small number of transport coefficients. The latter can be parametrized and constrained from measurements. They can also be studied using lattice techniques (see for example Ref. [34,Section 4] and Refs. [35,36] and references therein). From a practical point of view, a stochastic description of a large number of soft interactions can also be more efficient numerically than a rate-based approach.
A systematic hard-soft factorization of parton energy loss was proposed recently to describe parton propagation in a weakly-coupled QGP [34,37]. In this factorization, soft interactions are described as a stochastic process with drag and diffusion transport coefficients calculated perturbatively; hard interactions are solved with rates that are also calculated perturbatively. In the weakly-coupled regime, parton energy loss in this hardsoft factorizated scheme was shown to be equivalent to a fully rate-based treatment of parton energy loss [37]. Importantly parton energy loss in this hard-soft factorization can also be extended to next-to-leading order [37], a feature beyond the scope of this work which we shall explore in the future.
As discussed above, the drag and diffusion contribution to parton energy loss can be factorized systematically, and calculated non-perturbatively e.g. based on Electrostatic Quantum Chromodynamics (EQCD) [35], or fitted to data. These extractions will then depend on the separation scale µ, which appears in the approach. At higher order, the drag and diffusion coefficients will evolve with the scale µ ∼ πT , incorporating in a consistent way the running of the coupling. While this is beyond the scope of this work, we hope that this manuscript can provide a first step in that direction. Throughout the paper we will already study the dependence of various observables on the separation scale µ, and, encouragingly, find that this dependence is moderate in most cases.
The above work is based on the "effective kinetic theory" approach [25] derived for a weakly-coupled quarkgluon plasma. In a weakly-coupled plasma, quark and gluon excitations are described as quasi-particles with effective properties related to the local density of the plasma. In this effective kinetic approach, the dynamics of quasi-particles are described by Boltzmann transport equations. Leading order [O(α s )] realizations of this effective kinetic approach -extrapolated to large values of strong coupling constant α s -have been used widely to study parton energy loss (see e.g. Refs. [26,[38][39][40][41]).
In this work, we present the first numerical implementation of the hard-soft factorized parton energy loss model [37] discussed above. For our implementation we utilize the publicly available JETSCAPE framework [42], as it allows us a straightforward integration of our parton energy loss model with the other ingredients necessary for a full simulation of jet production in heavy ion collisions. We first test and validate this factorization of parton energy loss in the weak coupling regime for a static medium.
We introduce a dimensionless scale to quantify the kinematic range for which soft interactions can be described accurately with a stochastic approach. We use this scale to discuss a hard-soft factorization model for a strongly-coupled quark-gluon plasma, relevant for phenomenological applications in heavy ion collisions.
Finally, we present an application of our new factorized model of parton energy loss by calculating the energy and fermion-number cascade of an energetic parton propagating in a static medium, finding good agreement with known analytical approximations. The evolution of an energetic parton in a thermal medium of temperature T can be described by a Boltzmann transport equation [43]:
∂ ∂t + p p · ∇ δf a = −C[δf a , n a ](1)
where P = (p, p) is the four-momentum of the energetic parton and C is the collision kernel of the parton with the medium. The index a represents partons with a certain color and helicity state. We use the same notation for the parton momentum distributions as in Ref. [37]: the distribution of rare energetic partons of type a is δf a (p, x, t), to distinguish it from the quasi-thermal distribution of soft particles n a (p, T (x, t), u(x, t)), where u is the flow velocity. In this notation, the total phase space distribution of quasiparticle a is f a (p, x, T ) = n a (p, T (x, t), u(x, t)) + δf a (p, x, t). We assume p T and g ≡ αs 4π 1. Because interactions between energetic partons themselves are rare and can be neglected, the Boltzmann equation is effectively linear in δf a .
At leading order, the interactions between quasiparticles can be divided as 2 ↔ 2 elastic interactions and 1 ↔ 2 inelastic interactions. Elastic 2 ↔ 2 processes refer to elementary scatterings involving two incoming particles and two outgoing particles without any radiation. Multiple soft 2 ↔ 2 scatterings between the energetic parton and the plasma can induce a collinear radiation. In the effective kinetic approach, these multiple soft scatterings are resummed consistently, to account for interference between subsequent collisions which lead to the Landau-Pomeranchuk-Migdal (LPM) effect. This resummed collinear radiation is known as the effective 1 ↔ 2 process. The collision kernel of both 2 ↔ 2 and 1 ↔ 2 processes can be written as
C = C 1↔2 + C 2↔2 .(2)
Importantly, in our approach, we only follow the evolution of energetic partons with an energy above a cutoff p cut = 2 GeV. Our assumption is that we should focus our efforts on high-p T observables which are dominated by partons above this cutoff. After neglecting terms suppressed by exp(−p/T ), the collision kernels C 1↔2 and C 2↔2 read [37]: where the notation for the Lorentz-invariant integration is
C 1↔2 a [δf ] = (2π) 3 2|p| 2 ν a bc ∞ 0 dp dq γ a bc (p; p p, q p)δ(|p| − p − q ) × δf a (p) 1 ± n b (p ) ± n c (q ) − δf b (p p)n c (q ) + n b (p )δf c (k p) + (2π) 3 |p| 2 ν a bc ∞ 0 dqdp γ c ab (p p; p, qp)δ(|p| + q − p ) × δf a (p)n b (q) − δf c (p p) 1 ± n b (q) ,(3)C 2↔2 a [δf ] = 1 4|p|ν a bcd kp k |M ab cd (p, k; p , k )| 2 (2π) 4 δ (4) (P + K − P − K ) × δf a (p)n b (k) 1 ± n c (p ) ± n d (k ) − δf c (p )n d (k ) 1 ± n b (k) − n c (p )δf d (k ) 1 ± n b (k) ,(4)k · · · ≡ d 3 k 2k(2π) 3 . . .(5)
and ν a is the degeneracy of particle a.
For C 1↔2 , a is the incoming hard parton with the momentum p, and b, c are outgoing particles with the momentum p , k . γ a bc is the splitting kernel of a → bc, which can be calculated with the AMY integral equations [25,26]. For C 2↔2 , particle a is the incoming energetic parton with momentum p, particle b is the plasma particle with the momentum k interacting with a, and particle c, d are the outgoing particles with momentum p , k . M ab cd is the matrix element of the elementary process ab → cd [25].
B. Reformulating parton energy loss with hard-soft factorization
In the hard-soft factorized parton energy loss model introduced in Ref. [37], 1 ↔ 2 and 2 ↔ 2 processes are further divided into soft interactions and hard interactions. The collision kernel is rewritten as
C = C 1↔2 hard + C 1↔2 soft + C 2↔2 hard + C 2↔2 soft .(6)
In this hard-soft factorized model, soft interactions described by C 1↔2 soft and C 2↔2 soft are treated stochastically with the Langevin equation.
Hard inelastic interactions, C 1↔2 hard , are treated with an emission rate as calculated from the AMY integral equations [25]. 5 We refer to them as large-ω interactions.
The hard 2 ↔ 2 part C 2↔2 hard is further divided as (i) large-angle interactions, and (ii) splitting approximation processes, based on the energy transfer ω:
C 2↔2 hard = C 2↔2 large-angle + C 2↔2 split .(7)
The physical meaning of this separation is the following. Elastic collisions occur between an energetic parton (p T ) and a lower energy quasi-thermal quark or gluon (k ∼ T ). On rare occasions, the momentum transfer in these elastic collisions is sufficient to make the low-energy
> − FIG. 2.
Example of inelastic interaction, in which multiple soft scatterings induce the radiation of a soft gluon with energy ω. We denote the radiations with ω > µω as large-ω inelastic interactions. quark or gluon become an energetic parton with k T ; such partons are referred to in the literature as "recoil partons". The process through which a recoil parton is produced is akin to a splitting process: a single energetic particle splits in two energetic ones. The kinematic of this process simplifies and it benefits from being treated separately.
The factorization of the phase space for this reformulation is summarized in Fig. 1. An ensemble of cutoffs is used to divide the different regions of phase space. We discuss the details of the different treatments and these cutoffs in the following subsections. The diagram of a typical 1 ↔ 2 inelastic interaction is shown as Fig. 2. We assume the energy of the radiated particle is ω. We define a hard-soft cutoff µ ω based on the radiated energy ω, to divide C 1↔2 hard and C 1↔2 soft . In the weakly-coupled regime, the cutoff µ ω is limited to µ ω T , where T is the temperature of the thermal medium.
Collinear radiations with energy ω > µ ω are included into the hard part, C 1↔2 hard ; they are treated as usual with emission rates calculated from AMY's integral equations as in Eq. (3).
Treatment of hard interactions: elastic case (2 ↔ 2)
The diagram of a typical 2 ↔ 2 elastic interaction is shown in Fig. 3. We define the momentum transfer between the two incoming particles as Q = (ω, q); the four-momenta of the incoming and outgoing particles are P = (p, p), K = (k, p), P = P − Q, and K = K + Q. Usingq ⊥ ≡ q 2 − ω 2 and ω, we divide the phase space of elastic interactions as • Large-angle scattering C 2↔2 large-angle :q ⊥ > µq ⊥ and ω < Λ;
• "Splitting-like" process C 2↔2 split : ω > Λ with a. Large-angle scattering (C 2↔2 large-angle ) Hard scatterings withq ⊥ > µq ⊥ and ω < Λ are denoted as large-angle interactions, because the scattering angle
C 2↔2 hard = C 2↔2 large-angle + C 2↔2 split .(8)q ⊥ q z ≈q ⊥ ω(9)
is generally large in this region. The cutoff µq ⊥ is typically assumed to be gT µq ⊥ T [37], although we will see in Section III that this condition can be relaxed. We assume p ω in this region, and simplify the matrix elements accordingly.
We use vacuum matrix elements for C 2↔2 hard , because the screening effects are only significant for soft interactions (C 2↔2 soft ) in the weakly-coupled regime [37]. Since we are only interested in the evolution of energetic partons, we keep terms to the first order in T /p in the matrix elements.
The treatment of the regionq ⊥ > µq ⊥ and p−Λ < ω < p -which is handled differently for technical reasonsis discussed in Appendix B.
b. Splitting approximation (C 2↔2 split ) When both of the outgoing particles of a 2 ↔ 2 interaction are hard (p , k > p cut ), the interaction can be effectively considered as a splitting process. The splitting leads to a hard recoil parton which should be included in the calculation.
We use a cutoff Λ on ω to distinguish two hard outgoing particles from only one hard outgoing particles. In principle, this cutoff Λ should be 3T Λ p. In the numerical implementation, unless specified otherwise, we choose Λ = min( √ 3pT , p cut ) to divide C 2↔2 split and C 2↔2 large−angle . Recall that we use p cut = 2 GeV in this work. As shown in Fig. 3, splitting approximation process is the 2 ↔ 2 interactions with Λ < ω < p − Λ.
At the interface between the phase space of largeangle scattering (C 2↔2 large-angle ) and splitting-like processes (C 2↔2 split ), the two collision kernels should be consistent. We verified this in Fig. 4: the differential rates of C 2↔2 split and C 2↔2 large-angle are compatible in √ 3p cut T < ω < p cut . As long as we choose the cutoff Λ in this range, this division of C 2↔2 hard should be consistent. A detailed discussion of the splitting approximation The differential rate of splitting approximation interactions and large-angle interactions for gg ↔ gg process when αs = 0.3. The shaded area is the region of √ 3pcutT < ω < pcut. dΓvac/dω is the differential rate of vacuum matrix elements for 2 ↔ 2 interactions. The results are for p0 = 100 GeV. Note that in the numerical implementation, we double-count the large-angle interaction rate because we only sample in half of the phase space. Here, to compare with splitting approximation rate, we decrease the large-angle interaction rate in the numerical implementation by a factor of 1 2 to cancel out the double-count.
process is in Appendix D. The p T and ω T kinematic cuts lead to significant simplifications for the matrix elements entering into C 2↔2 split .
Treatment of soft interactions
In the hard-soft factorized approach, the large number of soft interactions are described stochastically with drag and diffusion coefficients. When the momentum transfer is small, the Boltzmann equation [Eq. (1)] can be approximated as a Fokker-Planck equation. The collision kernel of the Fokker-Planck equation is written as:
C 1↔2,2↔2 diff = C 1↔2 soft [δf ] + C 2↔2 soft [δf ] = − ∂ ∂p i η D,soft p i δf − 1 2 ∂ 2 ∂p i ∂p j × p ipjq L,soft + 1 2 δ ij −p ipj q soft δf ,(10)
where η D,soft is the drag coefficient of the soft interactions,q L,soft andq soft are the longitudinal and transverse momentum diffusion coefficients of the soft interactions.
In the diffusion process, the number and the identity of the particles are preserved. Since the soft radiations of the 1 ↔ 2 process are absorbed by the plasma, the number of particles is also preserved. We include both the soft 1 ↔ 2 and 2 ↔ 2 collisions in the diffusion process. The diffusion process can be solved using a Langevin equation [44] in the numerical implementation.
For soft 1 ↔ 2 process, we can obtain the perturbative longitudinal diffusion coefficients by expanding C 1↔2 and only keeping the soft radiation terms. At leading order in α s ,q
1↔2 L,soft = (2 − ln 2)g 4 C R C A T 2 µ ω 4π 3 ,(11)
where C R is the Casimir factor. For gluons, C R = C A , while for quarks, C R = C F . 6 The derivation of this value can be found in Appendix A. We assume that the radiation angle is zero for collinear radiations. Consequently the transverse diffusion coefficient of 1 ↔ 2 interactions is approximated as zero.
For soft 2 ↔ 2 processes, the diffusion coefficients can be calculated perturbatively; a modern derivation can be found in Ref. [37]. The transverse momentum diffusion coefficient due to soft scatterings iŝ
q 2↔2 soft = g 2 C R T m 2 D 4π ln 1 + µq ⊥ m D 2 ,(12)
where
m 2 D ≡ g 2 T 2 (N c /3+N f /6)
is the square of the leading order Debye mass, N c = 3 is the number of colors and N f is the number of flavors involved in the interactions.
The longitudinal diffusion coefficient at order O(α s ) iŝ q 2↔2 L,soft = g 2 C R T M 2 ∞ 4π ln 1 + µq ⊥ M ∞ 2 ,(13)
where M ∞ ≡ m 2 D /2 is the gluon asymptotic thermal mass [37,45].
Since detailed balance is preserved in the Fokker-Planck equation, as verified in Appendix E, the drag coefficient η D can be calculated from diffusion coefficients according to Einstein relation for both soft 1 ↔ 2 and 2 ↔ 2 processes:
η D,soft (E) =q L,soft 2T p 1 + O T p .(14)
Equations (11)(12)(13)(14) assume that the coupling α s is small. We discuss the range of validity of the perturbative coefficients in Section III A 1. Our long-term goal is to treat q soft andq L,soft as non-perturbative parameters, incorporating much more physics than leading order scattering. These parameters could then either be constrained with lattice inputs [35] or fitted to experimental data, e.g. with the Bayesian approach [46,47]. In either case, the results will depend on the separation scale µ, and this dependence would then have to match with the hard sector at LO (order g 2 ), NLO (order g 3 ), and NNLO (order g 4 , the first order the coupling runs). Ideally the hard sector, and thus the evolution with µ can be treated perturbatively. As a first step we will study the sensitivity to the scale separation µ in this manuscript.
Besides the identity preserving diffusion process, the identity of the particle can be converted through soft fermion exchange with the medium. This exchange must be screened with the non-perturbative HTL resummation scheme. In the hard-soft factorized approach adopted here, we separate the 2 ↔ 2 processes with fermion exchange into hard collisions withq ⊥ > µq ⊥ , and soft collisions withq ⊥ < µq ⊥ (see Fig. 1). The hard exchange collisions are treated with vacuum matrix elements, while the soft exchange collisions are incorporated into a conversion rate Γ conv q→g (p) for q → g:
Γ conv q→g (p) = g 2 C F m 2 ∞ 16πp log 1 + µ 2 q ⊥ m 2 ∞ .(15)
Here m 2 ∞ is the fermion asymptotic mass, m 2 ∞ = g 2 C F T 2 /4 [37,45]. In each time step there is a probability ∆t Γ conv for a quark to become a gluon, with the same momentum, and vice versa. Further details about the conversion rate C 2↔2 conv are given in Appendix C. In the future, the non-perturbative conversion coefficient Γ conv q→g can be taken from a next-to-leading order analysis [37], or can be determined from data in a Bayesian approach.
Summary
In summary, the collision kernel of hard-soft factorized model is reformulated as
C = C 2↔2 + C 1↔2 = C 1↔2 large-ω (µ ω ) + C 2↔2 large-angle (µq ⊥ , Λ) + C 2↔2 split (Λ) +C 1↔2,2↔2 diff (µ ω , µq ⊥ ) + C 2↔2 conv (µq ⊥ ) .(16)
The cutoff dependence of the stochastic description is cancelled in Eq. (16) by the cutoff dependence of the hard interactions. That is, each individual process in the hard-soft factorized model is dependent on the cutoff, but this dependence cancels out when all the processes are summed. We show this explicitly in Section III.
C. Running of the strong coupling αs
All discussions up to this point assumed that the strong coupling constant α s is fixed at a given small value. It is clear, however, that the strong coupling constant will be different for soft and hard interactions; this is in fact a key assumption of the present model: hard interactions are more perturbative than soft ones, because the coupling constant scales inversely with the momentum exchange between the energetic parton and the plasma (see Ref.
[29, Section V] for a discussion, for example). The running is slow (logarithmic in the momentum exchange), however, more studies will be necessary to understand the exact magnitude of loop corrections or nonperturbative effects on soft and hard collisions.
As a first step in introducing our model of parton energy loss, we keep the strong coupling constant α s fixed throughout the manuscript.
III. HARD-SOFT FACTORIZATION OF PARTON ENERGY LOSS IN THE WEAKLY-COUPLED REGIME: NUMERICAL STUDY
In the first part of this section, we compare the analytical equations for the soft-interaction parton transport coefficients [Eqs. (11)(12)(13)] with their numerical values evaluated from the matrix elements, and summarize the range of cutoff and coupling where they are consistent. We also compare (i) soft interactions modelled with matrix elements with (ii) soft interactions modelled with the Langevin equation. We perform this test in the weak coupling limit. We use this discussion to review the range of validity of the Fokker-Planck equation and its stochastic Langevin implementation.
In the second part of this section, we compute the energy loss of an energetic parton in a brick and discuss the dependence of the results on the soft-hard cutoffs introduced in Section II B.
A. Treatment of soft interactions
Soft interactions can be described either stochastically with transport coefficients, or microscopically with matrix elements. In what follows, we compare these two descriptions, with particular emphasis on the effect of the soft-hard cutoffs and of the coupling constant.
The tests performed in the present subsection are not expected to be related to exact composition of the plasma (number of quark flavors). Thus, for simplicity, the calculations are performed in the pure glue limit (N f = 0).
Analytical and numerical calculation of soft transport coefficients
In a weakly-coupled quark-gluon plasma, the drag and diffusion coefficients for soft interactions can be calculated analytically using perturbation theory [ Eqs. (11)(12)(13) ], as discussed in Section II B 3. The same drag and diffusion coefficients can be obtained by direct numerical integration of the parton energy loss rates; these rates are calculated from matrix elements screened by plasma effects [43].
The diffusion coefficients are defined as [37]
q(p) ≡ d dt (∆p ⊥ ) 2 , q L (p) ≡ d dt (∆p L ) 2 ,(17)
where ∆p ⊥ is the momentum change perpendicular to the direction of the energetic parton, and ∆p L is the longitudinal momentum change of the parton. The brackets represent an average over all interactions during the parton propagation. The numerical soft diffusion rates are thus calculated as [37] q 2↔2
soft (p) = µq ⊥ 0 dq ⊥ Λ −∞ dωq 2 ⊥ d 2 Γ(p, q) dωdq ⊥ 2↔2 , q 2↔2 L,soft (p) = µq ⊥ 0 dq ⊥ Λ −∞ dωω 2 d 2 Γ(p, q) dωdq ⊥ 2↔2 , q 1↔2 L,soft (p) = µω −µω dωω 2 dΓ(p, q) dω 1↔2 ,(18)
where dΓ(p, q)/dω and d 2 Γ(p, q)/dωdq ⊥ are the rates for an energetic parton with four-momentum (p, p) to undergo a four-momentum change (ω, q) calculated using screened matrix elements. The initial parton energy p is assumed to be much larger than all other energy scales in the problem, effectively p → ∞. The cutoffs µq ⊥ , Λ and µ ω are used to limit the phase space of interactions included in the transport coefficients, in the present case to limit the interactions to soft ones only. There are two important differences between Eq. (18) and the analytical diffusion coefficients Eqs. (11)(12)(13). First, Eq. (18) is formally valid for arbitrarily large cutoffs (µq ⊥ , Λ and µ ω ), while Eqs. (11)(12)(13) assume the cutoff to be at most of order T . Second, there is the question of the smallness of the coupling. Equations (11)(12)(13) are derived assuming α s 1. Equation (18) is valid at arbitrarily coupling, although the rates dΓ(p, q)/dω and d 2 Γ(p, q)/dωdq ⊥ themselves are typically calculated perturbatively. 7 A comparison of Eq. (18) and the analytical diffusion coefficients Eqs. (11)(12)(13) is shown in Fig. 5 as a function of the different cutoffs. This comparison is made at weak coupling (α s = 0.005) and yields the expected agreement between the two approaches, as long as the cutoffs are T . In Fig. 6 we compare the analytical soft diffusion coefficients Eqs. (11)(12)(13) with the numerical soft diffusion coefficients Eq. (18) at different values of the strong coupling 7 It is highlighted in Ref.
[29] that the AMY differential equation used to evaluate the inelastic collisions rate remains similar if interactions with the plasma are non-perturbative. One difference is the perturbative partonic collision kernel C(q) ∝ m 2 D / q 2 (q 2 + m 2 D ) that must be modified. Non-perturbative contributions to the thermal masses are another difference. 18)) and analytical (Eqs. (11)(12)(13)) momentum transport coefficients:
q 1↔2 L,soft ,q 2↔2 soft andq 2↔2 L,soft .
The numerical results are computed with exact 1 ↔ 2 or 2 ↔ 2 kinematics up to a cutoff µ/T . The analytical coefficients make kinematic approximations appropriate for µ/T 1. The results are shown for different values of the hard-soft cutoffs at αs = 0.005. We calculate these results using p0 = 100 GeV and T = 300 MeV in a pure glue medium (N f = 0). The cutoff µ in the figure denotes µq ⊥ for q 2↔2 soft andq 2↔2 L,soft , and µω forq 1↔2 L,soft . In the elastic case, the additional cutoff on ω is set to Λ = min(pcut, √ 3p0T ). 6. Comparison of the numerical and analyticalq 1↔2 L,soft , q 2↔2 soft andq 2↔2 L,soft with different coupling constants αs (see Fig. 5 for description). The solid curves denote analytical results, and the circles denote numerical results. For the kinematic cutoffs, we use µq ⊥ = µω = T and Λ = min( √ 3p0T , pcut). The numerical values of the transport coefficients were calculated assuming a T = 300 MeV pure glue medium (N f = 0) and an energetic parton with p0 = 100 GeV. constant α s . We find that the analytical soft diffusion coefficients agree well with the numerical calculations even at large coupling, except for a small tension inq 2↔2 L,soft at large α s . Tension between different calculations of the soft transport coefficients are in fact not unexpected: perturbative calculations can be equivalent at order g n yet be different at order g n+1 . These differences are negligible at weak coupling, but can become significant for larger values of the coupling. This is a natural consequence of pushing the calculations beyond their regimes of validity. There is a practical consequence: two parton energy loss calculations that use the exact same approach (weakly-coupled kinetic theory) can lead to different results, when used at large coupling; neither approach is more "correct" than the other. This is important to keep in mind when comparing the present soft-hard factorized energy loss model with other implementations such as Ref. [39].
q 2 2 soft /T 3 q 2 2 L, soft /T 3 q 1 2 L, soft /T 3 FIG.
Theoretical guidance on the range of applicability of the Fokker-Planck equation
The energy loss of energetic partons through soft interactions is described by solving the Fokker-Planck equation with a stochastic Langevin approach. The applicability of the stochastic description is limited to the regime where the Fokker-Planck equation holds. This regime of applicability depends partly on properties of the interactions rates. We can summarize the regime of validity of the Fokker-Planck equation by first expanding the Boltzmann equation for soft collisions (around ω = 0):
∂ t f (p, t) = ω f (1,0) (p, t) + 1 2 ω 2 f (2,0) (p, t) + 1 6 ω 3 f (3,0) (p, t) + . . . ,(19)
where f (p, t) is the momentum distribution of energetic partons at time t and
ω k = dωω k dΓ dω(20)
is the k-th moment of the differential collision rate dΓ/dω. 8 By keeping only the first two terms on the right-hand side, Eq. (19) simplifies to the Fokker-Planck equation.
Assuming a single initial energetic parton of energy p 0 ,
f (p, t = 0) = δ(p − p 0 ),(21)
the solution of the Fokker-Planck equation is
f F P (p, t) = exp − (p−(p0− ω t)) 2 2t ω 2 2π ω 2 t .(22)
The above solution simply describes the energy distribution of the energetic parton widening from scatterings withq L = ω 2 energy diffusion, and an average energy loss of ω t.
Using this solution, we can compute the ratio of the third and second terms in the expanded Boltzmann equation (Eq. (19)): 23)) around ∆p = 0, we obtain:
R = 1 6 ω 3 f (3,0) F P (p, t) 1 2 ω 2 f (2,0) F P (p, t) = − 2∆p ω 3 (∆p 2 − 3 ω 2 t) 3 ω 2 t(∆p 2 − ω 2 t) ,(23)R = − ∆p ω 3 ω 2 2 t + 2∆p 3 ω 3 3 ω 2 3 t 2 + O(∆p 5 ) .(24)
By taking the ratio of the second and first term of this expansion,
r ≡ −2∆p 3 ω 3 3 ω 2 3 t 2 ∆p ω 3 ω 2 2 t ,(25)
we can find the value of ∆p for which this ratio will be large:
∆p = 3 2 r ω 2 t ,(26)
with r a constant assumed to be smaller than 1. We can use this value of ∆p as the range of momentum around the mean energy loss that can reasonably be described by the Fokker-Planck equation. Using Eq. (26) and the first term of Eq. (24), we define the scale S as
S = ω 3 ω 2 3/2 1 √ t .(27)
When this scale S is much smaller than 1, the Fokker-Planck equation is expected to provide a good description of the Boltzmann equation in the relevant range of momentum. We emphasize that Eq. (27) was derived without any specific form for the rate dΓ/dω; in particular, the formula is the same for perturbative and nonperturbative calculations of the rate. a. Scale for inelastic rate For inelastic interactions at weak coupling, we can evaluate Eq. 27 analytically using the formula for the very soft inelastic differential rate described in Eq. (A9). In this soft inelastic limit, the scale is given by
S 1↔2 = π 3/2 3C A 2 − ln(2) µ 3/2 ω g 2 T 2 √ t .(28)
This implies that soft inelastic emissions with energy smaller than µ can be described with the Langevin equation as long as the evolution time t in the medium is sufficiently long:
t µ 3 ω g 4 T 4 .(29)
Assuming µ ω ∼ T results in t 1/[g 4 T ], while µ ω ∼ gT results in t 1/[gT ]. This implies that there is a very large difference between a stochastic description of soft interactions with ω T compared to soft interactions with ω gT : in the former case, one needs a plasma 1/g 3 larger. These values serve as a reminder that, while one can in principle increase the phase space of interactions described stochastically, one may need an unrealistically large plasma for this description to be valid.
b. Scale for elastic rate The dependence of the scale S [Eq. (27)] on the cutoff µq ⊥ is shown in Fig. 7, for a small and large value of the coupling constant: α s = 0.005 and 0.3. One can see that the dependence on the cutoff can be non-monotonic for small values of α s , unlike in the inelastic case. Numerical tests, as well as the analytical expression available for the second moment at small coupling [Eq. (13)], suggest that the second moment of the elastic rate is the origin of this non-monotonic dependence of the elastic scale S on µq ⊥ .
Comparison between the diffusion process and the collision rate
In this section, we verify numerically the conclusion from the previous section: we compare a stochastic and a microscopic evolution of energetic partons in a static medium. In the microscopic rate-based picture, we use kinematic cuts to forbid hard interactions of the energetic parton. Because we are comparing soft interactions, we must use screened elastic matrix elements [43] in the microscopic description. 9 The screened inelastic (1 ↔ 2) rate is obtained numerically by solving the AMY differential equation, except for very small ω values, in which case the analytical expression described in Appendix A (Eq. (A9)) is used.
We choose the hard-soft cutoffs (i.e. µ ω and µq ⊥ ) to be at the order of T in the following tests. We set the coupling to be α s = 0.005, which corresponds to g ≈ 0.25. We choose T = 300 MeV for the temperature of the fluid, and set the propagation time in the plasma to be t = (0.3/α s ) 2 = 3600 fm. 10 We perform the diffusion approach and the collision rate approach separately to calculate the single parton energy distribution of a hard 100 GeV gluon propagating in the static pure glue medium. We emphasize once again that we only include soft interactions in the test by introducing the following hard-soft cutoffs on radiation energy and momentum transfer: for C 1↔2 soft , we only include radiations with the radiation energy ω < µ ω ; while for C 2↔2 soft , we only include interactions withq ⊥ < µq ⊥ and the energy transfer ω < Λ.
According to Eq. (27), for inelastic interactions (C 1↔2 soft ) to be describable stochastically for a cutoff ∼ T , one needs t 1/[g 4 T ] ≈ 200 fm of propagation time in the conditions described above. As expected, we find in Figure 8 that for inelastic interactions, in the weakly-coupled regime, the diffusion process can reproduce the single parton energy distribution generated by the collision-rate process. The value of the scale S, shown for each cutoff µ ω , are indeed smaller than 1. As µ ω increases, small differences appear between the Langevin description and the microscopic collision approach; the scale S is correspondingly larger, though still smaller than 1.
The same results are shown for the elastic case (C 2↔2 soft ) in Fig. 9. This time, the scale S is somewhat larger, and somewhat larger differences can indeed be seen between the Langevin and collision rate descriptions. As for the elastic case, the scale S increases as the cutoff increases, where more and more collisions are described stochastically. 9 Note that this is for testing purpose only, and that this is different from the vacuum matrix elements used for C 2↔2 hard in the hard-soft factorized energy loss model. 10
B. Parton energy loss at small coupling in a static medium
Building on the validation from the previous section, we can combine our approaches for the hard and soft interactions to implement the entire hard-soft factorized parton energy loss model described in Section II B. Remember that in the following, we use vacuum matrix elements for C 2↔2 hard , since the screening effects are encoded in the drag and diffusion coefficients of soft interactions. We also extend this test to a full quark-gluon plasma, with N f = 3. We use once again α s = 0.005 (g ≈ 0.25), with a propagation time of t = (0.3/α s ) 2 = 3600 fm in a T = 300 MeV plasma.
As summarized by Eq. (16), the hard or soft processes alone are dependent on the cutoff, but their cutoff dependence cancels out when combined. We confirm that, for both the inelastic and elastic cases, the single parton energy distribution is independent on the hard-soft cut-offs at small coupling in Fig. 10, given a sufficiently long evolution time. These results are consistent with those obtained in the previous section.
IV. HARD-SOFT FACTORIZATION OF PARTON ENERGY LOSS BEYOND WEAK-COUPLING
Soft interactions between an energetic parton and a deconfined plasma are likely non-perturbative. Evaluating this non-perturbative rate from first principles is an ongoing challenge. In this section, we estimate this nonperturbative rate using a typical approach in the heavy ion literature: we use the perturbative rate and extrapolate it to large coupling.
Recall that we do not use a running coupling in this work. As such, we use the same value of α s for soft and hard interactions, with the understanding that the future introduction of a running coupling will indeed lead to smaller values of α s for hard interactions, as assumed in this work.
As discussed in Section III A 2, soft interactions can always be described stochastically, if propagation in the medium is sufficiently long. We quantified this duration as t ω 3 2 / ω 2 3 , or S 1 as defined in Eq. (27), with ω n given by Eq. (20). We emphasize once again that Eq. (27) is general, and not limited to the perturbative regime.
We can use inelastic interactions to get an estimate of the length of the medium required to describe soft interactions stochastically. When extrapolating the weaklycoupled inelastic rate to large coupling, the ω-dependence of the rate remains the same. This means that, within this approximation, the analytical expression for S -Eq. (28) -remains the same. Consequently, Eq. (29) remains the same as well, and it states that a stochastic description of inelastic interactions with ω < T requires a time t 1/[g 4 T ]. For temperatures of a few hundred MeV and a coupling g ∼ 1 − 2 encountered in heavy ion collisions, 1/[g 4 T ] < 1 fm. Under this estimate, it would be reasonable to describe stochastically soft interactions with µ T occurring in a heavy ion collision.
Note that the above conclusion is based on the estimate of the soft inelastic rate discussed above; should the non-perturbative rate differ significantly from it, it could lead to change the range of applicability of the Langevin equation. However, we do believe that the above estimates -based on extrapolations of the weakly-coupled rates to strong coupling -are encouraging.
In what follows, we use α s = 0.3 (g ≈ 2), and first compare a stochastic and a microscopic description of parton energy loss for soft interactions. We use a plasma of length 1 fm and temperature T = 300 MeV.
Note that, when the coupling is large, the analytical diffusion coefficients computed perturbatively are not necessarily consistent with numerical values obtained by direct integration of the rates (see Fig. 6 and surrounding discussion). For what follows, we use the numerical diffusion coefficients in the Langevin part of the hard-soft factorized model.
A. Comparison between diffusion process and collision rate
As in the weak-coupling case (Section III A 3), we perform this section's test in the pure glue limit (N f = 0).
We first study the inelastic interactions, and as discussed above, we expect inelastic interactions softer than ∼ T to be describable by the Langevin equation in a 1 fm brick. We show this explicitly in Fig. 11. We show calculations for three different cutoffs µ ω , and we plot the results for the scale S from Eq. (27). 11 As expected, agreement between the Langevin approach and the microscopic collision rate approach are best when S 1. In the current setting, agreement is still good for µ ω = 2T , for which S = 0.33. This is encouraging evidence that the effect of non-perturbative inelastic interactions (C 1↔2 soft ) can be treated stochastically in phenomenological applications such as heavy ion collisions. 27)). Only soft 2 ↔ 2 interactions with ω < Λ andq ⊥ < µq ⊥ are allowed. We choose Λ = min(pcut, √ 3p0T ).
The equivalent result for soft elastic interactions (C 2↔2 soft ) is shown in Fig. 12. The result is very different. On one hand, the mean energy and width of the parton distribution described with the Langevin equation is almost identical to that described with collision rates. However their shape are different, especially at smaller values of the cutoffs µq ⊥ . Agreement between the two approaches is improved when the cutoff is larger. This is also reflected in the values of the scale S, evaluated numerically with Eq. (27), which decreases with increasing µq ⊥ (see Fig. 7). This is different from what was observed (i) in the inelastic case (see Fig. 8, 11), and (ii) in the elastic case at weak coupling (see Fig. 9): both cases preferred smaller values of the cutoff. Yet this result is fully consistent with our discussion in Section III A 2 b of the scale S for the elastic rate: it is purely a consequence of the ω-dependence of the elastic rate. We verified in Fig. 13 that longer evolution times do lead to better agreement between the Langevin and the collision rate descriptions, reflected in smaller values of the scale S. Our tentative conclusion is that soft elastic collision may be more difficult to describe stochastically; it is possible that one needs a larger cutoff µq ⊥ to describe these elastic interaction stochastically, although more studies will be necessary to confirm this conclusion. Note, however, that observables which are mainly sensitive to the average energy loss and the width of the parton distribution may tolerate a wider range of soft interactions being described with the Langevin approach.
More generally, it is clear that the choice of cutoff is very important in stochastic descriptions: careful choices of cutoffs can broaden significantly the range of applicability of the factorized approach presented in this work. Importantly, the cutoff choice should be chosen based on the expected relative size of the third and second moments of the energy loss rates.
B. Parton energy loss at large coupling in a static medium
To close this section, we quantify the cutoff dependence of a 100 GeV parton propagating for 1 fm in a 300 MeV brick of plasma, with α s = 0.3. This "brick" is the same as in the previous section. The soft interactions are described with the Langevin equation, and hard interactions are included as in the full implementation of the hard-soft energy loss model (Section II B). We use N f = 3 in this test.
We plot the energy distributions with different values of the cutoff in Fig. 14. In this larger coupling regime, as expected from the results of the previous section, inelastic interactions (C 1↔2 ) are independent of the cutoff (panel (a)). For the elastic case (C 2↔2 ), the energy distributions with different values of the cutoff are slightly different in the large energy region, although the long tail of the distribution is not affected (panel (b)).
Note that we also performed a cutoff dependence test on the cutoff Λ for 2 ↔ 2 interactions. We found the energy distribution of a parton propagating in a static medium to be independent of the choice of Λ, as expected. The result and further discussion can be found in Appendix F.
V. APPLICATION: ENERGY AND FERMION-NUMBER CASCADE
In this section, we use the hard-soft factorized model to study the energy and fermion-number cascade resulting from inelastic interactions between an energetic parton and a thermal medium. This section thus focuses on C 1↔2 (Fig. 1-b) in the hard-soft factorized model; both the hard and soft inelastic interactions are included, with the soft inelastic interactions modeled by the Langevin evolution. The collision kernel C 2↔2 is switched off for this section.
A. Energy cascade of hard gluons
When a gluon propagates through a thermal QCD medium, successive medium-induced inelastic radiations result in a gluon cascade. An analytical approximation for the gluon cascade was introduced in Refs. [48,49]; it was argued that the successive medium-induced quasidemocratic emissions lead to the accumulation of gluons at zero energy and cause a power-law scaling in the small energy region. We will study this scaling in this section [50].
At leading order, the successive radiations can be assumed to be independent [51]. In the deep LPM region, where the time scale of the radiation process is much larger than the mean free path between multiple scatterings, the rate per unit time of a gluon with energy p splitting into two gluons with energy fractions z and 1−z can be approximated as 12 [51][52][53]
dΓ dz g↔gg = α s N c π 1 [z(1 − z)] 3/2 q eff p .(30)
Hereq eff is the average transverse momentum broadening of the radiated gluon, and z = ω/p with ω the energy of the radiated gluon. We have kept only the most singular parts of the splitting function at z ∼ 0. We will treatq eff as a fit parameter, and then relate it to the parameter q 2↔2 soft in Eq. (12). The energy of the initial gluon is p 0 , and we define x ≡ ω/p 0 . The evolution of the gluon spectrum D(x, τ ) = x(dN/dx) is governed by [48,50] ∂D(x, τ ) ∂τ
= 1 0 dz 1 [z(1 − z)] 3/2 × z x D x z , τ − z √ x D (x, τ ) ,(31)
where
τ ≡ α s N c π q eff p 0 t ,(32)
and t is the evolution time of the gluon. The exact solution for Eq. (31) can be calculated via Laplace transform:
D 0 (x, τ ) = τ √ x(1 − x) 3/2 e −π[τ 2 /(1−x)] .(33)
As remarked in Ref. [48], this power-law gluon spectrum Eq. (33) scales as 1/ √
x in the small-x region. In order to compare with Eq. (33), we first determine the approximate value ofq eff to use in the simplified rate Eq. (30); this value also enters Eqs. (31)(32)(33). We fixq eff by comparing Eq. (30) with the full leading-order inelastic rate, as shown in Fig. 15. With parameters given in Fig. 15, we findq eff 0.04 GeV 3 at ω/p 10 −2 . We will use this value ofq eff in our analysis of the cascade below.
It should be emphasized that Eq. (30) is an approximation to the full inelastic rates corresponding to Eq. (3). Indeed, a leading-log analysis of the full rates at small z in the deep LPM regime shows that [52]:
q eff =q 2↔2 soft (µ 2 ⊥ ) ,(34)
whereq 2↔2 soft is given in Eq. (12), and µ 2 ⊥ = C 0 √ 2ωq eff with C 0 ∼ 1. The cutoff µ 2 ⊥ scales with the accumulated transverse momentum of the radiated gluon over its formation time. A next-to-leading logarithmic analysis fixes the coefficient C 0 [54]:
µ 2 ⊥ = C 0 2ωq eff , C 0 = 2e 2−γ E +π/4 .(35)
For N f = 0, p = 1 TeV, T = 300 MeV, α s = 0.1 (same as in Fig. 15), and using ω/p = 10 −2 , we can solve Eqs. (34)(35) numerically. We findq eff ≈ 0.052 GeV 3 , which as expected is close to the value we found in Fig. 15.
We next perform the gluon cascade in a pure-glue medium (N f = 0) using the hard-soft factorized model, i.e we include both soft inelastic interactions described with the Langevin equation, and rate-based hard inelastic interactions which dominate this test. In Fig. 16, we compare this numerical result calculated by the current model with the analytical spectrum in Eq. (33). We find that the numerical solution for the medium-induced cascade is reasonably well described by the approximate analytic solution. In particular, the power law behavior, dN/dx ∝ x −3/2 , is nicely captured by this solution.
B. Fermion-number cascade of gluons and quarks
The fermion-number cascade was investigated in Ref. [55]. Given the power-law scaling in the small energy region, at small x, we can write the power-law spectrum of quarks and gluons as
D g ≡ x dN g dx = G √ x , D s ≡ N F i=1 (D qi + Dq i ) = Q √ x .(36)
As derived in Ref. [55], the quark-to-gluon ratio of the soft radiated partons is determined by the transformation rate between gluons and fermions. We have
Q 2N f G = 1 2N f 1 0 dzzK qg (z) 1 0 dzzK gq (z) ≈ 0.07,(37)
where K qg is the splitting function of g → qq, and K gq is the splitting function of q → gq. To test the quark-to-gluon ratio in the hard-soft factorized model, we numerically simulate the evolution of a gluon or a quark propagating through a static QGP medium (N f = 3) using the full leading order inelastic rate. We perform the calculation for both an energetic gluon and an energetic light quark with an initial energy of 10 TeV. The result is shown in Fig. 17; we find that it converges to the universal quark-to-gluon ratio when using the full QCD rates.
VI. SUMMARY AND OUTLOOK
This work introduces a new formulation of parton energy loss where soft and hard interactions with the underlying plasma are factorized and treated separately. The factorization is performed with cutoffs based on the momentum transfer of the interactions. Rare hard interactions are considered as independent successive interactions, and solved with collision rates (Sections II B 1 and II B 2); the larger momentum exchange with the medium make them more likely to be amenable to a perturbative description. On the other hand, frequent soft interactions are treated stochastically using a Langevin evolution with drag and diffusion coefficients encoding the effect of these soft interactions (Section II B 3); non-perturbative effects can thus be absorbed in these transport coefficients.
Our numerical implementation of this model (Section III) shows that this factorization works well in the weakly-coupled regime where the theory was derived [37]. In fact, by revisiting the conditions under which the Langevin equation can describe the Boltzmann equation (Section III A 2), we extended the region of phase space ("cutoffs") that can be described stochastically.
We used the dimensionless scale S (Eq. (27)) to quantify the length of a plasma necessary for soft collisions to be describable with the Langevin equation. Our numerical tests showed that this scale works very well in practice.
Because the scale S is a property of the Boltzmann equation and not a perturbative concept, we used it to extend our discussion of parton energy loss beyond the perturbative regime. We estimated that inelastic collisions resulting in parton energy loss of order T could be described stochastically in a QCD plasma of size ∼ 1 fm (Section IV). Given that inelastic interactions dominate parton energy loss for high-energy partons, this supports the applicability of the present energy loss model in heavy ion collisions.
This work paves the way to systematic phenomenological constraints on the soft transport coefficients of light partons. The key strength of our approach is that perturbative parton energy loss calculations are still being used for harder interactions -the regions of phase space where they are most likely to hold. Conversely, the interactions most sensitive to non-perturbative effectssoft interactions -are encoded in simple transport coefficients which can be constrained by comparison with measurements. A stochastic description of soft collisions can also be very efficient numerically, as a large number of soft interactions can be absorbed in the transport coefficients. These phenomenologically-constrained transport coefficients can eventually be compared with lattice results (e.g. Ref. [35]). A similar program is already being pursued for the energy loss of heavy quarks [56]; studies of light parton energy loss with a model that includes many features of soft-hard factorization, are also ongoing [47].
Future generalization of this framework includes improving the treatment of the radiation angle of collinear radiation, and the inclusion of a running coupling constant and of next-to-leading order effects; these additions will increase the type of observables that can be studied with this model. The inclusion of finite-size effects in this formalism will also be an important addition. These additions will be able to build on Ref. [47] and other works.
ACKNOWLEDGMENTS
We thank Weiyao Ke for his invaluable help in the early stage of this project, and Jacopo Ghiglieri for generously sharing notes on the splitting approximation that formed the basis of this work's discussion on the topic. We thank Sangyong Jeon, Chanwook Park, Abhijit Majumder and the other members of the JETSCAPE Collaboration for discussions regarding MARTINI and MATTER, and for their support with the JETSCAPE framework. We thank Yingru Xu for valuable discussions regarding the Langevin equation and its numerical implementation. This work was supported by the U. Appendix A: Inelastic rate at low ω At leading order, the differential rate of the 1 ↔ 2 process can be expressed using AMY's rate [26,29]:
dΓ(p, ω) dω 1↔2 = g 2 16πp 3 ω 2 (p − ω) 2 [1 ± n(ω)] [1 ± n(p − ω)] × P a bc (z) d 2 h (2π) 2 2h · ReF(h, p, ω)(A1)
where z = ω/p and P a bc (z) are the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting kernels of the radiation a → bc,
P a bc (z) = C F 1 + (1 − z) 2 z , q → gq C A 1 + z 4 + (1 − z) 4 z(1 − z) , g → gg d F C F d A z 2 + (1 − z) 2 , g → qq . (A2)
Very soft interactions (ω T ) are dominated by gluon radiations, i.e. g ↔ gg, q ↔ gq with a soft final state gluon (see footnote 6). In this case (ω T p), AMY's integral is symmetric and can be expanded in terms of the radiated energy ω [37]:
d 2 h (2π) 2 2h · ReF(h, p, ω) soft gluon = 8p 6 C A z 2 (1 − 2z) d 2 q ⊥ (2π) 2 × d 2 k ⊥ (2π) 2 C F (k ⊥ ) C F q ⊥ q 2 ⊥ + M 2 ∞ − q ⊥ + k ⊥ (k ⊥ + q ⊥ ) 2 + M 2 ∞ 2 ,(A3)
where the collision kernel is
C F (k ⊥ ) C F = g 2 T m 2 D k 2 ⊥ (k 2 ⊥ + m 2 D )
.
(A4)
We define the integral in Eq. (A3) as Combining the factors, we have
I = d 2 q ⊥ (2π) 2 d 2 k ⊥ (2π) 2 C F (k ⊥ ) C F × q ⊥ q 2 ⊥ + M 2 ∞ − q ⊥ + k ⊥ (k ⊥ + q ⊥ ) 2 + M 2I = g 2 T m 2 D dq 2 ⊥ 4π dk 2 ⊥ 4π 1 k 2 ⊥ (k 2 ⊥ + 2M 2 ∞ ) dφ q 2π dφ kq 2π q ⊥ q 2 ⊥ + M 2 ∞ − q ⊥ + k ⊥ (k ⊥ + q ⊥ ) 2 + M 2 ∞ 2 (A6)
By rescaling all the dimensional quantities by M ∞ , the integral I can be calculated as
I = g 2 T m 2 D M 2 ∞ dq 2 ⊥ 4π dk 2 ⊥ 4π 1 k 2 ⊥ (k 2 ⊥ + 2) × dφ q 2π dφ kq 2π q ⊥ q 2 ⊥ + 1 −q ⊥ +k ⊥ (k ⊥ +q ⊥ ) 2 + 1 2 = 2 − log(2) 8π 2 g 2 T.
(A7)
We therefore have an analytical approximation of the soft gluon radiation rates:
dΓ(p, ω) dω 1↔2 soft gluon = [2 − log(2)] g 4 C A T 16π 3 p × [1 ± n(ω)] [1 ± n(p − ω)] 1 − 2z (1 − z) 2 P a bc (z).(A8)
For the tests in Sections III and IV, we use this soft limit of differential rate when |ω| ≤ 0.2T . In Fig. (18), we compare this soft limit with AMY's full rate for g ↔ gg, and they agree well in the soft ω region.
With the soft radiation assumption ω T p, we can simplify Equation (A8) by neglecting the terms sup-pressed by ω/T and ω/p
dΓ(p, ω) dω 1↔2 soft gluon ≈ [2 − log(2)] g 4 C A C R T 2 8π 3 ω 2 . (A9)
Using the above expressions, we can calculate the perturbativeq 1↔2 L, soft . We find that the longitudinal momentum broadening of soft 1 ↔ 2 iŝ
q 1↔2 L, soft = µω −µω dωω 2 dΓ(p, ω) dω 1↔2 soft gluon = [2 − log(2)] 4π 3 g 4 C R C A T 2 µ ω .(A10)
Appendix B: Energy loss rate for hard 2 ↔ 2 interactions
The differential energy loss rate of a hard 2 ↔ 2 interaction is calculated using the vacuum matrix elements:
d 2 Γ ab↔cd vac dωdq ⊥ = ∞ q−ω 2 dk 2π 0 dφ 2π 1 4(2π) 3q ⊥ q Q ab cd (p, k, ω,q ⊥ , φ) 4p 2 , (B1) with Q ab cd (p, k, ω,q ⊥ , φ) = 1 ν a |M ab cd | 2 [n b (k)(1 ± n d (k + ω))] ,(B2)
where ν a = 2d a is the degeneracy of the particle a,q ⊥ = q 2 − ω 2 , M ab cd is the matrix element of a vacuum 2 ↔ 2 interaction as a hard parton a interacting with a thermal particle b and transforms into particles c and d. The expression of M ab cd can be found in Table II in [25]. The collision kernel of the 2 ↔ 2 large-angle interactions is
C 2↔2 large-angle = bcd Λ −∞ dω ∞ µq ⊥ dq ⊥ d 2 Γ ab↔cd vac dωdq ⊥ .(B3)
In Eq. (B3), if outgoing particles c and d are identical species, a symmetry factor of 1 2 should be included. However, this factor of 1 2 is canceled out to incorporate the interactions with p − Λ < ω < p, since symmetric 2 ↔ 2 interactions with p − Λ < ω < p are equivalent to interactions with ω < Λ. For c and d being distinct species, a factor of 1 2 is also necessary to cancel the double-count of the final states in cd . We eliminate this factor by constraining that the energy of particle c is larger than particle d. These asymmetric interactions with ω < Λ and p − Λ < ω < p are treated separately.
In Eq. (B1), the expression of Q ab cd (p, k, ω,q ⊥ , φ) is dependent on the types of particles a, b, c, and d. We summarize them using Mandelstam variables (s, t, u), Casimir factors (C A , C F ), and color degrees of freedom (d F , d A ) as follows, where C A = 3, C F = 4/3,
d A = N 2 c − 1, d F = N c .
We summarize the expression of Q ab cd (p, k, ω,q ⊥ , φ) for different interactions in Table I.
ab ↔ cd bcd Q ab cd /g 4 = bcd 1/νa|M ab cd | 2 [n b (1 ± n d )] /g 4 Gg ↔ Gg 4C 2 A s 2 +u 2 t 2 nB(k) [1 + nB(k + ω)] + O( T 2 p 2 ) Gq ↔ Gq 2N f · 4 d F d A CF CA s 2 +u 2 t 2 nF (k) [1 − nF (k + ω)] + O( T 2 p 2 ) Qq ↔ Qq 2N f · 4 d F d A C 2 F s 2 +u 2 t 2 nF (k) [1 − nF (k + ω)] + O( T 2 p 2 ) Qg ↔ Qg 4CF CA s 2 +u 2 t 2 nB(k) [1 + nB(k + ω)] + O( T 2 p 2 ) Gq ↔ Qg 2N f · 4 d F d A C 2 F u t nF (k) [1 + nB(k + ω)] + O( T 2 p 2 ) Qg ↔ Gq 4C 2 F u t nB(k) [1 − nF (k + ω)] + O( T 2 p 2 ) Gg ↔ Qq 2N f · 4C 2 F u t nB(k) [1 − nF (k + ω)] + O( T 2 p 2 ) Qq ↔ Gg 4C 2 F u t nF (k) [1 + nB(k + ω)] + O( T 2 p 2 )
TABLE I. In this table, we use capital letter G and Q to denote hard gluons and quarks (p > pcut), and lowercase letter g and q to denote soft gluons and quarks(p < pcut). To simplify the notation, we do not specify the quark species. Q and q include the conditions of various quark species, and can also be anti-quark.
Up to order T /p, we have the following kinematics:
t = −(−Q) 2 = −(P − P ) 2 = −q 2 ⊥ , s = −(P + K) 2 = −t 2q 2 (p + p )(k + k ) + q 2 − cos(φ) (4pp + t)(4kk + t) (2p) −t 2q 2 (k + k ) − cos φ √ 4kk + t 1 + T p , u = −(K − P ) 2 = −t − s −s.
(B4)
Appendix C: Soft conversion process
Soft conversion is a process where the identity of the hard parton is changed by its interaction with the medium. Diffusion processes only include the identity preserving soft interactions; a soft conversion process is necessary to consider identity non-preserving soft interactions.
The collision kernel of the soft conversion reads
C 2↔2 conv,qi [δf ] = δf qi (p)Γ conv q→g (p) − δf g (p) d A d F Γ conv g→q (p) C 2↔2 conv,qi [δf ] = δfq i (p)Γ conv q→g (p) − δf g (p) d A d F Γ conv g→q (p) C 2↔2 conv,g [δf ] = N f i=1 δf g (p) Γ conv g→qi (p) + Γ conv g→qi (p) − d F d A δf qi (p)Γ conv q→g (p) + δfq i (p)Γ conv q→g (p) } (C1)
As derived in in Section 3.3 of Ref. [37], at leading or-der, the parton identity exchange rate is
Γ conv q→g (p) = g 2 C F 4p µq ⊥ d 2 q ⊥ (2π) 2 m 2 ∞ q 2 ⊥ + m 2 ∞ , = g 2 C F m 2 ∞ 16πp ln 1 + μ 2 q ⊥ m ∞ 2 , Γ conv g→q (p) = d F d A Γ conv q→g (p),(C2)
where m 2 ∞ ≡ g 2 C F T 2 /4 is the asymptotic mass of quarks.
Given that the rate of these identity non-preserving soft interactions is suppressed by T /p and the energy exchange ω is small, we neglect the energy loss due to these soft conversion process, and only incorporate the identity exchange.
In the numerical implementation, at each time step, we change the identity of the leading parton according to the conversion rates in Eqs. C1 and C2.
Appendix D: Splitting approximation process
As discussed in the body of the text, the collision kernel for 2 ↔ 2 scattering processes can be simplified when the energy transfer is large. 13 For simplicity, we will begin the discussion with the pure glue theory. As we will show here, and as is obvious pictorially, the 2 ↔ 2 scattering rate with large ω can be written as an effective 1 → 2 rate, which takes the form
C 2↔2 split (Λ) = 1 2 p−Λ Λ dω dΓ(p, ω) dω ,(D1)
where
dΓ(p, ω) dω = g 4 8πp 3 P g gg (z) z 2 (1 − z) 2 × C A 2 1 − z + z 2 d 2 q ⊥ (2π) 2q (δE) δE 2 ,(D2)
Here we have defined
δE ≡ pq 2 ⊥ 2p k ,(D3)
and for comparison with other litterature we have defined q(δE) for the pure glue case [37] q(δE) expression we have used the thermodynamic integral, ∞ 0 dk kn B (k) = π 2 T 2 /6. The last remaining integral over z can be done and total rate for gluon absorption takes the form Γ a(g) bc
δE 2 ≡ d 3 k (2π) 3 k n B (k) 2πδ(k − − δE) . (D4) Γ a(g) bc cΛ cp c ln q → qg 2CF CA −CF CA + C 2 F /2 −2CF CA + C 2 F q g → qq 0 −N f (CA/3 + CF ) N f CF g → gg 4C 2 A 10 6 C 2 A −4C 2 A= g 4 T 2 96πp c Λ z 0 + c p − c ln log(z 0 ) ,(D34)
where z 0 = Λ/p. The coefficients, c Λ , c p , c ln are in tab-ular form as in Table II. We note (again) that the total rate for g → gg is Γ
g(g)
gg /2 to account for the symmetry of the final state. We also note that the second row in this table has been summed over quark flavors.
We will now consider the case when a soft quark is absorbed from the bath, and the hard particle of type a splits a → cd. The differential rate now takes the form dΓ a(q) cd
dω = g 4 F a cd (z) 32πp 3 d 2 q ⊥ (2π) 2 d 3 k (2π) 3 k n F (k) 2πδ(k − −δE) . (D35) where F a cd (z) ≡ |M aq cd | 2 /g 4 ν a z(1 − z) .(D36)
Evaluating the matrix elements (again using Table II. of [25] and Eq. (D21)), we find
F q1 q1q2 = 2C F z(1 − z) 1 + (1 − z) 2 z 2 , (D37a) F q1 q1q1 = 2C F z(1 − z) 1 + (1 − z) 2 z 2 + 1 + z 2 (1 − z) 2 + 4 C F − C A 2 1 z(1 − z) ,(D37b)F q1 q1q1 = 2C F z(1 − z) 1 + (1 − z) 2 z 2 + z 2 + (1 − z) 2 − 4 C F − C A 2 (1 − z) 2 z ,(D37c)F q1 q2q2 = 2C F z(1 − z) z 2 + (1 − z) 2 ,(D37d)F q1 gg =4C F z 2 + (1 − z) 2 z 2 (1 − z) 2 C F − C A 2 + C A 2 (z 2 + (1 − z) 2 ) ,(D37e)F g q1g = 4d F C F d A 1 + z 2 z 2 (1 − z) 3 C F − C A 2 (1 − z) 2 + C A 2 (1 + z 2 ) .(D37f)
Again integrating over the momentum fraction we find that the total rate takes the form Γ a(q) cd = g 4 32πp
T 2 24 c Λ z 0 + c p − c ln log(z 0 ) ,(D38)
where we used the integral, ∞ 0 dp p n F (p) = π 2 T 2 /12. The coefficients c Λ , c p and c ln are tabulated in Table III. The diffusion process as described by the Fokker-Planck equation (Eq. (10)) can be stochastically realized with the Langevin model. The stochastic Langevin equations solves the evolution of the space-time coordinates and the momentum of the particle [44,57]:
q 2 q1 → q1q2 +q1 → q1q2 4CF (2N f − 2) −2CF (2N f − 2) −4CF (2N f − 2) q1 → q1q1 8CF −4CF −8CF (1 + CA − 2CF ) q1 → q1q1 4CF 2 3 CF (−1 − 9CA + 18CF ) −4CF (1 − CA + 2CF ) q 2 q1 → q2q2 0 4 3 CF (N f − 1) 0 q1 → gg 0 − 8C F 3 (CA + 3CF ) 8C 2 F q 1 g → q1g +g →q1g 4CA(2N f ) (CF − 2CA)(2N f ) (2CF − 4CA)(2N f )∆x ∆t = p E ∆p ∆t = −η D,soft p + F thermal (t) ,(E1)
where x is the space coordinates of the parton, F thermal is a thermal random force satisfying the mean and the correlation function
F thermal i = 0 F thermal i F thermal j = − 1 ∆t p ipjqL + 1 2 (δ ij −p ipj )q .(E2)
The realization of the stochastic differential equation is dependent on the discretization scheme. We choose the pre-point Ito scheme in this work [58]. In the infinite medium limit, the initial energetic partons should eventually reach the thermal equilibrium via diffusion in the thermal plasma. The equilibrium distribution of the light parton δf (p) is proportional to exp(−p/T ) in the Fokker-Planck equation (Eq. (10)), and the time derivative of the equilibrium distribution is zero. We can thus obtain the drag coefficient η D,soft as Eq. (14).
We check the thermalization of the light partons in the QGP plasma using the Langevin model (Eq. (E2)) with the drag and diffusion coefficients in Equation (11)(12)(13)(14). As shown in Figure 19, after a long evolution time, the momentum distribution of the light parton approaches the Maxwell-Jüttner distribution [59] [60]:
δf (p) ∝ p 2 exp − E T . (E3)
Appendix F: Λ cutoff dependence
As described in Section II, the hard elastic interactions are divided as the large-angle process and the splitting approximation process. In Fig. 20, we show the evolution of a gluon in quark-gluon plasma (N f = 3) with only 2 ↔ 2 interactions. In Fig. 20(a), with only C 2↔2 large−angle and C 2↔2 diff , the tail of the energy distribution depends significantly on the value of Λ. In Fig. 20(b), with only C 2↔2 diff and C 2↔2 split , the interactions withq ⊥ > µq ⊥ and ω < Λ is missed, which result in a missing part of the energy distribution; inevitably, the energy distribution around the initial parton energy p 0 is found to depend on Λ. In Fig. 20(c), with all the types of the 2 ↔ 2 interactions combined (C 2↔2 large−angle +C 2↔2 diff +C 2↔2 split + C 2↔2 conv ), the result is found to be independent of the cutoff Λ, as expected.
II. HARD-SOFT FACTORIZATION OF PARTON ENERGY LOSS IN THE WEAKLY-COUPLED REGIME: THEORYA. Effective kinetic approach in weakly-coupled regime
FIG. 1 .
1Treatment of different processes in the hard-soft factorized parton energy loss model
1 .
1Treatment of hard interactions: inelastic case (1 ↔ 2)
FIG. 3 .
3(a) Example of large-angle elastic 2 ↔ 2 interactions, whereq ⊥ > µq ⊥ and ω < Λ; (b) example of elastic 2 ↔ 2 interactions with ω > Λ, which is treated with a splitting approximation (see text).
FIG. 4. The differential rate of splitting approximation interactions and large-angle interactions for gg ↔ gg process when αs = 0.3. The shaded area is the region of √ 3pcutT < ω < pcut. dΓvac/dω is the differential rate of vacuum matrix elements for 2 ↔ 2 interactions. The results are for p0 = 100 GeV. Note that in the numerical implementation, we double-count the large-angle interaction rate because we only sample in half of the phase space. Here, to compare with splitting approximation rate, we decrease the large-angle interaction rate in the numerical implementation by a factor of 1 2 to cancel out the double-count.
FIG. 5 .
5The ratio between the numerical (Eq. (
where ∆p = p − (p 0 − ω t) is the distance in momentum from the peak of the Fokker-Planck solution (Eq. (22)). Significant corrections to the Fokker-Planck solution Eq. (22) are expected unless R 1. As is clear from Eq. (23), the range of validity of the Fokker-Planck equation depends on properties of the rate (the second and third moments ω 2 and ω 3 ), as well as on time t and on the distance in momentum ∆p from the peak of the distribution. The Fokker-Planck equation describes the effect of soft interactions on an energetic parton. The soft interactions dominate for small values of ∆p. Expanding R (Eq. (
FIG. 7 .
7Dependence of the skewness scale S on the cutoff µq ⊥ , for the elastic parton energy loss rate. The top line is for αs = 0.3 and the bottom line for αs = 0.005. The points denote the values corresponding to µq ⊥ = 0.5, 1, 2T . This interaction rate is calculated assuming a pure glue medium (N f = 0).
We choose the evolution time t ∝ 1/α 2 s to keep the number of the collisions approximately the same for different values of αs. With the choice t = (0.3/αs) 2 , the evolution time is 1 fm when we use αs = 0.3 later in the manuscript.
FIG. 8 .FIG. 9 .
89The energy distribution of a 100 GeV gluon propagating through a 300 MeV pure glue medium (N f = 0) at αs = 0.005. The evolution time is t = (0.3/αs) 2 = 3600 fm. Only soft 1 ↔ 2 interactions with ω < µω are allowed. Three different values of the cutoff are shown: µω/T The energy distribution of a 100 GeV gluon propagating through a 300 MeV pure glue medium (N f = 0) at αs = 0.005. The evolution time is t = (0.3/αs) 2 = 3600 fm. Only soft 2 ↔ 2 interactions with ω < Λ andq ⊥ < µq ⊥ are allowed. Three different values of the cutoff µq ⊥ are shown: µq ⊥ /T = 0.5, 1, 2. We choose Λ = min(pcut, √ 3p0T ).
FIG. 10 .
10The energy distribution of a 100 GeV gluon propagating through 300 MeV QGP medium (N f = 3) at αs = 0.005 with different values of the cutoffs. The evolution time is t = (0.3/αs) 2 = 3600 fm. The subplot (a) only includes the C 1↔2 interactions and (b) only includes C 2↔2 interactions. In both cases the cutoff µ is varied: the soft interactions (those with momentum transfer less than µω and µq ⊥ respectively) are treated with a Langevin process, while the rest of the kinematic phase space is treated with rates. Results obtained when propagating an energetic light quark instead of a gluon can be found in Appendix G.
FIG. 11 .FIG. 12 .FIG. 13 .
111213The energy distribution of a 100 GeV gluon propagating through a 300 MeV pure glue medium (N f = 0) at αs = 0.3. The evolution time is t = (0.3/αs) 2 = 1 fm. Only soft 1 ↔ 2 interactions with ω < µω are allowed. Compare with the weak-coupling result from Fig. 8. The energy distribution resulting from a 100 GeV gluon propagating through a 300 MeV pure glue medium (N f = 0) at αs = 0.3. The evolution time is t = (0.3/αs) 2 = 1 fm. Only soft 2 ↔ 2 interactions with ω < Λ andq ⊥ < µq ⊥ are allowed. We choose Λ = min(pcut, √ 3p0T ). Compare with the weak-coupling result from Fig. 9. The energy distribution resulting from a 100 GeV gluon propagating through a 300 MeV pure glue medium (N f = 0) at αs = 0.3. The evolution time is 200, 50, 20 fm for µq ⊥ /T = 0.5, 1, 2; the times were chosen to obtain similarly small values of the skewness parameter S (Eq. (
FIG. 14 .
14The energy distribution of a 100 GeV gluon propagating through 300 MeV QGP medium (N f = 3) at αs = 0.3 with different values of the cutoff. The evolution time is t = (0.3/αs) 2 = 1 fm. The subplot (a) only includes C 1↔2 interactions and (b) only includes C 2↔2 interactions. See the weakly-coupled results in Fig. 10 for comparison and additional explanations. Results obtained when propagating an energetic light quark instead of a gluon can be found in Appendix G.
order rate (AMY) FIG. 15. Comparison between the full leading-order inelastic rate and the deep-LPM regime approximation of the rate from Eq. (30) withq eff = 0.04 GeV 3 for N f = 0 (p = 1 TeV, T = 300 MeV and αs = 0.1).
FIG.16. A comparison of the current numerical implementation of QCD kinetics and the analytical approximation of Ref.[48] for the energy cascade in the pure glue medium for different evolution times. The analytical solution is denoted by the dotted curve. In this test, we only include inelastic 1 ↔ 2 processes. We use N f = 0, αs = 0.1, T = 300 MeV and p0 = 1000 GeV.
FIG. 17 .
17The fermion number cascade of the numerical implementation in the QGP medium with different evolution times. In this test, we only include inelastic interactions (C 1↔2 ). We use N f = 3, αs = 0.3, T = 300 MeV and p0 = 10 TeV. The black horizontal line reflects the expected limiting value of DS/(2N f Dg) ≈ 0.07 (Eq.(37)).
S. Department of Energy Grant no. DE-FG02-05ER41367 (SAB, JFP and TD) and DE-FG-02-08ER41450 (DT). TD is also supported by NSF grant OAC-1550225. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.
rate full leading order rate (AMY) FIG. 18. Comparison of the g ↔ gg collision rate between the soft analytical expression (Eq. (A8)) and the full leading order rate. We used N f = 3, αs = 0.3, T = 300 MeV and p0 = 100 GeV.
FIG.19. Distribution of momentum from a gluon with initial energy E0 = 16 GeV evolved for a time t = 100 fm in a 300 MeV static medium, compared with the thermal distribution. We used N f = 3 and αs = 0.3.The full transition rate for collisionalboth the gluon and quark induced splittings.Appendix E: Detailed balance of the Langevin model
FIG. 20 .
20Momentum distributions of final gluons from an initial gluon with energy E0 = 200 GeV after 1 fm of evolution in a 300 MeV static medium. The system is evolved with elastic 2 ↔ 2 interactions only, for three different prescriptions for the cutoff Λ: Λ = {0.25, 1, 4} min( √ 3pT , pcut). The three panels show how the 2 ↔ 2 processes is divided into subprocesses: (a) a large-angle process with soft drag&diffusion; (b) a splitting process with soft drag&diffusion; and (c) the full 2 ↔ 2 rate including large-angle scattering, splitting, and soft drag&diffusion. The full rate shown in (c) is approximately independent of the prescription for Λ.
FIG. 21 .
21The energy distribution of a 100 GeV "up" quark propagating through 300 MeV QGP medium (N f = 3) at αs = 0.005 with different values of the cutoff. The evolution time is t = (0.3/αs) 2 = 3600 fm. The subplot (a) only includes C 1↔2 interactions and (b) only includes C 2↔2 interactions.
FIG. 22 .
22The energy distribution of a 100 GeV "up" quark propagating through 300 MeV QGP medium (N f = 3) at αs = 0.3 with different values of the cutoff. The evolution time is t = (0.3/αs) 2 = 1 fm. The subplot (a) only includes C 1↔2 interactions and (b) only includes C 2↔2 interactions. See the weakly-coupled results in Fig. 21 for comparison.
TABLE II .
IIThe total rates for gluon absorption.
TABLE III .
IIIThe total rates for quark absorption.
In particular, hydrodynamic simulations of this plasma's evolution do not rely on a quasiparticle picture of deconfined nuclear matter until hadronization.4 Note other works such as Refs.[32,33] assume that neither soft or hard interactions are perturbative, and consequently evaluate parton energy loss using gauge-field duality.
We thank Guy D. Moore for his numerical solver for AMY integral equations.
Note that the diffusion coefficientq 1↔2 L,soft does not depend on the number of the quark flavor, because very soft radiations are dominated by gluon scatterings.
The bounds on the integration are the same as in Eqs.(18), including the additional integration overq ⊥ necessary in the elastic case.
We verified that the result from Eq.(27)is close to that of Eq.(28). The values we quote are from Eq.(27)
Accounting for the identical particles in the final state, the total rate is 1/2 0 dΓ/dz dz.
We thank Jacopo Ghiglieri for sharing notes on this, which served as the basis for this appendix.
↔ 2 interactions is still independent on the hard-soft cutoff, while for 2 ↔ 2 interactions, there is a slightly larger cutoff dependence around the initial energy.
This is an approximation of the (unscreened) scattering rate given in Eq.(4)with the matrix element for the gg ↔ gg collisions given byIn this kinematic regime, we can neglect the population factors n c (p ) and n d (k ). We will writefor the k, p , and k integrals, with K = d 4 K/(2π)4and δ + (K 2 ) = θ(k 0 )δ(K 2 ). Next, we change variables to integrate over Q = P − P instead of P , and use the four momentum constraint to eliminate K = K + Q, yielding the phase space integralTo understand the kinematics of the process, it is convenient to use the light cone coordinates where q + = −q − = (q 0 + q z )/2 and q − = q 0 − q z and we take p along the z direction.while the outgoing onshell constraints read,Now, all four components of the momentum k are of order ∼ T . In order to satisfy the onshell constraints and energy-momentum conservation, we have the following scalings with the energy of the probe for the light cone momentaThus, the incoming transverse momentum k ⊥ ∼ T can be ignored, and transverse momentum conservation fixes thatPlus-coordinate momentum conservation yieldsMinus-coordinate momentum conservation yieldsTheand satisfy s + t + u = 0. Now we writeAssembling the ingredients we havewhereReorganizing terms one findsin agreement with Eq. (D2).The analysis can be extended to include quarks. Our starting point is again Eq. (4). As for the pure glue case it is our interest to describe the splitting process where p and k are both large. Then we have as beforeNow we distinguish two cases: (i) when a gluon is absorbed from the bath, and (ii) when a quark is absorbed from a bath.In the first case the gluon is absorbed from the bath and the hard particle splits into flavors cd. The differential rate takes the form dΓ a(g) cdwhere the effective splitting rate are the matrix elements (seeTable IIof[25]) evaluated using the kinematic approximations of Eq. (D21).The effective splitting function is for gluon absorption isHere for the process a → cd the momentum fraction z is associated with particle d, i.e. z = k /p = −q 2 ⊥ /u and 1 − z = p /p = −q 2 ⊥ /t. To find the total rate we must perform the integral over ω. The integration is straightforward and yields for gluon absorption Γ a(g) bc = g 4 32πpThe total rate for the splitting process through gluon absorption Γ a (g) = 1 2 bc Γ a(g) bc , where the factor of 1/2 is a symmetry factor. In practice this symmetry factor is handled by summing over only distinct processes, and, if the final state involves identical particles, by integrating over the distinct phase-space. In writing this Appendix G: Propagation of energetic light quarks In Sections III B and IV B, we presented the propagation of a hard gluon in a static quark-gluon plasma (N f = 3) at both small and large coupling. The energy distribution of this hard gluon evolution was presented for different values of hard-soft cutoffs inFigures 10 and 14. In this appendix, we perform the same tests for a hard "up" quark:Figures 21 and 22. The conclusion are the same for the evolution of a quark and that of a gluon. In the small coupling regime (α s = 0.005), both 1 ↔ 2 interactions and 2 ↔ 2 interactions are independent of the hard-soft cutoff. In the larger coupling regime (α s = 0.3),
. S Chatrchyan, CMS10.1016/j.physletb.2012.07.025arXiv:1205.6334Phys. Lett. B. 71566nucl-exS. Chatrchyan et al. (CMS), Phys. Lett. B 715, 66 (2012), arXiv:1205.6334 [nucl-ex].
. S Chatrchyan, CMS10.1016/j.physletb.2012.02.077arXiv:1201.3093Phys. Lett. B. 710256nucl-exS. Chatrchyan et al. (CMS), Phys. Lett. B 710, 256 (2012), arXiv:1201.3093 [nucl-ex].
. G Aad, ATLAS10.1103/PhysRevLett.110.022301arXiv:1210.6486Phys. Rev. Lett. 11022301hep-exG. Aad et al. (ATLAS), Phys. Rev. Lett. 110, 022301 (2013), arXiv:1210.6486 [hep-ex].
. G Aad, ATLAS)10.1140/epjc/s10052-014-3231-6arXiv:1408.4674Eur. Phys. J. C. 7523hep-exG. Aad et al. (ATLAS), Eur. Phys. J. C 75, 23 (2015), arXiv:1408.4674 [hep-ex].
. G Aad, ATLAS10.1103/PhysRevC.93.034914arXiv:1506.08552Phys. Rev. C. 9334914hep-exG. Aad et al. (ATLAS), Phys. Rev. C 93, 034914 (2016), arXiv:1506.08552 [hep-ex].
. S Chatrchyan, CMS10.1007/JHEP03(2015)022arXiv:1410.4825JHEP. 0322nucl-exS. Chatrchyan et al. (CMS), JHEP 03, 022 (2015), arXiv:1410.4825 [nucl-ex].
. S S Adler, PHENIX10.1103/PhysRevLett.94.232301arXiv:nucl-ex/0503003Phys. Rev. Lett. 94232301[7] S. S. Adler et al. (PHENIX), Phys. Rev. Lett. 94, 232301 (2005), arXiv:nucl-ex/0503003.
. M L Miller, K Reygers, S J Sanders, P Steinberg, 10.1146/annurev.nucl.57.090506.123020arXiv:nucl-ex/0701025Ann. Rev. Nucl. Part. Sci. 57205M. L. Miller, K. Reygers, S. J. Sanders, and P. Steinberg, Ann. Rev. Nucl. Part. Sci. 57, 205 (2007), arXiv:nucl- ex/0701025.
. N Armesto, E Scomparin, 10.1140/epjp/i2016-16052-4arXiv:1511.02151Eur. Phys. J. Plus. 131nucl-exN. Armesto and E. Scomparin, Eur. Phys. J. Plus 131, 52 (2016), arXiv:1511.02151 [nucl-ex].
. A Majumder, M Van Leeuwen, 10.1016/j.ppnp.2010.09.001arXiv:1002.2206Prog. Part. Nucl. Phys. 6641hep-phA. Majumder and M. Van Leeuwen, Prog. Part. Nucl. Phys. 66, 41 (2011), arXiv:1002.2206 [hep-ph].
. B Muller, J Schukraft, B Wyslouch, 10.1146/annurev-nucl-102711-094910arXiv:1202.3233Ann. Rev. Nucl. Part. Sci. 62361hep-exB. Muller, J. Schukraft, and B. Wyslouch, Ann. Rev. Nucl. Part. Sci. 62, 361 (2012), arXiv:1202.3233 [hep-ex].
. Y Mehtar-Tani, J G Milhano, K Tywoniuk, 10.1142/S0217751X13400137arXiv:1302.2579Int. J. Mod. Phys. A. 281340013hep-phY. Mehtar-Tani, J. G. Milhano, and K. Tywoniuk, Int. J. Mod. Phys. A 28, 1340013 (2013), arXiv:1302.2579 [hep-ph].
. G Roland, K Safarik, P Steinberg, 10.1016/j.ppnp.2014.05.001Prog. Part. Nucl. Phys. 7770G. Roland, K. Safarik, and P. Steinberg, Prog. Part. Nucl. Phys. 77, 70 (2014).
. M Connors, C Nattrass, R Reed, S Salur, 10.1103/RevModPhys.90.025005arXiv:1705.01974Rev. Mod. Phys. 9025005nuclexM. Connors, C. Nattrass, R. Reed, and S. Salur, Rev. Mod. Phys. 90, 025005 (2018), arXiv:1705.01974 [nucl- ex].
. W Busza, K Rajagopal, W Van Der Schee, 10.1146/annurev-nucl-101917-020852arXiv:1802.04801Ann. Rev. Nucl. Part. Sci. 68hep-phW. Busza, K. Rajagopal, and W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018), arXiv:1802.04801 [hep-ph].
. P Dorau, J.-B Rose, D Pablos, H Elfner, 10.1103/PhysRevC.101.035208arXiv:1910.07027Phys. Rev. C. 10135208nucl-thP. Dorau, J.-B. Rose, D. Pablos, and H. Elfner, Phys. Rev. C 101, 035208 (2020), arXiv:1910.07027 [nucl-th].
. R Baier, Y L Dokshitzer, A H Mueller, S Peigne, D Schiff, 10.1016/S0550-3213(96)00553-6arXiv:hep-ph/9607355Nucl. Phys. B. 483291R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff, Nucl. Phys. B 483, 291 (1997), arXiv:hep- ph/9607355.
. R Baier, Y L Dokshitzer, A H Mueller, S Peigne, D Schiff, 10.1016/S0550-3213(96)00581-0arXiv:hep-ph/9608322Nucl. Phys. B. 484265R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff, Nucl. Phys. B 484, 265 (1997), arXiv:hep- ph/9608322.
. B G Zakharov, 10.1134/1.567126arXiv:hep-ph/9607440JETP Lett. 63952B. G. Zakharov, JETP Lett. 63, 952 (1996), arXiv:hep- ph/9607440.
. B G Zakharov, 10.1134/1.567389arXiv:hep-ph/9704255JETP Lett. 65615B. G. Zakharov, JETP Lett. 65, 615 (1997), arXiv:hep- ph/9704255.
. M Gyulassy, P Levai, I Vitev, 10.1016/S0550-3213(99)00713-0arXiv:hep-ph/9907461Nucl. Phys. B. 571197M. Gyulassy, P. Levai, and I. Vitev, Nucl. Phys. B 571, 197 (2000), arXiv:hep-ph/9907461.
. M Gyulassy, P Levai, I Vitev, 10.1103/PhysRevLett.85.5535arXiv:nucl-th/0005032Phys. Rev. Lett. 855535M. Gyulassy, P. Levai, and I. Vitev, Phys. Rev. Lett. 85, 5535 (2000), arXiv:nucl-th/0005032.
. M Gyulassy, P Levai, I Vitev, 10.1016/S0550-3213(00)00652-0arXiv:nucl-th/0006010Nucl. Phys. B. 594371M. Gyulassy, P. Levai, and I. Vitev, Nucl. Phys. B 594, 371 (2001), arXiv:nucl-th/0006010.
. X.-F Guo, X.-N Wang, 10.1103/PhysRevLett.85.3591arXiv:hep-ph/0005044Phys. Rev. Lett. 853591X.-f. Guo and X.-N. Wang, Phys. Rev. Lett. 85, 3591 (2000), arXiv:hep-ph/0005044.
. P B Arnold, G D Moore, L G Yaffe, 10.1088/1126-6708/2003/01/030arXiv:hep-ph/0209353JHEP. 0130P. B. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 01, 030 (2003), arXiv:hep-ph/0209353.
. S Jeon, G D Moore, 10.1103/PhysRevC.71.034901arXiv:hep-ph/0309332Phys. Rev. C. 7134901S. Jeon and G. D. Moore, Phys. Rev. C 71, 034901 (2005), arXiv:hep-ph/0309332.
. N Armesto, C A Salgado, U A Wiedemann, 10.1103/PhysRevD.69.114003arXiv:hep-ph/0312106Phys. Rev. D. 69114003N. Armesto, C. A. Salgado, and U. A. Wiedemann, Phys. Rev. D 69, 114003 (2004), arXiv:hep-ph/0312106.
. S A Bass, C Gale, A Majumder, C Nonaka, G.-Y Qin, T Renk, J Ruppert, 10.1103/PhysRevC.79.024901arXiv:0808.0908Phys. Rev. C. 7924901nucl-thS. A. Bass, C. Gale, A. Majumder, C. Nonaka, G.-Y. Qin, T. Renk, and J. Ruppert, Phys. Rev. C 79, 024901 (2009), arXiv:0808.0908 [nucl-th].
. S Caron-Huot, C Gale, 10.1103/PhysRevC.82.064902arXiv:1006.2379Phys. Rev. C. 8264902hep-ph[29] S. Caron-Huot and C. Gale, Phys. Rev. C 82, 064902 (2010), arXiv:1006.2379 [hep-ph].
. N Armesto, 10.1103/PhysRevC.86.064904arXiv:1106.1106Phys. Rev. C. 8664904hep-phN. Armesto et al., Phys. Rev. C 86, 064904 (2012), arXiv:1106.1106 [hep-ph].
. E Shuryak, 10.1103/RevModPhys.89.035001arXiv:1412.8393Rev. Mod. Phys. 8935001hep-phE. Shuryak, Rev. Mod. Phys. 89, 035001 (2017), arXiv:1412.8393 [hep-ph].
. J Casalderrey-Solana, D C Gulhan, J G Milhano, D Pablos, K Rajagopal, 10.1007/JHEP09(2015)175arXiv:1405.3864JHEP. 1019hepphJ. Casalderrey-Solana, D. C. Gulhan, J. G. Milhano, D. Pablos, and K. Rajagopal, JHEP 10, 019 (2014), [Erratum: JHEP 09, 175 (2015)], arXiv:1405.3864 [hep- ph].
. J Casalderrey-Solana, D Gulhan, G Milhano, D Pablos, K Rajagopal, 10.1007/JHEP03(2017)135arXiv:1609.05842JHEP. 03135hep-phJ. Casalderrey-Solana, D. Gulhan, G. Milhano, D. Pab- los, and K. Rajagopal, JHEP 03, 135 (2017), arXiv:1609.05842 [hep-ph].
. J Ghiglieri, D Teaney, 10.1142/S0218301315300131arXiv:1502.03730Int. J. Mod. Phys. E. 241530013hep-phJ. Ghiglieri and D. Teaney, Int. J. Mod. Phys. E 24, 1530013 (2015), arXiv:1502.03730 [hep-ph].
. G D Moore, N Schlusser, 10.1103/PhysRevD.101.014505arXiv:1911.13127Phys. Rev. D. 10159903Phys.Rev.D. hep-latG. D. Moore and N. Schlusser, Phys. Rev. D 101, 014505 (2020), [Erratum: Phys.Rev.D 101, 059903 (2020)], arXiv:1911.13127 [hep-lat].
Transverse momentum broadening of highenergy partons from 3D lattice EQCD simulations. N Schlusser, 10.25534/tuprints-00013254Tech. Hochsch. Ph.D. thesisN. Schlusser, Transverse momentum broadening of high- energy partons from 3D lattice EQCD simulations, Ph.D. thesis, Darmstadt, Tech. Hochsch. (2020).
. J Ghiglieri, G D Moore, D Teaney, 10.1007/JHEP03(2016)095arXiv:1509.07773JHEP. 0395hep-phJ. Ghiglieri, G. D. Moore, and D. Teaney, JHEP 03, 095 (2016), arXiv:1509.07773 [hep-ph].
. G.-Y Qin, J Ruppert, C Gale, S Jeon, G D Moore, M G Mustafa, 10.1103/PhysRevLett.100.072301arXiv:0710.0605Phys. Rev. Lett. 10072301hep-phG.-Y. Qin, J. Ruppert, C. Gale, S. Jeon, G. D. Moore, and M. G. Mustafa, Phys. Rev. Lett. 100, 072301 (2008), arXiv:0710.0605 [hep-ph].
. B Schenke, C Gale, S Jeon, 10.1103/PhysRevC.80.054913arXiv:0909.2037Phys. Rev. C. 8054913hep-phB. Schenke, C. Gale, and S. Jeon, Phys. Rev. C 80, 054913 (2009), arXiv:0909.2037 [hep-ph].
. K M Burke, JET10.1103/PhysRevC.90.014909arXiv:1312.5003Phys. Rev. C. 9014909nucl-thK. M. Burke et al. (JET), Phys. Rev. C 90, 014909 (2014), arXiv:1312.5003 [nucl-th].
. S Cao, JETSCAPE10.1103/PhysRevC.96.024909arXiv:1705.00050Phys. Rev. C. 9624909nucl-thS. Cao et al. (JETSCAPE), Phys. Rev. C 96, 024909 (2017), arXiv:1705.00050 [nucl-th].
. P B Arnold, G D Moore, L G Yaffe, 10.1088/1126-6708/2003/05/051arXiv:hep-ph/0302165JHEP. 0551P. B. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 05, 051 (2003), arXiv:hep-ph/0302165.
. M He, H Van Hees, P B Gossiaux, R J Fries, R Rapp, 10.1103/PhysRevE.88.032138arXiv:1305.1425Phys. Rev. E. 8832138nucl-thM. He, H. van Hees, P. B. Gossiaux, R. J. Fries, and R. Rapp, Phys. Rev. E 88, 032138 (2013), arXiv:1305.1425 [nucl-th].
. J.-P Blaizot, E Iancu, 10.1016/S0370-1573(01)00061-8arXiv:hep-ph/0101103Phys. Rept. 359J.-P. Blaizot and E. Iancu, Phys. Rept. 359, 355 (2002), arXiv:hep-ph/0101103.
. S Cao, JETSCAPE10.1103/PhysRevC.104.024905arXiv:2102.11337Phys. Rev. C. 10424905nucl-thS. Cao et al. (JETSCAPE), Phys. Rev. C 104, 024905 (2021), arXiv:2102.11337 [nucl-th].
. W Ke, X.-N Wang, 10.1007/JHEP05(2021)041arXiv:2010.13680JHEP. 0541hep-phW. Ke and X.-N. Wang, JHEP 05, 041 (2021), arXiv:2010.13680 [hep-ph].
. J.-P Blaizot, E Iancu, Y Mehtar-Tani, 10.1103/PhysRevLett.111.052001arXiv:1301.6102Phys. Rev. Lett. 11152001hep-phJ.-P. Blaizot, E. Iancu, and Y. Mehtar-Tani, Phys. Rev. Lett. 111, 052001 (2013), arXiv:1301.6102 [hep-ph].
. J.-P Blaizot, Y Mehtar-Tani, 10.1016/j.aop.2016.01.002arXiv:1501.03443Annals Phys. 368148hep-phJ.-P. Blaizot and Y. Mehtar-Tani, Annals Phys. 368, 148 (2016), arXiv:1501.03443 [hep-ph].
A pedagogical introduction relating the AMY kinetic equations to the turbulent cascade presented here is given. 10.1146/annurev-nucl-101918-023825arXiv:1908.02113Ann. Rev. Nucl. Part. Sci. S. Schlichting and D. Teaney69nucl-thA pedagogical introduction relating the AMY kinetic equations to the turbulent cascade presented here is given in: S. Schlichting and D. Teaney, Ann. Rev. Nucl. Part. Sci. 69, 447 (2019), arXiv:1908.02113 [nucl-th].
. J.-P Blaizot, F Dominguez, E Iancu, Y Mehtar-Tani, 10.1007/JHEP06(2014)075arXiv:1311.5823JHEP. 0675hep-phJ.-P. Blaizot, F. Dominguez, E. Iancu, and Y. Mehtar- Tani, JHEP 06, 075 (2014), arXiv:1311.5823 [hep-ph].
. P B Arnold, W Xiao, 10.1103/PhysRevD.78.125008arXiv:0810.1026Phys. Rev. D. 78125008hep-phP. B. Arnold and W. Xiao, Phys. Rev. D 78, 125008 (2008), arXiv:0810.1026 [hep-ph].
. P Arnold, 10.1103/PhysRevD.79.065025Phys. Rev. D. 7965025P. Arnold, Phys. Rev. D 79, 065025 (2009).
. P B Arnold, C Dogan, 10.1103/PhysRevD.78.065008arXiv:0804.3359Phys. Rev. D. 7865008hep-phP. B. Arnold and C. Dogan, Phys. Rev. D 78, 065008 (2008), arXiv:0804.3359 [hep-ph].
. Y Mehtar-Tani, S Schlichting, 10.1007/JHEP09(2018)144arXiv:1807.06181JHEP. 09144hep-phY. Mehtar-Tani and S. Schlichting, JHEP 09, 144 (2018), arXiv:1807.06181 [hep-ph].
. W Ke, Y Xu, S A Bass, 10.1103/PhysRevC.98.064901arXiv:1806.08848Phys. Rev. C. 9864901nucl-thW. Ke, Y. Xu, and S. A. Bass, Phys. Rev. C 98, 064901 (2018), arXiv:1806.08848 [nucl-th].
. J Dunkel, P Hänggi, 10.1016/j.physrep.2008.12.001arXiv:0812.1996Phys. Rept. 471cond-mat.stat-mechJ. Dunkel and P. Hänggi, Phys. Rept. 471, 1 (2009), arXiv:0812.1996 [cond-mat.stat-mech].
. G D Moore, D Teaney, 10.1103/PhysRevC.71.064904arXiv:hep-ph/0412346Phys. Rev. C. 7164904G. D. Moore and D. Teaney, Phys. Rev. C 71, 064904 (2005), arXiv:hep-ph/0412346.
. F Jüttner, Annalen der Physik. 339856F. Jüttner, Annalen der Physik 339, 856 (1911).
. M See Also, N A Mendoza, S Araújo, H J Succi, Herrmann, Scientific reports. 2611and references thereinSee also M. Mendoza, N. A. Araújo, S. Succi, and H. J. Herrmann, Scientific reports 2, 611 (2012), and references therein.
| [] |
[
"Direct phase mapping of broadband Laguerre-Gaussian metasurfaces",
"Direct phase mapping of broadband Laguerre-Gaussian metasurfaces"
] | [
"Alexander Faßbender ",
"Jiří Babocký \nCentral European Institute of Technology\nBrno University of Technology\nPurkyňova 123612 00BrnoCzech Republic\n",
"Petr Dvořák \nCentral European Institute of Technology\nBrno University of Technology\nPurkyňova 123612 00BrnoCzech Republic\n",
"Vlastimil Křápek \nCentral European Institute of Technology\nBrno University of Technology\nPurkyňova 123612 00BrnoCzech Republic\n",
"Stefan Linden [email protected] ",
"\nUniversität Bonn\n1) Physikalisches Institut, Nussallee 1253115BonnGermany\n"
] | [
"Central European Institute of Technology\nBrno University of Technology\nPurkyňova 123612 00BrnoCzech Republic",
"Central European Institute of Technology\nBrno University of Technology\nPurkyňova 123612 00BrnoCzech Republic",
"Central European Institute of Technology\nBrno University of Technology\nPurkyňova 123612 00BrnoCzech Republic",
"Universität Bonn\n1) Physikalisches Institut, Nussallee 1253115BonnGermany"
] | [] | We report on the fabrication of metasurface phase plates consisting of gold nanoantenna arrays that generate Laguerre-Gaussian modes from a circularly polarized Gaussian input beam. The corresponding helical phase profiles with radial discontinuities are encoded in the metasurfaces by the orientation of the nanoantennas.A common-path interferometer is used to determine the orbital angular momentum of the generated beams. Additionally, we employ digital holography to record detailed phase profiles of the Laguerre-Gaussian modes. Experiments with different laser sources demonstrate the broadband operation of the metasurfaces. | 10.1063/1.5049368 | [
"https://arxiv.org/pdf/1807.06342v1.pdf"
] | 51,789,873 | 1807.06342 | 0067591b012c23c914ee5d1bb47d0f793e819dc3 |
Direct phase mapping of broadband Laguerre-Gaussian metasurfaces
17 Jul 2018
Alexander Faßbender
Jiří Babocký
Central European Institute of Technology
Brno University of Technology
Purkyňova 123612 00BrnoCzech Republic
Petr Dvořák
Central European Institute of Technology
Brno University of Technology
Purkyňova 123612 00BrnoCzech Republic
Vlastimil Křápek
Central European Institute of Technology
Brno University of Technology
Purkyňova 123612 00BrnoCzech Republic
Stefan Linden [email protected]
Universität Bonn
1) Physikalisches Institut, Nussallee 1253115BonnGermany
Direct phase mapping of broadband Laguerre-Gaussian metasurfaces
17 Jul 2018(Dated: 18 July 2018)1 arXiv:1807.06342v1 [physics.optics]numbers: 4230Rx (Phase retrieval), 4240Kw (Holographic interferometry), 4250Tx (Optical angular momentum and its quantum aspects), 4260Jf (Beam characteristics: profile, intensity, and powerspatial pattern formation), 7867Pt (Multilayerssuperlatticesphotonic structuresmetamaterials) Keywords: Metasurface, Laguerre-Gaussian beams, Interferometry
We report on the fabrication of metasurface phase plates consisting of gold nanoantenna arrays that generate Laguerre-Gaussian modes from a circularly polarized Gaussian input beam. The corresponding helical phase profiles with radial discontinuities are encoded in the metasurfaces by the orientation of the nanoantennas.A common-path interferometer is used to determine the orbital angular momentum of the generated beams. Additionally, we employ digital holography to record detailed phase profiles of the Laguerre-Gaussian modes. Experiments with different laser sources demonstrate the broadband operation of the metasurfaces.
Optical vortex beams have been the subject of intense research activities in recent years 1,2 and have found numerous applications in optical micromanipulation 3 , quantum optics 4 , imaging 3 , and communications 5 . Their characteristic feature is a helical phase distribution with a phase singularity on the optical axis. As a consequence of this, optical vortex beams possess annular intensity cross sections with strictly zero on-axis intensity. Moreover, they carry an orbital angular momentum (OAM) ofhl per photon, which is independent of the polarization state of the beam. Here, l is the so-called topological charge that determines the Optical vortex beams can be generated by a number of different methods, e.g., astigmatic mode converters 1 , spiral phase plates 6 , spatial light modulators 7,8 , and diffraction gratings 9 .
A promising new approach is based on metasurfaces [10][11][12][13][14][15] . A metasurface is an artificial ultrathin optical device that consists of a dense arrays of sub-wavelength building-blocks 16 .
These so-called meta-atoms serve as light scattering elements with properties that can be engineered by their geometry and material composition. A light beam impinging on the metasurface interacts with the meta-atoms and gives rise to a scattered wave. The resulting wave front is thereby determined by the spatial variation of the scattering properties of the metasurface. By encoding an azimuthal phase factor exp (ıϕl) into the metasurface, one can generate an optical vortex beam with topological charge l from a Gaussian input beam. In particular, geometric metasurfaces based on the Pancharatnam-Berry phase concept 12,13,15 are suited for this purpose, as they combine a simple design principle with broadband operation and ease of fabrication. This type of metasurface employs simple dipole antennas, e.g., plasmonic nanorods, as scattering elements to partially convert a circularly polarized input beam into light with the opposite circular polarization. The phase shift φ introduced by one of the dipoles is given by φ = 2σθ, where θ is the angle enclosed between the dipole axis and a reference axis (in our case the x-axis) and σ characterizes the circular polarization state of the incident light (right circular polarization (RCP): σ = 1, left circular polarization (LCP): σ = −1). Based on this simple rule, the desired phase distribution φ(x, y) can be directly translated into the required orientation θ(x, y) of the meta-atom at the position (x, y) in the metasurface.
In this article, we report on the generation of Laguerre-Gaussian beams using geometric metasurfaces consisting of gold nanorod antennas. A circularly polarized Gaussian input beam is transmitted through a metasurface to imprint the phase distribution φ l,p (r, ϕ) of the desired LG l,p beam onto the scattered wave with the opposite circular polarization:
φ l,p (r, ϕ) = ϕl + πu −L |l| p 2r 2 /w 2 0 .(1)
Here, u(x) is the unit step function, L l p (x) is a generalized Laguerre polynomial, and w 0 is the waist radius of the incident Gaussian beam. The first addend ϕl in equation (1) is responsible for the helical phase profile, while the second addend accounts for the phase jumps of the LG l,p beam in radial direction.
The geometric metasurface phase plates are fabricated on top of an indium tin oxide covered glass substrate by standard electron beam lithography in combination with thermal evaporation of gold. Each metasurface has a circular shape with a diameter of 100 µm and consists of a square array of gold nanorods with a period of 500 nm. The dimensions of a nanorod are 220 nm × 60 nm × 40 nm (length × width × height). For these parameters, the nanorods support a localized plasmon mode with a resonance wavelength of 1080 nm. The orientation of the nanorods in a given metasurface encodes the desired phase distribution.
LG 0,0
LG 0, 1 LG 0,2
LG 1,2 LG 1,1 LG 1,0 For the generation of a Laguerre-Gaussian beam with indices l and p, the angle θ(r, ϕ) between the long axis of the nanorod at the position (r, ϕ) and the x-axis is chosen to be θ = φ l,p (r, ϕ)/2. To record detailed phase profiles of the generated Laguerre-Gaussian beams directly behind the metasurface phase plates, we employ a coherence-controlled holographic microscope (CCHM). This microscope allows the use of spatial and temporal incoherent light source for illumination 17,18 . While typical off-axis systems cannot operate with incoherent light, the CCHM can handle the introduced incoherence. Therefore, it holds the advantage of an in-line system utilizing incoherent light for the suppression of coherent noise and combines LG 1,0
LG 2,0 LG 2,1 LG 2,2 Pol 1 λ/4 λ/4 Pol 2 L L L CCD MS Laser (a) h-pol v-pol RCP LCP (b)
LG -1,0
LG -2,0
LG 2,0 LG 3,0
LG LG 2,0 LG 1,0 LG 0,0
LG 0,1
LG 1,1
LG 2,1 LG 2,2
LG 1,2
LG transmitted through the sample arm still contains a contribution of the driving field. To retrieve solely the field scattered by the metasurface, we measured and reconstructed the field transmitted through a non-active part of the sample (i.e., without nanorods), which represents the portion of the driving field transmitted due to imperfect setup. This field was subsequently numerically subtracted from the total field transmitted through the active part of the sample. Figure 4b shows the phase reconstructed from CCHM measurements for metasurfaces generating various Laguerre-Gaussian modes. Clearly, there is very good agreement with designed phase distribution. The experiment can also be conducted at different wavelengths. We additionally chose wavelengths of λ 2 = 633 nm and λ 3 = 980 nm to illustrate the broadband working regime of the metasurface. None of these sources drives the rods in resonance. Figure
number and handedness of the intertwined helical phase fronts of the vortex. Prime examples of optical vortex beams are Laguerre-Gaussian modes (LG l,p ) with azimuthal index |l| ≥ 1 and arbitrary radial index p 1 .
FIG. 1 .
1The upper images show the phase distributions for a LG 1,0 mode with a sector of the metasurface captured by an electron micrograph. The lower images depict a LG 2,2 mode, designed for a beam waist radius of w 0 = 17 µm.
. 2. (a) Scheme of the setup used for the intensity measurements. Polarizer 2 can be tuned azimuthally to create a common-path interferometer. (b) Measured intensity distributions of different Laguerre-Gaussian modes.
Figure 1depicts scanning electron micrographs of sections of two metasurface phase plates together with the corresponding phase distributions.4 The intensity profiles of the generated Laguerre-Gaussian beams are characterized with the setup schematically shown inFig. 2a. As light source we use a continuous wave diode laser with a wavelength of λ = 780 nm that is spatially filtered by a single mode fiber to guarantee a high quality Gaussian beam. The input beam is sent through a circular polarizer consisting of a linear polarizer and a quarter wave plate and focused (f = 100 mm) onto the phase plate (waist radius w 0 = 17 µm). The scattered light as well as the transmitted Gaussian beam are collected with a second lens (f = 100 mm). A crossed circular analyzer blocks the input beam and the transmitted Laguerre-Gaussian beam is imaged onto a CCD camera.Figure 2b shows intensity distributions of several Laguerre-Gaussian beams. Intensity distributions of beams with the same azimuthal (radial) index are arranged in the same line (column). In this figure, the radial mode index p increases successively by one from left to right. As expected, the number of radial discontinuities in the corresponding intensity distributions (dark rings around the center) increases likewise. The azimuthal mode index l grows in steps of one from bottom to top. The associated increase in topological charge becomes noticeable by the growing low intensity region centered around the beam axis. To measure the topological charge of the generated Laguerre-Gaussian beams, we use the setup as a common-path interferometer. The transmitted Gaussian beam and the generated Laguerre-Gaussian beam have orthogonal linear polarizations after passing the second quarter-wave plate (see Fig. 2a). By adjusting the polarization axis of polarizer 2, we can overlap both fields on the CCD with comparable amplitudes. The resulting interference image allows for an easy and fast determination of the topological charge of the Laguerre-Gaussian beam. Since the Gaussian input has a flat phase front (l = 0), we can directly deduce the absolute value |l| of the topological charge of the generated Laguerre-Gaussian beam from the number of spiral fringes in the interference image. The sign of the topological charge follows from the handedness of the spirals. Exemplary interference images of Laguerre-Gaussian beams with p = 0 and l = ±1, ±2, ±3 are depicted in Fig. 3.
FIG. 3 .
3Results of the interferometric measurements, performed with a common-path interferometer. The OAM carried by these six different metasurfaces can be easily identified due to the overlap with a Gaussian beam of l = 0. The number in each upper right corner indicates the phase shift that is introduced in one azimuthal turn around the beam axis. it with the advantage of an off-axis system, which needs only one single interferographic image to extract the amplitude and phase distribution in detail. A halogen lamp with a bandpass filter (center wavelength 650 nm) can thus be used as light source. The beam is split up into reference and sample arm, see Fig. 4a. Polarization optics let only pass right handed circularly polarized light. Both arms are equipped with the same objectives for focusing onto the substrate and recollecting the light. The metasurface lies in the focal plane between the microscope objectives of the sample arm, while the reference arm contains a glass plate to adjust for the optical path through the substrate. Additionally, polarization optics in the sample arm behind the phase plate filter out the transmitted incident beam. A diffraction grating in the reference arm can correct for the incoherent illumination and is imaged onto the output plane. The hologram resulting from the overlap of sample and reference arm is recorded by a camera. The beams enclose an angle. Amplitude and phase of the beam behind the metasurface can be reconstructed by the method of digital holography 19 . Due to imperfect alignment and depolarization effects,
FIG. 4 .
4(a) Scheme of the CCHM, consisting of reference arm with reference sample (RS) and sample arm with metasurface (MS). (b) Detailed phase profiles of the Laguerre-Gaussian modes (0 ≤ l ≤ 2 and 0 ≤ p ≤ 2) recorded with a holographic microscope.
FIG. 5 .
5Broadband application of a LG 1,2 metasurface, demonstrated at wavelengths of 633 nm, 780 nm, and 980 nm, respectively.
5 depicts the same metasurface arrangement (LG 1,2 ) illuminated by the three different light sources. The measurements show no qualitative differences. In conclusion, we fabricate metasurfaces that imprint the phase profiles of Laguerre-Gaussian modes on a Gaussian beam. The nanorods forming the metasurface are plasmonic antennas that introduce the necessary phase shift by the scattering of light into the opposite circular polarization. Based on the principle of the geometric Pancharatnam-Berry phase, the phase profile is encoded in the orientation of the nanorods. The off-resonant scattering of the nanoantennas allows for a broadband application. We demonstrate the operation at three different wavelength: 633 nm, 780 nm, and 980 nm. The value and sign of the quantum number of orbital angular momentum are measured using a common-path 8 interferometer. Detailed phase profiles are obtained from a single holographic image using coherence-controlled holographic microscopy. The authors declare no competing financial interest. S.L. acknowledges financial support by the German Federal Ministry of Education and Research through the funding program Photonics Research Germany (project 13N14150). CCHM measurements were carried out with support of the Ministry of Education, Youth and Sports of the Czech Republic (projects CEITEC 2020, No. LQ1601, and CEITEC Nano RI, No. LM2015041).
Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. L Allen, M W Beijersbergen, R J C Spreeuw, J P Woerdman, Phys. Rev. A. 45L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
Orbital angular momentum: origins, behavior and applications. A M Yao, M J Padgett, Adv. Opt. Photon. 3A. M. Yao and M. J. Padgett, "Orbital angular momentum: origins, behavior and appli- cations," Adv. Opt. Photon. 3, 161-204 (2011).
Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity. H He, M Friese, N Heckenberg, H Rubinsztein-Dunlop, Phys. Rev. Lett. 75H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
Entanglement of the orbital angular momentum states of photons. A Mair, A Vaziri, G Weihs, A Zeilinger, Nature. 412A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
Optical communications using orbital angular momentum beams. A E Willner, H Huang, Y Yan, Y Ren, N Ahmed, G Xie, C Bao, L Li, Y Cao, Z Zhao, J Wang, M P J Lavery, M Tur, S Ramachandran, A F Molisch, N Ashrafi, S Ashrafi, Adv. Opt. Photon. 7A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, "Optical communications using orbital angular momentum beams," Adv. Opt. Photon. 7, 66-106 (2015).
Laguerre-gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses. K Sueda, G Miyaji, N Miyanaga, M Nakatsuka, Opt. Express. 12K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, "Laguerre-gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses," Opt. Express 12, 3548- 3553 (2004).
Universal generation of higher-order multiringed Laguerre-Gaussian beams by using a spatial light modulator. Y Ohtake, T Ando, N Fukuchi, N Matsumoto, H Ito, T Hara, Opt. Lett. 32Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, "Universal gener- ation of higher-order multiringed Laguerre-Gaussian beams by using a spatial light mod- ulator," Opt. Lett. 32, 1411-1413 (2007).
Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial 9 light modulators. N Matsumoto, T Ando, T Inoue, Y Ohtake, N Fukuchi, T Hara, J. Opt. Soc. Am. A. 25N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, "Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial 9 light modulators," J. Opt. Soc. Am. A 25, 1642-1651 (2008).
Screw dislocations in light wavefronts. V Bazhenov, M Soskin, M Vasnetsov, Journal of Modern Optics. 39V. Bazhenov, M. Soskin, and M. Vasnetsov, "Screw dislocations in light wavefronts," Journal of Modern Optics 39, 985-990 (1992).
Light propagation with phase discontinuities: generalized laws of reflection and refraction. N Yu, P Genevet, M A Kats, F Aieta, J.-P Tetienne, F Capasso, Z Gaburro, Science. 334N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, "Light propagation with phase discontinuities: generalized laws of reflection and refrac- tion," Science 334, 333-337 (2011).
Ultra-thin plasmonic optical vortex plate based on phase discontinuities. P Genevet, N Yu, F Aieta, J Lin, M A Kats, R Blanchard, M O Scully, Z Gaburro, F Capasso, Appl. Phys. Lett. 10013101P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, "Ultra-thin plasmonic optical vortex plate based on phase discontinuities," Appl. Phys. Lett. 100, 013101 (2012).
Dispersionless phase discontinuities for controlling light propagation. L Huang, X Chen, H Mühlenbernd, G Li, B Bai, Q Tan, G Jin, T Zentgraf, S Zhang, Nano Letters. 12L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, "Dispersionless phase discontinuities for controlling light propagation," Nano Letters 12, 5750-5755 (2012).
Photonic spin-controlled multifunctional shared-aperture antenna array. E Maguid, I Yulevich, D Veksler, V Kleiner, M L Brongersma, E Hasman, Science. 352E. Maguid, I. Yulevich, D. Veksler, V. Kleiner, M. L. Brongersma, and E. Hasman, "Photonic spin-controlled multifunctional shared-aperture antenna array," Science 352, 1202-1206 (2016).
Volumetric generation of optical vortices with metasurfaces. L Huang, X Song, B Reineke, T Li, X Li, J Liu, S Zhang, Y Wang, T Zentgraf, ACS Photonics. 4L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, "Volumetric generation of optical vortices with metasurfaces," ACS Photonics 4, 338-346 (2017).
On-chip generation of broadband high-order Laguerre-Gaussian modes in a metasurface. Y Wang, X Fang, Z Kuang, H Wang, D Wei, Y Liang, Q Wang, T Xu, Y Zhang, M Xiao, Opt. Lett. 42Y. Wang, X. Fang, Z. Kuang, H. Wang, D. Wei, Y. Liang, Q. Wang, T. Xu, Y. Zhang, and M. Xiao, "On-chip generation of broadband high-order Laguerre-Gaussian modes in a metasurface," Opt. Lett. 42, 2463-2466 (2017).
Flat optics with designer metasurfaces. N Yu, F Capasso, Nat. Mater. 13N. Yu and F. Capasso, "Flat optics with designer metasurfaces," Nat. Mater. 13, 139-150 (2014).
Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic microscope. T Slaby, P Kolman, Z Dostal, M Antos, M Lostak, R Chmelik, Opt. Express. 21T. Slaby, P. Kolman, Z. Dostal, M. Antos, M. Lostak, and R. Chmelik, "Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic mi- croscope," Opt. Express 21, 14747-14762 (2013).
Quantitative 3D phase imaging of plasmonic metasurfaces. J Babocký, A Křížová, L Štrbková, L Kejík, F Ligmajer, M Hrtoň, P Dvořák, M Týč, J Čolláková, V Křápek, R Kalousek, R Chmelík, T Šikola, ACS Photonics. 4J. Babocký, A. Křížová, L.Štrbková, L. Kejík, F. Ligmajer, M. Hrtoň, P. Dvořák, M. Týč, J.Čolláková, V. Křápek, R. Kalousek, R. Chmelík, and T.Šikola, "Quantitative 3D phase imaging of plasmonic metasurfaces," ACS Photonics 4, 1389-1397 (2017).
Digital holography for quantitative phasecontrast imaging. E Cuche, F Bevilacqua, C Depeursinge, Opt. Lett. 24E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase- contrast imaging," Opt. Lett. 24, 291-293 (1999).
| [] |
[
"Helical relativistic electron beam and THz radiation",
"Helical relativistic electron beam and THz radiation"
] | [
"S Son \n18 Caleb Lane, 28 Benjamin Rush Lane08540, 08540Princeton, PrincetonNJ, NJ\n",
"Sung Joon Moon \n18 Caleb Lane, 28 Benjamin Rush Lane08540, 08540Princeton, PrincetonNJ, NJ\n"
] | [
"18 Caleb Lane, 28 Benjamin Rush Lane08540, 08540Princeton, PrincetonNJ, NJ",
"18 Caleb Lane, 28 Benjamin Rush Lane08540, 08540Princeton, PrincetonNJ, NJ"
] | [] | A THz laser generation utilizing a helical relativistic electron beam propagating through a strong magnetic field is discussed. The initial amplification rate in this scheme is much stronger than that in the conventional free electron laser. A magnetic field of the order of Tesla can yield a radiation in the range of 0.5 to 3 THz, corresponding to the total energy of mJ and the duration of tens of pico-second, or the temporal power of the order of GW. PACS numbers: 42.88, 41.85.L, 52.25.Xz A THz electromagnetic (E&M) wave has a range of practical applications[1][2][3][4]. In particular, the light wave of the frequency of 1 to 10 THz is under increased attention[5,6]. Around the frequency range of 100 GHz, there exist appropriate technologies such as gyrotron[7][8][9]. However, these technologies cannot be extended to the range above a few hundred GHz, due to the wellknown scaling problem[10]. Other technologies such as the quantum cascade laser[11,12]and the free electron laser [13] have their own limitations in generating an intense E&M wave. There have been preliminary proposals for generating a THz radiation based on the recent advances in the intense visible laser[14][15][16], in the context of the inertial confinement fusion[17][18][19].In this paper, we propose a scheme to generate a THz radiation, where the energy is extracted from the perpendicular kinetic energy of an relativistic electron beam, in the presence of a strong magnetic field. In this scheme, a relativistic electron beam gets launched to a slightly skewed direction with respect to the magnetic field. The electrons gyrate around the magnetic field and the perpendicular velocity of the electron exhibits a periodic structure(Fig 1). When a certain resonance condition is satisfied, a specific E&M THz wave becomes amplified. In particular, the rate at which the electron energy is extracted is proportional to the electric field strength, as opposed to the energy intensity as in the conventional free electron laser (FEL). This difference leads to a much more explosive amplification compared to the conventional FEL. The goal of this paper is to estimate the amplification efficiency.We start by describing the motion of the helical electrons and the FEL amplification, and then describe a new scheme enabling an explosive amplification. A relativistic electron moving in the presence of the magnetic field is described bywhere m e is the electron mass, v is the electron velocity, z v x v y FIG. 1: The helical velocity structure of the relativistic electron beam propagating with a constant velocity along a slightly skewed direction to the z-axis. This schematic diagram is not drawn to scale.γ −2 0 = 1 − (v 2 0z + v 2 p )/c 2 is the relativistic factor, v 0z (v p is the parallel (perpendicular) velocity relative to the zaxis, and the magnetic field is given asConsider a linearly-polarized E&M wave propagating along the z-direction so that E x (z, t) = E 1 cos(kz − ckt), E y = E z = 0, B y (z, t) = E 1 cos(kz − ckt), and B x = B z = 0. The perturbed motion of the electron iswhere E 1 = E 1 cos(kz − ckt)x, and B 1 = E 1 cos(kz − ckt)ŷ is the magnetic field of the E&M wave. Consider the first case when v (0) z ≫ v (0) x and v (0) z ≫ v (0) y so that the terms of v (0) x and v (0) y in the linearized first order | null | [
"https://arxiv.org/pdf/1111.1452v1.pdf"
] | 117,937,711 | 1111.1452 | 3dc80fa71b629a8a005514643a1199cc13cfb37a |
Helical relativistic electron beam and THz radiation
6 Nov 2011 (Dated: November 8, 2011)
S Son
18 Caleb Lane, 28 Benjamin Rush Lane08540, 08540Princeton, PrincetonNJ, NJ
Sung Joon Moon
18 Caleb Lane, 28 Benjamin Rush Lane08540, 08540Princeton, PrincetonNJ, NJ
Helical relativistic electron beam and THz radiation
6 Nov 2011 (Dated: November 8, 2011)
A THz laser generation utilizing a helical relativistic electron beam propagating through a strong magnetic field is discussed. The initial amplification rate in this scheme is much stronger than that in the conventional free electron laser. A magnetic field of the order of Tesla can yield a radiation in the range of 0.5 to 3 THz, corresponding to the total energy of mJ and the duration of tens of pico-second, or the temporal power of the order of GW. PACS numbers: 42.88, 41.85.L, 52.25.Xz A THz electromagnetic (E&M) wave has a range of practical applications[1][2][3][4]. In particular, the light wave of the frequency of 1 to 10 THz is under increased attention[5,6]. Around the frequency range of 100 GHz, there exist appropriate technologies such as gyrotron[7][8][9]. However, these technologies cannot be extended to the range above a few hundred GHz, due to the wellknown scaling problem[10]. Other technologies such as the quantum cascade laser[11,12]and the free electron laser [13] have their own limitations in generating an intense E&M wave. There have been preliminary proposals for generating a THz radiation based on the recent advances in the intense visible laser[14][15][16], in the context of the inertial confinement fusion[17][18][19].In this paper, we propose a scheme to generate a THz radiation, where the energy is extracted from the perpendicular kinetic energy of an relativistic electron beam, in the presence of a strong magnetic field. In this scheme, a relativistic electron beam gets launched to a slightly skewed direction with respect to the magnetic field. The electrons gyrate around the magnetic field and the perpendicular velocity of the electron exhibits a periodic structure(Fig 1). When a certain resonance condition is satisfied, a specific E&M THz wave becomes amplified. In particular, the rate at which the electron energy is extracted is proportional to the electric field strength, as opposed to the energy intensity as in the conventional free electron laser (FEL). This difference leads to a much more explosive amplification compared to the conventional FEL. The goal of this paper is to estimate the amplification efficiency.We start by describing the motion of the helical electrons and the FEL amplification, and then describe a new scheme enabling an explosive amplification. A relativistic electron moving in the presence of the magnetic field is described bywhere m e is the electron mass, v is the electron velocity, z v x v y FIG. 1: The helical velocity structure of the relativistic electron beam propagating with a constant velocity along a slightly skewed direction to the z-axis. This schematic diagram is not drawn to scale.γ −2 0 = 1 − (v 2 0z + v 2 p )/c 2 is the relativistic factor, v 0z (v p is the parallel (perpendicular) velocity relative to the zaxis, and the magnetic field is given asConsider a linearly-polarized E&M wave propagating along the z-direction so that E x (z, t) = E 1 cos(kz − ckt), E y = E z = 0, B y (z, t) = E 1 cos(kz − ckt), and B x = B z = 0. The perturbed motion of the electron iswhere E 1 = E 1 cos(kz − ckt)x, and B 1 = E 1 cos(kz − ckt)ŷ is the magnetic field of the E&M wave. Consider the first case when v (0) z ≫ v (0) x and v (0) z ≫ v (0) y so that the terms of v (0) x and v (0) y in the linearized first order
A THz laser generation utilizing a helical relativistic electron beam propagating through a strong magnetic field is discussed. The initial amplification rate in this scheme is much stronger than that in the conventional free electron laser. A magnetic field of the order of Tesla can yield a radiation in the range of 0.5 to 3 THz, corresponding to the total energy of mJ and the duration of tens of pico-second, or the temporal power of the order of GW. A THz electromagnetic (E&M) wave has a range of practical applications [1][2][3][4]. In particular, the light wave of the frequency of 1 to 10 THz is under increased attention [5,6]. Around the frequency range of 100 GHz, there exist appropriate technologies such as gyrotron [7][8][9]. However, these technologies cannot be extended to the range above a few hundred GHz, due to the wellknown scaling problem [10]. Other technologies such as the quantum cascade laser [11,12] and the free electron laser [13] have their own limitations in generating an intense E&M wave. There have been preliminary proposals for generating a THz radiation based on the recent advances in the intense visible laser [14][15][16], in the context of the inertial confinement fusion [17][18][19].
In this paper, we propose a scheme to generate a THz radiation, where the energy is extracted from the perpendicular kinetic energy of an relativistic electron beam, in the presence of a strong magnetic field. In this scheme, a relativistic electron beam gets launched to a slightly skewed direction with respect to the magnetic field. The electrons gyrate around the magnetic field and the perpendicular velocity of the electron exhibits a periodic structure (Fig 1). When a certain resonance condition is satisfied, a specific E&M THz wave becomes amplified. In particular, the rate at which the electron energy is extracted is proportional to the electric field strength, as opposed to the energy intensity as in the conventional free electron laser (FEL). This difference leads to a much more explosive amplification compared to the conventional FEL. The goal of this paper is to estimate the amplification efficiency.
We start by describing the motion of the helical electrons and the FEL amplification, and then describe a new scheme enabling an explosive amplification. A relativistic electron moving in the presence of the magnetic field is described by
m e dγ 0 v dt = −e v c × B,(1)
where m e is the electron mass, v is the electron velocity, γ −2 0 = 1 − (v 2 0z + v 2 p )/c 2 is the relativistic factor, v 0z (v p is the parallel (perpendicular) velocity relative to the zaxis, and the magnetic field is given as
B = B 0ẑ . The solution is v (0) z (t) = v 0z , v (0) x (t) = v p cos(ω ce t + φ 0 ), and v (0) y (t) = −v p sin(ω ce t + φ 0 ), where ω ce = eB 0 /γ 0 m e c. Consider a linearly-polarized E&M wave propagating along the z-direction so that E x (z, t) = E 1 cos(kz − ckt), E y = E z = 0, B y (z, t) = E 1 cos(kz − ckt), and B x = B z = 0. The perturbed motion of the electron is m e dγ 0 v (1) dt + dγ 1 v (0) dt = −e v (1) c × B 0 equation, compared to v (0) z = v 0z
, can be ignored. The linearized equation is given as
m e γ 0 dv (1) x dt = −eE 1 1 − v 0z c − eB 0 v (1) y c , m e γ 0 dv (1) z dt + γ 3 0 v 2 0z c 2 dv (1) z dt = −eE 1 v (0) x c , m e γ 0 dv (1) y dt = eB 0 v (1) x c , where E 1 = E 1 cos(kz − ckt) and v (0) x (t) = v p cos(ω ce t + φ 0 ), v (0) z = v 0z = const.
The solution in the perpendicular direction can be obtained by using the complex
coordinate v p (t) = v (1) x + iv (1) y such that v p (t)
is the solution of the following equation:
dv p dt + iω ce v p = − eE 1 γ 0 m e 1 − v 0z c .(2)
Then, the electron energy loss rate by the E&M wave per unit volume is
dǫ dt = n e m e c 2 dγ dt = n e m e γ 2 0 dv (1) dt · v (0) = n e γ 3 0 v0z c γ 0 + γ 3 0 (v0z ) 2 c 2 + (1 − v0z c ) γ 0 eE 1 v (0) x ,(3)
where is the ensemble average over the phase φ 0 , and eE 1 v 0x is, from Eq. (1), given as
eE 1 v 0x = eE 1 v p c cos(kz − ckt + ω ce t + φ 0 ).(4)
Consider the second case v 0z ≪ v 0x and v 0z ≪ v 0y , where the computation is more complicated, in the absence of a closed-form solution. Retaining only the resonance term, the energy loss rate becomes
dǫ dt = n e γ 3 0 1 γ 0 + γ 3 0 v 2 p 2c 2 eE 1 v (0) x .(5)
The resonance condition for the FEL amplification is v 0z k − ck + ω ce = 0, or k = ω ce /(c − v 0z ). With the resonance condition being satisfied, the ensemble average of the energy loss or gain cancels out in the first order of E 1 if the distribution over the phase angle φ 0 is uniform. The ensemble average in the second order of E 1 provides the conventional FEL amplification. Now, let us dicuss the difference between the conventional electron beam and the helical beam. We note that there exists circumstances where the ensemble average
eE 1 v (0) x
does not cancel out in the first order of E 1 .
Consider the time slice at t = 0, where the helical structure of the electron velocity is given as in Fig. 1:
v 0x (t = 0, z) = v p cos(k h z) v 0y (t = 0, z) = −v p sin(k h z) v 0z (t = 0, z) = v 0z ,
where k h = ω ce /v 0z is the helix wave vector. This helical structure, formed by the electron gun, has zero phase velocity in the laboratory frame or is a static wave. An electron initially (t = 0) located at z = z 0 evolves as
v 0x (t) = v p cos(k h z 0 + ω ce t) v 0y (t) = −v p sin(k h z 0 + ω ce t) v 0z (t) = v 0z .
Since E 1 (z, t) = E 1 cos(kz − ckt), the ensemble average over the electrons, EV = e E z (v 0x + iv 0y ) , is given as
EV = E 1 v p cos (k(v 0z t + z 0 ) − ckt + ω ce t − k h z 0 ) dz 0 = E 1 v p cos ((kv 0z − ck + ω ce )t + (k h + k)z 0 ) dz 0 .
The phase angle φ 0 (z) is given as φ 0 (z) = (k h + k)z. While the ensemble average over the entire beam does cancel out, it does not locally for a fixed value of z. The electrons of the same phase, located in the range z 0 − δz < z < z 0 + δz where δ = π/2(k + k H ), contribute coherently to EV so that the local E&M wave is amplified or damped by the these coherently phased electrons. The E&M wave that itnitially gets amplified by the coherent electrons gets damped by different but coherently phased electrons as it propagates. The frequency at which the E&M wave experiences the change between the amplification and the damping is estimated to be Ω ∼ = 1/δt, where δt(c − v 0 ) = δz. Since (c − v 0z )k = ω ce , δt can be estimate as 1/ω ce so that Ω ∼ = ω ce . This local amplification is in contrast with the electrons with random phases for fixed z = z 0 .
The above argument suggests that there exists a local amplification mechanism for the helical plasma, that could be used as a THz generation. Denoting the relativistic factors γ m = (1 − v 2 0z /c 2 ) −1/2 and γ = (1 − v 2 /c 2 ) −1/2 , the frequency of the amplified wave can be drived from the resonance condition k(c − v 0z ) = ω ce as 2γ 2 m ω ce . Consider the case γ m = 7, γ = 10 and B 0 = 1 T, which correspond to 2γ 2 m ω ce = 300 GHz. Consider another when γ m = 30, γ = 40 and B 0 = 1 T, 2γ 2 m ω ce = 1.4 THz. Let us estimate the E&M wave growth rate for the amplification. For simplicity, let us use the reference frame where we move with the electron beam with the same velocity in the z-direction. Assume that γ m > 1, v 0zm ≪ v 0xm and v 0zm ≪ v 0ym . If v 0x (v 0y ) is the perpendicular velocity in the laboratory frame, v 0xm = γ m v 0x (v 0ym = γ m v 0y ) is the perpendicular velocity in the moving frame. If the electron density in the laboratory frame is n e , then it is n em = n e /γ m in the moving frame. The electron energy loss rate in the moving frame given in Eq. (5) is
dǫ dt ∼ = αn em eE 1 v 0xm ,(6)
where α is a constant of order of 1. By considering the local E&M wave and the local amplification, we obtain from dǫ/dt = (1/8π
)(dE 2 1 /dt) that dE 1 dt ∼ = α4πen em v 0xm cos(Ωt),(7)
where Ω ∼ = eB 0 /m e c is in the moving frame. Eq. (7) shows that the initial growth rate, (dE 1 /dt)/E 1 , is infinite. During the time duration of 1/Ω, E 1 grows to E 1 (T ) = αn em ev 0xm /Ω, and the ratio of the E&M energy intensity to the particle kinetic energy becomes
E1(T ) 2 8π n em m e (v 0xm ) 2 ∼ = α 2 2 ω pem Ω 2 ,(8)
where ω 2 pem = 4πn em e 2 /m e and cos(Ωt) 2 = 1/2 is used. Eq. (8) suggests that the THz E&M wave gets amplified to the energy intensity comparable to the perpendicular electron kinetic energy intensity times the ratio ω 2 pem /Ω 2 during the time duration of Ω, which is the maximum energy that could be extracted. In the moving frame, the perpendicular kinetic energy of an electron is N m e v 2 0xm = N γ 2 m m e v 2 0x . The maximum total energy radiating into the THz wave is E max = N γ 2 m m e v 2 0x (ω 2 pem /Ω 2 ) so that the maximum THz energy in the laboratory frame is γ m E max = N γ 3 m m e v 2 0x (ω 2 pem /Ω 2 ), where N is the total number of electrons in the beam. In order to extract the appreciable fraction of the electron kinetic energy, the ratio ω pem /Ω needs to be maximized. As shown in the nonneutral plasma beam analysis, it is theoretically possible to get ω pem /Ω ∼ = 1.
Let us give a few examples of the practically relevant beam parameters. Consider a 10 pico-second electron beam with γ = 35, n e = 10 13 cm −3 and the total number of electrons being 10 10 , and assume that the magnetic field is order of 1 T. If the beam gets launched with v p /v 0z = 0.03, the parallel relativistic factor is γ m = 25. The resonant frequency for the THz radiation is roughly 1 THz. In the moving frame, the electron density becomes roughly 4 × 10 11 cm −3 , and ω pem /Ω ∼ = 0.1; the beam duration is 250 pico-second. The total energy of the electron is 7 × 10 15 eV, and at the maximum, a few percents of the total electron kinetic energy can be radiated into the THz E&M wave. As another example, consider a 10 picosecond electron beam with γ = 14. Assume that the electron density is 10 14 cm −3 , the total number of electrons is 10 10 , and the beam of v p /v z0 = 0.06 gets launched (γ m = 11). Assuming the magnetic field is order of 1 T, the resonant frequency is roughly 0.5 THz. In the moving frame, the electron density is roughly 10 13 cm −3 , ω pem /Ω ∼ = 0.1, and the beam duration is 100 pico-second. The total energy of the electron is 10 16 eV, and as much as tens of percents of the total electron kinetic energy can be radiated into the THz E&M wave.
To summarize, a scheme of THz generation is discussed, where the spatial helical structure of the relativistic electron beam is used for the amplification, via a physical mechanism similar to that of the FEL. In contrast to the FEL with the magnets, the energy extraction rate from the electrons is not proportional to the intensity, rather it is proportional to the electric field of the E&M wave. This property makes this scheme advantageous, as the THZ field can be explosively amplified up to certain amplitude. The overall efficiency is another advantage. A THz radiation with the total energy of a few tens of percents of the total electron beam energy can be as high as gyrotron or magnetron; the only difference is the operating regime, the THz range.
PACS numbers: 42.88, 41.85.L, 52.25.Xz
FIG. 1 :
1The helical velocity structure of the relativistic electron beam propagating with a constant velocity along a slightly skewed direction to the z-axis. This schematic diagram is not drawn to scale.
. M Nagel, P H Bolivar, M Brucherseifer, H Kurz, A Bosserhoff, R Buttner, Appl. Phys. Lett. 80154M. Nagel, P. H. Bolivar, M. Brucherseifer, H. Kurz, A. Bosserhoff, and R. Buttner, Appl. Phys. Lett. 80, 154 (2002).
. K B Cooper, R J Dengler, G Chattopadhyay, E Schlecht, J Gill, A Skalare, I Mehdi, P H Siegel, IEEE, Microwave and Wireless Components Letters. 1864K. B. Cooper, R. J. Dengler, G. Chattopadhyay, E. Schlecht, J. Gill, A. Skalare, I. Mehdi, and P. H. Siegel, IEEE, Microwave and Wireless Components Letters 18, 64 (2008).
Antennas and Propagation. P H Siegel, IEEE Transactions on. 552957P. H. Siegel, Antennas and Propagation, IEEE Transac- tions on 55, 2957 (2007).
. K Yamamoto, M Yamaguchi, F Miyamaru, M Tani, M Hangyo, T Ikeda, A Matsushita, K Koide, M Tatsuno, Y Minami, Jpn. J. Appl. Phys. 43414K. Yamamoto, M. Yamaguchi, F. Miyamaru, M. Tani, M. Hangyo, T. Ikeda, A. Matsushita, K. Koide, M. Tat- suno, and Y.Minami, Jpn. J. Appl. Phys. 43, L414 (2004).
Microwave Theory and Techniques. P H Siegel, IEEE Transaction on. 50910P. H. Siegel, Microwave Theory and Techniques, IEEE Transaction on 50, 910 (2002).
Microwave Theory and Techniques. P H Siegel, IEEE Transaction on. 522438P. H. Siegel, Microwave Theory and Techniques, IEEE Transaction on 52, 2438 (2004).
. K R Chu, H Y Chen, C L Hung, T H Chang, L R Barnett, S H Chen, T T Yang, Phys. Rev. Lett. 814760K. R. Chu, H. Y. Chen, C. L. Hung, T. H. Chang, L. R. Barnett, S. H. Chen, and T. T. Yang, Phys. Rev. Lett. 81, 4760 (1998).
. K E Kreischer, R J Temkin, Phys. Rev. Lett. 595471987K. E. Kreischer and R. J. Temkin, Phys. Rev. Lett. 59, 1987 (547).
. V L Bratman, Y L Kalynov, V N Manuilov, Phys. Rev. Lett. 102245101V. L. Bratman, Y. L. Kalynov, and V. N. Manuilov, Phys. Rev. Lett. 102, 245101 (2009).
. J H Booske, Physics of Plasmas. 1555502J. H. Booske, Physics of Plasmas 15, 055502 (2008).
. J Faist, F Capasso, D L Sivco, C Sirtori, A L Hutchinson, A Y Cho, Science. 264553J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).
. M Tonouchi, Nature Photonics. 197M. Tonouchi, Nature Photonics 1, 97 (2007).
. W B Colson, Nucl. Inst. Meth. Phys. 2371W. B. Colson, Nucl. Inst. Meth. Phys. A237, 1 (1985).
. V M Malkin, N J Fisch, Phys. Rev. Lett. 99205001V. M. Malkin and N. J. Fisch, Phys. Rev. Lett. 99, 205001 (2007).
. S Son, S Ku, Phys. Plasmas. 1710703S. Son and S. Ku, Phys. Plasmas 17, 010703 (2009).
. S Son, S Ku, S J Moon, Phys. Plasmas. 17114506S. Son, S. Ku, and S. J. Moon, Phys. Plasmas 17, 114506 (2010).
. M Tabak, J Hammer, M E Glinsky, W L Kruerand, S C Wilks, J Woodworth, E M Campbell, M J Perry, R J Mason, Physics of Plasmas. 11626M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruerand, S. C. Wilks, J. Woodworth, E. M. Campbell, M. J. Perry, and R. J. Mason, Physics of Plasmas 1, 1626 (1994).
. S Son, N J Fisch, Phys. Lett. A. 32916S. Son and N. J. Fisch, Phys. Lett. A 329, 16 (2004).
. S Son, N J Fisch, Phys. Rev. Lett. 95225002S. Son and N. J. Fisch, Phys. Rev. Lett. 95, 225002 (2005).
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[] | [] | [] | [] | The multi-period dynamics of energy storage (ES), intermittent renewable generation and uncontrollable power loads, make the optimization of power system operation (PSO) challenging. A multi-period optimal PSO under uncertainty is formulated using the chance-constrained optimization (CCO) modeling paradigm, where the constraints include the nonlinear energy storage and AC power flow models. Based on the emerging scenario optimization method which does not rely on preknown probability distribution functions, this paper develops a novel solution method for this challenging CCO problem. The proposed method is computationally effective for mainly two reasons. First, the original AC power flow constraints are approximated by a set of learning-assisted quadratic convex inequalities based on a generalized least absolute shrinkage and selection operator. Second, considering the physical patterns of data and motived by learning-based sampling, the strategic sampling method is developed to significantly reduce the required number of scenarios through different sampling strategies. The simulation results on IEEE standard systems indicate that 1) the proposed strategic sampling significantly improves the computational efficiency of the scenario-based approach for solving the chance-constrained optimal PSO problem, 2) the data-driven convex approximation of power flow can be promising alternatives of nonlinear and nonconvex AC power flow. | 10.1109/tsg.2021.3127922 | [
"https://arxiv.org/pdf/2107.10013v1.pdf"
] | 236,154,941 | 2107.10013 | 4603b34ab287d48bbccfcf1936d3bbf0e61d6439 |
1Index Terms-chance-constrainedpower flowscenario opti- mizationLASSOdata-driven
The multi-period dynamics of energy storage (ES), intermittent renewable generation and uncontrollable power loads, make the optimization of power system operation (PSO) challenging. A multi-period optimal PSO under uncertainty is formulated using the chance-constrained optimization (CCO) modeling paradigm, where the constraints include the nonlinear energy storage and AC power flow models. Based on the emerging scenario optimization method which does not rely on preknown probability distribution functions, this paper develops a novel solution method for this challenging CCO problem. The proposed method is computationally effective for mainly two reasons. First, the original AC power flow constraints are approximated by a set of learning-assisted quadratic convex inequalities based on a generalized least absolute shrinkage and selection operator. Second, considering the physical patterns of data and motived by learning-based sampling, the strategic sampling method is developed to significantly reduce the required number of scenarios through different sampling strategies. The simulation results on IEEE standard systems indicate that 1) the proposed strategic sampling significantly improves the computational efficiency of the scenario-based approach for solving the chance-constrained optimal PSO problem, 2) the data-driven convex approximation of power flow can be promising alternatives of nonlinear and nonconvex AC power flow.
I. INTRODUCTION
nergy storage (ES) has been well-recognized for dealing with the challenges in power systems, such as shaving peak-load and filling valley-load. However, the current cost of battery ES is still expensive. According to the roadmap of ES issued by the U.S. department of Energy in 2020, by 2030 the levelized cost of battery ES may be reduced to only 10% of the current cost [1]. This probably makes ES widely used in power systems. However, the inter-temporal property of ES may couple the multi-period power system operation (PSO). Moreover, the intermittence of renewable energy (RE) brings the uncertainty to PSO. Hence, the exploration on optimizing the multi-period PSO with ES (PSO-ES) under the uncertainty The authors are with the Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL 32816 USA (e-mail: [email protected], Qifeng.li@ucf,edu). of RE is rather challenging. Unfortunately, the current deterministic approaches are incapable of capturing the uncertainty in the context of optimization. There exist some approaches of modeling optimization problems under uncertainty, such as stochastic, robust, and chance-constraint optimization methods [2]- [7]. The stochastic optimization [2], [3] attempts to find solutions with the best expected objective values based on the predefined probability distributions. The robust optimization [3], [4] enforces strict feasibility under the worst case, resulting in high conservativeness. Unlike the two methods above, the chance-constrained optimization (CCO) [5]- [7] can guarantee that the satisfactory probability of a solution is above a certain level if properly implemented. Generally, power system operators put higher weight in security than in costsaving. As a tradeoff, power operators may be more interested in solutions with the low probability of constraint violation. Therefore, in this paper, CCO is adopted to model the multiperiod PSO-ES problem under uncertainty (CC-PSO-ES). Despite its widespread applications in engineering disciplines, the original CCO is generally computationally expensive for large-scale systems like power grids. Additionally, the conventional solution methods of CCO overly depend on actual joint probability distribution function (PDF) of random variables which is hard to access [5]- [7].
As an alternative to the PDF-based methods to solve the CCO problems, the scenario-based solution method, called scenario optimization has been applied in probabilistic optimization problems [8], [9], learning models and artificial intelligence (AI) [10], [11]. The key to scenario optimization aims at how to determine the minimum sample size (MSS) required to satisfy the specific probability level [8], [12], [13]. Reference [8] illustrates a random sampling (RS)-based method (RSM) to estimate the MSS associated with the number of decision variables under the convex program. However, for complex systems with numerous decision variables, the MSS estimated by RSM may explode as the MSS is proportional to the size of decision variables [8]. To tackle this issue, the fast algorithm for scenario technique (FAST), a two-stage method [10] has been proposed to cut down the sample size and applied in computing optimal power flow with uncertainty [14]. As stated in RS-based methods [9], [10], there may have a small size of 'active scenarios' that essentially decides the optimal solution. The number of active scenarios is proven to be at most E the decision variable size, which is far smaller than the sample size determined by RS-based methods. In other words, most of scenarios selected by RS-based methods are 'inactive' and removable. However, these active scenarios are unknown before solving the RS-based optimization problems. Inspired by the resounding sequential sampling [10], [11] used in machine learning and the existence of physical patterns in data, a concept of strategic sampling is developed in this paper to find a much smaller size of scenarios that can approximate the effect of the active scenarios before optimization, through physicsguided sampling [32], [33], dissimilarity-based learning [15] and reinforcement learning [16] methods. The above-mentioned scenario-related optimization methods are currently only applicable to convex program problems [9], [10]. However, the constraints of AC power flow (ACPF) in CC-PSO-ES problem are inherently nonlinear and nonconvex [6]. Currently, the approximations of ACPF have been discussed from the perspectives of linearization and convexification. The linear approximations like the DC model [7], [17] and other linear ACPF [6], [18], are generally easy to solve, however, many of which ignore the quadratic terms of voltages resulting in inaccuracy of model. The typical convex approximations have been widely studied, such as the secondorder cone (SOC) [19], semi-definite programming (SDP) [20], convex DistFlow [21], quadratic convex (QC) [22], momentbased [23], convex hull relaxation [24], and the learning-based convex approximation [25], [26]. References [26]- [28] found that the SDP relaxation may not guarantee the exactness of solutions and its exactness greatly depends on the critical assumptions of network topologies and physical parameter settings. In [26], the authors developed an ensemble learningbased data-driven convex quadratic approximation (DDCQA) of ACPF with higher accuracy and efficiency than the SDP relaxation. This paper introduces the generalized least absolute shrinkage and selection operator (LASSO) [29], [30] to improve the DDCQA developed in [26] from both aspects of computing time and space used.
To solve the CC-PSO-ES which is a complex multi-period nonlinear nonconvex optimization problem, this paper proposes a novel scenario-based solution method based on the DDCQA of ACPF and strategic sampling. The proposed approach is more computationally effective using only few effective scenarios, compared with RSM. The contributions of this paper are written as below:
1) The strategic sampling is developed based on physicsguided sampling and learning-based sampling methods to select a small size of scenarios for solving CC-PSO-ES problem.
2) The DDCQA of ACPF is improved by generalized LASSO from the aspects of computational time and space complexity, then applied to convert the originally intractable nonconvex CC-PSO-ES problem into a tractable convex quadratic optimization problem.
The rest of this paper is organized as follows: Section II illustrates the formulations of deterministic and chanceconstrained multi-period PSO-ES problems. In Section III, scenario optimization is introduced, and the novel scenariobased solution method for CC-PSO-ES problem is developed through strategic sampling and the DDCQA of ACPF modified by generalized LASSO. The empirical IEEE case analyses and conclusions are displayed in Section IV and V, respectively.
II. PROBLEM FORMULATION
This section formulates the optimal multi-period operation for power systems with energy storage under the modeling paradigm of chance-constrained optimization step-by-step.
A. Deterministic Multi-period PSO with Battery Energy Storage
In an n-bus power system, the deterministic formulation of the multi-period PSO-ES is given as following which can also be considered as a multi-period AC optimal power flow (ACOPF) with adjustable generation and battery energy storage:
∑ ∑ ( 1 , + 2 ( , ) 2 ) ∈ (1a) s.t. , ∑ ( , =1 − , ) + , ∑ ( , + , ) =1 = , − , + , ,(1b)
, , 2
+ , , 2 = , 2 (1h) , ≤ , ,0 + ∆ ∑ , , =1 ≤ , (1i) , 2 ≤ , 2 + , 2 ≤ , 2 (1j) , ≤ , ≤ , (1k) , ≤ , ≤ , (1l) , 2 + , 2 ≤ ,(1m)
where , = 1,2, . . , ; T is the index set of time periods; the subscript denotes the t-th hour; 1 , 2 are the generator cost coefficients; , , , are the generator active and reactive power; , , , are the net active and reactive power inputs of power loads and renewable energy outputs; , , , , , , , are the lower and upper limits of the generator active and reactive power; , , , are the lower and upper limits of bus voltage;
, is the branch transmission capacity; , , , represent the real and imaginary parts of voltage; , , , denote the active and reactive line flow;
, are the real and imaginary parts of the line admittance;
, , , , , are the active and reactive power of energy storage;
, , is the active power loss of energy storage; = + , are the equivalent resistances of the battery and converter [24]; , 2 is the squared magnitude of voltage that approximates to 1.0 pu; , , is the net active power of energy storage; , , , are the capacity limits of energy storage;
, ,0 is the initial energy status of energy storage; , is the maximum apparent power of energy storage.
B. Chance-Constrained Formulation of Multi-period PSO with Adjustable Generation and Battery Energy Storage
Considering that the net load inputs of power loads and renewable energy generations (PLRES) are random variables, the power generation may be fluctuating, which consists of the base and adjustable parts. The base part meets the forecast net demand injection of PLRES. The gap between the forecast and actual net demand injections is satisfied by the adjustable part. According to the affine control scheme [31], then
, , = , ( ) + ∆ (2a) , , = , ( ) + ∆ (2b) , , = , ( ) − ∑ ∆ =1 (2c) , , = , ( ) − ∑ ∆ =1 (2d) , , = , , ( ) − ∑ ∆ =1 (2e) , , = , , ( ) − ∑ ∆ =1 (2f) ∑ + ∑ = 1 (2g) ∑ + ∑ = 1 (2h)
Where S is the scenario sets, ∈ ; , ( ) , , ( ) are the base parts of the generator's active and reactive power; , are the participation factors of the generator's or energy storage's active and reactive power, respectively ; , ( ) , , ( ) denote the forecast values of the net active and reactive load inputs of PLRES, respectively; ∆ , ∆ denote the corresponding forecast errors of , , and , , , respectively. Note that we assume the difference between the base case and real-time power loss can be compensated by the reference generator and it is typically negligible. Suppose that the chance-constraint method is used to model the problem (1) under uncertainty. Then, by substituting (2a)~(2h) into (1) and updating each variable in (1) with the superscript s, the deterministic problem (1) is reformulated into a CC-PSO-ES problem in (2):
min ∈ [∑ ∑ ( 1 , , + 2 ( , , ) 2 ] ∈ (2i) s.t. (ℎ( , ) = 0) ≥ 1 − (2j) ( ( , ) ≤ 0) ≥ 1 − (2k)
where ℎ( , ), ( , ) compacts the constraints of (1b)~(1h) and (1i)~(1m), respectively; is the variable vector consisting of the decision variables , , , , , , and the state variables such as the bus voltage; is the random variable vector such as the power loads and renewable energy generations, , , and , , ; ( • ) enforces each constraint at a specific confidence level; is the probability level.
III. A SCENARIO-BASED SOLUTION METHOD WITH STRATEGIC SAMPLING AND DATA DRIVEN CONVEX APPROXIMATION
A. Scenario Optimization
Scenario optimization has been widely used in machine learning [10], [11], whose general idea is to use a finite number of scenarios to approximate the probabilistic constraints (2f) and (2g) with a specific confidence level. The mathematical formulation can be represented as
(3a) s.t. ( , ( ) ) ≤ 0, ( = 1,2, . . , ′ ) (3b)
where (3a) is a linear objective function related to the decision variable vector ; ( ) denotes the s-th scenario sampled from the uncertainty set; is a convex function on ; ′ is the estimated number of scenarios. In the existing applications of scenario optimization, the constraints depend upon the random samples, and the sample size discussed in the statistical learning [12], [13] has a conversative estimate unrelated to the number of decision variables. Until in reference [8], the lower bound of the sample size related to the decision variable size under convex program settings is derived from the aspect of binomial distribution. A theorem in [8] states that, if ′ is sufficiently large, the optimal solution of (3) can satisfy the chance constraints (2j)~(2k). Scenario optimization is still an emerging solution method for chance-constrained optimization that does not rely on pre-known PDFs [8]- [11]. The RS-based method in [8] provides some discussions on how to determine the required number of scenarios, shown in (3c):
′ ≥ 2 ( 1 + ′ ) (3c)
where ′ is the dimension of decision variable vector; ∈ (0,1) , 1 − ∈ (0,1) are the violation probability level and confidence level, respectively. However, in CC-PSO-ES problem, the number of scenarios required by the RS-based method may be large, which results in significant challenge in computation. Moreover, scenario optimization is designed for convex optimization, while there have nonconvex constraints in current CC-PSO-ES problem. Hence, the following sections will discuss how to tackle the two issues through the strategic sampling and DDCQA, respectively.
B. Strategic Sampling
Instead of random sampling with plentiful inactive scenarios [8], we attempt to develop a framework of strategic sampling to find out a smaller number of effective scenarios that include the active ones. According to different selection strategies, there may have diverse specific strategic sampling methods shown in Fig.1, such as physics-guided sampling (PGS), learning-based sampling, and hybrid sampling, etc. [15], [16], [32], [33]. The PGS is designed considering there might have specific patterns in PLRES data. The patterns may be related to the temporal, spatial, and meteorological conditions [32], [33]. The learning-based sampling is based on machine learning methods, such as dissimilarity-based learning and reinforcement learning (RL) [15], [16]. The hybrid one may be the combination of any two sampling methods. In this research, two types of two-stage hybrid sampling methods are developed and named as HS1 and HS2. The first stage is the physics-guided sampling (PGS). Then, at the second stage, i.e., the stage of learning-based sampling, one of dissimilaritybased sampling (DBS) and RL-based sampling (RLS) will be chosen to select the d dissimilar samples. As a rule of thumb, in IEEE-5 system, d is suggested to be the number of the decision variables; all other systems can set d to be 10% of the decision variable size. Assume there are regional power systems like IEEE-5, -9, -57, -118 systems. The gist of PGS for a specific regional power system can be described in Fig.2 [32], [33].
Dissimilarity-based Sampling
The dissimilarity-based sampling (DBS) is defined to use a learning function to select the scenarios. The purpose of using a learning function is to ensure that the scenarios selected should be dissimilar enough to maximize the scenario difference in a specific data subset obtained after PGS. To measure the dissimilarity of samples, in machine learning the distance metrics are commonly used, such as the Euclidean distance [10], [11], [15]. Assume that , are the i-th and j-th samples where = ( 1 , 2 , . . , ); is the dimension of each sample. The dissimilarity between two samples can be measured by the Euclidean distance calculated as
= √∑ ( − ) 2 =1 (4a)
where denotes the dissimilarity of two samples, used for determining the new samples. The specific hybrid sampling that consists of PGS and DBS is described as HS1 in Fig.3 where N is computed by FAST.
Reinforcement Learning-based Sampling
The RL-based sampling (RLS) [16] considers each sample as a state , selecting each sample as an action , and the probability of each transition from the i-th state to j-th state as defined as:
( | , ) = ∑ ∈ (4b)
where the indices i and j correspond to the current and next state; H is the potential state set for next state. The reward of each transition is defined to be proportional to , and the discount factor of reward is set to one in this research. The implementation of RLS is shown as HS2 in Fig.4. The main difference between DBS and RLS lies in that RLS only consider the dissimilarity between the current selected sample and the candidate sample, while DBS computes the average dissimilarity between the previous all selected samples and the candidate sample.
C. Data-Driven Convex Quadratic Approximation of Power Flow
Note that the scenario-based solution methods discussed so far are designed for the convex program [8], [9], while, in the original CC-PSO-ES problem, the constraints of ACPF are nonconvex. To deal with the issue of nonconvexity, we improve the data-driven convex quadratic approximation (DDCQA) of ACPF [26] using the generalized LASSO [29], [30]. For the ease of analysis, the constraints (1b)-(1e) at t-th hour are reformulated as the following matrix form:
, = (5a) , = (5b) , = , ,(5c)
Where , , , are symmetric indefinite matrices consisting of elements of the admittance matrix, implying all de-pendent variables , , , , , , and , are nonconvex functions of the independent variables or , . Hence, a convex quadratic mapping (5g)-(5j) between power, i.e., , , , , , , and , , and voltage or , , is defined as:
, = + + (5g) , = + + (5h) , = , , + , + (5i) , = , , + , + (5j)
Where * , * denote positive semi-definite (PSD) coefficient matrices of the quadratic terms, respectively; * , * denote coefficient vectors of the linear terms; * , * denote constant terms; the upper index (*) includes the set { , } indicating the active or reactive power. According to [26], the PSD matrices in (5g)-(5j) can be obtained via training historical data using the polynomial regression as a basic learner to learn the convex relationships between the voltage and the active or reactive power. Then, ensemble learning methods are used to assemble all basic learners, to boost the performance of model. However, the PSD matrices in (5g)-(5j) are dense and high-dimensional, which is an obstacle in computing the complex CC-PSO-ES problem. Therefore, the generalized LASSO [29], [30] is introduced to learn more compact and sparser PSD matrices for the purposes of speeding up the computational efficiency and saving the storage space. The following illustration of generalized LASSO takes , as an example based on dataset { , } =1 where , are the i-th observed voltage input and active power output at t-th hour; M is the training sample size. The detailed formulation is written as
1 ∑ ( − − − ) 2 =1 + ∑ | | ′ =1 (5k) s.t. ≽ 0 (5l)
where is the j-th coefficient constituted by the entries of ; '≽' means is a PSD matrix; ′ is the number of coefficients; ≥ 0 is a tunable regularization parameter that controls the degree of shrinkage. By the shrinkage, some coefficients may be zero, which means the matrix becomes sparse.
D. Convex Hull Relaxation of Energy Storage Model
As the magnitude of voltage has little change, we assume that , 2 ≈ 1.0 pu in (1f). Then, the convex hull relaxation [24] of the energy storage model can be formulated as the followings:
‖ , ‖ 2 − , ≤ 0 (6a) ‖ , ‖ 2 − , ≤(6b
E. Scenario Programming Formulation of CC-PSO-ES
Based on the DDCQA of power flow constraints in (5g)-(5j) and the convex hull relaxation of energy storage model in (6a)-(6c), the corresponding convex constraints are written as Then, we introduce an auxiliary variable z used to reformulate the objective function (2i) into a linear formulation (7f) with a convex constraint (7g) as Minimize:
+ + ≤ , − , + , , (7a) + + ≤ , − , + , ,(7b)
(7f)
s.t. ∑ ∑ ∑ ( 0 + 1 , , + 2 ( , , ) 2 ) ≤ | | ∈ ∈ (7g)
where | | is the number of scenarios considered. As the constraints above are all convex, the scenario programming, i.e., the strategic sampling-based solution approach for the CC-PSO-ES problem can be rewritten in (8) as
Minimize: (7f) (8a) s.t. ′ ( , ( ) ) ≤ 0, ( = 1,2, . . , )(8b)
where ′ ( , ) compacts all constraints in (7a)-(7e), (7g), (1k)-(1m), and (6a)-(6c) at each scenario. The corresponding sampling procedure for the problem (8) is illustrated in the section of strategic sampling.
IV. SIMULATION ANALYSIS
A. Case selection and Data Collection
The real-world power systems and their data are expected to use in this research. However, they are not available in public. As empirical alternatives, some IEEE standard test systems such as IEEE-5, -9, -57, -118 systems and the relevant simulating data are applied. The net active and reactive power loads at each bus are based on the hourly load curves of ISO new England and set up at the range of [0.7, 1.3] of their true values to simulate the uncertainty of power loads and renewable energy generations and to generate the 24-hour simulating data. The settings of energy storage units are summarized in TABLE I. Considering sampling in the whole sample space may be computationally expensive, for each test system, the sample size determined by the RS-based method [8] is treated as the sample space. The goal is to demonstrate the efficacy of the proposed solution method with fewer effective scenarios via the DDCQA of ACPF and the strategic sampling methods. The simulations are performed in Matlab with cvx package.
B. Computational Complexity Comparison of DDCQA
To compare the computational efficiency of the DDCQA of ACPF before and after improvement, we explore the average training time for the active and reactive power at each bus and the average computation time for solving the CC-PSO-ES problem on test systems, using the method in [26] For the IEEE-118 system, the average training time and the average computing time of CC-PSO-ES used now are only about 2% and 5% of the ones before, respectively. Moreover, the storage space usage of the matrix before and after improving DDCQA on all test systems has been displayed in Fig.7. Similarly, Fig.7 shows, that on all test systems, the storage space consumed now is reduced by over 75% compared with the one consumed before. For IEEE-57 and -118 systems, the storage space used now may only account for 1%-2% of the one used before. Overall, the improved DDCQA based on the generalized LASSO greatly degrades the computational complexity in training and computing the optimization problem.
C. Performance Comparison of Solution Methods
In this numerical experiment, we set the allowed violation probability level and the confidence level to be =0.05 and 1 − =0.9999, respectively. The estimated number of scenarios required by the RS-based method [8] and FAST, and the decision variable size can be shown in II indicates that the number of active scenarios only accounts for at most 2.5% of the sample size computed by the RS-based method. In other words, the majority of N' samples may be useless for solving the CC-PSO-ES problem. Compared with RS-based method, FAST greatly reduces the number of scenarios to from N' to N. Especially, for IEEE-57 and -118 systems, the sample sizes in multi-period situation required by RS-based method are more than 230k and 690k, respectively. It will be difficult to solve the CC-PSO-ES problems with such large number of scenarios in practice. The ratio by the objective cost and base cost in TABLE III demonstrates that in this research the (±30%) uncertainty of REPL may increase the total cost by 2%~15%.
To solve the CC-PSO-ES problems, the strategic sampling methods, i.e., two-stage hybrid sampling methods proposed are used. In practice, the first stage of physics-guided sampling (PGS) should be implemented at first. Then, the second stage of learning-based sampling methods, i.e., DBS and RLS, will be applied directly to determine the effective sample size. The difference between two hybrid sampling methods resides in the stage of learning-based sampling. As the initial sample can affect the sample size selected by RLS and DBS, the experimental simulation starting with different initial samples is explored. The best and worst sample selections by RLS and DBS, i.e., their minimum and maximum sample sizes for solving the CC-PSO-ES problems, are computed, and shown in TABLE IV. From TABLE IV, we can infer that: 1) The hybrid sampling methods through RLS and DBS can further reduce the effective sample size, i.e., the sample size required by the scenario-based methods can be far less than the ones with the RS-based method and FAST. For instance, in IEEE-5 system the solution methods through RLS and DBS find the optimal solution within 890 and 450 scenarios, respectively, which is smaller than 1050 scenarios determined by FAST. In a similar fashion, in IEEE-9, -57 and -118 systems, 120, 800 and 1800 scenarios are large enough for the solution methods through RLS and DBS to reach the optimal solution.
2) Under both best and worst sample selections, DBS outperforms RLS almost on four test systems. More specifically, the solution method through DBS finds the optimal solution of CC-PSO-ES problem more efficiently with less scenarios than the one through RLS, except that for the best cases in IEEE-9 and -118 systems DBS and RLS perform equally well. The main reason may be that DBS selects each sample based on the maximum average dissimilarity between the candidate sample and the previous all selected ones, while RLS selects samples through the maximum dissimilarity between the candidate sample and the most recently selected one.
D. Verification of Learning-based Sampling Methods
To verify the DBS and RLS, at each round a new scenario is added sequentially until the number of scenarios reaches to N. In the actual application of DBS and RLS, there is no need to repeat the verification process. The corresponding results of CC-PSO-ES problems on four IEEE test cases are computed and compared based on DBS and RLS, displayed in Fig. 8~11. For the figures above, the x-axis denotes the number of scenarios required at each round of solving CC-PSO-ES, and the y-axis is the objective cost of CC-PSO-ES. At each round of computation in verification process, a new scenario is added sequentially. In other words, each figure depicts the relationship between the objective cost and number of scenarios, i.e., how the objective cost changes as the number of scenarios are added sequentially. Both DBS and RLS have two lines, i.e., the best and worst cases, denoted as 'DBS_best' and 'DBS_worst', 'RLS_best' and 'RLS_worst', respectively. The dashed line represents the total cost without considering the uncertainty of REPL, denoted as 'Base_cost'. The objective cost is normalized by 'Base_cost'. The lines may become flat finally as the new scenarios are incorporated sequentially, which indicates that the optimal solution of CC-PSO-ES problem has been achieved.
V. CONCLUSION AND FUTURE WORK
This paper presents a novel solution method for solving the chance-constrained multi-period optimal power system operation (PSO) with battery energy storage (CC-PSO-ES), which is originally nonconvex and computationally intractable. The proposed method, which is based on the data-driven convex quadratic approximation (DDCQA) of ACPF and the strategic sampling, i.e., hybrid sampling methods, only uses a small number of scenarios without the pre-known PDF of the uncertainty of REPL. The DDCQA is modified through the generalized LASSO and applied to address the nonconvex problem of ACPF constraints in CC-PSO-ES problem. Unlike the RSbased methods, the hybrid sampling methods (HSMs) are proposed with dissimilarity-based learning and reinforcement learning methods. HSMs determine a smaller sample size than the RS-based methods. Eventually, the originally intractable CC-PSO-ES is converted to a tractable convex quadratic optimization problem with few effective scenarios. In our future work, we intend to test the proposed method in real-life largescale power systems.
)
are the base parts of the energy storage's active and reactive power;
Fig. 1
1Fig.1 Framework of Strategic Sampling 1. Physics-guided Sampling
Fig. 2
2The Flowchart of Physics-guided Sampling
Fig. 3
3The Flowchart of Hybrid Sampling Type 1 (HS1) Fig.4 The Flowchart of Hybrid Sampling Type 2 (HS2)
1,t 2,t , . . , 2 , ] = [ 1,t 1,t , . . , , , ] (5e) , = [ 2 −1,t 2 , 2 −1,t 2 , ] = [ , , , , ]
named as 'old' and the one improved by the generalized LASSO named as 'new', shown in Figs.5 and 6. Figs.5 and 6 indicate that there exist significant improvements in both the training time and computing time of CC-PSO-ES problem, before and after using generalized LASSO. Particularly, on IEEE-57 system, it only takes about 25% of the original average training time to train the improved DDCQA and about 40% of the original average computing time of CC-PSO-ES to obtain the solution.
Fig. 5
5Comparison of Training Time Fig.6 Comparison of Computing Time of CC-PSO-ES Fig.7 Storage Space Usage Comparison of Matrix
Fig. 8
8Performance Comparison of RLS and DBS in IEEE-5 System Fig.9 Performance Comparison of RLS and DBS in IEEE-9 System Fig.10 Performance Comparison of RLS and DBS in IEEE-57 System Fig.11 Performance Comparison of RLS and DBS in IEEE-118 System
Optimal Operation of Power Systems with Energy Storage under Uncertainty: A Scenariobased Method with Strategic Sampling Ren Hu and Qifeng Li, Senior Member, IEEE
TABLE I The
ISettings of Energy Storage UnitsCase
IEEE-5
IEEE-9
IEEE-57
IEEE-118
Units
2
2
3
3
Bus No.
3, 5
5, 7
8, 9, 12
59, 90, 116
Capacity
1MVA,
2MWh
0.75MVA,
1.5MWh
0.75MVA,
1.5MWh
1MVA,
2MWh
TABLE II as below. Meanwhile, the total costs without and with considering the uncertainty of renewable energy and power load (REPL) are computed respectively, as 'Base Cost' (BC) and 'Objective Cost' (OC) in TABLE III. The 'Objective Cost' is computed through FAST as the benchmark.
TABLE II Sample
IISizes of FAST and RSM and the Size of Decision VariableCase
IEEE-5
IEEE-9
IEEE-57
IEEE-118
d'
864
1104
5904
17328
N
1050
1290
6090
17514
N'
34929
44529
236529
693489
Ratio1= d'/N'
0.02474
0.02479
0.02496
0.02499
TABLE III
Total Costs without and with the Uncertainty of REPL
Case
IEEE-5
IEEE-9
IEEE-57
IEEE-118
Base Cost ($/h)
451710
229074
280175
3509250
Objective Cost ($/h)
477487
262187
296260
3603450
Ratio2=OC/BC
1.0570
1.1445
1.0574
1.0268
TABLE
TABLE IV The
IVBest and Worst Sample Selections by RLS and DBSCase
IEEE-5
IEEE-9
IEEE-57
IEEE-118
RLS(best)
223
2
4
2
RLS(worst)
883
114
729
1780
DBS(best)
3
2
3
2
DBS(worst)
432
96
289
637
Energy Storage Grand Challenge. Washington DCThe U.S. Department of EnergyEnergy Storage Grand Challenge, 1st ed., The U.S. Department of Ener- gy, Washington DC, 2020, pp. 12-20
Stochastic Unit Commitment in Isolated Systems With Renewable Penetration Under CVaR Assessment. M Asensio, J Contreras, 10.1109/TSG.2015.2469134IEEE Trans. Smart Grid. 73M. Asensio and J. Contreras, "Stochastic Unit Commitment in Isolated Systems With Renewable Penetration Under CVaR Assessment," IEEE Trans. Smart Grid, vol. 7, no. 3, pp. 1356-1367, May 2016, doi: 10.1109/TSG.2015.2469134.
Unified Stochastic and Robust Unit Commitment. C Zhao, Y Guan, 10.1109/TPWRS.2013.2251916IEEE Trans. Power Syst. 283C. Zhao and Y. Guan, "Unified Stochastic and Robust Unit Commit- ment," IEEE Trans. Power Syst., vol. 28, no. 3, pp. 3353-3361, Aug. 2013, doi: 10.1109/TPWRS.2013.2251916.
Optimal Wind Farm Allocation in Multi-Area Power Systems Using Distributionally Robust Optimization Approach. F Alismail, P Xiong, C Singh, 10.1109/TPWRS.2017.2695002IEEE Trans. Power Syst. 331F. Alismail, P. Xiong and C. Singh, "Optimal Wind Farm Allocation in Multi-Area Power Systems Using Distributionally Robust Optimization Approach," IEEE Trans. Power Syst., vol. 33, no. 1, pp. 536-544, Jan. 2018, doi: 10.1109/TPWRS.2017.2695002.
Data-Driven Chance Constrained Programs over Wasserstein Balls. Zhi Chen, Daniel Kuhn, Wolfram Wiesemann, arXiv:1809.00210Zhi Chen, Daniel Kuhn, Wolfram Wiesemann, "Data-Driven Chance Constrained Programs over Wasserstein Balls", 2018, arXiv:1809.00210.
Chance Constrained Programming for Optimal Power Flow Under Uncertainty. H Zhang, P Li, 10.1109/TPWRS.2011.2154367IEEE Trans. Power Syst. 264H. Zhang and P. Li, "Chance Constrained Programming for Optimal Power Flow Under Uncertainty," IEEE Trans. Power Syst., vol. 26, no. 4, pp. 2417-2424, Nov. 2011, doi: 10.1109/TPWRS.2011.2154367.
DC Optimal Power Flow with Joint Chance Constraints. A Pena-Ordieres, D K Molzahn, L Roald, A Waechter, 10.1109/TPWRS.2020.3004023IEEE Trans. Power Syst. A. Pena-Ordieres, D. K. Molzahn, L. Roald and A. Waechter, "DC Op- timal Power Flow with Joint Chance Constraints," IEEE Trans. Power Syst., doi: 10.1109/TPWRS.2020.3004023.
General Scenario Program: Application in Smart Grid Optimization under Endogenous Uncertainty. Q Li, arXiv:2104.13494IEEE Trans. Power Syst. Under review. Q. Li, "General Scenario Program: Application in Smart Grid Optimiza- tion under Endogenous Uncertainty." IEEE Trans. Power Syst. Under review. arXiv:2104.13494 (2021).
The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs. M C Campi, S Garatti, 10.1137/07069821XSIAM J. Optim. 193M.C. Campi, S. Garatti, "The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs", SIAM J. Optim., vol. 19, no. 3, pp:1211-1230, 2008. https://doi.org/10.1137/07069821X.
FAST-Fast Algorithm for the Scenario Technique. Algo Carè, Simone Garatti, Marco C Campi, Operations Research. 623Algo Carè, Simone Garatti,Marco C. Campi, "FAST-Fast Algorithm for the Scenario Technique", Operations Research, vol. 62, no. 3, pp. 662-671, June 2014.
Adaptive sequential sampling for surrogate model generation with artificial neural networks. John Eason, Selen Cremaschi, Computers & Chemical Engineering. 68John Eason, Selen Cremaschi, "Adaptive sequential sampling for surro- gate model generation with artificial neural networks", Computers & Chemical Engineering, vol. 68, pp: 220-232, September 2014.
Exact computation of minimum sample size for estimation of binomial parameters. Xinjia Chen, Journal of Statistical Planning and Inference. 1418Chen, Xinjia. "Exact computation of minimum sample size for estima- tion of binomial parameters," Journal of Statistical Planning and Infer- ence, vol. 141, no.8, pp: 2622-2632, 2011.
Randomized Strategies for Probabilistic Solutions of Uncertain Feasibility and Optimization Problems. T Alamo, R Tempo, E F Camacho, 10.1109/TAC.2009.2031207IEEE Trans. Autom. Control. 5411T. Alamo, R. Tempo, and E. F. Camacho, "Randomized Strategies for Probabilistic Solutions of Uncertain Feasibility and Optimization Prob- lems," IEEE Trans. Autom. Control, vol. 54, no. 11, pp. 2545-2559, Nov. 2009, doi: 10.1109/TAC.2009.2031207.
Efficient Solution Strategy for Chance Constrained Optimal Power Flow based on FAST and Data driven Convexification. R Hu, Q F Li, arXiv:2105.05336R. Hu, Q. F. Li, "Efficient Solution Strategy for Chance Constrained Optimal Power Flow based on FAST and Data driven Convexification," arXiv:2105.05336, 2021. https://arxiv.org/abs/2105.05336.
A survey of manifold learning for images. Robert Pless, Richard Souvenir, IPSJ Transactions on Computer Vision and Applications. 1Pless, Robert, and Richard Souvenir. "A survey of manifold learning for images." IPSJ Transactions on Computer Vision and Applications, vol. 1, pp: 83-94, 2009.
Deep reinforcement learning-based sampling method for structural reliability assessment. Zhengliang Xiang, Yuequan Bao, Zhiyi Tang, Hui Li, Reliability Engineering & System Safety. 199106901Xiang, Zhengliang, Yuequan Bao, Zhiyi Tang, and Hui Li. "Deep rein- forcement learning-based sampling method for structural reliability as- sessment." Reliability Engineering & System Safety, vol. 199, pp:106901, 2021.
Introducing a Novel DC Power Flow Method with Reactive Power Considerations. S M Fatemi, S Abedi, G B Gharehpetian, IEEE Trans. Power Syst. 306S. M. Fatemi, S. Abedi, G. B. Gharehpetian, et al., "Introducing a Novel DC Power Flow Method with Reactive Power Considerations." IEEE Trans. Power Syst., vol. 30, no. 6, pp. 3012-3023, Nov. 2015.
Ensemble Learning based Linear Power Flow. R Hu, Q Li, S Lei, 10.1109/PESGM41954.2020.92817932020 IEEE Power & Energy Society General Meeting (PESGM), Montreal. QCR. Hu, Q. Li and S. Lei, "Ensemble Learning based Linear Power Flow," 2020 IEEE Power & Energy Society General Meeting (PESGM), Mon- treal, QC, 2020, pp. 1-5, doi: 10.1109/PESGM41954.2020.9281793.
Radial distribution load flow using conic programming. R Jabr, IEEE Trans. Power Syst. 213R. Jabr, "Radial distribution load flow using conic programming." IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1458-1459, Aug. 2006.
Semidefinite programming for optimal power flow problems. X Bai, H Wei, K Fujisawa, Y Wang, Int. J. Elect. Power Energy Syst. 3067X. Bai, H. Wei, K. Fujisawa, and Y. Wang, "Semidefinite programming for optimal power flow problems." Int. J. Elect. Power Energy Syst., vol. 30, no. 67, pp. 383-392, 2008.
Convex optimization for DES planning and operation in radial distribution systems with high penetration of photovoltaic resources. Q Li, R Ayyanar, V Vittal, IEEE Transactions on Sustainable Energy. 73Q. Li , R. Ayyanar, and V. Vittal. "Convex optimization for DES plan- ning and operation in radial distribution systems with high penetration of photovoltaic resources." IEEE Transactions on Sustainable Energy 7, no. 3 (2016): 985-995.
The QC Relaxation: A Theoretical and Computational Study on Optimal Power Flow. C Coffrin, H Hijazi, IEEE Trans. Power Syst. 314C. Coffrin, H. Hijazi, et al., "The QC Relaxation: A Theoretical and Computational Study on Optimal Power Flow." IEEE Trans. Power Syst., vol. 31, no. 4, pp. 3008-3018, July. 2016.
Sparsity-exploiting moment-based relaxations of the optimal power flow problem. D Molzahn, I Hiskens, IEEE Trans. Power Syst. 306D. Molzahn and I. Hiskens, "Sparsity-exploiting moment-based relaxa- tions of the optimal power flow problem." IEEE Trans. Power Syst., vol. 30, no. 6, pp. 3168-3180, Nov. 2015.
Convex hull of the quadratic branch AC power flow equations and its application in radial distribution networks. Q Li, V Vittal, IEEE Trans. Power Syst. 331Q. Li, and V. Vittal. "Convex hull of the quadratic branch AC power flow equations and its application in radial distribution networks." IEEE Trans. Power Syst., vol.33, no. 1, pp.839-850, Jan. 2017.
Uncertainty-Aware Three-Phase Optimal Power Flow Based on Data-Driven Convexification. Q Li, IEEE Transactions on Power Systems. 362Q. Li, "Uncertainty-Aware Three-Phase Optimal Power Flow Based on Data-Driven Convexification." IEEE Transactions on Power Systems 36, no. 2 (2021): 1645-1648.
Ensemble Learning Based Convex Approximation of Three-Phase Power Flow. R Hu, Q Li, F Qiu, 10.1109/TPWRS.2021.3055481IEEE Trans. Power Syst. R. Hu, Q. Li and F. Qiu, "Ensemble Learning Based Convex Approxi- mation of Three-Phase Power Flow," IEEE Trans. Power Syst., doi: 10.1109/TPWRS.2021.3055481.
Inexactness of SDP Relaxation and Valid Inequalities for Optimal Power Flow. B Kocuk, S S Dey, X A Sun, IEEE Trans. Power Syst. 311B. Kocuk, S. S. Dey, X. A. Sun, "Inexactness of SDP Relaxation and Valid Inequalities for Optimal Power Flow." IEEE Trans. Power Syst., vol. 31, no. 1, pp. 642-651, Jan. 2016.
Accurate Semidefinite Programming Models for Optimal Power Flow in Distribution Systems. Z Y Wang, D S Kirschen, B S Zhang, arXiv:1711.07853arXiv preprintZ.Y. Wang, D.S. Kirschen,B.S. Zhang, "Accurate Semidefinite Pro- gramming Models for Optimal Power Flow in Distribution Systems." arXiv preprint arXiv:1711.07853, 2017.
Term Selection for a Class of Separable Nonlinear Models. M Gan, G Chen, L Chen, C L P Chen, 10.1109/TNNLS.2019.2904952IEEE Transactions on Neural Networks and Learning Systems. 31M. Gan, G. Chen, L. Chen, and C. L. P. Chen, "Term Selection for a Class of Separable Nonlinear Models," in IEEE Transactions on Neural Networks and Learning Systems, vol. 31, no. 2, pp. 445-451, Feb. 2020, doi: 10.1109/TNNLS.2019.2904952.
The generalized LASSO. V Roth, 10.1109/TNN.2003.809398IEEE Transactions on Neural Networks. 15V. Roth, "The generalized LASSO," in IEEE Transactions on Neural Networks, vol. 15, no. 1, pp. 16-28, Jan. 2004, doi: 10.1109/TNN.2003.809398.
Chance-Constrained Optimal Power Flow: Risk-Aware Network Control under Uncertainty. Daniel Bienstock, Michael Chertkov, Sean Harnett, SIAM Rev. 563Daniel Bienstock, Michael Chertkov, and Sean Harnett, "Chance- Constrained Optimal Power Flow: Risk-Aware Network Control under Uncertainty," SIAM Rev., vol.56(3), pp:461-495, 2014.
A comprehensive review on uncertainty modeling techniques in power system studies. Morteza Aien, Ali Hajebrahimi, Mahmud Fotuhi-Firuzabad, Renewable and Sustainable Energy Reviews. 575Aien, Morteza, Ali Hajebrahimi, and Mahmud Fotuhi-Firuzabad. "A comprehensive review on uncertainty modeling techniques in power sys- tem studies." Renewable and Sustainable Energy Reviews, vol. 57, no.5, pp: 1077-1089, May 2016.
Day-ahead wind power forecasting based on the clustering of equivalent power curves. Mao Yang, Chaoyu Shi, Huiyu Liu, Energy. 2185119515Yang, Mao, Chaoyu Shi, and Huiyu Liu. "Day-ahead wind power fore- casting based on the clustering of equivalent power curves." Energy, vol.218, no.5, pp: 119515, 2021.
| [] |
[
"Absolutely special subvarieties and absolute Hodge cycles",
"Absolutely special subvarieties and absolute Hodge cycles"
] | [
"Tobias Kreutz "
] | [] | [] | We introduce the notion of dR-absolutely special subvarieties in motivic variations of Hodge structure as special subvarieties cut out by (de Rham-)absolute Hodge cycles and conjecture that all special subvarieties are dR-absolutely special. This is implied by Deligne's conjecture that all Hodge cycles are absolute Hodge cycles, but is a much weaker conjecture. We prove our conjecture for subvarieties satisfying a simple monodromy condition introduced in [KOU20]. We study applications to typical respectively atypical intersections andQ-bialgebraic subvarieties. Finally, we show that Deligne's conjecture as well as ours can be reduced to the case of special points in motivic variations. | null | [
"https://arxiv.org/pdf/2111.00216v2.pdf"
] | 240,353,950 | 2111.00216 | 27cd3a8dbaac9636225a7330aca7a46aa4eafa83 |
Absolutely special subvarieties and absolute Hodge cycles
27 May 2022
Tobias Kreutz
Absolutely special subvarieties and absolute Hodge cycles
27 May 2022
We introduce the notion of dR-absolutely special subvarieties in motivic variations of Hodge structure as special subvarieties cut out by (de Rham-)absolute Hodge cycles and conjecture that all special subvarieties are dR-absolutely special. This is implied by Deligne's conjecture that all Hodge cycles are absolute Hodge cycles, but is a much weaker conjecture. We prove our conjecture for subvarieties satisfying a simple monodromy condition introduced in [KOU20]. We study applications to typical respectively atypical intersections andQ-bialgebraic subvarieties. Finally, we show that Deligne's conjecture as well as ours can be reduced to the case of special points in motivic variations.
Introduction
Let f : X → S be a smooth projective morphism of smooth (irreducible) quasiprojective algebraic varieties over C. For any integer k ≥ 0 the k-th primitive cohomology of this family gives rise to a polarizable Q-variation of Hodge structure (V, V, ∇, F • ), where V = R k prim f an * Q is a Q-local system on S an , (V = R k prim f * Ω • X/S , ∇) is the associated algebraic vector bundle with flat connection and F • is the Hodge filtration of the family. We will usually drop the additional data from the notation and simply denote the variation of Hodge structure by V. Let Z ⊂ S be a closed irreducible subvariety. After choosing a point s ∈ Z(C), one defines the generic Mumford-Tate group G Z of Z as the subgroup of GL(V s ) defined by the condition that it fixes all generic Hodge tensors on Z (a different choice of s defines a conjugated subgroup).
The variation V gives rise to interesting algebraic subvarieties of S, the so-called special subvarieties of S.
Definition 0.1 ( [KO21], Definition 1.2). A closed irreducible subvariety Z ⊂ S is called special for V if it is maximal among the closed irreducible algebraic subvarieties of S having the same generic Mumford-Tate group as Z.
Thus special subvarieties are varieties which are cut out by Hodge cycles. Although the definition of Hodge cycles is purely analytic, the Hodge conjecture asserts that all Hodge cycles come from algebraic cycles. We therefore expect that Hodge cycles, seen as classes in the algebraic vector bundle V, are preserved under algebraic field automorphisms of C. This motivates the definition of absolute Hodge cycles by Deligne. For any automorphism σ ∈ Aut(C/Q) we can form the conjugate S σ := S ⊗ C,σ C. The projection defines a map σ −1 : S ⊗ C,σ C → S. Similarly as above, from the conjugated family f σ : X σ → S σ we obtain a variation of Hodge structure (V σ , V σ , ∇ σ , F •,σ ). The fact that de Rham cohomology can be defined algebraically provides comparison isomorphisms ι σ : (V σ , ∇ σ , F •,σ ) ∼ = σ −1 * (V, ∇, F • ) of the algebraic filtered vector bundles with connection over S σ . We call a collection of variations of Hodge structure (V σ ) σ satisfying the above comparison an absolute variation of Hodge structure. An absolute variation of Hodge structure over S is called of geometric origin if it arises from a smooth projective family f : X → S as above. When given an absolute variation of Hodge structure, following Deligne, one can define a notion of dR-absolute Hodge tensor as a Hodge tensor α ∈ V ⊗ s such that, for any σ ∈ Aut(C/Q), the σ-conjugate of the de Rham component of α comes from a Hodge tensor in (V σ s σ ) ⊗ via the comparison isomorphism ι σ . Remark 0.2. The notion of absolute Hodge cycle introduced in Deligne's work [Del82] is stronger than the version we use here, in the sense that his notion includes a condition on the ℓ-adic components of Hodge cycles as well. Here we only consider de Rham absolute Hodge cycles in the terminology of ( [Voi06], Definition 0.1). In the following, we will write dR-absolute Hodge for this notion.
For a subvariety Z ⊂ S we define the generic dR-absolute Mumford-Tate group G AH Z to be the subgroup of GL(V s ) defined by the condition that it fixes all generic dR-absolute Hodge tensors over Z.
Since the Hodge conjecture asserts that all Hodge cycles come from algebraic cycles, and an automorphism σ ∈ Aut(C/Q) maps algebraic cycles to algebraic cycles, the Hodge conjecture implies that all Hodge cycles are dR-absolute Hodge.
Conjecture 0.3 ( [Del82]). All Hodge cycles are dR-absolute Hodge, i.e. G Z = G AH Z . Suppose the family f : X → S is defined over a number field K. Then the filtered vector bundle with connection (V, ∇, F • ) is also defined over K. In this case, we call the absolute variation aQ-absolute variation. It is evident from the definition that the de Rham comparison induces a canonical isomorphism
G AH Z ⊗ Q C ∼ = G AH Z σ ⊗ Q C
of group schemes over Q. Therefore, assuming Conjecture 0.3, if Z ⊂ S is special then Z σ ⊂ S σ is also a special subvariety for every σ. Hence Deligne's conjecture 0.3 together with the fact that there are only countably many special subvarieties implies that special subvarieties are defined overQ and all of their finitely many Gal(Q/K)conjugates are again special. Still, Deligne's conjecture is much stronger: as explained in ( [Voi06], Lemma 1.4), it is equivalent to the statement that the locus of Hodge classes in the algebraic vector bundle V is defined overQ and its Galois conjugates are also contained in the locus of Hodge classes. We will now formulate a much weaker conjecture which still implies the definability of special subvarieties overQ.
The group-theoretic nature of Definition 0.1 invites us to make a variant of it, replacing the generic Mumford-Tate group by the generic dR-absolute Mumford-Tate group.
Definition 0.4 (see Definition 3.1). A closed irreducible algebraic subvariety Z is called dR-absolutely special if it is maximal among the closed irreducible algebraic subvarieties of S with generic dR-absolute Mumford-Tate group G AH Z . In particular, we can think of dR-absolutely special subvarieties as subvarieties cut out by dR-absolute Hodge cycles (this is literally true, see the interpretation in terms of period maps in Proposition 3.6).
Note that dR-absolutely special subvarieties satisfy all the good arithmetic properties that are conjectured to hold for all special subvarieties: they are defined over number fields, and their Galois translates are again special subvarieties.
Deligne's conjecture 0.3 immediately implies Conjecture 0.5. If Z ⊂ S is a special subvariety, then Z is dR-absolutely special.
Except for Deligne's proof in the case of (families of) abelian varieties ([Del82], Theorem 2.11), Conjecture 0.3 seems completely out of reach, and we have very little knowledge about the existence of dR-absolute Hodge cycles in general variations. Considering that, it is perhaps surprising that we can prove Conjecture 0.5 in a number of cases. We do so for subvarieties which satisfy a simple monodromy condition introduced in [KOU20]. For a closed irreducible subvariety Z ⊂ S we define the algebraic monodromy group H Z as the connected component of the identity of the Zariski closure of the image of the monodromy representation ρ Z : π 1 (Z an , s) → GL(V s ) corresponding to the restriction of the local system V to Z an .
Definition 0.6 ([KOU20], Definition 1.10). A closed irreducible subvariety Z ⊂ S is called weakly non-factor if it is not contained in a closed irreducible Y ⊂ S such that H Z is a strict normal subgroup of H Y .
Theorem 0.7 (see Corollary 3.10). Assume Z ⊂ S is a special subvariety which is weakly non-factor. Then Z is dR-absolutely special.
This justifies the result ([KOU20], Theorem 1.12) that weakly non-factor special subvarieties are defined overQ and their Galois conjugates are special. Examples of weakly non-factor special subvarieties were given in ([KOU20], Corollary 1.13). An irreducible subvariety Z is called of positive period dimension if H Z = 1, or equivalently, the image of Z under the period map is not a point.
Corollary 0.8 (see Corollary 3.12). Suppose that the adjoint group G ad S is simple. Let Z ⊂ S be a strict special subvariety which is of positive period dimension and maximal for these properties. Then Z is dR-absolutely special.
One might wonder about the difference between Conjecture 0.5 and Deligne's conjecture that Hodge classes are dR-absolute Hodge. We want to emphasize the idea that Conjecture 0.5 is a more geometric statement than Conjecture 0.3. While Deligne's conjecture predicts that there is no group G with G Z G ⊂ G AH Z , Conjecture 0.5 only implies a corresponding geometric statement, namely that no such group G can arise as the generic Mumford-Tate group of a closed irreducible algebraic subvariety Y containing Z. Proposition 0.9 (see Proposition 3.3). Let Z be dR-absolutely special. Then there does not exist a closed irreducible algebraic subvariety Y ⊃ Z with the property
G Z G Y ⊂ G AH Z .
The above philosophy suggests that Conjecture 0.5 is particularly strong in situations where the geometry of Z is very closely related to the group-theoretic properties of the period domain. These cases are known in Hodge theory as typical intersections (cf. [BKU21], Definition 1.4). In Definition 4.4 we introduce an integer δ AH (Z), called the absolute Hodge defect of Z, which measures the difference of the dimensions of the Mumford-Tate domains attached to G Z and G AH Z . Conjecture 0.3 predicts that δ AH (Z) = 0 for all subvarieties Z. We show in Theorem 4.5 that the absolute Hodge defect can be bounded in terms of the Hodge codimension of S and Z as introduced in ([BKU21], Definition 4.1). This shows that the closer Z is to being a typical intersection, the stronger Conjecture 0.5 becomes in relation to Conjecture 0.3. Theorem 0.10 (see Theorem 4.5). Assume that G S = G AH S , i.e. all generic Hodge cycles on S are dR-absolute Hodge. Let Z be a dR-absolutely special subvariety. Then
δ AH (Z) ≤ Hcd(S) − Hcd(Z).
In particular, if Z is a typical special subvariety then δ AH (Z) = 0.
Let us give a few applications of dR-absolutely special subvarieties. First we use a recent result of Baldi-Klingler-Ullmo (cf. [BKU21], Proposition 2.4) to give new examples of dR-absolutely special subvarieties. The level of a variation of Hodge structure V as defined in ( [BKU21], Definition 3.13) is a measure for the length of the Hodge filtration on the Lie algebra of the adjoint generic Mumford-Tate group.
Theorem 0.11 (see Theorem 4.9). Suppose G ad S is simple and the level of V is greater or equal to two. Then any maximal atypical special subvariety Z ⊂ S of positive period dimension is dR-absolutely special.
As a generalization of ([UY11], Theorem 1.4) one conjectures (cf. [Kli21], 4.1) that Q-bialgebraic subvarieties in aQ-absolute variation of Hodge structure of geometric origin are special (see Definition 4.15 for a definition ofQ-bialgebraic subvarieties). We prove that this is indeed the case for maximalQ-bialgebraic subvarieties of positive period dimension in case the adjoint group of G S is simple.
Theorem 0.12 (see Theorem 4.18). Suppose G ad S is simple. Let Z ⊂ S be a strict max-imalQ-bialgebraic subvariety of positive period dimension. Then Z is a dR-absolutely special subvariety (and in particular a special subvariety).
We define a class of motivic variations of Hodge structure as those absolute variations which look as if they come from geometry.
Definition 0.13. An absolute variation of Hodge structure (V σ ) σ on S is called a motivic variation of Hodge structure if there exists a dense open subvariety U ⊂ S such that for all points x ∈ U (C), the collection of Hodge structures (V σ x σ ) σ together with their comparison isomorphisms is the realization of a motive for dR-absolute Hodge cycles. AQ-absolute variation of Hodge structure with this property is calledQ-motivic variation of Hodge structure.
Remark 0.14. We refer to ( [DM82], §6) for the construction of the category of motives for absolute Hodge cycles. Again, we consider a variant of it, using only dR-absolute Hodge cycles. Every absolute variation of geometric origin is an example of such a motivic variation.
It is reasonable to expect that Deligne's Conjecture 0.3, and therefore also Conjecture 0.5 should hold for subvarieties inQ-motivic variations. As a final application, we use monodromy arguments to show that these conjectures forQ-absolute variations of geometric origin can be reduced to the case of special points inQ-motivic variations. Then it holds for all subvarieties inQ-absolute variations of geometric origin.
(ii) Suppose that Conjecture 0.5 holds for special points inQ-motivic variations. Then it holds for all special subvarieties inQ-absolute variations of geometric origin.
(iii) Suppose that special points inQ-motivic variations are defined overQ and all their Galois conjugates are special. Then the same holds for all special subvarieties in Q-absolute variations of geometric origin.
(iv) Suppose that allQ-bialgebraic points inQ-motivic variations are dR-absolutely special. Then allQ-bialgebraic subvarieties inQ-absolute variations of geometric origin are dR-absolutely special.
Remark 0.16. The part of Theorem 0.15(iii) concerning definability overQ was already proven in ([KOU20], Corollary 1.14), but the part on Galois conjugates was left open, cf. ( [KOU20], Remark 3.5).
Notations. All varieties in this paper are assumed to be reduced. Unless stated otherwise, by a subvariety Z of a complex algebraic variety S we mean a closed irreducible complex algebraic subvariety. If S is defined over a number field K ⊂ C and Z is defined over an extension L of K, we denote by Z L the associated L-variety. If a distinction is necessary, we will refer to Z considered as a complex variety by Z C = Z L ⊗ L C. Throughout this paper, by a Hodge cycle we mean a (0, 0)-cycle.
Variations of Hodge structure and special subvarieties
In this section we recall fundamental facts from Hodge theory that will be used throughout the paper. Let S be a smooth irreducible quasi-projective algebraic variety over C, and V Z a (pure, polarizable) Z-variation of Hodge structure on S. Denote by V := V Z ⊗ Z Q the associated Q-variation of Hodge structure. (i) A priori, our definition of G Z depends on the choice of s ∈ Z(C). However, as we vary the point s, the associated groups form a local system of algebraic groups over Z, which shows that the group G Z is independent of the choice of s up to monodromy. In fact, for s ∈ Z(C) away from a countable union of closed algebraic subvarieties we can identify G Z with the Mumford-Tate group of the Hodge structure V s . From this we see that G Z is a connected reductive group over Q.
Hodge varieties and special subvarieties
(ii) The generic Mumford-Tate group G Z is often defined by restricting to the smooth locus of Z. We show in Remark 1.5 that our definition gives the same group.
An important feature of variations of Hodge structure is the fact that they have a geometric interpretation in terms of period maps: to the variation of Hodge structure V one can naturally attach a holomorphic period map Φ : S an → Γ\D with target a so-called Hodge variety. We now recall the most important aspects of this notion. Let Z ⊂ S be a closed irreducible subvariety. The inclusion G Z ⊂ G S gives rise to a morphism of Hodge data (G Z , D Z ) → (G S , D), and the image X Z is a special subvariety of X. Then Φ(Z an ) ⊂ X Z and X Z is the smallest special subvariety of X containing Φ(Z an ).
Remark 1.5. We can now justify our definition of the generic Mumford-Tate group in the following way: Let G Z sm be the generic Mumford-Tate group in restriction to the smooth locus Z sm of Z and X Z sm the associated special subvariety of X. Then Φ(Z sm,an ) ⊂ X Z sm and since the smooth locus is dense in Z an (even for the analytic topology) and X Z sm is a closed analytic subspace of X, we get
Φ(Z an ) ⊂ X Z sm .
From this we see that every generic Hodge tensor on Z sm extends to Z, i.e. G Z = G Z sm .
The subvarieties of S which are maximal with a given generic Mumford-Tate group are of particular Hodge-theoretic importance.
Definition 1.6 ([KO21], Definition 1.2). A closed irreducible algebraic subvariety Z ⊂ S is called special for V if it is maximal among the closed irreducible algebraic subvarieties of S having the same generic Mumford-Tate group as Z.
A famous result of Cattani-Deligne-Kaplan (and recently reproved using o-minimal methods by Bakker-Klingler-Tsimerman in [BKT20]) describes the special subvarieties of S using the period map.
Monodromy
When we forget about the Hodge filtration and just consider the underlying local system of the variation, we enter the world of monodromy.
Definition 1.8. The algebraic monodromy group H Z of Z is defined to be the identity component of the Zariski closure of the image of the monodromy representation ρ Z : π 1 (Z an , s) → GL(V s ).
Note that our subvariety Z ⊂ S will often be singular, and that usually the fundamental group of a singular analytic space is not particularly well-behaved. In our case however, we will use that Z is a closed subvariety of the smooth variety S and the local system V moreover supports a variation of Hodge structure. In this situation we see that in the definition of the algebraic monodromy group H Z we may always replace Z by its smooth locus. Lemma 1.9. Let Z ⊂ S be a closed irreducible subvariety and let Z sm be the smooth locus of Z. Then H Z = H Z sm , i.e. every global section of some V ⊗(m,n) over a finité etale cover of Z sm extends uniquely to a global section over a finiteétale cover of Z.
Proof. Let ι : X G Z := Γ Z \D Z → X = Γ\D
denote the morphism of Hodge varieties induced from the morphism of Hodge data
(G Z , D Z ) → (G S , D). By ([And92], Theorem 1) the group H Z sm is a normal subgroup of G Z sm = G Z .
The projection to the quotient gives rise to a morphism of Hodge data
(G Z , D Z ) → (G Z /H Z sm , D Z ),
and we denote the associated morphism of Hodge varieties by
π : X G Z = Γ Z \D Z → X G Z /H Z sm := Γ Z \D Z .
After possibly replacing Z sm by a finiteétale cover, we get a period map Φ ′ : Z sm,an → X G Z . By ( [KO21], Lemma 4.12) the projection of Φ ′ (Z sm,an ) to the quotient Hodge variety
X G Z /H Z sm = Γ Z \D Z is a single point y. Denoting by X H Z sm ,y = π −1 ({y})
the preimage of y in X G Z , the smooth locus of Z is thus contained in the preim-
age of ι(X H Z sm ,y ) under Φ. Now since Z sm is dense in Z, we deduce that also Z ⊂ Φ −1 (ι(X H Z sm ,y )). Since Z is irreducible, under the lifted period map Φ :S → D (1)
for the universal coverS of S, the universal coverZ is mapped to D Z and its projection to D Z is a single pointỹ ∈ D Z . The lifted period map (1) is π 1 (S an , s)-equivariant. This shows that up to a finite subgroup, the image of the monodromy representation
ρ Z : π 1 (Z an , s) → GL(V s ) is contained in G Z (Q)
, and the projection
π 1 (Z an , s) → G Z (Q) ։ G Z (Q)/H Z sm (Q) (2)
is contained in the stabilizer of the pointỹ. Since this stabilizer is a compact real Lie group, we conclude that the image of (2) is finite. Therefore a finite index subgroup of
ρ Z (π 1 (Z an , s)) is contained in H Z sm , which shows that H Z = H Z sm .
The following Theorem is a well-known result due to André (in the smooth case, but we can easily reduce to that case using Lemma 1.9 and Remark 1.5):
Proposition 1.10 ([And92], Theorem 1). The algebraic monodromy group H Z is a normal subgroup of the derived group of the generic Mumford-Tate group G Z .
Just like the generic Mumford-Tate group, the algebraic monodromy group of a variation of Hodge structure has an interpretation in terms of period maps. The description in the proof of Lemma 1.9 motivates the following definition.
Definition 1.11. Let X ι ← X 1 π → X 2 be a diagram of morphisms of Hodge varieties and x 2 ∈ X 2 . An irreducible component of a variety of the form ι(π −1 ({x 2 })) is called a weakly special subvariety of X.
Definition 1.12. A closed irreducible algebraic subvariety Z ⊂ S is called weakly special for V if it is maximal among the closed irreducible algebraic subvarieties of S having the same algebraic monodromy group as Z.
Weakly special subvarieties admit an interpretation via the period map.
Theorem 1.13 ([KO21], Corollary 4.14). The weakly special subvarieties of S are precisely the irreducible components of the preimages of weakly special subvarieties in X under the period map.
We will frequently use the Theorem of the fixed part. The following is a slight generalization of ([Sch73], Corollary 7.23) to the non-smooth case.
Theorem 1.14 ([Sch73], Corollary 7.23). Let ξ be a global section of V ⊗(m,n) over a finiteétale cover Z ′ of a closed irreducible subvariety Z ⊂ S such that ξ s is a Hodge class for some point s ∈ Z ′ (C). Then ξ t is Hodge for every point t ∈ Z ′ (C).
Proof. By the Theorem of the fixed part applied to a suitable finiteétale cover of the smooth locus of Z, we see as in ( [KO21], Lemma 4.12) that the period map restricted to Z sm factors as Φ :
Z sm,an → ι(X H Z ,y ) ⊂ ι(X G Z )
where X H Z ,y := π −1 ({y}) and ι : X G Z → X and π : X G Z → X G Z /H Z are morphisms of Hodge varieties. As above, since Z sm is dense in Z, the image of Z under the period map is also contained in ι(X H Z ,y ). In particular, up to replacing Z by some finiteétale cover we can form the period map
Φ ′ : Z an → X G Z → X G Z /H Z(3)
and the above inclusion shows that it is constant. Consider the natural representation
of G Z /H Z on (V ⊗(m,n) s ) H Z .
Under this representation, the period map (3) corresponds to the subvariation of Hodge structure (V ⊗(m,n) Z ) H Z over Z, which is therefore constant. Hence every global section that is a Hodge class at one point is a Hodge class everywhere.
The following Lemma is a geometric application of the Theorem of the fixed part.
Lemma 1.15. If Z ⊂ Y are two closed irreducible subvarieties satisfying H Y ⊂ G Z , then G Z = G Y . Proof. Let s ∈ Z(C). Since G Z is reductive, it is enough to show that every fixed tensor v ∈ V ⊗ s of G Z is also fixed by G Y .
The condition H Y ⊂ G Z shows that after possibly replacing Y by a finiteétale cover, there is a global section on Y extending v. Since v is Hodge at the point s ∈ Z(C), the Theorem of the fixed part (Theorem 1.14) allows us to conclude that this global section is a generic Hodge tensor over Y . So G Y fixes v, as desired.
As can be easily seen from the description using the period map, every special subvariety is weakly special. We show how Lemma 1.15 can be applied to give a more direct argument.
Lemma 1.16. Any special subvariety is weakly special.
Proof. Suppose Z is special and Y ⊃ Z is a closed irreducible subvariety of S which satisfies H Y = H Z ⊂ G Z . By Lemma 1.15 we get G Z = G Y which implies Z = Y since Z is special. Hence Z is weakly special.
Fields of definition of weakly special subvarieties
Suppose S is defined overQ. In this section, we prove that weakly special subvarieties of S are defined overQ once they contain aQ-point. This recovers and generalizes a result of Saito-Schnell ( [SS16]) for special subvarieties. Surprisingly, as in their paper, we merely require the base S to be defined overQ, there is no condition on the variation V.
Choose aQ-point s ∈ S(Q) and a prime number ℓ. The monodromy representation π 1 (S an C , s) → GL(V Z,s ⊗ Z ℓ ) corresponding to the (analytic) Z ℓ -local system V Z ⊗ Z ℓ factors through theétale fundamental group, giving rise to a continuous representation
ρ ℓ : πé t 1 (SQ, s) ∼ = πé t 1 (S C , s) → GL(V Z,s ⊗ Z ℓ ), hence to anétale Q ℓ -local system V ℓ on SQ.
Here the isomorphism πé t 1 (SQ, s) ∼ = πé t 1 (S C , s) comes from the fact that theétale fundamental group of a smooth (quasiprojective) variety is invariant under base change of algebraically closed fields ( [GR03] Exposé VIII, Proposition 4.6).
For a closed irreducible complex subvariety Z ⊂ S, choose a point s ∈ Z(C) and define H ℓ,Z to be the identity component of the Zariski closure of the image of πé t 1 (Z, s) in GL(V ℓ,s ).
Proposition 1.17. The comparison isomorphism V ℓ,s ∼ = V s ⊗ Q ℓ induces an isomor- phism H ℓ,Z ∼ = H Z ⊗ Q ℓ . Proof. We recall the argument from ([Moo17], Lemma 4.3.4). Theétale fundamental group πé t 1 (Z, s) is the profinite completion of π 1 (Z an , s) (cf. [GR03] Exposé V, Corollary 5.2). It is enough to show that ρ ℓ (πé t 1 (Z, s)) ⊂ H Z ⊗ Q ℓ .
This follows from the fact that the image of π 1 (Z an , s) is dense in ρ ℓ (πé t 1 (Z, s)) for the ℓ-adic topology, so a fortiori for the Zariski topology.
Lemma 1.18. We have a natural isomorphism H Z σ ⊗ Q ℓ ∼ = H Z ⊗ Q ℓ for any σ ∈ Aut(C/Q).
Proof. Following the above Proposition 1.17, it is enough to prove H ℓ,Z ∼ = H ℓ,Z σ . The projection σ −1 : Z σ = Z ⊗ C,σ C ∼ = Z is an isomorphism of abstract schemes (but not of varieties over C) and induces the following diagram
πé t 1 (Z, s) / / πé t 1 (S C , s) / / GL(V ℓ,s ) πé t 1 (Z σ , s σ ) / / ∼ = O O πé t 1 (S C , s σ ) / / ∼ = O O GL(V ℓ,s σ ) . ∼ = O O
Now the fact that the local system V ℓ is defined overQ translates into the commutativity of the right hand square of the diagram. It follows easily that the entire diagram commutes. As a consequence, there is a natural isomorphism between the Zariski closure of the image of πé t 1 (Z, s) in GL(V ℓ,s ) and the Zariski closure of the image of πé t 1 (Z σ , s σ ) in GL(V ℓ,s σ ).
Lemma 1.19. If Z is weakly special for V, then Z σ is weakly special for V for any σ ∈ Aut(C/Q).
Proof. Let Y ⊃ Z σ be closed irreducible such that H Y = H Z σ . From Lemma 1.18 we see that the subvariety Y σ −1 containing Z satisfies H Y σ −1 ⊗ Q ℓ = H Z ⊗ Q ℓ . This forces H Y σ −1 = H Z . Now the fact that Z is weakly special implies that Z = Y σ −1 , hence Z σ = Y .
We conclude that Z σ is weakly special.
In contrast to the case of special subvarieties, we cannot expect weakly special subvarieties to be defined overQ in general. For example, if the period map does not contract positive dimensional subvarieties to a point, every point s ∈ S(C) is weakly special. However, we now show that they are defined overQ once they contain a singlē Q-point: Theorem 1.20. Let Z be a weakly special subvariety of S containing aQ-point. Then Z is defined overQ.
Proof. We need to show that the set {Z σ } σ∈Aut(C/Q) of conjugates of Z is countable. For z aQ-point contained in Z, we have z ∈ Z σ (C) for all σ. Note that all Z σ are again weakly special by Lemma 1.19. By the characterization of weakly special subvarieties in Proposition 1.13, for every σ there is a diagram of morphisms of Hodge varieties X ισ ← X σ 1 πσ → X σ 2 and a point x σ ∈ X σ 2 such that Z σ is one of the finitely many irreducible components of the preimage of ι σ (π −1 σ ({x σ })) under the period map. In particular, Z σ is determined (up to choosing one of finitely many irreducible components) by the diagram of Hodge morphisms and the point x σ . There are only countably many choices for the diagram of Hodge morphisms, and we need to show that for a given diagram (ι, π) there are only countably many choices for ι(π −1 ({x σ })). Since z ∈ Z σ (C) for all σ, the ι(π −1 ({x σ })) all contain a common point. By noting that the π −1 ({x σ }) are pairwise disjoint as x σ varies, and the group Γ ⊂ G S (Q) is countable, one sees that this leaves only countably many choices for ι(π −1 ({x σ })).
As a corollary we obtain a new proof of the following result, proven in ([SS16], Theorem 1):
Corollary 1.21. Let Z be a special subvariety of S containing aQ-point. Then Z is defined overQ.
Absolute variations of Hodge structure
Let S be a smooth irreducible quasi-projective complex algebraic subvariety. For every σ ∈ Aut(C/Q) we can form the conjugate S σ = S ⊗ C,σ C. We use the notation σ −1 for the projection σ −1 : S σ = S ⊗ C,σ C → S. In this section we define an absolute variation of Hodge structure on S as a collection of variations of Hodge structure V σ on each Aut(C/Q)-conjugate of S with certain compatibilities.
Definition 2.1. An absolute variation of Hodge structure is the datum of a Z-variation of Hodge structure (V σ Z , V σ , ∇ σ , F •,σ ) on S σ for each σ ∈ Aut(C/Q) together with an isomorphism
ι σ : (V σ , ∇ σ , F •,σ ) ∼ = σ −1 * (V id , ∇ id , F •,id )(4)
of the associated filtered algebraic vector bundles with connection. A polarization of the absolute variation of Hodge structure is a polarization α σ on V σ Z for each σ ∈ Aut(C/Q) such that the polarizations correspond to each other under the isomorphism ι σ .
From now on, all absolute variations are assumed to be polarizable. We will often forget the Z-structure and write (V σ ) σ for the associated collection of Q-variations of Hodge structure, which we also call an absolute variation of Hodge structure. Note however that we require that all our Q-variations admit a Z-structure. We use the notation (V, V, ∇,
F • ) := (V id , V id , ∇ id , F •,id ).
Remark 2.2. Given a Z-variation of Hodge structure V Z , using Deligne's canonical extension (cf. [Del70]) one can show that the holomorphic vector bundle (V an = V Z ⊗ Z O S an , ∇ an ) together with the holomorphic connection defined by V arises as the analytification of an algebraic vector bundle with regular algebraic connection (V, ∇). Moreover, making use of Schmidt's nilpotent orbit theorem, one can see that the Hodge filtration arises from an algebraic filtration on V ([Sch73], 4.13). Therefore the comparison isomorphism (4) makes sense for general variations of Hodge structure.
Example 2.3. Let f : X → S be a smooth projective morphism of irreducible smooth quasi-projective algebraic varieties over C. For any automorphism σ ∈ Aut(C/Q) we consider the base change f σ : X σ → S σ . The collection of variations of Hodge structure V σ := R k prim f σ,an * Q on S σ is an absolute variation of Hodge structure on S.
Suppose that S is defined over a subfield K ⊂ C and V, ∇ and F • are all defined over K. In this case, an absolute variation of Hodge structure satisfying V σ = V for all σ ∈ Aut(C/K) is called a K-absolute variation. For example, if K is a number field, we only need to specify V σ for the finitely many σ ∈ Gal(K/Q). If the morphism f is defined over K, then Example 2.3 produces a K-absolute variation.
We call an absolute variation of Hodge structure on a point an absolute Hodge structure. It is a collection of Q-Hodge structures V σ together with isomorphisms ι σ : V σ ⊗ C ∼ = V ⊗ C respecting the filtration. Absolute Hodge structures naturally form a Tannakian category where the morphisms are morphisms of Hodge structures which are dR-absolute Hodge in the sense of the definition in the next section.
dR-absolute Hodge cycles
In this section we recall a slightly weaker version considered in [Voi06] Theorem 2.5. Let ξ be a global section of V ⊗(m,n) over a finiteétale cover Z ′ of a closed irreducible subvariety Z ⊂ S such that ξ s is dR-absolute Hodge for some point s ∈ Z ′ (C). Then ξ t is dR-absolute Hodge for every point t ∈ Z ′ (C).
Proof. To the global section ξ over Z ′ we can attach a flat section ξ dR of V ⊗(m,n) Z ′ . Conjugation by σ ∈ Aut(C/Q) produces a global section ξ σ dR of the vector bundle (V σ ) ⊗(m,n) Z ′σ . Since the connection is also algebraic, ∇ σ ξ σ dR = 0. Hence ξ σ dR is a flat section and applying the Riemann-Hilbert correspondence to the restriction ξ σ dR | Z ′σ,sm to Z ′ σ,sm gives a global section of the C-local system (V σ C ) ⊗(m,n) Z ′σ,sm over the smooth locus of Z ′ σ . By Lemma 1.9, this global section uniquely extends to a global section ξ σ on Z ′ σ . The associated flat section of (V σ ) ⊗(m,n) Z ′σ is ξ σ dR because this is true over the dense open subset Z ′ σ,sm . By assumption, ξ σ s σ is a Hodge cycle, so by Theorem 1.14 the fiber ξ σ t σ is a Hodge cycle for every t ∈ Z ′ (C). We conclude that ξ t is dR-absolute Hodge for every t ∈ Z ′ (C).
Let AH Z ⊂ V ⊗ s denote the subset of all tensors v ∈ V ⊗ s such that the translates of v by parallel transport are dR-absolute Hodge at every point t ∈ Z(C).
Definition 2.6. We define the generic dR-absolute Mumford-Tate group G AH Z to be the subgroup of GL(V s ) fixing all tensors in AH Z .
Remark 2.7. If Z = {s} is a point, then we may define G AH Z as the Tannakian group of the subcategory generated by the absolute Hodge structure (V σ s σ ) σ and its dual inside the category of absolute Hodge structures. To see this, note that since the absolute Hodge structure is polarized the Tannakian group is a (possibly non-connected) reductive group and therefore characterized by the tensors it fixes. Since these are exactly the dR-absolute Hodge tensors, we recover the above definition. In the geometric situation of Example 2.3, the group G AH s is then the motivic Galois group of H k prim (X s ) in terms of motives for dR-absolute Hodge cycles (cf. [DM82], §6 for the variant which uses the stronger notion of absolute Hodge cycles).
Since every dR-absolute Hodge cycle is in particular a Hodge cycle we see that we have the inclusion G Z ⊂ G AH Z . We make crucial use of the following Lemma, which is a geometric incarnation of Deligne's Principle B:
Lemma 2.8. Suppose Z ⊂ Y are closed irreducible subvarieties that satisfy H Y ⊂ G AH Z . Then G AH Z = G AH Y . Proof. We show that G AH Y ⊂ G AH Z , the other inclusion being clear. Let s ∈ Z(C). The condition H Y ⊂ G AH Z translates into AH Z ⊂ (V ⊗ s ) H Y .
Now every element of v ∈ AH Z is fixed by H Y and is dR-absolute Hodge at the point s ∈ Z(C). We apply Theorem 2.5 to show that v extends to a global section over a finiteétale cover Y ′ of Y which is dR-absolute Hodge at every point of Y ′ , and so AH Z ⊂ AH Y .
Proposition 2.9. The group G AH Z is a (possibly non-connected) reductive group. The algebraic monodromy group H Z is a normal subgroup of G AH Z .
Proof. Again, the argument in the proof of Theorem 2.5 shows that G AH Z does not change when we replace Z by its smooth locus and we may therefore assume that Z is smooth. We claim that there exists a point s ∈ Z(C) with G AH forms a sub-absolute Hodge structure of (V σ s σ ) ⊗(m,n) σ and is therefore preserved by G AH s .
dR-absolutely special subvarieties
In this section we introduce the notion of dR-absolutely special subvariety, study its properties and prove that weakly non-factor special subvarieties are dR-absolutely special.
Definition and first properties
Let (V σ ) σ be an absolute variation of Hodge structure on a smooth irreducible quasiprojective complex algebraic variety S and Z ⊂ S a closed irreducible subvariety. Since we do not know whether G Z = G AH Z , it is natural to define a notion of dR-absolutely special subvariety simply by replacing G Z in Definition 1.6 by G AH Z .
Definition 3.1. A closed irreducible algebraic subvariety Z ⊂ S is called dR-absolutely special if it is maximal among the closed irreducible algebraic subvarieties of S with generic dR-absolute Mumford-Tate group G AH Z .
We mention a few properties of dR-absolutely special subvarieties:
Proposition 3.2. Let Z be a dR-absolutely special subvariety.
(i) Z is special.
(ii) Z σ ⊂ S σ is dR-absolutely special for all σ ∈ Aut(C/Q).
(iii) If S is defined over a finite extension K of Q and the variation is K-absolute, then Z is defined overQ and all its Gal(Q/K)-conjugates are special.
Proof. Suppose that Y ⊃ Z is closed irreducible such that G Y = G Z . It follows that H Y ⊂ G Z ⊂ G AH Z ,
and therefore G AH Y = G AH Z by Lemma 2.8. We conclude that Z = Y because Z is dR-absolutely special. This shows that Z is special. Assertion (ii) follows from the fact that we have a natural isomorphism G AH Z ⊗ C ∼ = G AH Z σ ⊗ C. By the fact that there are only countably many special subvarieties of S for V, this implies (iii).
Note that dR-absolutely special subvarieties have the good arithmetic properties that conjecturally hold for special subvarieties.
Deligne's conjecture that Hodge classes are dR-absolute Hodge states that we have an equality of groups G Z = G AH Z . For dR-absolutely special subvarieties we can at least show that there does not exist any closed irreducible subvariety Y ⊃ Z whose generic Mumford-Tate group contradicts this equality.
G Z G Y ⊂ G AH Z .
Proof. Let Y ⊃ Z be a closed irreducible subvariety with G Y ⊂ G AH Z . It follows that H Y ⊂ G AH Z , and therefore Lemma 2.8 implies G AH Y = G AH Z . We conclude that Z = Y because Z is dR-absolutely special.
For the next Proposition, suppose that S is defined overQ and the variation is ā Q-absolute variation. Proof. Let W ⊃ Z be a maximal subvariety with G AH W = G AH Z . Then W is dRabsolutely special and therefore defined overQ by Proposition 3.2. In particular, Y ⊂ W and thus G AH
Y = G AH Z . It follows from Proposition 2.9 that H Z is a normal subgroup of G AH Y . Now the inclusions H Z ⊂ H Y ⊂ G Y ⊂ G AH Y show that H Z is also normal in H Y and G Y .
Remark 3.5. The normality of H Z in H Y was observed in the course of the proof of ([KOU20], Proposition 3.2).
dR-absolutely special subvarieties and period maps
Let Z ⊂ S be a closed irreducible subvariety. Denote by (G AH Z , D AH Z ) and (G AH S , D AH S ) the Hodge data defined by G AH Z and G AH S . We want to describe the dR-absolutely special subvarieties of S in terms of the period map Φ : S an → Γ\D AH S .
The inclusion G AH Z ⊂ G AH S induces a morphism of Hodge varieties
ι : Γ AH Z \D AH Z → Γ\D AH S .
Here Γ AH Z := Γ ∩ G AH Z (Q).
Proposition 3.6. The subvariety Z is dR-absolutely special if and only if Z is a (complex analytic) irreducible component of Φ −1 (ι(Γ AH Z \D AH Z )).
Proof. Clearly the restriction of the period map Φ to any subvariety
Y ⊃ Z with G Y ⊂ G AH Y = G AH Z factors through ι(Γ AH Z \D AH Z ). Conversely, let W be a complex analytic irreducible component of Φ −1 (ι(Γ AH Z \D AH Z )) containing Z.
It follows from ([BKT20], Theorem 1.6) that W is algebraic, and it satisfies G W ⊂ G AH Z . Applying Lemma 2.8, we see that G AH W = G AH Z . As Z is dR-absolutely special this gives Z = W .
Note that Φ −1 (ι(Γ AH Z \D AH Z )) may have several (but finitely many) irreducible components and therefore Z is not necessarily contained in a unique smallest dR-absolutely special subvariety. Still, Z is contained in a unique irreducible component of an intersection of irreducible components of Φ −1 (ι(Γ AH Z \D AH Z )). Components of this form will be called dR-absolutely special intersections.
Corollary 3.7. Every subvariety Z is contained in a unique smallest dR-absolutely special intersection, called the dR-absolutely special closure of Z.
Weakly non-factor subvarieties
Weakly non-factor subvarieties are a class of subvarieties that satisfy a certain monodromy condition introduced in ([KOU20], Definition 1.10). Their significance for the arithmetic properties of special subvarieties was already established in [KOU20]. In this section, we prove a slight strengthening of their result: any special subvariety which is weakly non-factor is in fact dR-absolutely special. Theorem 3.9. Assume Z ⊂ S is weakly special and weakly non-factor. Then Z is dR-absolutely special.
Proof. Assume Y ⊃ Z is a closed irreducible subvariety such that G AH Y = G AH Z . By Proposition 2.9, the groups H Z and H Y are both normal in G AH Y = G AH Z , and therefore H Z is a normal subgroup of H Y . The fact that Z is weakly non-factor now implies H Z = H Y . We conclude that Z = Y since Z is weakly special. Thus Z is dR-absolutely special.
As a consequence we see that Conjecture 0.5 holds true for weakly non-factor subvarieties.
Corollary 3.10. Assume Z ⊂ S is a special subvariety which is weakly non-factor. Then Z is dR-absolutely special.
Suppose S is defined overQ and the variation (V σ ) σ isQ-absolute. In this case, Corollary 3.10 and Proposition 3.2 imply that a special, weakly non-factor subvariety is defined overQ and all its Galois conjugates are special. This was already proven in [KOU20].
Write G ad S = G 1 × G 2 × ... × G n as a product of simple factors. This gives rise to a product decomposition of the Hodge variety
Γ\D = Γ 1 \D 1 × Γ 2 \D 2 × ... × Γ n \D n .(5)
Corollary 3.11. Let Z ⊂ S be a maximal strict special subvariety with the property that the projection of Φ(Z an ) to each simple factor Γ i \D i is positive dimensional. Then Z is dR-absolutely special.
Proof. We prove that Z is weakly non-factor to apply Theorem 3.9. Let Y ⊃ Z be such that H Z is a strict normal subgroup of H Y . Then as Z is a maximal strict special subvariety we have the equality G Y = G S . It follows that H Y H S and the fact that the projection is positive dimensional on each simple factor implies that H Y = H S . Thus H Z is a proper normal subgroup of H S . This contradicts the fact that the projection of Z to each simple factor is positive dimensional.
If G ad S is simple, this reduces to the following example taken from ([KOU20], Corollary 1.13). An irreducible subvariety Z ⊂ S is called of positive period dimension if Φ(Z an ) is not a point, or equivalently, if the algebraic monodromy group H Z is nontrivial.
Corollary 3.12. Suppose that G ad S is simple. Let Z ⊂ S be a strict special subvariety which is of positive period dimension, and maximal for these properties. Then Z is dR-absolutely special.
Remark 3.13. Instead of the adjoint group G ad S one can also assume that the derived group G der S is simple. Since G der S is an extension of G ad S by a finite group, the simpleness of G der S is equivalent to that of G ad S . As in [KOU20], we use the adjoint group because it appears naturally in the decomposition (5).
Denote by HL pos the Hodge locus of positive period dimension, which is defined as the union of all strict special subvarieties of S which are of positive period dimension ([KO21], Definition 1.4). Then the Corollary shows that if G ad S is simple, then HL pos is a countable union of dR-absolutely special subvarieties. In particular, HL pos is cut out by dR-absolute Hodge cycles.
Applications
We give applications to several arithmetic questions in Hodge theory. As before, we let (V σ ) σ be an absolute variation of Hodge structure on a smooth irreducible quasiprojective complex algebraic variety S.
dR-absolutely special subvarieties and typical intersections
The property that all generic Hodge cycles on a special subvariety Z are dR-absolute Hodge (i.e. G Z = G AH Z ) implies that Z is dR-absolutely special, but the converse does not hold. For instance, note that for the examples of dR-absolutely special subvarieties given by Corollary 3.12, we cannot determine the group G AH Z . In this section we will see that for a dR-absolutely special subvariety we can give an upper bound for the "difference" between G Z and G AH Z in terms of the Hodge codimension. The strength of this bound depends on how close Z is to being a typical intersection.
Hcd(Z) := dim Γ Z \D Z − dim Φ(Z an ),
where Γ Z \D Z is the Hodge variety for the generic Mumford-Tate group G Z of Z. Otherwise, Z is called typical.
Note that if Z is special, the non-strict inequality Hcd(S) ≥ Hcd(Z) always holds. Proof. It follows the description of dR-absolutely special subvarieties in Proposition 3.6 that dim
Γ\D S − dim Φ(S an ) ≥ dim Γ AH Z \D AH Z − dim Φ(Z an ). We get δ AH (Z) = dim Γ AH Z \D AH Z − dim Γ Z \D Z ≤ (dim Γ\D S − dim Φ(S an )) − (dim Γ Z \D Z − dim Φ(Z an )) = Hcd(S) − Hcd(Z).
Remark 4.6. In general, without assuming that all Hodge cycles on S are dR-absolute Hodge, we get the inequality δ AH (Z) − δ AH (S) ≤ Hcd(S) − Hcd(Z).
Corollary 4.7. Assume that G S = G AH S . Let Z be a dR-absolutely special subvariety which is a typical intersection. Then δ AH (Z) = 0.
If S is a Shimura variety, then every special subvariety is a typical intersection. In general, the question whether special subvarieties are typical or atypical is closely related to the level of the variation of Hodge structure as defined in [BKU21]. For every Hodge generic point x ∈ S(C), the representation S → G S,R of the Deligne torus defining the Hodge structure at the point x induces a Q-Hodge structure of weight zero on the Lie algebra g S , and the adjoint Lie algebra g ad S can be viewed as a sub-Hodge structure by identifying it with the derived Lie algebra g der S := [g S , g S ]. There is a compatibility between the Hodge structure and the Lie algebra structure, and g ad S is called a Q-Hodge Lie algebra in ( [BKU21], Definition 3.8).
Maximal atypical special subvarieties
Under suitable assumptions, maximal atypical special subvarieties of positive period dimension are dR-absolutely special.
Theorem 4.9. Suppose G ad S is simple and the level of V is greater or equal to two. Then any maximal atypical special subvariety of positive period dimension is dR-absolutely special.
Proof. If the level of V is greater or equal to three, it follows from ([BKU21], Theorem 2.3) that all strict special subvarieties are atypical. Hence in this case the statement follows from Corollary 3.12. It remains to handle the case of level two. Suppose Z ⊂ S is a maximal atypical special subvariety of positive period dimension. If Y Z is a dRabsolutely special subvariety with G AH Y = G AH Z , then Y is a typical special subvariety. By ( [BKU21], Proposition 2.4), the adjoint generic Mumford-Tate group G ad Y is simple, and thus also the normal subgroup H Y . Since H Z G AH Z = G AH Y , the group H Z is a normal subgroup of H Y . Now the fact that Z is of positive period dimension implies H Z = H Y , which is a contradiction because Z is weakly special.
Remark 4.10. In the case that V is of level one, the target of the period map is a Shimura variety, so using Deligne's result ( [Del82], Theorem 2.11) that Hodge classes on abelian varieties are absolute Hodge one can prove in many cases that special subvarieties are dR-absolutely special.
4.3Q-bialgebraic subvarieties
Let S be an irreducible smooth quasi-projective complex algebraic variety. •X is a complex algebraic variety,
• h : π 1 (S an ) → Aut(X) is a group homomorphism to the algebraic automorphisms ofX,
• D is a holomorphic map which is h-equivariant.
For bialgebraic structures we define a notion of bialgebraic subvarieties. In order to detect special subvarieties by the bialgebraic formalism, we need a more refined definition of bialgebraicity that takes arithmetic properties into account. Let S be defined overQ and (V σ ) σ aQ-absolute variation on S. Then, as the compact dualĎ S = G S,C /P can be defined overQ, the period mapΦ :S →Ď S defines aQ-bialgebraic structure.
Proposition 4.16. Any dR-absolutely special subvariety isQ-bialgebraic.
Proof. If Z ⊂ S is dR-absolutely special, then Z is defined overQ. Moreover, since Z is special, π −1 (Z) will be a union of irreducible components of the preimage of the compact dualĎ Z = G Z,C /P Z ⊂Ď S of D Z . As this inclusion is induced by the inclusion G Z ⊂ G S of algebraic groups over Q and the cocharacter defining the parabolic subgroup P Z can be defined overQ (cf. [Mil05], Lemma 12.1), the subvarietyĎ Z is defined overQ.
As a generalization of ([UY11], Theorem 1.4) one conjectures the following:
Conjecture 4.17. TheQ-bialgebraic subvarieties of S are exactly the dR-absolutely special subvarieties.
We prove this conjecture for maximal subvarieties of positive period dimension when the adjoint group of G S is simple.
Theorem 4.18. Suppose G ad S is simple. Let Z ⊂ S be a strict maximalQ-bialgebraic subvariety of positive period dimension. Then Z is a dR-absolutely special subvariety (and in particular a special subvariety).
Proof. Let Y ⊃ Z be a dR-absolutely special subvariety such that G AH Y = G AH Z . By Proposition 4.16, the variety Y isQ-bialgebraic. By the maximality, either Y = Z or Y = S. We have to exclude the latter case. Suppose Y = S, then H Z G AH Z = G AH S . It follows that H Z H S . As H S is simple, this forces H Z = H S since Z is of positive period dimension. But Z is weakly special by Proposition 4.13, so this would mean that Z = S which is a contradiction to the fact that Z is a strict subvariety.
Reduction to the case of points
Corollary 3.11 suggests that in some sense special points are the hardest case for Conjectures 0.3 and 0.5. In this section we prove that both can in fact be reduced to the case of special points.
Let (V σ ) σ be aQ-absolute variation of Hodge structure on S of geometric origin. For a closed irreducible subvariety Z ⊂ S we denote by Y the union of all dR-absolutely special subvarieties of S containing Z with generic dR-absolute Mumford-Tate group G AH Z . These are only finitely many by Proposition 3.6. We claim that Z σ is special in
S σ for V σ if and only if Z σ is special in Y σ for V σ | Y σ . Indeed, if W ⊃ Z σ is such that G W = G Z σ ,
then Lemma 2.8 shows that G AH W = G AH Z σ and hence W is contained in Y σ . Replacing S by Y , we may therefore assume that G AH S = G AH Z . It follows that H Z is a normal subgroup of H S , G S and G AH S . As H Z is the stabilizer of finitely many tensors in V ⊗ s for some s ∈ Z(C), there exists a finite collection of integers (a i , b i ) i≤n such the variation of Hodge structure V ′ := i≤n (V ⊗(a i ,b i ) ) H Z has algebraic monodromy H S /H Z , generic Mumford-Tate group G S /H Z and generic dR-absolute Mumford-Tate group G AH S /H Z . Indeed, for this to be the case we need to ensure that G AH S /H Z acts faithfully on the fiber V ′ s , which is guaranteed if V ′ s contains the finitely many tensors defining H Z .
The collection (V ′ σ ) σ defined by
V ′ σ := i≤n (V σ ) ⊗(a i ,b i ) H Z σ
forms an absolute variation on S. After replacing S by a desingularizationS → S, we may assume that S is smooth. The period map attached to V ′ is
Φ ′ : S an → Γ ′ \D ′ ,
where Γ ′ \D ′ is a Hodge variety associated with the quotient G S /H Z . LetS be a smooth compactification of S overQ by a normal crossings divisor. Denoting byS ⊂S the subset where the local monodromy is finite, by ([CMSP17], Corollary 13.7.6) the period map Φ ′ extends to a proper mapΦ
′ :S an → Γ ′ \D ′ .
Hence again replacing S byS we may assume that the period map Φ ′ is proper. By [BBT19] there is a factorization Φ ′ = Ψ • f , where f : S → B is a proper surjective map of algebraic varieties with connected fibers and Ψ : B an → Γ ′ \D ′ is a quasi-finite period map.
Proposition 4.19. For every σ ∈ Aut(C/Q), the period map
Φ ′ σ : S σ,an → Γ ′ σ \D ′ σ
corresponding to the variation of Hodge structure V ′ σ also factors as
Φ ′ σ = Ψ σ • f σ ,
where f σ : S σ → B σ is the σ-conjugate of f : S → B and Ψ σ : B σ,an → Γ ′ σ \D ′ σ is a quasi-finite period map. In particular, there exists a variation of Hodge structure
V ′ σ B on B σ such that V ′ σ = f σ, * V ′ σ B .
Proof. The first map f : S → B in the Stein factorization of Φ ′ is characterized by the fact that the kernel S × B S is the connected component of the diagonal of the kernel the period map Φ ′ : S an → Γ ′ \D ′ . Thus it suffices to prove that for every σ, the kernel S σ × B σ S σ equals the connected component of the diagonal of the kernel of the period map Φ ′ σ : S σ,an → Γ ′ σ \D ′ σ . This follows from the fact that the Hodge cycle defining this kernel is dR-absolute Hodge (as it is the identity, and hence dR-absolute Hodge on the diagonal). Proof. We claim that V ′ σ B = f σ * V ′ σ , where V ′ σ B and V ′ σ are the associated algebraic vector bundles. Following the argument in ( [KOU20], Lemma 3.4), the projection formula gives
f σ * V ′ σ = f σ * (f σ, * V ′ σ B ⊗ O S σ O S σ ) = V ′ σ B ⊗ O B σ f σ * O S σ = V ′ σ B ,(6)
and similarly for the filtration and the connection. Here we use that by the Stein factorization f σ * O S σ = O B σ . Now the comparison isomorphisms for the absolute variation of Hodge structure (V ′ σ B ) σ follow from those for V ′ σ by applying f σ * . We prove that the absolute variation in question is aQ-motivic variation. By construction, there is a dense open subvariety U ⊂ S such that for each point x ∈ U (C), the collection (V σ x σ ) σ comes from a motive for dR-absolute Hodge cycles. Since f is proper, the image of S \ U under f is closed subvariety of B, and we claim that for b = f (x) ∈ B(C) chosen away from this image, the collection (V ′ σ B,b σ ) σ comes from a motive for dR-absolute Hodge cycles. As V ′ = f * V ′ B , we have
V ′ B,b = V ′ x = i≤n V ⊗(a i ,b i ) x H Z .
From the normality of H Z in G AH S we see that the variation of Hodge structure V ′ on S is the kernel of an idempotent operator on i≤n V ⊗(a i ,b i ) which is dR-absolute Hodge. In particular, if (V σ x σ ) σ comes from a motive for dR-absolute Hodge cycles, then (V ′ σ B,b σ ) σ = (V ′ σ x σ ) σ is the realization of a motive lying in the Tannakian subcategory generated by this motive. We claim that the filtered algebraic vector bundle with connection (V ′ , ∇ ′ , F ′ ,• ) on S is defined overQ. Indeed, the de Rham component of the idempotent dR-absolute Hodge operator on i≤n V ⊗(a i ,b i ) defining V ′ is defined overQ, and its kernel is V ′ . By the remark following ([BBT19], Theorem 1.1), the morphism f : S → B is defined overQ. Using (6) we conclude that (V ′ σ B ) σ is ā Q-motivic variation.
For every closed subvariety W of S, the image W ′ := f (W ) is a closed subvariety of B. Then W is a special subvariety of S for V ′ if and only if W ′ is a special subvariety of B. We denote by G ′ W the generic Mumford-Tate group of W with respect to the variation of Hodge structure V ′ and by G ′ AH W the generic dR-absolute Mumford-Tate group of W with respect to the absolute variation (V ′ σ ) σ .
Proposition 4.21. For every closed irreducible subvariety W ⊂ S, we have
G AH W ′ = G ′ AH W .
Proof. By definition, the variation of Hodge structure V ′ is constant on the fibers of f : S → B. Since these fibers are connected it follows that also the absolute variation of Hodge structure is constant, which proves the proposition. (ii) Suppose that Conjecture 0.5 holds for special points inQ-motivic variations. Then it holds for all special subvarieties inQ-absolute variations of geometric origin.
(iii) Suppose that special points inQ-motivic variations are defined overQ and all their Galois conjugates are special. Then the same holds for all special subvarieties in Q-absolute variations of geometric origin.
(iv) Suppose that allQ-bialgebraic points inQ-motivic variations are dR-absolutely special. Then allQ-bialgebraic subvarieties inQ-absolute variations of geometric origin are dR-absolutely special.
Proof. Let (V σ ) σ be aQ-variation on S of geometric origin. For a subvariety Z ⊂ S, we construct a morphism f : S → B overQ and aQ-motivic variation (V ′ σ B ) σ as described above. By construction, the image of Z under the period map Φ ′ is a point. As the map Ψ is quasi-finite, we see that x = f (Z) is a point of B. We show that the various conjectures for Z can be reduced to the ones for the point x ∈ B(C).
(i) Let Z ⊂ S be a closed irreducible subvariety. If Y is a special subvariety containing Z with G Y = G Z , then Lemma 2.8 shows that G AH Y = G AH Z . Hence we may assume that Z is special. By assumption, Deligne's conjecture holds for the special point x in theQ-motivic variation (V ′ σ B ) σ on B. Thus we have the equality G x = G AH
x . Proposition 4.21 now implies that G ′ Z = G ′ AH Z . It follows that
G Z = H Z · G ′ Z = H Z · G ′ AH Z = G AH Z .
(ii) Let Z ⊂ S be a special subvariety. We have to prove that if the special point x ∈ B(C) is dR-absolutely special for (V ′ σ B ) σ , then Z is dR-absolutely special for (V σ ) σ . Indeed, if Z ⊂ W ⊂ S is such that G AH Z = G AH W then setting W ′ := f (W ) we obtain G AH W ′ = G AH x . We may assume that W is special. As x is dR-absolutely special by assumption, we get W ′ = {x}. Since W was assumed to be special, it is thus an irreducible component of f −1 ({x}), as is Z. We conclude that W = Z and Z is dR-absolutely special.
(iii) Let Z ⊂ S be a special subvariety. By assumption, the special point x = f (Z) is defined overQ and its Galois conjugates x σ ∈ B σ are special points. Since f is defined overQ, so is Z. Similarly, Z σ is an irreducible component of (f σ ) −1 (x σ ) and since x σ is special it follows from Proposition 4.19 that Z σ ⊂ S σ is a special subvariety.
(iv) It follows from the factorization of the period map for V ′ that if Z ⊂ S isQbialgebraic, then x ∈ B(Q) is aQ-bialgebraic point. By assumption, x is dRabsolutely special. Arguing as in (ii), we see that Z ⊂ S is dR-absolutely special.
Remark 4.23. The first part of Theorem 4.22(iii) was already proven in [KOU20]. We emphasize that we follow the same strategy, except that their proof uses an argument where one is forced to change theQ-structure on V in order to prove that V ′ is defined overQ. By doing so, one also changes the notion of dR-absolute Hodge cycles. As a consequence, their proof does not allow the conclusion for Galois conjugates in the second part of Theorem 4.22(iii), let alone a proof of the other parts of Theorem 4.22. The formalism of dR-absolutely special subvarieties allows us to show that V ′ is defined overQ without affecting theQ-structure.
(i) Suppose Deligne's conjecture 0.3 holds for special points inQ-motivic variations.
any point s ∈ S(C) the fiber V s carries a polarizable Hodge structure. For every m, n ≥ 0 we define the tensor Hodge structure V closed irreducible algebraic subvariety Z ⊂ S one can attach an important Hodge-theoretic invariant: its generic Mumford-Tate group with respect to the variation V. After choosing a point s ∈ Z(C), let H Z ⊂ V ⊗ s be the subspace of those v ∈ V ⊗ s satisfying the condition that every parallel transport of v is a Hodge cycle at every point of Z.Definition 1.1. The generic Mumford-Tate group G Z of Z is defined to be the subgroup of GL(V s ) fixing the tensors in H Z .Remark 1.2.
(i) A (connected) Hodge datum is a pair (G, D) consisting of a reductive group G over Q and a connected component D of the G(R)-conjugacy class of a Hodge cocharacter h : S → G R satisfying the following conditions: (a) the weight homomorphism w : G m,R ⊂ S h → G R is defined over Q and factors through the center of G, (b) the involution induced by h(i) is a Cartan involution of G ad R . (ii) A morphism of Hodge data (G, D) → (G ′ , D ′ ) is a morphism of reductive groups G → G ′ such that D maps to D ′ . Definition 1.4 ([Kli17], 3.3).
(i) A (connected) Hodge variety attached to the Hodge datum (G, D) is a quotient Γ\D of D by an arithmetic subgroup Γ ⊂ G(Q). It is naturally a complex analytic space, and becomes a complex manifold after possibly replacing Γ by a finite index subgroup.(ii) A morphism of Hodge varieties Γ\D → Γ ′ \D ′ is a morphism induced by a morphism of Hodge data (G, D) → (G ′ , D ′ ) under which Γ gets mapped to Γ ′ .(iii) A special subvariety of a Hodge variety Γ\D is the image of a morphism of Hodge varieties.From a variation of Hodge structure V, one can construct (after possibly replacing S by a finiteétale cover) a period map Φ : S an → X := Γ\D, where (G S , D) is the generic Hodge datum on S.
Theorem 1.7 ([CDK95], Corollary 1.3). The special subvarieties of S are precisely the irreducible components of the preimages of special subvarieties in X under the period map.
of Deligne's notion of absolute Hodge cycle. Choose a point s ∈ S(C). For every tensor α ∈ V an absolute variation of Hodge structure (V σ ) σ on S, conjugation by any element σ ∈ Aut(C/Q) gives a conjugated de Rham tensor α σ dR ∈ (V σ s σ ) ⊗(m,n) .Definition 2.4 ([Voi06], Definition 0.1). The tensor α ∈ V ⊗(m,n) s is called dR-absolute Hodge if for every σ ∈ Aut(C/Q) the conjugate α σ dR ∈ (V σ s σ ) ⊗(m,n) is induced by a Hodge tensor in (V σ s σ ) ⊗(m,n) .A powerful tool in the study of absolute Hodge classes is Deligne's Principle B (cf.[Del82], Theorem 2.12).
s = G AH Z . In fact, if we choose any Hodge generic point s, then H Z ⊂ G s ⊂ G AH s and the above Lemma 2.8 gives G AH s = G AH Z . Now the reductivity follows from Remark 2.7. Following the proof of ([And92], Theorem 1), in order to show that H Z G AH s . This follows from the fact that the collection (V σ s σ ) ⊗(m,n) H Z σ σ
Proposition 3 . 3 .
33Let Z be dR-absolutely special. Then there does not exists a closed irreducible subvariety Y ⊃ Z with the property
Proposition 3. 4 .
4Let Z be a closed irreducible subvariety and Y := Z Zar,Q the smallest Q-subvariety containing Z. Then H Z is a normal subgroup of H Y , G Y and G AH Y .
Definition 3.8 ([KOU20], Definition 1.10). A closed irreducible subvariety Z ⊂ S is called weakly non-factor if it is not contained in a closed irreducible Y ⊂ S such that H Z is a strict normal subgroup of H Y .
.
Let Z ⊂ S be a closed irreducible subvariety. The Hodge codimension of Z is defined to be
Definition 4.2 ([BKU21], Definition 4.2). A closed irreducible subvariety Z ⊂ S is called atypical ifHcd(S) > Hcd(Z).
Remark 4. 3 .
3Strictly speaking, one has to modify Definition 4.2 slightly to take possible singularities of the image Φ(S an ) of the period mapping into account, compare ([BKU21], Definition 4.2).
Definition 4. 4 .
4For a closed irreducible subvariety Z, we define the absolute Hodge defect to beδ AH (Z) := dim Γ AH Z \D AH Z − dim Γ Z \D Z .Theorem 4.5. Let Z be a dR-absolutely special subvariety and assume that G AH S = G S , i.e. all generic Hodge cycles on S are dR-absolute Hodge. Then δ AH (Z) ≤ Hcd(S) − Hcd(Z).
(i) The level of an irreducible real Hodge structure V of weight zero is the largest integer k such that V k,−k = 0 in the decomposition V C = ⊕ k∈Z V k,−k . We define the level of an irreducible Q-Hodge structure V of weight zero as the maximum of the levels of the irreducible factors of V R , and the level of a Q-Hodge structure of weight zero as the minimum of the levels of its irreducible Q-factors.(ii) The level of the variation of Hodge structure V is defined to be the level of the Q-Hodge structure g ad S .One can show that the level is independent of the choice of the Hodge generic point x ([BKU21], 3.6).For Shimura varieties, the variation V has level one. For variations of Hodge structure of level ≥ 3 however, it was proved in ([BKU21], Theorem 2.3) that all strict special subvarieties are atypical.
. A bialgebraic structure on S is a pair D :S →X an , h : π 1 (S an ) → Aut(X) where • π :S → S an denotes the universal cover of S an ,
Definition 4 .
412 ([KUY18], Definition 4.2 and 4.3).(i) A closed irreducible analytic subvariety W ⊂S is called an irreducible algebraic subvariety ofS if W is an analytic irreducible component of D −1 (W ), whereW is a closed algebraic subvariety ofX. (ii) A closed irreducible algebraic subvariety Z ⊂ S is called bialgebraic if one (equiv. any) irreducible analytic component of π −1 (Z) is an algebraic subvariety ofS in the above sense. Consider a variation of Hodge structure V on S. IfΦ :S → D S denotes the lifted period map and ι : D S ⊂Ď S the embedding of D S into its compact dual, let D = ι •Φ :S →Ď an S denote the composition. Then D :S →Ď an S , h : π 1 (S an ) → Aut(Ď S )is a bialgebraic structure. Here h is given by the natural map π 1 (S an ) → G S (C).
Proposition 4 .
413 ([Kli17], Proposition 7.4). The bialgebraic subvarieties of S for this bialgebraic structure are exactly the weakly special subvarieties.
Definition 4 .
414 ([KUY18], 4.11). A bialgebraic structure D :S →X an , h : π 1 (S an ) → Aut(X)is calledQ-bialgebraic if the following holds:(i) The base variety S is defined overQ.(ii) The varietyX is defined overQ and the homomorphism h takes values in the automorphisms ofX overQ.
Definition 4 .
415 ([KUY18]). A closed irreducible subvariety Z ⊂ S is calledQ-bialgebraic if it is defined overQ and an (resp. any) analytic irreducible component of π −1 (Z) is an analytic irreducible component of D −1 (W ) for aQ-subvarietyW ofX.
Proposition 4 . 20 .
420The map f : S → B can be defined overQ, and the collection (V ′ σ B ) σ forms aQ-motivic variation of Hodge structure on B.
Theorem 4. 22 .
22(i) Suppose Deligne's conjecture 0.3 holds for special points inQmotivic variations. Then it holds for all subvarieties inQ-absolute variations of geometric origin.
Acknowledgments. The author would like to thank Bruno Klingler for many helpful discussions and comments.
Mumford-tate groups of mixed Hodge structures and the theorem of the fixed part. Yves André, Compositio Mathematica. 821Yves André. Mumford-tate groups of mixed Hodge structures and the the- orem of the fixed part. Compositio Mathematica, 82(1):1-24, 1992.
o-minimal GAGA and a conjecture of Griffiths. Benjamin Bakker, Yohan Brunebarbe, Jacob Tsimerman, arXiv:1811.12230Benjamin Bakker, Yohan Brunebarbe, and Jacob Tsimerman. o-minimal GAGA and a conjecture of Griffiths, 2019. arXiv:1811.12230.
Tame topology of arithmetic quotients and algebraicity of Hodge loci. Benjamin Bakker, Bruno Klingler, Jacob Tsimerman, J. Amer. Math. Soc. 334Benjamin Bakker, Bruno Klingler, and Jacob Tsimerman. Tame topology of arithmetic quotients and algebraicity of Hodge loci. J. Amer. Math. Soc., 33(4):917-939, 2020.
On the distribution of the Hodge locus. Gregorio Baldi, Bruno Klingler, Emmanuel Ullmo, arXiv:2107.08838Gregorio Baldi, Bruno Klingler, and Emmanuel Ullmo. On the distribution of the Hodge locus, 2021. arXiv:2107.08838.
On the locus of Hodge classes. Eduardo Cattani, Pierre Deligne, Aroldo Kaplan, J. Amer. Math. Soc. 82Eduardo Cattani, Pierre Deligne, and Aroldo Kaplan. On the locus of Hodge classes. J. Amer. Math. Soc., 8(2):483-506, 1995.
Cambridge Studies in Advanced Mathematics. James Carlson, Stefan Müller-Stach, Chris Peters, Cambridge University PressPeriod Mappings and Period Domains. 2 editionJames Carlson, Stefan Müller-Stach, and Chris Peters. Period Mappings and Period Domains. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2 edition, 2017.
Équations différentiellesà points singuliers réguliers. Pierre Deligne, Lecture Notes in Mathematics. 163Springer-VerlagPierre Deligne.Équations différentiellesà points singuliers réguliers. Lec- ture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin-New York, 1970.
Hodge cycles on abelian varieties. Pierre Deligne, Hodge cycles, motives, and Shimura varieties. Berlin-New YorkSpringer-Verlag900Pierre Deligne. Hodge cycles on abelian varieties. In Hodge cycles, motives, and Shimura varieties, volume 900 of Lecture Notes in Mathematics, pages 9-100. Springer-Verlag, Berlin-New York, 1982.
Tannakian categories. In Hodge cycles, motives, and Shimura varieties. Pierre Deligne, James S Milne, Lecture Notes in Mathematics. 900Springer-VerlagPierre Deligne and James S. Milne. Tannakian categories. In Hodge cycles, motives, and Shimura varieties, volume 900 of Lecture Notes in Mathemat- ics, pages 101-228. Springer-Verlag, Berlin-New York, 1982.
. Alexander Grothendieck, Michel Raynaud, Documents Mathématiques. 3Revêtementsétales et groupe fondamental (SGA 1Alexander Grothendieck and Michel Raynaud. Revêtementsétales et groupe fondamental (SGA 1), volume 3 of Documents Mathématiques (Paris).
Séminaire de géométrie algébrique du Bois Marie 1960-61. Société Mathématique De France, Lecture Notes in Math. 224SpringerSociété Mathématique de France, Paris, 2003. Séminaire de géométrie algébrique du Bois Marie 1960-61. [Lecture Notes in Math., 224, Springer, Berlin].
Hodge loci and atypical intersections: conjectures. Bruno Klingler, arXiv:1711.09387Bruno Klingler. Hodge loci and atypical intersections: conjectures, 2017. arXiv:1711.09387.
Hodge theory, between algebraicity and transcendence. Bruno Klingler, arXiv:2112.138142021Bruno Klingler. Hodge theory, between algebraicity and transcendence, 2021. arXiv:2112.13814.
On the closure of the Hodge locus of positive period dimension. Inventiones mathematicae. Bruno Klingler, Ania Otwinowska, Bruno Klingler and Ania Otwinowska. On the closure of the Hodge locus of positive period dimension. Inventiones mathematicae, 03 2021.
On the fields of definition of Hodge loci. Bruno Klingler, Ania Otwinowska, David Urbanik, arXiv:2010.03359Bruno Klingler, Ania Otwinowska, and David Urbanik. On the fields of definition of Hodge loci, 2020. arXiv:2010.03359.
Bi-algebraic geometry and the André-Oort conjecture. Bruno Klingler, Emmanuel Ullmo, Andrei Yafaev, Algebraic geometry: Salt Lake City. Providence, RIAmer. Math. Soc97Bruno Klingler, Emmanuel Ullmo, and Andrei Yafaev. Bi-algebraic geom- etry and the André-Oort conjecture. In Algebraic geometry: Salt Lake City 2015, volume 97 of Proc. Sympos. Pure Math., pages 319-359. Amer. Math. Soc., Providence, RI, 2018.
Introduction to Shimura varieties. J S Milne, Harmonic analysis, the trace formula, and Shimura varieties. Providence, RIAmer. Math. Soc4J. S. Milne. Introduction to Shimura varieties. In Harmonic analysis, the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 265-378. Amer. Math. Soc., Providence, RI, 2005.
Families of motives and the Mumford-Tate conjecture. Ben Moonen, Milan J. Math. 852Ben Moonen. Families of motives and the Mumford-Tate conjecture. Milan J. Math., 85(2):257-307, 2017.
Variation of Hodge structure: The singularities of the period mapping. Inventiones mathematicae. Wilfried Schmid, 22Wilfried Schmid. Variation of Hodge structure: The singularities of the period mapping. Inventiones mathematicae, 22:211-320, 1973.
Fields of definition of Hodge loci. Morihiko Saito, Christian Schnell, Recent advances in Hodge theory. CambridgeCambridge Univ. Press427Morihiko Saito and Christian Schnell. Fields of definition of Hodge loci. In Recent advances in Hodge theory, volume 427 of London Math. Soc. Lecture Note Ser., pages 275-291. Cambridge Univ. Press, Cambridge, 2016.
A characterization of special subvarieties. Emmanuel Ullmo, Andrei Yafaev, Mathematika. 572Emmanuel Ullmo and Andrei Yafaev. A characterization of special subva- rieties. Mathematika, 57(2):263-273, 2011.
Hodge loci and absolute Hodge classes. Claire Voisin, Compositio Mathematica. 143Claire Voisin. Hodge loci and absolute Hodge classes. Compositio Mathe- matica, 143, 06 2006.
| [] |
[
"Shear Viscosity Coefficient from Microscopic Models Typeset using REVT E X",
"Shear Viscosity Coefficient from Microscopic Models Typeset using REVT E X"
] | [
"Azwinndini Muronga \nSchool of Physics and Astronomy\nInstitut für Theoretische Physik, J.W. Goethe-Universität\nUniversity of Minnesota\n55455, D-60325Minneapolis, Frankfurt am MainMinnesotaUSA, Germany\n"
] | [
"School of Physics and Astronomy\nInstitut für Theoretische Physik, J.W. Goethe-Universität\nUniversity of Minnesota\n55455, D-60325Minneapolis, Frankfurt am MainMinnesotaUSA, Germany"
] | [] | The transport coefficient of shear viscosity is studied for a hadron matter through microscopic transport model, the Ultra-relativistic Quantum Molecular Dynamics (UrQMD), using the Green-Kubo formulas. Moleculardynamical simulations are performed for a system of light mesons in a box with periodic boundary conditions. Starting from an initial state composed of π, η , ω , ρ , φ with a uniform phase-space distribution, the evolution takes place through elastic collisions, production and annihilation. The system approaches a stationary state of mesons and their resonances, which is characterized by common temperature. After equilibration, thermodynamic quantities such as the energy density, particle density, and pressure are calculated. From such an equilibrated state the shear viscosity coefficient is calculated from the fluctuations of stress tensor around equilibrium using Green-Kubo relations. We do our simulations here at zero net baryon density so that the equilibration times depend on the energy density. We do not include hadron strings as degrees of freedom so as to maintain detailed balance. Hence we do not get the saturation of temperature but this leads to longer equilibration times. | 10.1103/physrevc.69.044901 | [
"https://arxiv.org/pdf/nucl-th/0309056v2.pdf"
] | 16,211,757 | nucl-th/0309056 | 62a30da37aad4b436ab0708530d6b2df5b894001 |
Shear Viscosity Coefficient from Microscopic Models Typeset using REVT E X
2 Dec 2003
Azwinndini Muronga
School of Physics and Astronomy
Institut für Theoretische Physik, J.W. Goethe-Universität
University of Minnesota
55455, D-60325Minneapolis, Frankfurt am MainMinnesotaUSA, Germany
Shear Viscosity Coefficient from Microscopic Models Typeset using REVT E X
2 Dec 2003(August 4, 2021)* Present address: 1
The transport coefficient of shear viscosity is studied for a hadron matter through microscopic transport model, the Ultra-relativistic Quantum Molecular Dynamics (UrQMD), using the Green-Kubo formulas. Moleculardynamical simulations are performed for a system of light mesons in a box with periodic boundary conditions. Starting from an initial state composed of π, η , ω , ρ , φ with a uniform phase-space distribution, the evolution takes place through elastic collisions, production and annihilation. The system approaches a stationary state of mesons and their resonances, which is characterized by common temperature. After equilibration, thermodynamic quantities such as the energy density, particle density, and pressure are calculated. From such an equilibrated state the shear viscosity coefficient is calculated from the fluctuations of stress tensor around equilibrium using Green-Kubo relations. We do our simulations here at zero net baryon density so that the equilibration times depend on the energy density. We do not include hadron strings as degrees of freedom so as to maintain detailed balance. Hence we do not get the saturation of temperature but this leads to longer equilibration times.
I. INTRODUCTION
High energy heavy ion reactions are studied experimentally and theoretically to obtain information about the properties of nuclear matter under the extreme conditions of high densities and /or temperatures. One of the most important aspects of studying nucleusnucleus reactions at these extreme conditions is the possibility that normal nuclear matter can undergo a phase transition into a new state of matter, the quark-gluon plasma [?]. In this state the degrees of freedom are partons (quarks and gluons).
In this work we study only the the thermodynamic and transport properties of hadron matter. Hence the relevant degrees of freedom are hadrons. We study the equilibration of the system in infinite hadron matter using UrQMD [2]. We restrict ourselves to a a system that contains only meson resonance degrees of freedom. The infinite hadron matter is modelled by initializing the system by light mesons only. We fix baryon density and energy density of the system in a cubic box and impose periodic boundary conditions. We then propagate the system in time until we obtain equilibration.
The equation of state and transport coefficients of hot, dense hadron gases are quite important quantities in high energy nuclear physics. In the ultra-relativistic heavy ion experiments at CERN and BNL, the final state of interactions is dominated by hadrons and hence the observables are mainly hadrons. Therefore knowledge of the equation of state and transport coefficients of a hadron gas is necessary for a better understanding of the observables. Phenomenologically both the transport properties and the equation of state of hadron gas are the major source of uncertainties in dissipative fluid dynamics.
In spite of their importance, the equation of state and transport coefficients of hot, dense hadron gases are still poorly known because of the nonperturbative nature of the strong interaction. Progress in the study of hadron matter transport coefficients is very slow, and only a calculation of transport coefficients in the variational method [3,4] and relaxation time approximation [5] has been done. From those previous studies a lot has been learned about the transport coefficients of binary mixtures such as ππ system. However in a more realistic situation we need to describe transport properties of a many-body system. This in turn would require taking into account various interaction processes and in-medium effects. Thus, we need to investigate the thermodynamic and transport properties of a hadron gas by using a microscopic model that includes realistic interactions among hadrons. In this work, we adopt a relativistic microscopic model, UrQMD and perform molecular-dynamical simulations for a hadron gas of mesons.
We focus on the hadronic scale temperature (100 MeV < T < 200 MeV) and zero baryon number density which are expected to be realized in the central high energy nuclear collisions. Thermodynamic properties and transport coefficients of hadronic matter in this region should play important roles in dissipative fluid dynamical models. Sets of statistical ensembles are prepared for the system of fixed energy density and baryon number density. Using these ensembles, the equation of state is investigated. The statistical ensembles is then applied in calculating the shear viscosity coefficient of a hadron gas of mesons.
The equation of state of a hot and dense hadron gas had been investigated using UrQMD [2,6]. The work has provided valuable information regarding the nature of the hadron gas. In those simulations the temperature reaches a limiting value with increasing energy density. This is because in those simulations the detailed balance is broken. This in turn leads to the irreversibility of the equilibrated system. And without the reversal process of multiparticle production energy balance between the forward and backward reactions is no longer realized and hence the saturation of the temperature occurs. Although it is interesting and important to formulate these multi-particle interaction processes exactly in the present simulation, straightforward implementation of them is not easy. In this work, avoiding this complicated problem, we disabled three or many-body interactions in UrQMD. We have also disabled decays or interactions that involves photons.
The rest of the paper is organized as follows: In section II we study the equilibration and thermodynamics of the system. In section III we study the thermodynamic of a pure resonance meson gas for comparison with the results from UrQMD. In section IV we calculate the shear viscosity coefficient from stress tensor fluctuations around the equilibrium state through UrQMD using Green-Kubo relations. Finally in section V we summarize our results.
II. EQUILIBRATION OF INFINITE MATTER IN A BOX
To investigate the equilibration of the system we performed microscopic calculation using UrQMD. UrQMD is designed to simulate ultra-relativistic heavy ion collision experiments. The description of the model can be found in [2]. In studying the equilibration of the hadron gas we would like to maintain detailed balance in the simulations. Multi-particle productions plays an important role in the equilibration of the hadron gas. However in UrQMD their inclusion in the simulations breaks detailed balance due to the absence of reverse processes. In order to avoid this problem in the present simulations we consider only up to two-body absorption/annihilation and decay processes. Thus the fundamental processes in the UrQMD version we use here are two-body elastic and quasi-elastic collisions between hadrons, and strong decays of resonances. Even though we started with light mesons in the initial state we consider all the mesons and meson-resonance included into the UrQMD model, in the final state.
When studying the equilibration of hadron gas it is important to maintain detailed balance in the microscopic model. Though the contributions of the multi-particle productions dominate the system at early stages of the non-equilibrated system, the reverse process plays an important role in the latter, equilibration stage. The absence of reverse processes leads to one-way conversion of the energy to particles. However, the exact treatment of multi-particle absorption processes is very difficult. In order to treat them effectively in our case, we only consider up to 2-body decays.
In this work, we focus our investigation on the thermodynamic and transport properties of a hadronic system. For this purpose, we consider a system in a cubic box and impose periodic boundary conditions in configuration space. Thus if a particle leaves the box, another one with the same momentum enters from the opposite side. This calculation is similar to the one done in [6] but with different degree of freedom and included processes in the system. A further similar analysis was done in [7,8] using different cascade models with different degrees of freedom.
The energy density ε and the baryon number density n B in the box are fixed as input parameters, and these quantities are conserved throughout the simulation. The initial distributions of mesons are given by uniform random distributions in phase space. The energy is defined as ε = E/V , where E is the energy of N particles:
E = N i=1 m 2 i + p 2 i (1)
The 3-momenta p i of the particles in the initial state are randomly distributed in the center of mass system of the particles:
N i=1 p i = 0 .(2)
The time evolution is now described by UrQMD. Though the initial particles are only π , η , ω , ρ , φ, many mesons and meson-resonances are produced through interactions. We now propagate all particles in the box using periodic boundary conditions, that is, particles moving out of the box are reinserted at the opposite side with the same momentum. The phase-space distribution of mesons then can change due to elastic collisions, resonance production and their decays to lighter mesons again. We recall that we include all the mesons and meson-resonances in UrQMD.
To investigate the equilibration phenomena of the system we look at the particle densities and energy distributions of each particle. As time increases the system tends towards an equilibrium state. When the system is in thermal equilibrium, the slope parameters of the energy distributions for all particles should have the same value, and that value is the inverse of temperature. To investigate this, we study the time evolution of the inverse slopes of various particles.
Running UrQMD many times with the same input parameters and taking the stationary configuration in equilibrium, we can obtain statistical ensembles with fixed temperature. By using these ensembles, we can calculate thermodynamic quantities, such as the particle density, pressure, and so on, as functions of temperature and baryon number density. We extract the shear viscosity coefficient by finding the energy-momentum tensor correlations and then employ the Green-Kubo relations..
We specify the initial input parameters: the volume of the box V , the net baryon number density n B , and the total energy density ε. We consider the input parameters which will give the temperature range 100 -200 MeV. Here n B = 0.0 fm −3 is taken as the net baryon number density of the system. We generated a statistical ensemble of 200 events. Figure 1 shows the time evolution of the various particles densities (π, η, ρ, K) at zero net baryon number density and energy density ε = 0.3 GeV/fm 3 . After several fm/c the number of pions decreases first due to inelastic collisions and annihilation that produces other meson resonances. The pion density then increases due to decay of heavier meson resonances to an equilibration. The number of kaons (in general strange mesons) increases to equilibration value in much longer times than other particles. In figure 2 we show the same situation but with different initial energy density of the box, ε = 0.9 GeV/fm 3 . For large initial energy densities the equilibration times are much larger. 2. The time evolution of particle densities for each particle with V = 1000 fm 3 and ε = 0.9 GeV/fm 3 Figures 1 and 2 display the time evolution of particle densities. These figures show that the system approaches a stationary state with time. The saturation of particle densities indicates the realization of chemical equilibrium. We conclude that chemical equilibrium in our system is realized.
A. Chemical Equilibration
B. Thermal Equilibration and Temperature
In this subsection we investigate the approach to thermal equilibrium. This is driven by the momentum equilibration of the system. That is, when the momentum anisotropy of the system has dropped to a limiting value such that the system can be described by simple global thermodynamic variables like temperature. The thermal equilibration times have to be contrasted to those for chemical equilibrium.
dN i d 3 p = dN 4πEpdE = C exp(−βE i ),(3)
as time increases, where β is the slope parameter of the distribution. Here E i = (p 2 i + m 2 i ) 1/2 is the energy of particle i. Moreover, the slopes of the energy distributions converge to a common value. These results indicate realization of thermal equilibrium. Figure 4 displays the time evolution of the inverse slopes of different particle species that were calculated by fitting the energy distributions to a Boltzmann distribution. The solid curves correspond to the time evolution of the inverse slope of pions. From this figure, it is seen that the difference between the pion inverse slope and other particles' inverse slopes become zero for times latter than 350 fm/c. Therefore, we conclude that thermal equilibrium is established at about t = 350 fm/c; the values of the inverse slope parameters of the energy distribution for all particles become equal for latter times. Thus we can regard this value as the temperature of the system. The equilibration time is large. If we allow for multi-particle production and absorption the equilibration time would be shorten significantly.
III. HADRONIC GAS MODEL
In this subsection we compare the UrQMD box calculations with a simple statistical model for an ideal hadron gas where the system is described by a grand canonical ensemble of noninteracting bosons in equilibrium at temperature T . All meson species considered in UrQMD are also been used in the statistical model. In hadron gas model we use as input the same energy density and net baryon density to obtain the temperature of the system.
In hadron gas we find that the temperature increases continuously with energy density. Figures 5 and 6 show the relations between the temperature and thermodynamic quantities such as energy density,
ε = 1 V all particles i=1 E i ,(4)
particle density, and pressure,
P = 1 3V all particles i=1 p 2 i E i .(5)
In these figures, all curves correspond to the relativistic Bose-Einstein gas
ε(T, µ) = k g k d 3 p (2π) 3 E k e E k −µ T − 1 ,(6)n(T, µ) = k g k d 3 p (2π) 3 1 e E k −µ T − 1 ,(7)p(T, µ) = k g k d 3 p (2π) 3 p 2 3E k 1 e E k −µ T − 1 ,(8)
where g k is a degeneracy factor. In these calculations the meson chemical potential µ is fixed to zero. Figure 5 shows the energy density versus temperature for mesons. In this figure, the difference between UrQMD results and those for the calculation of the free gas model is negligible. Figure 6 shows the pressure versus temperature for mesons. There is deviation of UrQMD results from the free gas model results especially at high temperatures. The influence of interactions is clear above T ∼ m π . Enhancement of heavy meson resonances grows as the temperature increases.
In a previous study [6] the limiting value of temperature with increasing energy appeared. As already mentioned this is because of the lack of reversal process of multi-particle production in that study. In this calculation where we try to maintain detailed balanced in UrQMD, this limiting temperature does not appear. This is an important result of taking detailed balance into account.
However, in this simulation the lack of multi-particle production leads to long equilibration times. This is also because we do not have meson-baryon interactions, such as πN → R and their inverse processes. The enhancement of heavy baryon resonances causes an increase in the abundances of mesons, and vice versa. Heavy resonances readily produce two pions, and thus the enhancement of heavy baryon resonances promotes meson production. Therefore, interactions between mesons and baryons are very important in the study of the properties of a mixed hadron gas. Inclusion of multi-particle interactions would also shorten the equilibration time considerably.
IV. SHEAR VISCOSITY COEFFICIENT
Transport coefficients such as viscosities, diffusivities and conductivities characterizes the dynamics of fluctuations of dissipative fluxes in a medium. Transport coefficients can be measured, as in the case of condensed matter applications. However in principle they should be calculable theoretically from first principles.
In a weakly coupled theory transport coefficients can be computed in a perturbative expansion, employing either kinetic theory or field theory using Kubo formulas [9][10][11][12][13][14][15]. The resulting Kubo relations [16] express transport coefficients in terms of the zero-frequency slope of spectral densities of current-current, or stress tensor-stress tensor correlation functions, Monte Carlo simulations for transport coefficients is a powerful tool when studying transport coefficients using Green-Kubo relations. For calculation of transport coefficients of shear viscosity, thermal conductivity, thermal diffusion and mutual diffusion for a binary mixture of hard spheres see [17] and for the calculation of diffusion coefficient of a hadron gas see [8] Knowledge of various transport coefficients is important in dissipative fluid dynamical models [18]. In this paper we consider the evaluation of shear viscosity coefficient of a hadron gas of mesons and their resonances.
In trying to stay close to the extended irreversible thermodynamic processes we will, however, use the Kubo formulas in fluctuation theory to extract transport coefficients.
In the longitudinal boost-invariant flow the important coefficient is the shear viscosity. In dissipative fluids the expression for the entropy 4-current is governed by transport coefficients and relaxation coefficients. These coefficients determine the strength of the fluctuations of dissipative fluxes about the equilibrium state. The generalized entropy plays an important role in the description of the fluctuations of conserved quantities and of the dissipative fluxes. Now we calculate the coefficient of shear viscosity. First, the fluctuation-dissipation theorem tells us that shear viscosity η is given by the stress tensor correlations [16]
η = V T ∞ 0 π ij (t) · π ij (t + t ′ ) dt ′ ,(9)
where π ij ≡ T ij −δ ij P denotes the traceless part of the stress tensor and P ≡ 1 3 T i i the (local) pressure. The angular brackets stand for equilibrium average, i.e., average over the number of ensemble states and average over the number of particles. The correlation functions are damped exponentially with time (see Fig. 7):
π ij (t) · π ij (t + t ′ ) ∝ exp − t ′ τ π .(10)
The solid lines in Fig. 7 are the fits to the correlations and the inverse slope corresponds to the relaxation time. The shear viscosity coefficient can be rewritten in the simple form
η = V T π ij (t) · π ij (t) τ π ,(11)
where τ π is the relaxation time of the shear flux. In this work we used a box of volume V = 1000 fm 3 . The results are insensitive to the box length greater than 6 fm.
To this end, we have to remark that the transport coefficients represents the fluctuations of the dissipative fluxes around an equilibrium state. In terms of fluctuations the Green-Kubo relation (at zero frequency) for shear viscosity can be written as
η = V T ∞ 0 δπ ij (0)δπ ij (t) dt (i = j) ,(12)
In the above equation the fluctuations of shear flux are exponentially damped. They are obtained found from the second differential of the generalized entropy expression [18] δπ
ij (0)δπ kl (t) = ηT (τ π V ) −1 △ ijkl exp(−t/τ π ) .(13)
with △ ijkl = (δ ik δ jl + δ il δ jk − (2/3)δ ij δ kl ). In the limit of vanishing relaxation times, we recover the formulae of Landau and Lifshitz, since in this limit τ −1 exp(t/τ ) → 2δ(t) with δ(t) the Dirac delta function. Equation (13) relates the dissipative coefficient η to the fluctuations of the fluxes with respect to equilibrium. We see that fluctuations determine the dissipative coefficients. Conversely, transport coefficients determine the strength of the fluctuations.
If the evolution of the fluctuations on the fluxes is described by the Maxwell-Cattaneo (see [18]) relation equations then after integration the above expression for the shear viscosity coefficient reduces to
η = τ π V T δπ ij (0)δπ ij (0) .(14)
In what follows we will use the existing microscopic model, namely UrQMD, to extract the shear viscosity coefficient. Figure 8 shows the shear viscosity coefficient results from UrQMD using Kubo relations. As in the variational approach the coefficient grows with temperature. The UrQMD results are about twice those from the variational method. This might be due to the many meson resonances included in UrQMD while in the variational method we only have pions. Also the cross section parameterizations are different in the two approaches. Figure 9 shows the relaxation time for shear flux in a hot pion gas calculated from UrQMD by fitting the shear stress correlations. The dependence of the shear relaxation time on temperature is similar to the one obtained using variational method. The results obtained here are about a factor of two less than variational method results. The reasons are similar to the ones given above for the shear viscosity coefficient.
V. CONCLUSIONS AND OUTLOOK
The transport coefficients for a hadron gas can be obtained easily from microscopic transport models such as UrQMD. The study of fluctuations of dissipative fluxes around equilibrium yields Green-Kubo relations which are more easily applied. The use of fluctuations through Kubo relations has the advantage of finding not only the transport coefficients but also the corresponding relaxation times. In addition it is also possible to obtain the relaxation coefficients such as β 2 used in [18].
Since the shear viscosity coefficient for QCD has been calculated by many authors using either kinetic theory or pertubative expansion, it will be interesting to calculate the shear viscosity coefficient for quark gluon plasma using microscopic models in the form of parton cascade models such as VNI/BMS [19]. This is currently under investigation [20].
FIG. 1 .
1The time evolution of particle densities for each particle with V = 1000 fm 3 and ε = 0
FIG. 3 .
3Energy distributions of π, η, ρ and K at four different values of time, t = 100 fm/c, t = 200 fm/c, t = 300 fm/c and t = 400 fm/c. The lines are the fitted results that are given by Boltzmann distributions, C exp(−βE). The calculation was done with V = 1000 fm 3 , n B = 0.0 fm −3 and ε = 0.9 GeV/fm 3 .
Figure 3
3displays energy distributions of π, η, ρ and K at time t = 100, 200, 300 and 400 fm/c. For equilibrated system the energy distributions approach the Boltzmann distribution,
FIG. 4 .
4The time evolution of the inverse slopes β −1 for π, η, ρ and K with V = 1000 fm 3 , n B = 0.0 fm −3 , ε = 0.9 GeV/fm 3 . The value of β −1 was calculated from the fitting of energy distributions. Here the solid curves represent the time evolution of β −1 for π.
FIG. 5 .FIG. 6 .
56The equation of state of a mixed hadron gas at finite temperature (100 MeV < T < 200 MeV) and zero baryon density (0.0 fm −3 ). The energy density of mesons is plotted as functions of the temperature. The curve corresponds to the free gas model represented by Eq. The equation of state of a mixed hadron gas at finite temperature (100 MeV < T < 200 MeV) and zero baryon density (0.0 fm −3 ). The pressure of pions is plotted as functions of the temperature. The curve corresponds to the free gas model represented by Eq. (8).
FIG. 7 .FIG. 8 .FIG. 9 .
789Stress-tensor correlation of the mesons as a function of time. The curves are the exponential fits to extract relaxation times Shear viscosity of meson gas as a function of temperature. The relaxation time for the shear flux of meson gas as a function of temperature.
ACKNOWLEDGMENTSI would like to thank Joe Kapusta and Horst Stöcker for valuable comments. This work was supported by the US Department of Energy grant DE-FG02-87ER40382.
A compilation of current RHIC results can be found in: Quark Matter '01, Proceedings of the Fifteenth International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions at. Stony-Brook, NY, USA698A compilation of current RHIC results can be found in: Quark Matter '01, Proceedings of the Fifteenth International Conference on Ultra- Relativistic Nucleus-Nucleus Collisions at Stony-Brook, NY, USA, Nucl. Phys. A 698 (2002);
Quark Matter '02, Proceedings of the Sixteenth International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions at. Nantes, FranceQuark Matter '02, Proceedings of the Sixteenth International Conference on Ultra- Relativistic Nucleus-Nucleus Collisions at Nantes, France, to be published in Nucl. Phys. A.
. S A Bass, M Belkacem, M Bleicher, M Brandstetter, L Bravina, C Ernst, L Gerland, M Hofmann, S Hofmann, J Konopka, G Mao, L Neise, S Soff, C Spieles, H Weber, L A Winckelmann, H Stöcker, W Greiner, C Hartnack, J. Aichelin and N. Amelin, Prog. Part. Nucl. Phys. 41225S. A. Bass, M. Belkacem, M. Bleicher, M. Brandstetter, L. Bravina, C. Ernst, L. Ger- land, M. Hofmann, S. Hofmann, J. Konopka, G. Mao, L. Neise, S. Soff, C. Spieles, H. Weber, L. A. Winckelmann, H. Stöcker, W. Greiner, C. Hartnack, J. Aichelin and N. Amelin, Prog. Part. Nucl. Phys. 41 (1998) 225.
. M Prakash, M Prakash, R Venugopalan, G Welke, Phys. Rep. 227321M. Prakash, M. Prakash, R. Venugopalan, and G. Welke, Phys. Rep. 227 (1993) 321.
. D Davesne, Phys. Rev. C. 533069D. Davesne, Phys. Rev. C 53 (1996) 3069.
. S Gavin, Nucl. Phys. A. 435826S. Gavin, Nucl. Phys. A 435 (1985) 826.
. M Belkacem, M Brandstetter, S A Bass, M Bleicher, L Bravina, M I Gorenstein, J Konopka, L Neise, C Spieles, S Soff, H Weber, H Stöcker, W Greiner, Phys. Rev. C. 581727M. Belkacem, M. Brandstetter, S. A. Bass, M. Bleicher, L. Bravina, M. I. Gorenstein, J. Konopka, L. Neise, C. Spieles, S. Soff, H. Weber, H. Stöcker and W. Greiner, Phys. Rev. C 58 (1998) 1727.
Mossel and A. Sibirtsev. E L Bratkovskaya, W Cassing, C Greiner, M Effernberger, U , Nucl. Phys. A. 675661E. L. Bratkovskaya, W. Cassing, C. Greiner, M. Effernberger, U. Mossel and A. Sibirt- sev, Nucl. Phys. A 675, (2000) 661.
. N Sasaki, O Miyamura, S Muroya, C Nonaka, Europhys. Lett. 5438N. Sasaki, O. Miyamura, S. Muroya, C. Nonaka, Europhys. Lett. 54 (2001) 38.
. G Baym, H Monien, C J Pethick, D G , Phys. Rev. Lett. 641867G. Baym, H. Monien, C.J. Pethick and D.G. Ravenhall, Phys. Rev. Lett. 64, (1990) 1867;
. Nucl. Phys. 525415Nucl. Phys. A525, 415C (1991);
. G Baym, H Heiselberg, Phys. Rev. D. 565254G. Baym and H. Heiselberg, Phys. Rev. D 56 (1997) 5254.
. S Jeon, Phys. Rev. 523591S. Jeon, Phys. Rev. D52, (1995) 3591.
. P Arnold, G D Moore, L G Yaffe, JHEP. 00111P. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0011 (2000) 001;
. JHEP. 030551JHEP 0305 (2003) 051.
. A Hosoya, K Kajantie, Nucl. Phys. 250666A.Hosoya and K. Kajantie, Nucl. Phys. B250, (1985) 666.
. H Heiselberg, G Baym, C J Pethick, J Popp, Nucl. Phys. A. 544569H. Heiselberg, G. Baym, C.J. Pethick and J. Popp, Nucl. Phys. A 544 (1992) 569c;
. H Heiselberg, Phys. Rev. Lett. 723013H. Heiselberg, Phys. Rev. Lett. 72 (1994) 3013;
. H Heiselberg, Phys. Rev. D. 494739H. Heiselberg, Phys. Rev. D 49 (1994) 4739.
. G Aarts, J M M Resco, JHEP. 020453G. Aarts and J.M.M. Resco, JHEP 0204 (2002) 053.
. M E Carrington, Hou Defu, R Kobes, Phys. Rev. 6425001M.E. Carrington, Hou Defu and R. Kobes, Phys. Rev. D64 (2001) 025001.
. R Kubo, Rep. Prog. Phys. 29R. Kubo, Rep. Prog. Phys. 29 (1966), Part I, 255.
. J J Erpenbeck, Phys. Rev. A. 394716J. J. Erpenbeck, Phys. Rev. A 39, (1989) 4716.
. A Muronga, nucl-th/0309055A. Muronga, nucl-th/0309055.
. S A Bass, B Mueller, D K Srivastava, Phys. Lett. 551277S. A. Bass, B. Mueller and D.K. Srivastava, Phys. Lett. B551 (2003) 277.
. A Muronga, work in progressA. Muronga, work in progress.
| [] |
[
"ORDER REDUCTION OF NONLINEAR QUASI-PERIODIC SYSTEMS SUBJECTED TO EXTERNAL EXCITATIONS",
"ORDER REDUCTION OF NONLINEAR QUASI-PERIODIC SYSTEMS SUBJECTED TO EXTERNAL EXCITATIONS"
] | [
"Sandesh G Bhat [email protected] \nArizona State University\n6075 S. Innovation Way West101D, 85212MesaAZ\n",
"Cherangara Susheelkumar \nArizona State University\n6075 S. Innovation Way West, 10185212MesaAZ\n",
"Subramanian \nArizona State University\n6075 S. Innovation Way West, 10185212MesaAZ\n",
"Sangram Redkar [email protected] \nArizona State University\n6075 S. Innovation Way West, 101B, Mesa85212AZ\n"
] | [
"Arizona State University\n6075 S. Innovation Way West101D, 85212MesaAZ",
"Arizona State University\n6075 S. Innovation Way West, 10185212MesaAZ",
"Arizona State University\n6075 S. Innovation Way West, 10185212MesaAZ",
"Arizona State University\n6075 S. Innovation Way West, 101B, Mesa85212AZ"
] | [] | In his paper, we present order reduction techniques for nonlinear quasi-periodic systems subjected to external excitations. The order reduction techniques presented here are based on the Lyapunov-Perrone (L-P) Transformation. For a class of non-resonant quasi-periodic systems, the L-P transformation can convert a linear quasi-periodic system into a linear time-invariant one. This Linear Time-Invariant (LTI) system retains the dynamics of the original quasi-periodic system. Once this LTI system is obtained, the tools and techniques available for analysis of LTI systems can be used, and the results could be obtained for the original quasi-periodic system via the L-P transformation. This approach is similar to using the Lyapunov-Floquet (L-F) transformation to convert a linear time-periodic system into an LTI system and perform analysis and control.Order reduction is a systematic way of constructing dynamical system models with relatively smaller states that accurately retain large-scale systems' essential dynamics. In this work, reduced-order modeling techniques for nonlinear quasi-periodic systems subjected to external excitations are presented. The methods proposed here use the L-P transformation that makes the linear part of transformed equations time-invariant. In this work, two order reduction techniques are suggested. The first method is simply an application of the well-known Guyan like reduction method to nonlinear systems. The second technique is based on the concept of an invariant manifold for quasi-periodic systems.The 'quasi-periodic invariant manifold' based technique yields' reducibility conditions.' These conditions help us to understand the various types of resonant interactions in the system. These resonances indicate energy interactions between the system states, nonlinearity, and external excitation. To retain the essential dynamical characteristics, one has to preserve all these 'resonant' states in the reducedorder model. Thus, if the 'reducibility conditions' are satisfied then only, a nonlinear order reduction based on the quasi-periodic invariant manifold approach is possible. It is found that the invariant manifold approach yields good results. These methodologies are general and can be used for parametric study, sensitivity analysis, and controller design.(2) that captures the essential dynamics of the large-scale system. It is noted that ( ) t A the matrix is quasiperiodic and contains incommensurate frequencies. To the best of the author's knowledge, no current techniques would allow direct order reduction from the equation (1) to the equation(2).The reducibility of a linear quasi-periodic system has been the subject of research in the scientific community. Many excellent references discuss the reducibility of quasi-periodic systems[20][21][22][23][24][25][26]. Recently, Waswa and Redkar presented a technique based on L-F transformation, state augmentation, and normal form to reduce linear quasi-periodic system into an LTI system[21]. Very recently, Subramanian and Redkar presented a method to compute L-P transformation based on intuitive state z z F w z z J | 10.1016/j.ijnonlinmec.2022.103994 | [
"https://arxiv.org/pdf/2012.00837v1.pdf"
] | 125,678,573 | 2012.00837 | 4ffdf2ca461e583a522fa155e854fccb4fad5c2d |
ORDER REDUCTION OF NONLINEAR QUASI-PERIODIC SYSTEMS SUBJECTED TO EXTERNAL EXCITATIONS
Sandesh G Bhat [email protected]
Arizona State University
6075 S. Innovation Way West101D, 85212MesaAZ
Cherangara Susheelkumar
Arizona State University
6075 S. Innovation Way West, 10185212MesaAZ
Subramanian
Arizona State University
6075 S. Innovation Way West, 10185212MesaAZ
Sangram Redkar [email protected]
Arizona State University
6075 S. Innovation Way West, 101B, Mesa85212AZ
ORDER REDUCTION OF NONLINEAR QUASI-PERIODIC SYSTEMS SUBJECTED TO EXTERNAL EXCITATIONS
In his paper, we present order reduction techniques for nonlinear quasi-periodic systems subjected to external excitations. The order reduction techniques presented here are based on the Lyapunov-Perrone (L-P) Transformation. For a class of non-resonant quasi-periodic systems, the L-P transformation can convert a linear quasi-periodic system into a linear time-invariant one. This Linear Time-Invariant (LTI) system retains the dynamics of the original quasi-periodic system. Once this LTI system is obtained, the tools and techniques available for analysis of LTI systems can be used, and the results could be obtained for the original quasi-periodic system via the L-P transformation. This approach is similar to using the Lyapunov-Floquet (L-F) transformation to convert a linear time-periodic system into an LTI system and perform analysis and control.Order reduction is a systematic way of constructing dynamical system models with relatively smaller states that accurately retain large-scale systems' essential dynamics. In this work, reduced-order modeling techniques for nonlinear quasi-periodic systems subjected to external excitations are presented. The methods proposed here use the L-P transformation that makes the linear part of transformed equations time-invariant. In this work, two order reduction techniques are suggested. The first method is simply an application of the well-known Guyan like reduction method to nonlinear systems. The second technique is based on the concept of an invariant manifold for quasi-periodic systems.The 'quasi-periodic invariant manifold' based technique yields' reducibility conditions.' These conditions help us to understand the various types of resonant interactions in the system. These resonances indicate energy interactions between the system states, nonlinearity, and external excitation. To retain the essential dynamical characteristics, one has to preserve all these 'resonant' states in the reducedorder model. Thus, if the 'reducibility conditions' are satisfied then only, a nonlinear order reduction based on the quasi-periodic invariant manifold approach is possible. It is found that the invariant manifold approach yields good results. These methodologies are general and can be used for parametric study, sensitivity analysis, and controller design.(2) that captures the essential dynamics of the large-scale system. It is noted that ( ) t A the matrix is quasiperiodic and contains incommensurate frequencies. To the best of the author's knowledge, no current techniques would allow direct order reduction from the equation (1) to the equation(2).The reducibility of a linear quasi-periodic system has been the subject of research in the scientific community. Many excellent references discuss the reducibility of quasi-periodic systems[20][21][22][23][24][25][26]. Recently, Waswa and Redkar presented a technique based on L-F transformation, state augmentation, and normal form to reduce linear quasi-periodic system into an LTI system[21]. Very recently, Subramanian and Redkar presented a method to compute L-P transformation based on intuitive state z z F w z z J
INTRODUCTION
Order reduction is constructing small order systems from the large-scale structure, which captures the dominant dynamics [1]. In the design and development process, engineers are often faced with analyzing complex dynamical systems governed by a large set of integrodifferential (ordinary/partial) equations. These systems are complicated to solve analytically, and one has to resort to numerical techniques [2]. While solving these dynamical systems numerically, one has to consider various issues like convergence, numerical truncation errors, and, most importantly, the limited computational resources and time. To simulate the dynamical system's response accurately within a reasonable amount of time, one can try to construct an equivalent small-scale system known as the 'reduced order model' that will approximate the large-scale system dynamics [3,4]. This approach simplifies a sizeable dynamical system by replacing it with an equivalent small-scale system known as the 'Model Order Reduction' (MOR).
Researchers have used order reduction of linear systems using multiple techniques. Some of the techniques include error minimization [5], pole clustering [6], transfer function based [7], and Pade approximation [8], to mention a few. For a comprehensive overview, we refer to reference [9]. Nonlinear order reduction is studied by researchers from a structural point of view in the second-order and state-space form [10][11][12]. For order reduction of time-periodic system, researchers utilized the L-F transformation and performed order reduction. For details on this approach, we refer to references [13][14][15][16][17].
Order Reduction procedure comprises the following steps. 1. A study of the large-scale system and identifying the dominant states pertaining to the dominant dynamics. 2. Elimination of the non-dominant states either by simply neglecting their contribution or replacing them with the appropriate functions of dominant states. 3. Formulation of an equivalent reduced-order system consisting only of the dominant states.
In this work, we concentrate on the 'automatic' order reduction of an important class of engineering systems known as the 'nonlinear quasi-periodic systems.' [18][19][20]. These systems have quasi-periodic coefficients [7,8] and in the state-space form expressed as
( ) ( , ) ( ) t t t x A x f x F (1) where ( ) t A
is an n n matrix with quasi-periodic coefficients and ( , ) t f x is a nonlinear function with monomials of x , ( ) t F is the external deterministic excitation, and x is an n vector of appropriate dimensions. The objective of order reduction is to construct a reduced-order system
( ) ( , ) ( ) r r r r r r t t t x A x f x F
augmentation and normal forms [27]. This technique can be utilized to calculate a closed-form expression for the L-P transformation. In this paper, we use the L-P transformation-based approach presented by Subramanian and Redkar to obtain the LTI representation of the quasi-periodic system and perform the order reduction. For clarity, it is noted that the reducibility of a quasi-periodic system means converting a linear quasi-periodic system into an LTI system of the same dimension, and order reduction means reducing the size (or the number of states) of the original quasi-periodic system or its LTI representation obtained via the L-P transformation. This paper briefly reviews the L-P transformation computation and its application. The linear and nonlinear order reduction techniques are outlined in section two. In section three, we present an application-Mathieu Hill-type equation for which the L-P transformation can be determined using the state augmentation and normal forms. For this system, the reduced-order models are constructed using linear and nonlinear techniques. In the end, in section five, discussions and conclusions are presented.
MATHEMATICAL PRELIMINARIES
Computation of L-P transformation
A quasi-periodic dynamical system, without any external excitation, can be expressed as
( ) ; t y A y (3) where A(t) is a n n matrix containing a finite number (k) of incommensurable frequencies (k ≥ 2) 1 ( ) ( , ......, ) 2 k t A t t k A (4)
It is noted that A(t) is continuous and periodic in each argument, but the ratio of any two frequencies is irrational [28]. It can be observed that the dynamical system given by the equation (4) is linear, and the method of normal forms [29] is inapplicable. However, the parametric excitation terms can be considered fictitious states, and the linear nonautonomous system given by the equation can be expressed as a nonlinear autonomous system as follows.
Consider the most general form of the equation (3) with quasi-periodicity provided by
0 ( ( )) t x B B x (5) where ( ) t A expressed1 2 n J ω , 1 n n m ω (8)
The system shown in the equation (7) is amenable to an application of Normal Forms [15,16]. A near identity transformation [30,31] (of the form given by equation (9)) is applied to the equation (7). (9) where ( ) r h v is a formal power series in v of degree r with T periodic coefficients that leads to
( ) r z v h v( ) ( ) ( ) ( ) r r r h v v Jv Jv Jh v f v v (10)
The higher-order nonlinear terms in the equation (10) are eliminated by considering the following condition
( ) ( ) ( ) 0 ( ) r r r h v Jv Jh v f v v (11)v v v h v v e f v v e v
and j e is the th j member of the natural basis After solving the equation (11), the solvability expression for a given degree of nonlinearity can be expressed as
. jm r j jm f h m λ(12)
where
In case the solvability condition [32] given by the equation (13) is satisfied, one can obtain the linear equation given by equation (14) and the near identify transformation given by equation (9).
v Jv (14)
The near identity transformation given by the equation (9) is wholly known in the non-resonant case and ( ) r h v contains the terms that explicitly depend upon fictitious states p and q . One can substitute these fictitious state in terms of their closed-form expression
Cos( ) i i q t , and Sin( ) i i p t that yields ( , ) r t
h v leading to the following form of the near identity transformation
[ ( )] ( ) t t z I Q v Q v (15)
This transformation is similar to the Lyapunov-Floquet Transformation [33] but for quasi-periodic systems.
Limitations:
One important aspect is that this technique uses the normal form technique, which can be viewed as an extension of the higher-order averaging method [30,32] and has the same limitations as the averaging. This approach may not yield accurate results when the nonlinearity is very strong (i.e., very strong parametric excitation in the present case) or the linear term is absent (i.e., 0 B 0 c.f. equation (5)). On the other hand, to the best of the author's knowledge, this approach is the only approach that yields the L-P transformation given by the equation (15) in a closed-form. The authors have successfully used this approach to analyze linear and nonlinear quasi-periodic systems.
Computation of the inverse of the Lyapunov-Perrone Transformation:
For parametrically excited quasi-periodic linear systems of the equation (3) form, the L-P transformation is sufficient for carrying out analysis. The inverse of the L-P transformation is needed for quasi-periodic nonlinear systems or quasi-periodic linear/nonlinear systems with deterministic or stochastic excitations. The L-P transformation is a matrix where the matrix elements contain a truncated quasi-periodic Fourier series. Inverting a quasi-periodic matrix is not a trivial problem. In this section, we present two possible approaches to obtain the inverse L-P transformation.
Symbolic Computation:
In minimal cases, when the L-P transformation matrix (given by equation (15)) is small ( 2 2 ) and contains only a few terms, Symbolic computation software like Mathematica or Maple may be able to find the inverse. However, the inverse computed with this direct approach should be checked for the following conditions.
1 1 (0) ( ) ( ) t t Q I Q Q I (16)
The expression provided for
1 ( ) t Q
may need further simplification for ease in future use.
Neural Network: One can also use a dynamical method using a recurrent neural network proposed for inversion of the time-varying matrix. One could use the gradient method [34], Zhang dynamics [35][36][37], or Chen dynamics [38] to find an inverse. In this section, we briefly present the Zhang dynamics approach [39] that could be used for inverting the L-P transformation.
Consider a time-varying matrix ( ) t Y with inverse 1 ( ) ( ) t t W Y so that the equation (17) is valid ( ) ( ) ( ) ( ) t t t t Y W I Y W I 0(17)
We assume ( ) t Y is known and
( ) d t dt Y
exists. The objective is to find ( ) t W using the following equation
( ( ), ) ( ) ( ) t t t t E W Y W I (18) where ( ( ), ) t t E W
is a matrix-valued error function. The derivative of the error function ( ( ), )
t t E W should be selected such that ( ( ), ) t t E W 0 . Thus, ( ( ), ) t t E W can be chosen as ( ( ), ) ( ( ( ), )) d t t t t dt E W F E W(19)
where is a scaling factor for the convergence and ( ( ( ), )) t t F E W is called an activation function or matrix mapping recurrent neural network.
Differentiating equation (18) w.r.t. time and substituting equation (19) and (18) yields
( ) ( ) ( ) ( ) ( ( ( ), )) ( ) ( ) ( ) ( ) ( ( ) ( ) ) t t t t t t t t t t t t Y W Y W F E W Y W Y W F Y W I (20)
The equation (20) is a matrix differential equation that can be solved for
( ) t W using an appropriate initial condition. In the current paper ( ) t Y is the L-P transformation matrix ( ) t Q and ( ) t W is the inverse of L-P transformation 1 ( ) t Q
. Thus equation (20) can be written as
1 1 1 ( ) ( ) ( ) ( ) ( ( ) ( ) ) t t t t t t Q Q Q Q F Q Q I (21)
One has to select an appropriate activation function and scaling constant () F to achieve convergence.
The equation (21) can be numerically integrated with the initial condition
1 (0) Q I to determine 1 ( ) t Q
. For more details on the Zhang Neural Network, its application, and proof of convergence, we refer to reference [37].
ORDER REDUCTION TECHNIQUES
Order reduction via linear projection
Consider a nonlinear quasi-periodic system described by the equation (1). Applying the L-P transformation ( )
( ) ( ) t t t x Q z produces ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) t = t + t t t t t + ,t t -1 -1 z Jz f z F Jz w z F Q Q (22)
where J is the constant matrix and ( ) ,t w z represents an appropriately defined nonlinear quasi-periodic vector consisting of monomials of j z .
Again, the objective of order reduction is to replace the nonlinear quasi-periodic system given by equation (22) with an equivalent system provided by
( ) ( ) ( ) ( ) r r r r r r t = t + ,t t z J z w z F (23)
We partition equation (22) In the linear technique, the contribution of the non-dominant states is considered insignificant and hence neglected. Thus, the reduced-order model is given by
( ) ( ) ( 0, ) ( ) r r r r r r t = t + , t t z J z w z F (25)
The equation (25) is the reduced-order model of the actual large-scale system described by the equation (24). The equation (25) can be integrated numerically and using the transformation ( )
( ) ( ) r t t t x Q Tz , where ( ) T r r r n r T I 0
all the states in x can be recovered.
This linear projection technique is simple and easy to implement. It may or may not provide accurate results. The selection of dominant states depends upon the judgment of the analyst. It does not give a clear insight into system dynamics if the system dynamics are complex and involves internal and parametric resonance.
Order Reduction Using Invariant Manifold.
This methodology is based on the ' Invariant Manifold Theory.' According to this theory, there exists a relationship between the dominant "master" and the non-dominant "slave" states of the system, and it is possible to replace (under certain conditions) the non-dominant states with dominant states. Thus, the order of the system can be reduced.
We assume that the frequency of forcing is incommensurate with the frequency of quasi-periodic parametric excitation. The constraint (or manifold governing) equations relating 'master' and 'slave' states are considerably complex but admit the solution in the form of asymptotic expansion. The relationship between the dominant and the non-dominant states of the system will involve contributions from the forcing and nonlinearity. If there are no resonances, then it is possible to replace the non-dominant states with the dominant states.
Once again, consider a nonlinear quasi-periodic system given by the equation (22) in the L-P transformed domain that is further partitioned as the equation (24). After ordering and expanding the nonlinear terms, we obtain.
i i i si r s s t t t t t a t t t t t b z J z w z z w z z w z z w z z Ο z F z J z w z z w z z w z z w z z Ο z F (26)
where 1 ( , ) n rn t w z include the terms of monomials of order n in 'master' dynamics and 1 ( , ) n sn t w z include terms of monomials of order n in 'slave' dynamics. In this approach, we assume a nonlinear relationship between the dominant ( ) (27) Here ( , ) ij r t h z are the unknown quasi-periodic vector coefficients. Substitution of the equation (27) into (26)
t t t t t t t t t t t t t t t t t t t h h z h z w h h h h h h h h h h h h h h h F(29)
Substituting equation (26) and (27) into the equation (28) and equating it to the equation (29) yields a complex partial differential equation involving various orders of . By correlating the terms of the same order of , we obtain the equations, which need to be solved to determine
t t t h J h F (30)
The equation (30) is a linear equation involving pure temporal arguments. The solution of the equation (30) can be determined using the convolution integral [29,40] as
( ) 01 01 0 ( ) (0) ( ) s s t t t s t d J J h e h e F(31)
If the forcing ( ) t F is harmonic with frequency f k , then after the L-P transformation, the frequency of harmonic excitation ( ) t F becomes 1 2 ( )
p f p p =- k p ω , where p
ω is the vector containing quasiperiodic frequencies in the L-P transformation
1 2 1 2 { }, { } T p p p ω p .
Expressing forcing in the most general form as
1 2 1 2 ( ) ( ) p f i k t s p p k k p p =- t p ω F C e(32)
If the eigenvalues of s J are purely imaginary and given by ; 1, 2,...
( ) ( ) ( ) ( ) p f j i n k t i t s jp p k j jp p k j j p p k p f j p f j t C e C e i k i k e e h
p ω p ω (33) It can be seen that if
0 p f j k p ω
for any combination, 01 ( ) t h cannot be found out, and the system is said to be in 'linear resonance.' This resonance is referred to as a 'primary' or a 'main resonance' in perturbation analysis.
Collecting the terms at the order of 1 (36) It can be seen that equations (34), (35) and (36) are coupled equations. However, the equation (36) can be solved independently. Assuming the most general form of nonlinearity and expanding the known and unknown terms in multiple Fourier series as
yields 0 12 2 02 02 ( ) ( ) ( ) s s r r t t t h h J h F w z (34) m z ω p
Collecting the terms to solve for the unknowns yields the 'reducibility condition' given by the equation (39), (39) where all the terms appearing in the equation (39) are defined before.
1 ( ) ( ) 0 r l l p l i m p ω
In the absence of resonances 22 ( , ) r t h z
can be obtained. It should be noted at this stage that forcing frequency does not appear in the equation (39) , implying that there is no direct interaction between the nonlinearity and the external excitation. However, as we construct the solution using equations (34) and (35) forcing interacts with the nonlinearity, giving rise to additional 'resonance conditions.'
To find out the solution to the equation (35), which contains the contribution from the nonlinearity 2 ( , ) s r t w z , we expand the known and the unknown terms in the multiple Fourier series of the form
a h = i p ω (42)
The 'combined reducibility condition' can be expressed as
( ) 0 l p i p ω (43)
It can be observed that all the terms appearing in the equation (34) are free from spatial arguments, and it can be solved using the convolution, as discussed before. The forcing terms 20 , which can be written as
1 2 3 s p nl f p p p (45)
The exact combination will be determined by the kind of terms present in the forcing.
As we collect the terms at the order of 2 transformation. This solution is called nonlinear reduced-order system response. Similar to the linear reduced-order system analysis, the nonlinear reduced-order system response can be compared with the original system's response calculated via numerical integration of the equation (58). The time trace comparison is shown in Figures 5 and 6. Figure 7 compares phase planes for the original and the reduced-order system via the nonlinear technique. It can be noticed that the nonlinear reduced-order model captures the dynamics of the original system quite well. For additional insight, the Welch power spectrum for the original system response is compared with the Welch power spectrum for the nonlinearly reduced system in Figure 8. These power spectrums match, indicating that the original system's dynamics (frequency content) are captured in the nonlinearly reduced-order system. These symbolic computations were performed using Mathematica™.
i i C C i i
Welch Power Spectral Density Estimate
Numerical Solution OR nonlinear
,
, .... , ...
of z (of order i ) with quasi-periodic coefficients.
r z and the non-dominant ( )
Figure 1 :Figure 2 :Figure 3 :Figure 4 :
1234Time trace comparisons of original and the reduced-order system via the linear approach ( ) Time trace comparisons of original and the reduced-order system via the linear approach ( ) Phase plane comparisons of original and the reduced-order system via the linear approach( ) x t v/s ( )x t Welch Power Spectral density comparison of original and the reduced-order system via the linear approach
Figure 5 :Figure 6 :Figure 7 :Figure 8 :
5678Time trace comparisons of original and the reduced-order system via the nonlinear approach ( ) x t v/s time Time trace comparisons of original and the reduced-order system via the nonlinear approach Phase plane comparisons of original and the reduced-order system via the nonlinear approach( ) x t v/s ( )x t Welch Power Spectral density comparison of original and the reduced-order system via the nonlinear approach
as a constant matrix 0 B and a quasi-periodic matrix( ) t
B
of appropriate
dimensions. Typically, ( )
t
B
comprises of
1
( Cos( )
Sin( ))
n
i
i
i
i
i
a
t b
t
type terms where i
is the
frequency of quasi-periodic excitation (c.f. equation (4)). Assuming
Cos( )
i
i
q
t
and
Sin( )
i
i
p
t
the
equation (5) can be expressed as
0
( )
x B x f x
(6)
where
[ , , ]
T
x x p q ,
1
2
[
]
, ,... T
n
p p
p
p
,
1
2
[
]
, ,... T
n
q q q
q
and
Applying the modal transformation
x Mz , if 0
B has semi-simple eigenvalues, the equation (6) is
transformed into
1 ( )
z Jz M f z
(7)
where J is the Jordan form of 0
B (assumed to have semi-simple eigenvalues). The diagonal elements of
the J matrix contain the linear matrix's ( 0
B ) eigenvalues and frequencies of parametric excitations
1
[ ... ]
n
n m
that are incommensurate.
The equation (46) has only temporal arguments; the equation (47) is linear in spatial arguments; the equation (48) depends upon quadratic spatial arguments, and the equation (49) involves cubic spatial arguments. As before, one has to solve these equations sequentially.It can be observed that the equation (49) can be solved independently and involves contribution from . To solve this equation, we expand the known terms and unknown terms in the multiple Fourier series (c.f. equation(36))if the following' reducibility condition' is satisfied.is known, we can solve the equation(48). This equation contains terms arising from the , we obtain
0
30
0
2
01
02
01
3
13
12
2
2
3
03
03
[
( )]
( )
( )
[ ( )]
( )
( )
( )
( )
( )
( )
m
s
s
s
r
r
r
t
t
t
t
t
t
t
t
t
t
01
h
h
h
h
h
h
h
J h
F
w
w
w
z
z
(46)
0
31
1
01
2
01
2
02
01
12
13
13
23
22
2
2
3
13
[ ( )]
[ ( )]
( )
( )
( , )
( , )
( , )
( , )
r
r
r
m
s
s
r
r r
r
s
r
r
r
r
r
t
t
t
t
t
t
t
t
t
h
z
h
h
z h
h z
h
h
h
h
z
J z
F J h
z
z
w
w
w
z
z
z
(47)
32
2
01
12
01
22
23
23
33
2
3
23
( )
( , )
( )
( , )
( , )
( , )
r
r
r
r
s
s
r r
r
s
r
r
r
r
t
t
t
t
t
t
t
h
z
h z
z h
h z
h
h
h
J z
F J h
z
z
w
w
z
z
(48)
33
3
22
33
33
2
3
33
( , )
( , )
( , )
r
r
s
s
r r
s
r
r
r
t
t
t
t
h z
z
h
h J z J h
z
z
w
w
z
(49)
2 ( , )
s
r t
w z
denoted by
33
2 ( , )
s
r t
w
z
1
2
(
)
33
1
( , )
s
i
t
r
jm
j
r
p -
j=
p =-
t =
h
e
e
m
p ω
m
h z
z
(50)
3
1
2
(
)
3
1
( , )
s
i
t
s
r
jm
j
r
p
j=
p =
t =
a
e
e
m
p ω
m
z
z
w
(51)
1
2
1
2
1
1
2
1
2
... ,
1,
...
3;
{
},
{
}
r
m m
m
T
r
r
r
z z
z i
m
... m
p p
m
z
ω
p
It is possible to determine 33 ( , )
r t
h z
1
(
)
(
)
0
r
l l
p
l
i
m
p ω
(52)
Once 33 ( , )
r t
h z
product of
33 ( , ) ( )
r
r
r
t
t
h z
F
z
(where ( )
r t
F
3,
5,
2.5,
2
/ ,
7
/ ,
1
/ ,
1,
1
a
a
b b c c
rad s
rad s
rad s A
A
the 1
J , 2
J , 1
C and 2
C are given as
2
2
1
2
1
2
1.78
0
2.29
0
,
,
,
(3.5)
0
1.78
0
2.29
is the forcing on the master states), the contribution from As before, we can obtain 23 ( , ) r t h z via term-by-term comparison if and only if the following 'combined reducibility condition' is satisfied.It can be observed that this 'combined reducibility condition' involves a contribution from the forcing. ) andTo determine13 ( , )r t h z , we expand the known and the unknown terms in the multiple Fourier series of the form given by equation(40)and equation(41), respectively and obtain the 'combined reducibility condition' given by equation(43). 1, ;It is possible to continue the procedure discussed above to construct the relationship between 'slave' and 'master' states to the desired order and recover various 'resonance conditions' involving contributions from external excitation and nonlinearity at multiple orders.APPLICATIONSConsider a coupled undamped Mathieu Hill-type nonlinear quasi-periodic system subjected to external excitation given byThe equation (58) can be expressed asApplying the L-P transformation ( ) ( , ) ( )C and 2 C are constant depending upon the initial conditions of the fictitious states. For more details on the computation of J , we refer to reference[27].In this particular case,One can apply order reduction techniques discussed in section 3.a) Order reduction using the linear methodEquation(61)The equation(63)is the reduced-order model of the system described by the equation(61). This reduced-order system is integrated numerically with typical initial conditions, and all the states in x are obtained using the L-P ( )transformation. This solution is called linear reduced-order system response. This response can be compared with the original system's response calculated via numerical integration of the equation (58). The time trace comparison is shown inFigures 1 and 2.Figure 3compares phase planes for the original and the reduced-order system via the linear technique. It can be noticed that the linear reduced-order model fails to capture the dynamics of the original system. One reason for this failure is that the slave states are also excited by forcing ( ) s t F that is completely ignored in the reduced-order model. The linear order reduction technique may yield acceptable results when the eigenvalues corresponding to slave states have negative real parts or no forcing on slave states. However, in general, the linear order reduction approach for nonlinear quasi-periodic systems subjected to external excitation may not yield accurate results. For clarity, the Welch power spectrum for the original system response is compared with the Welch power spectrum for the linearly reduced system inFigure 4. These power spectrums do not match, indicating that the original system's dynamics (frequency content) are not captured in the linearly reduced-order system.b) Order reduction using an invariant manifoldAs discussed earlier, we try to relate the non-dominant states to the dominant states by a quasi-periodic nonlinear transformation. If the system does not exhibit any resonances (like the case under consideration), then the 'reducibility condition' is satisfied, and the system order can be reduced.We start with the equation (61) and select the same states [] as the dominant states and try to find a nonlinear quasi-periodic relationship of the form given by the equation(27). For this particular example, the relationship between s z and r z areThe equation(67)is the reduced-order model of the system described by the equation (61). As before, this reduced-order system is integrated numerically with typical initial conditions, and all the states in x are obtained using the L-P ( )DISCUSSION AND CONCLUSIONSThis paper presents a technique for obtaining a reduced-order model of a nonlinear quasi-periodic system subjected to external excitation. The central idea here is to assume a quasi-periodic transformation with unknown coefficients between the master and the slave states. This transformation can be determined by collecting the terms of the same order and solving them using harmonic balance.In the solution process, we obtain reducibility conditions that indicate resonances between system states, nonlinearity, and external excitation. The linear resonance condition is also obtained as we find the solution of quasi-periodic transformation. One crucial point here is how one can decide which states to retain and which ones to eliminate. Initially, one could start with the states corresponding to eigenvalues close to external excitation frequencies and start the order reduction process. The resonance interactions in the nonlinear quasi-periodic system are complex, and in the course of order reduction, one may see a "small deviser" problem. Such a case indicates resonant interaction, and these resonant states must be included in the master states. It can be noted that with the advent of symbolic software like Mathematica and Maple, the procedure for order reduction can be automated[41]. One can consider quasi-periodic and external excitation as fictitious states and carry out the order reduction. This approach is presented in reference[42], and further simplification via the method of form can be achieved as discussed in reference[43]for autonomous systems.The reduced-order model will contain all the essential dynamics and responses of the reduced-order system quantitatively and qualitatively, similar to the original system. This reduced-order system can be simplified using the method of normal forms. One can study this simplified system for bifurcation and control. The reduced-order system can be used for the optimization of essential parameters, study sensitivity, and design controllers.
Large Scale Systems Modeling. M S Mahmoud, M G Singh, Pergamon PressOxford, UK.M. S. Mahmoud and M. G. Singh, Large Scale Systems Modeling, Pergamon Press, Oxford, UK. 1981
Modal Analysis-Based Reduced-Order Models for Nonlinear Structures -An Invariant Manifold Approach. S W Shaw, C Pierre, E Pesheck, The Shock and Vibration Digest. 31S. W. Shaw, C. Pierre and E. Pesheck "Modal Analysis-Based Reduced-Order Models for Nonlinear Structures -An Invariant Manifold Approach.", The Shock and Vibration Digest, vol. 31(1), pp. 3-16, 1999.
Model Reduction and Nonlinear Normal Modes in Structural Dynamics. T D Burton, M E Young, ASME Winter Annual Meeting. Chicago, IL, ASMET. D. Burton and M. E. Young, "Model Reduction and Nonlinear Normal Modes in Structural Dynamics.", ASME Winter Annual Meeting, Chicago, IL, ASME, 1994.
The Reduction of Nonlinear Structural Dynamic Models. T D Burton, R W Rhee, Journal of Vibration and Control. 64T. D. Burton and R.W. Rhee, "The Reduction of Nonlinear Structural Dynamic Models.", Journal of Vibration and Control, vol. 6(4), pp. 531-556, 2000.
Order reduction of linear systems using an error minimization technique. S Mukherjee, R N Mishra, 10.1016/0016-0032(87)90037-80016-0032Journal of the Franklin Institute. 3231S. Mukherjee, R.N. Mishra, "Order reduction of linear systems using an error minimization technique," Journal of the Franklin Institute, Volume 323, Issue 1, Pages 23-32, ISSN 0016-0032, 1987 https://doi.org/10.1016/0016-0032(87)90037-8.
Order reduction of linear systems with an improved pole clustering. R Komarasamy, N Albhonso, G Gurusamy, Journal of Vibration and Control. 1812Komarasamy R, Albhonso N, Gurusamy G. "Order reduction of linear systems with an improved pole clustering." Journal of Vibration and Control.18(12):1876-1885, 2012
Reduction of transfer functions by the stability-equation method. T C Chen, C Y Chang, Han, Kw, Journal of Franklin Institute. 3084Chen, TC, Chang, CY, Han, KW "Reduction of transfer functions by the stability-equation method." Journal of Franklin Institute 308(4): 389-404,1979
Clustering method for reducing the order of linear system using a Pade approximation. C B Vishwakarma, Prasad , R " , IETE) Journal of Research. 545Institution of Electronics and Telecommunication EngineersVishwakarma, CB, Prasad, R "Clustering method for reducing the order of linear system using a Pade approximation". Institution of Electronics and Telecommunication Engineers (IETE) Journal of Research 54(5): 326-330.,2008
Model Order Reduction: Theory, Research Aspects and Applications. H A Wilhelmus, A Schildershenk, Van Der Vorstjoost Rommes, SpringerWilhelmus H. A. SchildersHenk A. van der VorstJoost Rommes, 'Model Order Reduction: Theory, Research Aspects and Applications', Springer 2008.
Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition. Boris Kramer, Karen E Willcox, AIAA Journal. 576Boris Kramer and Karen E. Willcox' Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition', AIAA Journal,57:6, 2297-2307,2019
A quadratic manifold for model order reduction of nonlinear structural dynamics. Shobhit Jain, Paolo Tiso, Johannes B Rutzmoser, Daniel J Rixen, 10.1016/j.compstruc.2017.04.0050045-7949Computers & Structures. 188Shobhit Jain, Paolo Tiso, Johannes B. Rutzmoser, Daniel J. Rixen, A quadratic manifold for model order reduction of nonlinear structural dynamics, Computers & Structures, Volume 188, Pages 80-94, ISSN 0045-7949, 2017 https://doi.org/10.1016/j.compstruc.2017.04.005.
A reduction method for nonlinear structural dynamic analysis. S R Idelsohn, A Cardona, 10.1016/0045-7825(85)90125-2Comput Methods Appl Mech Eng. 493S.R. Idelsohn, A. Cardona' A reduction method for nonlinear structural dynamic analysis.' Comput Methods Appl Mech Eng, 49 (3) , pp. 253-279, 10.1016/0045-7825(85)90125-2,1985
Order reduction of nonlinear systems with time-periodic coefficients using invariant manifolds. S C Sinha, Sangram Redkar, Eric A Butcher, 10.1016/j.jsv.2004.07.0270022-460XJournal of Sound and Vibration. 284Issues 3-5, Pages 985-1002S.C. Sinha, Sangram Redkar, Eric A. Butcher, Order reduction of nonlinear systems with time-periodic coefficients using invariant manifolds, Journal of Sound and Vibration, Volume 284, Issues 3-5, Pages 985-1002, 2005, ISSN 0022-460X, https://doi.org/10.1016/j.jsv.2004.07.027.
Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications. S C Sinha, 10.1007/s11071-005-2822-zNonlinear Dyn. 41Sinha S. C. et al. Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications. Nonlinear Dyn 41, 237-273 ,2005. https://doi.org/10.1007/s11071-005-2822-z
On macromodeling of nonlinear systems with timeperiodic coefficients. S C Sinha, Sangram Redkar, Eric A Butcher, 10.1016/j.cnsns.2004.11.0051007-5704Communications in Nonlinear Science and Numerical Simulation. 114S.C. Sinha, Sangram Redkar, Eric A. Butcher, On macromodeling of nonlinear systems with time- periodic coefficients, Communications in Nonlinear Science and Numerical Simulation, Volume 11, Issue 4, 2006, Pages 510-530, ISSN 1007-5704, https://doi.org/10.1016/j.cnsns.2004.11.005.
Reduced-Order Modeling of Parametrically Excited Micro-Electro-Mechanical Systems (MEMS). S Redkar, 10.1155/2010/632831Advances in Mechanical EngineeringRedkar S. Reduced-Order Modeling of Parametrically Excited Micro-Electro-Mechanical Systems (MEMS). Advances in Mechanical Engineering. January 2010. doi:10.1155/2010/632831
Reduced-order modeling of nonlinear time-periodic systems subjected to external periodic excitations. S C Sangram Redkar, Sinha, 10.1016/j.cnsns.2011.02.0021007-5704Communications in Nonlinear Science and Numerical Simulation. 1610Sangram Redkar, S.C. Sinha, Reduced-order modeling of nonlinear time-periodic systems subjected to external periodic excitations, Communications in Nonlinear Science and Numerical Simulation, Volume 16, Issue 10, 2011, Pages 4120-4133, ISSN 1007-5704, https://doi.org/10.1016/j.cnsns.2011.02.002.
Globally and locally attractive solutions for quasiperiodically forced systems. M V Bartuccelli, J H B Deane, G Gentile, J. Math. Anal. Appl. 328699Bartuccelli, M. V., Deane, J. H. B., and Gentile, G., "Globally and locally attractive solutions for quasi- periodically forced systems," J. Math. Anal. Appl. 328, 699,2007.
Lyapunov Stability of Quasiperiodic Systems. Sangram Redkar, 10.1155/2012/721382ID 721382Mathematical Problems in Engineering. 2012Sangram Redkar, "Lyapunov Stability of Quasiperiodic Systems," Mathematical Problems in Engineering, vol. 2012, Article ID 721382, 10 pages, 2012. https://doi.org/10.1155/2012/721382
Stability of a 2-dimensional Mathieu-type system with quasi-periodic coefficients. T J Waters, Nonlinear Dynamics. 603T. J. Waters, "Stability of a 2-dimensional Mathieu-type system with quasi-periodic coefficients," Nonlinear Dynamics, vol. 60, no. 3, pp. 341-356, 2010.
Reducibility and Analysis of Linear Quasi-Periodic Systems Via Normal Forms. P M B Waswa, S Redkar, 10.1115/1.4046899ASME. J. Comput. Nonlinear Dynam. 15991010Waswa, P. M. B., and Redkar, S. "Reducibility and Analysis of Linear Quasi-Periodic Systems Via Normal Forms." ASME. J. Comput. Nonlinear Dynam. September 2020; 15(9): 091010. https://doi.org/10.1115/1.4046899,
On the reducibility of linear quasi-periodic systems with Liouvillean basic frequencies and multiple eigenvalues. Dongfeng Zhang, Junxiang Xu, Hao Wu, Xindong Xu, Journal of Differential Equations. 26912Dongfeng Zhang, Junxiang Xu, Hao Wu, Xindong Xu, "On the reducibility of linear quasi-periodic systems with Liouvillean basic frequencies and multiple eigenvalues", Journal of Differential Equations, Volume 269, Issue 12, 2020,
On the effective reducibility of a class of Quasi-periodic nonlinear systems near the equilibrium, Dynamical Systems. Jia Li & Chunpeng, Zhu, 10.1080/14689367.2018.153736234Jia Li & Chunpeng Zhu, "On the effective reducibility of a class of Quasi-periodic nonlinear systems near the equilibrium, Dynamical Systems", 34:3, 385-398, 2019 DOI: 10.1080/14689367.2018.1537362
On the reducibility of linear differential equations with quasi-periodic coefficients. Àngel Jorba, Carles Simó, 10.1016/0022-0396(92)90107-X0022- 0396Journal of Differential Equations. 981Àngel Jorba, Carles Simó, "On the reducibility of linear differential equations with quasi-periodic coefficients", Journal of Differential Equations, Volume 98, Issue 1, 1992, Pages 111-124, ISSN 0022- 0396. https://doi.org/10.1016/0022-0396(92)90107-X.
Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. X Hou, J You, 10.1007/s00222-012-0379-2Invent. math. 190Hou, X., You, J. "Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems." Invent. math. 190, 209-260 (2012). https://doi.org/10.1007/s00222-012-0379-2
On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schrödinger Equation. Nina Xue, Wencai Zhao, 10.1155/2020/6260253ID 6260253Journal of Function Spaces. 2020Nina Xue, Wencai Zhao, "On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schrödinger Equation," Journal of Function Spaces, vol. 2020, Article ID 6260253, 11 pages, 2020. https://doi.org/10.1155/2020/6260253
Proceedings of the ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. S C Subramanian, S Redkar, P Waswa, 10.1115/DETC2020-2223016th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC). Virtual, Online. 2Lyapunov Perron Transformation for Linear Quasi-Periodic SystemsSubramanian, SC, Redkar, S, & Waswa, P. "Lyapunov Perron Transformation for Linear Quasi- Periodic Systems." Proceedings of the ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC). Virtual, Online. August 17-19, 2020. V002T02A003. ASME. https://doi.org/10.1115/DETC2020-22230
Transition curves for the quasi-periodic Mathieu equation. R S Zounes, R H Rand, SIAM Journal on Applied Mathematics. 584R. S. Zounes and R. H. Rand, "Transition curves for the quasi-periodic Mathieu equation," SIAM Journal on Applied Mathematics, vol. 58, no. 4, pp. 1094-1115, 1998.
A H Nayfeh, Perturbation Methods. John Wiley & SonsA. H. Nayfeh, Perturbation Methods, John Wiley & Sons., 2008.
Averaging methods in nonlinear dynamical systems. J A Sanders, V Ferdinand, M James, SpringerNew YorkJ. A. Sanders, V. Ferdinand, and M. James, Averaging methods in nonlinear dynamical systems, New York: Springer, 2007.
Method of Normal Forms. A H Nayfeh, Wiley-VCH2nd editionA. H. Nayfeh, Method of Normal Forms, Wiley-VCH; 2nd edition 2011.
Normal forms and unfoldings for local dynamical systems. J Murdock, Springer Science & Business MediaJ. Murdock, Normal forms and unfoldings for local dynamical systems, Springer Science & Business Media, 2006.
Comparison of Poincaré Normal Forms and Floquet Theory for Analysis of Linear Time Periodic Systems. S C Subramanian, S Redkar, 10.1115/1.4048715ASME. J. Comput. Nonlinear Dynam. 16114502Subramanian, S. C., and Redkar, S. "Comparison of Poincaré Normal Forms and Floquet Theory for Analysis of Linear Time Periodic Systems." ASME. J. Comput. Nonlinear Dynam.; 16(1): 014502, January 2021. https://doi.org/10.1115/1.4048715
A Novel Recurrent Neural Network-Based Ultra-Fast, Robust, and Scalable Solver for Inverting a "Time-Varying Matrix. Vahid Tavakkoli, 10.3390/s19184002Sensors. 194002Tavakkoli, Vahid et al. "A Novel Recurrent Neural Network-Based Ultra-Fast, Robust, and Scalable Solver for Inverting a "Time-Varying Matrix. " Sensors (Basel, Switzerland) vol. 19,18 4002. 16 Sep. 2019, doi:10.3390/s19184002
Design and analysis of a general recurrent neural network model for time-varying matrix inversion. Y Zhang, S Ge, 10.1109/TNN.2005.857946IEEE Trans. Neural Netw. 16Zhang Y., Ge S. "Design and analysis of a general recurrent neural network model for time-varying matrix inversion." IEEE Trans. Neural Netw. 2005;16:1477-1490. DOI: 10.1109/TNN.2005.857946.
Zhang neural network, Getz-Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots' kinematic control. D Guo, Y Zhang, 10.1016/j.neucom.2012.05.012Neurocomputing. 97Guo D., Zhang Y. "Zhang neural network, Getz-Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots' kinematic control." Neurocomputing. 2012;97:22-32. DOI: 10.1016/j.neucom.2012.05.012.
Comprehensive Analysis of A New Varying Parameter Zeroing Neural Network for Time-Varying Matrix Inversion. L Xiao, Y Zhang, J Dai, Q Zuo, S Wang, 10.1109/TII.2020.2989173IEEE Transactions on Industrial Informatics. L. Xiao, Y. Zhang, J. Dai, Q. Zuo, and S. Wang, "Comprehensive Analysis of A New Varying Parameter Zeroing Neural Network for Time-Varying Matrix Inversion," in IEEE Transactions on Industrial Informatics, DOI: 10.1109/TII.2020.2989173.
Recurrent implicit dynamics for online matrix inversion. Ke Chen, Appl. Math. Comput. 219Chen, Ke. "Recurrent implicit dynamics for online matrix inversion." Appl. Math. Comput. 219, (2013): 10218-10224.
Performance Analysis of Gradient Neural Network Exploited for Online Time-Varying Matrix Inversion. Y Zhang, K Chen, H Tan, 10.1109/TAC.2009.2023779IEEE Trans. Autom. Control. 54Zhang Y., Chen K., Tan H. "Performance Analysis of Gradient Neural Network Exploited for Online Time-Varying Matrix Inversion." IEEE Trans. Autom. Control. 2009;54:1940-1945. DOI: 10.1109/TAC.2009.2023779.
Modern Control Theory. W Brogan, PearsonW. Brogan Modern Control Theory, Pearson 1990.
Scientific Computing with Mathematica®: Mathematical Problems for Ordinary Differential Equations. M Addolorata, A Romano, Springer Science & Business Mediawith a CD-ROMM. Addolorata and A. Romano, Scientific Computing with Mathematica®: Mathematical Problems for Ordinary Differential Equations; with a CD-ROM, Springer Science & Business Media, 2001
A plain approach for center manifold reduction of nonlinear systems with external periodic excitations. P M Waswa, S Redkar, S C Subramanian, 10.1177/1077546319890213Journal of Vibration and Control. 26Waswa PM, Redkar S, Subramanian SC. "A plain approach for center manifold reduction of nonlinear systems with external periodic excitations." Journal of Vibration and Control. 2020;26(11- 12):929-940. doi:10.1177/1077546319890213
A direct approach for simplifying nonlinear systems with external periodic excitation using normal forms. P M Waswa, S Redkar, Nonlinear Dynamics. P. M. Waswa and S. Redkar, "A direct approach for simplifying nonlinear systems with external periodic excitation using normal forms," Nonlinear Dynamics, p. 1065-1088, 2020.
| [] |
[
"A theorem about two-body decay and its application for a doubly-charged boson H ±± going to τ ± τ ±",
"A theorem about two-body decay and its application for a doubly-charged boson H ±± going to τ ± τ ±"
] | [
"Li-Gang Xia \nDepartment of Physics\nTsinghua University\n100084BeijingPeople's Republic of China\n"
] | [
"Department of Physics\nTsinghua University\n100084BeijingPeople's Republic of China"
] | [] | In a general decay chain A → B1B2 → C1C2 . . ., we prove that the angular correlation function I(θ1, θ2, φ+) in the decay of B1,2 is irrelevant to the polarization of the mother particle A at production. This guarantees that we can use these angular distributions to determine the spin-parity nature of A without knowing its production details. As an example, we investigate the decay of a potential doubly-charged boson H ±± going to same-sign τ lepton pair. | null | [
"https://arxiv.org/pdf/1702.08186v2.pdf"
] | 119,532,236 | 1702.08186 | 21e307ddc3ccec804ab9b6a6fa7bffe2a68b129a |
A theorem about two-body decay and its application for a doubly-charged boson H ±± going to τ ± τ ±
Li-Gang Xia
Department of Physics
Tsinghua University
100084BeijingPeople's Republic of China
A theorem about two-body decay and its application for a doubly-charged boson H ±± going to τ ± τ ±
In a general decay chain A → B1B2 → C1C2 . . ., we prove that the angular correlation function I(θ1, θ2, φ+) in the decay of B1,2 is irrelevant to the polarization of the mother particle A at production. This guarantees that we can use these angular distributions to determine the spin-parity nature of A without knowing its production details. As an example, we investigate the decay of a potential doubly-charged boson H ±± going to same-sign τ lepton pair.
I. INTRODUCTION
After the discovery of the higgs boson h(125) [1,2], we are more and more interested in searching for high-mass particles, such as doubly-charged higgs bosons [3][4][5], denoted by H ±± . Once we observe any unknown particle, it is crucial to determine its spin-parity (J P ) nature to discriminate different theoretic models. A good means is to study the angular distributions in a decay chain where the unknown particle is involved [6][7][8][9][10]. For the Standard Model (SM) higgs, its spin-parity nature can be probed in the decay modes h(125) → W + W − /ZZ/τ + τ − [11][12][13][14][15]. The validity of this method relies on that the correlation of the decay planes of W/Z/τ does not depend upon the polarization of h(125) at production. This is proved in a general case in this paper. As an example, we also investigate the decay H ++ → τ + τ + , where the spin-statistic relation provides more interesting constraints as the final state is two identical fermions.
II. PROOF OF THE THEOREM
Let us consider a general decay chain A → B 1 B 2 with B 1 → C 1 X 1 and B 2 → C 2 X 2 , where B 1 and B 2 can be different particles and C 1 X 1 and C 2 X 2 can be different decay modes even if B 1 and B 2 are identical particles. Here we prove a theorem, which states that the angular correlation function I(θ 1 , θ 2 , φ + ) (defined in Eq. 9) in the decay of the daughter particles B 1,2 is independent upon the polarization of the mother particle A. Let φ + denote the angle between two decay planes B i → C i X i (i = 1, 2). Therefore, we can measure the φ + distribution to determine the spin-parity nature of the mother particle A without knowing its production details 1 .
Before calculating the amplitude, we introduce the definition of the coordinate system to describe the decay chain as illustrated in Fig. 1. For the decay A → B 1 B 2 , we take the flight direction of A as the +z axis (if it is still, we take its spin direction as the +z direction), denoted byẑ(A). θ and φ are the polar angle and azimuthal angle of B 1 in the center-of-mass (c.m.) frame of A. For the decay B 1 → C 1 X 1 , we take the flight direction of B 1 in the c.m. frame of A as the +z axis, denoted byẑ(B 1 ) and the direction ofẑ(A)×ẑ(B 1 ) as the +y axis, denoted byŷ(B 1 ). The +x axis in this decay system is then defined asŷ(B 1 ) ×ẑ(B 1 ). θ 1 and φ 1 are the polar angle and azimuthal angle of C 1 in the c.m. frame of B 1 . The same set of definitions holds for the decay B 2 → C 2 X 2 . φ + is defined in Eq. 1. It represents the angle between the two decay planes of B i → C i X i (i = 1, 2). Here φ 1 , φ 2 and φ + are constrained in the range [0, 2π).
φ + ≡ φ 1 + φ 2 , if φ 1 + φ 2 < 2π φ 1 + φ 2 − 2π , if φ 1 + φ 2 > 2π(1)
According to the helicity formalism developed by Jacob and Wick [16], the amplitude is
A= λ1,λ2 F J λ1λ2 D J * M,λ1−λ2 (Ω) × G j1 ρ1σ1 D j1 * λ1,ρ1−σ1 (Ω 1 ) ×G j2 ρ2σ2 D j2 * λ2,ρ2−σ2 (Ω 2 ) .(2)
Here the spin of A, B 1 and B 2 is J, j 1 and j 2 respectively. M is the third spin-component of A. The indices λ 1,2 , 1 After finishing this work, I was informed that the same statement had been verified in Ref. [6] in the case that B 1,2 are spin-1 particles and C 1,2 and X 1,2 are spin-1 2 particles. I also admit that it is of no difficulty to generalize it to any allowed spin values for B, C and X as shown in this work. 1. The definition of the coordinate system in the decay chain A → B1B2 with B1 → C1X1 and B2 → C2X2. The horizontal arrow represents the flight direction of the mother particle A. The red arrows represent the flight directions of B1,2 in the rest frame of A. The blue arrows represent the flight directions of C1,2 in the rest frame of B1,2 respectively. φ+ defined in Eq. 1 thus represents the angle between the decay plane of B1 and that of B2. ρ 1,2 and σ 1,2 denote the helicity of B 1,2 , C 1,2 and X 1,2 respectively.
) φ , θ ( A (A) z 1 B 1 C 1 X ) 1 (B z ) 1 (B x ) 1 (B y ) 1 φ , 1 θ ( 2 B 2 C 2 X ) 2 (B z ) 2 (B x ) 2 (B y ) 2 φ , 2 θ ( FIG.D J mn (Ω) ≡ D J mn (φ, θ, 0) = e −imφ d J mn (θ) and D J mn (d J mn ) is the Wigner D (d) function. F J λ1λ2
is the helicity amplitude for A → B 1 B 2 and defined as
F J λ1λ2 ≡ JM ; λ 1 , λ 2 |M|JM ,(3)
with M being the transition matrix derived from the S matrix. It is worthwhile to note that F J λ1λ2 does not rely on M because M is rotation-invariant. Similarly, G ji ρiσi is the helicity amplitude for B i → C i X i (i = 1, 2).
Taking the absolute square of A and summing over all possible initial and final states, the differential cross section can be written as
dσ dΩdΩ 1 dΩ 2 ∝ M,λ1,λ 1 ,λ2,λ 2 F J λ1λ2 F J * λ 1 λ 2 e i((λ1−λ 1 )φ1+(λ2−λ 2 )φ2) ×d J M,λ1−λ2 (θ)d J M,λ 1 −λ 2 (θ)f j1,j2 λ1λ 1 ;λ2λ 2 (θ 1 , θ 2 ) ,(4)
with f j1,j2
λ1λ 1 ;λ2λ 2 (θ 1 , θ 2 ) ≡ ρ1,σ1,ρ2,σ2 |G j1 ρ1σ1 | 2 |G j2 ρ2σ2 | 2 d j1 λ1,ρ1−σ1 (θ 1 )d j1 λ 1 ,ρ1−σ1 (θ 1 ) ×d j2 λ2,ρ2−σ2 (θ 2 )d j2 λ 2 ,ρ2−σ2 (θ 2 ) .(5)
Here the summation on M is over the polarization state of A at production. If we do not know the detailed production information, the summation cannot be performed.
Defining
δλ ( ) ≡ λ ( ) 1 − λ ( ) 2 , the exponential term in Eq. 4 is equivalent to e i[(λ1−λ 1 )φ+−(δλ−δλ )φ2] .
Performing the integration on φ 2 and using the definition of φ + , we have (keeping only the terms related with
φ 2 ) 2π 0 dφ 2 e i((λ1−λ 1 )φ1+(λ2−λ 2 )φ2) = φ+ 0 dφ 2 e i[(λ1−λ 1 )φ+−(δλ−δλ )φ2] + 2π φ+ dφ 2 e i[(λ1−λ 1 )(φ++2π)−(δλ−δλ )φ2] .(6)
Noting that (λ 1 −λ 1 ), δλ and δλ are integers, the integration gives the requirement δλ = δλ . Then the differential cross section in terms of λ
( ) 1 , δλ ( ) and φ + is λ1,λ 1 ,δλ F J λ1,λ1−δλ F J * λ 1 ,λ 1 −δλ e i(λ1−λ 1 )φ+ × M d J M,δλ (θ) 2 f j1,j2 λ1λ 1 ;λ1−δλ,λ 1 −δλ (θ 1 , θ 2 ) .(7)
According to the orthogonality relations of the Wigner D functions, we obtain
d J mn (θ) 2 d cos θ = 2 2J + 1 ,(8)
which is independent upon the indices m, n. Using this property, we find that integration over θ of the terms related with M in Eq. 7 only provides a constant factor M 2 2J+2 , which is irrelevant to the normalized angular distributions in the B 1,2 decays. So we finalize the proof of this theorem in Eq. 9.
I(θ 1 , θ 2 , φ + ) ≡ 1 σ dσ d cos θ 1 d cos θ 2 dφ + ∝ λ1,λ 1 ,δλ F J λ1,λ1−δλ F J * λ 1 ,λ 1 −δλ ×e i(λ1−λ 1 )φ+ f j1,j2 λ1λ 1 ;λ1−δλ,λ 1 −δλ (θ 1 , θ 2 ) .(9)
Experimentally, we are interested in the φ + distribution, which can be used to measure the spin-parity nature of A. We integrate out θ 1 and θ 2 and rewrite F J mn ≡ R J mn e iϕ J mn , where R J mn and ϕ J mn are real. The φ + distribution turns out to be
dσ σdφ + ∝ λ1,δλ R J λ1,λ1−δλ 2 F j1,j2 λ1λ1;λ1−δλ,λ1−δλ + λ1 =λ 1 δλ R J λ1,λ1−δλ R J λ 1 ,λ 1 −δλ F j1,j2 λ1λ 1 ;λ1−δλ,λ 1 −δλ × cos[(λ 1 − λ 1 )φ + + (ϕ J λ1,λ1−δλ − ϕ J λ 1 ,λ 1 −δλ )] ,(10)
with F j1,j2
λ1λ 1 ;λ2,λ 2 ≡ f j1,j2 λ1λ 1 ;λ2,λ 2 (θ 1 , θ 2 )d cos θ 1 d cos θ 2 . (11)
Here the second term in Eq. 10 is obtained using the fact that the summation is invariant with the exchange λ 1 ↔ λ 1 .
If the parity is conserved in the decay A → B 1 B 2 (namely, P −1 MP = M with P being the parity operator), we have
R J mn = P A P B1 P B2 (−1) J−j1−j2 R J −m,−n , ϕ J mn = ϕ J −m,−n ,(12)
where P A/B1/B2 is the parity of A/B 1 /B 2 and the factor −1 is absorbed in R J mn (namely, we require 0 ≤ ϕ J mn < π). Noting that the second summation in Eq. 10 is invariant with the index exchange (λ 1 , λ 1 , δλ) ↔ (−λ 1 , −λ 1 , −δλ), thus we have
λ1 =λ 1 δλ · · · = 1 2 λ1 =λ 1 δλ · · ·+ 1 2 −λ1 =−λ 1 −δλ · · · .(13)
Using the symmetry relation in Eq. 12, this summation turns out to be
1 2 λ1 =λ 1 δλ R J λ1,λ1−δλ R J λ 1 ,λ 1 −δλ × F j1,j2 λ1λ 1 ;λ1−δλ,λ 1 −δλ × cos[(λ 1 − λ 1 )φ + + (ϕ J λ1,λ1−δλ − ϕ J λ 1 ,λ 1 −δλ )] +F j1,j2 −λ1,−λ 1 ;−λ1+δλ,−λ 1 +δλ × cos[(λ 1 − λ 1 )φ + − (ϕ J λ1,λ1−δλ − ϕ J λ 1 ,λ 1 −δλ )] .(14)
Focusing on the expressions of Eq. 11 and Eq. 5, we are able to show that
F j1,j2 λ1λ 1 ;λ1−δλ,λ 1 −δλ = F j1,j2 −λ1,−λ 1 ;−λ1+δλ,−λ 1 +δλ ,(15)
using the following property of the Wigner d function
d j mn (π − θ) = (−1) j−n d j −m,n (θ) .(16)
With Eq. 14 and Eq. 15, Eq. 10 can be simplified as
dσ σdφ + ∝ λ1,δλ R J λ1,λ1−δλ 2 F j1j2 λ1λ1;λ1−δλ,λ1−δλ + λ1 =λ 1 δλ R J λ1,λ1−δλ R J λ 1 ,λ 1 −δλ F j1j2 λ1λ 1 ;λ1−δλ,λ 1 −δλ × cos(ϕ J λ1,λ1−δλ − ϕ J λ 1 ,λ 1 −δλ ) cos[(λ 1 − λ 1 )φ + ] ,(17)
This expression is actually the Fourier series for a 2πperiodic even function. Comparing Eq. 10 and Eq. 17, we can see that the terms which are odd with respective to φ + are forbidden due to parity conservation in the decay A → B 1 B 2 .
Now we consider the special case that B 1 and B 2 are identical particles and B 1,2 decay to the same final state, for example, we will study a doubly charged boson decay H ++ → τ + τ + → π + π +ν τντ . For identical particles, the state with the spin J and the third component M is
|JM ; λ 1 λ 2 S = |JM ; λ 1 λ 2 + (−1) J |JM ; λ 2 λ 1 ,(18)
which satisfies the spin-statistics relation. Here the normalization factor is omitted. The helicity amplitude F J λ1λ2 = S JM ; λ 1 λ 2 |M|JM has the symmetryF J λ1λ2 = (−1) J F J λ2λ1 . This symmetry relation will further constrain the helicity states, namely, the indices λ 1 , λ 1 and δλ in the summation in Eq. 9, 10 and 17.
III. STUDY OF H ++
→ τ + τ + → π + π +ν τντ
Ref. [17] is an example of the application of this theorem. It studies the decay Z → ZZ → l + l − l + l − , where B 1,2 are identical bosons. Here we consider the decay chain H ++ → τ + τ + → π + π +ν τντ . For two spin-1 2 identical fermions, we write down all states explicitly. The helicity index λ = + 1 2 ( −1 2 ) is denoted by R (L). The third state is already a parity eigenstate. The first two states can be combined to have a definite parity.
(1 + (−1) J )(|JM ; LL ± |JM ; RR ) , P = ∓1 (22)
In addition, the angular momentum conservation requires |λ 1 − λ 2 | ≤ J. Now we can give the selection rules, which are summarized in Table I. We can see that the states with odd spin and even parity are forbidden. For comparison, the selection rules for a neutral particle decaying to spin-1 2 fermion anti-fermion pair are summarized in Table II. In future electron-electron colliders, H −− may be produced in the process e − e − → H −− . However, the reaction rate for a spin-1 H −− will be highly suppressed because the vector coupling requires that both electrons have the same handness while the only allowed state is |LR − |RL . Similarly, the production rate for a scalar H −− is also highly suppressed. This is called "helicity suppression". Replacing A, B 1,2 and C 1,2 by H ++ , τ + and π + respectively in Eq. 2, the amplitude is
A =G 1 2 0 1 2 G 1 2 0 1 2 e iM φ F J RR d J M 0 (θ)e i( 1 2 φ1+ 1 2 φ2) sin θ 1 2 sin θ 2 2 + F J LL d J M 0 (θ)e −i( 1 2 φ1+ 1 2 φ2) cos θ 1 2 cos θ 2 2 − F J LR d J M,−1 (θ)e i(− 1 2 φ1+ 1 2 φ2) cos θ 1 2 sin θ 2 2 − F J RL d J M,1 (θ)e i( 1 2 φ1− 1 2 φ2) sin θ 1 2 cos θ 2 2 .(23)
Here we have only one decay helicity amplitude, G the τ + decay. This is because π + is a pseudo-scalar and ν τ is right-handed. The angular correlation function is I(θ 1 , θ 2 , φ + ) ∝1 + cos θ 1 cos θ 2 , for odd J I(θ 1 , θ 2 , φ + ) ∝1 + a 2 J + (1 − a 2 J ) cos θ 1 cos θ 2 −P H sin θ 1 sin θ 2 cos φ + , for even J (24)
Here for even J, a J is defined as a J ≡ |F J LR |/|F J RR |. P H is the parity of H ++ . We can see that the polarization information of H ++ does not appear in the angular distributions. The φ + distribution is
dσ σdφ + ∝ 1 for odd J 1 − P H π 2 16 1 1+a 2 J cos φ + for even J .
The φ + distributions for different J P s are shown in Fig. 2, where a J = 1 is assumed for illustration.
[rad]
+ φ 0 /2 π π /2 π 3 π 2
Angular distribution 0 0.5 Here are a few conclusions.
1. The φ + distribution is uniform for odd J.
2. For J = 0, the helicity amplitudes F J LR and F J RL are forbidden due to angular momentum conservation. Thus a J = 0 and the φ + distribution becomes
dσ σdφ + ∝ 1 − P H π 2 16 cos φ + ,(25)
which is the same as that in the decay h(125) → τ + τ − → π + π − ν τντ .
3. For nonzero even J, the φ + distribution depends upon J through the amplitude ratio a J .
Experimentally, it is difficult to reconstruct the τ lepton information due to the invisible neutrinos [18,19]. But we are able to obtain the decay plane angle φ + in some ways (see a most recent review Ref. [20] and references therein). The so-called impact parameter method [21] is suitable for the decay τ + → π +ν τ studied here. It requires that final π + s have significant impact parameters, which condition can be satisfied at high-energy colliders such as the Large Hadron Collider (LHC).
IV. CONCLUSIONS
In summary, for a general decay chain A → B 1 B 2 → C 1 C 2 . . ., we have proved that the angular correlation function I(θ 1 , θ 2 , φ + ) in the decay of the daughter particles B 1,2 is independent upon the polarization of the mother particle A at production. It guarantees that the spin-parity nature of the mother particle A can be determined by measuring the angular correlation of the two decay planes B i → C i . . . (i = 1, 2) without knowing its production details. This theorem has a simple form if the parity is conserved in the decay A → B 1 B 2 . Taking a potential doubly-charged particle decay H ++ → τ + τ + as example, we present the selection rules for various spin-parity combinations. It is found that this decay is forbidden for the H ++ with odd spin and even parity. Furthermore, we show that the angle between the two τ decay plans is an effective observable to determine the spin-parity nature of H ++ .
V. ACKNOWLEDGEMENT
Li-Gang Xia would like to thank Fang Dai for many helpful discussions. The author is also indebted to Yuan-Ning Gao for enlightening discussions. This work is supported by the General Financial Grant from the China Postdoctoral Science Foundation (Grant No. 2015M581062).
|JM ; LL S =(1 + (−1) J )|JM ; LL P|JM ; LL S = −|JM ; RR S (19) |JM ; RR S =(1 + (−1) J )|JM ; RR P|JM ; RR S = −|JM ; LL S (20) |JM ; LR S =|JM ; LR + (−1) J |JM ; RL P|JM ; LR S = −|JM ; LR S (21)
FIG. 2 .
2The φ+ distributions for different J P s. The black line represents odd J. The red solid (dashed) curve represents J P = 0 + (0 − ). The green solid (dashed) curve represents even J > 0 with even (odd) parity assuming aJ = 1.
arXiv:1702.08186v2 [hep-ph] 19 May 2017
TABLE I .
ISelection rules for a particle decaying to two spin-12
identical fermions.
Parity
J = 0
J = 2, 4, 6, . . .
J = 1, 3, 5, . . .
even
|LL − |RR
|LL − |RR
forbidden
odd
|LL + |RR
|LL + |RR
|LR − |RL
|LR + |RL
TABLE II. Selection rules for a particle decaying to spin-1
2
fermion anti-fermion pair.
Parity
J = 0
J = 2, 4, 6, . . .
J = 1, 3, 5, . . .
even
|LL + |RR
|LL + |RR
|LL − |RR
|LR + |RL
|LR − |RL
odd
|LL − |RR
|LL − |RR
|LL + |RR
|LR − |RL
|LR + |RL
. G Aad, ATLAS CollaborationPhys. Lett. B. 7161G. Aad et al., ATLAS Collaboration, Phys. Lett. B 716, 1 (2012).
. S Chatrchyan, CMS CollaborationPhys. Lett. B. 71630S. Chatrchyan et al., CMS Collaboration, Phys. Lett. B 716, 30 (2012).
. G Aad, ATLAS CollaborationJHEP. 150341G. Aad et al., ATLAS Collaboration, JHEP 1503, 041 (2015).
. G Aad, ATLAS CollaborationEur. Phys. J. C. 722244G. Aad et al., ATLAS Collaboration, Eur. Phys. J. C, 72, 2244 (2012).
. S Chatrchyan, CMS CollaborationEur.Phys.J.C. 722189S. Chatrchyan et al., CMS Collaboration, Eur.Phys.J.C, 72, 2189 (2012).
. J R Dell'aquila, C A Nelson, Phys. Rev. D. 3380J. R. Dell'Aquila and C. A. Nelson, Phys. Rev. D 33, 80 (1986).
. J R Dell'aquila, C A Nelson, Phys. Rev. D. 3393J. R. Dell'Aquila and C. A. Nelson, Phys. Rev. D 33, 93 (1986).
. J R Dell'aquila, C A Nelson, Phys. Rev. D. 33101J. R. Dell'Aquila and C. A. Nelson, Phys. Rev. D 33, 101 (1986).
. M R Buckley, H Murayama, W Klemm, V Rentala, Phys. Rev. D. 7814028M. R. Buckley, H. Murayama, W. Klemm, and V. Rentala, Phys. Rev. D 78, 014028 (2008).
. F Boudjema, R K Singh, JHEP. 090728F. Boudjema and R. K. Singh, JHEP 0907, 028 (2009).
. G Aad, ATLAS CollaborationEur. Phys. J. C. 75152Eur. Phys. J. CG. Aad et al., ATLAS Collaboration, Eur. Phys. J. C, 75, 476 (2015), Eur. Phys. J. C, 76, 152 (2016).
. G Aad, ATLAS CollaborationEur. Phys. J. C. 75231G. Aad et al., ATLAS Collaboration, Eur. Phys. J. C, 75, 231 (2015).
. G Aad, ATLAS CollaborationPhys. Lett. B. 726120G. Aad et al., ATLAS Collaboration, Phys. Lett. B, 726, 120 (2013).
. V Khachatryan, CMS CollaborationPhys. Rev. D. 9212004V. Khachatryan et al., CMS Collaboration, Phys. Rev. D 92, 012004 (2012).
. S Chatrchyan, CMS CollaborationPhys. Rev. Lett. 11081803S. Chatrchyan et al., CMS Collaboration, Phys. Rev. Lett. 110, 081803 (2013).
. M Jacob, G C Wick, Annals Phys. 281M. Jacob and G.C. Wick, Annals Phys. 281, 774-799 (2000).
. W.-Y Keung, I Low, J Shu, Phys. Rev. Lett. 10191802W.-Y. Keung, I. Low, and J. Shu, Phys. Rev. Lett. 101, 091802 (2008).
. A Elagin, P Murat, A Pranko, A Safonov, Nucl. Instrum. Meth. A. 654481A. Elagin, P. Murat, A. Pranko, and A. Safonov, Nucl. Instrum. Meth. A 654, 481 (2011).
. Li-Gang Xia, Chin. Phys. C. 40113003Li-Gang Xia, Chin. Phys. C 40, 113003 (2016).
. S Berge, W Bernreuther, S Kirchner, Phys. Rev. D. 9296012S. Berge, W. Bernreuther, and S. Kirchner, Phys. Rev. D 92, 096012 (2015).
. S Berge, W Bernreuther, Phys. Lett. B. 671470S. Berge and W. Bernreuther, Phys. Lett. B 671, 470 (2009).
| [] |
[
"Reduced basis methods-an application to variational discretization of parametrized elliptic optimal control problems",
"Reduced basis methods-an application to variational discretization of parametrized elliptic optimal control problems"
] | [
"Ahmad Ahmad Ali ",
"Michael Hinze "
] | [] | [] | We consider a class of parameter-dependent optimal control problems of elliptic PDEs with constraints of general type on the control variable. Applying the concept of variational discretization, [4], together with techniques from the Reduced basis method, we construct a reduced basis surrogate model for the control problem. We establish estimators for the greedy sampling procedure which only involve the residuals of the state and the adjoint equation, but not of the gradient equation of the optimality system. The estimators are sharp up to a constant, i.e. they are equivalent to the approximation erros in control, state, and adjoint state. Numerical experiments show the performance of our approach. | 10.1137/18m1227147 | [
"https://arxiv.org/pdf/1808.05687v1.pdf"
] | 119,134,571 | 1808.05687 | 6e6e342d8ffe2c28cdf535c7f8bd3d0144988dc7 |
Reduced basis methods-an application to variational discretization of parametrized elliptic optimal control problems
Ahmad Ahmad Ali
Michael Hinze
Reduced basis methods-an application to variational discretization of parametrized elliptic optimal control problems
We consider a class of parameter-dependent optimal control problems of elliptic PDEs with constraints of general type on the control variable. Applying the concept of variational discretization, [4], together with techniques from the Reduced basis method, we construct a reduced basis surrogate model for the control problem. We establish estimators for the greedy sampling procedure which only involve the residuals of the state and the adjoint equation, but not of the gradient equation of the optimality system. The estimators are sharp up to a constant, i.e. they are equivalent to the approximation erros in control, state, and adjoint state. Numerical experiments show the performance of our approach.
Introduction
The research in this work is motivated by the reduced basis approaches of [1] applied to approximate the solution manifold of the parameter dependent control constrained optimal control problem (1). The approach taken there uses a fully discrete treatment of the optimal control problem (1), so that the constructed a posteriori error estimators involve the residuals of the state, of the adjoint and of the gradient equation of the corresponding optimality conditions. Since the gradient equation in the control constrained case is nonsmooth one expects large contributions of the control residual in the estimation process. Our approach uses variational discretization [4] of (1) which avoids explicit discretization of the control variable, see problem (7). This approach then allows us to construct reliable and effective a posteriori error bounds only involving the residuals of the state and the adjoint state, respectively, see Theorem 4.2. Moreover, in Corollary 4.3 we propose an estimator for the relative error in the controls which only involves the residuals of the state and the adjoint state. We test our approach at the numerical examples presented in [1]. It is one important result of our work that the reduced basis spaces constructed with our approach for a given error level have much smaller dimensions than the respective spaces constructed with the approaches of [1]. In the present work we focus on the approximation quality of the reduced spaces constructed with our approach from the dimensionality point of view. We do not discuss questions related to offline-online decomposition in our approach.
We note that our numerical analysis related to the error equivalence of Theorem 4.2 is motivated by techniques frequently used in the convergence analysis of adaptive finite element methods for optimal control problems, see e.g. [2]. For excellent introductions to the reduced basis method for approximations of parameter dependent elliptic PDEs we refer the reader to [3,6]. For a discussion of reduced basis approaches to approximate parameter dependent optimal control problems we refer the reader to [1], where also further literature can be found, and also detailed discussions related to offlineonline decomposition in the numerical implementation are provided.
General setting
Let P ⊂ R p , p ∈ N, be a compact set of parameters, and for a given parameter µ ∈ P we consider the variational discrete ( [4]) control problem (P) min (u,y)∈U ad ×Y J(u, y) := 1 2 y − z 2 L 2 (Ω 0 ) + α 2 u 2 U subject to (1) a(y, v; µ) = b(u, v; µ)
+ f (v; µ) ∀ v ∈ Y.(2)
Here (2) represents a finite element discrete elliptic PDE in a bounded domain Ω ⊂ R d for d ∈ {1, 2, 3} with boundary ∂Ω. Y denotes the space of piecewise linear and continuous finite elements. We assume the approximation process is conforming. The space Y is equipped by the inner product (·, ·) Y and the norm · Y := (·, ·) Y , in addition, there exist constants ρ 1 , ρ 2 > 0 such that there holds
ρ 1 y H 1 (Ω) ≤ y Y ≤ ρ 2 y H 1 (Ω) ∀ y ∈ Y,(3)
with · H 1 (Ω) being the norm of the classical Sobolev space H 1 (Ω).
The controls are from a real Hilbert space U equipped by the inner product (·, ·) U and the norm · U := (·, ·) U , and the set of admissible controls U ad ⊆ U is assumed to be non-empty, closed and convex.
We denote by Ω 0 ⊆ Ω an open subset, and L 2 (Ω 0 ) the classical Lebesgue space endowed with the standard inner product (·, ·) L 2 (Ω 0 ) and the norm · L 2 (Ω 0 ) := (·, ·) L 2 (Ω 0 ) .The desired state z ∈ L 2 (Ω 0 ) and the parameter α > 0 are given data.
The parameter dependent bilinear form a(·,
·; µ) : Y × Y → R is contin- uous γ(µ) := sup y,v∈Y \{0} |a(y, v; µ)| y Y v Y ≤ γ 0 < ∞ ∀ µ ∈ P,
and coercive
β(µ) := inf y∈Y \{0} a(y, y; µ) y 2 Y ≥ β 0 > 0 ∀ µ ∈ P,
where γ 0 and β 0 are real numbers independent of µ. The parameter depen-
dent bilinear form b(·, ·; µ) : U × Y → R is continuous κ(µ) := sup (u,v)∈U ×Y \{(0,0)} |b(u, v; µ)| u U v Y ≤ κ 0 < ∞ ∀ µ ∈ P,
where κ 0 is a real number independent of µ. Finally, f (·; µ) ∈ Y * is a parameter dependent linear form, where Y * denotes the topological dual of Y with norm · Y * defined by
l(·; µ) Y * := sup v Y =1 l(v; µ),
for a give functional l(·; µ) ∈ Y * depending on the parameter µ. We assume that there exists a constant σ 0 independent of µ such that
sup v∈Y \{0} |f (v; µ)| v Y ≤ σ 0 < ∞ ∀ µ ∈ P.
We find it convenient to introduce here for the upcoming analysis the Riesz isomorphism R : Y * → Y which is defined for a given f ∈ Y * by the unique element Rf ∈ Y such that
f (v) = (Rf, v) Y ∀ v ∈ Y.
Under the previous assumptions one can verify that the problem (P) admits a unique solution for every µ ∈ P. The corresponding first order necessary conditions, which are also sufficient in this case, are stated in the next result. For the proof see for instance [5,Chapter 3]. Theorem 2.1 Let u ∈ U ad be the solution of (P) for a given µ ∈ P. Then there exist a state y ∈ Y and an adjoint state p ∈ Y such that there holds
a(y, v; µ) = b(u, v; µ) + f (v; µ) ∀ v ∈ Y, (4) a(v, p; µ) = (y − z, v) L 2 (Ω 0 ) ∀ v ∈ Y, (5) b(v − u, p; µ) + α(u, v − u) U ≥ 0 ∀ v ∈ U ad .(6)
The varying parameter µ in the state equation (2) could represent physical or/and geometrical quantities, like diffusion or convection speed, or the width of the spacial domain Ω. Considering the problem (P) in the context of realtime or multi-query scenarios can be very costly when the dimension of the finite element space Y is very high. In this work we adopt the reduced basis method, see for instance [3], to obtain a surrogate for (P) that is relatively cheaper to solve and at the same time delivers acceptable approximation for the solution of (P) at a given µ. To this end, we first define a reduced problem for (P), and establish a posterior error estimators that predict the expecting approximation error when using the reduced problem. Then, we apply a greedy procedure (see Algorithm 1) to improve the approximation quality of the reduced problem.
The reduced problem and the greedy procedure
Let Y N ⊂ Y be a finite dimensional subspace. We define the reduced counterpart of the problem (P) for a given µ ∈ P by
(P N ) min (u,y N )∈U ad ×Y N J(u, y N ) := 1 2 y N − z 2 L 2 (Ω 0 ) + α 2 u 2 U subject to (7) a(y N , v N ; µ) = b(u, v N ; µ) + f (v N ; µ) ∀ v N ∈ Y N .(8)
We point out that in (P N ) the controls are still sought in U ad . In a similar way to (P), one can show that (P N ) has a unique solution for a given µ, and it satisfies the following optimality conditions. Theorem 3.1 Let u N ∈ U ad be the solution of (P N ) for a given µ ∈ P. Then there exist a state y N ∈ Y N and an adjoint state p N ∈ Y N such that there holds
a(y N , v N ; µ) = b(u N , v N ; µ) + f (v N ; µ) ∀ v N ∈ Y N , (9) a(v N , p N ; µ) = (y N − z, v N ) L 2 (Ω 0 ) ∀ v N ∈ Y N , (10) b(v − u N , p N ; µ) + α(u N , v − u N ) U ≥ 0 ∀ v ∈ U ad .(11)
The space Y N shall be constructed successively using the following greedy procedure.
Algorithm 1 (Greedy procedure) 1. Choose S train ⊂ P, µ 1 ∈ S train arbitrary, ε tol > 0, and N max ∈ N.
2. Set N = 1, Φ 1 := {y(µ 1 ), p(µ 1 )}, and Y 1 := span(Φ 1 ).
3. while max µ∈S train ∆(Y N , µ) > ε tol and N ≤ N max do 4. µ N +1 := arg max µ∈S train ∆(Y N , µ) 5. Φ N +1 := Φ N ∪ {y(µ N +1 ), p(µ N +1 )} 6. Y N +1 := span(Φ N +1 ) 7. N ← N + 1 8. end while
Here S train ⊂ P is a finite subset, called a training set, which assumed to be rich enough in parameters to well represent P. N max is the maximum number of iterations, and ε tol is a given error tolerance. In the iteration of index N , the pair {y(µ N ), p(µ N )} is the optimal state and adjoint state, respectively, corresponding to the problem (P) at µ N , and Φ N is the reduced basis which assumed to be orthonormal. If it is not, one can apply an orthonormalization process like Gram-Schmidt. An orthonormal reduced basis guarantees algebraic stability when N increases, see [3]. The quantity ∆(Y N , µ) is an estimator for the expected error in approximating the solution of (P) by the one of (P N ) for a given µ when using the space Y N . The maximum of ∆(Y N , µ) over S train is obtained by linear search.
We note that at line 5 in the previous algorithm one should implement a condition testing if the dimension of Φ N +1 differs from the one of Φ N . If it does not, the while loop should be terminated.
One choice for ∆(Y N , µ) could be
∆(Y N , µ) := u(µ) − u N (µ) U ,
i.e. the error between the solution of (P) and of (P N ). However, considering this choice in a linear search process over a very large training set S train is computationally too costly, since the solution of the highly dimensional problem (P) is needed. In the next section we establish a choice for ∆(Y N , µ) that does not depend on the solution of (P).
A posteriori error analysis
We start by associating to the solution (u N , y N , p N ) of (9)-(11) at a given µ ∈ P the functionỹ ∈ Y that satisfies
a(ỹ, v; µ) = b(u N , v; µ) + f (v; µ) ∀ v ∈ Y,(12)
and the functionp ∈ Y such that
a(v,p; µ) = (y N − z, v) L 2 (Ω 0 ) ∀ v ∈ Y.(13)
Furthermore, we introduce the linear form r y (·; µ) ∈ Y * defined by
r y (v; µ) := b(u N , v; µ) + f (v; µ) − a(y N , v; µ) ∀ v ∈ Y, and r p (·; µ) ∈ Y * by r p (v; µ) := (y N − z, v) L 2 (Ω 0 ) − a(v, p N ; µ) ∀ v ∈ Y.
We provide some estimates forỹ andp that will be utilized in the upcoming analysis.
Lemma 4.1 Suppose that (u, y, p) is the solution of (4)-(6), and (u N , y N , p N ) the solution of (9)-(11). Letỹ,p be as defined in (12), (13), respectively. Then there holds
y −ỹ Y ≤ κ(µ) β(µ) u − u N U ,(14)p −p Y ≤ 1 ρ 2 1 β(µ) y − y N Y ,(15)1 γ(µ) r y (·; µ) Y * ≤ ỹ − y N Y ≤ 1 β(µ) r y (·; µ) Y * (16) 1 γ(µ) r p (·; µ) Y * ≤ p − p N Y ≤ 1 β(µ) r p (·; µ) Y *(17)
Proof. The proof is divided into four parts for clarity. In part (III) and (IV) of the proof we shall apply the estimating techniques from [3] for linear elliptic PDEs.
(I) Estimating y −ỹ Y : Using the coercivity of a, the continuity of b, together with (4) and (12) gives
β(µ) y −ỹ 2 Y ≤ a(y −ỹ, y −ỹ; µ) = b(u − u N , y −ỹ; µ) ≤ κ(µ) u − u N U y −ỹ Y ,
from which (14) follows after dividing both sides by β(µ) y −ỹ Y .
(II) Estimating p −p Y : Similarly, but this time with (5) and (13) we have
β(µ) p −p 2 Y ≤ a(p −p, p −p; µ) = (y − y N , p −p) L 2 (Ω 0 ) ≤ y − y N H 1 (Ω) p −p H 1 (Ω) ≤ 1 ρ 2 1 y − y N Y p −p Y ,
where we used (3). Dividing both sides by β(µ) p −p Y gives (15).
(III) Estimating ỹ − y N Y : From the coercivity of a and the definition of r y , we have
β(µ) ỹ − y N 2 Y ≤ a(ỹ − y N ,ỹ − y N ; µ) = a(ỹ,ỹ − y N ; µ) − a(y N ,ỹ − y N ; µ) = b(u N ,ỹ − y N ; µ) + f (ỹ − y N ; µ) − a(y N ,ỹ − y N ; µ) = r y (ỹ − y N ; µ) ≤ r y (·; µ) Y * ỹ − y N Y ,
which gives the upper bound in (16) after dividing both sides by β(µ) ỹ − y N Y . On the other hand, let v := Rr y (·; µ) be the Riesz representative of r y (·; µ). Then using the continuity of a it follows that
r y (·; µ) 2 Y * = v 2 Y = (v, v) Y = r y (v; µ) = a(ỹ − y N , v; µ) ≤ γ(µ) ỹ − y N Y v Y = γ(µ) ỹ − y N Y r y (·; µ) Y * .
Dividing both sides of the previous inequality by γ(µ) r y (·; µ) Y * yields the lower bound in (16).
(IV) Estimating p − p N Y : From the coercivity of a and the definition of r p we have
β(µ) p − p N 2 Y ≤ a(p − p N ,p − p N ; µ) = a(p − p N ,p; µ) − a(p − p N , p N ; µ) = (y N − z,p − p N ) L 2 (Ω 0 ) − a(p − p N , p N ; µ) = r p (p − p N ; µ) ≤ r p (·; µ) Y * p − p N Y ,
from which we deduce the upper bound in (17) after dividing both sides by β(µ) p − p N Y . On the other hand, let v := Rr p (·; µ) be the Riesz representative of r p (·; µ). Then using the continuity of a we get
r p (·; µ) 2 Y * = v 2 Y = (v, v) Y = r p (v; µ) = a(v,p − p N ; µ) ≤ γ(µ) p − p N Y v Y = γ(µ) p − p N Y r p (·; µ) Y * ,
Dividing both sides of the previous inequality by γ(µ) r p (·; µ) Y * gives the lower bound in (17). This completes the proof.
We now state our main result. It provides a posteriori estimator for the error in approximating the solution of (P) by the one of (P N ). The estimator is sharp up to a constant.
Theorem 4.2 Suppose that (u, y, p) is the solution of (4)-(6), and (u N , y N , p N ) the solution of (9)-(11). Then there holds
δ uyp (µ) ≤ u − u N U + y − y N Y + p − p N Y ≤ ∆ uyp (µ), where ∆ uyp (µ) :=c 1 (µ) r y (·; µ) Y * + c 2 (µ) r p (·; µ) Y * , δ uyp (µ) :=c 3 (µ) r y (·; µ) Y * + c 4 (µ) r p (·; µ) Y * , c 1 (µ) := 1 β(µ) 1 ρ 1 √ α + 1 + 1 ρ 2 1 β(µ) κ(µ) β(µ)ρ 1 √ α + 1 , c 2 (µ) := 1 β(µ) κ(µ) α + κ(µ) 2 β(µ)α + κ(µ) 2 ρ 2 1 β 2 (µ)α + 1 , c 3 (µ) := 1 2γ(µ) max κ(µ) β(µ) , 1 −1 , c 4 (µ) := 1 2γ(µ) max 1 ρ 2 1 β(µ) , 1 −1 .
Proof. The proof falls into two parts, and we shall follow the ideas of [2, Theorem 3.2] for adaptive finite element method for elliptic control problems. (6), and v := u in (11), and adding the resulting inequalities, we get
(I) Establishing an upper bound for u − u N U + y − y N Y + p − p N Y : Taking v := u N inα u − u N 2 U ≤ b(u N − u, p − p N ; µ) = b(u N − u, p −p; µ) + b(u N − u,p − p N ; µ) =: S 1 + S 2 .(18)
Recalling (3), an upper bound for S 1 can be obtained as follows.
S 1 = b(u N − u, p −p; µ) = a(ỹ − y, p −p; µ) = (y − y N ,ỹ − y) L 2 (Ω 0 ) = (y − y N ,ỹ − y N ) L 2 (Ω 0 ) − y − y N 2 L 2 (Ω 0 ) ≤ 1 2 ỹ − y N 2 L 2 (Ω 0 ) ≤ 1 2 ỹ − y N 2 H 1 (Ω) ≤ 1 2ρ 2 1 ỹ − y N 2 Y ,
On the other hand, for S 2 we have
S 2 = b(u N − u,p − p N ; µ) ≤ α 2 u N − u 2 U + 1 2α κ(µ) 2 p − p N 2 Y .
Using the bounds of S 1 and S 2 in (18) yields
u − u N U ≤ 1 ρ 1 √ α ỹ − y N Y + 1 α κ(µ) p − p N Y .(19)
Applying the triangle inequality, (14), together with (19) results in
y − y N Y ≤ y −ỹ Y + ỹ − y N Y ≤ κ(µ) β(µ) u − u N U + ỹ − y N Y ≤ κ(µ) β(µ)ρ 1 √ α + 1 ỹ − y N Y + κ(µ) 2 β(µ)α p − p N Y .(20)
Again the triangle inequality, (15), and (20) yields to
p − p N Y ≤ p −p Y + p − p N Y ≤ 1 ρ 2 1 β(µ) y − y N Y + p − p N Y ≤ 1 ρ 2 1 β(µ) κ(µ) β(µ)ρ 1 √ α + 1 ỹ − y N Y + κ(µ) 2 ρ 2 1 β 2 (µ)α + 1 p − p N Y .(21)
Combining (19), (20), (21), and recalling (16), (17), we get
u − u N U + y − y N Y + p − p N Y ≤ c 1 (µ) r y (·; µ) Y * + c 2 (µ) r p (·; µ) Y * , where c 1 (µ) := 1 β(µ) 1 ρ 1 √ α + 1 + 1 ρ 2 1 β(µ) κ(µ) β(µ)ρ 1 √ α + 1 , c 2 (µ) := 1 β(µ) κ(µ) α + κ(µ) 2 β(µ)α + κ(µ) 2 ρ 2 1 β 2 (µ)α + 1 .
(II) Establishing a lower bound for u − u N U + y − y N Y + p − p N Y : From (16), the triangle inequality, and (14) we have
1 γ(µ) r y (·; µ) Y * ≤ ỹ − y N Y ≤ ỹ − y Y + y − y N Y ≤ κ(µ) β(µ) u − u N U + y − y N Y ≤ max κ(µ) β(µ) , 1 u − u N U + y − y N Y .(22)
Similarly, but this time with (17) and (15) we get
1 γ(µ) r p (·; µ) Y * ≤ p − p N Y ≤ p − p Y + p − p N Y ≤ 1 ρ 2 1 β(µ) y − y N Y + p − p N Y ≤ max 1 ρ 2 1 β(µ) , 1 y − y N Y + p − p N Y .(23)
From (22) and (23) one can easily deduce that
c 3 (µ) r y (·; µ) Y * + c 4 (µ) r p (·; µ) Y * ≤ u − u N U + y − y N Y + p − p N Y , where c 3 (µ) := 1 2γ(µ) max κ(µ) β(µ) , 1 −1 , c 4 (µ) := 1 2γ(µ) max 1 ρ 2 1 β(µ) , 1 −1 .
This concludes the proof.
Next, we establish a posteriori estimator for the relative error of the controls.
u − u N U u U ≤ 2∆ u (µ) u N U (24) provided that 2∆u(µ) u N U ≤ 1, where ∆ u (µ) := 1 ρ 1 √ αβ(µ) r y (·; µ) Y * + κ(µ) αβ(µ) r p (·; µ) Y * .
Proof. From the estimate (19) combined with (16) and (17) we obtain
u − u N U ≤ ∆ u (µ).(25)
Let 2∆u(µ) u N U ≤ 1, then we have
u U − u N U ≤ u − u N U ≤ ∆ u (µ) ≤ 1 2 u N U .
It follows from the previous inequality that if u N U ≥ u U , then
1 2 u N U ≤ u U(26)
which is clearly valid also when u N U < u U . Thus, from (25) and (26) the desired estimate (24) can be deduced.
Remark 1
The term S 1 in the proof of Theorem 4.2 is over estimated by dropping the term − 1 2 y − y N 2 L 2 (Ω 0 ) , consequently, so is the term (19). By this, a gap of a noticeable size should be expected between the relative error of controls and its a posteriori estimator in (24).
Remark 2
To consider the upper bound ∆ u (µ) from Corollary 4.3 or ∆ uyp (µ) from Theorem 4.2 for ∆(Y N , µ) in Algorithm 1, the constants κ(µ) and β(µ) should be generally replaced by other ones, sayκ(µ) andβ(µ), respectively, that are computationally cheaper to evaluate. In particular, we assume that
β(µ) ≥β(µ) ≥ β 0 ∀ µ ∈ P, κ(µ) ≤κ(µ) ≤ κ 0 ∀ µ ∈ P.
Such constantsκ(µ) andβ(µ) can be obtained using, for instance, the mintheta approach after assuming parameter-separability for the bilinear forms a and b, see [3] for the details.
Convergence analysis
In this section we are concerned with the question of whether the solution of the reduced control problem (P N ) converges to the solution of (P) as N → ∞. For this purpose, we need to investigate the continuity with respect to the parameter µ and the uniform boundedness with respect to N for the quantities that appear during the analysis.
For a given u ∈ U , we introduce the mapping
S u : P → Y(27)
such that the function y ∈ Y , y := S u (µ), is the solution of the variational problem (2) corresponding to u ∈ U and µ ∈ P. By Lax-Milgram's lemma, the mapping (27) is well defined.
In what follows we set
a(·, ·, µ) − a(·, ·, ξ) A := sup v Y =1, w Y =1 |a(v, w, µ) − a(v, w, ξ)|, f (·, µ) − f (·, ξ) F := sup v Y =1 |f (v, µ) − f (v, ξ)|, and b(·, ·, µ) − b(·, ·, ξ) B := sup v U =1, w Y =1 |b(v, w, µ) − b(v, w, ξ)|.
Lemma 5.1 For a given u ∈ U , let S u be the mapping defined in (27). Then the following estimates hold;
S u (µ) Y ≤ c 0 ( u U + 1) ∀ µ ∈ P,(28)
where c 0 := 1 β 0 max(κ 0 , σ 0 ), and
S u (µ 2 ) − S u (µ 1 ) Y ≤ 1 β 0 c 0 a(·, ·; µ 2 ) − a(·, ·; µ 1 ) A ( u U + 1) + b(·, ·; µ 2 ) − b(·, ·; µ 1 ) B u U + f (·; µ 2 ) − f (·; µ 1 ) F ,(29)
for any µ 1 , µ 2 ∈ P.
Proof. To prove (28), we denote y := S u (µ). From the coerciveness of the bilinear form a(·, ·; µ) together with the boundedness of b(·, ·; µ) and f (·; µ), one obtains
β 0 y 2 Y ≤ a(y, y; µ) = b(u, y; µ) + f (y; µ) ≤ κ 0 u U y Y + σ 0 y Y , ≤ max(κ 0 , σ 0 )( u U + 1) y Y ,
which gives (28) after dividing in the previous inequality both sides by β 0 y Y .
To verify (29) we define y 1 := S u (µ 1 ) and y 2 := S u (µ 2 ). Employing the coerciveness of a(·, ·; µ 1 ) and the estimate (28), we get
β 0 y 1 − y 2 2 Y ≤ a(y 1 − y 2 , y 1 − y 2 ; µ 1 ) = b(u, y 1 − y 2 ; µ 1 ) + f (y 1 − y 2 ; µ 1 ) − a(y 2 , y 1 − y 2 ; µ 1 ) = b(u, y 1 − y 2 ; µ 1 ) + f (y 1 − y 2 ; µ 1 ) − a(y 2 , y 1 − y 2 ; µ 1 ) + a(y 2 , y 1 − y 2 ; µ 2 ) − b(u, y 1 − y 2 ; µ 2 ) − f (y 1 − y 2 ; µ 2 ) ≤ a(·, ·; µ 2 ) − a(·, ·; µ 1 ) A y 2 Y y 1 − y 2 Y + b(·, ·; µ 2 ) − b(·, ·; µ 1 ) B u U y 1 − y 2 Y + f (·; µ 2 ) − f (·; µ 1 ) F y 1 − y 2 Y ≤ c 0 a(·, ·; µ 2 ) − a(·, ·; µ 1 ) A ( u U + 1) y 1 − y 2 Y + b(·, ·; µ 2 ) − b(·, ·; µ 1 ) B u U y 1 − y 2 Y + f (·; µ 2 ) − f (·; µ 1 ) F y 1 − y 2 Y ,
from which one deduces (29) after dividing both sides of the inequality by β 0 y 1 − y 2 Y .
We associate to the reduced variational problem (8) the mapping
S N,u : P → Y N ,(30)
where the function y N ∈ Y N , y N := S N,u (µ), is the solution of (8) corresponding to the given u ∈ U and µ ∈ P.
Lemma 5.2 For a given u ∈ U , let S N,u be the mapping defined in (30). Then the following estimates hold
S N,u (µ) Y ≤ c 0 ( u U + 1) ∀ µ ∈ P,
where c 0 := 1 β 0 max(κ 0 , σ 0 ), and
S N,u (µ 2 ) − S N,u (µ 1 ) Y ≤ 1 β 0 c 0 a(·, ·; µ 2 ) − a(·, ·; µ 1 ) A ( u U + 1) + b(·, ·; µ 2 ) − b(·, ·; µ 1 ) B u U + f (·; µ 2 ) − f (·; µ 1 ) F ,
for any µ 1 , µ 2 ∈ P.
Proof. A long the lines of Lemma 5.1's proof.
Theorem 5.3 Letū(µ) ∈ U ad be the solution of (P) for an arbitrary µ ∈ P.
Then, there exists a constant c > 0 independent of µ such that there holds
ū(µ) U ≤ c z L 2 (Ω 0 ) + u U + 1 ∀ u ∈ U ad .(31)
Proof. For a given µ ∈ P, let u ∈ U ad be an arbitrary feasible control with the corresponding state y(µ), and letȳ(µ) ∈ Y denote the state associated with the optimal controlū(µ). Then, the optimality ofū implies
α 2 ū 2 U ≤ J(ū) = 1 2 ȳ − z 2 L 2 (Ω 0 ) + α 2 ū 2 U ≤ J(u) = 1 2 y − z 2 L 2 (Ω 0 ) + α 2 u 2 U ≤ y 2 L 2 (Ω 0 ) + z 2 L 2 (Ω 0 ) + α 2 u 2 U ≤ y 2 H 1 (Ω) + z 2 L 2 (Ω 0 ) + α 2 u 2 U ≤ 1 ρ 2 1 y 2 Y + z 2 L 2 (Ω 0 ) + α 2 u 2 U ≤ c 2 0 ρ 2 1 u U + 1 2 + z 2 L 2 (Ω 0 ) + α 2 u 2 U ,
where (3) and (28) are used in the last two inequalities, respectively. Taking the square root of both sides of the previous inequality gives the desired result.
Theorem 5.4 Letū N (µ) ∈ U ad be the solution of (P N ) for an arbitrary µ ∈ P. Then, there exists a constant c > 0 independent of µ or N such that there holds
ū N (µ) U ≤ c z L 2 (Ω 0 ) + u U + 1 ∀ u ∈ U ad .
Proof. A long the lines of Theorem 5.3's proof.
Theorem 5.5 Let u(µ) ∈ U ad be the solution of (P) corresponding to some given µ ∈ P. Then, for any µ 1 , µ 2 ∈ P the following estimate holds
u(µ 1 ) − u(µ 2 ) U ≤ c a(µ 2 ) − a(µ 1 ) A + b(µ 2 ) − b(µ 1 ) B + f (µ 2 ) − f (µ 1 ) F
for some c > 0 independent of µ 1 and µ 2 . Here a(µ) := a(·, ·; µ), b(µ) := b(·, ·; µ) and f (µ) := f (·; µ) for any µ ∈ P.
Proof. Let u 1 := u(µ 1 ) and u 2 := u(µ 2 ). According to Theorem 2.1, the optimal triple (u 1 , y 1 ,
p 1 ) ∈ U ad × Y × Y satisfies a(y 1 , v; µ 1 ) = b(u 1 , v; µ 1 ) + f (v; µ 1 ) ∀ v ∈ Y, (32) a(v, p 1 ; µ 1 ) = (y 1 − z, v) L 2 (Ω 0 ) ∀ v ∈ Y, (33) b(v − u 1 , p 1 ; µ 1 ) + α(u 1 , v − u 1 ) U ≥ 0 ∀ v ∈ U ad ,(34)while (u 2 , y 2 , p 2 ) ∈ U ad × Y × Y satisfies a(y 2 , v; µ 2 ) = b(u 2 , v; µ 2 ) + f (v; µ 2 ) ∀ v ∈ Y, (35) a(v, p 2 ; µ 2 ) = (y 2 − z, v) L 2 (Ω 0 ) ∀ v ∈ Y, (36) b(v − u 2 , p 2 ; µ 2 ) + α(u 2 , v − u 2 ) U ≥ 0 ∀ v ∈ U ad .(37)
We shall utilize the auxiliary functionsỹ 1 ,ỹ 2 ∈ Y satisfying
a(ỹ 1 , v; µ 2 ) = b(u 1 , v; µ 2 ) + f (v; µ 2 ) ∀ v ∈ Y, a(ỹ 2 , v; µ 1 ) = b(u 2 , v; µ 1 ) + f (v; µ 1 ) ∀ v ∈ Y.
Testing (34) against u 2 , and (37) against u 1 , and adding the resulting inequalities yields
α u 1 − u 2 2 U ≤ b(u 2 − u 1 , p 1 ; µ 1 ) + b(u 1 − u 2 , p 2 ; µ 2 ) = b(u 2 , p 1 ; µ 1 ) − b(u 1 , p 1 ; µ 1 ) + b(u 1 , p 2 ; µ 2 ) − b(u 2 , p 2 ; µ 2 ) = b(u 2 , p 1 ; µ 1 ) − a(y 1 , p 1 ; µ 1 ) + f (p 1 ; µ 1 ) + b(u 1 , p 2 ; µ 2 ) − a(y 2 , p 2 ; µ 2 ) + f (p 2 ; µ 2 ) = a(ỹ 2 − y 1 , p 1 ; µ 1 ) + a(ỹ 1 − y 2 , p 2 ; µ 2 ) = (y 1 − z,ỹ 2 − y 1 ) L 2 (Ω 0 ) + (y 2 − z,ỹ 1 − y 2 ) L 2 (Ω 0 ) = (y 1 − z,ỹ 2 − y 2 ) L 2 (Ω 0 ) + (y 2 − z,ỹ 1 − y 1 ) L 2 (Ω 0 ) − y 1 − y 2 2 L 2 (Ω 0 ) ≤ y 1 − z L 2 (Ω 0 ) ỹ 2 − y 2 L 2 (Ω 0 ) + y 2 − z L 2 (Ω 0 ) ỹ 1 − y 1 L 2 (Ω 0 ) ≤ c u 1 U + z L 2 (Ω 0 ) + 1 ỹ 2 − y 2 L 2 (Ω 0 ) + c u 2 U + z L 2 (Ω 0 ) + 1 ỹ 1 − y 1 L 2 (Ω 0 ) ,
where (28) is used in the last inequality. We proceed by utilizing (29)
≤ c u 1 U + z L 2 (Ω 0 ) + 1 a(µ 2 ) − a(µ 1 ) A ( u 2 U + 1) + b(µ 2 ) − b(µ 1 ) B u 2 U + f (µ 2 ) − f (µ 1 ) F + c u 2 U + z L 2 (Ω 0 ) + 1 a(µ 2 ) − a(µ 1 ) A ( u 1 U + 1) + b(µ 2 ) − b(µ 1 ) B u 1 U + f (µ 2 ) − f (µ 1 ) F ≤ c a(µ 2 ) − a(µ 1 ) A + b(µ 2 ) − b(µ 1 ) B + f (µ 2 ) − f (µ 1 ) F
Recalling (31) and taking the square root of the both sides gives the desired result.
Theorem 5.6 Let u N (µ) ∈ U ad be the solution of (P N ) corresponding to some given µ ∈ P. Then, for any µ 1 , µ 2 ∈ P the following estimate holds
u N (µ 1 ) − u N (µ 2 ) U ≤ c a(µ 2 ) − a(µ 1 ) A + b(µ 2 ) − b(µ 1 ) B + f (µ 2 ) − f (µ 1 ) F
for some c > 0 independent of µ 1 , µ 2 or N . Here a(µ) := a(·, ·; µ), b(µ) := b(·, ·; µ) and f (µ) := f (·; µ) for any µ ∈ P.
Proof. A long the lines of Theorem 5.5's proof.
Recall that the space Y N considered in (P N ) is constructed from the snapshots {y(µ 1 ), p(µ 1 ), . . . , y(µ N ), p(µ N )} taken from (P) at the sample parameters {µ 1 , . . . , µ N } =: P N ⊂ P. We denote
h N := max µ∈P min µ ∈P N µ − µ
with · being the Euclidean norm in R p . We shall assume that 0 < h N ≤ 1 and that as N → ∞, h N → 0, i.e. the set P N gets denser in P as N increases. Furthermore, it is natural to assume that for any µ ∈ P N there holds
u N (µ) = u(µ),
where u N (µ) and u(µ) denote the solutions of (P N ) and (P), respectively, at the given µ since the mapping P µ → u N (µ) ∈ U is supposed to interpolate the mapping P µ → u(µ) ∈ U at the set of parameters P N . Finally, we assume that for any µ 1 , µ 2 ∈ P we have
a(µ 2 ) − a(µ 1 ) A ≤ c µ 2 − µ 1 qa , b(µ 2 ) − b(µ 1 ) B ≤ c µ 2 − µ 1 q b , f (µ 2 ) − f (µ 1 ) F ≤ c µ 2 − µ 1 q f ,
for some c, q a , q b , q f > 0 independent of µ 1 or µ 2 where · denotes the Euclidean norm in R p , i.e. the bilinear forms a and b and the linear form f are continuous in µ. Under these assumptions, we formulate the next theorem.
Theorem 5.7 Let u N (µ), u(µ) ∈ U denote the solutions of (P N ) and (P), respectively, for a given µ ∈ P. Then, the following estimate holds
u N (µ) − u(µ) U ≤ ch t N
where t := 1 2 min{q a , q b , q f } and c > 0 is a constant independent of h N or µ. Proof. Let µ ∈ P be given, and let µ * := arg min µ ∈P N µ − µ . Then, recalling Theorem 5.5, Theorem 5.6, the fact that u N (µ * ) = u(µ * ), and the continuity of a, b and f gives
u(µ) − u N (µ) U ≤ u(µ) − u(µ * ) U + u(µ * ) − u N (µ * ) U + u N (µ * ) − u N (µ) U ≤ c a(µ) − a(µ * ) A + b(µ) − b(µ * ) B + f (µ) − f (µ * ) F ≤ c µ − µ * qa + µ − µ * q b + µ − µ * q f ≤ c h qa N + h q b N + h q f N ≤ ch t N where t := 1 2 min{q a , q b , q f }.
Numerical examples
In this section we apply our theoretical findings to construct numerically reduced surrogates for two examples, namely a thermal block problem and a Graetz flow problem, which are taken from [1]. In particular, we discretize those two examples using variational discretization, then we build their reduced counterparts using the greedy procedure from Algorithm 1, where we use the bound 2∆ u (µ)/ u N U from Corollary 4.3 for the estimator ∆(Y N , µ). Finally, we compare the solutions of the reduced problems to their corresponding ones from the highly dimensional problems to asses the quality of the obtained reduced models.
Example 1 (Thermal block) We consider the control problem From the previous given data, it is an easy task to see that (3) holds with ρ 2 = 1 and
min (u,y)∈U ad ×Y J(u, y) = 1 2 y − z 2 L 2 (Ω 0 ) + α 2 u 2 U subject to µ Ω 1 ∇y · ∇v dx + Ω 2 ∇y · ∇v dx = Ω uv dx ∀ v ∈ Y,ρ 1 = 1 √ c 2 p +1 where c p = 1 √ 2π is the Poincaré's constant in the inequality v L 2 (Ω) ≤ c p ∇v L 2 (Ω) ∀ v ∈ H 1 0 (Ω)
. Furthermore, we takeκ(µ) = c p andβ(µ) = min(µ, 1).
We use a uniform triangulation for Ω such that dim(Y ) ≈ 8300. The solution of both the variational discrete control problem and the reduced control problem for a given parameter µ is achieved by solving the corresponding optimality conditions using a semismooth Newton's method with the stopping criteria
1 α p (k) − p (k+1) L 2 (Ω) ≤ 10 −11 ,
where p (k) is the adjoint variable at the k-th iteration. The reduced space Y N for the considered problem was constructed employing the greedy procedure introduced in Algorithm 1 with the choice S train := {s j } 100 j=1 , s j := 0.5×(3/0.5) (j−1)/99 , µ 1 := 0.5, ε tol = 10 −8 , N max = 30, and ∆(Y N , µ) := 2∆ u (µ)/ u N L 2 (Ω) .
The algorithm terminated before reaching the prescribed tolerance ε tol and that was after 22 iterations as it could not enrich the reduced basis with any new linearly independent samples. To investigate the quality of the obtained reduced basis and the sharpness of the employed upper bound ∆(Y N , µ), we compute the maximum of the relative error u − u N L 2 (Ω) / u L 2 (Ω) and of the corresponding bound 2∆ u (µ)/ u N L 2 (Ω) over the set S test := {s j } 125 j=1 , s j := 0.503 × (2.99/0.503) (j−1)/125 for the greedy algorithm iterations N = 1, . . . , 22. The graphical illustration is presented in Figure 1. We see that the error decays dramatically in the first nine iterations, namely it drops from 1 to slightly above 10 −6 , then the decay becomes very slow and the error almost stabilizes at 10 −6 in the last four iterations.
As predicted in Remark 1, we can see a gap between the relative error and the used estimator ∆(Y N , µ). This plot compares to Fig.1(b) of [1]. We observe that four iterations of the greedy algorithm with our approach deliver the same error reduction as thirty iterations of the greedy algorithm in [1]. A similar behaviour is observed for Example 2 with the Graetz flow in Figure 4, which compares to the results documented in Fig. 3(b) of [1]. For this example six iterations of the greedy algorithm with our approach deliver the same error reduction as thirty iterations of the greedy algorithm in [1].
Example 2 (Graetz flow) We consider the problem and Y (µ 2 ) ⊂ {v ∈ H 1 (Ω(µ 2 )) ∩ C(Ω(µ 2 )) : v| Γ D (µ 2 ) = 1} is the space of piecewise linear and continuous finite elements. The underlying PDE has the homogeneous Neumann boundary condition ∂ η y| Γ N (µ 2 ) = 0 on the portion Γ N (µ 2 ) of the boundary of the domain Ω(µ 2 ) , and the Dirichlet boundary condition y| Γ D (µ 2 ) = 1 on the portion Γ D (µ 2 ). An illustration for the domain Ω(µ 2 ) and the boundary segments Γ D (µ 2 ) and Γ N (µ 2 ) is given in Figure 2. We introduce the lifting functionỹ(x) := 1 to handle the nonhomogeneous Dirichlet boundary condition, and reformulate the problem over the reference domain Ω := Ω(µ ref 2 ), and endow the state space Y := Y (µ ref 2 ) by the inner product (·, ·) Y given by is endowed with a parameter dependent inner product (·, ·) U (µ 2 ) from the affine geometry transformation, see [6]. After transforming the problem over Ω we deduce that (3) holds with ρ 1 = max(µ ref 1 (1 + c p ), 1) −2 , where the constant c p is from the Poincaré's inequality
min (u,y)∈U ad (µ 2 )×Y (µ 2 ) J(u, y) = 1 2 y − z 2 L 2 (Ω 0 (µ 2 )) + α 2 u 2 U (µ 2 ) subject to 1 µ 1 Ω(µ 2 ) ∇y · ∇v dx + Ω(µ 2 ) β(x) · ∇yv dx = Ω(µ 2 ) uv dx ∀ v ∈ Y (µ 2 ),(v, w) Y := 1 µ ref 1 Ω ∇w · ∇v dx + 1 2 Ω β(x) · ∇wv dx + Ω β(x) · ∇vw dxΩ v 2 dx ≤ c p Ω |∇v| 2 dx ∀ v ∈ H 1 (Ω) : v| Γ D (µ ref 2 ) = 0. In addition, we takẽ β(µ 1 , µ 2 ) = min µ ref 1 min( 1 µ 1 µ 2 , µ 2 µ 1 , 1 µ 1 ), 1 , andκ(µ 1 , µ 2 ) = 1 ρ 1 ( √ µ 2 + 1).
The domain Ω is partitioned via a uniform triangulation such that dim(Y ) ≈ 4900. The optimality conditions corresponding to the variational discrete control problem and the reduced control problem are solved using a semismooth Newton's method with the stopping criteria
1 α p (k) − p (k+1) U (µ 2 ) ≤ 10 −11 ,
where p (k) is the adjoint variable at the k-th iteration.
The optimal controls and their active sets for the parameter values (µ 1 , µ 2 ) = (5, 0.8), (18, 1.2) computed on the reference domain are presented in Figure 3.
The reduced basis for the space Y N is constructed applying the Algorithm 1 with the choice S train := {(s 1 j , s 2 k )} for j, k = 1, . . . , 30 where s 1 j := 5 × (18/5) (j−1)/29 and s 2 k := (0.4/29) × (k − 1) + 0.8. Furthermore, we take µ 1 := (5, 0.8), ε tol = 10 −8 , N max = 30, and ∆(Y N , µ) := 2∆ u (µ)/ u N U (µ 2 ) .
The algorithm terminated at N max = 30 before reaching the tolerance ε tol . To asses the quality of the resulting reduced basis and the sharpness of the bound ∆(Y N , µ), we compare the maximum of the relative error u − u N U (µ 2 ) / u U (µ 2 ) to the bound 2∆ u (µ)/ u N U (µ 2 ) computed over the test set S test := {(s 1 j , s 2 k )}, for j = 1, . . . , 10 and k = 1, . . . , 5 where s j := 5.2 × (17.5/5.2) (j−1)/9 , and s 2 k := (0.35/4) × (k − 1) + 0.82 for the greedy algorithm iterations N = 1, . . . , 30. The outcome of the experiment is presented in Figure 4. The error decay is of moderate speed in comparison to the previous example. It could be because the current problem has more parameters and one of which stems from the geometry of the domain. We again see the gap between the bound and the error, which supports the prediction of Remark 1.
Conclusions
With present a reduced basis method for the approximation of optimal control problems with control constraints. We use variational discretization from [4] for the numerical approximation of the optimal control problems. This allows us to use methods from [2] to prove an error equivalence for our residual based error estimator, which finally is one of the key ingredients for the convergence proof of our approach in Theorem 5.7. Our numerical results indicate that the reduced basis method combined with variational discretization for a prescribed error tolerance seems to deliver reduced basis spaces of much smaller dimension than in the existing approaches reported in the literature, compare e.g. the numerical results reported in [1]. However, this comes along with a more sophisticated numerical implementation of the variational discretization approach in the case of control constraints, for which the classical offline-online decomposition techniques are not applicable in a straightforward manner.
Corollary 4. 3
3Under the hypothesis of Theorem 4.2, there holds
Ω 1 ∪
1Ω 2 , Ω 0 := Ω, z(x) = 1 in Ω, U := L 2 (Ω), (·, ·) U := (·, ·) L 2 (Ω) ,U ad := {u ∈ L 2 (Ω) : u(x) ≥ u a (x) a.e x ∈ Ω}, u a (x) := 2 + 2(x 1 − 0.5), µ ∈ P := [0.5, 3], α = 10 −2 , and the space Y ⊂ H 1 0 (Ω) ∩ C(Ω)is the space of piecewise linear and continuous finite elements endowed with the inner product (·, ·) Y := (∇·, ∇·) L 2 (Ω) . The underlying PDE admits a homogeneous Dirichlet boundary condition on the boundary ∂Ω of the domain Ω.
2 ) := (0, 1.5 + µ 2 ) × (0, 1), Ω 1 (µ 2 ) := (0.2µ 2 , 0.8µ 2 ) × (0.3, 0.7), Ω 2 (µ 2 ) := (µ 2 + 0.2, µ 2 + 1.5) × (0.3, 0.7), Ω 0 (µ 2 ) := Ω 1 (µ 2 ) ∪ Ω 2 (µ 2 ) β(x) = (x 2 (1 − x 2 ), 0) T in Ω(µ 2 ), z(x) = 0.5 in Ω 1 (µ 2 ), z(x) = 2 in Ω 2 (µ 2 ) U (µ 2 ) := L 2 (Ω(µ 2 )), (·, ·) U (µ 2 ) := (·, ·) L 2 (Ω(µ 2 )) , U ad (µ 2 ) := {u ∈ L 2 (Ω(µ 2 )) : u(x) ≥ u a (x) a.e x ∈ Ω(µ 2 )}, u a (x) := −0.5, (µ 1 , µ 2 ) ∈ P := [5, 18] × [0.8, 1.2], α = 10 −2 ,
Figure 1
1Example 1: The maximum of u − u N L 2 (Ω) / u L 2 (Ω) the relative error of controls and the corresponding upper bounds 2∆ u (µ)/ u N L 2 (Ω) over S test versus the greedy algorithm iterations N = 1, . . . , 22.
Figure 2
2Example 2: The domain Ω(µ 2 ) for the Graetz flow problem.
Figure 3 Figure 4
34Example 2: The optimal controls, and their active sets (enclosed by the curves) for (µ 1 , µ 2 ) = (5, 0.8), and (18, 1.2) computed on the reference domain Ω. Example 2: The maximum of u − u N U (µ 2 ) / u U (µ 2 ) the relative error of controls and the upper bound 2∆ u (µ)/ u N U (µ 2 ) over S test versus the greedy algorithm iterations N = 1, . . . , 30.
Certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints. Eduard Bader, Mark Kärcher, A Martin, Karen Grepl, Veroy, SIAM Journal on Scientific Computing. 386Eduard Bader, Mark Kärcher, Martin A Grepl, and Karen Veroy. Certi- fied reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints. SIAM Journal on Scientific Computing, 38(6):A3921-A3946, 2016.
Adaptive finite element method for elliptic optimal control problems: convergence and optimality. Wei Gong, Ningning Yan, Numerische Mathematik. 1354Wei Gong and Ningning Yan. Adaptive finite element method for ellip- tic optimal control problems: convergence and optimality. Numerische Mathematik, 135(4):1121-1170, 2017.
Reduced basis methods for parametrized pdes-a tutorial introduction for stationary and instationary problems. Model reduction and approximation: theory and algorithms. Bernard Haasdonk, 1565Bernard Haasdonk. Reduced basis methods for parametrized pdes-a tutorial introduction for stationary and instationary problems. Model reduction and approximation: theory and algorithms, 15:65, 2017.
A variational discretization concept in control constrained optimization: the linear-quadratic case. Michael Hinze, Computational Optimization and Applications. 301Michael Hinze. A variational discretization concept in control constrained optimization: the linear-quadratic case. Computational Optimization and Applications, 30(1):45-61, 2005.
Optimization with PDE constraints. Michael Hinze, René Pinnau, Michael Ulbrich, Stefan Ulbrich, Springer Science & Business Media23Michael Hinze, René Pinnau, Michael Ulbrich, and Stefan Ulbrich. Opti- mization with PDE constraints, volume 23. Springer Science & Business Media, 2008.
Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Gianluigi Rozza, Dinh Bao, Phuong Huynh, Anthony T Patera, Archives of Computational Methods in Engineering. 1531Gianluigi Rozza, Dinh Bao Phuong Huynh, and Anthony T Patera. Re- duced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives of Computational Methods in Engineering, 15(3):1, 2007.
| [] |
[
"arXiv:math-ph/0506016v1 7 Jun 2005 Rotation Numbers, Boundary Forces and Gap Labelling",
"arXiv:math-ph/0506016v1 7 Jun 2005 Rotation Numbers, Boundary Forces and Gap Labelling"
] | [
"Johannes Kellendonk [email protected] \nInstitute Girard Desargues\nUniversité\nClaude Bernard Lyon 1F-69622Villeurbanne\n",
"Ioannis Zois [email protected] \nSchool of Mathematics\nCardif University\nPO Box 926CF24 4YHCardiffUK\n"
] | [
"Institute Girard Desargues\nUniversité\nClaude Bernard Lyon 1F-69622Villeurbanne",
"School of Mathematics\nCardif University\nPO Box 926CF24 4YHCardiffUK"
] | [] | We review the Johnson-Moser rotation number and the K 0 -theoretical gap labelling of Bellissard for one-dimensional Schrödinger operators. We compare them with two further gap-labels, one being related to the motion of Dirichlet eigenvalues, the other being a K 1 -theoretical gap label. We argue that the latter provides a natural generalisation of the Johnson-Moser rotation number to higher dimensions. | 10.1088/0305-4470/38/18/005 | [
"https://arxiv.org/pdf/math-ph/0506016v1.pdf"
] | 16,400,733 | math-ph/0506016 | 5bb31c34478151eb1bc38db31e2a42d47d14004a |
arXiv:math-ph/0506016v1 7 Jun 2005 Rotation Numbers, Boundary Forces and Gap Labelling
January 31, 2005
Johannes Kellendonk [email protected]
Institute Girard Desargues
Université
Claude Bernard Lyon 1F-69622Villeurbanne
Ioannis Zois [email protected]
School of Mathematics
Cardif University
PO Box 926CF24 4YHCardiffUK
arXiv:math-ph/0506016v1 7 Jun 2005 Rotation Numbers, Boundary Forces and Gap Labelling
January 31, 2005
We review the Johnson-Moser rotation number and the K 0 -theoretical gap labelling of Bellissard for one-dimensional Schrödinger operators. We compare them with two further gap-labels, one being related to the motion of Dirichlet eigenvalues, the other being a K 1 -theoretical gap label. We argue that the latter provides a natural generalisation of the Johnson-Moser rotation number to higher dimensions.
Introduction
It is an interesting and well known observation that the boundary of a domain plays a prominent role both in mathematics and in physics. A case that comes immediately into mind is the theory of differential equations where the boundary conditions determine quite a lot of the whole solution. In a purely topological context the boundary may even determine the behaviour of the system in the bulk completely. A case like this was studied in [KS04a,KS04b] where a correspondance between bulk and boundary topological invariants for certain physical systems arising in solid state physics was found. This was mathematically based on K-theoretic and cyclic cohomological properties of the Wiener-Hopf extension of the C * -algebra of observables. In most applications we have in mind, this C * -algebra is obtained by considering the Schrödinger operator and its translates describing the 1-particle approximation of the solid. In this article we consider a simple example, a Schrödinger operator on the real line, where such a correspondance can be established more directly with the help of the Sturm-Liouville theorem. The K 0 -theory gap labels (below referred to also as even K-gap labels) introduced by Bellissard et al. [BLT85,Be92] are bulk invariants. It is known that these are equal to the Johnson-Moser rotation numbers [JM82] the existing proof being essentially a corollary of the Sturm-Liouville theorem by which they are identified with the integrated density of states on the gaps. In the first part of the paper (Sections 2,3) we provide a direct identification of the Johnson-Moser rotation number (for energies in gaps) with a boundary invariant, here called the Dirichlet rotation number. This boundary invariant has a physical interpretation, namely as boundary force per unit energy. Moreover, it can be interpreted as a K 1 -theory gap label (or odd K-gap label).
In the second part (Sections 4,5) we indicate how the equality between the K 0 and the K 1 -theory gap labels also follows from the above-mentioned noncommutative topology of the Wiener Hopf extension. The advantage of this approach is that, unlike the definition of the geometrical rotation numbers and the Sturm-Liouville theorem, it is not restricted in dimension. We tend to think of the K 1 -theory gap label, which is naturally defined in any dimension, as the operator algebraic formulation of the Johnson-Moser rotation number.
Whereas the first part is based on a single operator, although its translates play a fundamental role, we consider in the second part covariant families of operators indexed by the hull of the potential. This is the right framework for the use of ergodic theorems and noncommutative topology. The last section is mainly based on [Kel] and therefore held briefly.
Preliminaries
In this article we consider as in [Jo86] a one-dimensional Schrödinger operator H = −∂ 2 +V with (real) bounded potential which we assume (stricter as in [Jo86]) to be bounded differentiable. We also consider its translates H ξ := −∂ 2 + V ξ , V ξ (x) = V (x + ξ), and lateron its hull. The differential equation HΨ = EΨ for complex valued functions Ψ over R has for all E two linear independent solutions but not all E belong to the spectrum σ(H) of H as an operator acting on L 2 (R). In this situation the following property of solutions holds [CL55].
Theorem 1 If E / ∈ σ(H) there exist two real solutions Ψ + and Ψ − of (H − E)Ψ = 0, Ψ + vanishing at ∞ and Ψ − vanishing at −∞. These solutions are linear independent and unique up to multiplication by a factor.
We mention as an aside that Johnson proves even exponential dichotomy for such energies [Jo86]. Clearly σ(H ξ ) = σ(H) for all ξ.
We consider also the action of H ξ on L 2 (R ≤0 ) with Dirichlet boundary conditions at the boundary. If we need to emphazise this we will also writeĤ ξ for the half-sided operator. The spectrum is then no longer the same. Whereas the essential part of the spectrum ofĤ ξ is contained in that of H ξ [Jo86] the half sided operator may have isolated eigenvalues in the gaps in σ(H ξ ). Here a gap is a connected component of the complement of the spectrum, hence in particular an open set. E is an eigenvalue ofĤ ξ if (Ĥ ξ − E)Ψ = Ψ for Ψ ∈ L 2 (R ≤0 ) which for E in a gap of σ(H ξ ) amounts to saying that the solution Ψ − of (H ξ − E)Ψ − = 0 from Theorem 1 satisfies in addition Ψ − (0) = 0.
Definition 1 We call E ∈ R a right Dirichlet value of H ξ if it is an eigenvalue ofĤ ξ .
We recall the important Sturm-Liouville theorem:
Theorem 2 Consider H := −∂ 2 + V with (real) bounded continuous potential acting on L 2 ([a, b]) with Dirichlet boundary conditions. The spectrum is discrete and bounded from below.
A real eigenfunction to the nth eigenvalue (counted from below) has exactly n − 1 zeroes in the interior (a, b) of [a, b].
Rotation numbers
The winding number of a continuous function f : R/Z → R/Z is intuitively speaking the number of times its graph wraps around the circle R/Z. This is counted relative to the orientations induced by the order on R. Let Λ = {Λ n } n be an increasing chain of compact intervals Λ n = [a n , b n ] ⊂ Λ n+1 ⊂ R whose union covers R. The quantity Λ(f ) := lim n→∞ 1 b n − a n bn an f (x)dx is called the Λ-mean of the function f : R → R, existence of the limit assumed. Now let f : R → R/Z be continuous and choose a continuous extensionf : R → R. To define the rotation number of f we consider the expression
rot Λ (f ) = lim n→∞f (b n ) −f (a n )
b n − a n which becomes the winding number of f if f is periodic of period 1. The limit does not exist in general but if it does it is independent of the extensionf . If f is piecewise differentiable then rot Λ (f ) = Λ(f ′ ). Moreover, if U : R → C is a nowhere vanishing continuous piecewise differentiable function then we can consider the rotation number of its argument function which becomes
rot Λ ( arg(U) 2π ) = lim n→∞ 1 2πi(b n − a n ) bn an U |U| U |U| ′ dx
(1)
The Johnson-Moser rotation number
Johnson and Moser in [JM82] have defined rotation numbers for the Schrödinger operator H = −∂ 2 + V on the real line where V is a real almost periodic potential. They are defined as follows: Let Ψ(x) be the nonzero real solution of (H − E)Ψ = 0 which vanishes at −∞, then Ψ ′ + iΨ : R → C is nowhere vanishing and
α Λ (H, E) := 2 rot Λ ( arg(Ψ ′ + iΨ) 2π
).
(2) (Our normalisation differs from that in [JM82] for later convenience.) For the class of potentials considered here the limit is indeed defined and even independent on the choice of Λ, we will come back to that in Section 4. Note that α Λ (H, E) has the following interpretations. If N(a, b; E) denotes the number of zeroes of the above solution Ψ in [a, b] then α Λ (H, E) is the Λ-mean of the density of zeroes of Ψ, namely one has
α Λ (H, E) = lim n→∞ N(a n , b n ; E) b n − a n .
The integrated density of states of H at E is
IDS Λ (H, E) = lim n→∞ 1 |Λ n | Tr(P E (H Λn ))(3)
provided the limit exists. Here |Λ n | = b n − a n is the volume of Λ n , H Λn the restriction of H to Λ n with Dirichlet boundary conditions and, for self-adjoint A, P E (A) is the spectral projection onto the spectral subspace of spectral values smaller or equal to E. It will be important that P (A) is a continuous function of A if E is not in the spectrum of A. Since Tr(P E (H Λn )) is the number of eigenfunctions of H Λn to eigenvalue smaller or equal E Theorem 2 implies
Corollary 1 α Λ (H, E) = IDS Λ (H, E).
In particular, like the integrated density of states α Λ (H, E) is monotonically increasing in E and constant on the gaps of the spectrum of H. It is moreover the same for all H ξ .
The Dirichlet rotation number
We now consider the continuous 1-parameter family of operators This set depends actually only on µ, since Ψ is unique up to a multiplicative factor and we have:
{H ξ } ξ with ξ ∈ R and H ξ = −∂ 2 + V ξ , where V ξ (x) = V (x + ξ). WeLemma 1 Let ξ ∈ R such that D ξ (∆) = ∅ and µ ∈ D ξ (∆). Then S(µ) = Z(µ, ξ). Proof: Let Ψ be a non-zero solution (H ξ − µ)Ψ = 0 satisfying Ψ(0) = Ψ(−∞) = 0 and define Ψ η (x) = Ψ(x + (η − ξ)). Then (H η − µ)Ψ η = 0 and Ψ η (−∞) = 0 for all η. Hence Z(µ, ξ) = {η|Ψ(η − ξ) = 0} = {η|Ψ η (0) = 0} ⊂ S(µ). For the opposite inclusion if µ ∈ D η (∆), then there exists Φ such that (H η − µ)Φ = 0 with Φ(0) = Φ(−∞) = 0. Define Φ ξ (x + (η − ξ)) = Φ(x). Then (H ξ − µ)Φ ξ = 0 with Φ ξ (−∞) = 0.
By Theorem 1, Ψ = λΦ ξ for some λ ∈ C * , which implies Ψ(η − ξ) = λΦ(0) = 0 and hence η ∈ Z(µ, ξ), thus S(µ) ⊆ Z(µ, ξ). ✷ Let ξ ∈ S(µ), µ ∈ ∆. Since the spectrum ofĤ ξ in the gap ∆ consists of isolated eigenvalues which are non-degenerate by Theorem 1 we can use perturbation theory to find a neighbourhood (ξ − ǫ, ξ + ǫ) and a differentiable function ξ → µ(ξ) on this neighbourhood which is uniquely defined by the property that µ(ξ) ∈ D ξ (∆). In fact, level-crossing of right Dirichlet values cannot occur in gaps, since it would lead to degeneracies. As in [Ke04] we see that its first derivative is strictly negative:
dµ(ξ) dξ = 0 −∞ dx|Ψ ξ (x)| 2 V ′ ξ = −|Ψ ′ ξ (0)| 2 < 0.
Here Ψ ξ is a normalised eigenfunction ofĤ ξ . Thus around each value ξ for which we find a right Dirichlet value in ∆ we have locally defined curves µ(ξ) which are strictly monotonically decreasing and non-intersecting. SinceĤ ξ is norm-continuous in ξ in the generalised sense, its spectrum σ(Ĥ ξ ) is lower semi-continuous [K] in ξ so that the curves µ(ξ) can be continued until they reach the boundary of ∆ or their limit at +∞ or −∞, if it exists. Let K be the circle of complex numbers of modulus 1. We define the functionμ :
R → K byμ (ξ) = exp 2πi µ∈D ξ µ − E 0 |∆| where E 0 = inf ∆ and |∆| is the width of ∆.
Thenμ is a continuous function which is differentiable at all points where none of the curves µ(ξ) touches the boundary.
Definition 2 The Dirichlet rotation number is
β Λ (H, ∆) := −rot Λ ( argμ 2π ) .
Lemma 2 If, for some µ ∈ ∆, |S(µ)| > 1 then ∆ contains at most one right Dirichlet value of H ξ .
Proof: We first remark that the same discussion can be performed for the left Dirichlet values of H ξ , namely values E for which exist Ψ solving (H ξ − E)Ψ = 0 with Ψ(0) = Ψ(+∞) = 0. These similarily define locally curves µ * (ξ) whose first derivative are now strictly positive. They can't intersect with any of the curves µ(ξ), because a right Dirichlet value which is at the same time a left Dirichlet value must be a true eigenvalue of H. Let S * (µ) and Z * (µ) be defined as S(µ) and Z(µ) but for left Dirichlet values. We claim that between two points of S(µ) lies one point of S * (µ). This then implies the lemma, because if D ξ contained two points an elementary geometric argument shows that the curves defined by right Dirichlet values through these points necessarily have to intersect a curve defined by left Dirichlet values. To prove our claim we consider the analogous statement for Z(µ) and Z * (µ) and let Ψ ± be a real solution of (H 0 −µ)Ψ = 0 with Ψ ± (±∞) = 0. Since µ is not an eigenvalue the Wronskian [Ψ + , Ψ − ] which is always constant does not vanish. Furthermore, if Ψ + (x) = 0 then Ψ − (x) = −[Ψ + , Ψ − ]/Ψ ′ + (x). This expression changes sign between two consecutive zeroes of Ψ + and hence Ψ − must have a zero in between. ✷
Remark 1 Under the hypothesis of the lemma the sum in the definition ofμ contains at most one element. We believe that the result of the lemma is true under all circumstances.
Theorem 3 α Λ (H, E) = β Λ (H, ∆).
Proof: By Lemma 1 α Λ (H, µ) is the Λ-mean of the density of S(µ). Suppose the hypothesis of the Lemma 2 holds. Then S(µ) can be identified with the set of intersection points between the constant curve ξ → exp 2πi E−E 0 |∆| andμ(ξ). Since µ ′ (ξ) < 0 the Λ-mean of the density of these intersection points is minus the rotation number of argμ 2π . Now suppose that S(µ) contains at most one element. Then α Λ (H, µ) = 0. On the other hand, there can only be finitely many curves defined by right Dirichlet values. Since they intersect the constant curve ξ → exp 2πi µ−E 0 |∆| only once, β Λ (H, ∆) must be 0. ✷ Remark 1 An even nicer geometric picture arrises if we take into account also the left Dirichlet values of H ξ for the definition ofμ. For this purpose redefineμ : R → K bỹ
µ(ξ) = exp πi µ∈D ξ µ − E 0 |∆| − µ∈D * ξ µ − E 0 |∆| where D ξ (∆) * is the set of left Dirichlet values of H ξ in ∆.
Thenμ is as well a continuous piecewise differentiable function and rot Λ ( argμ 2π ) is the same number as before except that it yields the Λ-mean of the winding per length of the Dirichlet values around a circle which is obtained from two copies of ∆ by identification of their boundary points. For periodic systems, this circle can be identified with the homology cycle corresponding to a gap in the complex spectral curve of H [BBEIM] and so β Λ (H, ∆) is the winding number of the Dirichlet values around it. This is similar to Hatsugai's interpretation of the edge Hall conductivity as a winding number (see [Ha93]). There the role of the parameter ξ is played by the magnetic flux.
Odd K-gap labels and Dirichlet rotation numbers
We define another type of gap label which is formulated using operator traces and derivations instead of curves on topological spaces. It has its origin in an odd pairing between K-theory and cyclic cohomology.
We fix a gap ∆ in the spectrum of H of length |∆| and set E 0 = inf(∆). Let P ∆ = P ∆ (Ĥ ξ ) be the spectral projection ofĤ ξ onto the energy interval ∆. Then
U ξ := P ∆ e 2iπĤ ξ −E 0 |∆| + 1 − P ∆(4)
acts essentially as the unitary of time evolution by time 1 |∆| on the eigenfunctions ofĤ ξ in ∆. These eigenfunctions are all localised near the edge and therefore is the following expression a boundary quantity.
Definition 3 The odd K-gap label is
Π Λ (H, ∆) = − lim n→∞ 1 2iπ|b n − a n | bn an Tr[(U * ξ − 1)∂ ξ U ξ ]dξ
Where Tr is the standard operator trace on L 2 (R).
Theorem 4 Π Λ (H, ∆) = β Λ (H, ∆).
Proof: Note that the rank of P ∆ is equal to |D ξ (∆)|, the number of elements in D ξ (∆). Let us first suppose that this is either 1 or 0 which would be implied under the conditions of Lemma 2.
Since U * ξ − 1 = P ∆ (e 2iπĤ ξ −E 0 |∆| − 1) we can express the trace using the normalised eigenfunctions Ψ ξ ofĤ ξ to µ(ξ), provided |D ξ (∆)| = 1,
Tr[(U * ξ − 1)∂ ξ U ξ ] ξ U ξ )|Ψ ξ = Ψ ξ |U * ξ − 1|Ψ ξ Ψ ξ |∂ ξ U ξ |Ψ ξ . (5) Substituting Ψ ξ |∂ ξ U ξ |Ψ ξ = ∂ ξ Ψ ξ |U ξ |Ψ ξ = ∂ ξ e 2iπ µ(ξ)−E 0 |∆|
in the previous expression we arrive at
Tr[(U * ξ − 1)∂ ξ U ξ ] = (e −2iπ µ(ξ)−E 0 |∆| − 1)∂ ξ e 2iπ µ(ξ)−E 0 |∆| . Since U * ξ − 1 = 0 if D ξ (∆) = ∅ we have Π Λ (H, ∆) = − lim n→∞ 1 2iπ|b n − a n | bn an (μ(ξ) − 1)μ ′ (ξ)dξ = − 1 2iπ Λ(μμ ′ ) (6)
which is the expression for β Λ (H, ∆). If |D ξ | > 1 one has to replace the r.h.s. of (5) by a sum over eigenfunctions ofĤ ξ and the calculation will be similar. ✷
Interpretation as boundary force per unit energy
We assume for simplicity |D ξ | ≤ 1. Then we obtain from (6)
Π Λ (H, ∆) = − lim n→∞ 1 |b n − a n | bn an µ ′ (ξ) |D ξ (∆)| |∆| dξ .
The r.h.s. is 1 |∆| times the Λ-mean of the expectation value of the gradient force w.r.t. the density matrix associated with the egde states in the gap. Since translatingĤ ξ in ξ is unitarily equivalent to translating the position of the boundary, Π can be seen as the force per unit energy the edge states in the gap of the system exhibit on the boundary [Kel].
Hulls and ergodic theorems
So far we have worked with a single potential and its translates. When completed w.r.t. a natural metric topology this set of translates yields a topological space, called the hull of the potential. As it has become apparent in recent years, many topological invariants of the physical system depend mainly on the topology of this hull with its R action by translation of the potential. Besides, the use of invariant ergodic probability measures on the hull allows to tackle the problem of existence of the Λ-means in a probabilistic sense. It is therefore most natural to interprete the results of the last section in the framework of R-actions on hulls. This allows for a generalisation to higher dimensional systems, to which the theorems of Section 2 do not extend.
Given a potential V consider its hull
Ω = {V ξ |ξ ∈ R} ,
which is the compactification of the set of translates of V in the sense of [Jo86,Be92]. The action of R by translation of the potential extends to an action on Ω by homeomorphisms which we denote by ω → x · ω. The elements of Ω may be identified with those real functions (potentials) which may be obtained as limits of sequences of translates of V . We shall write V ω for the potential corresponding to ω ∈ Ω. If ω 0 is the point of Ω corresponding to V then
V ξ = V −ξ·ω 0 . Also V y·ω (x) = V ω (x − y)
and so the family of Hamiltonians
H ω = −∂ 2 + V ω is covariant in the sense that H x·ω = U(x)H ω U * (x) were U(x) is the operator of translation by x.
The bulk spectrum is by definition the union of their spectra. The valididty of the following theorem, namely that Ω carries an R-invariant ergodic probability measure, can be verified for many situations, see [BHZ00] for considerations relating it to the Gibbs measure.
Theorem 5 Suppose that (Ω, R) carries an invariant ergodic probability measure P. Let ∆ be a gap in the bulk spectrum and E ∈ ∆. Then almost surely (w.r.t. this measure) the limits to define α Λ (H ω , E) and Π Λ (H ω , ∆) exist and are independent of Λ and ω ∈ Ω. The almost sure value of Π Λ is the P-average
Π(∆) = 1 2iπ Ω dP(ω)Tr((U * ω − 1)δ ⊥ U ω ) where (δ ⊥ f )(ω) = df (t·ω) dt t=0
and U ω is defined as in (4) withĤ ω in place ofĤ ξ .
Proof: The crucial input is Birkhoff's ergodic theorem which allows to replace
lim n→∞ 1 |Λ n | Λn F (x · ω)dx = Ω dPF (ω)
for almost all ω and any F ∈ L 1 (Ω, P). The corresponding construction for the rotation number α has been carried out in [JM82] for almost periodic potentials and for the more general set up in [Jo86,Be92]. For Π Λ the relevant function is F (ω) = Tr((U * ω − 1)δ ⊥ U ω ) which leads to the expression of the almost sure value of Π Λ . ✷
K-theoretic interpretation
The dynamical system (Ω, R) does not depend on the details of V , but only on its spatial structure (or what may be called its long range order). In fact, for systems whose atomic positions are described by Delone sets there are methods to construct the hull directly from this set, c.f. [BHZ00,FHK02]. The detailled form of the potential is rather encoded in a continuous function v : Ω → R so that V ω (x) = v(−x · ω) is the potential corresponding to ω. C(Ω) is thus the algebra of continuous potentials for a given spatial structure.
If one combines this algebra with the Weyl-algebra of rapidly decreasing functions of momentum operators one obtains the algebra of continuous observables which is the C * -crossed product C(Ω)⋊ ϕ R. It is the C * -closure of the convolution algebra of functions f :
R → C(Ω) with prod- uct f 1 f 2 (x) = R dyf 1 (y)ϕ y f 2 (x−y) and involution f * (x) = ϕ x f (−x), where ϕ y (f )(ω) = f (y·ω).
It has a faithful family of representations {π ω } ω∈Ω on L 2 (R) by integral operators,
x|π ω (f )|y = f (y − x)(−x · ω).
It has the following important property. For each continuous function F : R → C vanishing at 0 and ∞ there exists an elementF ∈ C(Ω) ⋊ ϕ R such that F (H ω ) = π ω (F ). Some of the topological properties of the family of Schrödinger operators {H ω } ω∈Ω are therefore captured by the topology of the C * -algebra. The invariant measure P over Ω gives rise to a trace T :
C(Ω) ⋊ ϕ R → C, T (f ) = Ω dPf (0).
Theorem 6 ( [Be92]) Let E be in a gap of the bulk spectrum of {H ω } ω∈Ω so that in particular there exists a projectionP E ∈ C(Ω) ⋊ ϕ R such that π ω (P E ) = P E (H ω ) is the projection onto the spectral subspace of H ω to energies below the gap. Suppose that the potential which gave rise to the hull Ω is smooth. Then the almost sure value of
IDS Λ (H, E) is IDS(E) := T (P E ).
We mention that this result is more subtle than just an application of Birkhoff's theorem and interpretating the result in C * -algebraic terms as it needs a Shubin type argument which holds for smooth potentials, namely lim n→∞ 1 |Λ n | (Tr(P E (H Λn ) − Tr(χ Λn P E (H))) = 0.
The elementP E is a projection. As any trace on a C * -algebra, T depends only on the homotopy class ofP E in the set of projections of C(Ω) ⋊ ϕ R. The even K-group K 0 (C(Ω ⋊ ϕ R) is constructed from homotopy classes of projections and the map on projections P → T (P ) induces a functional on this group, or stated differently, the elements of the K 0 -group pair with T . It is therefore reasonable to refer to T (P E ) as an even K-gap label (or K 0 -theory gap label) of the gap. This is the K 0 -theoretical gap labelling of [BLT85,Be92].
There is a similar identification of the odd K-gap label as the result of a functional applied to the odd K-group of a C * -algebra. This C * -algebra is the C * -algebra of observables on the half space near 0, the position of the boundary. It turns out to be convenient to consider also the cases in which the boundary is at s = 0. We therefore consider the space Ω × R with the product topology. This topological space, whose second component denotes the position of the boundary, carries an action of R by translation of the potential and the boundary (so that their relative position remains the same). The relevant C * -algebra is then the crossed product (constructed as above) C 0 (Ω × R) ⋊φ R withφ y (f )(ω, s) = f (y · ω, s + y). It has a family of representations {π ω,s } ω∈Ω,s∈R on L 2 (R) by integral operators,
x|π ω,s (f )|y = f (y − x)(−x · ω, s − x).
It has the following important property: for each continuous function F : R → C vanishing at 0 and ∞ and such that F (H ω ) = 0 for all ω, there exists an elementF ∈ C 0 (Ω × R) ⋊φ R such that F (H ω,s ) = π ω,s (F ), where H ω,s is the restriction of H ω to R ≤s with Dirichlet boundary conditions at s. Let U = {U ω,s }, U ω,s := P ∆ e 2iπ Hω,s−E 0
|∆| + 1 − P ∆ ,(7)
similar to (4). The product measure of P with the Lebesgue measure is an R-invariant measure on Ω × R and defines a traceT (f ) = Ω R dPdsf (0).
Π Λ (H, ∆) = Π(∆) := 1 2iπT ( U * − 1δ ⊥ U − 1).
The expression of the theorem depends only on the homotopy class of U − 1 + 1 in the set of unitaries of (the unitization of) C 0 (Ω × R) ⋊φ R. The odd K-group K 1 (C 0 (Ω × R) ⋊φ R) is constructed from homotopy classes of unitaries and the map on unitaries U →T ((U * − 1)δ ⊥ U) induces a functional on this group. It is therefore that we refer to 1 2iπT ( U * − 1δ ⊥ U − 1) as an odd K-gap label of the gap.
The proof of the following theorem is based on the topology of the above C * -algebras.
Theorem 8 ( [Kel]) T (P E ) = 1 2iπT ( U * − 1δ ⊥ U − 1). In other words, IDS(E) = Π(∆), E ∈ ∆.
Conclusion and final remarks
We have discussed four quantities which serve as gap-labels for one-dimensional Schrödinger operators. They are all equal but their definition relies on different concepts. The Johnson-Moser rotation number α measures the mean oscillation of a single solution. The Dirichlet rotation number β counts the mean winding of the eigenvalues of the halfsided operators around a circle compactification of the gap. Π and IDS are operator algebraic expressions with concrete physical interpretations, the boundary force per energy and the integrated density of states. Whereas the identities α = β = Π are rather elementary, their identity with IDS is based on a fundamental theorem, the Sturm-Liouville theorem. We tend to think therefore of Π as the natural operator algebraic formulation of the Johnson-Moser rotation number and of Theorem 8 as an operator analog of the Sturm-Liouville theorem. The advantage is that Π, IDS and Theorem 8 generalise naturally to higher dimensions [Kel]. In fact, the expression for IDS is the same as in (3) if one uses Føllner sequences {Λ n } n for R d . The expression of Π Λ in R d requires a choice of a d − 1-dimensional subspace, the boundary, and soĤ ξ is the restriction of the Schrödinger operator H ξ = −Σ j ∂ 2 j + V ξ , V ξ (x) = V (x + ξe d ), to the half space R d−1 × R ≤0 with Dirichlet boundary conditions. Then Π Λ = − lim n→∞ 1 |Σ n |(b n − a n ) bn an Tr((U * ξ,Σn − 1)∂ ξ U ξ,Σn )dξ , U ξ,Σn = P ∆ (Ĥ ξ,Σn )e 2πiĤ ξ,Σn −E 0 |∆| + 1 − P ∆ (Ĥ ξ,Σn ) .
Here Σ n is a Føllner sequence for the boundary andĤ ξ,Σn is the restriction of H ξ to Σ n × R ≤0 with Dirichlet boundary conditions. We do not know of a direct link between this expression and the generalisation proposed by Johnson [Jo91] for odd-dimensional systems.
shall prove that the Johnson-Moser rotation number is a rotation number which is defined by right Dirichlet values as a function of ξ. We choose a gap ∆ in σ(H ξ ) = σ(H) for this section and define the set of right Dirichlet values in ∆ D ξ (∆) := {µ ∈ ∆|∃Ψ : (H ξ − µ)Ψ = 0 and Ψ(0) = Ψ(−∞) = 0} . Thus with respect to this choice of gap we can define S(µ) := {η|µ ∈ D η (∆)}. Suppose µ ∈ D ξ (∆) for some ξ (in particular, D ξ (∆) = ∅). Then there exists a non-zero solution (H ξ − µ)Ψ = 0 satisfying Ψ(0) = Ψ(−∞) = 0. Let Z(µ, ξ) := {x|Ψ(x − ξ) = 0}.
Theorem 7 ([Kel]) Let ∆ be a gap in the bulk spectrum of {H ω } ω∈Ω . The almost sure value of Π(∆) is
Acknowledgements:The second author would like to thank EPSRC for financial support (contract number GR/R64995/01) and the University of Lyon I, Institute Girard Desargues, for its hospitality.
E D Belokolos, A I Bobenko, V Z Enol'skii, A R Its, V B Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer-VerlagE.D. Belokolos, A.I. Bobenko, V.Z. Enol'skii, A.R. Its, V.B. Matveev, Algebro- Geometric Approach to Nonlinear Integrable Equations, Springer-Verlag 1995.
J Bellissard, R Lima, D Testard, Almost periodic Schrödinger operators, 1-64 in Mathematics + Physics. SingaporeWorld Scientific Publishing1J. Bellissard, R. Lima, D. Testard, Almost periodic Schrödinger operators, 1-64 in Mathematics + Physics. Vol. 1, World Scientific Publishing, Singapore 1985.
Gap labelling theorems for Schrödinger operators. J Bellissard, From Number Theory to Physics. BerlinSpringerJ. Bellissard, Gap labelling theorems for Schrödinger operators, 538-630, in From Num- ber Theory to Physics, (Springer, Berlin, 1992).
Hulls of aperiodic solids and gaplabelling theorems. J Bellissard, D J L Herrmann, M Zarrouati, Directions in Mathematical Quasicrystals. R.V. MoodyRIAmer. Math. Soc. ProvidenceJ. Bellissard, D.J.L. Herrmann, M. Zarrouati, Hulls of aperiodic solids and gap- labelling theorems, 217-258, in Directions in Mathematical Quasicrystals, M. Baake and R.V. Moody, editors, Amer. Math. Soc. Providence, RI, 2000.
E A Coddington, N Levinson, Theory of Ordinary Differential Equations. McGraw-HillE.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw- Hill 1955.
Cohomology of canonical projection tilings. A H Forrest, J Hunton, J Kellendonk, Comm. Math. Phys. 2262A.H. Forrest, J. Hunton, and J. Kellendonk, Cohomology of canonical projection tilings, Comm. Math. Phys. 226 (2002), no. 2, 289-322.
Edge States in the integer quantum Hall effect and the Riemann surface of the Bloch function. Y Hatsugai, Phys. Rev. Lett. 48Phys. Rev. BY. Hatsugai, Edge States in the integer quantum Hall effect and the Riemann surface of the Bloch function, Phys. Rev. B 48, 11851-11862 (1993). The Chern Number and Edge States in the Integer Quantum Hall Effect, Phys. Rev. Lett. 71, 3697-3700 (1993).
The rotation number for almost periodic potentials, Commun. R Johnson, J Moser, Math. Phys. 84R. Johnson, J. Moser, The rotation number for almost periodic potentials, Com- mun. Math. Phys. 84, 403-438 (1982).
Exponential Dichotomy, Rotation Number, and Linear Differential Operators with Bounded Coefficients. R Johnson, J. Differential Equations. 61R. Johnson, Exponential Dichotomy, Rotation Number, and Linear Differential Opera- tors with Bounded Coefficients, J. Differential Equations 61, 54-78 (1986).
Oscillation theory for the odd-dimensional Schrödinger operator. R Johnson, J. Differential Equations. 92R. Johnson, Oscillation theory for the odd-dimensional Schrödinger operator, J. Differ- ential Equations 92, 145-162 (1991).
Perturbation Theory for Linear Operators. T Kato, Springer-VerlagT. Kato Perturbation Theory for Linear Operators, Springer-Verlag 1995.
Quantization of edge currents for continuous magnetic operators. J Kellendonk, H Schulz-Baldes, J. Funct. Anal. 209J. Kellendonk, H. Schulz-Baldes, Quantization of edge currents for continuous mag- netic operators, J. Funct. Anal. 209, 388-413 (2004).
Boundary maps for C * -crossed products with R with an application to the Quantum Hall Effect. J Kellendonk, H Schulz-Baldes, Commun. Math. Phys. to appear inJ. Kellendonk, H. Schulz-Baldes, Boundary maps for C * -crossed products with R with an application to the Quantum Hall Effect, to appear in Commun. Math. Phys..
Topological quantization of boundary forces and the integrated density of states. J Kellendonk, J. Phys. A. 37[Ke04] J. Kellendonk, Topological quantization of boundary forces and the integrated density of states, J. Phys. A. 37 L161-L166 (2004).
Gap Labelling and the pressure on the boundary. J Kellendonk, Preprint mpJ. Kellendonk, Gap Labelling and the pressure on the boundary, Preprint mp-arc 04-213.
| [] |
[
"Aggregation dynamics of active rotating particles in dense passive media",
"Aggregation dynamics of active rotating particles in dense passive media"
] | [
"Juan L Aragones ",
"Joshua P Steimel ",
"Alfredo Alexander-Katz "
] | [] | [] | Active matter systems are able to exhibit emergent non-equilibrium behavior due to activity-induced effective interactions between the active particles. Here we study the aggregation and dynamical behavior of active rotating particles, spinners, embedded in 2D passive colloidal monolayers. Using both experiments and simulations we observe aggregation of active particles or spinners whose behavior resembles classical 2D Cahn-Hilliard coarsening. The aggregation behavior and spinner attraction depend on the mechanical properties of the passive monolayer and the activity of spinners. Spinner aggregation only occurs when the passive monolayer behaves elastically and when the spinner activity exceeds a minimum activity threshold. Interestingly, for the spinner concentrations investigated here, the spinner concentration does not seem to change the dynamics of the aggregation behavior. There is a characteristic cluster size which maximizes spinner aggregation by minimizing the drag through the passive monolayer and maximizing the stress applied on the passive medium. We also show a ternary mixture of passive particles and co-rotating and counter-rotating spinners that aggregate into clusters of co and counter-rotating spinners respectively. | 10.1039/c8sm02207k | [
"https://pubs.rsc.org/en/content/articlepdf/2019/sm/c8sm02207k"
] | 118,931,444 | 1701.06930 | 058e17405afaa13a0a352cf46f9100041966bbbf |
Aggregation dynamics of active rotating particles in dense passive media
Juan L Aragones
Joshua P Steimel
Alfredo Alexander-Katz
Aggregation dynamics of active rotating particles in dense passive media
10.1039/c8sm02207kThis journal is Soft Matter, 2019, 15, 3929--3937 | 3929 Cite this: Soft Matter, 2019, 15, 3929
Active matter systems are able to exhibit emergent non-equilibrium behavior due to activity-induced effective interactions between the active particles. Here we study the aggregation and dynamical behavior of active rotating particles, spinners, embedded in 2D passive colloidal monolayers. Using both experiments and simulations we observe aggregation of active particles or spinners whose behavior resembles classical 2D Cahn-Hilliard coarsening. The aggregation behavior and spinner attraction depend on the mechanical properties of the passive monolayer and the activity of spinners. Spinner aggregation only occurs when the passive monolayer behaves elastically and when the spinner activity exceeds a minimum activity threshold. Interestingly, for the spinner concentrations investigated here, the spinner concentration does not seem to change the dynamics of the aggregation behavior. There is a characteristic cluster size which maximizes spinner aggregation by minimizing the drag through the passive monolayer and maximizing the stress applied on the passive medium. We also show a ternary mixture of passive particles and co-rotating and counter-rotating spinners that aggregate into clusters of co and counter-rotating spinners respectively.
Introduction
Attractive interactions between particles in a homogeneous mixture induce the formation of clusters. These clusters grow in time until the mixture separates into two distinct phases. The dynamics of phase separation in binary mixtures is well characterized [1][2][3] and depends on the dimensionality, thermodynamic conditions, and type of cluster growth, diffusion or source/interface limited. Often, these attractive interactions are induced by direct chemical interactions. Alternatively, they can also be induced by electromagnetic, phoretic, 4 collision 5 or hydrodynamic forces. [6][7][8] Non-equilibrium interactions can also promote particle aggregation. Due to the non-equilibrium nature of active matter systems, these are excellent candidates to study novel mechanisms of particle aggregation and subsequent phase separation that may deviate from traditional Cahn-Hilliard models.
Active matter systems are composed of active agents that consume energy from their environment and convert it into motion or mechanical forces. The most prominent examples of these active systems are living organisms, which exhibit striking emergent non-equilibrium behaviors such as swarming, lining, vortexes, etc. 9-23 Synthetic active systems that are able to mimic and reproduce some of the emergent behavior exhibited by living organisms can be used as model systems to study the underlying physical principles which govern their behavior. These synthetic active systems are composed of active units that locally convert energy into motion. This activity perpetually drives these systems out-of-equilibrium. Alternatively, activity can also be induced via an externally applied field or stimuli, some examples of which include magnetic or electric fields, light-catalyzed chemical reactions, vibrating granular beds and optical tweezers. [24][25][26] One particular mode of activity, rotation in plane, has recently aroused a great deal of interest for synthetic and simulation active matter studies. These systems of micro-rotors or spinning particles have been studied in theory and simulations [27][28][29][30][31] using magnetic particles, 24,32 Janus particles, 25,33 rotating robots 34 and even some biological systems 35 as the active units. Almost all of these studies report some type of emergent nonequilibrium steady state where aggregation or phase separation occurs and most of the analysis tends to focus on the dynamics of the active units, the structure or order of the aggregate or active matter system, the origin of the aggregation (typically via some type of cooperative alignment of the active units) and developing a phase diagram of the emergent non-equilibrium steady states as a function of the activity or density of the active units. There have been very few studies that investigate and characterize the dynamics of these active rotating aggregates in terms of the evolution of the cluster size as a function of time and as a function of activity, active particle concentration, and area fraction of the monolayer in order to compare the phase separation behavior of active systems to traditional Cahn-Hilliard behavior. Additionally, in most research works these spinning active systems contain only active components. This is problematic as most biological systems or processes are not composed of purely active components. In biological systems the active or motile components, i.e. cells, are often surrounded by immobile, passive, or even abiotic interfaces. Investigating emergent non-equilbrium behavior and characterizing the aggregation dynamics in such an artificial model system composed of active and passive components can potentially help distinguish which biological interactions can be attributed to purely physical phenomena and which interactions require presumably physical and biological/biochemical stimuli.
Here, we study a model active matter system that is composed of both passive and active components to study the aggregation of active particles. The active component in this system is superparamagnetic particles embedded in a dense monolayer of passive particles. The superparamagnetic particles are made active upon actuation of an externally rotating magnetic field (around the axis perpendicular to the monolayer) causing the particles to spin, henceforth referred to as spinners and schematically shown in Fig. 1A. In this system, active particles exhibit a long-range attractive interaction, 36,37 which emerges from the non-equilibrium nature of the system and is mediated by the mechanical properties of the passive medium. We observe that spinners embedded in a dense passive monolayer tend to aggregate forming clusters which grow in time, as schematically shown in Fig. 1. By means of experiments and numerical simulations we demonstrate that this spinner aggregation is driven by the non-equilibrium attractive interaction induced by the mechanical properties of the passive matrix. Moreover, we show that the spinner aggregation process follows a dynamics that resembles spinodal decomposition in passive liquids and coalescence. The dynamics of the aggregation process depends on the mechanical properties of the monolayer as well as the activity of the spinners. This type of non-equilibrium attractive interaction opens the door to controlling the state of the system via control of the mechanical properties of the medium, the activity of the spinners, and the density of the spinners.
Materials and methods
Experiments
Our synthetic model system is composed of active spinning particles and passive particles. The spinners are superparamagnetic polymer-based magnetite particles purchased from Bangs Laboratories, while the passive particles are composed of polystyrene purchased from Phosphorex; both, active and passive, are 3 mm in diameter. We use a concentrated solution of spinners, E4 mg mL À1 , and passive particles, E0.8 mg mL À1 , to study the aggregation and dynamical behavior of the spinners. The spinners are made active by externally applying a magnetic field which rotates around the axis perpendicular to the plane of the monolayer. The solution of spinners and passive particles is inserted into a channel (22 mm (L) Â 3 mm (W) Â 300 mm (H)), fabricated using a glass slide, spacer, and cover slip. Once the solution is inserted into the channel it is sealed with epoxy and allowed to sediment for 10 minutes to form a dense monolayer, before being magnetically actuated. Using particles rolling on a glass slide we demonstrate that there is no interaction between the spinner and the wall. These particles rolling on a glass substrate translate at much lower velocities, close to the limit where the only friction that occurs is due to hydrodynamics. The strength of the magnetic field is 5 mT, which is large enough to maintain alignment of the superparamagnetic particles with the rotational frequency of the field. The magnetic field is actuated at an angular frequency, o, of 5 Hz for approximately 10 minutes, with the rotational sense switched every 2 minutes. This rotational frequency corresponds to Re = 1.25 Â 10 À6 . The combination of the strength of the applied field and the rotational frequency used imparted enough activity to the superparamagnetic particles to induce the emergent attractive interaction mediated by the elastic passive medium. Ferromagnetic particles can also be used as the active unit alternatively and similar aggregation behavior is observed.
Simulations
In addition, we carry out numerical simulations of this system. In particular our coarse grained model consists of pseudo-hard sphere particles, 38 N = 324, suspended in a fluid of density r = 1 and kinetic viscosity n = 1/6 modeled using the lattice-Boltzmann method. We use the fluctuating lattice-Boltzmann equation 39 with k B T = 2  10 À5 and the solver D3Q19. We discretized the simulation box in a three dimensional grid of N x  N y  N z = 101  101  20 bounded in the z direction by no-slip walls and periodic boundary conditions in the x and y directions. We set the grid spacing, Dx, and time step, Dt, equal to unity. The particles are treated as real solid objects 40 of diameter s = 4Dx. The momentum exchange between the fluid and solid particles is calculated following a moving boundary condition, 41 which allows us to calculate the forces and torques exerted on the particles from the momentum transferred from the fluid. The particles settle on the bottom wall of the channel forming a monolayer under the action of a gravitational force, F G = 0.005. The activity is achieved by imposing a constant torque, which in general corresponds to Re = 0.72, unless otherwise noted.
Results and discussion
To investigate the emergent aggregation behavior of active rotating particles embedded in monolayers of passive particles, we created a dense passive monolayer of inactive polystyrene particles with a particle area fraction, f A , of approximately f A E 0.7. The passive monolayer was doped with active superparamagnetic particles so that the particle fraction of the active component was approximately 0.5%. Upon actuation of the magnetic field, we observe that spinners aggregate forming nearly circular actively rotating clusters, whose average radius, hR cluster i, grows with time, as shown in Fig. 2A. These clusters of spinners can be seen growing over time in the experimental snapshots at the top of Fig. 2A, where the spinners are the darker regions in the snapshots. This behavior is markedly different from that observed in a system of purely active superparamagnetic and ferromagnetic particles, [42][43][44][45][46][47] as shown in Fig. 2B. At similar particle area fractions and rotational frequencies in the purely active system the aggregation of particles is dominated by magnetic dipole-dipole interactions. 36,37 This magnetic force decays as r À4 so aggregation will only occur between clusters that are initially very close together. We observe the formation of small chains or random aggregates that slightly grow with time due to fluid flow induced motion, as seen in Fig. 2B. However, there is no emergent long range attractive interaction that drives spinner aggregation in the purely active case and as a result the spinner aggregation is much slower; a phenomenon observed in other similar systems. 46,47 This emergent increase in aggregation growth is counter-intuitive because in pure equilibrium arguments one would expect the monolayer to impede aggregation because of the high viscosity of the media.
We conducted numerical simulations of this system to study the role of the passive matrix in the spinner aggregation process, and consider whether magnetic dipole-dipole interactions play a major role in the spinner aggregation process, particularly at large cluster sizes. Therefore, our coarse grained model neglects the dipole-dipole interactions to isolate the effect of the passive matrix in the spinner aggregation process. In agreement with the experimental results we observe that spinners aggregate forming circular clusters when embedded in dense monolayers of passive particles of f A = 0.8. We calculate the time evolution of the area of the active clusters, A(t), for the experimental and simulation trajectories, as shown in Fig. 3. We again observe that the size of the clusters grows with time. Hence, the presence of the passive monolayer promotes aggregation of the spinners, even in the absence of magnetic dipole-dipole interactions. Interestingly, we find that A(t) p t 1/2 for the simulations and A(t) p t 0.37 for the experimental system. For a classical Cahn-Hilliard spinodal decomposition scenario in 2D, one would expect that the characteristic length of the clusters of active particles will scale as t 1/3 in the case of diffusion-limited coarsening or as t 1/2 in the case of interface/source-limited coarsening. In diffusion-limited coarsening, the cluster-matrix interface acts as a sink and maintains the concentration of particles at the interface and thus the aggregation is determined by the rate of diffusion of particles towards the clusters. On the contrary, in the interface-limited case the diffusion of particles is fast and the limiting factor is the interface, which acts as a poor sink, not accommodating incoming particles quickly enough. Our simulation results appear to place us in the interface or source-limited scenario while our experimental scaling law is closer to the diffusion limited case, which may be due to the lack of magnetic dipole-dipole interactions in the simulation. The absence of dipole-dipole interactions induces fluctuations in the time evolution of the domain length, A(t). When two clusters collide, the clusters split into pieces and while the new merged cluster is re-configuring the size of the clusters fluctuates.
There have been a number of purely active simulation systems that have found the interface limited scenario scaling of t 1/2 , although the origin of this scaling behavior is still unclear. 48,49 Recently, a micro-rotor system with spinning robots as the active unit exhibited a characteristic length which scaled as t 1/3 , the diffusion limited scenario. 34 However, as a difference from previous studies that only considered purely active systems, our system is composed of both active and passive components. What seems clear is that the aggregation dynamics of spinners is enhanced by the presence of the dense passive monolayer when compared to the purely active spinner system. The scaling behavior in the purely active system is about 0.012, in agreement with previous results. 46,47 The difference is even more pronounced with another system at similar field strengths but higher rotational frequencies where the scaling factor of t 0.37 is orders of magnitudes larger. 46,47 This stark difference in aggregation dynamics between the purely active and hybrid system is due to a recent result where we showed that an attractive interaction emerges between two co-rotating particles, or spinners, if embedded in dense passive monolayers. 36,37 This emergent attractive interaction and the subsequent non-equilibrium phase separation are thus mediated by the elasticity of the medium and the ability of the spinners to stress that medium. Under the actuation of the rotating magnetic field, the spinners rotate around the axis perpendicular to the substrate generating a rotational fluid flow. [50][51][52] This causes the surrounding passive particles to rotate due to the momentum transferred through the fluid in which the particles are suspended. In addition, at small but finite Re, the spinner's rotational motion produces a so-called secondary flow due to the fluid inertia, which pushes away the nearest shell of passive particles, effectively compressing the passive monolayer. Thus, two co-rotating spinners apply compressive and shear stresses on the passive particles located in between the spinners, referred to as the bridge. This produces a stochastic, but steady degradation of the bridge, which allows the spinners to approach, resulting in an attractive interaction. 36 Moreover, depending on the mechanical properties of the passive monolayer, this attractive interaction between active rotating particles may be of a very long-range nature. 37 The solid-like character of the passive monolayer induces an attractive interaction between the active particles resulting in the aggregation of the spinners embedded in passive matrixes. However, we do not observe the complete phase separation of the system into passive and active domains for the actuation time period investigated. Instead, spinners aggregate forming clusters which grow with time embedded within the passive matrix. The dynamics of this aggregation process resembles spinodal decomposition, in which active clusters coalesce. From the experimental trajectories we compute the time evolution of the number of active clusters, N(t), as shown in Fig. 4A. We observe two different dynamical regimes within the experimental time scale. At short time scales (less than 100 s) the clusters exhibit an initial regime of slow cluster growth, or an almost constant number of clusters. This makes sense as we previously determined that the characteristic time scale of continuous activity, at a rotational frequency of 5 Hz, to induce the emergent long range attractive interaction between spinners is on the order of 10-100 s. So this slow growth regime appears because the spinners must first stress the elastic medium before the attractive interaction that drives aggregation emerges. This is followed by another regime (after 100 s) where the spinners aggregate at a much faster rate. As it can be seen in Fig. 4A, the scaling of the number of clusters with time in this regime is characterized by an exponent of EÀ0.7. We also analyze the dynamic scaling of the aggregation of spinners in our simulation model. In this case, we also observe two dynamical regimes: an initial slow decrease in the number of clusters (i.e. growth of the cluster sizes), followed by a regime with dynamic scaling of exponent EÀ0.5, as shown in Fig. 4B. Moreover, this dynamic scaling seems independent of the spinner concentration. The t 1/2 dynamical scaling of the cluster growth has been observed in simulations conducted by Vicsek and several distinct purely active systems, although the origin of this scaling behavior is still unclear. 48,49 However, we believe that the origin of this scaling behavior is similar to that observed in traditional 2D coarsening 53,54 in the source or interface limited case. Interestingly the slow growth regime scaling seems to correspond to the diffusion limited coarsening case but we actually see enhanced scaling beyond the source
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limited scenario which must be due to the emergent interaction induced by the presence of the dense passive monolayer. We hypothesize that the difference between the exponents of the dynamic scaling observed in experiments and simulations is due to the magnetic interaction between clusters of spinners, which increases the strength of the spinner-spinner attraction at short distances. Additionally, this also helps to stabilize spinner clusters. If we also assume a dynamic scaling factor for the slower initial regime in the spinner aggregation process, we obtain for experiments and simulations exponents E0.29 (2) and E0.04 (5), respectively. In this first aggregation regime the differences between the experimental and simulation dynamic scaling exponents are much greater than for the second regime. This behavior is in agreement with our hypothesis that the effect of the magnetic dipole-dipole interactions significantly increases the spinner aggregation dynamics. At the beginning of the aggregation process, active particles that are close to one another (less than 4 particle diameters apart) will form a cluster almost instantaneously upon application of the external magnetic field. Also, when active clusters are small the dipoles of the particles are more easily aligned, which results in higher magnetization of the clusters. In contrast, as the size of the clusters increases, some of the dipoles of the particles are frustrated by the cluster structure, which results in a smaller magnetization of the clusters as their size increases. Therefore, the magnetic interaction is more relevant between smaller clusters than between bigger clusters.
The mechanical properties of the passive media determine the interaction between the spinners and thus the dynamics of the spinner aggregation. The mechanical properties of the monolayer can be calculated by measuring the mean square displacement (MSD) of the particles in the monolayer in the absence of active particles, specifically the storage and loss moduli, G 0 and G 00 , respectively. 55,56 In simulations, we observe that monolayers of hard-sphere particles at area fractions f A 4 0.7 respond as viscoelastic materials, behaving as a viscous system at low frequencies and as a solid-like material at high frequencies. 36 However, for f A o 0.7 the monolayer behaves as a viscous material over the entire frequency range.
To study the effect of the mechanical properties of the monolayer on the dynamics of the spinner aggregation, we investigate, by means of our simulation model, spinners embedded in passive monolayers at different particle area fractions f A = 0.5, 0.7, and 0.8. As can be seen in Fig. 5, spinners in monolayers of a particle area fraction of f A = 0.8 and 0.7 follow similar scaling laws. However, at an area fraction of f A = 0.5, the spinners do not aggregate within the simulation time scale, as shown by the green diamonds in Fig. 5. The small amount of spinner aggregation observed in Fig. 5 is due to spinners being initially positioned together or close enough so that the removal of a single passive row of particles was required. It should also be noted that for a more dilute concentration of spinners no aggregation is observed on the simulation timescale (data not shown). In addition, at these particle area fractions the monolayer is unable to maintain spinners within a cluster and thus the number of clusters exhibits large fluctuations. Thus, the presence of a passive monolayer that behaves as a solid-like material induces an attractive interaction between the active rotating particles, which results in aggregation of spinners. Interestingly, the dynamics of the spinner aggregation seems to be independent of the storage modulus, G 0 , which is higher for a monolayer of f A = 0.8 than for a monolayer of f A = 0.7. This might be due to canceling of two competing effects. On one hand, the effective interaction grows with G 0 , but on the other hand the motion of the medium is controlled by Z, which also grows.
For spinners in a passive monolayer with a packing fraction f A 4 0.7 the mechanics of the passive monolayer also plays an important role in keeping the cluster of active particles together. We have previously reported that for a system composed of purely active particles the spinners will repel due to the secondary flows generated by the spinners. 36,57 The spinners within the cluster should then repel, but the passive monolayer exerts a force on the spinners that serves to stabilize the active cluster. This is evident from the stochastic fluctuations shown in the cluster size time evolution, as shown in Fig. 3B and 5, which correspond to clusters breaking and reforming during the aggregation process. The size of the fluctuations increases with the spinners' concentration, due to the bigger size of the clusters. Aside from the mechanical properties of the monolayer, the other requisite for spinner attraction is the ability to stress the passive monolayer. Therefore, we also explore the effect of the spinners' activity on the aggregation dynamics by applying different rotational frequencies, Re = 0.1, 0.72 and 3.58, to spinners embedded in passive matrixes of f A = 0.8. In agreement with our previous observations for the spinner-spinner interaction in passive environments, 36,37 we observe there exists a minimum threshold of loading stress, or spinner activity, for the spinner attractive interaction to be important. Spinners rotating at Re smaller than 0.1 do not aggregate. At these activities, the stress applied to the passive monolayer is not large enough to promote the occurrence of yielding events, which ultimately result in spinner aggregation. 36 On the contrary, spinners rotating at Re Z 0.72 do aggregate, and the higher the rotational frequency, the faster the evolution of the system. Interestingly, the dynamic scaling exponent of the spinner aggregation seems to be independent of the rotational frequency, as shown in Fig. 6. However, the range of the initial dynamical regime, which probably corresponds to the fastest growing unstable composition mode, shifts towards shorter times. The spinner-spinner attraction in passive matrixes follows activated dynamics. 37 The monolayer region in between the two spinners (i.e. the bridge) needs to be loaded before it yields, which has an associated time scale. This time scale depends on the mechanical properties and configuration of the monolayer. If the stress applied by the spinners overcomes this time scale, the passive particle mobility increases, resulting in yielding events, which leads to the erosion or degradation of the bridge. 36 Therefore, the higher the rotational frequency of the spinners (i.e. Re), the shorter time required to stress the bridge and thus as the frequency of the spinners increases the faster the growth of the clusters at short time scales. The differences observed between the experiment and simulations on Re come from the approximations made in our simulation model. For example, in our simulations the momentum transfer between the spinner and neighboring particles comes exclusively from the fluid, while in the experiment friction and collision between particles may play an important role in transferring momentum.
We further investigate the microscopic details of the spinner aggregation process by tracking the active clusters over time, noting when clusters collide, initial separation distances between clusters which merge, and the velocity at which clusters approach, as shown in Fig. 7. In Fig. 7A, the velocity at which spinner clusters approach as a function of cluster size is presented. We observe that there is a maximum velocity associated with a cluster size of approximately 45 mm. This behavior of the cluster velocity as a function of size reveals two competing effects involved in the mobility of the clusters, and therefore their aggregation. The effect which opposes spinner aggregation is the effective drag, which opposes the movement of the clusters through the monolayer, and the drag increases with cluster size. Meanwhile the stress that the spinner cluster can exert on the monolayer increases with the size of the cluster. This increases the frequency of the yielding events resulting in the degradation of the bridge and spinner aggregation. In addition, we observe that the range of the attractive interaction between clusters, R i , increases with the cluster size, as shown in Fig. 7B. Individual spinner clusters were tracked and as the clusters collide and form bigger clusters the initial distance between colliding clusters was calculated and plotted as a function of the average radius of the two colliding clusters. As discussed above, the stress exerted on the monolayer increases with the cluster size. Therefore, this increase in the applied stress on the monolayer results in higher mobility of the passive particles of the monolayer, which results in longer range interactions.
Finally, we explore the behavior of a ternary mixture in which a passive monolayer is doped with a symmetric mixture of spinners rotating in opposite senses, clockwise and counter-clockwise. In our previous work, we demonstrated that while two co-rotating spinners embedded in a passive matrix experience an attractive interaction, counter-rotating spinners exhibit a repulsive interaction in dense passive environments. 36 We observe that spinners in dense passive monolayers tend to form clusters of co-rotating particles and thus we observe the formation of three different phases: (i) passive particles, (ii) spinners rotating clockwise and (iii) spinners rotating counter-clockwise, as shown in Fig. 8. This is the result of the attractive interaction between spinners rotating in the same direction and the repulsive interaction between spinners rotating in opposite directions. Interestingly, again we see a scaling factor of t 1/2 for this system as well which is reminiscent of source limited coarsening. A similar system composed of a ternary mixture of co-rotating, counter-rotating, and passive particles has been previously investigated; 27,29 however the aggregation dynamics were not reported for this system so there is no comparison to be made.
Conclusions
We studied the aggregation of active rotating particles embedded in a passive monolayer. We demonstrate that the non-equilibrium attractive interaction between spinners within dense passive matrixes 36,37 results in their aggregation. This aggregation resembles 2D coarsening, 53 which has also been described for other pure active systems. 48,49 The experimental system seems to exhibit coarsening behavior reminiscent of the diffusion limited scenario, while the simulation appears similar to the source limited case. This difference may be due to the lack of magnetic interactions in the simulations. Although the system size we can reach does not allow us to unambiguously determine the dynamic scaling exponent of the spinner aggregation process, we explore the effect of the particle area fraction of the monolayer, spinner concentration and spinner activity on the aggregation behavior. We observe that the monolayer must behave as a solid, f A 4 0.7, in order to observe spinner aggregation. In addition, for the spinners to stress the monolayer and thus produce yielding events that result in the attraction of spinners, there is a minimum activity threshold, Re 4 0.1, in simulations. Interestingly, the aggregation dynamics seems to be independent of the spinner concentration. We also study the microscopic details of the cluster aggregation. We observe that spinner clusters move faster as the size increases up to a velocity maximum at around R cluster = 45 mm. Finally, we show that a ternary mixture of passive particles and co-rotating and counter-rotating spinners results in the formation of clusters of spinners with the same sense of rotation. This is the result of the attractive interaction between spinners rotating in the same direction, and the repulsive interaction between spinners rotating in opposite directions. Interestingly, again we see a scaling factor of t 1/2 for this system as well which is reminiscent of source limited coarsening.
Conflicts of interest
There are no conflicts to declare.
Fig. 1
1Schematic representation of the system. (A) Co-rotating spinners randomly distributed within a monolayer of passive particles of f A = 0.8, which under the action of the magnetic field rotate around the axis perpendicular to the monolayer plane (i.e. the z-axis). (B) Spinner clusters form due to attraction between active particles.
Fig. 2
2Average spinner cluster radius, hR cluster i, as a function of time for the hybrid active-passive (A) and the purely active system (B). The experimental snapshots show the aggregation of spinners as a function of time for each system. The spinners correspond to the darker spots. The scale bar in the left snapshot corresponds to 100 mm. Soft Matter Paper Open Access Article. Published on 15 April 2019. Downloaded on 8/29/2020 10:14:36 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence.
Fig. 3 (
3A) log-log scale for the spinner cluster size (area), A(t), as a function of time in a passive monolayer of f A = 0.8 for four different spinner concentrations: 1.8% (black circles), 4.94% (red squares), 9.88% (green diamonds) and 20.1% (blue triangles) in simulations. (B) log-log scale for the spinner cluster length from experiments as a function of time in a passive monolayer of f A E 0.7. t* corresponds to the dimensionless time t Áo and the cluster area is scaled by the square particle diameter, Article. Published on 15 April 2019. Downloaded on 8/29/2020 10:14:36 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence.
Fig. 4 (
4A) log-log scale for the number of clusters as a function of time as obtained from the experiments at f A E 0.7. The two sets of data correspond to two independent experiments. (B) log-log scale for the number of clusters as a function of time as obtained from the simulation model for four different spinner concentrations at f A = 0.8: 1.8% (black circles), 4.94% (red squares), 9.88% (green diamonds) and 20.1% (blue triangles). t* corresponds to the dimensionless time tÁo.
Fig. 5 (
5A) log-log scale of the time evolution of the number of active clusters at Re = 0.72 and a spinner concentration of 4.94% within passive monolayers of f A = 0.8 (blue squares), 0.7 (red circles) and 0.5 (green diamonds). (B) log-log scale of the time evolution of the number of active clusters at Re = 0.72 and a spinner concentration of 9.88% within passive monolayers of f A = 0.8 (blue squares), 0.7 (red circles) and 0.5 (green diamonds). t* corresponds to the dimensionless time tÁo.
Fig. 6
6Aggregation dynamics of spinners within passive monolayers of f A = 0.8 at two different rotational frequencies Re = 0.72 (black and red symbols) and 3.58 (blue and green symbols) and spinner concentrations 4.94% (circles and diamonds) and 9.88% (squares and triangles). t* corresponds to the dimensionless time tÁo.
Fig. 7 (
7A) Average approach velocity of the active clusters as a function of the cluster size (i.e. radius of the cluster). (B) Range of the attractive interaction between active clusters as a function of the cluster size. Error bars correspond to the standard deviation of the trials.
Fig. 8
8Aggregation dynamics of spinners rotating clockwise (black circles) at a concentration of 2.47%, and counter-clockwise (red squares) at a concentration of 2.47% within a passive monolayer of f A = 0.8. t* corresponds to the dimensionless time tÁo.
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AcknowledgementsThis work was supported by Department of Energy BES award #ER46919 (theoretical and simulation work) and the Chang Family (experimental work).
J D Gunton, M Miguel, P Sahni, Phase Transition and Critical Phenomena. Academic, London8J. D. Gunton, M. San Miguel and P. Sahni, Phase Transition and Critical Phenomena, Academic, London, 1983, vol. 8.
. J W Cahn, Acta Metall. 9J. W. Cahn, Acta Metall., 1961, 9, 795-801.
. J W Cahn, J E Hilliard, J. Chem. Phys. 28J. W. Cahn and J. E. Hilliard, J. Chem. Phys., 1958, 28, 258-267.
. J Palacci, S Sacanna, A P Steinberg, D J Pine, P M Chaikin, Science. 339J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine and P. M. Chaikin, Science, 2013, 339, 936-940.
. L Corté, P M Chaikin, J P Gollub, D J Pine, Nat. Phys. 4L. Corté, P. M. Chaikin, J. P. Gollub and D. J. Pine, Nat. Phys., 2008, 4, 420-424.
. T Ishikawa, T J Pedley, Phys. Rev. Lett. 88103T. Ishikawa and T. J. Pedley, Phys. Rev. Lett., 2008, 100, 088103.
. A P Berke, L Turner, H C Berg, E Lauga, Phys. Rev. Lett. 38102A. P. Berke, L. Turner, H. C. Berg and E. Lauga, Phys. Rev. Lett., 2008, 101, 038102.
. K Drescher, K C Leptos, I Tuval, T Ishikawa, T J Pedley, R E Goldstein, Phys. Rev. Lett. 168101K. Drescher, K. C. Leptos, I. Tuval, T. Ishikawa, T. J. Pedley and R. E. Goldstein, Phys. Rev. Lett., 2009, 102, 168101.
. J Buhl, D J T Sumpter, I D Couzin, J J Hale, E Despland, E R Miller, S J Simpson, Science. 312J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, Science, 2006, 312, 1402-1406.
. N C Darnton, L Turner, S Rojevsky, H C Berg, Biophys. J. 98N. C. Darnton, L. Turner, S. Rojevsky and H. C. Berg, Biophys. J., 2010, 98, 2082-2090.
. A Ordemann, G Balazsi, F Moss, Phys. A. 325A. Ordemann, G. Balazsi and F. Moss, Phys. A, 2003, 325, 260-266.
. Y Wu, A D Kaiser, Y Jiang, M S Alber, Proc. Natl. Acad. Sci. U. S. A. 106Y. Wu, A. D. Kaiser, Y. Jiang and M. S. Alber, Proc. Natl. Acad. Sci. U. S. A., 2009, 106, 1222-1227.
. H P Zhang, A Be'er, E.-L Florin, H L Swinney, Proc. Natl. Acad. Sci. U. S. A. 107H. P. Zhang, A. Be'er, E.-L. Florin and H. L. Swinney, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 13626-13630.
. S.-N Lin, W.-C Lo, C.-J Lo, Soft Matter. 10S.-N. Lin, W.-C. Lo and C.-J. Lo, Soft Matter, 2014, 10, 760-766.
. J Toner, Y Tu, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 58J. Toner and Y. Tu, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 58, 4828-4858.
. M Ballerini, N Cabibbo, R Candelier, A Cavagna, E Cisbani, I Giardina, V Lecomte, A Orlandi, G Parisi, A Procaccini, M Viale, V Zdravkovic, Proc. Natl. Acad. Sci. U. S. A. 105M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Proc. Natl. Acad. Sci. U. S. A., 2008, 105, 1232-1237.
. A Cavagna, A Cimarelli, I Giardina, G Parisi, R Santagati, F Stefanini, M Viale, Proc. Natl. Acad. Sci. U. S. A. 107A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini and M. Viale, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 11865-11870.
. J L Silverberg, M Bierbaum, J P Sethna, I Cohen, Phys. Rev. Lett. J. L. Silverberg, M. Bierbaum, J. P. Sethna and I. Cohen, Phys. Rev. Lett., 2013, 110, 228701.
. B Szabó, G J Szöllösi, B Gönci, Z Jurányi, D Selmeczi, T Vicsek, Nonlinear, Soft Matter Phys. 61908Phys. Rev. E: Stat.B. Szabó, G. J. Szöllösi, B. Gönci, Z. Jurányi, D. Selmeczi and T. Vicsek, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2006, 74, 061908.
. D F Hinz, A Panchenko, T.-Y Kim, E Fried, Soft Matter. 10D. F. Hinz, A. Panchenko, T.-Y. Kim and E. Fried, Soft Matter, 2014, 10, 9082-9089.
. H Wioland, F G Woodhouse, J Dunkel, J O Kessler, R E Goldstein, Phys. Rev. Lett. H. Wioland, F. G. Woodhouse, J. Dunkel, J. O. Kessler and R. E. Goldstein, Phys. Rev. Lett., 2013, 110, 268102.
. T Sanchez, D T N Chen, S J Decamp, M Heymann, Z Dogic, Nature. 491T. Sanchez, D. T. N. Chen, S. J. DeCamp, M. Heymann and Z. Dogic, Nature, 2012, 491, 431-434.
. J.-F Joanny, S Ramaswamy, Nature. 467J.-F. Joanny and S. Ramaswamy, Nature, 2010, 467, 33-34.
. J E Martin, A Snezhko, Rep. Prog. Phys. 126601J. E. Martin and A. Snezhko, Rep. Prog. Phys., 2013, 76, 126601.
. J Yan, S C Bae, S Granick, Soft Matter. 11J. Yan, S. C. Bae and S. Granick, Soft Matter, 2015, 11, 147-153.
. N Kumar, H Soni, S Ramaswamy, A K Sood, Nat. Commun. 54688N. Kumar, H. Soni, S. Ramaswamy and A. K. Sood, Nat. Commun., 2014, 5, 4688.
. K Yeo, E Lushi, P M Vlahovska, Soft Matter. 12K. Yeo, E. Lushi and P. M. Vlahovska, Soft Matter, 2016, 12, 5645-5652.
. K Yeo, E Lushi, P M Vlahovska, Phys. Rev. Lett. 114188301K. Yeo, E. Lushi and P. M. Vlahovska, Phys. Rev. Lett., 2015, 114, 188301.
. N H P Nguyen, D Klotsa, M Engel, S C Glotzer, Phys. Rev. Lett. 75701N. H. P. Nguyen, D. Klotsa, M. Engel and S. C. Glotzer, Phys. Rev. Lett., 2014, 112, 075701.
. M Spellings, M Engel, D Klotsa, S Sabrina, A M Drews, N H P Nguyen, K J M Bishop, S C Glotzer, Proc. Natl. Acad. Sci. U. S. A. 112M. Spellings, M. Engel, D. Klotsa, S. Sabrina, A. M. Drews, N. H. P. Nguyen, K. J. M. Bishop and S. C. Glotzer, Proc. Natl. Acad. Sci. U. S. A., 2015, 112, E4642-E4650.
. B C V Zuiden, J Paulose, W T M Irvine, D Bartolo, V Vitelli, Proc. Natl. Acad. Sci. U. S. A. 113B. C. V. Zuiden, J. Paulose, W. T. M. Irvine, D. Bartolo and V. Vitelli, Proc. Natl. Acad. Sci. U. S. A., 2016, 113, 12919-12924.
. G Kokot, S Das, R G Winkler, G Gompper, I S Aranson, A Snezhko, Proc. Natl. Acad. Sci. U. S. A. 114G. Kokot, S. Das, R. G. Winkler, G. Gompper, I. S. Aranson and A. Snezhko, Proc. Natl. Acad. Sci. U. S. A., 2017, 114, 12870-12875.
. M Han, J Yan, S Granick, E Luijten, Proc. Natl. Acad. Sci. U. S. A. 114M. Han, J. Yan, S. Granick and E. Luijten, Proc. Natl. Acad. Sci. U. S. A., 2017, 114, 7513-7518.
. C Scholz, M Engel, T Pöschel, Nat. Commun. 931C. Scholz, M. Engel and T. Pöschel, Nat. Commun., 2018, 9, 931.
. A P Petroff, X.-L Wu, A Libchaber, Phys. Rev. Lett. 114158102A. P. Petroff, X.-L. Wu and A. Libchaber, Phys. Rev. Lett., 2015, 114, 158102.
. J L Aragones, J P Steimel, A Alexander-Katz, Nat. Commun. J. L. Aragones, J. P. Steimel and A. Alexander-Katz, Nat. Commun., 2016, 7, 11325.
. J P Steimel, J L Aragones, H Hu, N Qureshi, A Alexander-Katz, Proc. Natl. Acad. Sci. U. S. A. 113J. P. Steimel, J. L. Aragones, H. Hu, N. Qureshi and A. Alexander-Katz, Proc. Natl. Acad. Sci. U. S. A., 2016, 113, 4652-4657.
. J Jover, A J Haslam, A Galindo, G Jackson, E A Muller, J. Chem. Phys. 144505J. Jover, A. J. Haslam, A. Galindo, G. Jackson and E. A. Muller, J. Chem. Phys., 2012, 137, 144505.
. B Dünweg, A Ladd, Adv. Polym. Sci. B. Dünweg and A. Ladd, Adv. Polym. Sci., 2008, 1-78.
. E.-J Ding, C K Aidun, J. Stat. Phys. 112685E.-J. Ding and C. K. Aidun, J. Stat. Phys., 2003, 112, 685.
. A J C Ladd, J. Fluid Mech. 285A. J. C. Ladd, J. Fluid Mech., 1994, 271, 285.
. M Kalontarov, M T Tolley, H Lipson, D Erickson, Microfluid Nanofluid, 9M. Kalontarov, M. T. Tolley, H. Lipson and D. Erickson, Microfluid. Nanofluid., 2010, 9, 551-558.
. J H E Promislow, A P Gast, M Fermigier, J. Chem. Phys. 5492J. H. E. Promislow, A. P. Gast and M. Fermigier, J. Chem. Phys., 1995, 102, 5492.
. J Richardi, J.-J Weis, J. Chem. Phys. J. Richardi and J.-J. Weis, J. Chem. Phys., 2011, 135, 124502.
. Y Gao, M A Hulsen, T G Kang, J M J Toonder, Nonlinear, Soft Matter Phys. 8641503Phys. Rev. E: Stat.Y. Gao, M. A. Hulsen, T. G. Kang and J. M. J. den Toonder, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 86, 041503.
. C Reynolds, D Robinson, D Aarts, M Wilson, W Sampson, R Dullens, Europhys. Lett. 28001C. Reynolds, D. Robinson, D. Aarts, M. Wilson, W. Sampson and R. Dullens, Europhys. Lett., 2016, 116, 28001.
. C P Reynolds, University of OxfordMSc thesisC. P. Reynolds, MSc thesis, University of Oxford, 2016.
. S Dey, D Das, R Rajesh, Phys. Rev. Lett. S. Dey, D. Das and R. Rajesh, Phys. Rev. Lett., 2012, 108, 238001.
. G S Redner, M F Hagan, A Baskaran, Phys. Rev. Lett. 55701G. S. Redner, M. F. Hagan and A. Baskaran, Phys. Rev. Lett., 2013, 110, 055701.
. E Climent, K Yeo, M R Maxey, G E Karniadakis, J. Fluids Eng. 129E. Climent, K. Yeo, M. R. Maxey and G. E. Karniadakis, J. Fluids Eng., 2007, 129, 379-387.
. B A Grzybowski, X Jiang, H Stone, G M Whitesides, Nonlinear, Soft Matter Phys. 11603Phys. Rev. E: Stat.B. A. Grzybowski, X. Jiang, H. Stone and G. M. Whitesides, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2001, 64, 011603.
. B Grzybowski, H Stone, G Whitesides, Nature. 405B. Grzybowski, H. Stone and G. Whitesides, Nature, 2000, 405, 1033-1036.
. K Binder, D Stauffer, Phys. Rev. Lett. 33K. Binder and D. Stauffer, Phys. Rev. Lett., 1974, 33, 1006-1009.
. B Meerson, Rev. Mod. Phys. 68B. Meerson, Rev. Mod. Phys., 1996, 68, 215-257.
. T Mason, D Weitz, Phys. Rev. Lett. 74T. Mason and D. Weitz, Phys. Rev. Lett., 1995, 74, 1250-1253.
. T G Mason, Rheol. Acta. 39T. G. Mason, Rheol. Acta, 2000, 39, 371-378.
. Y Goto, H Tanaka, Nat. Commun. 6Y. Goto and H. Tanaka, Nat. Commun., 2016, 6, 1-10.
| [] |
[
"Graph Algorithms for Topology Identification using Power Grid Probing",
"Graph Algorithms for Topology Identification using Power Grid Probing"
] | [
"Guido Cavraro ",
"Senior Member, IEEEVassilis Kekatos "
] | [] | [] | To perform any meaningful optimization task, power distribution operators need to know the topology and line impedances of their electric networks. Nevertheless, distribution grids currently lack a comprehensive metering infrastructure. Although smart inverters are widely used for control purposes, they have been recently advocated as the means for an active data acquisition paradigm: Reading the voltage deviations induced by intentionally perturbing inverter injections, the system operator can potentially recover the electric grid topology. Adopting inverter probing for feeder processing, a suite of graph-based topology identification algorithms is developed here. If the grid is probed at all leaf nodes but voltage data are metered at all nodes, the entire feeder topology can be successfully recovered. When voltage data are collected only at probing buses, the operator can find a reduced feeder featuring key properties and similarities to the actual feeder. To handle modeling inaccuracies and load nonstationarity, noisy probing data need to be preprocessed. If the suggested guidelines on the magnitude and duration of probing are followed, the recoverability guarantees carry over from the noiseless to the noisy setup with high probability. | 10.1109/lcsys.2018.2846801 | [
"https://arxiv.org/pdf/1803.04506v2.pdf"
] | 49,538,778 | 1803.04506 | d7d91c812efaadf057429e4a42886ce055c81362 |
Graph Algorithms for Topology Identification using Power Grid Probing
10 Jun 2018
Guido Cavraro
Senior Member, IEEEVassilis Kekatos
Graph Algorithms for Topology Identification using Power Grid Probing
10 Jun 2018arXiv:1803.04506v2 [math.OC]Index Terms-Energy systemsidentificationsmart grid
To perform any meaningful optimization task, power distribution operators need to know the topology and line impedances of their electric networks. Nevertheless, distribution grids currently lack a comprehensive metering infrastructure. Although smart inverters are widely used for control purposes, they have been recently advocated as the means for an active data acquisition paradigm: Reading the voltage deviations induced by intentionally perturbing inverter injections, the system operator can potentially recover the electric grid topology. Adopting inverter probing for feeder processing, a suite of graph-based topology identification algorithms is developed here. If the grid is probed at all leaf nodes but voltage data are metered at all nodes, the entire feeder topology can be successfully recovered. When voltage data are collected only at probing buses, the operator can find a reduced feeder featuring key properties and similarities to the actual feeder. To handle modeling inaccuracies and load nonstationarity, noisy probing data need to be preprocessed. If the suggested guidelines on the magnitude and duration of probing are followed, the recoverability guarantees carry over from the noiseless to the noisy setup with high probability.
I. INTRODUCTION
P OWER distribution grids will be heavily affected by the penetration of distributed energy resources. To comply with network constraints, system operators need to know the topologies of their electric networks. Often utilities have limited information on their primary or secondary networks. Even if they know the line infrastructure and impedances, they may not know which lines are energized.
This explains the recent interest on feeder topology processing. Several works capitalize on the properties of grid data covariance matrices to reconstruct feeder topologies; see e.g., [1], [2]. Graphical models have been used to fit a spanning tree relying on the mutual information of voltage data [3]. Tree recovery methods operating on a bottom-up fashion have been devised in [4]; yet they presume non-metered buses have degree larger than two, fail in buses with constant power factor, and lack practical guidelines for handling noisy setups. All the previous approaches build on second-order statistics of grid data. However, since sample statistics converge to their ensemble counterparts only asymptotically, a large number of grid data is typically needed to attain reasonable performance thus rendering topology estimates obsolete.
When the line infrastructure is known, the problem of finding the energized lines can be cast as a maximum likelihood detection problem in [5], [6]. Given power readings at all terminal nodes and selected lines, topology identification has also The been posed as a spanning tree recovery exploiting the concept of graph cycles [7]. Line impedances are estimated via a total least-squares fit in [8]. Presuming phasor measurements and sufficient input excitation, a Kron-reduced admittance matrix is recovered via a low rank-plus-sparse decomposition [9].
Rather than passively collecting data, an active data acquisition paradigm has been suggested in [10]: Inverters are commanded to instantaneously vary their power injections so that the operator can infer non-metered loads by processing the incurred voltage profiles. Perturbing the primary controls of inverters to identify topologies in DC microgrids has also been suggested in [11]. Line impedances have been estimated by having inverters injecting harmonics in [12]. Instead of load learning, grid probing has been adopted towards topology inference in [13], which analyzes topology recoverability via grid probing and estimates the grid Laplacian via a convex relaxation followed by a heuristic to enforce radiality.
The current work extends [13] on three fronts. First, it provides a graph algorithm for recovering feeder topologies using the voltage deviations induced at all nodes upon probing a subset of them (Section IV). Second, topology recoverability is studied under partially observed voltage deviations, an algorithm is devised, and links between the revealed grid and the actual grid are established (Section V). Third, noisy data setups are handled by properly modifying the previous schemes and by providing probing guidelines to ensure recoverability with high probability (Section VI).
II. MODELING PRELIMINARIES Let G = (N , L) be an undirected tree graph, where N is the set of nodes and L the set of edges L := {(m, n) : m, n ∈ N }. A tree is termed rooted if one of its nodes is designated as the root. This root node will be henceforth indexed by 0. In a tree graph, a path is the unique sequence of edges connecting two nodes. The nodes adjacent to the edges forming the path between node m and the root are the ancestors of node m and form the set A m . Reversely, if n ∈ A m , then m is a descendant of node n. The descendants of node m comprise the set D m . By convention, m ∈ A m and m ∈ D m . If n ∈ A m and (m, n) ∈ E, node n is the parent of m. A node without descendants is called a leaf or terminal node. Leaf nodes are collected in the set F , while non-leaf nodes will be termed internal nodes. For each node m, define its depth d m := |A m | as the number of its ancestors. The depth of the entire tree is d G := max m∈N d m . If n ∈ A m and d n = k, node n is the unique k-depth ancestor of node m and will be denoted by α k m for k = 0, . . . , d m . Let also T k m denote the subset of the nodes belonging to the subtree of G rooted at the k-depth node m and containing all the descendants of m.
Our analysis will be built on the concept of the level sets of a node. The k-th level set of node m is defined as [13]
N k m := D α k m \ D α k+1 m , k = 0, . . . , d m − 1 D m , k = d m .(1)
In essence, the level set N k m consists of node α k m and all the subtrees rooted at α k m excluding the one containing node m. Since by definition N k m ⊆ D α k m , the level sets satisfy the ensuing properties that will be needed later.
Lemma 1 ( [13]): Let m be a node in a tree graph. (i) The node α k m is the only node in N k m at depth k; the remaining nodes in N k m are at larger depths;
(ii) if n, s ∈ N k m , then α k n = α k s = α k m ∈ N k m ; (iii) if m ∈ F , then N dm m = {m};
(iv) if s ∈ D n and n ∈ N , then N k n = N k s for k < d n ; and (v) if d m = k, then m ∈ N k m and m / ∈ N ℓ m for ℓ < k. A radial single-phase distribution grid having N + 1 buses can be modeled by a tree graph G = (N , L)rooted at the substation. The nodes in N := {0, . . . , N } denote grid buses, and the edges in L lines. Define v n as the deviation of the voltage magnitude at node n from the substation voltage, and p n + jq n as the power injected through node n. The voltage deviations and power injections at all buses in N \ {0} are stacked in v, p, and q. Let r ℓ + jx ℓ be the impedance of line ℓ, and collect all impedances in r + jx. The so termed linearized distribution flow (LDF) model approximates nodal voltage magnitudes as [14], [1]
v = Rp + Xq(2)
where (R, X) are the inverses of weighted reduced Laplacian matrices of the grid [5]. Let r m be the m-th column of R, and R m,n its (m, n)-th entry that equals [1] R m,n = ℓ=(c,d)∈L c,d∈Am∩An
r ℓ .(3)
The entry R m,n can be equivalently interpreted as the voltage drop between the substation and bus m when a unitary active power is injected as bus n and the remaining buses are unloaded. Leveraging this interpretation, the entries of R relate to the levels sets in G as follows. III. GRID PROBING USING SMART INVERTERS Solar panels and energy storage units are interfaced to the grid via inverters featuring advanced communication, actuation, and sensing capabilities. An inverter can be commanded to shed solar generation, or change its power factor within msec. The distribution feeder as an electric circuit responds within a second and reaches a different steady-state voltage profile. Upon measuring the bus voltage differences incurred by probing, the feeder topology may be identified. Rather than processing smart meter data on a 15-or 60-min basis, probing actively senses voltages on a per-second basis, over which conventional loads are assumed to be invariant.
The buses hosting controllable inverters are collected in P ⊆ N with P := |P|. Consider the probing action at time t. Each bus m ∈ P perturbs its active injection by δ m (t) for one second or so. All inverter perturbations {δ m (t)} m∈P at time t are stacked in the P -length vector δ(t). Based on the model in (2), perturbations in active power injections incur voltage differencesṽ
(t) := v(t) − v(t − 1) = R P δ(t)(4)
where R P ∈ R N ×C is the submatrix obtained by keeping only the columns of R indexed by P. The grid is perturbed over T probing periods. Stacking the probing actions {δ(t)} T t=1 and voltage differences {ṽ(t)} T t=1 respectively as columns of matrices ∆ andṼ yields
V = R P ∆.(5)
The data model in (4)- (5) presumes that injections at nonprobing buses remain constant during probing and ignores modeling inaccuracies and measurement noise. The practical setup of noisy data is handled in Section VI.
Knowing ∆ and measuringṼ in (5), the goal is to recover the grid connectivity along with line resistances; line reactances can be found by reactive probing likewise. This task of topology identification can be split into three stages: s1) finding R P from (5); s2) recovering the level sets for all buses in P; and s3) finding topology and resistances.
At stage s1), if the probing matrix ∆ ∈ R C×T is full rowrank, then matrix R P can be uniquely recovered as R P = V∆ + , where ∆ + is the right pseudo-inverse of ∆. Under this setup, probing for T = P times suffices to find R P .
At stage s2), using Lemma 2 we can recover the level sets for each bus m ∈ P as follows:
1) Append a zero entry at the top of the vector r m .
2) Group the entries of r m to find the level sets of node m; see Lemma 2-(ii).
3) The number of unique values in the entries of r m yields the depth d m . 4) Rank the unique values of r m in increasing order, to find the depth of each level set; see Lemma 2-(iii). Given the level sets for all m ∈ P, stage s3) recovers the grid topology as detailed next.
IV. TOPOLOGY RECOVERY
By properly probing the nodes in P, the matrix R P can be found at stage s1). Then, the level sets for all buses in P can be recovered at stage s2). Nevertheless, knowing these level sets may not guarantee topology recoverability. Interestingly, probing a radial grid at all leaf nodes has been shown to be sufficient for topology identification [13, Th. 1]. To comply with this requirement, the next setup will be henceforth assumed; see also [4].
Assumption 1: All leaf nodes are probed, that is F ⊆ P. Albeit Assumption 1 ensures topology recoverability, it does not provide a solution for stage s3). We will next devise a recursive graph algorithm for grid topology recovery. The input to the recursion is a depth k and a maximal subset of probing nodes P k n having the same (k − 1)-depth and k-depth ancestors. The (k − 1)-depth ancestor is known and is denoted by α k−1 n . The k-depth ancestor is known to exist, is assigned the symbol n, yet the value of n is unknown for now. We are also given the level sets N k m for all m ∈ P k n . The recursion proceeds in three steps.
The first step finds the k-depth ancestor n by intersecting the sets N k m for all m ∈ P k n . The existence and uniqueness of this intersection are asserted next as shown in the appendix.
Proposition 1: Consider the subset P k n of probing nodes located on the subtree rooted at an unknown k-depth node n ∈ N . The node n can be found as the unique intersection
{n} = m∈P k n N k m .(6)
At the second step, node n is connected to node α k−1 n . Since
n = α k m ∈ N k m and α k−1 n = α k−1 m ∈ N k−1 m , the resistance of line (n, α k−1 n ) can be found as [Lemma 2-(iii)] r α k−1 n ,n = r α k−1 m ,α k m = R α k m ,m − R α k−1 m ,m(7)
for any m ∈ P k n . The third step partitions P k n \ {n} into subsets of buses sharing the same (k + 1)-depth ancestor. This can be easily accomplished thanks to the next result.
Proposition 2: For nodes m and m ′ in a tree graph, it holds that α k+1
m = α k+1 m ′ if and only if N k m = N k m ′ .
Based on Prop. 2 (shown in the appendix), the set P k n \ {n} can be partitioned by grouping buses with identical N k m 's. The buses forming one of these partitions P k+1 s have the same kdepth and (k + 1)-depth ancestors. Node n was found to be the k-depth ancestor. The (k + 1)-depth ancestor is known to exist and is assigned the symbol s. The value of s is found by invoking the recursion with new inputs the depth (k + 1), the set of buses P k+1 s along with their (k + 1)-depth level sets, and their common k-depth ancestor (node n).
Algorithm 1 Topology Recovery with Complete Data
Input: N , {N k m } dm k=0
for all m ∈ P. 1: Run Root&Branch(P, ∅, 0). Output: Radial grid and line resistances over N . Function Root&Branch(P k n , α k−1 n , k) 1: Identify the node n serving as the common k-depth ancestor for all buses in P k n via (6). 2: if k > 0, then 3: Connect node n to α k−1 n with the resistance of (7). 4 To initialize the recursion, set P 0 n = P since every probing bus has the substation as 0-depth ancestor. At k = 0, the second step is skipped as the substation does not have any ancestors to connect. The recursion terminates when P k n is a singleton {m}. In this case, the first step identifies m as node n; the second step links m to its known ancestor α k−1 m ; and the third step has no partition to accomplish. The recursion is tabulated as Alg. 1.
V. TOPOLOGY RECOVERY WITH PARTIAL DATA
Although the scheme described earlier probes the grid only through a subset of buses P, voltage responses are collected across all buses. This may be unrealistic in distribution grids with limited real-time metering infrastructure, where the operator reads voltage data only at a subset of buses. To simplify the exposition, the next assumption will be adopted.
Assumption 2: Voltage differences are metered only in P. Under this assumption, the probing model (5) becomes
V = R PP ∆(8)
where nowṼ is of dimension P × T and R PP is obtained from R upon maintaining only the rows and columns in P. Similar to (5), R PP is identifiable if ∆ is full row-rank. This is the equivalent of stage s1) in Section III under the partial data setup. Towards the equivalent of stage s2), since column r m is partially observed, only the metered level sets of node m ∈ P defined as M k m := N k m ∩ P can be recovered. The metered level sets for node m can be obtained by grouping the indices associated with the same values in the observed subvector of r m . Although, the grid topology cannot be fully recovered based on M k m 's, one can recover a reduced grid relying on the concept of internal identifiable nodes; see Fig. 1.
Definition 1: The set I ⊂ N of internal identifiable nodes consists of all buses in G having at least two children with each of one of them being the ancestor of a probing bus.
The reduced grid induced by P can now be defined as the graph G r := (N r , L r ) with • node set N r := P ∪ I; • ℓ = (m, n) ∈ L r if m, n ∈ N r and all other nodes on the path from m to n in G do not belong to N r ; and • the resistance of line ℓ = (m, n) ∈ L r equals the effective resistance between m and n in G, that is r eff mn := (e m − e n ) ⊤ R(e m − e n ), where e m is the m-th canonical vector [15]. In fact, for radial G, the resistance r eff mn equals the sum of resistances across the m − n path in G; see [15].
Let R r be the inverse reduced Laplacian associated with G r . From the properties of effective resistances, it holds [15] R r PP = R PP .
In words, the grid G is not the only electric grid having R PP as the top-left block of its R matrix. The reduced grid G r ; the (meshed) Kron reduction of G given P; and even grids having nodes additional to N can yield the same R PP ; see Fig. 1. Lacking any more detailed information, the grid G r features desirable properties: i) it connects the actuated and possibly additional identifiable nodes in a radial fashion; ii) it satisfies (9) with the minimal number of nodes; and iii) its resistances correspond to the effective resistances of G. Actually, this reduced grid conveys all the information needed to solve an optimal power flow task [16]. The next lemma shows that the number of metered level sets M k m coincides with the number of level sets N k m for all m ∈ P, so the degrees of probing buses can be reliably recovered even with partial data.
Lemma 3: Let G r = (N r , L r ) be the reduced grid of a radial graph G, and let Assumption 1 hold true. Then, N k m ∩M k m = ∅ for all m ∈ F and k = 1, . . . , d m .
Proof: Arguing by contradiction, suppose there exists m ∈ F such that N k m ∩ M k m = ∅ for some k ≤ d m . Since by definition α k m ∈ N k m , the hypothesis N k m ∩ M k m = ∅ implies that α k m / ∈ M k m . Therefore, α k m / ∈ P and the degree of α k m is g α k m ≥ 3. The latter implies that α k m has at least one child w / ∈ A m . Let the leaf node s ∈ D w . Observe that s belongs to both N k m and M k m , contradicting the hypothesis. The next result proved in the appendix guarantees that the topology of G r is identifiable under Assumption 1.
Proposition 3: Given a tree G = (N , L) with leaf nodes F ⊆ N and under Assumption 1, its reduced graph G r = (N r , L r ) is uniquely characterized by {M k m } dm k=0 for all m ∈ P, up to different labellings for non-probing nodes.
Moving forward to the equivalent of stage s3) in Section III, a three-step recursion operating on metered rather than ordinary level sets is devised next. Suppose we are given the set of probing nodes P k n having the same (k − 1)-depth and k-depth ancestors (known and unknown, respectively), along with their k-depth metered level sets.
At the first step, if there exists a node m ∈ P k n such that M k m = P k n , then the k-depth ancestor n is set as m. Otherwise, a non-probing node is added and assigned to be the k-depth ancestor. This is justified by the next result.
Proposition 4: The root n of subtree T k n is a probing node if and only if M k n = P k n . Proof: Proving by contradiction, suppose there exists a node m ∈ T k n with M k m = T k n ∩ P = P k n and m = n. Since m is not the root of T k n , it holds that d m > k, m / ∈ M k n , and so m / ∈ P k n . If n is a probing node and the root of T k n , then d n = k and so N k n = D n . Because of this, it follows that M k n = N k n ∩ P = D n ∩ P = T k n ∩ P = P k n . At the second step, node n = α k m is connected to node α k−1 n = α k−1 m . The line resistance can be found through a modified version of (7). Given any bus m ∈ P k n , Lemma 3 ensures that there exist at least two probing buses s ∈ N k−1 m and s ′ ∈ N k m . Moreover, Lemma 2-(ii) guarantees that R α k−1 m ,m = R s,m and R α k m ,m = R s ′ ,m . Since nodes m, s, and s ′ are metered, both R s,m and R s ′ ,m can be retrieved from R PP . Thus, the sought resistance can be found as
r α k−1 m ,α k m = R α k m ,m − R α k−1 m ,m = R s ′ ,m − R s,m .(10)
At the third step, the set P k n \ {n} is partitioned into subsets of buses having the same (k + 1)-depth ancestor. This can be accomplished by comparing their k-depth metered level sets, as asserted by the next result.
Proposition 5: Let m, m ′ ∈ P k n . It holds that α k+1
m = α k+1 m ′ if and only if M k m = M k m ′ . Proof: If α k+1 m = α k+1 m ′ , then Proposition 2 ensures that N k m = N k m ′ and so M k m = M k m ′ . We will show that if α k+1 m = α k+1 m ′ , then M k m = M k m ′ . Since m, m ′ ∈ P k n , it holds that n = α k m = α k m ′ and M k m = (D n \D α k+1 m ) ∩ P, M k m ′ = (D n \D α k+1 m ′ ) ∩ P. Because D α k+1 m = D α k+1 m ′ , D α k+1 m ∩P = 0, and D α k+1 m ′ ∩P = 0, it follows that M k m = M k m ′ .
The recursion is tabulated as Alg. 2. It is initialized at k = 1, since the substation is not probed and M 0 m does not exist; and is terminated as in Section IV.
Algorithm 2 Topology Recovery with Partial Data
Input: M, {M k m } dm k=1 for all m ∈ P. 1: Run Root&Branch-P(P, ∅, 1). Output: Reduced grid G r and resistances over L r . Function Root&Branch-P(P k n , α k−1 n , k) 1: if ∃ node n such that M k n = P k n , then Partition P k n \{n} into groups of buses {P k+1 s } having identical k-depth metered level sets. 11: Run Root&Branch-P(P k+1 s , n, k + 1) for all s. 12: end if VI. TOPOLOGY RECOVERY WITH NOISY DATA So far, matrices R P and R PP have been obtained using the noiseless model of (4). Under a more realistic setup, inverter and voltage perturbations are related as v(t) = R P δ(t) + n(t) (11) where n(t) captures possible deviations due to non-probing buses, measurement noise, and modeling errors. Stacking {n(t)} T t=1 as columns of matrix N, model (5) translates tõ
V = R P ∆ + N.(12)
Under this setup, a least-square estimate can be found aŝ
R P := arg min Θ Ṽ − Θ∆ 2 F =Ṽ∆ + .(13)
To facilitate its statistical characterization and implementation, a simplified probing protocol is advocated: p1) Every probing bus m ∈ P perturbs its injection by an identical amount δ m over T m consecutive periods. p2) During these T m probing periods, the remaining probing buses do not perturb their injections. Under this protocol, the probing matrix takes the form
∆ = δ 1 e 1 1 ⊤ T1 δ 2 e 2 1 ⊤ T2 · · · δ P e P 1 ⊤ TP .(14)
If at time t node m is probed, the collectedṽ(t) is simplỹ v(t) = δ m r m + n(t).
Under (14)- (15), it is not hard to see that the minimization in (13) decouples over the columns of R P . In fact, the mth column of R P can be found as the scaled sample mean of voltage differences collected only over the times T m :=
m−1 τ =1 T τ + 1, . . . , m τ =1 T τ node m was probed r m = 1 δ m T m t∈Tmṽ (t).(16)
To statistically characterizer m , we will next postulate a model for the error term n(t) in (15) as
n(t) := Rp(t) + Xq(t) + w(t)(17)
wherep(t) + jq(t) are the injection deviations from nonactuated buses, and w(t) captures approximation errors and measurement noise. If {p(t),q(t), w(t)} are independent zero-mean with respective covariance matrices σ 2 p I, σ 2 q I, and σ 2 w I; the random vector n(t) is zero-mean with covariance matrix Φ := σ 2 p R 2 + σ 2 q X 2 + σ 2 w I. Invoking the central limit theorem, the estimater m can be approximated as zero-mean Gaussian with covariance matrix 1 δ 2 m Tm Φ. By increasing T m and/or δ m , the estimater m can go arbitrarily close to the actual r m , and this distance can be bounded probabilistically using Φ. Note however, that Φ depends on the unknown (R, X). To resolve this issue, we resort to an upper bound on Φ based on minimal prior information: Suppose the spectral radii ρ(R) and ρ(X), and the variances (σ 2 p , σ 2 q , σ 2 w ) are known; see [16] for upper bounds. Then, it is not hard to verify that ρ(Φ) ≤ σ 2 , where σ 2 := σ 2 p ρ 2 (R) + σ 2 q ρ 2 (X) + σ 2 w . The standard Gaussian concentration inequality bounds the deviation of the n-th entry ofr m from its actual value as
Pr |R n,m − R n,m | ≥ 4σ δ m √ T m ≤ π 0 := 6 · 10 −5 . (18)
Let us now return to stage s2) of recovering level sets from the columns of R P . In the noiseless case, level sets were identified as the indices of r m related to equal values. Almost surely though, there will not be any equal entries in the noisŷ r m . Instead, the entries ofr m will be concentrated around the actual values. To identify groups of similar values, first sort the entries ofr m in increasing order, and then take the differences of successive sorted entries. A key fact stemming from Lemma 2-(iii) guarantees that the minimum difference between the entries of r m is larger or equal to the smallest line resistance r min . Hence, if all estimates were confined within |R n,m − R n,m | ≤ r min /4, a difference of sortedR n,m 's larger than r min /2 would safely pinpoint the boundary between two node groups. In practice, if the operator knows r min a priori and selects
δ m T m ≥ 16σ/r min(19)
the requirement |R n,m −R n,m | ≤ r min /4 will be satisfied with probability higher than 99.95%. In such a case and taking the union bound, the probability of correctly recovering all level sets is larger than 1 − N 2 π 0 . The argument carries over to R PP under the partial data setup.
VII. NUMERICAL TESTS
Our algorithms were validated on the IEEE 37-bus feeder converted to its single-phase equivalent [5]. Figures 1a-1b show the actual and reduced topologies that can be recovered under a noiseless setup if all leaf nodes are probed. Setups with complete and partial noisy data were tested. Probing was performed on a per-second basis following the protocol p1)-p2) of Sec. VI. Probing buses were equipped with inverters having the same rating as the related load. Loads were generated by adding a zero-mean Gaussian variation to the benchmark data, with standard deviation 0.067 times the average of nominal loads. Voltages were obtained via a power flow solver, and then corrupted by zero-mean Gaussian noise with 3σ deviation of 0.01% per unit (pu). Although typical voltage sensors exhibit accuracies in the range of 0.1-0.5%, here we adopt the highaccuracy specifications of the micro-phasor measurement unit of [17].
For the 37-bus feeder r min = 0.0014 pu. From the rated δ m 's; the r min ; and (19), the number of probing actions was set as T m = 90. In the partial data case, the smallest effective resistance was 0.0021 pu, yielding T m = 39. Level sets were obtained using the procedure described in Sec. VI, and given as inputs to Alg. 1 and 2. The algorithms were tested through 10,000 Monte Carlo tests. Table I demonstrates that the probability of error in topology recovery and the mean percentage error (MPE) of line resistances in correctly detected topologies decay gracefully for increasing T m .
VIII. CONCLUSIONS
To conclude, this letter has put forth an active sensing paradigm for topology identification of inverter-enabled grids. If all lead nodes are probed and voltage responses are metered at all nodes, the grid topology can be unveiled via a recursive algorithm. If voltage responses are metered only at probing buses, a reduced topology can be recovered instead. Guidelines for designing probing actions to cope with noisy data have been tested on a benchmark feeder. Generalizing to multiphase and meshed grids; coupling (re)active probing strategies; incorporating prior line information; and exploiting voltage phasors are exciting research directions.
APPENDIX
Proof of Proposition 1. We will first show that
m∈P k n N k m = m∈T k n ∩F N k m .(20)
By definition P k n = T k n ∩ P. Consider a node w ∈ P k n with w / ∈ F . Two cases can be identified. In case i), w equals the root n of subtree T k n and hence N k w = D n = T k n by the second branch in (1). Note that N k s ∩ D n = N k s for all s ∈ T k n ∩ P. In case ii), node w is different than n and thus implies that N k w = N k s for all s ∈ F ∩ D w . Either way, it holds
m∈P k n N k m = m∈(T k n ∩P)\{w} N k m .(21)
By recursively applying (21) For the sake of contradiction, assume there exists another reduced gridĜ r = (N r ,L r ) withL r = L r such that F (G r ) = F (Ĝ r ) = F (G) and {M k w (G r ) = M k w (Ĝ r )} dw k=0 for all w ∈ P. Note that Lemma 3 and the latter equality imply that d w (G r ) = d w (Ĝ r ) for all w ∈ P.
SinceĜ r = G r up to different labelling for non-probing nodes, there exists a subtree T k n (Ĝ r ) with the properties: 1) It appears both inĜ r and G r . 2) Node n has different parent nodes inĜ r and G r , that is m = α k−1 n (Ĝ r ) = α k−1 n (G r ). 3) At least one of α k−1 n (Ĝ r ) and α k−1 n (G r ) belongs to P. Such a T k n (Ĝ r ) exists and it may be the singleton T k n (Ĝ r ) = {n} for n ∈ F . Assume without loss of generality n ∈ P. Based on p3), two cases are identified for m.
Case I: m ∈ P. We will next show that d m (Ĝ r ) = d m (G r ). From p2) and Lemma 1-(i), it follows m ∈ M k−1 n (Ĝ r ).
On the other hand, property p2) along with the hypothesis M k−1 s (Ĝ r ) = M k−1 s (G r ) imply that d m (G r ) > k − 1 = d m (Ĝ r ). Lemma 1-(v) ensures then that m ∈ N k−1 m (Ĝ r ), but m / ∈ N k−1 m (G r ). Case II: m / ∈ P. Since non-probing buses have degree greater than two in reduced grids, there exists at least one probing node s, such that s ∈ D m , but s / ∈ T k n (Ĝ r ). Observe that m, s ∈ N k−1 n (Ĝ r ) and s ∈ M k−1 n (Ĝ r ). Let now w be the parent of n in G r . Due to p3), it holds that w ∈ P. Using Lemma 1-(i), node w ∈ M k−1 n (G r ) and so w ∈ M k−1 n (Ĝ r ) with possibly w = s. Therefore, d w (G r ) < d w (Ĝ r ) and Lemma 1-(v) ensures w / ∈ N k−1 w (Ĝ r ) and w ∈ N k−1 w (G r ).
Lemma 2 (
2[13]): Let m, n, s be nodes in a radial grid. (i) if m ∈ F , then R m,m > R n,m for all n = m; (ii) n, s ∈ N k m for a k if and only if R n,m = R s,m ; and (iii) if n ∈ N k−1 m , s ∈ N k m , then R s,m = R n,m + r α
Fig. 1 .
1a) the original IEEE 37-bus feeder; b) its reduced equivalent; and c) another feeder with the same R PP . Red nodes are probed; black and blue are not. Blue nodes are internal identifiable nodes comprising I.
1
1n ∈ I and set it as the root of T k n . 5: end if 6: if k >
=
for each non-leaf probing bus m, the equivalence in (20) follows. From the definition of level sets in (1), N k m = D α k m \ D α k+1 m but D α k m = D n since n is the common k-depth ancestor for all m ∈ P k n . The intersection in the RHS of (D n \ {n}. Proof of Proposition 2. If α k+1 m = α k+1 m ′ , it follows that α k m = α k m ′ . Then D α k+1 m = D α k+1 m ′ and D α k m = D α k m ′ . By the definition of the level sets in (1), it follows that N k m = N k m ′ . Conversely, assume that N k m = N k m ′ . Since α k m and α k m ′ are the only nodes at depth k respectively in N k m and N k m ′ (see Lemma 1, claim (i)), it follows that α k m = α k m ′ . By the definition of the level sets in (1), it holds that N k m = D α k m \ D α k+1 m , while D α k+1 m ⊂ D α k m . Similarly for node m ′ , it holds that N k m ′ = D α k m ′ \ D Proof of Proposition 3.
authors are with the Bradley Dept. of ECE, Virginia Tech, Blacksburg, VA 24061, USA. Emails: {cavraro,kekatos}@vt.edu. Work partially supported by the NSF-CAREER grant 1751085.
: end if 5: if P k n \ {n} = ∅, then Run Root&Branch(P k+1 s , n, k + 1) for all s. 8: end if6:
Partition P k
n \{n} into groups of buses {P k+1
s
} having
identical k-depth level sets.
7:
TABLE I NUMERICAL
ITESTS UNDER FULL AND PARTIAL NOISY DATATm
1
10
20
40
90
Alg. 1
Error Prob. [%]
98.5
55.3
20.9
3.1
0.2
MPE [%]
35.1
32.5
31.2 30.9
28.5
Tm
1
5
10
20
39
Alg. 2
Error Prob. [%]
97.2
45.8
26.3 18.9
0.1
MPE [%]
18.6
16.4
15.4 14.8
13.2
Structure learning in power distribution networks. D Deka, M Chertkov, S Backhaus, early accessD. Deka, M. Chertkov, and S. Backhaus, "Structure learning in power distribution networks," 2017, (early access).
Identification of power distribution network topology via voltage correlation analysis. S Bolognani, N Bof, D Michelotti, R Muraro, L Schenato, Proc. IEEE Conf. on Decision and Control. IEEE Conf. on Decision and ControlFlorence, ItalyS. Bolognani, N. Bof, D. Michelotti, R. Muraro, and L. Schenato, "Iden- tification of power distribution network topology via voltage correlation analysis," in Proc. IEEE Conf. on Decision and Control, Florence, Italy, Dec. 2013.
Distributed energy resources topology identification via graphical modeling. Y Weng, Y Liao, R Rajagopal, IEEE Trans. Power Syst. 324Y. Weng, Y. Liao, and R. Rajagopal, "Distributed energy resources topology identification via graphical modeling," IEEE Trans. Power Syst., vol. 32, no. 4, pp. 2682-2694, Jul. 2017.
Exact topology and parameter estimation in distribution grids with minimal observability. S Park, D Deka, M Chertkov, S. Park, D. Deka, and M. Chertkov, "Exact topology and parameter estimation in distribution grids with minimal observability," Oct. 2017. [Online]. Available: https://arxiv.org/abs/1710.10727
Voltage analytics for power distribution network topology verification. G Cavraro, V Kekatos, S Veeramachaneni, IEEE Trans. Smart Grid. early accessG. Cavraro, V. Kekatos, and S. Veeramachaneni, "Voltage analytics for power distribution network topology verification," IEEE Trans. Smart Grid, 2017, (early access).
Topology identification in distribution network with limited measurements. Y Sharon, A M Annaswamy, A L Motto, A Chakraborty, Proc. IEEE Conf. on Innov. Smart Grid Tech. IEEE Conf. on Innov. Smart Grid TechY. Sharon, A. M. Annaswamy, A. L. Motto, and A. Chakraborty, "Topol- ogy identification in distribution network with limited measurements," in Proc. IEEE Conf. on Innov. Smart Grid Tech., Jan 2012.
Distribution system topology detection using consumer load and line flow measurements. R Sevlian, R , R. Sevlian and R. Rajagopal, "Distribution system topology detection using consumer load and line flow measurements," Sep. 2017. [Online].
PaToPa: A data-driven parameter and topology joint estimation framework in distribution grids. J Yu, Y Weng, R Rajagopal, IEEE Trans. Power Syst. 99J. Yu, Y. Weng, and R. Rajagopal, "PaToPa: A data-driven parameter and topology joint estimation framework in distribution grids," IEEE Trans. Power Syst., vol. PP, no. 99, 2017.
On the inverse power flow problem. Y Yuan, O Ardakanian, S Low, C Tomlin, Y. Yuan, O. Ardakanian, S. Low, and C. Tomlin, "On the inverse power flow problem," Dec. 2017. [Online]. Available: https://arxiv.org/abs/1610.06631
Enhancing observability in distribution grids using smart meter data. S Bhela, V Kekatos, S Veeramachaneni, IEEE Trans. Smart Grid. early accessS. Bhela, V. Kekatos, and S. Veeramachaneni, "Enhancing observability in distribution grids using smart meter data," IEEE Trans. Smart Grid, 2017, (early access).
Topology identification for multiple-bus DC microgrids via primary control perturbations. M Angjelichinoski, C Stefanovic, P Popovski, A Scaglione, F Blaabjerg, IEEE Intl. Conf. on DC Microgrids. Nurnberg, GermanyM. Angjelichinoski, C. Stefanovic, P. Popovski, A. Scaglione, and F. Blaabjerg, "Topology identification for multiple-bus DC microgrids via primary control perturbations," in IEEE Intl. Conf. on DC Micro- grids, Nurnberg, Germany, Jun. 2017.
On-line grid impedance estimation based on harmonic injection for grid-connected PV inverter. M Ciobotaru, R Teodorescu, F Blaabjerg, IEEE Intl. Symp. on Industrial Electronics. M. Ciobotaru, R. Teodorescu, and F. Blaabjerg, "On-line grid impedance estimation based on harmonic injection for grid-connected PV inverter," in IEEE Intl. Symp. on Industrial Electronics, Jun. 2007.
Inverter probing for power distribution network topology processing. G Cavraro, V Kekatos, IEEE Trans. Control Netw. Syst. submitted). [OnlineG. Cavraro and V. Kekatos, "Inverter probing for power distribution network topology processing," IEEE Trans. Control Netw. Syst., Feb. 2018, (submitted). [Online]. Available: https://arxiv.org/pdf/1802.06027.pdf
Network reconfiguration in distribution systems for loss reduction and load balancing. M Baran, F Wu, IEEE Trans. Power Del. 42M. Baran and F. Wu, "Network reconfiguration in distribution systems for loss reduction and load balancing," IEEE Trans. Power Del., vol. 4, no. 2, pp. 1401-1407, Apr. 1989.
Kron reduction of graphs with applications to electrical networks. F Dorfler, F Bullo, IEEE Trans. Circuits Syst. I. 601F. Dorfler and F. Bullo, "Kron reduction of graphs with applications to electrical networks," IEEE Trans. Circuits Syst. I, vol. 60, no. 1, pp. 150-163, Jan. 2013.
A Distributed Feedback Control Approach to the Optimal Reactive Power Flow Problem. S Bolognani, G Cavraro, S Zampieri, Springer Intl. PublishingHeidelbergS. Bolognani, G. Cavraro, and S. Zampieri, A Distributed Feedback Control Approach to the Optimal Reactive Power Flow Problem. Hei- delberg: Springer Intl. Publishing, 2013, pp. 259-277.
Microgrids: PQube 3 MicroPMU. Power Sensors Ltd. Alameda, CASynchrophasors for DistributionSynchrophasors for Distribution, Microgrids: PQube 3 MicroPMU. Power Sensors Ltd. Alameda, CA. [Online]. Available: https://powerstandards.com/Download/MicroPMU%20Data%20Sheet%20Rev1 3.pdf
| [] |
[
"Collective excitations of an imbalanced fermion gas in a 1D optical lattice",
"Collective excitations of an imbalanced fermion gas in a 1D optical lattice"
] | [
"R Mendoza \nPosgrado en Ciencias Físicas\nInstituto de Física\nInstituto de Física\nUNAM\nUNAM, Apdo. Postal 20-36401000MéxicoD.FUNAM, México\n",
"Mauricio Fortes \nPosgrado en Ciencias Físicas\nInstituto de Física\nInstituto de Física\nUNAM\nUNAM, Apdo. Postal 20-36401000MéxicoD.FUNAM, México\n",
"M A Solís \nPosgrado en Ciencias Físicas\nInstituto de Física\nInstituto de Física\nUNAM\nUNAM, Apdo. Postal 20-36401000MéxicoD.FUNAM, México\n"
] | [
"Posgrado en Ciencias Físicas\nInstituto de Física\nInstituto de Física\nUNAM\nUNAM, Apdo. Postal 20-36401000MéxicoD.FUNAM, México",
"Posgrado en Ciencias Físicas\nInstituto de Física\nInstituto de Física\nUNAM\nUNAM, Apdo. Postal 20-36401000MéxicoD.FUNAM, México",
"Posgrado en Ciencias Físicas\nInstituto de Física\nInstituto de Física\nUNAM\nUNAM, Apdo. Postal 20-36401000MéxicoD.FUNAM, México"
] | [] | The collective excitations that minimize the Helmholtz free energy of a population-imbalanced mixture of a 6 Li gas loaded in a quasi one-dimensional optical lattice are obtained. These excitations reveal a rotonic branch after solving the Bethe-Salpeter equation under a generalized random phase approximation based on a single-band Hubbard Hamiltonian. The phase diagram describing stability regions of Fulde-Farrell-Larkin-Ovchinnikov and Sarma phases is also analyzed. | 10.1007/s10909-013-0926-2 | [
"https://arxiv.org/pdf/1307.2655v1.pdf"
] | 118,652,368 | 1307.2655 | 5b1edbfb7e95d0692d69219fb5405cff247a723d |
Collective excitations of an imbalanced fermion gas in a 1D optical lattice
10 Jul 2013
R Mendoza
Posgrado en Ciencias Físicas
Instituto de Física
Instituto de Física
UNAM
UNAM, Apdo. Postal 20-36401000MéxicoD.FUNAM, México
Mauricio Fortes
Posgrado en Ciencias Físicas
Instituto de Física
Instituto de Física
UNAM
UNAM, Apdo. Postal 20-36401000MéxicoD.FUNAM, México
M A Solís
Posgrado en Ciencias Físicas
Instituto de Física
Instituto de Física
UNAM
UNAM, Apdo. Postal 20-36401000MéxicoD.FUNAM, México
Collective excitations of an imbalanced fermion gas in a 1D optical lattice
10 Jul 2013arXiv:1307.2655v1 [cond-mat.quant-gas]numbers: 7470Tx7425Ha7520Hr Keywords: SuperfluidityBethe-SalpeterCollective excitationsPhase diagram
The collective excitations that minimize the Helmholtz free energy of a population-imbalanced mixture of a 6 Li gas loaded in a quasi one-dimensional optical lattice are obtained. These excitations reveal a rotonic branch after solving the Bethe-Salpeter equation under a generalized random phase approximation based on a single-band Hubbard Hamiltonian. The phase diagram describing stability regions of Fulde-Farrell-Larkin-Ovchinnikov and Sarma phases is also analyzed.
I. INTRODUCTION
Optical lattices are tailored made to study stronglycorrelated Fermi systems, the stability of different phases and the effects of dimensionality in population-imbalan ced mixtures of different species of ultra cold gases under attractive interactions [1][2][3][4] . In the latter case, the Fermi surfaces of each species are no longer aligned and Cooper pairs have non-zero total momenta 2q. Such phases were first studied by Fulde and Ferrell (FF) 5 , who used an order parameter that varies as a single plane wave, and by Larkin and Ovchinnikov (LO) 6 , who suggested that the order parameter is a superposition of two plane waves.
The mean-field treatment of the FFLO phase in a variety of systems, such as atomic Fermi gases with population imbalance loaded in optical lattices [7][8][9][10][11][12][13] , shows that the FFLO phase competes with a number of other phases, such as the Sarma (q = 0) states 14,15 , but in some regions of momentum space the FFLO phase is more stable as it provides the minimum of the mean-field expression of the Helmholtz free energy. In addition, recent calculations on the FFLO phase of the same system in two-and threedimensional 12,13 optical lattices suggest that the region of stability of this phase as a function of polarization increases when the dimensionality of the periodic lattice is lowered.
In this work, we use a Bethe-Salpeter approach to obtain the collective excitations of the two-particle propagator of a polarized mixture of two hyperfine states |↑> and |↓> of a 6 Li atomic Fermi gas with attractive interactions loaded in a quasi one-dimensional optical lattice described by a single-band Hubbard Hamiltonian.
II. HUBBARD MODEL
The Hamiltonian of a two-component Fermi gas under an attractive contact interaction in a periodic lattice with constant a is 12
H = −J x i,j x ,σĉ † i,σĉ j,σ − J y i,j y ,σĉ † i,σĉ j,σ −J z i,j z ,σĉ † i,σĉ j,σ − i µ † ↑ĉ † i,↑ĉ i,↑ + µ ↓ĉ † i,↓ĉ i,↓ +U iĉ † i,↑ĉ † i,↓ĉ i,↓ĉi,↑ ,(1)
where J ν is the tunneling strength of the atoms between nearest-neighbor sites in the ν-direction; U is the onsite attractive interaction strength; µ ↑,↓ is the chemical potential of species | ↑>, | ↓>, and the Fermi operatorĉ † i,σ (ĉ i,σ ) creates (destroys) an atom on site i. We assume a system with a total number of atoms M = M ↑ + M ↓ distributed along N sites of the opticallattice potential. For a quasi one-dimensional (1D) system the tunneling strengths satisfy J x ≫ J y = J z and the usual tight-binding lattice dispersion energies are
ξ ↑,↓ (k) = 2 ν J ν (1 − cos k ν a) − µ ↑,↓ .
The order parameter ∆ i = U ĉ i,↓ĉi,↑ of the FFLO states is assumed to vary as a single plane wave, ∆ i = ∆ exp (2ıq · r i ), where 2q is the pair center-of-mass momentum, r i the coordinate of site i, and ∆ is the usual BCS gap.
III. PHASE DIAGRAMS
Within the mean field approximation and using a Bogoliubov transformation to diagonalize the Hamiltonian, the grand canonical partition function Z can be obtained in terms of both, electronlike and hole-
like dispersion ω ± = E q (k) ± η q (k), where η q (k) = 1 2 [ξ ↑ (k + q) − ξ ↓ (q − k)] and E q (k) = χ 2 q (k) + ∆ 2 .
The thermodynamic potential Ω = − 1 β ln Z is
Ω = 1 N k χ q (k) + ω − (k, q) + ∆ 2 U − 1 β k ln 1 + e −βω+(k,q) ) + ln(1 + e βω−(k,q) , (2) where χ q (k) = 1 2 [ξ ↑ (k + q) + ξ ↓ (q − k)] and β = 1/k B T. The Helmholtz free energy F (∆, q, f ↑ , f ↓ , T ) = Ω + µ ↑ f ↑ + µ ↓ f ↓ , where f ↑,↓ ≡ M ↑,↓ /N is the spin-up (spin-down)
filling fraction, can now be minimized with respect to ∆, q, µ ↑ and µ ↓ . This leads to a set of four equations that determine the stable phases of this system as a function of temperature and polarization P ≡
f ↑ −f ↓ f ↑ +f ↓ . f ↑ = 1 N k u 2 q (k)f (ω + (k, q)) + v 2 q (k)f (−ω − (k, q)) , f ↓ = 1 N k u 2 q (k)f (ω − (k, q)) + v 2 q (k)f (−ω + (k, q)) , 1 = U N k 1 − f (ω − (k, q)) − f (ω + (k, q)) 2E q (k) 0 = 1 N k ∂η q (k) ∂q x [f (ω + (k, q)) − f (ω − (k, q))] + ∂χ q (k) ∂q x × 1 − χ q (k) E q (k) [1 − f (ω + (k, q)) − f (ω − (k, q))] ,(3)
where u q (k) = 1
2 1 + χq (k) Eq(k) , v q (k) = 1 2 1 − χq (k) Eq (k)
and f (x) is the Fermi distribution function. In Fig. 1 we show the gap ∆, pair momentum q x and chemical potentials for each species µ ↑ , µ ↓ as a function of polarization that minimize the Helmholtz free energy for a total filling factor f = f ↑ + f ↓ = 0.4685; hopping strengths J x = 0.078 E R , J y = J z = 10 −4 E R , and U/Jx = 2.64, where E R = 2 (2π/λ) 2 /2m is the recoil energy and λ = 1030 nm is the lattice wavelength. When P = 0, q x = 0 and ∆ = 0 the system is in the FFLO phase which becomes the BCS state when both P → 0 and q x → 0; when P = 0, but q x = 0 the system is in the Sarma 14 or breached-pair phase 15 and the transition to the normal state occurs when the gap ∆ vanishes. The phase diagram of the quasi 1D system is shown in Fig 2. It is interesting to note that the FFLO phase is dominant over quite a large region of the phase-diagram and is stable at higher values for the polarization (up to P ≃ 0.64) compared to our previous work in 2D 12 and 3D 13 systems with the same composition and dynamical parameters. In addition, the mixed phase or phase separation regime in which an unpolarized BCS core and a polarized normal fluid (in momentum space) coexist at very low temperatures and moderate polarizations 16 is no longer present in this system in contrast to the 3D system (and to a lesser extent in the 2D optical lattice).
IV. COLLECTIVE STATES
The spectrum of the collective modes can be obtained from the poles of the two-particle Green's function K (1, 2; 3, 4), where we use the compact notation 1 = {σ 1 , r 1 , t 1 }, 2 = {σ 2 , r 2 , t 2 }, ... with σ i denoting the spin variables, r i the vector for lattice site i, and t i , the time variable. K satisfies the following Dyson equation:
K = K 0 + K 0 IK,(4)
where K 0 (1, 2; 3, 4) is the two-particle free propagator which is defined by a pair of fully dressed single-particle Green´s function, K 0 (1, 2; 3, 4) = G(1; 3)G(4; 2), and the interaction kernel I is given by functional derivatives of the mass operator.
Using the generalized random phase approximation, we replace the single-particle excitations with those obtained by diagonalizing the Hartree-Fock (HF) Hamiltonian while the collective modes are obtained by solving the Bethe-Salpeter (BS) equation in which the singleparticle Green's functions are calculated in the HF approximation, and the BS kernel is obtained by summing ladder and bubble diagrams 17 . The resulting equation for the BS amplitudesΨ q (k, Q) iŝ
Ψ q (k, Q) = −UD pΨ q (p, Q) + UM pΨ q (p, Q),(5)
whereΨ q (k, Q) is a vector with four components and the 4 × 4 matrices UD and UM represent the contribution resulting from the direct and exchange interactions, respectively. The dispersion ω(Q) for the collective excitations is obtained from the solutions of the 4 × 4 secular determinant defined by (5). In Fig. 3 we show the col- lective excitations of the 1D system. For small Q the Goldstone mode is clearly present with a sound velocity of v s = 4.92 mm/s. For larger Q, a rotonlike minimum appears with a gap ∆ r = 0.0157E R and a critical flow velocity v f = 1.05 mm/s.
V. CONCLUSIONS
We have shown that superfluid phases of the FFLO and Sarma types are present in ultra cold Fermi gases loaded in quasi one-dimensional optical lattices. The region of stability of the FFLO states in the phase diagram is larger and supports higher population imbalances than identical systems in 2D and 3D. The energy dispersion of collective excitations have the usual Goldstone-mode behavior with a sound velocity of 4.92 mm/s. In addition, for higher momenta a rotonic branch is also present.
FIG. 1 :
1Gap (dots), chemical potentials and pair momentum that minimize the Helmholtz free energy.
FIG. 2 :
2(Color online) Phase diagram of an imbalanced Fermi gas in a quasi 1D optical lattice.
Li gas in a quasi 1D optical lattice with λ = 1030 nm and total filling factor f = 0.4685. The Hubbard parameters are Jx = 0.078 ER, Jy = Jz = 10 −4 ER and the attractive on-site interaction is U/J = 2.64.
AcknowledgmentsWe acknowledge the partial support from UNAM-DGAPA grants IN105011, IN-111613 and CONACyT grant 104917.
. T Esslinger, Ann. Rev. Condensed Matter Phys. 1129T. Esslinger, Ann. Rev. Condensed Matter Phys. 1, 129 (2010).
. Y Shin, C H Schunck, A Schirotzek, W Ketterle, Nature. 451689Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle, Nature 451, 689 (2008).
. W Ketterle, Y Shin, A Schirotzek, C H Schunk, J. Phys, Condensed Matter. 21164206W. Ketterle, Y. Shin, A. Schirotzek and C. H. Schunk, J. Phys, Condensed Matter 21, 164206 (2009).
. Yean-An Liao, A S C Rittner, T Paprotta, W Li, G B Partridge, R G Hulet, S K Baur, E J Mueller, Nature. 467567Yean-an Liao, A.S.C. Rittner, T. Paprotta, W. Li, G.B. Partridge, R.G. Hulet, S.K. Baur and E.J. Mueller, Nature 467, 567 (2010).
. P Fulde, R A Ferrell, Phys. Rev. 135550P. Fulde, and R. A. Ferrell, Phys. Rev. 135, A550 (1964).
. A I Larkin, Y N Ovchinnikov, Zh. Eksp. Teor. Fiz. 471136Sov. Phys. JETPA. I. Larkin, and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz., 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)].
. T Koponen, New Journal of Physics. 8179T. Koponen et al., New Journal of Physics 8, 179 (2006)
. T Koponen, Phys. Rev. Lett. 99120403T. Koponen et al., Phys. Rev. Lett. 99, 120403 (2007);
Martikainen. T Paananen, T K Koponen, P Törma, J P , Phys. Rev. A. 7753602T. Paananen, T. K. Koponen, P. Törma, and J.P. Mar- tikainen, Phys. Rev. A 77, 053602, (2008).
. Tung-Lam Dao, A Georges, M Capone, Phys. Rev. B. 76104517Tung-Lam Dao, A. Georges, and M. Capone, Phys. Rev. B 76, 104517 (2007);
. Q Chen, Phys. Rev. B. 7514521Q. Chen et al., Phys. Rev. B 75, 014521 (2007);
. Xia-Ji Liu, H Hu, P D Drummond, Phys. Rev. A. 7643605Xia-Ji Liu, H. Hu, and P. D. Drummond, Phys. Rev. A 76, 043605 (2007);
. M Rizzi, Phys. Rev. B. 77245105M. Rizzi, et al., Phys. Rev. B 77, 245105 (2008);
. Xia-Ji Liu, Hui Hu, P , Xia-Ji Liu, Hui Hu, and P.
. D Drummond, Phys. Rev. A. 7823601D. Drummond, Phys. Rev. A 78, 023601 (2008);
. M Reza Bakhtiari, M J Leskinen, P Törma, Phys. Rev. Lett. 101120404M. Reza Bakhtiari, M. J. Leskinen, and P. Törma, Phys. Rev. Lett. 101, 120404 (2008);
. A Lazarides, B Van Schaeybroec, Phys. Rev. A. 7741602A. Lazarides and B. Van Schaeybroec, Phys. Rev. A 77, 041602 (2008);
. T Paananen, J. Phys. B: At. Mol. Opt. Phys. 42165304T Paananen, J. Phys. B: At. Mol. Opt. Phys. 42 165304 (2009);
. X Cui, Y Wang, Phys. Rev. B. 79180509X. Cui and Y. Wang, Phys. Rev. B 79, 180509(R) (2009);
. A Mishra, H Mishra, Eur. Phys. J. D. 5375A. Mishra and H. Mishra, Eur. Phys. J. D 53, 75 (2009);
. B Wang, Han-Dong Chen, S. Das Sarma, Phys. Rev. A. 7951604B. Wang, Han-Dong Chen, and S. Das Sarma, Phys. Rev. A 79, 051604(R) (2009);
. Y Yanase, Phys. Rev. B. 80R220510Y. Yanase, Phys. Rev. B 80, 220510(R) (2009);
. A Ptok, M Máska, M Mierzejewski, J. Phys.: Condens. Matter. 21295601A. Ptok, M. Máska, and M. Mierzejewski, J. Phys.: Condens. Matter 21, 295601 (2009);
. Yan Chen, Phys. Rev. B. 7954512Yan Chen et al., Phys. Rev. B 79, 054512 (2009);
. Yen Lee Loh, N Trivedi, Phys. Rev. Lett. 104165302Yen Lee Loh and N. Trivedi, Phys. Rev. Lett. 104, 165302 (2010);
. A Korolyuk, F Massel, P Törma, Phys. Rev. Lett. 104236402A. Korolyuk, F. Mas- sel, and P. Törma, Phys. Rev. Lett. 104, 236402 (2010);
. F , F.
. Heidrich-Meisner, Phys. Rev. A. 8123629Heidrich-Meisner et al., Phys. Rev. A 81, 023629 (2010);
. S K Baur, J Shumway, E J Mueller, Phys. Rev. A. 8133628S. K. Baur, J. Shumway, and E. J. Mueller, Phys. Rev. A 81, 033628 (2010);
. A Korolyuk, F Massel, P Törmä, Phys. Rev. Lett. 104236402A. Korolyuk, F. Massel, and P. Törmä, Phys. Rev. Lett. 104, 236402 (2010);
. M J Wolak, Phys. Rev. A. 8213614M. J. Wolak et al., Phys. Rev. A 82, 013614 (2010);
. L Radzihovsky, D Sheehy, Rep. Prog. Phys. 7376501L. Radzihovsky and D. Sheehy, Rep. Prog. Phys. 73, 076501 (2010).
. K Machida, T Mizushima, M Ichioka, Phys. Rev. Lett. 97120407K. Machida, T. Mizushima and M. Ichioka, Phys. Rev. Lett. 97 120407 (2006).
. T L Dao, M Ferrero, A Georges, M Capone, O Parcollet, Phys. Rev. Lett. 101236405T.L. Dao, M. Ferrero, A. Georges, M. Capone and O. Par- collet, Phys. Rev. Lett. 101 236405 (2008).
. Z G Koinov, R Mendoza, M Fortes, Phys. Rev. Lett. 106100402Z. G. Koinov, R. Mendoza and M. Fortes, Phys. Rev. Lett. 106, 100402 (2011).
. R Mendoza, M Fortes, M A Solís, Z G Koinov, arXiv:1306.4706R. Mendoza, M. Fortes, M.A. Solís and Z.G. Koinov, arXiv:1306.4706 (2013).
. G Sarma, J. Phys. Chem. 241029G. Sarma, J. Phys. Chem. 24, 1029 (1963).
. W V Liu, F Wilczek, Phys. Rev. Lett. 9047002W.V. Liu and F. Wilczek, Phys. Rev. Lett., 90, 047002 (2003).
. P F Bedaque, H Caldas, G Kupak, Phys. Rev. Lett. 91247002P. F. Bedaque, H. Caldas, and G. Kupak, Phys. Rev. Lett. 91, 247002 (2003);
. H Caldas, Phys Rev. A. 6963602H. Caldas, Phys Rev. A 69, 063602 (2004);
. H Caldas, C W Morais, A L Mota, Phys. Rev. D. 7245008H. Caldas, C. W. Morais and A. L. Mota, Phys. Rev. D 72, 045008 (2005);
. S Sachdev, K Yang, Phys. Rev. B. 73174504S. Sachdev and K. Yang, Phys. Rev. B 73, 174504 (2006). (1994).
. Z G Koinov, Physica C. 407470Z. G. Koinov, Physica C 407, 470 (2010);
. Physica Status Solidi (B). 247140Physica Status Solidi (B) 247, 140 (2010);
. cond-mat/1010.1200Ann. Phys. (Berlin). 522693Ann. Phys. (Berlin) 522, 693 (2010); cond-mat/1010.1200.
. K V Samokhin, Phys. Rev. B. 81224507K. V. Samokhin, Phys. Rev. B 81, 224507 (2010).
. Y.-P Shim, R A Duine, A H Macdonald, Phys. Rev. A. 7453602Y.-P. Shim, R. A. Duine, and A. H. MacDonald, Phys. Rev. A 74, 053602 (2006).
| [] |
[
"Equation of state of strongly coupled Hamiltonian lattice QCD at finite density",
"Equation of state of strongly coupled Hamiltonian lattice QCD at finite density"
] | [
"Yasuo Umino ",
"Ect ",
"\nStrada delle Tabarelle\n286 I-38050VillazzanoItaly\n",
"\nDepartamento de Física Teòrica\nInstituto de Física Corpuscular -C.S.I.C\nUniversitat de València\nE-46100Burjassot, ValènciaSpain\n"
] | [
"Strada delle Tabarelle\n286 I-38050VillazzanoItaly",
"Departamento de Física Teòrica\nInstituto de Física Corpuscular -C.S.I.C\nUniversitat de València\nE-46100Burjassot, ValènciaSpain"
] | [] | We calculate the equation of state of strongly coupled Hamiltonian lattice QCD at finite density by constructing a solution to the equation of motion corresponding to an effective Hamiltonian using Wilson fermions. We find that up to and beyond the chiral symmetry restoration density the pressure of the quark Fermi sea can be negative indicating its mechanical instability. This result is in qualitative agreement with continuum models and should be verifiable by future numerical simulations. | 10.1142/s0217732302009234 | [
"https://arxiv.org/pdf/hep-ph/0012071v2.pdf"
] | 8,760,696 | hep-ph/0109132 | 91a6b6c036141f7c0639f4d0ce5f9a7a3c9553c4 |
Equation of state of strongly coupled Hamiltonian lattice QCD at finite density
12 May 2002 January 24, 2019
Yasuo Umino
Ect
Strada delle Tabarelle
286 I-38050VillazzanoItaly
Departamento de Física Teòrica
Instituto de Física Corpuscular -C.S.I.C
Universitat de València
E-46100Burjassot, ValènciaSpain
Equation of state of strongly coupled Hamiltonian lattice QCD at finite density
12 May 2002 January 24, 2019numbers: 1115H1238 Keywords: Lattice Field TheoryStrong Coupling QCD Submitted to Physics Letters B
We calculate the equation of state of strongly coupled Hamiltonian lattice QCD at finite density by constructing a solution to the equation of motion corresponding to an effective Hamiltonian using Wilson fermions. We find that up to and beyond the chiral symmetry restoration density the pressure of the quark Fermi sea can be negative indicating its mechanical instability. This result is in qualitative agreement with continuum models and should be verifiable by future numerical simulations.
Simulating Quantum Chromodynamics (QCD) at finite density is one of the outstanding problems in lattice gauge theory [1]. In fact, because of the sign probelm there are currently no reliable numerical simulations of finite density QCD with three colors even in the strong coupling limit [2]. This is a rather frustrating situation in view of the current intense interest in finite density QCD fueled by the phenomenology of heavy ion collisions, neutron stars, early universe and color superconductivity. Therefore even a qualitative description of finite density lattice QCD is welcome.
One method of studying finite density lattice QCD is to invoke the strong coupling approximation where analytical methods are applicable. Strongly coupled lattice QCD at finite quark chemical potential µ and temperature T has previously been studied analytically both in the Euclidean [3,4,5] and in the Hamiltonian [6,7,8,9] formulations. One of the main objectives of these studies was to investigate the nature of chiral symmetry restoration at finite T and/or µ. This has been accomplished by constructing some effective action or Hamiltonian for strongly coupled lattice QCD using Kogut-Susskind fermions. Except for [8] these effective descriptions involve composite meson and baryon fields which are treated in the mean field approximation. 1 The consensus is that at zero or low T , strongly coupled lattice QCD at finite µ undergoes a first order chiral phase transition from the broken symmetry phase below a critical chemical potential µ C to a chirally symmetric phase above µ C .
In this letter we present a calculation of the equation of state of strongly coupled lattice QCD at finite density in the Hamiltonian formulation using Wilson fermions. We find that up to and beyond the chiral symmetry restoration density the pressure of the many body system can be negative indicating its mechanical instability. This new result is in qualitative agreement with those obtained using continuum effective QCD models at finite density [10,11] and should be verifiable by future numerical simulations.
As in previous studies on this subject we begin with an effective description of strongly coupled lattice QCD. We shall use Smit's effective Hamiltonian [12] which involves only the quark field Ψ with a nearest neighbour interaction. This effective Hamiltonian has been studied in free space by Smit [12] and by Le Yaouanc et al. [13] who subsequently extended their analysis to finite T and µ using Kogut-Susskind fermions [8]. A similar effective Hamiltonian has recently been derived by Gregory et al. [9] to study strongly coupled lattice QCD at finite µ, again using Kogut-Susskind fermions.
Henthforth we shall adopt the notation of Smit [12], set the lattice spacing to unity and work in momentum space. Then the charge conjugation symmetric form of Smit's Hamiltonian using Wilson fermions may be written as
H eff = 1 2 M 0 (γ 0 ) ρν p (Ψ † aα ) ρ ( p ), (Ψ aα ) ν (− p ) − − K 8N c p 1 ,..., p 4 l δ p 1 +···+ p 4 , 0 e i(( p 1 + p 2 )·n l ) + e i(( p 3 + p 4 )·n l ) ⊗ (Σ l ) ρν (Ψ † aα ) ρ ( p 1 )(Ψ bα ) ν ( p 2 ) − (Σ l ) † ρν (Ψ aα ) ν ( p 1 )(Ψ † bα ) ρ ( p 2 ) ⊗ (Σ l ) † γδ (Ψ † bβ ) γ ( p 3 )(Ψ aβ ) δ ( p 4 ) − (Σ l ) γδ (Ψ bβ ) δ ( p 3 )(Ψ † aβ ) γ ( p 4 ) (1) where (Σ l ) = −i (γ 0 γ l − irγ 0 )
with the Wilson parameter r taking on values between 0 and 1. In the above Hamiltonian color, flavor and Dirac indices are denoted by (a, b), (α, β) and (ρ, ν, γ, δ), respectively, and summation convention is implied. N c is the number of colors. The three parameters in H eff are the Wilson parameter r, the current quark mass M 0 and the effective coupling constant K = 2N c /(N 2 c − 1) 1/g 2 where g is the QCD coupling constant. When r = M 0 = 0 the Hamiltonian possesses a U(4N f ) symmetry with N f being the number of flavors. This symmetry is spontaneously broken down to U(2N f ) ⊗ U(2N f ) accompanied by the appearance of 8N 2 f Goldstone bosons [12]. A finite current quark mass also breaks the original U(4N f ) symmetry, albeit explicitly, down to (2N f ) ⊗ U(2N f ). Introduction of a finite Wilson parameter further breaks the latter symmetry explicitly down to U(N f ) thereby solving the fermion doubling problem.
The above Hamiltonian has been derived in the temporal gauge using second order degenerate perturbation theory, and provides an effective description of only the ground state of strongly coupled lattice QCD [12]. This ground state is the one in which no links are excited by the color electric flux. In the strong coupling limit the energy of one excited color electric flux link is
E = 1 2N c (N 2 c − 1) g 2 = 1 K(2)
Therefore an extension of H eff to finite T and/or µ will be valid as long as T, µ < 1/K [8]. 2 We shall see that this condition is satisfied in the present work.
Our method for obtaining the equation of state of strongly coupled lattice QCD at finite density using H eff does not involve composite fields. Instead we explicitly construct a solution to the equation of motion corresponding to H eff for all densities and use it to calculate the equation of state. For free space such a solution has been found in [14]. This solution has the same structure as the free lattice Dirac field and exactly diagonalizes H eff to second order in field operators. It obeys the free lattice Dirac equation with a dynamical quark mass which is determined by solving a gap equation.
Temporarily dropping color and flavor indices this solution is given by
Ψ ν (t, p ) = b( p )ξ ν ( p )e −iω( p )t + d † (− p )η ν (− p )e +iω( p )t(3)
with ν denoting the Dirac index. The annihilation operators for particles b and anti-particles d annihilate an interacting vacuum state and obey the free fermion anti-commutation relations. The properties of the spinors ξ and η are given in [14]. The equation of motion for a free lattice Dirac field fixes the excitation energy ω( p ) to be
ω( p ) = l sin 2 ( p ·n l ) + M 2 ( p ) 1/2(4)
where M( p ) is the dynamical quark mass. The extension of the method developed in [14] to finite T and µ is accomplished in two steps. The first one is to make the following trivial replacement of the current quark mass term in H eff Eq. (1)
M 0 (γ 0 ) ρν → M 0 (γ 0 ) ρν − µ 0 δ ρν(5)
where µ 0 is the quark chemical potential. Note that µ 0 should not be identified with the total chemical potential µ tot of the interacting many body system. As we shall see below the interaction will induce a correction to µ 0 which in general is momentum dependent. We shall therefore refer to µ 0 as the "bare" quark chemical potential and treat it as a parameter. The second step is to observe that the annihilation operators b and d in Eq. (3) no longer annihilate the interacting vacuum state at finite T and µ denoted as | G(T, µ) . In order to construct operators that annihilate | G(T, µ) we apply a generalized thermal Bogoliubov transformation to the b and d operators following the formalism of thermal field dynamics [15] b
( p ) = α p B( p ) − β pB † (− p ) (6) d( p ) = γ p D( p ) − δ pD † (− p )(7)
The thermal field operators B andB † annihilate a quasi-particle and create a quasi-hole at finite T and µ, respectively, while D andD † are the annihilation operator for a quasi-anti-particle and creation operator for a quasi-anti-hole, respectively. These thermal annihilation operators annihilate the interacting thermal vacuum state for each T and µ.
B( p )| G(T, µ) =B( p )| G(T, µ) = D( p )| G(T, µ) =D( p )| G(T, µ) = 0 (8)
The thermal doubling of the Hilbert space accompanying the thermal Bogoliubov transformation is implicit in Eq. (8) where the vacuum state which is annihilated by thermal operators B,B, D andD is defined. Since we shall be working only in the space of quantum field operators it is not necessary to specify the structure of | G(T, µ) .
The thermal operators also satisfy the fermion anti-commutation relations
δ p, q = B † ( p ), B( q ) + = B † ( p ),B( q ) + = D † ( p ), D( q ) + = D † ( p ),D( q ) + (9)
with vanishing anti-commutators for the remaining combinations. The coefficients of the transformation are α p = 1 − n − p , β p = n − p , γ p = 1 − n + p and δ p = n + p , where n ± p = [e (ωp±µ)/(k B T ) + 1] −1 are the Fermi distribution functions for particles and anti-particles. They are chosen so that the total particle number densities are given by
n − p = G(T, µ)| b † ( p )b( p )| G(T, µ) (10) n + p = G(T, µ)| d † ( p )d( p )| G(T, µ)(11)
Hence in this approach temperature and chemical potential are introduced simultaneously through the coefficients of the thermal Bogoliubov transformation and are treated on an equal footing. We stress that the chemical potential appearing in the Fermi distribution functions is the total chemical potential of the interacting many body system. In addition to these changes, we demand that our ansatz satisfies the equation of motion corresponding to the free lattice Dirac Hamiltonian with a chemical potential term given by
H 0 = 1 2 p − l sin( p ·n l )(γ 0 γ l ) ρν + M( p )(γ 0 ) ρν − µ tot δ ρν ⊗ Ψ † ρ (t, p ), Ψ ν (t, p ) −(12)
As in [14] the mass M( p ) is identified with the dynamical quark mass. Thus our ansatz at finite T and µ is
Ψ ν (t, p ) = α p B( p ) − β pB † (− p ) ξ ν ( p )e −i[ω( p )−µtot]t + γ p D † (− p ) − δ pD ( p ) η ν (− p )e +i[ω( p )+µtot]t (13)
The spinors ξ and η obey the same properties as in free space and the excitation energy ω( p ) has the same form as in Eq. (4). The unknown quantities in Eq. (13) are the dynamical quark mass and the total chemical potential.
In this work we shall take the T → 0 limit which amounts to setting γ p = 1 and δ p = 0 in Eq. (7) thereby suppressing the excitation of anti-holes. In this limit β 2 p becomes the Heaviside function β 2 p = θ(µ tot − ω( p )) defining the Fermi momentum p F where
µ tot = l sin 2 ( p F ·n l ) + M 2 ( p F ) 1/2(14)
One of the simplest quantities to calculate using the ansatz of Eq. (13) in the T → 0 limit is the quark number density n given by
n = 1 2V N f N c Ψ γ 0 Ψ = p θ(µ tot − ω( p ))(15)
Therefore, above a sufficiently large value of µ tot the quark number density becomes a constant which with the present normalization will equal unity. This saturation effect is purely a lattice artifact originating from the sin 2 ( p·n l ) term in ω( p ). Another quantity that may be readily calculated using the T → 0 ansatz is the chiral condensate. It is found to be proprotional to the dynamical quark mass
1 2V N f N c Ψ Ψ = − p α 2 p M( p ) ω( p )(16)
Below we shall derive a gap equation for M( p ) and show that for a given physically reasonable set of parameters there exists a critical chemical potential above which M( p ) = 0. Thus the chiral condensate may be identified as being the order parameter for the chiral phase transition at finite density. However before deriving the gap equation we shall demonstrate that in the T → 0 limit the ansatz shown in Eq. (13) exactly diagonalizes the effective Hamiltonian to second order in field operators for all densities. We make use of the fact that our ansatz satisfies the equation of motion corresponding to the free lattice Dirac Hamiltonian H 0 given in Eq. (12). Therefore we have the relation
: (Ψ aα ) µ (t, q ), H 0 − : = : (Ψ aα ) µ (t, q ), H eff − :(17)
where the symbol : : denotes normal ordering with respect to the vacuum at zero temperature | G(T = 0, µ) . Evaluating both sides of Eq. (17) we obtain
l sin( q ·n l )(γ 0 γ l ) ρδ + M( q )(γ 0 ) ρδ − µ tot δ ρδ (Ψ aα ) δ (t, q ) = M 0 (γ 0 ) ρδ − µ 0 δ ρδ + 1 N c K p l α 2 p Λ + νγ ( p ) ⊗ cos ( p − q ) ·n l (Σ l ) γν (Σ l ) † ρδ + (Σ l ) † ρν (Σ l ) γδ + cos ( p + q ) ·n l (Σ l ) † γν (Σ l ) † ρδ + (Σ l ) ρν (Σ l ) γδ − 1 N c K 4 p, q l 2α 2 p Λ + νγ ( p ) − δ νγ ⊗ N c (Σ l ) ρν (Σ l ) † γδ + (Σ l ) † ρν (Σ l ) γδ + cos ( p + q ) ·n l (Σ l ) † ρν (Σ l ) † γδ + (Σ l ) ρν (Σ l ) γδ (Ψ aα ) δ (t, q ) (18)
with Λ + ( p ) ≡ ξ( p ) ⊗ ξ † ( p ) being the positive energy projection operator defined in [14]. To second order in field operators the off-diagonal Hamiltonian is given by
H off | G(0, µ) = q α q ξ † ρ ( q ) M 0 (γ 0 ) ρδ − µ 0 δ ρδ + 1 N c K p, q l α 2 p α q Λ + νρ ( p ) ⊗ ξ † γ ( q ) cos ( p − q ) ·n l (Σ l ) ρν (Σ l ) † γδ + (Σ l ) † ρν (Σ l ) γδ + cos ( p + q ) ·n l (Σ l ) † ρν (Σ l ) † γδ + (Σ l ) ρν (Σ l ) γδ − 1 N c K 4 p, q l α q 2α 2 p Λ + νγ ( p ) − δ νγ ⊗ ξ † ρ ( q ) N c (Σ l ) ρν (Σ l ) † γδ + (Σ l ) † ρν (Σ l ) γδ + cos ( p + q ) ·n l (Σ l ) † ρν (Σ l ) † γδ + (Σ l ) ρν (Σ l ) γδ η δ (− q ) ⊗B † α,a ( q )D † α,a (− q )| G(0, µ)(19)H off | G(0, µ) = q α q ξ † ν ( q ) − l sin( q ·n l )(γ 0 γ l ) νδ −M( q )(γ 0 ) νδ + µ tot δ νδ η δ (− q ) ⊗B † α,a ( q )D † α,a (− q )| G(0, µ) = q α q ξ † ν ( q ) ω( q ) + µ tot η ν (− q ) ⊗B † α,a ( q )D † α,a (− q )| G(0, µ) = 0 (20)
Therefore our ansatz exactly diagonalizes the effective Hamiltonian to second order in field operators for all densities. We now derive the equations for the dynamical quark mass and the total chemical potential and solve them to determine our solution Eq. (13) for each density. To accomplish this we explicitly evaluate the right hand side of Eq. (18) to reveal its Dirac structure. The result may be cast in the following compact form
l sin( q ·n l )(γ 0 γ l ) νδ + M( q )(γ 0 ) νδ − µ tot δ νδ (Ψ aα ) δ (t, q ) = A( q )(γ 0 γ l ) νδ + B( q )(γ 0 ) νδ + C( q )δ νδ (Ψ aα ) δ (t, q ) (21)
The equations for M( p ) and µ tot are obtained by equating the coefficents of the γ 0 operator and the Kronecker delta function, respectively.
The gap equation determining M( p ) is given by the coefficient B( q )
M( q ) = B( q ) = M 0 + 3 2 K(1 − r 2 ) p 1 − β 2 p M( p ) ω( p ) + K N c p,l 1 − β 2 p M( p ) ω( p ) 8r 2 cos( p ·n l ) cos( q ·n l ) − 1 2 (1 + r 2 ) cos( p + q ) ·n l(22)
The structure of this gap equation is very similar to the one in free space (β 2 p = 0) found in [14]. The dynamical quark mass is a constant to lowest order in N c but becomes momentum dependent once 1/N c correction is taken into account.
Similarly, the total chemical potential is given by the coefficient C( q )
µ tot = −C( q ) = µ 0 + 1 4 K N c l p β 2 p 2N c 1 + r 2 − 2 1 − r 2 cos ( p + q ) (23)
Thus µ tot is a sum of the bare chemical potential µ 0 and an interaction induced chemical potential which is proportional to the effective coupling constant K. Furthermore, the latter contribution to µ tot is momentum dependent and this dependence is a 1/N c correction just as in the case of the gap equation. It should be noted that the above shifting of the bare chemical potential by the interaction is not a new effect. For example, in the wellknown and well-studied Nambu-Jona-Lasinio model [16] at finite T and µ the interaction induces a contribution to the total chemical potential which is proportional to the number density [17,18]. The two equations (22) and (23) are coupled and therefore solutions for M and µ tot must be found self-consistently for each value of the input parameter µ 0 . In Figure 1 we present M as a function of µ tot for two values of K determined by solving Eqs. 22) and (23) self-consistently to lowest order in N c using M 0 = 0, r = 0.25 and N c = 3. There is a second order chiral phase transition when the effective coupling constant K is 0.9. The order of the phase transition changes to first order when K is increased to 1.0. potential of (µ tot ) C ≈ 0.716, while if the coupling constant is increased to K = 1.0 the phase transition becomes first order with a larger critical chemical potential of (µ tot ) C ≈ 0.871. Furthermore, we find that when K = 0.9 lattice saturation sets in around µ tot ≈ 0.898 while this effect takes place immediately above (µ tot ) C for K = 1.0. These values of chemical potentials are smaller than the energy E = 1/K required to excite one color electric flux link as given in Eq. (2). Therefore with a reasonable set of parameters it is possible to extend Smit's effective Hamiltonian to finite density as was first pointed out in [8].
Having solved for the dynamical quark mass and the total chemical potential we have constructed a solution to the equation of motion for H eff in the T → 0 limit to lowest order in N c . In addition we have shown that this solution exactly diagonalizes the effective Hamiltonian to second order in field operators for all densities. Therefore we may use it to evaluate the vacuum energy density which to lowest order in N c is given by
1 V G(0, µ) |H eff | G(0, µ) = −2N c p α 2 p 3 2 K(1 + r 2 ) + ω( p ) + M ω( p ) M 0 − µ tot + (1 + β 2 p )µ 0 (24)
Numerically we find that the difference of the vacuum energy densities in the symmetry restored (M = 0) and broken (M = 0) phases of the theory is positive
1 V G(0, µ) |H eff | G(0, µ) | M =0 − 1 V G(0, µ) |H eff | G(0, µ) | M =0 > 0(25)
Therefore the phase with broken chiral symmetry is the energetically favored phase.
The equation of state is obtained by numerically evaluating the thermodynamic potential density using Eq. (24). In Figure 2 we plot the pressure as a function of µ tot for K = 0.9 and 1.0 with M 0 = 0, r = 0.25 and N c = 3. In both cases we find that the pressure of the quark Fermi sea is negative and monotonically decreasing in the broken symmetry phase. For K = 0.9 the pressure remains negative but increasing in the symmetry restored phase, at least until the lattice saturation point, and has a cusp where the two phases meet. Unfortunately, we can not make a definite quantitative statement on the behaviour of the pressure in the symmetry restored phase for K = 1.0 due to lattice saturation, except to mention that there is a discontinuity when going from one phase to another. However, we may conclude that up to and beyond the chiral symmetry restoration point the quark Fermi sea can have negative pressure and therefore can be mechanically unstable with an imaginary speed of sound.
Our conclusion regarding the (strongly coupled) quark matter stability at finite density is consistent with similar studies using the Nambu-Jona-Lasinio model [10] and the effective instanton induced 't Hooft interaction model [11]. These mean field calculations show that cold and dense quark matter may be unstable in the phase with spontaneously broken chiral symmetry, but can become stable in the symmetry restored phase at high enough density. In particular, the result for the pressure of cold and dense quark matter obtained in [11] is qualitatively the same as the one shown in Figure 2. 3 The possibility of unstable quark mattter lead the authors of [10] and [11] to speculate the formation of nucleon droplets, reminiscent of the MIT bag model, in the broken symmetry phase. We shall not dwell on such a speculation here since we are working in an artificial strong coupling regime. Nevertheless, it would be interesting to compare our result concerning the negative pressure with future lattice simulations of finite density QCD at strong coupling.
In this work we studied the equation of state of strongly coupled lattice QCD in the Hamiltonian formulation using Wilson fermions. This was accomplished by constructing a solution of the equation of motion correponding The results were obtained using the same parameters as in Figure 1. For K = 0.9, the critical chemical potential is (µ tot ) C ≈ 0.716 and lattice saturation sets in around µ tot ≈ 0.898, while this effect takes place right above (µ tot ) C ≈ 0.871 for K = 1.0.
to an effective Hamiltonian which exactly diagonalizes the Hamiltonian to second order in field operators for all densities. We found that: the dynamical quark mass is in general momentum dependent; the interaction induces a momentum dependent contribution to the total chemical potential making it necessary to solve for the dynamical quark mass self-consistently with µ tot ; the elementary excitations of the theory consist of color singlet quark-antiquark pairs coupled to zero total three momentum; and the broken symmetry phase is the energetically favored phase. To leading order in N c we find that the chiral phase transition can be either first or second order depending on the value of the effective coupling constant. In addition, the pressure of the strongly interacting many body system is found to be negative up to and beyond the chiral phase transition density. A similar behaviour for the pressure has been obtained with r = 0 which corresponds to using Kogut-Susskind fermions. Therefore our result concerning the negative pressure seems to be robust, at least to leading order in N c , and should be verifiable by future numerical simulations of strongly coupled lattice QCD at finite density.
Figure 1 :
1(22) and (23) to O(N 0 c ), which is the same order in the 1/N c expansion used to obtain results in all previous studies of strongly coupled lattice QCD. At this order in N c both the dynamical mass and the total chemical potential are momentum independent. The values of input parameters are M 0 = 0, r = 0.25 and N c =3.From the figure we see that the chiral phase transition can be either first or second order depending on the value of the effective coupling constant. When K = 0.9 we find a second order phase transition with a critical chemical Dynamical quark mass M as a function of total chemical potential µ tot for two values of effective coupling constant K. These results were obtained by solving Eqs. (
Figure 2 :
2Pressure P as a function total chemical potential µ tot for two values of effective coupling constant K.
The work of[8] does not involve composite fields but the approach is equivalent to the mean field approximation.
Note that in[8] E has been approximated by E ≈ N c g 2 .
Compare Figure 1 of [11] with Figure 2 of this letter.
AcknowledgementsI am indebted to M.-P. Lombardo for comments and suggestions which lead to an improvement of this manuscript, as well as to O.W. Greenberg for reading the first draft. Part of this work was completed while I was at ECT * as a Junior Visiting Scientist. I thank the Center for its hospitality and generous support.
. M Creutz, Nucl. Phys. Proc. Suppl. 94219M. Creutz, Nucl. Phys. Proc. Suppl. 94 (2001) 219.
. R Aloisio, V Azcoiti, G Di Carlo, A Galante, A F Grillo, Nucl. Phys. B. 564489R. Aloisio, V. Azcoiti, G. Di Carlo, A. Galante and A.F. Grillo, Nucl. Phys. B 564 (2000) 489.
. P H Damgaard, D Hochberg, N Kawamoto, Phys. Letts. B. 158239P.H. Damgaard, D. Hochberg, N. Kawamoto, Phys. Letts. B 158 (1985) 239.
. E.-M Ilgenfritz, J Kripfganz, Z. Phys. C. 2979E.-M. Ilgenfritz and J. Kripfganz, Z. Phys. C 29 (1985) 79.
. N Bilić, K Demeterfi, B Petersson, Nucl. Phys. B. 377651N. Bilić, K. Demeterfi and B. Petersson, Nucl. Phys. B 377 (1992) 651.
. A Patel, Phys. Letts. B. 141244A. Patel, Phys. Letts. B 141 (1984) 244.
. C P Van Den, Doel, Phys. Letts. B. 143210C.P. Van Den Doel, Phys. Letts. B 143 (1984) 210.
. A Le Yaouanc, L Oliver, O Pène, J.-C Raynal, M Jarfi, O Lazrak, Phys Rev D. 373691A. Le Yaouanc, L. Oliver, O. Pène and J.-C. Raynal, M. Jarfi and O. Lazrak, Phys Rev D 37 (1986) 3691; 3702.
. E B Gregory, S.-H Guo, H Kröger, X.-Q Luo, Phys. Rev. D. 6254508E.B. Gregory, S.-H. Guo, H. Kröger and X.-Q. Luo, Phys. Rev. D 62 (2000) 054508.
. M Buballa, Nucl. Phys. A. 611393M. Buballa, Nucl. Phys. A 611 (1996) 393.
. M Alford, K Rajagopal, F Wilczek, Phys. Letts. B. 422247M. Alford, K. Rajagopal and F. Wilczek, Phys. Letts. B 422 (1998) 247.
. J Smit, Nucl. Phys. B. 175307J. Smit, Nucl. Phys. B 175 (1980) 307.
. A Le Yaouanc, L Oliver, O Pène, J.-C , Phys. Rev D. 333098A. Le Yaouanc, L. Oliver, O. Pène and J.-C. Raynal, Phys. Rev D 33 (1986) 3098.
Y Umino, arXiv:hep-lat/0007356v.2The corrected version of Figure 1b is available at Los Alamos Archives. 492385Y. Umino, Phys. Letts. B 492 (2000) 385; The corrected version of Fig- ure 1b is available at Los Alamos Archives arXiv:hep-lat/0007356 v.2.
H Umezawa, H Matsumoto, M Tachiki, Thermo Field Dynamics and Condensed States. North-Holland, AmsterdamH. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dy- namics and Condensed States, (North-Holland, Amsterdam, 1982);
H Umezawa, Advanced Field Theory. New YorkAIPH. Umezawa, Advanced Field Theory, (AIP, New York, 1992).
. Y Nambu, G Jona-Lasinio, Phys. Rev. 122246Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246.
. M Asakawa, K Yazaki, Nucl. Phys. A. 504668M. Asakawa and K. Yazaki, Nucl. Phys. A 504 (1989) 668.
. S P Klevansky, Rev. Mod. Phys. 64649S.P. Klevansky, Rev. Mod. Phys. 64 (1992) 649.
| [] |
[
"Herschel ⋆ -ATLAS: Rapid evolution of dust in galaxies over the last 5 billion years",
"Herschel ⋆ -ATLAS: Rapid evolution of dust in galaxies over the last 5 billion years"
] | [
"L Dunne \nSchool of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK\n",
"H L Gomez \nSchool of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK\n",
"E Da Cunha \nMax Planck Institute for Astronomy\nKonigstuhl 1769117HeidelbergGermany\n\nDepartment of Physics\nUniversity of Crete\nPO Box 220871003HeraklionGreece\n",
"S Charlot \nUMR 7095\nInstitut d'Astrophysique de Paris\nCNRS\nUniversité Pierre & Marie Curie\n98bis bd Arago75014ParisFrance\n",
"S Dye \nSchool of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK\n",
"S Eales \nSchool of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK\n",
"S J Maddox \nSchool of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK\n",
"K Rowlands \nSchool of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK\n",
"D J B Smith \nSchool of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK\n",
"R Auld \nSchool of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK\n",
"M Baes \nSterrenkundig Observatorium\nUniversiteit Gent\nKrijgslaan 281 S9B-9000GentBelgium\n",
"D G Bonfield \nCentre for Astrophysics\nScience & Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfieldHertsUK\n",
"N Bourne \nSchool of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK\n",
"S Buttiglione \nINAF-Osservatorio Astronomico di Padova\nVicolo Osservatorio 5I-35122PadovaItaly\n",
"A Cava \nDepartmento de Astrofísica\nFacultad de CC. Físicas\nUniversidad Complutense de Madrid\nE-28040MadridSpain\n",
"D L Clements \nDepartment of Physics and Astronomy\nUniversity of California\n92697IrvineCAUSA\n",
"K E K Coppin \nDepartment of Physics\nMcGill University\nErnest Rutherford Building, 3600 Rue UniversityH3A 2T8MontrealQuebecCanada\n\nInstitute for Computational Cosmology\nDurham University\nSouth RoadDH1 3LEDurhamUK\n",
"A Cooray \nPhysics Department\nImperial College\nPrince Consort RoadSW7 2AZLondon\n",
"A Dariush \nSchool of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK\n",
"M J Jarvis \nCentre for Astrophysics\nScience & Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfieldHertsUK\n",
"L Kelvin \nSchool of Physics and Astronomy\nSUPA\nUniversity of St\nAndrews\n\nNorth Haugh\nKY16 9SSSt. AndrewsUK\n",
"E Pascale \nSchool of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK\n",
"M Pohlen \nSchool of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK\n",
"C Popescu \nJeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPrestonUK\n",
"E E Rigby \nSchool of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK\n",
"A Robotham \nSchool of Physics and Astronomy\nSUPA\nUniversity of St\nAndrews\n\nNorth Haugh\nKY16 9SSSt. AndrewsUK\n",
"G Rodighiero \nDepartment of Astronomy\nUniversity of Padova\nVicolo Osservatorio 3I-35122PadovaItaly\n",
"A E Sansom \nJeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPrestonUK\n",
"S Serjeant \nAstrophysics Branch\nNASA Ames Research Center\n2456, 94035Mail Stop, Moffett FieldCAUSA\n",
"P Temi \nDept. of Physics and Astronomy\nThe Open University\nMilton KeynesMK7 6AA\n",
"M Thompson \nCentre for Astrophysics\nScience & Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfieldHertsUK\n",
"R Tuffs \nMax Planck Institut fuer Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany\n",
"P Van Der Werf \nUk Astronomy Technology Centre\nRoyal Observatory\nEH9 3HJEdinburghUK\n\nLeiden Observatory\nLeiden University\nP.O. Box 9513NL -2300 RALeiden\n",
"C Vlahakis \nDepartamento de Astronomia\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n",
"\nSISSA\nVia Bonomea 265I-34136TriesteItaly\n",
"\nInstitute for Astronomy\nSUPA\nUniversity of Edinbugh\nRoyal Observatory\nBlackford HillEH9 3HJEdinbughUK\n"
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"Max Planck Institute for Astronomy\nKonigstuhl 1769117HeidelbergGermany",
"Department of Physics\nUniversity of Crete\nPO Box 220871003HeraklionGreece",
"UMR 7095\nInstitut d'Astrophysique de Paris\nCNRS\nUniversité Pierre & Marie Curie\n98bis bd Arago75014ParisFrance",
"School of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK",
"School of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK",
"School of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK",
"School of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK",
"School of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK",
"School of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK",
"Sterrenkundig Observatorium\nUniversiteit Gent\nKrijgslaan 281 S9B-9000GentBelgium",
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"School of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK",
"INAF-Osservatorio Astronomico di Padova\nVicolo Osservatorio 5I-35122PadovaItaly",
"Departmento de Astrofísica\nFacultad de CC. Físicas\nUniversidad Complutense de Madrid\nE-28040MadridSpain",
"Department of Physics and Astronomy\nUniversity of California\n92697IrvineCAUSA",
"Department of Physics\nMcGill University\nErnest Rutherford Building, 3600 Rue UniversityH3A 2T8MontrealQuebecCanada",
"Institute for Computational Cosmology\nDurham University\nSouth RoadDH1 3LEDurhamUK",
"Physics Department\nImperial College\nPrince Consort RoadSW7 2AZLondon",
"School of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK",
"Centre for Astrophysics\nScience & Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfieldHertsUK",
"School of Physics and Astronomy\nSUPA\nUniversity of St\nAndrews",
"North Haugh\nKY16 9SSSt. AndrewsUK",
"School of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK",
"School of Physics & Astronomy\nCardiff University\nQueen Buildings, The ParadeCF24 3AACardiffUK",
"Jeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPrestonUK",
"School of Physics & Astronomy\nNottingham University\nUniversity Park CampusNG7 2RDNottinghamUK",
"School of Physics and Astronomy\nSUPA\nUniversity of St\nAndrews",
"North Haugh\nKY16 9SSSt. AndrewsUK",
"Department of Astronomy\nUniversity of Padova\nVicolo Osservatorio 3I-35122PadovaItaly",
"Jeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPrestonUK",
"Astrophysics Branch\nNASA Ames Research Center\n2456, 94035Mail Stop, Moffett FieldCAUSA",
"Dept. of Physics and Astronomy\nThe Open University\nMilton KeynesMK7 6AA",
"Centre for Astrophysics\nScience & Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfieldHertsUK",
"Max Planck Institut fuer Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany",
"Uk Astronomy Technology Centre\nRoyal Observatory\nEH9 3HJEdinburghUK",
"Leiden Observatory\nLeiden University\nP.O. Box 9513NL -2300 RALeiden",
"Departamento de Astronomia\nUniversidad de Chile\nCasilla 36-DSantiagoChile",
"SISSA\nVia Bonomea 265I-34136TriesteItaly",
"Institute for Astronomy\nSUPA\nUniversity of Edinbugh\nRoyal Observatory\nBlackford HillEH9 3HJEdinbughUK"
] | [
"Mon. Not. R. Astron. Soc"
] | We present the first direct and unbiased measurement of the evolution of the dust mass function of galaxies over the past 5 billion years of cosmic history using data from the Science Demonstration Phase of the Herschel-ATLAS. The sample consists of galaxies selected at 250µm which have reliable counterparts from SDSS at z < 0.5, and contains 1867 sources. Dust masses are calculated using both a single temperature grey-body model for the spectral energy distribution and also using a model with multiple temperature components. The dust temperature for either model shows no trend with redshift. Splitting the sample into bins of redshift reveals a strong evolution in the dust properties of the most massive galaxies. At z = 0.4 − 0.5, massive galaxies had dust masses about five times larger than in the local Universe. At the same time, the dust-to-stellar mass ratio was about 3-4 times larger, and the optical depth derived from fitting the UV-sub-mm data with an energy balance model was also higher. This increase in the dust content of massive galaxies at high redshift is difficult to explain using standard dust evolution models and requires a rapid gas consumption timescale together with either a more top-heavy IMF, efficient mantle growth, less dust destruction or combinations of all three. This evolution in dust mass is likely to be associated with a change in overall ISM mass, and points to an enhanced supply of fuel for star formation at earlier cosmic epochs. | 10.1111/j.1365-2966.2011.19363.x | [
"https://arxiv.org/pdf/1012.5186v3.pdf"
] | 18,446,956 | 1012.5186 | d95e9bde4ef8bab4189c0c3af9c9634ad23cced4 |
Herschel ⋆ -ATLAS: Rapid evolution of dust in galaxies over the last 5 billion years
19 Jul 2011. 2002
L Dunne
School of Physics & Astronomy
Nottingham University
University Park CampusNG7 2RDNottinghamUK
H L Gomez
School of Physics & Astronomy
Cardiff University
Queen Buildings, The ParadeCF24 3AACardiffUK
E Da Cunha
Max Planck Institute for Astronomy
Konigstuhl 1769117HeidelbergGermany
Department of Physics
University of Crete
PO Box 220871003HeraklionGreece
S Charlot
UMR 7095
Institut d'Astrophysique de Paris
CNRS
Université Pierre & Marie Curie
98bis bd Arago75014ParisFrance
S Dye
School of Physics & Astronomy
Cardiff University
Queen Buildings, The ParadeCF24 3AACardiffUK
S Eales
School of Physics & Astronomy
Cardiff University
Queen Buildings, The ParadeCF24 3AACardiffUK
S J Maddox
School of Physics & Astronomy
Nottingham University
University Park CampusNG7 2RDNottinghamUK
K Rowlands
School of Physics & Astronomy
Nottingham University
University Park CampusNG7 2RDNottinghamUK
D J B Smith
School of Physics & Astronomy
Nottingham University
University Park CampusNG7 2RDNottinghamUK
R Auld
School of Physics & Astronomy
Cardiff University
Queen Buildings, The ParadeCF24 3AACardiffUK
M Baes
Sterrenkundig Observatorium
Universiteit Gent
Krijgslaan 281 S9B-9000GentBelgium
D G Bonfield
Centre for Astrophysics
Science & Technology Research Institute
University of Hertfordshire
AL10 9ABHatfieldHertsUK
N Bourne
School of Physics & Astronomy
Nottingham University
University Park CampusNG7 2RDNottinghamUK
S Buttiglione
INAF-Osservatorio Astronomico di Padova
Vicolo Osservatorio 5I-35122PadovaItaly
A Cava
Departmento de Astrofísica
Facultad de CC. Físicas
Universidad Complutense de Madrid
E-28040MadridSpain
D L Clements
Department of Physics and Astronomy
University of California
92697IrvineCAUSA
K E K Coppin
Department of Physics
McGill University
Ernest Rutherford Building, 3600 Rue UniversityH3A 2T8MontrealQuebecCanada
Institute for Computational Cosmology
Durham University
South RoadDH1 3LEDurhamUK
A Cooray
Physics Department
Imperial College
Prince Consort RoadSW7 2AZLondon
A Dariush
School of Physics & Astronomy
Cardiff University
Queen Buildings, The ParadeCF24 3AACardiffUK
M J Jarvis
Centre for Astrophysics
Science & Technology Research Institute
University of Hertfordshire
AL10 9ABHatfieldHertsUK
L Kelvin
School of Physics and Astronomy
SUPA
University of St
Andrews
North Haugh
KY16 9SSSt. AndrewsUK
E Pascale
School of Physics & Astronomy
Cardiff University
Queen Buildings, The ParadeCF24 3AACardiffUK
M Pohlen
School of Physics & Astronomy
Cardiff University
Queen Buildings, The ParadeCF24 3AACardiffUK
C Popescu
Jeremiah Horrocks Institute
University of Central Lancashire
PR1 2HEPrestonUK
E E Rigby
School of Physics & Astronomy
Nottingham University
University Park CampusNG7 2RDNottinghamUK
A Robotham
School of Physics and Astronomy
SUPA
University of St
Andrews
North Haugh
KY16 9SSSt. AndrewsUK
G Rodighiero
Department of Astronomy
University of Padova
Vicolo Osservatorio 3I-35122PadovaItaly
A E Sansom
Jeremiah Horrocks Institute
University of Central Lancashire
PR1 2HEPrestonUK
S Serjeant
Astrophysics Branch
NASA Ames Research Center
2456, 94035Mail Stop, Moffett FieldCAUSA
P Temi
Dept. of Physics and Astronomy
The Open University
Milton KeynesMK7 6AA
M Thompson
Centre for Astrophysics
Science & Technology Research Institute
University of Hertfordshire
AL10 9ABHatfieldHertsUK
R Tuffs
Max Planck Institut fuer Kernphysik
Saupfercheckweg 1D-69117HeidelbergGermany
P Van Der Werf
Uk Astronomy Technology Centre
Royal Observatory
EH9 3HJEdinburghUK
Leiden Observatory
Leiden University
P.O. Box 9513NL -2300 RALeiden
C Vlahakis
Departamento de Astronomia
Universidad de Chile
Casilla 36-DSantiagoChile
SISSA
Via Bonomea 265I-34136TriesteItaly
Institute for Astronomy
SUPA
University of Edinbugh
Royal Observatory
Blackford HillEH9 3HJEdinbughUK
Herschel ⋆ -ATLAS: Rapid evolution of dust in galaxies over the last 5 billion years
Mon. Not. R. Astron. Soc
0001719 Jul 2011. 2002Printed (MN L A T E X style file v2.2)Galaxies: LocalInfraredStar-formingLFMFISM
We present the first direct and unbiased measurement of the evolution of the dust mass function of galaxies over the past 5 billion years of cosmic history using data from the Science Demonstration Phase of the Herschel-ATLAS. The sample consists of galaxies selected at 250µm which have reliable counterparts from SDSS at z < 0.5, and contains 1867 sources. Dust masses are calculated using both a single temperature grey-body model for the spectral energy distribution and also using a model with multiple temperature components. The dust temperature for either model shows no trend with redshift. Splitting the sample into bins of redshift reveals a strong evolution in the dust properties of the most massive galaxies. At z = 0.4 − 0.5, massive galaxies had dust masses about five times larger than in the local Universe. At the same time, the dust-to-stellar mass ratio was about 3-4 times larger, and the optical depth derived from fitting the UV-sub-mm data with an energy balance model was also higher. This increase in the dust content of massive galaxies at high redshift is difficult to explain using standard dust evolution models and requires a rapid gas consumption timescale together with either a more top-heavy IMF, efficient mantle growth, less dust destruction or combinations of all three. This evolution in dust mass is likely to be associated with a change in overall ISM mass, and points to an enhanced supply of fuel for star formation at earlier cosmic epochs.
† E-mail:[email protected]
INTRODUCTION
The evolution of the dust content of galaxies is an important and poorly understood topic. Dust is responsible for obscuring the UV and optical light from galaxies and thus introduces biases into our measures of galaxy properties based on their stellar light (Driver et al. 2007). The energy absorbed by dust is re-emitted at longer infrared and sub-millimetre (sub-mm) wavelengths, providing a means of recovering the stolen starlight. Dust emission is often used as an indicator of the current star formation rate in galaxies -although this calibration makes the assumption that young, massive stars are the main source of heating for the dust and that the majority of the UV photons from the young stars are absorbed and re-radiated by dust (Kennicutt et al. 1998(Kennicutt et al. , 2009Calzetti et al. 2007). Many surveys of dust emission from 24-850µm (Saunders et al. 1990;Blain et al. 1999;Le Floc'h et al. 2005;Gruppioni et al. 2010;Dye et al. 2010;Eales et al. 2010) have noted the very strong evolution present in these bands and this is usually ascribed to a decrease in the star formation rate density over the past 8 billion years of cosmic history (z ∼ 1: Madau et al. 1996, Hopkins 2004. The interpretation of this evolution is complicated by the fact that the dust luminosity of a galaxy is a function of both the dust content and the temperature of the dust. It is pertinent to now ask the question "What drives the evolution in the FIR luminosity density?", is it an increase in dust heating (due to enhanced star formation activity) or an increase the dust content of galaxies (due to their higher gas content in the past) -or both?
Dust is thought to be produced by both low-intermediate mass AGB stars (Gehrz 1989;Ferrarotti & Gail 2006;Sargent et al. 2010) and by massive stars when they explode as supernovae at the end of their short lives (Rho et al. 2008;Dunne et al. 2009; Barlow et al. 2010). Thus, the dust mass in a galaxy should be related to its current and past star formation history. Dust is also destroyed through astration and via supernovae shocks (Jones et al. 1994), and may also reform through accretion in both the dense and diffuse ISM (Zhukovska et al. 2008;Inoue 2003;Tielens 1998). The life cycle of dust is thus a complicated process which many have attempted to model Dwek et al. 1998;Calura et al. 2008, Gomez et al. 2010Gall, Anderson & Hjorth 2011) and yet the basic statistic describing the dust content of galaxies -the dust mass function (DMF) -is not well determined.
The first attempts to measure the dust mass function were made by Dunne et al. (2000; hereafter D00) and Dunne & Eales (2001;hereafter DE01) as part of the SLUGS survey using a sample of IRAS bright galaxies observed with SCUBA at 450 and 850µm . Vlahakis, Dunne & Eales (2005; hereafter VDE05) improved on this by adding an optically selected sample with sub-mm observations. These combined studies, however, comprised less than 200 objects -none of which were selected on the basis of their dust mass. These studies were also at very low-z and did not allow for a determination of evolution. A high-z dust mass function was estimated by hereafter DEE03) using data from deep sub-mm surveys. This showed considerable evolution with galaxies at the high mass end requiring an order of magnitude more dust at z ∼ 2.5 compared to today (for pure luminosity evolution), though with generous caveats due to the difficulties in making this measurement. Finally, Eales et al. (2009) used BLAST data from 250-500µm and also concluded that there was strong evolution in the dust mass function between z = 0 − 1 but were also limited by small number statistics and confusion in the BLAST data due to their large beam size.
In this paper, we present the first direct measurement of the space density of galaxies as a function of dust mass out to z = 0.5. Our sample is an order of magnitude larger than previous studies, and is the first which is near 'dust mass' selected. We then use this sample to study the evolution of dust mass in galaxies over the past ∼ 5 billion years of cosmic history in conjunction with the elementary dust evolution model of Edmunds (2001).
The new sample which allows us to study the dust mass function in this way comes from the Herschel-Astrophysical Terahertz Large Area Survey (H-ATLAS; Eales et al., 2010), which is the largest open-time key project currently being carried out with the Herschel Space Observatory (Pilbratt et al., 2010). H-ATLAS will survey in excess of 550 deg 2 in five bands centered on 100, 160, 250, 350 and 500µm, using the PACS (Poglitsch et al., 2010) and SPIRE instruments (Griffin et al., 2010). The observations consist of two scans in parallel mode reaching 5σ point source sensitivities of 132, 126, 32, 36 and 45 mJy in the 100, 160, 250, 350 & 500µm bands respectively, with beam sizes of approximately 9 ′′ , 13 ′′ , 18 ′′ , 25 ′′ and 35 ′′ . The SPIRE and PACS map-making are described in the papers by Pascale et al. (2011) and Ibar et al. (2010), while the catalogues are described in Rigby et al. (2011). One of the primary aims of the Herschel-ATLAS is to obtain the first unbiased survey of the local Universe at sub-mm wavelengths, and as a result was designed to overlap with existing large optical and infrared surveys. These Science Demonstration Phase (SDP) observations are centered on the 9 h field of the Galaxy And Mass Assembly (GAMA; Driver et al. 2011) survey. The SDP field covers 14.4 sq. deg and comprises approximately one thirtieth of the eventual full H-ATLAS sky coverage.
In section 2 we describe the sample that we have chosen to use for this analysis and the completeness corrections required. In section 3 we describe how we have derived luminosities and dust masses from the Herschel data, while in section 4, we present the dust mass function and evaluate its evolution. Section 6 compares the DMF to models of dust evolution in order to explain the origin of the strong evolution. Throughout we use a cosmology with Ωm = 0.27, ΩΛ = 0.73 and Ho = 71 km s −1 Mpc −1 .
SAMPLE DEFINITIONS
The sub-mm catalogue used in this work is based on the > 5σ at 250µm catalogue from Rigby et al. (2011), which contains 6610 sources. The 250µm fluxes of sources selected in this way have been shown to be unaffected by flux boosting, see Rigby et al. (2011) for a thorough description. Sources from this catalogue are matched to optical counterparts from SDSS DR7 (Abazajian et al. 2009) down to a limiting magnitude of r-modelmag =22.4 using a Likelihood Ratio (LR) technique (e.g. Sutherland & Saunders 1992). The method is described in detail in Smith et al. (2011). Briefly, each optical galaxy within 10 ′′ of a 250µm source is assigned a reliability, R, which is the probability that it is truly associated with the 250µm emission. This method accounts for the possibility that true IDs are below the optical flux limit, the positional uncertainties of both samples, and deals with sharing the likelihoods when there are multiple counterparts. For our study we have used a reliability cut of R 0.8 as this ensures a low contamination rate (< 5 percent) which leaves 2423 250µm sources with reliable counterparts. The LR method tells us that ∼ 3800 counterparts should be present in the SDSS catalogue, however we can only unambiguously associate around 64 percent of these. Our sample is thus low in contamination but incomplete (we will deal specifically with the incompleteness of the ID process in the next Templates for three galaxies showing the range of optical fluxes expected for galaxies which are at the SPIRE flux limit of S 250 = 32 mJy at z = 0.5; the limit of our study. The templates are for M82 (a typical starburst), a Herschel-ATLAS template derived from our survey data by Smith et al. in prep and Arp 220, a highly obscured local ULIRG. The SDSS limit of r = 22.4 is shown as a horizontal dotted line and even a galaxy as obscured as Arp 220 is still visible as an ID to our optical limit at z = 0.5. The yellow shape represent the SDSS-r band filter which was used to compute the optical flux section). A further cut was made to this sample to remove any stars or unresolved objects, this was done using a star-galaxy separation technique based on optical/IR colour and size, similar to that used by Baldry et al. (2010). Only six objects in the final reliable ID catalogue have 'stellar or QSO IDs' and so required removal. We also removed the five sources which were identified as being lensed by Negrello et al. (2010). We then used the GAMA database (Driver et al. 2011) to obtain spectroscopic redshifts for as many of the sources as possible (GAMA target selection is based on SDSS so no further matching is required). These are supplemented by public redshifts from SDSS DR7 (Abazajian et al., 2009), 2SLAQ-LRG (Cannon et al., 2006), 2SLAQ-QSO (Croom et al., 2009) and 6dFGRS (Jones et al., 2009). Where spectroscopic redshifts were not available we used photometric redshifts which were produced for H-ATLAS using SDSS and UKIDSS-LAS (Lawrence et al. 2007) data and the ANNz method (Collister & Lahav 2004). This method is described fully in Smith et al. (2011).
Section 2.1 shows that we can quantify the statistical completeness of the IDs out to z = 0.5 and we choose this as the redshift limit of the current study. The total number of sources in the final sample is 1867 with 1095 spectroscopic redshifts. With this sample, the number of expected false IDs (summing 1 − R, see Smith et al. 2011) is 60 (or 3.2 percent).
Completeness corrections
There are three sources of incompleteness in this current sample.
(i) Sub-mm Catalogue Incompleteness (Cs): This is due to the 250µm flux limit of the survey and the efficacy of the source ex- There is no strong correlation apart from at the brightest fluxes. Only 4 galaxies lie within 0.4 mag of the flux limit used for IDs (r < 22.4) at z < 0.5 and so we consider that optical incompleteness is not a serious problem for this sample. Right: r-mag versus redshift for all sources in GAMA-9 (pale blue squares) and SPIRE IDs with R 0.8 (black triangles). Herschel sources tend to be larger mass optical galaxies and so the SDSS flux-limit does not affect our ability to optically identify H-ATLAS sources until z ∼ 0.5. Note that the right panel uses the brighter limit of r < 19 appropriate for the GAMA redshift survey. traction process. The catalogue number density completeness has been estimated through simulations and presented by Rigby et al. (2011). Apart from the very small range of flux near to the limit, at 32 − 34 mJy the catalogue is > 80 percent complete. Correction factors are applied to each source in turn based on its flux following Tables 1 and 2 in Rigby et al. (2011). The largest correction is in the flux range 32-32.7 mJy and is a factor 2.17, this applies to 124 sources out of a total of 1867 at z < 0.5.
(ii) ID Incompleteness (Cz): The LR method measures in an empirical way a quantity Qo, which is the fraction of SPIRE sources with counterparts above the flux limit in the optical survey. However, it is not possible to unambiguously identify all these counterparts with > 80 percent confidence due to positional uncertainties, close secondaries and the random probability of finding a background source within that search radius. Smith et al. (2011) have estimated a completeness for reliable IDs as a function of redshift. This allows us to make a statistical number density correction in redshift slices for the sources which should have a counterpart above the SDSS limit in that redshift slice, but which do not have R 0.8. This correction is applied to each source and is listed in Table 1. The ID incompleteness is a function of redshift (not unexpectedly) with corrections of a factor ∼ 2 needing to be applied in the highest redshift bins.
(iii) Optical catalogue incompleteness (Cr): This correction is required because the SDSS catalogue from which we made the identifications is itself incomplete as we approach the optical flux limit of r = 22.4. We ascertained the completeness using the background source catalogue used in the ID analysis of Smith et al. (2011), containing all sources which passed the star-galaxy separation at r-modelmag < 22.4 in the primary SDSS DR7 catalogue in a region of ∼ 35 degrees centered on the SDP field. We fitted a linear slope to the logarithmic number counts in the range r = 19 − 21.5 and extrapolated this to fainter magnitudes. We then used the difference between observed and expected number counts to estimate completeness. The results are presented in Table 2 and show that completeness is above 80 percent to r = 21.8, falling to 50 percent by r = 22.2. By restricting our analysis to z < 0.5 we keep 97 percent of the sources below r ∼ 22 and so in the range of acceptable completeness. It is possible, in principle, for there to be some form of optical incompleteness in the sample which is not corrected for with the above prescription, e.g. a population of objects which begin to appear at high redshifts in the H-ATLAS sample but which are not well represented in SDSS. Such a population could conceivably consist of very obscured star-bursts. To test our susceptibility to this, we estimate the SDSS r magnitude of a highly obscured galaxy with an SED like that of Arp 220 (Av = 15) at our 250µm flux limit at the redshift limit of z = 0.5 and find that it would still be detected in our sample. Figure 1 shows three different SED templates normalised to S250 = 32 mJy at z = 0.5: M82, an H-ATLAS based template appropriate for sources at z = 0.5 from Smith et al. in prep, and Arp 220. All templates less obscured than Arp 220 are easily visible at our optical flux limit. We will therefore proceed on the assumption that no such new populations exist below the optical limit in our highest z bins. Figure 2a plots r-mag as a function of 250µm flux. A galaxy with S250 below ∼ 100mJy can have a wide range of optical magnitude (r-mag = 16.5-22.0), and while optical magnitude is a strong function of redshift this is not the case for the sub-mm flux. Figure 2b shows r-mag as a function of redshift for all galaxies in the GAMA 9hr (Driver et al. 2011) spectroscopic sample (cyan), as well as the reliable SPIRE IDs (black). This shows a lack of Herschel sources at the fainter magnitudes at low redshifts (i.e. the lowest absolute magnitudes or stellar masses). 1 It appears that H-ATLAS is less sensitive to low stellar mass galaxies than the SDSS (due to them having lower dust masses) and so only at high-z does the r-band limit preclude the identification of Herschel sources.
DUST MASS AND LUMINOSITY
The Herschel fluxes are translated into monochromatic rest-frame 250µm luminosities following
L250 = 4πD 2 (1 + z) S250 K(1)
where L250 is in WHz −1 , D is the co-moving distance, S250 is the observed flux density at 250µm and K is the K-correction which is given by:
K = ν obs ν obs(1+z) 3+β e (hν obs(1+z) /kT iso ) − 1 e (hν obs /kT iso ) − 1(2)
where ν obs is the observed frequency at 250µm , ν obs(1+z) is the rest-frame frequency and Tiso and β are the temperature and emissivity index describing the global SED shape. In order to derive the values required for the K-correction, a simple grey body SED of the form S ∝ ν β B(ν, T ) was fitted to the PACS and SPIRE fluxes as described in Dye et al. (2010), with a fixed dust emissivity index of β = 1.5 and a temperature range of 10-50 K. Where insufficient data points are available for the fit (300/1867), the median temperature of 26 K from the galaxies which could be fitted was used. With only SPIRE data on the Rayleigh-Jeans side of the SED (as is the case for most sources), only the combination of β and Tiso is well constrained, with the two parameters being inversely correlated by the fit; good fits are obtained with β = 1.5 − 2.0. These simple grey body fits can be performed for the majority of sources and are accurate at representing the flux between rest-frame 250-166µm (relevant for our redshift range) and so are suitable for applying the K-correction.
A dust mass can also be calculated from the observed 250µm flux density and the grey body temperature as:
1 The limit of GAMA is r ∼ 19 which is brighter than the SDSS limit used for H-ATLAS IDs (r ∼ 22.4).
Miso
= S250 D 2 (1 + z) K κ250 B(ν250, Tiso)(3)
where κ250 is the dust mass absorption coefficient which we take to be equal to 0.89 m 2 kg −1 at 250µm (equivalent to scaling κ850 = 0.077 m 2 kg −1 , as used by D00, James et al. 2002, da Cunha, Charlot & Elbaz 2008 with a β = 2). It also lies within the range of values found for the diffuse ISM in the Milky Way and other nearby galaxies (Boulanger et al. 1996;Sodroski et al. 1997;Bianchi et al. 1999;Planck Collaboration 2011a). The dust mass via Eq. 3 scales as M d ∝ T −2.4 at z ∼ 0 for temperatures around 20 K; changing the temperature from 20-30 K results in a reduction in mass by a factor 2.6. At z = 0.5 this dependence is steeper since the peak of the dust emission is shifted to longer wavelengths so the observed frame is even further from the Rayleigh-Jeans regime. Changing from β = 1.5 to 2.0 reduces the temperatures by ∼ 3K and increases the dust masses by ∼ 30 − 50 percent.
This isothermal dust mass estimate can be biased low as it is now well established that dust exists at a range of temperatures in galaxies. Only dust in close proximity to sources of heating (e.g. star forming regions) will be warm enough to emit at λ 100µm but this small fraction of dust (by mass) can strongly influence the temperature of the isothermal fits. The bulk of the ISM (and therefore the dust) resides in the diffuse phase which is heated by the interstellar radiation field to a cooler temperature typically in the range of 15-20 K (Helou et al. 1986; DE01 and references within; Popescu et al. 2002;VDE05;Draine et al. 2007;Willmer et al. 2009;Bendo et al. 2010;Boselli et al. 2010;Kramer et al. 2010;Bernard et al. 2010;Planck Collaboration 2011b). For more accurate dust mass estimates we require the mass-weighted temperature of the dust emitting at 250µm which requires fitting a model with multiple (at least two) temperature components. This is not to say that the FIR fluxes for most of the H-ATLAS galaxies are not fitted adequately by the single temperature model; an isothermal model and a more realistic multi-temperature model are often degenerate in their ability to describe the SED shape with a limited number of data points. To illustrate this, we show in Fig 3 an example of isothermal and 2-component SED fits to a H-ATLAS source with a well sampled SED. Although the 2-component fit is formally better, there is nothing to choose between them as descriptions of the fluxes of the H-ATLAS source between 60-500µm . DE01 studied this issue for a sample of SLUGS galaxies with 450µm detections, and concluded that the best overall description of that sample was a two-temperature model with β = 2 and a cold component temperature of ∼ 20K.
To deal with the cold dust component, we now introduce a more sophisticated SED model which includes dust in several physically motivated components, following the prescription of Charlot & Fall (2000). The results of this fitting are presented and described in detail in Smith et al. in prep, and outlined here in brief. This simple, but empirically motivated, SED model fits broadband photometry from the UV-sub-mm to estimate a wide variety of parameters (da Cunha et al. 2008 -hereafter DCE08;da Cunha et al. 2010a). The method uses libraries of optical and infrared models (25,000 optical and 50,000 infrared) and fits those optical-IR combinations which satisfy an energy balance criteria to the data. The dust mass in this model is computed from the sum of the masses in various temperature components contributing to the SED, including cool dust in the diffuse ISM, warm dust in birth clouds, hot dust (transiently heated small grains emitting in the mid-IR) and PAHs. In the fits to H-ATLAS sources (and SINGS galaxies; DCE08) around 90 percent of the dust mass is in the cold diffuse ISM component and this is also the best constrained component due to the better sampling of the FIR and sub-mm part of the SED with Herschel. Many other studies also find that the cold dust component dominates the overall mass, and so it is the most important one to constrain when measuring the dust mass function (e.g. DE01; VDE05; Draine et al. 2007;Willmer 2009;Liu et al. 2010). The priors used by DCE08 for the temperatures of the grains in equilibrium are 30-60 K for the warm component and 15-25 K for the cold component. These values agree well with temperatures measured for local galaxies (Braine et al. 1997;Alton et al. 1998;Hippelein et al. 2003;Popescu et al. 2002;Meijerink et al. 2005, DE01, V05, Stevens et al. 2005, Stickel et al. 2007Draine et al. 2007;Willmer et al. 2009;Planck Collaboration 2011b) and also with temperatures measured from stacking optically selected galaxies with the same stellar mass and redshift range as our sample into Herschel-ATLAS maps (Bourne et al. in prep). The value of κ used in the DCE08 model is (by design) comparable with that used in the isothermal fits here.
The prior space of the parameters is sampled by fitting to several million optical-FIR model combinations and returns a probability density function (PDF) for the dust mass and other parameters (e.g. dust temperature, stellar mass, dust luminosity, optical depth and star formation rate) from which the median and 68 percent confidence percentiles are taken as the estimate of the quantity and its error.
This model was fitted to the 60 percent of the galaxies in our sample for which useful optical and NIR data were available from GAMA. We fitted only to galaxies which have matched aperture photometry in r-defined apertures as this best represents the total flux of the galaxy in each band as described in Hill et al. (2011);Driver et al. (2011). The distribution of sources with and without these SED fits as a function of redshift is shown in Figure 4. Those without fits dominate only in the highest redshift bin, from z = 0.4 − 0.5.
The errors on the dust mass range from ±0.05 − 0.27 dex and this error budget includes all uncertainties in the fitting from flux errors to changes in temperature and contribution of the various dust components. Some typical SED fits and PDFs for the dust mass and cold temperature parameters are shown in Figure 5. The dust mass is generally a well constrained parameter of these model fits; the PDF is narrower when more IR wavelengths are available and so the cold temperature is then better constrained.
A comparison of the isothermal dust masses ( Miso ) and the full SED based masses ( M sed ) is shown in Figure 6a and there is generally poor agreement between the two, with the scatter of a factor 2-3 related to the difference between the temperature of the isothermal fit and that of the DCE08 model fit. This sensitivity is because at 250µm we are near to the peak of the black body function for the cold temperatures appropriate to the bulk of the dust mass (15-20 K). At longer sub-mm wavelengths (such as 850µm ), this temperature sensitivity is less severe, but the choice of dust temperature used when estimating masses at rest wavelengths close to those of Herschel is clearly important.
For sources which have insufficient UV-sub-mm data to use the DCE08 model, we need to extrapolate dust masses by comparing L250 with M sed for those sources which do have fits. The relationship is linear, with some scatter introduced by the range in dust temperature for the cold ISM component (Figure 6b):
log M sed = log L250 − 16.47
(4) Figure 5. SED fits and probability distribution functions for dust mass and diffuse ISM dust temperature for a range of H-ATLAS sources. The black curve on the SED plot is the total attenuated starlight and re-radiated dust emission. Blue curve is the unattenuated starlight. Green is the attenuated starlight and red is the dust emission. The red squares show the observed photometry and errors or upper limits. The limit to the dust mass accuracy is our ability to determine the cold temperature, which is better constrained when there are more FIR data points available. The best constraints on the dust mass are ∼ 0.05 dex and the worst are ∼ 0.27 dex.
Equation 4 is used to convert L250 to dust mass in cases where the full SED could not be fitted (747 sources out of 1867). The relationship between M sed and the cold temperature of the diffuse ISM (which dominates the dust mass in these galaxies) is similar to that in Eqn. 3, since the DCE08 model fits the sum of grey-bodies at different temperatures to the photometry. The colder the temperature fitted, the higher the dust mass will be for a given L250 . This is clearly demonstrated in Fig. 6 (bottom). In using this relationship for sources without SED fits we are making the assumption that they will also fall on this relationship and that there is no systematic trend in those sources without fits (e.g. if only the highest redshift or most/least luminous sources did not have fits). Since the galaxies without fits span a range of redshift and luminosity and as we also find no correlation of temperature with L250 for our sample (see Section 4.2 and Figure 14), there should be no bias introduced in using the relationship in Figure 6b to estimate masses for those sources without fits.
The scatter in this figure is influenced by our choice of prior for the temperature of the diffuse dust component (15-25 K). Al-lowing a wider prior will broaden the scatter if a significant number of sources are best fitted by hotter or colder temperatures. This issue is explored further by Smith et al. in prep who conclude that for a sample of galaxies with well constrained cold temperatures this prior encompasses ¿80 percent of the population. Further study of the temperatures of the populations requires a larger sample with good 5 band FIR/sub-mm photometry, which will be possible with the next Phase of H-ATLAS data comprising 10 times the area of SDP. The difficulty in using a wider prior is that when we lack full coverage of the FIR/sub-mm SED (as is the case where we have only limits at PACS wavelengths and 500µm ) there is only a weak constraint on the cold temperature (σTc ∼ 2.5 − 3 K). This can place a galaxy with a real temperature of 15 K down at 12 K, and produces quite a bias in the fitted dust temperature (since at 12 K the mass is very sensitive to temperature). The model will fit arbitrarily high masses of very cold dust since this contributes very little to the overall energy balance. Our choice to restrict the temperature prior to the parameter space which is preferred by observations of nearby galaxies, and by those galaxies well sampled in Figure 6. Top: Comparison of dust masses from the DCE08 model ( M sed ) versus the dust mass obtained by using the isothermal grey body fit ( M iso ) using Eq. 3. Points are colour-coded by the isothermal temperature. The one-to-one line is shown in black. There is a large difference between the two mass estimates which is a strong function of fitted isothermal temperature. Bottom: Comparison of M sed to L 250 showing a reasonably tight and linear correlation. The best fit relationship (Eq. 4) is over-plotted. The scatter in this relationship is driven by the diffuse ISM dust temperature, which is used to colour-code the points.
H-ATLAS, potentially means we underestimate the masses of some cooler sources, but we would prefer to be conservative at this point.
THE DUST MASS FUNCTION
Estimators
To calculate the dust mass function we use the method of Page & Carrera (2000; hereafter PC00) who describe a method to estimate binned luminosity functions that is less biased than the 1/Vmax method (Schmidt 1968). To begin with, we produce measurements of the 250µm luminosity function since this is more directly related to the flux measurements from Herschel and enables us to discuss the method without the added complication of translating 250µm luminosity into dust mass. The PC00 estimator is given by :
φ = N i=1 Cs Cz Cr Lmax L min z max(L) z min dV dz dz dL(5)
where Cs, Cz Cr are the completeness corrections for each object as described in Section 2.1 and the sum is over all galaxies in a given slice of redshift and luminosity bin. Lmax and Lmin are the maximum and minimum luminosities of the bin. zmin is the mini-mum redshift of the slice and z max(L) is the maximum redshift to which an object with luminosity, L, can be observed given the flux limit and K-correction, or the redshift slice maximum, whichever is the smaller. The PC00 method has the advantage of properly calculating the available volume for each L − z bin and, in particular, it does not overestimate the volume for objects near to the flux limit. This prevents the artificial turn-down produced by 1/Vmax in the first luminosity bin of each redshift slice. We compare to the 1/Vmax estimator in Figure 7 and confirm that the 1/Vmax estimate of the 250µm luminosity function suffers from the bias noted by PC00 due to slicing in redshift bins.
In the PC00 formalism described above, the accessible volume is not calculated individually for each source (as for 1/Vmax) but is instead calculated for each bin in the L − z plane using a global K-correction. However, we know that each object in our sample has a different K-correction because they have different grey body SED fits. We therefore modified the estimator such that the accessible volume for a given L − z bin is calculated for each galaxy in that bin in turn using its grey body SED fit to generate its limiting redshift zmax,i = z(Li, Smin, T d ) across the bin. These individual contributions are then summed within the bin such that:
φ = N i=1 Cs Cz Cr Lmax L min z max,i z min dV dz dz dL (6)
Note that this is not the same as reverting to the 1/Vmax estimator as we are still calculating the volume available for each L−z bin, however we are now being more precise about the shape of the limiting curve for each source based on its individual SED. This is clear from the difference in the LF calculated this way, as shown in Figure 7(b) compared to the PC00 and 1/Vmax methods shown in Figure 7(a). This change affected the highest redshift bins most as expected.
In this case, the error on the space density is given by
σ φ = N i=1 (φi) 2(7)
where φi is the individual φ contribution of a galaxy to a particular redshift and luminosity bin, and the sum is over all galaxies in that bin. The error bars in Figure 7 show these errors.
This 250µm luminosity function differs slightly from that presented in Dye et al. (2010) in that the ID sample has since been updated to include extra redshifts (1867 compared to 1688) and also to remove stars, for which there were 130 contaminating the previous sample 2 . While Dye et al. (2010) did attempt to correct for incompleteness in the optical IDs of the sub-mm sample, we are now able to extend this to correct for incompleteness as a function of redshift, r-mag and sub-mm flux which was not previously possible. The results are, however, comparable in that strong evolution in the 250µm LF is evident out to z ∼ 0.4. There is then seemingly a halt, with little evolution between z = 0.4 and z = 0.5. This is still consistent with Dye et al. (2010) within the error bars of both estimators.
We suspect that this behaviour in the highest redshift bins is a result of a bias in the ANNz photo-z we are using. Figure 8a shows a comparison between spectroscopic and photometric redshifts of and PC00 method (solid circles and lines) in five redshift slices of ∆z = 0.1 out to z = 0.5. Colours denote the redshifts as: black (0 < z < 0.1), red (0.1 < z < 0.2), green (0.2 < z < 0.3), blue (0.3 < z < 0.4) and cyan (0.4 < z < 0.5). The bias in 1/Vmax in the lowest luminosity bin in each redshift slice is apparent from the turn-down in this bin. Bottom: 250µm luminosity function calculated using the modified PC00 estimator which includes an individual K-correction for each object in an L − z bin. Using individual K-corrections has a more significant effect in the highest redshift slices as expected. There is a bias in the photo-z at redshifts greater than ∼ 0.3 where photo-z tend to underestimate the true redshift. A 1-to-1 correlation is shown by the solid line. Bottom: The fraction of photo-z used in the luminosity function as a function of redshift. Photo-z start to dominate the LF at the same redshift where the photo-z bias begins.
H-ATLAS sources in the SDP region. There is a bias above z ∼ 0.3 − 0.35 where the photometric redshifts tend to underestimate the true redshift (see Fleuren et al. in prep). This issue is further exacerbated by the fact that this is also the redshift at which the LF becomes dominated by photo-z (Fig. 8b).
There is another potential bias in the highest-z slice due to the optical flux limit approaching the main body of galaxies in the sample. While we correct for the incompleteness in space density due to the r-band limit, we are not able to deal with any accompanying bias which might allow only those galaxies with lower dustto-stellar mass ratios into the sample at the highest redshifts (see Fig 19 and Section 6 for more discussion). Greater depth in optical/IR ancillary data will be required to test the continuing evolution of the luminosity and dust mass functions beyond z = 0.5 and this will soon be available with VISTA-VIKING and other deep optical imaging for the H-ATLAS regions from VST-KIDS, INT and CFHT.
Having demonstrated that our modified version of the PC00 estimator produces sensible results on the 250µm luminosity function, we now turn to the estimate of the dust mass function (DMF). We again use Eqn 6 however we now sum all galaxies in a bin of the M d − z plane. We use the ratio of M d to L250 to estimate the Lmax and Lmin for each galaxy, which is required to compute the individual K-correction. The results for both single temperature masses and SED based masses are shown in Figure 9.
Both estimates of the dust mass function show a similar evolutionary trend as the 250µm LF, with the same apparent slow down at higher redshift which we believe may be related to issues with the photo-z. The evolution is present whichever estimate of the dust mass is used, however, we will continue the discussion using the DMF from the DCE08 SED based dust masses (Fig. 9a) as we believe that this is the best possible estimate at this time.
The dust mass function also shows a down-turn in some redshift slices at the low mass end. We do not believe that this represents a true dearth of low mass sources at higher redshifts but rather reflects the more complex selection function in dust mass compared to L250 . While there is a strong linear relationship between our dust mass and L250 (Figure 6b) there is still scatter on this relationship due to the variation in the temperature of the cold dust in the ISM. At fixed L250 warm galaxies will have smaller dust masses than cooler ones, which leads to a sort of 'Eddington' bias in the dust masses. At the limiting L250 for a given redshift bin we are not as complete as we think for low dust masses, since we can only detect galaxies with low dust masses if their dust is warmer than average. This in turn leads to the apparent drop in space density. In the two highest redshift bins, the fraction of sources without SED fits increases and so the dust masses are then directly proportional to the 250µm luminosity. To improve on this, we would need to use a bi-variate dust mass/ L250 approach for which the current data are insufficient, however this analysis will be possible with the complete H-ATLAS data-set.
Dust temperatures and evolution of the LF
An important ingredient in our estimate of the dust mass is the dust temperature. In order to interpret the increase in sub-mm luminosity as an increase in the dust content of galaxies, we have to be wary of potential biases in our measurements of the dust temperature. The dust temperature is most accurately constrained when PACS data, which span the peak of the dust SED, are available in conjunction with the longer sub-mm data from SPIRE, which also constrains the cold temperature. For the SDP field, the PACS data is shallow and this results in PACS detections for only 262 galaxies. The fraction of PACS detections as a function of redshift are 41, 21, 8, 7, 4 percent respectively from the lowest to highest redshift bin. A comparison of the temperatures from fits with PACS detections and without in the lowest redshift bin (where PACS samples a representative fraction of the population) is shown in Figure 10. Fits without PACS detections are included only if the 350µm flux was greater than 3σ in addition to the > 5σ 250µm flux. The left panel shows the cold dust temperature from the DCE08 fits and it can be seen that PACS sources have a range of cold ISM temperatures, however, those without PACS detections tend to have mostly cooler dust in their ISM. Smith et al. in prep show that the DCE08 fitting does tend to underestimate the cold temperature slightly when PACS data are removed from a fit, but this effect is of order 1-2 K and does not fully account for the difference in these distributions. When we consider instead the peak temperature of the SED, that given by the isothermal fit (right panel in Fig. 10), we see a different trend. Now the PACS detections are found only at the higher end of the temperature range while those without PACS detections span a wider range of temperature. There is no bias when the PACS data are removed from the fits (the temperatures vary randomly by Figure 9. Dust mass functions using the modified PC00 estimator calculated in 5 redshift slices of ∆z = 0.1. Top SED based dust masses. Bottom Isothermal dust masses. The relation between dust mass and L 250 has some scatter due to variation in the temperature of the cold ISM dust, which results in down-turns in the lowest mass bins in each redshift slice. The broader error on M d acts to convolve the true DMF with a Gaussian of width approximately 0.2 dex. Schechter functions are plotted in the top panel with the faint-end slope fixed to that which fits best in the z < 0.1 slice. Parameters for the fits are given in Table 3. ∼ ±3 K when the PACS data are removed). The sources with the coldest isothermal temperatures (Tiso < 20K) are not detected by PACS, even in the lowest redshift bin, as they either do not contain enough warm dust, or are not massive enough, to be detected in our shallow PACS data. We also note that where PACS does not provide a > 5σ detection, we use the upper limit in the fitting which provides useful constraints on dust temperature for many more H-ATLAS sources.
The trend for PACS to detect the warmer sub-population of H-ATLAS sources becomes more pronounced at higher redshift, as the galaxies must be intrinsically more and more luminous to be detected by PACS. Figure 11 shows the fitted temperature versus redshift for sources which are detected by PACS (black points) and those not-detected by PACS but which have at least 2 good quality sub-mm points for the fit. If we used only the PACS detected sample, we would infer an evolution in dust temperature in this redshift range -but this is a selection bias due to the sensitivity of the PACS bands to warm dust and the shallow survey limit for PACS.
We can also look at the dust colour temperatures of the H-ATLAS sources in comparison to other samples without the complications of fitting models. In Figure 12 we compare the FIR/submm colours of the 35 H-ATLAS sources which have 60, 100 and 500µm detections at 3σ with the colours of SLUGS galaxies from DE01 and VDE05 to see how these sub-mm selected sources compare to those selected at 60µm from the IRAS BGS (Soifer et al. 1989) To allow a comparison between 500µm fluxes from H-ATLAS and 450µm fluxes from SLUGS, we reduce the SLUGS 450µm values by 37 percent using a standard template suitable for SLUGS sources from DE01 (approximately ∝ ν 3 at these wavelengths). All of these sources are local. Figure 12 shows that the H-ATLAS sources are significantly colder in their colours than the warmest end of the IRAS sample; they overlap rather better with the optically selected SLUGS sample. This is not surprising given our selection at 250µm is more sensitive to the bulk dust mass of a galaxy while that at 60µm from IRAS is more sensitive to warm dust (either large, warm grains in star forming regions or small transiently heated grains). We note that, since only a very small number (35) of H-ATLAS sources are detected by IRAS, these few sources shown in Fig. 12 are also likely to have 'warmer' colours than the overall H-ATLAS sample. This agrees with the findings that PACS is sensitive to only the warmer H-ATLAS sources at higher redshifts and the SED fitting results which show that the H-ATLAS sources contain relatively cooler dust.
If the evolution in the 250µm LF were due simply to an increase in the 'activity' of galaxies of the same dust mass, then we should see a corresponding increase in dust temperature with redshift and no evolution in the DMF. To explain the amount of evolution in the 250µm LF without any increase in dust mass would require an increase in the average dust temperature of order a factor 2 over the period 0 < z < 0.5. We investigated the relationship between both the cold ISM dust temperature from the DCE08 fits and the isothermal grey-body temperature with redshift and found no trend for either ( Figure 13) at z < 0.5, similar to the results from Amblard et al. (2010) and inconsistent with the temperature evolution required to explain the increase in the 250µm luminosity density.
The temperature of nearby (z < 0.1) dusty galaxies has been shown to be correlated with their IR luminosity (the so-called Lir − T d relation; e.g. D00, Dale et al. 2001). A natural explana-tion for this observation might be that a galaxy which has hotter dust (for a given mass) will have a larger IR luminosity than a similar mass galaxy with cooler dust. Recent work which extends to more sensitive surveys and samples selected at longer wavelengths suggests that this does not hold at higher redshifts and that galaxies are in general cooler at a given IR luminosity than previously believed (Coppin et al. 2008;Amblard et al. 2010;Rex et al. 2010, Symeonidis et al. 2009, Seymour et al. 2010. Symeonidis et al. (2011) suggest that this is due to more rapid evolution of "cold" galaxies over the period 0.1 < z < 1 than "warm" ones. Recent studies at other wavelengths (70-160µm from Spitzer and PACS) seem to support this interpretation, finding that cold galaxies are responsible for most of the increase in the IR luminosity density over the range 0 < z < 0.4 (Seymour et al. 2010;Gruppioni et al. 2010). This is in agreement with the evolution seen in H-ATLAS galaxies which are largely comprised of this 'cold' population. Despite our average luminosity increasing with redshift, we see no increase in the average temperature (either isothermal or cold ISM temperature) and indeed we also see no correlation of either temperature with luminosity (either dust luminosity from the DCE08 model or L250 ) for this sample (see Figure 14).
To summarise, while we are subject to uncertainties in our ability to derive the dust masses and the exact scale of any evolution, we are nevertheless confident that:
• The evolution in the 250µm LF out to z = 0.5 cannot be driven by dust temperature increases; there must be some evolution in the mass of dust as well.
• The H-ATLAS sources at z 0.5 are colder than previous samples based on IRAS data and therefore most of the evolution at low redshift is driven by an increase in the luminosity or space density of such cooler galaxies.
• H-ATLAS sources show no trend of increase in dust temperature with either redshift or luminosity at z < 0.5
Comparison to low redshift dust mass functions
We can compare the lowest redshift bin in the DMF (0 < z < 0.1) to previous estimates from the SCUBA Local Universe and Galaxy Survey (D00; VDE05), which used SCUBA to observe samples of galaxies selected either at 60µm from the IRAS Bright Galaxy Sample (Soifer et al. 1989) or in the B-band from the CfA redshift survey (Huchra et al. 1983). The IRAS SLUGS galaxies were mostly luminous star-bursts, and in principle this should have produced an unbiased estimate of the local dust mass function as long as there was no class of galaxy unrepresented in the original IRAS BGS sample. However, it was argued in D00 and VDE05 that this selection at bright 60µm fluxes quite likely missed cold but dusty galaxies, given the small sample size of ∼ 100, thus may have produced a DMF which was biased low. The optically selected SLUGS sample overcame the dust temperature bias and did indeed show that there were very dusty objects which were not represented as a class in the IRAS BGS (similarly confirmed by the ISO Serendipity Survey: Stickel et al. 2007). The directly measured DMF presented by VDE05 suffered from small number statistics, and instead V05 followed the work of Serjeant & Harrison (2005) in extrapolating the IRAS PSCz (Saunders et al. 2000) out to longer wavelengths (850µm ) using the empirical colour-colour relations derived from the combination of IRAS and optically selected SLUGS galaxies. This set of 850µm estimates for all IRAS PSCz sources was then converted to a dust mass assuming a temperature of 20 K (the average cold component temperature found by DE01 and VDE05) and The DMFs are compared in Figure 15, where the black solid line and points are from H-ATLAS at z < 0.1, the blue dot-dash line and filled triangles is the SLUGS IRAS directly-measured DMF (D00) and the red dashed line and open triangles is the DMF based on the extrapolation of the IRAS PSCz by VDE05. In this figure, the H-ATLAS DMF has been corrected for the known under-density of the GAMA-9hr field relative to SDSS as required when comparing to an all-sky measurement such as SLUGS or IRAS PSCz. This correction is a factor of 1.4 (Driver et al. 2011). The SLUGS DMFs have been corrected to the cosmology used in this paper, however these corrections are small at low-z.
It is remarkable that despite the considerable differences in sample size, area and selection wavelength, the SLUGS estimate from VDE05 based on extrapolating the IRAS PSCz gives a very good agreement to our measure. This implies that there is not a significant population of objects in the PSCz sample, or the H-ATLAS sample which is not represented by the combined optical and 60µm selected SLUGS samples (which comprised only 200 objects). Note that had VDE05 used the IRAS data alone to measure dust masses, the results would be extremely different. It is only that SLUGS allowed an empirical statistical translation between IRAS colours and sub-mm flux and from there, assumed a mass-weighted cold temperature for the bulk of the dust that they were able to obtain such a good measure of the DMF.
The original direct measure of the DMF from the bright IRAS SLUGS sample (blue line in Fig 15; D00) dramatically underestimates the dust content in the local Universe (this was also noted by VDE05 once the optically selected sample was included). The dust masses were derived for those objects in an identical way to the VDE05 DMF (and very similar to our current method which has an average measured cold temperature of between 15-19 K), however the IRAS BGS simply missed objects which were dusty but did not have enough warm dust to make it above the 60µm selection. Herschel is able to select sources based on their total dust content, rather than simply the small fraction of dust heated to > 30 K. Herschel samples are therefore likely to contain a far wider range of galaxies in various states of activity, so long as they have enough material in their ISM.
Evolution of the dust mass function
For illustration, we now fit Schechter functions (Schechter 1976) to the dust mass functions in each redshift slice. Only in the first redshift bin do we fit to the faint end slope α, for other redshift bins we keep this parameter fixed at the value which best fits the lowest redshift bin (α = −1.01) to avoid the incompleteness problem mentioned above with the lowest mass bins at high redshift. The best-fitting parameters for the slope α, characteristic mass M * d and normalisation φ * are given in Table 3, where the errors are calculated from the 68 percent confidence interval from the χ 2 contours. For the lowest redshift bin, we include errors which reflect the marginalisation over the un-plotted parameter. The χ 2 contours for M * d and φ * are shown in Figure 16.
There is a strong evolution in the characteristic dust mass M * d with redshift, from M * = 3.8 × 10 7 M⊙ at z < 0.1 to M * d = 3.0 × 10 8 M⊙ at z = 0.4 − 0.5. There is seemingly a decline in φ * over the same redshift range, from 0.0059 − 0.0018 Mpc −3 dex −1 (however this could also be due to sample incompleteness which is not corrected for despite our best attempts). The drop in φ * and increase in M * d are correlated (see Fig 16), and therefore we caution against using the increase in the fitted M * d alone as a measure of the dust mass evolution. If we keep φ * fixed at 0.005 Mpc −3 dex −1 (which is the average of that for the first two redshift bins) then the M * d of the highest redshift bins decreases to 1.8 × 10 8 M⊙ giving an evolution in M * d over the range z = 0 − 0.5 of a factor ∼ 5 rather than ∼ 8 as is the case if the normalisation is allowed to drop.
We calculate the dust mass density in redshift slices using Eqn. 8.
ρ d = Γ(2 + α) M * d φ *(8)
This assumes that we can extrapolate the Schechter function beyond the range over which it has been directly measured. Given the low value of α used (∼ −1) the resulting integral is convergent and so whether we extrapolate or not has negligible effect on the resulting mass density values. The values for ρ d are listed in Table 3 and shown as a function of redshift in Figure 17. There is clearly evolution in the cosmic dust mass density out to z ∼ 0.4 of a factor ∼ 3 which can be described by the relationship ρ d ∝ (1 + z) 4.5 . In the highest redshift bin the dust mass density appears to drop (despite the increase in M * d ), but we again caution that this may be due to incompleteness/photo-z bias in the final redshift bin. This measure of the dust mass density at low redshift can be compared to that made by Driver et al. (2007). They used the optical B-band disk luminosity density from the Millennium Galaxy Catalogue scaled by a fixed dust mass-to-light ratio from Tuffs et al. (2004). Their quoted value for the dust density is ρ d = 3.8 ± 1.2 × 10 5 Mpc −3 at z < 0.1 but this is for a κ value from Draine & Li (2001) which is lower than that used here by 70 percent. Scaling their result to our κ, and correcting the density of our lowest redshift bin by the factor 1.4 from Driver et al. (2011) (to allow for the underdensity of the GAMA-9hr field relative to SDSS at z < 0.1) we have values of ρ d = 2.2 ± 0.7 × 10 5 Mpc −3 (optical based) and ρ d = 1.4 ± 0.2 × 10 5 Mpc −3 (DMF) which are in rather good agreement given the very different ways in which these estimates have been made.
We can also calculate the dust mass density parameter Ω dust from Ω dust = ρ d ρcrit where ρcrit = 1.399 × 10 11 M⊙ Mpc −3 is the critical density for h = 0.71. This gives values of Ω dust = 0.7 − 2 × 10 −6 depending on redshift. Fukugita & Peebles (2004) estimated a theoretical value of Ω dust = 2.5 × 10 −6 today based on the estimated density of cold gas, the metallicity weighted luminosity function of galaxies and a dust to metals ratio of 0.2. This is a little higher than our (density corrected) lowest redshift estimate of 1.0 ± 0.14 × 10 −6 but not worryingly so. Ménard et al. (2010) also estimate a dust The first line of the table is the fit to all three parameters for the lowest redshift bin with associated errors from the 68 percent confidence interval derived from the χ 2 contours. The following entries are where α is fixed to the best-fitting value in the lowest redshift bin. The final entry is the fit to the z ∼ 2.5 DMF from DEE03 corrected to this cosmology and κ 250 . Cos. Var. is the cosmic variance estimated using the calculator from Driver & Robotham (2010). N bin is the number of sources in that redshift bin and z phot /ztot is the fraction of photometric redshifts in that bin. Figure 17. Integrated dust mass density as a function of redshift for H-ATLAS calculated using Eqn 8. The best fitting relationship excluding the higher redshift point is over-plotted, which is ρ d ∝ (1 + z) 4.5 . density in the halos of galaxies through a statistical measurement of reddening in background quasars when cross-correlated with SDSS galaxies. They estimate a dust density of Ω halo dust = 2.1 × 10 −6 for a mean redshift of z ∼ 0.35 and suggest that this is dominated by 0.5 L * galaxy halos. Comparing this to our measure of the dust within galaxies at the same redshift (Ω gals dust = 2 × 10 −6 ) we see that at this redshift there is about the same amount of dust outside galaxies in their halos as there is within. We note here that dust in the halos of galaxies will be so cold and diffuse that we will not be able to detect it in emission with H-ATLAS and so it is not included in our DMF. The decrease in ρ d at recent times could be due to dust being depleted in star formation, destroyed in galaxies by shocks or also lost from galaxies (and from our detection) to the halos. We will return to this interesting observation in Section 6.
Redshift α M * d φ * ρ d χ 2 ν Cos. Var. N bin z phot /ztot (×10 7 M ⊙ ) (×10 −3 Mpc −3 dex −1 ) (×10 5 M ⊙ Mpc −3 ) 0.0 − 0
We can compare the DMF from H-ATLAS to that at even higher redshifts, as traced by the 850µm selected SMG population. An estimate of the DMF for these sources at a median redshift of z ∼ 2.5 was presented in DEE03, using the 1/Vmax method. In Fig 18 we show this higher-z DMF alongside the H-ATLAS Figure 18. Comparison of the H-ATLAS dust mass function in five redshift slices (as in Fig 9) and the high redshift, z ∼ 2.5, DMF from D03 (magenta dashed line). data, where the z ∼ 2.5 DMF is the magenta solid line with filled triangles. The DEE03 higher-z DMF has been scaled to the same cosmology and value of κ250 as used here. At z ∼ 2.5, observed 850µm corresponds to rest-frame ∼ 250µm and so our lower-z H-ATLAS sample and the one at z ∼ 2.5 are selected in a broadly similar rest-frame band. DEE03 used a dust temperature of 25 K to estimate the dust mass, which allowed for some evolution over the low-z SLUGS value of 20 K. The z ∼ 2.5 sources from DEE03 are all ULIRGS and these higher luminosity sources do show enhanced dust temperatures in the local Universe (Clements, Dunne & Eales 2010;da Cunha et al. 2010b). It is also consistent with the cold, extended dust and gas component (T = 25 − 30 K) of the highly lensed SMG at z = 2.3 (Swinbank et al. 2010;Danielson et al. 2011) and other lensed sources discovered by Herschel (Negrello et al. 2010). If we were to recompute the z ∼ 2.5 DMF using a temperature of 20 K instead, this would shift the points along the dust mass axis by a factor ∼ 1.7.
For either temperature assumption, the z ∼ 2.5 DMF is broadly consistent with the H-ATLAS DMF in the two highest redshift bins (z = 0.3 − 0.5). The fits to the high-z DMF are shown in Table 3 and the dust density at z ∼ 2.5 is also consistent with that in the z = 0.3 − 0.5 range from H-ATLAS. If true, this implies that the rapid evolution in dust mass may be confined to the most recent 4-6 billion years of cosmic history. Notwithstanding the earlier statement that this trend needs to be confirmed with a larger sample, dust masses are unlikely to continue rising at this pace because the dust masses at very high redshifts (Michałowski et al. 2010;Pipino et al. 2010) are not very different to those we see here.
This implies that the evolution in the 250µm LF is due at least in part to a larger interstellar dust content in galaxies in the past as compared to today, at least out to z ∼ 0.4 (corresponding to a look-back time of 4 Gyr). However, an increase in star-formation rate is also an important factor as if the dust mass increased at a constant SFR we would expect to see a decline in dust temperature with redshift. Our observations thus point to an increase in both dust mass and star formation activity. If the evolution in the DMF is interpreted as pure luminosity (or mass) evolution (as opposed to number density evolution), then this corresponds to a factor 4-5 increase in dust mass at the high mass end over the past 4 Gyr. Since dust is strongly correlated to the rest of the mass in the interstellar medium (ISM) (particularly the molecular component), this also implies a similar increase in the gas masses over this period. In contrast, we know that the stellar masses of galaxies do not increase with look-back time, showing very little evolution in the mass range we are dealing with (predominantly L * or higher) (Pozzetti et al. 2007;Wang & Jing 2010). The evolution of the DMF is therefore telling us something quite profound about the evolution of the dust content of galaxies, and by inference, the gas fractions of galaxies over this period.
THE DUST CONTENT OF H-ATLAS GALAXIES
There are two ways in which we can quantify the dust content: the amount of light absorbed by dust (or opacity), and the dustto-stellar mass ratio. Both of these are derived from the DCE08 SED model fits for galaxies which were bright enough (r 20.5) that aperture matched photometry was extracted by GAMA (Hill et al. 2011). Due to this being shallower than the depth to which we can ID the H-ATLAS sources we have to take care not to introduce selection biases when making these comparisons. Figure 19 shows r-mag as a function of redshift for the H-ATLAS sources and again highlights that H-ATLAS does not detect low stellar mass (or low absolute Mr) sources. The panels in Fig. 19 have colour coded points for sources where SED fits were made, and the colours represent either the V-band optical depth (top) or the dust-to-stellar mass ratio (bottom). At z ∼ 0.35 the optical sample which has SED fits becomes incomplete, with only the brighter fraction of the galaxies having SED fits at a given redshift. This can lead to a lowering of the average optical depth, or dust-to-stellar mass ratio in bins at z > 0.35, since the brighter galaxies (higher stellar masses) tend to have lower values of optical depth or dust-to-stellar mass. Thus in the following discussion we limit our model comparisons to the data with z < 0.35. We hope to extend the SED fitting to the fainter sources in future work.
First we plot the amount of optical light obscured by dust: the V-band opacity. This is derived from the DCE08 SED model fits, and is calculated both in the birth clouds where stars are born (τV from DCE08) and also in the diffuse ISM (µτV from DCE08). ure 20 shows the evolution of both forms of V-band optical depth from the model fits, indicating that galaxies are becoming more obscured back to z ∼ 0.4. Choi et al. (2006), Villar et al. (2008) and Garn et al. (2010) also find a higher dust attenuation in high redshift star forming galaxies. This is sometimes attributed to an increase of SFR with look-back time (Garn et al. 2010) and an attendant increase in dust content rather than to a change in dust properties. It is also possible that the apparent increase of optical depth with increasing redshift is related to the correlation between IR luminosity and dust attenuation (Choi et al. 2006), whereby more IR luminous galaxies tend to be more obscured. The average IR luminosity of our sample increases strongly with redshift (due both to the flux limit of the survey and the strong evolution of the LF) and it is currently not possible for us to disentangle the effects of redshift from those of luminosity since we do not have a large enough sample to make cuts in redshift at fixed luminosity. Regardless of which is the driver, the observational statement remains that a sub-mm selected sample will contain more highly attenuated galaxies at higher redshifts. This is in contrast to some UV selected samples which show either no trend with redshift or a decline of attenuation at higher-z, due to their selection effects (Burgarella et al. 2007;Xu et al., 2007;Buat et al. 2009). This just highlights the obvious -that FIR and UV selected samples are composed of quite different objects.
Our relationships with redshift are as follows:
birth clouds : τV = 3.43z + 1.56 diffuse ISM : µτV = 1.50z + 0.36 which implies that the attenuation from the birth clouds is rising faster with increasing redshift than that in the diffuse ISM. At higher redshifts we are therefore finding that the birth clouds are producing a larger fraction of the attenuation in the galaxy than at low redshift. We find this trend interesting but further work is required to explain and confirm it, firstly ensuring in a larger sample that it is not again related to the luminosity (more luminous sources also have higher relative attenuation from the birth clouds). Including Balmer line measurements in the DCE08 fits will also better constrain the optical depth in the birth clouds. Secondly, we can look at dust and stellar mass together us- Figure 19. r-mag versus redshift for the H-ATLAS sources. Black open circles represent H-ATLAS sources which are too faint for an SED fit using the DCE08 model at the current time, or which were not in the region covered by GAMA photometric catalogues. Coloured points denote the values of either V-band optical depth (top) or dust-to-stellar mass ratio (bottom) from the DCE08 fits. The limit of reasonable completeness in the optical for the SED fits is z ∼ 0.35. Beyond this redshift, averaged values of optical depth or dust-to-stellar mass ratio will be biased low because only the brightest optical galaxies in that redshift bin will have SED fits (and these tend to have less obscuration).
ing the stellar masses from the DCE08 SED fits. Figure 21 shows the variation of dust and stellar mass with redshift, where the dust mass has been scaled up by a factor 178 in order to roughly make M d and M * equivalent at the lower boundary at low-z. Magenta points show stellar mass, open black squares are the scaled dust mass. The stellar mass remains fairly constant with redshift, while there is a distinct lack of high dust mass objects in the local Universe (as is shown also by the DMF). The dust-to-stellar mass ratio as a function of redshift is shown in Figure 23 and discussed in more detail in the next section.
MODELLING THE EVOLUTION OF DUST
In this Section we will attempt to explain the evolution we see in the dust content of H-ATLAS sources and in the DMF. We do this using a chemical and dust evolution model which traces the yield of heavy elements and dust in a galaxy as its gas is converted into stars. A full treatment of the evolution of galaxies will be considered in Gomez et al. in prep. Here we will consider the elementary model of Edmunds (2001; see also Edmunds & Eales 1998) in which one assumes that the recycling of gas and dust in the interstellar medium is instantaneous. Details of the model are given in Figure 21. Stellar mass (magenta) and dust mass scaled by 178 (black open squares) versus redshift. The dust mass is scaled to make the dust and stellar lower limits approximately coincide at low-z. This illustrates the different trends of dust and stellar mass with redshift, with the dust mass evolving more rapidly than the stellar mass (as is also evident from the DMF). At lower redshifts there are many galaxies with higher stellar masses than the scaled dust mass, while at high redshifts both stellar and dust masses are comparable with the same scaling.
Appendix A, but in brief, a galaxy is considered to be a closed box with no loss or addition of gas during its evolution. The evolution of the galaxy is measured in terms of f , its gas fraction, which represents the fraction of the baryonic mass in the form of gas. Gas is converted into stars using a star formation prescription ψ(t) = kg(t) 1.5 , where g is the gas mass and k is the star formation efficiency (inversely proportional to the star formation time-scale). We define an effective yield p = p ′ /α ∼ 0.01 where α ∼ 0.7 is the mass fraction of the ISM locked up in stars (Eq.10) and p ′ is the yield returned from stars for a given initial mass function (IMF). We can interpret p as being the true mass fraction of heavy elements returned per stellar generation, since some fraction of the generated heavy elements is locked up in low mass stars and remnants. In the first instance, we use the Scalo form of the IMF (Scalo 1986) for Milky Way evolution (e.g. Calura et al. 2008). The metal mass fraction of a galaxy is tracked through p and therefore follows metals incorporated into long lived stars and remnants or cycled through the ISM where they are available to be made into dust. The parameters which determine how many of the available metals are in the form of dust relate to the sources of dust in a galaxy and we consider three of these:
(i) Massive stars and SNe: χ1 is the efficiency of dust condensation from new heavy elements made in massive star winds or supernovae.
(ii) Low-intermediate mass stars (LIMS): χ2 is the efficiency of dust condensation from the heavy elements made in the stellar winds of stars during their RG/AGB phases.
(iii) Mantle growth in the ISM: We can also assume that grains accrete at a rate proportional to the available metals and dust cores in dense interstellar clouds (Edmunds 2001). ǫ is the fraction of the ISM dense enough for mantle growth, ηc is the efficiency of interstellar depletion in the dense cloud (i.e. if all the metals in the dense clouds are accreted onto dust grains then ηc = 1). (2003) used observations of dust in lowintermediate mass stars to show that χ2 ∼ 0.16 yet theoretical models following grain growth in stellar atmospheres (e.g. Zhukovska et al. 2008) suggest higher values of χ2 ∼ 0.5. We adopt the higher value here, but note that there is some considerable uncertainty on χ2. For core-collapse supernovae (using theoretical models of dust formation e.g. Todini & Ferrara et al. 2001) Morgan & Edmunds suggest that χ1 ∼ 0.2; this agrees with the highest range of dust masses published for Galactic supernova remnants Gomez et al. 2009). If core-collapse SNe are not significant producers of dust (e.g. Barlow et al. 2010) or if most of their dust is then destroyed in the remnant (Bianchi & Schneider 2007) then this fraction decreases to χ1 0.1, making it difficult to explain the dust masses we see in our Galaxy or in high-redshift submillimetre bright galaxies with stellar sources of dust (e.g. Dwek et al. 2007;Michałowski et al. 2010).
Morgan & Edmunds
For mantles we arbitrarily set ǫ = 0.3 and from interstellar depletion levels in our Galaxy and following Edmunds (2001), we set ηc ∼ 0.7 (that is, we assume that if the clouds are dense, then it is likely that the dust grains accrete mantles). In this scenario, the dust is formed during the later stages of stellar evolution and uses up the available metals in dense clouds. The addition of accretion of metals onto grain cores with the parameters described here will double the peak dust mass reached by a galaxy. Assuming no destruction of grains, a closed box model and mantle growth gives the highest dust mass attainable for galaxies.
Dust destruction can be added to this elementary model by assuming some fraction δ of interstellar grains are removed from the ISM as a mass ds is forming stars. We use two destruction scenarios: one with a constant destruction rate δ = 0.3 (Edmunds 2001) and the second where δ is proportional to the Type-II SNe rate (which gives a similar result to Dwek's approximation for MW IMF; Dwek et al. 2011). We also allow a mantle growth proportional to SFR since one would expect that the efficiency will depend on the molecular fraction of the ISM (which in turn is related to the SFR; Papadopoulos & Pelupessy 2010).
Finally, we relax the closed-box assumption and include outflows in the model (Appendix A) since galactic-scale outflows are thought to be ubiquitous in galaxies (Menard et al. 2010 made a remarkable detection of dust reddening in the halos of galaxies which implies at least as much dust is residing in the halos as in the disks).
Here we test outflows in which enriched gas is lost at a rate proportional to one and four times the SFR (more powerful outflows are unlikely, since in the latter case, the galaxy would only retain approximately 20 per cent of its initial gas mass).
Evolution of Dust to Stellar Mass
The dust-to-stellar mass ratio of the models discussed here is shown in Figure 22 over the life-time of the galaxy as measured by the gas fraction, f . The shaded region shows the range of values of M d /M * estimated for the H-ATLAS galaxies, which have a peak value of 7 × 10 −3 at z = 0.31 and then decreases as the galaxy evolves in time (to lower gas fractions) to 2 × 10 −3 . This global trend is reproduced by the closed box model where dust is contributed by both massive stars and LIMS, or via mantle growth, however the models struggle to produce values of M d /M * as high as observed. We also plot in Figure 22, the variation of M d /M * if low-intermediate mass star-dust is the only stellar contributor to the dust budget (χ1 = 0, χ2 = 0.5). It is clear that the LIMS dust source cannot reproduce the values of dust/stellar mass seen in the H-ATLAS sources alone. Either significantly more dust is contributed to the ISM via massive stars/SNe than currently inferred, or a significant contribution from accretion of mantles in the ISM is required (indeed we would need significantly more dust accretion in the ISM than dust produced by LIMS). The simple model also suggests that the H-ATLAS galaxies must be gas rich (f > 0.4) in order to have dust-to-stellar mass ratios this high. (Typical gas fractions for spiral galaxies today are f ∼ 0.1 − 0.2.)
We can also consider the evolution of dust-to-stellar mass as a function of time (Eq. 21). This is shown in Fig 23a using dust production and yield parameters appropriate for spiral galaxies like the Milky Way (p = 0.01, α = 0.7, χ1 = 0.1, χ2 = 0.5, ǫηc = 0.24, k = 0.25 Gyr −1 ). We compare the model for two formation times of z = 0.6 and z = 1, where formation time in this model can simply mean the time of the last major star formation event. In this scenario, we would expect any previous star formation to have already pre-enriched the ISM with some metallicity Zi, therefore increasing the available metals for grain growth in the ISM.
From Fig 23a, we see that the MW model does not match the variation of dust/stellar mass from H-ATLAS observations even if we increase the mantle growth or the amount of dust formed by stars, since the increase in dust-to-stellar content with gas fraction (as we look back to larger redshifts and earlier times in the evolution of the galaxy) is simply not rapid enough. Fig 23b shows the same two formation times but now we have tuned the parameters to match the data for a formation at z = 0.6. In order to do this we have to increase the SF efficiency parameter (k = 1.5 Gyr −1 ) to produce a steeper relationship as observed. An increase in k compared to the MW model is hardly surprising, since these higher values are typical of star-forming spirals with initial SFRs 3 of ψ ∼ 50 M⊙ yr −1 which is in agreement with the observations of H-ATLAS sources at higher redshifts. However, increasing k then dramatically reduces the actual dust content at any epoch due to removal of the ISM through the increase in star formation efficiency. To explain the high M d /M * values for the H-ATLAS sample, we would then need to increase the dust condensation efficiencies (i.e. the amount of metals which end up in dust) to a minimum of 60 percent and the effective yield p of heavy elements from stars would need increase by at least a factor of two. This is much higher than observed condensation efficiencies for LIMS or massive stars/SNe although the difference could come from mantle growth. An increase in the effective yield can only be achieved through the IMF. The stellar masses of H-ATLAS galaxies are based on the Chabrier IMF (Chabrier 2003), which has α ∼ 0.6 (compared to α ∼ 0.7 for Scalo). However, to significantly increase the yield from the stellar populations, we would require a top-heavy IMF (e.g. Harayama, Eisenhauer & Martins 2008). In comparison to the MW-Scalo IMF, the effective yield p can increase by a factor of 4 and more material is returned to the ISM (α < 0.5). A model with these 'top-heavy' parameters is shown in Figure 23b (solid blue), and reproduces the H-ATLAS observations without the need for extremely efficient mantle growth or higher dust contribution from SNe. A top-heavy IMF also frees up more gas and metals in the ISM throughout the evolution of the galaxy with time, i.e. f ∼ 0.5 at z = 0.4 compared to the f ∼ 0.3 for a Scalo IMF, providing a consistent picture with the observed high dust-to-stellar mass ratios and the expected high gas fraction for H-ATLAS sources.
If we assume an earlier formation time, or time since last star formation phase, the model cannot reproduce the H-ATLAS observations and would require even more extreme values for the dust condensation efficiency and/or yield. This suggests a time for the last major star formation episode for H-ATLAS galaxies to be 3 depending on the initial gas mass of the galaxies Gas Fraction, f Closed, ε=δ=0 Closed,χ 1 =0, χ 2 =0.5 Closed ε=0.3, δ=0.3
Closed ε α sfr Outflow λ/α = 1 Outflow λ/α = 4 Figure 22. Variation of dust-to-stellar mass ratio as a function of gas fraction. The shaded box region is the range of values observed for the H-ATLAS galaxies. The models are (i) a closed box with no gas entering/leaving the system with dust from both supernovae χ 1 = 0.1 and LIMS stars χ 2 = 0.5 (thick solid; black); (ii) with dust from LIMS only χ 1 = 0, χ 2 = 0.5 (thin solid; black); (iii) model (i) now including mantle growth (dot-dashed; black); (iv) A model with mantle growth, where the mantle rate is proportional to the SFR (solid; red); (v) and (vi) a model which has outflow with gas lost at a rate proportional to one or four times the SFR (λ/α) (dashed; blue). somewhere in the past 5-6 Gyr (which is consistent with the detailed SED modelling of Rowlands et al. in prep).
In summary, from this simple model, it is difficult to explain the high dust-to-stellar mass ratios in the H-ATLAS data even by assuming we are observing these galaxies at their peak dust mass unless (i) the fraction of metals incorporated into dust is higher (although we would require χ > 70 per cent of all metals to be incorporated into dust) or χ > 50 per cent with pre-enrichment; (ii) The yield is significantly increased via a top heavy IMF. An IMF of the form φ(m) ∝ m −1.7 would increase the yield and hence dust mass by a factor of four, easily accounting for the highest M d /M * ratios. Such IMFs have been postulated to explain observations of high-z sub-mm galaxies, highly star-forming galaxies in the local Universe and galaxies with high molecular gas densities (Baugh et al. 2005;Papadolpoulos 2010;Gunawardhana et al. 2011). (iii) H-ATLAS galaxies are rapidly consuming their gas following a relatively recent major episode of star formation (at z ∼ 0.6).
Evolution of the DMF
We now turn to the evolution of the dust mass itself as evidenced from the DMF (Fig 9) which shows an increase in the dust mass of the most massive sources of a factor 4-5 in a relatively small timescale (0 < z < 0.5, ∆t < 5 Gyr). To show the maximum change in dust mass in galaxies in the model, we plot the ratio (R) of dust mass at time t to that at the present day, assuming a gas fraction of f ∼ 0.1 today (Figure 24). For a closed box model, there is little evidence for the dust mass in a given galaxy changing by more than a factor of 1.5 in the past compared to its present day value.
It is clear that including outflows produces a better fit to the variation of dust mass observed in the DMF, with the maximum change in dust mass approaching the observed change in DMF with R ∼ 4 for the extreme outflow model. However, in this case, the peak M d /M * is at least an order of magnitude below the observed values predicting only 2 × 10 −4 (see Fig 22). In this scenario, we The dust-to-stellar mass ratio as a function of redshift. Stellar and dust masses are derived from the SED fits using the models of DCE08 and are discussed in detail in Section 3 and Smith et al. (2011b). Black points show those sources with spectroscopic redshifts, while red points include photometric redshifts. Each sample is limited in redshift to the point where the optical flux limit is not biasing the selection to low dust-to-mass ratios. The model lines for the dust model (Section. 6.1) corresponding to the Milky Way including mantle growth and destruction are over-plotted with formation redshifts of z = 0.6 (dot-dashed) and z = 1 (dotted). A model including pre-enrichment of Z i ∼ 0.1Z ⊙ with formation timescale at z = 0.6 is also shown (solid; black). Right: Same as left including pre-enrichment, but models are now tuned to match the data for the z = 0.6 formation time. With pre-enrichment, we require χ 1 = 0.1, χ 2 = 0.5, p = 0.02, ǫ = 0.9 and SF efficiency k = 1.5 Gyr −1 to 'fit' the data points (black dot-dashed) or χ 1 = χ 2 = ǫ = 0.5, p = 0.02 (not shown). Also shown is a model with mantle growth varying with SFR and a top-heavy IMF described by α = 0.5, p = 0.03 (solid; blue). Adding outflow or destruction rates which vary with SFR would make the decline in M d /M * more pronounced at lower redshifts (later evolutionary times). Closed ε α SFR Outflow λ/α = 1 Outflow λ/α = 4 Figure 24. Ratio (R) of dust mass at gas fraction f to that at f = 0.1 (today). The models are (i) a closed box with no gas entering/leaving the system with dust from both supernovae χ 1 = 0.1 and LIMS stars χ 2 = 0.5 (thick solid; black); (ii) with dust from LIMS only χ 1 = 0, χ 2 = 0.5 (thin solid; black); (iii) including mantle growth (dot-dashed; black); (iv) A model with mantle growth proportional to the SFR (solid; red). (v) and (vi) A model which has outflow with gas lost at a rate proportional to one or four times the SFR (λ/α) (dotted; blue). It is worth noting that for higher returned fraction from stars to the ISM (i.e. α = 0.5), the ratio decreases for all models (R < 3 for the extreme outflow).
would require χ > 0.8, ǫη > 0.8 and p > 0.03. Such high dust condensation efficiencies from stellar sources are not observed in the MW, and a yield as high as p = 0.03 would again, imply a top heavy IMF. For an outflow model with λ/α = 1.0, the parameters χ > 0.6, ǫη > 0.3 and p > 0.02 would be required to produce the H-ATLAS dust-to-stellar mass ratios, these are more reasonable values yet this outflow rate is not sufficient to account for the increase in dust mass seen in the DMF (reaching a maxi-mum R ∼ 1.5; Fig 24). We believe that outflows must be present at some level (Alton, ) and the observation made earlier that there is as much dust in galaxy halos as there is in galaxies themselves is strong circumstantial evidence for some outflow activity. Given that there are other ways (e.g. radiation pressure on grains; Davies et al. 1998) to remove dust from disks, we can attempt to derive a rough upper limit for the outflow required to produce as much dust in halos at z ∼ 0.35 as found by Ménard et al. (2010). We integrate the dust mass lost from outflows during the evolution of the galaxy and compare this to the dust mass in the galaxy at z = 0.3 − 0.4 for various values of outflow and star formation efficiency k. The results are shown in Table 4. This assumes no dust destruction in either the halo or the disk, and as such is a very simple model. Equality in dust mass inside and outside galaxies can be achieved by z = 0.3 by having moderate outflow < 4 × SFR and 0.25 < k < 1.5Gyr −1 . This is not to say that all galaxies need have similar evolution; it is quite likely that H-ATLAS sources are more active and dusty and as such may contain more dust in their halos than the average SDSS galaxy probed by Ménard. This simple exercise merely gives some idea of what sort of 'average' chemical evolution history is required to reproduce the observation.
We now have a conundrum in that the observed evolution in dust mass requires significant outflow of material, however such outflow leads to the lowest values of dust-to-stellar mass ratio and cannot be reconciled to the observations without extreme alterations to the condensation efficiencies for dust or the stellar yields. Including dust destruction and mantle growth models which vary with the SFR alleviates this somewhat since both decrease the dust mass more significantly at later times. The change in dust mass over the same period compared to the elementary model with constant ǫ and δ is then more pronounced, but not enough to explain the evolution in the DMF.
One solution to this is if the galaxies with the highest dust masses at z ∼ 0.4 − 0.5 are not the progenitors of the H-ATLAS Table 4. t is the age since formation of the galaxy at z = 0.6. Outflow = 1 and 4 is outflow proportional to 1 and 4 times the star formation rate. 'Halo/Disk' is the ratio of the integrated dust mass lost in outflow from t form to t divided by the dust mass in the galaxy at t. sources at z ∼ 0.1. We speculate on a scenario where the low redshift spiral galaxies (z < 0.15) which do fit the MW model in Fig 23a comprise one population and the higher redshift (more dusty) objects are a rapidly evolving star-burst population with much higher star formation efficiencies (higher k), higher dust condensation efficiencies and/or top-heavy IMFs. The fate of the high redshift dusty population is that they rapidly consume their gas (and dust) in star formation and by low redshift they are no longer detected in H-ATLAS as their gas and dust is exhausted (f < 0.05).
Today they would lie in the faint end of the DMF, mostly below the limits to which we can currently probe. They would need to be large stellar mass objects (since their stellar masses are already large at z = 0.5) but have little gas and dust today. They could plausibly be intermediate mass (log M * = 10.5 − 11.5) early type galaxies (ETG) in the local Universe, although they would still be relatively young since they were forming stars actively at z = 0.4−0.6. Such depleted objects could have had much more dust in the past with ratios of > 4 for the closed box scenario and the model with mantle growth proportional to SFR. In fact, the dust content of such galaxies in the past could be even higher since the build up of a hot X-ray ISM in ETG rapidly destroys any remaining dust (e.g. Jones et al. 1994). This is an attractive solution as severe outflows are then no longer required to reproduce the strong dust mass evolution seen in the DMF. Such a scenario predicts a population of early type galaxies with moderate dust content and moderate ages (< 5 − 6 Gyr) as the last remnants of their ISM is depleted and the dust gradually destroyed. H-ATLAS has in fact discovered some promising candidates for this transitional phase which are discussed in detail by Rowlands et al. in prep. Although a closed model does not reproduce the complexity of dust and metal growth within galaxies, we note that this elementary model including mantle growth predicts the highest dust masses for galaxies with the same initial gas mass and SFRs. Inflows and outflows of material simply reduce the dust fraction in the ISM. A full treatment of the build up of metals in galaxies from stars of different initial masses further compounds this since relaxing the instantaneous approximation would produce less dust at earlier times (at larger values of f ). The difficulties we have in producing the observed dust evolution with this elementary treatment are thus only going to be exacerbated once a more complex treatment is adopted and therefore our conclusions about the requirements for higher yields and condensation efficiencies are conservative. To address the issues above, in particular, the importance of the star formation history and the role of the IMF, a more complex model of dust and chemical evolution is required which allows mantle growth, destruction and even the shape of the IMF to depend on the star formation rate of galaxies. This is beyond the scope of this paper and the reader is referred to Gomez et al. (in prep) for a more complete investigation of the origin and evolution of dust in galaxies.
Final caveat
There is one important way in which the observed dust masses could be over-estimated; through the dust mass absorption coefficient κ. This normalises the amount of emission from dust to the mass of material present and is dependent on the optical properties and shapes of the dust grains (for a more thorough review of the literature see Alton et al. 2004). The value of κ used here is based on that measured in the diffuse ISM of the Milky Way (Boulanger et al. 1996;Sodroski et al. 1997;Planck Collaboration 2011a) and also on nearby galaxies by James et al. (2002). This value is some 70 percent higher than that predicted by some models of dust, including the silicate-graphite-PAH model of Li & Draine (2001), but lower than those measured in environments where dust may be aggregated, icy mantles or 'fluffy' (Matthis & Whiffen 1989;Ossenkopf & Henning 1994;Krugel & Siebenmorgen 1994). Latest results from Planck (Planck Collaboration 2011a) do see a variation in the dust emissivity with temperature which is expected if there is grain growth in the ISM. It is thus not inconceivable that κ could be globally higher in galaxies with larger fractions of their ISM in states which lend themselves to the growth of grains, or where larger fractions of grains have a SNe origin, or are undergoing destruction by shocks. For example Ossenkopf & Henning (1994) show that in only 10 5 years of grain evolution in dense environments (10 6 − 10 8 cm −3 ) the dust emissivity can increase by a factor ∼ 5 due to the freeze out of molecular ice mantles and coagulation. The same authors also show that changing the ratio of carbon to silicate dust can change the emissivity by ∼ 40 percent. Such a change in global dust composition could reflect the time dependence of evolution of various dust sources (e.g. SN-II dominating in early time) or metallicity changes favouring O or C-rich AGB phases. The mechanism for changing the fraction of the ISM in the densest phases conducive to mantle growth could be triggered star formation and feedback (e.g. following an interaction). The fraction of gas in dense clumps has been found to increase markedly in parts of GMCs which are affected by feedback from recently formed OB stars (Moore et al. 2007). Draine et al. (2007) find that for local SINGS galaxies there is no need to consider ice-mantles in the modelling of the dust emission, but similar modelling has not been attempted for higher redshift and more sub-mm luminous sources such as the H-ATLAS sources.
A measurement of κ at Herschel wavelengths (but for local normal galaxies) has been attempted by Weibe et al. (2009) and Eales et al. (2010b). Both works, however, suggest a much lower value for κ, which would increase the dust masses estimated here by a factor ∼ 3. Given the already difficult task in modelling the dust masses, we do not believe that κ250 can be significantly lower than the values assumed here. A determination of κ for H-ATLAS galaxies is ideally required (as these are sub-mm selected sources which may preferentially have higher κ). Should an enhanced κ at higher redshifts be the explanation for the large sub-mm luminosities of H-ATLAS galaxies then this has important implications for the interpretation of all high-z SMG and Herschel observations. A change in κ will lead to a change in the opacity of galaxies since the interaction of the grains with optical/UV photons will be altered. A strong test is to look at the effects of different κ on the attenuationinclination relation in the optical as differing values of κ in the sub-mm will (for a fixed observed sub-mm flux) produce different values for the dust opacity in the optical-UV (see Popescu et al. 2011 for further details). For galaxies in the Millennium Survey (Driver et al. 2007) the Li & Draine (2001) values of κ (which are lower than those used here by 70 percent) gave the best consistency with the observed attenuation-inclination relation, however it will be interesting to see the results of similar modelling for H-ATLAS sub-mm selected sources (Andrae et al. in prep). One result of an increasing κ with redshift would be a flattening of the attenuationinclination relation with redshift.
A thorough investigation of all the implications using radiative transfer modelling is required but a change in κ is likely to affect dust masses and the outputs of semi-analytic models which try to predict the SMG populations. If the FIR luminosity of high-z galaxies is not dominated by obscured star formation (i.e. there is a contribution from low opacity diffuse ISM or 'leaky' star forming regions) then a change in κ may also lead to a bias in SFR estimated via FIR luminosities. Very high dust masses and sub-mm fluxes for SMG in the early Universe have proved challenging for dust formation models and semi-analytic models of galaxy formation. In addition to exploring additional sources of dust and IMF variations to explain the SMG populations, it is worth considering of the possibility of dust grain property evolution as well.
CONCLUSIONS
We have estimated the dust mass function for the Science Demonstration Phase data from the Herschel-ATLAS survey, and investigated the evolution of the dust mass in galaxies over the past 5 billion years. We find that:
• There is no evidence for evolution of dust temperature out to z = 0.5 in this 250µm selected sample.
• The dust mass function and dust mass density shows strong evolution out to z = 0.4 − 0.5. In terms of pure mass evolution this corresponds to a factor 4-5 increase in the dust masses of the most massive galaxies over the past 5 billion years
• Similar strong evolution is found in the ratio of dust-to-stellar mass and V-band optical depth -Herschel-selected galaxies were more dusty and more obscured at z = 0.4 compared to today.
• In order to account for the evolution of the dust content we need to radically alter chemical and dust evolution models. We cannot reproduce these trends with Milky Way metal or dust yields or star formation efficiencies.
• H-ATLAS 250µm selected sources are highly efficient at converting metals into dust, either through mantle growth or through a bias in the IMF towards higher mass stars. They must also be observed following an episode of star formation (either recent formation or recent major burst) where the gas has been consumed at a much faster rate than galaxies like the Milky Way today.
• As dust and gas (particularly molecular gas associated with SF) are tightly correlated in galaxies, this increase in dust content is suggestive of galaxies being more gas rich at z = 0.5. According to the simple chemical model, we are possibly witnessing the period of growth toward peak dust mass when gas fractions are ∼ 0.5 or higher. This strong decline in gas and dust content may be an explanation for the decrease in star-formation rate density in recent times as measured in many multi-wavelength surveys.
This study uses only 3 percent of the area of the H-ATLAS data. Future improvements will come from the wider area coverage of the full survey, reducing uncertainties due to cosmic variance and small number statistics. Use of deeper optical/IR data from forthcoming surveys such as VISTA-VIKING, pan-STARRS, DES and VST-KIDS will also allow us to push to earlier times and higher redshifts to find the epoch of maximum dust content in the Universe.
Mtot(outflow) = 1 + (λ/α)g 1 + λ/α . (20)
Dust and Stellar Mass
The dust mass per unit stellar mass for the elementary model for equal χ with no mantles, destruction or outflow, is given by Eq. 21:
M d αs = −χpgln(g) 1 − g(21)
We can rewrite Eq. 21 as a function of time, since SFR ψ(t) is related to the gas mass via is related to the gas mass via ψ(t) = kg(t) 1.5
where k is the star formation efficiency measured in Gyr −1 and the variation of g with time is g = 1.5 αkt + 1.5 2 (23) High values of k will result in a higher SFR and a more rapid build up of the final stellar mass for the same initial gas mass.
For outflow models, the dust mass fraction and the gas mass is reduced as described in Eqs. ?? -19.
Figure 1 .
1Figure 1. Templates for three galaxies showing the range of optical fluxes expected for galaxies which are at the SPIRE flux limit of S 250 = 32 mJy at z = 0.5; the limit of our study. The templates are for M82 (a typical starburst), a Herschel-ATLAS template derived from our survey data by Smith et al. in prep and Arp 220, a highly obscured local ULIRG. The SDSS limit of r = 22.4 is shown as a horizontal dotted line and even a galaxy as obscured as Arp 220 is still visible as an ID to our optical limit at z = 0.5. The yellow shape represent the SDSS-r band filter which was used to compute the optical flux
Figure 2 .
2Left: SDSS r-modelmag as a function of 250µm flux.
The optical libraries have stochastic star formation histories and the stellar outputs are computed using the latest version of the Bruzual & Charlot (2003) population synthesis code (Charlot & Bruzual in prep) libraries and a Chabrier (2003) Galactic-disc Initial Mass Function (IMF).
Figure 3 .
3Top row: Isothermal and 2-component SEDs for an H-ATLAS sources with a well sampled SED. Redshifts and fitted parameters are shown in each panel.For the isothermal fits T and β were free to vary while for the 2-component fits β was fixed to be 2. The parameter Nc/Nw is the ratio of cold/warm mass.
Figure 4 .
4The distribution of sources with DCE08 SED fits as a function of redshift (red), those without fits are shown in blue (dashed).
Figure 7 .
7Top: 250µm luminosity functions calculated via the 1/Vmax method (open triangles /dashed)
photo−z fraction
Figure 8 .
8Top: Comparison of spectroscopic versus photometric redshifts for galaxies in the H-ATLAS SDP.
or in the optical. H-ATLAS fluxes at 60µm are from the IIFSCz catalogue of Wang & Rowan-Robinson (2009), 100µm fluxes are from PACS and 500µm from SPIRE (Rigby et al. 2011).
Figure 10 .
10Left: Cold ISM temperature from the DCE08 fits for the lowest redshift bin which has 41 percent PACS detections. The sources with PACS detections are shown by the red dashed line while those without PACS detections but which do have a 350µm flux above 3σ in addition to the 5σ 250µm point are shown in black. Right: Same but for the isothermal temperature
Figure 11 .
11Left: Cold ISM temperature from the DCE08 fits versus redshift for sources with PACS detections (black filled) and which have 350µm fluxes above 3σ in additions to the 5σ 250µm flux (red open). Right: Same but for the isothermal temperature. Here there is a correlation between T iso and redshift for the PACS detections (r = 0.4).
Figure 12 .
12Colour plots for the 35 H-ATLAS galaxies with detections at 60, 100 and 500µm compared to those for SLUGS sources detected at 450µm from an IRAS 60µm selected sample (IRAS) and an optically selected sample (OS). The SLUGS points have had their 450µm fluxes adjusted downward by 37 percent to make them equivalent to 500µm
Figure 13 .
13Left: The temperature of the cold interstellar dust component as a function of redshift z. Only sources with either a PACS detection or a 350µm flux above 3σ, in addition to the 250µm flux, are plotted. Mean values and 1-σ errors on the mean are shown as black points. The data points in magenta show the full distribution of the temperatures. The large error bar in the top right shows the average 68 percent confidence range on the temperature for an individual fit.Right:The isothermal temperature estimated from a grey body fit versus redshift, same coding as before. The line plotted shows the evolution in temperature required in order to explain the evolution in the 250µm LF without any increase in the dust masses. Neither method for estimating the dust temperature shows any evolution with redshift.
Figure 14 .
14Left: Cold dust temperature and L 250 showing no correlation. The points are colour coded by redshift. Right: Same as (left) but for the isothermal dust temperature a mass opacity coefficient of κ850 = 0.077 m 2 kg −1 . From this set of masses they then produced an estimate of the DMF.
Figure 15 .
15Comparison of the local dust mass functions at z < 0.1 from H-ATLAS (black solid line and points) along with estimates from SLUGS. Blue dot-dash line and solid triangles -directly measured DMF from IRAS SLUGS sample (D00, DE01). Red dashed line and open triangles -extrapolated DMF from IRAS PSCz using sub-mm colours from the optical SLUGS sample (VDE05). The H-ATLAS points have been corrected for the factor 1.4 under-density in the GAMA-9hr field for this redshift range compared to SDSS at large.
Figure 16 .
16χ 2 confidence intervals at 68, 90, 99 percent for M * d and φ * with fixed α for the five redshift bins. This shows the clear evolution of M * d over the interval 0 < z < 0.5.
Figure 20 .
20Upper red points: Mean V-band optical depth in the birth clouds (from the DCE08 SED fits of Smith et al. in prep) as a function of redshift with the best linear fit. Lower black points: V-band optical depth in the ISM (µτv from DCE08).
Figure 23 .
23Left:
Table 1 .
1The percentage completeness of our reliable ID catalogue as a function of redshift, as taken fromSmith et al. (2011). The correction factor used in the luminosity function is denoted by Cz.z
Completeness (%)
Cz
0.0 -0.1
93.2
1.07
0.1 -0.2
83.2
1.20
0.2 -0.3
74.2
1.35
0.3 -0.4
55.6
1.80
0.4 -0.5
53.1
1.88
Table 2 .
2The percentage completeness as a function of r magnitude for the catalogue used to make the identifications to H-ATLAS sources. The correction factor used in the luminosity function is denoted by Cr.r mag
Completeness (%)
Cr
21.6
91.1
1.10
21.7
87.6
1.14
21.8
82.8
1.21
21.9
77.7
1.29
22.0
70.5
1.42
22.1
61.6
1.62
22.2
52.5
1.90
22.3
42.8
2.33
22.4
17.0
5.88
Table 3 .
3The Schechter parameters fitted to the dust mass function
Due to using an earlier version of the LR estimate which combined stars and galaxies together
ACKNOWLEDGMENTSHLG acknowledges useful discussions with Mike Edmunds. The Herschel-ATLAS is a project with Herschel, which is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. The H-ATLAS web-site is http://www.h-atlas.org. GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programs including GALEX MIS, VST KIDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is: http://www.gama-survey.org/. This work received support from the ALMA-CONICYT Fund for the Development of Chilean Astronomy (Project 31090013) and from the Center of Excellence in Astrophysics and Associated Technologies (PBF06)APPENDIX A: CHEMICAL EVOLUTION MODELLINGThis simple chemical evolution model describes the star, gas, metal and dust content of a galaxy making the instantaneous recycling approximation. The mass fraction of metals, Z in this model changes as a mass ds of the ISM is formed into stars assuming no inflows or outflows via the following equation(Edmunds 2001):where g is the gas mass and α (Eq. 10) is the fraction of mass from a generation of star formation which is locked up in long-lived stars or remnants mR as determined by the initial mass function (φ(m)):p is the effective yield of heavy elements from stars p = p ′ /α ∼ 0.01 where α ∼ 0.7 in agreement with Milky Way values for a Scalo IMF.In a closed box model (i.e. no inflow or outflow of material), the total mass of the system (Mtot = gas + stars) is unity so that the fraction of gas in a galaxy (the ratio of gas mass to total baryonic mass) is f = g. In this scenario, the initial conditions are: Z = 0 at g = f = 1 and the gas mass of the galaxy is given by g = 1 − αs. The analytic solution for the metal mass fraction Z is (Eq. 11):An early episode of star-formation prior to the evolution of the closed box would pre-enrich the gas and increase the interstellar metallicity (pre-enrichment is often invoked to explain the metallicities of globular clusters in the Milky Way). We can include preenrichment of the ISM with metals Zi usingwhereHarris 2004). Correspondingly, the dust mass fraction y varies with ds via:where χ is a parameter to describe the fraction of the mass of interstellar metals in dust grains from supernovae remnants or their massive star progenitors (χ1), and/or from the stellar atmospheres of low-intermediate mass stars(LIMS: χ2). The analytic solution is given in Eq. 14 for y = 0 at g = 1 and for α = 0.7 (typical locked up fraction for a Scalo IMF):For the special case where χ1 = χ2 = χ, Eq. 14 reduces to:We can add an additional term to the dust mass from stars by assuming that grains accrete at a rate proportional to the available metals and dust cores in dense interstellar clouds (following Edmunds 2001):where ǫ is the fraction of the ISM dense enough for mantle growth (here we set this arbitrarily to 0.3), ηc is the efficiency of interstellar depletion in the dense cloud (i.e. if all the metals in the dense clouds are accreted onto dust grains then ηc = 1).Dust destruction via supernova shocks can be added to this elementary model by assuming some fraction δ of interstellar grains are removed from the ISM as a mass ds is forming stars (therefore adding a term −δds to Eq. 13). In this work, we test both a constant fraction with δ = 0.3 (appropriate for MW-type galaxies and therefore provides a testcase with a minimum destruction level expected for the H-ATLAS spirals) and a function that varies proportionally to the SFR (since a higher SFR equates to a higher Type II SN rate).OutflowWe include a simple model for outflow of gas, in which gas is added or lost from the system at rates proportional to the star formation rate. For large galaxies this outflow rate is assumed to be less than four times the SFR (λ/α 4; see Eales & Edmunds 1996 for discussion; this corresponds to a galaxy which retains only ∼ 20 per cent of its original mass). We do not consider inflow of unenriched material since this only slightly reduces the dust mass w.r.t. the closed box model and doesn't significantly change the evolution of a galaxy(Edmunds 2001). One can imagine a scenario with inflow of pre-enriched material (e.g. merger), providing new material for star formation, even at later times when the original gas mass of the galaxy has been consumed through the star formation efficiency parameter k. Modelling the effects of this on the dust mass is beyond the scope of the simple model presented here.Outflows remove dust from the interstellar medium via −λyds. The solution is given by Eq. 17 if destruction δ = 0:The gas mass g is related to the gas fraction f in this model by:the metallicity mass fraction, Z:and the total mass of the system is:
. K N Abazajian, ApJS. 182543Abazajian K.N., et al., 2009, ApJS, 182, 543
. J Adouze, B M Tinsley, Ann. Rev. Astron. Astro. 1443Adouze J, Tinsley B.M. 1976, Ann. Rev. Astron. Astro. 14, p43
. Alton P B Bianchi, S Rand, R J Xilouris, E M Davies, J I Trewhella, M , ApJ. 507125Alton P. B., Bianchi S., Rand R. J., Xilouris E. M., Davies J. I., Trewhella M., 1998, ApJ, 507, L125
. Alton P B Bianchi, S Davies, J I , A&A. 34351Alton P. B., Bianchi S. & Davies J. I., 1999, A&A, 343, 51
. Alton P B Xilouris, E M Misiriotis, A Dasrya, K M Dumke, M , A&A. 425109Alton P. B., Xilouris E. M., Misiriotis A., Dasrya K. M., Dumke M., 2004, A&A, 425, 109
. S Bianchi, J I Davies, P B Alton, A&A. 3441Bianchi S.,Davies J. I. & Alton P. B., 1999, A&A, 344, L1
. S Bianchi, R Schneider, MNRAS. 378973Bianchi S., Schneider R., 2001, MNRAS, 378, 973
. I Baldry, MNRAS. 40486Baldry I. et al., 2010, MNRAS, 404, 86
. M Barlow, A&A. 518138Barlow M. et al., 2010, A&A, 518, L138
. C M Baugh, MNRAS. 3561191Baugh C. M. et al., 2005, MNRAS, 356, 1191
. G Bendo, A&A. 51865Bendo G. et al., 2010, A&A, 518, L65
. A W Blain, I Smail, R J Ivison, J.-P Kneib, MNRAS. 303632Blain A. W., Smail I., Ivison R. J., Kneib J.-P., 1999, MNRAS, 303, 632
. Boselli, A&A. 51861Boselli et al., 2010, A&A, 518, L61
. F Boulanger, A&A. 312256Boulanger F. et al., 1996, A&A, 312, 256
. J Braine, M Guelin, M Dumke, N Brouillet, F Herpin, R Wielebinski, A&A. 326963Braine J., Guelin M.,Dumke M., Brouillet N., Herpin F., Wielebin- ski R., 1997, A&A, 326, 963
. G Bruzual, S Charlot, MNRAS. 3441000Bruzual G. & Charlot S., 2003, MNRAS, 344, 1000
. V Buat, T Takeuhci, D Burgarella, E Giovannoli, K L Murata, A&A. 507693Buat V., Takeuhci T., Burgarella D., Giovannoli E., Murata K. L., 2009, A&A, 507, 693
. D Burgarella, Le Floch, E Takeuchi, Y Huang, J S Buat, V Rieke, G H Tyler, K D , MNRAS. 380968Burgarella D., Le Floch E., Takeuchi Y., Huang J. S., Buat V., Rieke G. H., Tyler K. D., 2007, MNRAS, 380, 968
. F Calura, A Pipino, F Matteucci, A&A. 479669Calura F., Pipino A. & Matteucci F., 2008, A&A, 479, 669
. D Calzetti, ApJ. 666870Calzetti D. et al., 2007, ApJ, 666, 870
. R Cannon, MNRAS. 372425Cannon R. et al., 2006, MNRAS, 372, 425
. G Chabrier, PASP. 115763Chabrier G., 2003, PASP, 115, 763
. Charlot S Fall, M , ApJ. 539718Charlot S. & Fall M., 2000, ApJ, 539, 718
. P I Choi, ApJ. 637227Choi P. I. et al., 2006, ApJ, 637, 227
. D L Clements, L Dunne, S A Eales, MNRAS. 403274Clements D. L., Dunne L. & Eales S. A, 2010, MNRAS, 403, 274
. A A Collister, O Lahav, PASP. 116345Collister A.A. & Lahav O., 2004, PASP, 116, 345
. K Coppin, MNRAS. 3841597Coppin K. et al., 2008, MNRAS, 384, 1597
. S M Croom, MNRAS. 39219Croom S.M. et al., 2009, MNRAS, 392, 19
. E Da Cunha, S Charlot, D Elbaz, MNRAS. 3881595da Cunha E., Charlot S. & Elbaz D., 2008, MNRAS, 388, 1595
. E Da Cunha, C Eminian, S Charlot, J Blaizot, MNRAS. 4031894da Cunha E., Eminian C., Charlot S., Blaizot J., 2010a, MNRAS, 403, 1894
. E Da Cunha, V Charmandaris, T Diaz-Santos, L Armus, J A Marshall, D Elbaz, arXiv:1008.2000A&A. in pressda Cunha E., Charmandaris V., Diaz-Santos T., Armus L., Marshall J. A., Elbaz D., 2010b, A&A, in press (arXiv:1008.2000)
. D A Dale, G Helou, A Contursi, N A Silbermann, S Kolhatkar, ApJ. 549215Dale D. A., Helou G., Contursi A., Silbermann N. A., Kolhatkar S., 2001, ApJ, 549, 215
. A L R Danielson, MNRAS. 4101687Danielson A. L. R. et al., 2011, MNRAS, 410, 1687
. J I Davies, P B Alton, S Bianchi, M Trewhella, MNRAS. 3001006Davies J. I., Alton P. B., Bianchi S., Trewhella M., 1998, MNRAS, 300, 1006
. S Driver, C C Popescu, R J Tuffs, J Liske, A W Graham, P D Allen, R De Propris, MNRAS. 3791022Driver S., Popescu C. C., Tuffs R. J., Liske J., Graham A. W., Allen P. D., de Propris R., 2007, MNRAS, 379, 1022
. S Driver, MNRAS. 413971Driver S. et al., 2011, MNRAS, 413, 971
. S Driver, A Robotham, MNRAS. 4072131Driver S. & Robotham A., 2010., MNRAS, 407, 2131
. L Dunne, MNRAS. 315115Dunne L. et al., 2000, MNRAS, 315, 115
. L Dunne, S A Eales, MNRAS. 327697Dunne L. & Eales S. A., 2001, MNRAS, 327, 697
. L Dunne, S A Eales, M G Edmunds, MNRAS. 341589Dunne L., Eales S. A. & Edmunds M. G., 2003, MNRAS, 341, 589
. L Dunne, S A Eales, R J Ivison, H Morgan, Nature. 424285Dunne L., Eales S. A., Ivison R. J., Morgan H., Edmunds M. G., 2003, Nature, 424, 285
. L Dunne, MNRAS. 3941307Dunne L. et al., 2009, MNRAS, 394, 1307
. E Dwek, ApJ. 501643Dwek E. 1998, ApJ, 501, 643
. E Dwek, F Galliano, A P Jones, ApJ. 662927Dwek E., Galliano F. & Jones A. P., 2007, ApJ, 662, 927
. E Dwek, I Cherchneff, ApJ. 72763Dwek E., Cherchneff I., 2011, ApJ, 727, 63
. S Dye, A&A. 51810Dye S. et al., 2010, A&A, 518, L10
. S A Eales, M G Edmunds, MNRAS. 29929Eales S. A. & Edmunds M. G., 1996, MNRAS, 299, L29
. S A Eales, ApJ. 7071779Eales S. A. et al., 2009, ApJ, 707, 1779
. S Eales, PASP. 122499Eales S., et al., 2010a, PASP, 122, 499
. S A Eales, A&A. 51862Eales S. A. et al., 2010b, A&A, 518, L62
. M G Edmunds, MNRAS. 328223Edmunds M. G., 2001, MNRAS, 328, 223
. A S Ferrarotti, H.-P Gail, A&A. 447553Ferrarotti A. S. & Gail H.-P., 2006, A&A, 447, 553
. M Fukugita, P J E Peebles, ApJ. 616643Fukugita M. & Peebles P. J. E., 2004, ApJ, 616, 643
. C Gall, A C Andersen, J Hjorth, A&A. 52813Gall C., Andersen A. C. & Hjorth J., 2011, A&A, 528, 13
. T Garn, MNRAS. 402Garn T. et al., 2010, MNRAS, 402, 2017
. R Gehrz, IAUS. 135445Gehrz R., 1989, IAUS, 135, 445
. H Gomez, MNRAS. 3971621Gomez H. et al., 2009, MNRAS, 397, 1621
. M Griffin, A&A. 5183Griffin M. et al., 2010, A&A, 518, L3
. C Gruppioni, A&A. 51827Gruppioni C. et al., 2010, A&A, 518, L27
. M Gunawardhana, arXiv:1104.2379MNRAS. in pressGunawardhana M. et al., 2011, MNRAS, in press (arXiv:1104.2379)
. Y Harayama, F Eisenhauer, F Martins, ApJ. 6751319Harayama Y., Eisenhauer F. & Martins F., 2008, ApJ, 675 1319
. G Helou, ApJ. 31133Helou G. 1986, ApJ, 311, L33
. D Hill, MNRAS. 412765Hill D. et al., 2011, MNRAS, 412, 765
. H Hippelein, M Haas, R J Tuffs, D Lemke, M Stickel, U Klaas, H J Volk, A&A. 407137Hippelein H., Haas M., Tuffs R. J., Lemke D., Stickel M., Klaas U., Volk H. J., 2003, A&A, 407, 137
. A M Hopkins, ApJ. 615209Hopkins A. M., 2004, ApJ, 615, 209
. J Huchra, M Davis, D Latham, J Tonry, ApJS. 5289Huchra J., Davis M., Latham D., Tonry J., 1983, ApJS, 52, 89
. E Ibar, MNRAS. 40938Ibar E. et al., 2010, MNRAS, 409, 38
. A K Inoue, PASJ. 55901Inoue A. K., 2003, PASJ, 55, 901
. A James, MNRAS. 335753James A. et al., 2002, MNRAS, 335, 753
. A P Jones, A Tielens, D J Hollenbacj, C F Mckee, ApJ. 433797Jones A. P., Tielens A., Hollenbacj D. J., McKee C. F., 1994, ApJ, 433, 797
. D H Jones, MNRAS. 399683Jones D.H. et al., 2009, MNRAS, 399, 683
. R C Kennicutt, ApJ. 498541Kennicutt R. C., 1998, ApJ, 498, 541
. R C Kennicutt, ApJ. 7031672Kennicutt R. C. et al., 2009, ApJ, 703, 1672
. O Krause, Nature. 432596Krause O. et al., 2004, Nature, 432, 596
. E Krugel, R Siebenmorgen, A&A. 288929Krugel, E., & Siebenmorgen, R. 1994, A&A, 288, 929
. A Lawrence, MNRAS. 3791599Lawrence A. et al., 2007, MNRAS, 379, 1599
. Le Floc, ' , E , ApJ. 632169Le FLoc'h E. et al., 2005, ApJ, 632, 169
. A Li, B T Draine, ApJ. 554778Li A. & Draine B. T., 2001, ApJ, 554, 778
. P Madau, H C Ferguson, M E Dickinson, M Giavalisco, C C Steidel, A Fruchter, MNRAS. 2831388Madau P., Ferguson H. C., Dickinson M. E., Giavalisco M., Steidel C. C., Fruchter A., 1995, MNRAS, 283, 1388
. F Mannucci, G Cresci, R Maiolino, A Marconi, A Gnerucci, MNRAS. 4182115Mannucci, F., Cresci, G., Maiolino, R., Marconi, A., Gnerucci, A., 2010, MNRAS, 418, 2115
. J S Mathis, G Whiffen, ApJ. 341808Mathis, J.S., & Whiffen, G., 1989, ApJ, 341, 808
. R Meijerink, R P J Tilanus, C P Dullemond, F P Israel, P P Van Der Werf, A&A. 430427Meijerink R., Tilanus R. P. J., Dullemond C. P., Israel F. P., van der Werf P. P., 2005, A&A, 430, 427
. B Ménard, R Scranton, M Fukugita, G Richards, MNRAS. 4051025Ménard B., Scranton R., Fukugita M., Richards G., 2010, MNRAS, 405, 1025
. M J Michałowski, E J Murphy, J Hjorth, D Watson, J Hjorth, C Gall, J S Dunlop, A&A. 52215Michałowski M.J., Murphy E.J., Hjorth J., Watson D., Hjorth J., Gall C., Dunlop J.S., 2010, A&A, 522, 15
. T Moore, MNRAS. 379663Moore T. et al., 2007, MNRAS, 379, 663
. H L Morgan, M G Edmunds, MNRAS. 343427Morgan H. L. & Edmunds M. G., 2003, MNRAS, 343, 427
. H Morgan, L Dunne, S A Eales, R J Ivison, M G Edmunds, ApJ. 59733Morgan H., Dunne L., Eales S. A., Ivison R. J., Edmunds M. G., 2003, ApJ, 597, L33
. M Negrello, Science. 330800Negrello M. et al., 2010, Science, 330, 800
. V Ossenkopf, Th Henning, A&A. 291943Ossenkopf, V., & Henning, Th., 1994, A&A, 291, 943
. M J Page, F J Carrera, MNRAS. 311433Page M. J. & Carrera F. J., 2000, MNRAS, 311, 433
. P P Papadopoulos, ApJ. 720226Papadopoulos P. P., 2010, ApJ, 720, 226
. P P Papadopoulos, F I Pelupessy, ApJ. 7171037Papadopoulos P. P. & Pelupessy F. I., 2010, ApJ, 717, 1037
. E Pascale, arXiv:1010.5782MNRAS. 5181A&APascale E. et al., 2011, MNRAS, in press (arXiv:1010.5782) Pilbratt G. et al., 2010, A&A, 518, L1
. A Poglitsch, Planck Collaboration ; Planck CollaborationarXiv:1011.2036arXiv:1011.2045A&A accepted. 2A&APlanck Collaboration, 2011a, A&A, accepted (arXiv:1011.2036) Planck Collaboration, 2011b, A&A accepted (arXiv:1011.2045) Poglitsch A. et al., 2010, A&A, L2
. C C Popescu, R J Tuffs, H J Volk, D Pierini, B F Madore, ApJ. 567221Popescu C. C., Tuffs R. J., Volk H. J., Pierini D., Madore B. F., 2002, ApJ, 567, 221
. C C Popescu, R J Tuffs, M A Dopita, J Fischer, N D Kylafis, B F Madore, A&A. 527109Popescu C. C., Tuffs R. J., Dopita M. A., Fischer J., Kylafis N. D., Madore B. F., 2011, A&A, 527, 109
. L Pozzetti, A&A. 474443Pozzetti L. et al., 2007, A&A, 474, 443
. E E Rigby, arXiv:1010.5787MNRAS. in pressRigby E.E. et al., 2011, MNRAS, in press (arXiv:1010.5787)
. J Rho, ApJ. 673271Rho J. et al., 2008, ApJ, 673, 271
. D Romano, C Chiappini, F Matteucci, M ; Tosi, B A Sargent, A&A. 430878ApJRomano D., Chiappini C., Matteucci F., Tosi M., 2005, A&A., 430, 491, Sargent B. A. et al., 2010, ApJ, 716, 878
. W Saunders, M Rowan-Robinson, A Lawrence, G Efstathiou, N Kaiser, R S Ellis, C S Frenk, MNRAS. 242318Saunders W., Rowan-Robinson M., Lawrence A., Efstathiou G., Kaiser N., Ellis R. S., Frenk C. S., 1990, MNRAS, 242, 318
. W Saunders, MNRAS. 31755Saunders W. et al., 2000, MNRAS, 317, 55
. J M Scalo, Fund. Cosmic Phys. 111Scalo, J. M. 1986, Fund. Cosmic Phys., 11, 1
. P Schechter, ApJ. 203297Schechter P., 1976, ApJ, 203, 297
. M Schmidt, ApJ. 151393Schmidt M., 1968, ApJ, 151, 393
. S Serjeant, D Harrison, MNRAS. 356192Serjeant S. & Harrison D., 2005, MNRAS, 356, 192
. N Seymour, M Symeonidis, M J Page, M Huynh, T Dwelly, I Mchardy, G Rieke, MNRAS. 4022666Seymour N., Symeonidis M., Page M. J., Huynh M., Dwelly T., McHardy I., Rieke G., 2010, MNRAS, 402, 2666
. T J Sodroski, N Odegard, R G Arendt, E Dwek, J L Weiland, M G Hauser, T Kelsall, ApJ. 480173Sodroski T. J., Odegard N., Arendt R. G., Dwek E., Weiland J. L., Hauser M. G., Kelsall T., 1997, ApJ, 480, 173
. B T Soifer, L Boehmer, G Neugebauer, D B Sanders, AJ. 98766Soifer B. T., Boehmer L., Neugebauer G., Sanders D. B., 1989, AJ, 98, 766
. D J B Smith, arXiv:1007.5260MNRAS. in pressSmith D.J.B. et al., 2011, MNRAS, in press (arXiv:1007.5260)
. J A Stevens, M Amure, W K Gear, MNRAS. 357361Stevens J. A., Amure M. & Gear W. K., 2005, MNRAS, 357, 361
. M Stickel, U Klaas, D Lemke, A&A. 466831Stickel M., Klaas, U. & Lemke D., 2007, A&A, 466, 831
. R Sutherland, R Saunders, MNRAS. 259413Sutherland R. & Saunders R., 1992, MNRAS, 259, 413
. M Swinbank, Nature. 464733Swinbank M. et al., 2010, Nature, 464, 733
. M Symeonidis, M J Page, N Seymour, T Dwelly, K Coppin, I Mchardy, G H Rieke, M Huynh, ApJL. 66073Symeonidis M., Page M. J., Seymour N., Dwelly T., Coppin K., McHardy I., Rieke G. H., Huynh M., 2009, ApJL, 660, L73
. M Symeonidis, M J Page, N Seymour, MNRAS. 411983Symeonidis M., Page M. J. & Seymour N., 2011, MNRAS, 411, 983
. A G G M Tielens, ApJ. 499267Tielens A. G. G. M., 1998, ApJ, 499, 267
. P & Todini, A Ferrara, MNRAS. 325726Todini P & Ferrara A., 2001, MNRAS, 325, 726
. R J Tuffs, C C Popescu, H J Volk, N D Kylafis, M A Dopita, A&A. 419835Tuffs R.J, Popescu C.C., Volk H.J., Kylafis N.D., Dopita M.A., 2004, A&A, 419, 835
. M L Vandalfsen, W E Harris, ApJ. 127368VanDalfsen M.L., & Harris W.E., 2004, ApJ, 127, 368
. V Villar, ApJ. 677169Villar V. et al., 2008, ApJ, 677, 169
. C Vlahakis, L Dunne, S A Eales, MNRAS. 3641253Vlahakis C., Dunne L. & Eales S. A., 2005, MNRAS, 364, 1253
. L Wang, M Rowan-Robinson, MNRAS. 398109Wang L., & Rowan-Robinson M., 2009, MNRAS, 398, 109
. L Wang, Y P Jing, MNRAS. 4021796Wang L. & Jing Y. P., 2010, MNRAS, 402, 1796
. D V Weibe, ApJ. 7071809Weibe D. V. et al., 2009, ApJ, 707, 1809
. C N A Willmer, ApJ. 138146Willmer C. N. A. et al., 2009, ApJ, 138, 146
. K C Xu, ApJS. 173432Xu K. C. et a., 2007, ApJS, 173, 432
. S Zhukovska, H.-P Gail, M Trieloff, A&A. 479453Zhukovska S., Gail H.-P. & Trieloff M., 2008, A&A, 479, 453
| [] |
[
"DIFFERENTIABILITY INSIDE SETS WITH UPPER MINKOWSKI DIMENSION ONE",
"DIFFERENTIABILITY INSIDE SETS WITH UPPER MINKOWSKI DIMENSION ONE"
] | [
"Michael Dymond ",
"Olga Maleva "
] | [] | [] | We show that every finite-dimensional Euclidean space contains compact universal differentiability sets of upper Minkowski dimension one. In other words, there are compact sets S of upper Minkowski dimension one such that every Lipschitz function defined on the whole space is differentiable inside S. Such sets are constructed explicitly. | null | [
"https://arxiv.org/pdf/1305.3154v1.pdf"
] | 119,649,893 | 1305.3154 | d3537d8edb898b74420b94059d92e8562ff06844 |
DIFFERENTIABILITY INSIDE SETS WITH UPPER MINKOWSKI DIMENSION ONE
14 May 2013
Michael Dymond
Olga Maleva
DIFFERENTIABILITY INSIDE SETS WITH UPPER MINKOWSKI DIMENSION ONE
14 May 2013arXiv:1305.3154v1 [math.FA]
We show that every finite-dimensional Euclidean space contains compact universal differentiability sets of upper Minkowski dimension one. In other words, there are compact sets S of upper Minkowski dimension one such that every Lipschitz function defined on the whole space is differentiable inside S. Such sets are constructed explicitly.
Introduction
Lipschitz functions on Banach spaces have somewhat strong differentiability properties. Rademacher's Theorem is a classical result and states that a Lipschitz function on a Euclidean space is differentiable almost everywhere with respect to the Lebesgue measure [7, p. 100]. A more recent theorem, due to Preiss and published in 1990, asserts that every Lipschitz function on a Banach space X, with norm differentiable away from the origin, is differentiable on a dense subset of X [8].
For Lipschitz functions on R, a converse statement to Rademacher's Theorem is true; for every subset N of R with Lebesgue measure zero, there exists a Lipschitz function f = f N that is nowhere differentiable on N. This fact is proved in [9], where a full characterisation of the possible sets of non-differentiability of a Lipschitz function on R is given.
The converse statement to Rademacher's Theorem fails in higher dimensions. In [8], it is proved that there exists a subset of R 2 , with Lebesgue measure zero, which contains a point of differentiability of every Lipschitz function on R 2 . Sets containing a point of differentiability of every Lipschitz function are said to have the universal differentiability property and are called universal differentiability sets. This terminology was introduced by Doré and Maleva and first appeared in [3].
As alluded to above, subsets of R d can be distinguished according to their Lebesgue measure. The Lebesgue null, universal differentiability set given in [8] contains every line segment between pairs of points with rational co-ordinates. In some sense, this set is still rather large; the closure of this set is the whole of R 2 . Recent work of Doré and Maleva has uncovered much smaller sets in R d with the universal differentiability property. A compact universal differentiability set with Lebesgue measure zero is constructed in [2].
In order to detect smaller universal differentiability sets, we must appeal to some other notion of size, rather than the Lebesgue measure. In the theory of fractal geometry, the size of a set is often ascertained by dimension. The Hausdorff dimension, based on a construction of Carathéodory is the oldest and perhaps the most important example of a dimension [5, p. 27]. It is defined for all subsets of a Euclidean space according to the Hausdorff measure; see [7] or [5] for a complete definition. In [3], a compact universal differentiability set with Hausdorff dimension one is constructed. This result is optimal in the sense that any universal differentiability set in a Euclidean space must have Hausdorff dimension at least one [3,Lemma 1.5]. Finally, it is proved in [4] that every non-zero Banach space with separable dual contains a closed and bounded universal differentiability set of Hausdorff dimension one. This is a generalisation of the main result of [3] for infinite dimensional Banach spaces.
The Minkowski dimensions of a bounded subset of R d are closely related to the Hausdorff dimension. Whilst the Hausdorff dimension of a set is based on coverings by sets of arbitrarily small diameter, the Minkowski dimensions are defined similarly according to coverings by sets of the same small diameter. For this reason, the Minkowski dimension is often referred to as the box-counting dimension [5, p. 41]. The definition below follows [7, p. 76-77]. Definition 1.1. Given a bounded subset A of R d and ǫ > 0, we define N ǫ (A) to be the minimal number of balls of radius ǫ required to cover A. That is, N ǫ (A) is the smallest integer n for which there exists balls
B 1 , . . . , B n ⊆ R d , each of radius ǫ, such that A ⊆ ∪ i B i .
The lower Minkowski dimension of A is then defined by
(1.1) dim M (A) = inf s > 0 : lim inf ǫ→0 N ǫ (A)ǫ s = 0 ,
and the upper Minkowski dimension of A is given by
(1.2) dim M (A) = inf s > 0 : lim sup ǫ→0 N ǫ (A)ǫ s = 0 .
Writing dim H for the Hausdorff dimension, it is readily verified that
dim H (A) ≤ dim M (A) ≤ dim M (A) for all bounded A ⊆ R d .
The Hausdorff dimension and Minkowski dimensions can be very different: For example, a countable dense subset of a ball in R d has Hausdorff dimension 0 whilst having the maximum upper and lower Minkowski dimension d. A construction of a set having lower Minkowski dimension strictly less than its upper Minkowski dimension is given in [7, p. 77]. It is worth noting that the Minkowski dimension behaves nicely with respect to closures; we have that dim M (A) = dim M (Clos(A)) and dim M (A) = dim M (Clos(A)).
In the present paper, we verify the existence of a compact universal differentiability set with upper and lower Minkowski dimension one in R d for d ≥ 2. Such a set is constructed explicitly. This is an improvement on the result of [3], where a compact universal differentiability set of Hausdorff dimension one is given. The construction in [3] involves considering a G δ set O of Hausdorff dimension one, containing all line segments between points belonging to a countable dense subset R of the unit ball in R d (and hence the Minkowski dimension of O is equal to d). The set O can be expressed as
O = ∞ k=1 O k , where each O k is an open subset of R d and O k+1 ⊆ O k .
For each k ≥ 1, a set R k is defined consisting of a finite union of line segments between points from R. Since R k is then a closed subset of O, it is possible to choose
w k > 0 such that B w k (R k ) ⊆ O k . The final sets U λ are then defined by U λ = ∞ k=1 k≤n≤(1+λ)k B λw k (R k ) for λ ∈ [0, 1]. Observe that, for each k ≥ 1 and λ ∈ [0, 1] the closed set k≤n≤(1+λ)k B λw k (R k ) is contained in O k . Consequently, dim H (U λ ) ≤ dim H (O) = 1.
There is no non-trivial upper bound for the Minkowski dimensions of the sets U λ constructed in [3]. For constructing a universal differentiability set with upper or lower Minkowski dimension one, the approach of [3] fails because the set O has the maximum upper and lower Minkowksi dimension, d.
To get a set of lower Minkowski dimension one it would be enough to control the number of δ-cubes (this will refer to a cube with side equal 2δ) for a specific sequence δ n ց 0. Assume p > 1 is a fixed number and we want to make sure that the set to be constructed has lower Minkowski dimension less than p. Imagine that we have reached the nth step of the construction where we require that C n is an upper estimate for a number of δ n -cubes needed to cover the final set W , and C n δ p n < 1. The idea for the next step is to divide each δ n -cube by a K n × · · · × K n grid into smaller δ n+1 = δ n /K n -cubes. If K n is big enough, then as δ p n /δ p n+1 = K p n , we are free to choose inside the given δ n -cube any number of δ n+1 -boxes up to K p n . This could, for example, be K n (log K n ) Mn ≪ K p n for any fixed p > 1, see inequality (3.28); K n = Q sn and |E n | ≤ s 2d n , M n ≪ sn log sn . We then have that the product of the total number of δ n+1 -cubes needed to cover W by δ p n+1 is bounded by 1 from above as well. Since this is satisfied for all n, we conclude that dim M (W ) ≤ p. Since this is true for every p > 1, we obtain a set of lower Minkowski dimension 1.
Getting dim M (W ) ≤ 1 is less clear. As n grows, the number K n has to tend to infinity or otherwise we would get many points of porosity inside W , see below for the definition and discussion of porosity. In order to prove that dim M (W ) ≤ p we should be able to show that there exists δ 0 > 0 such that for every δ ∈ (0, δ 0 ) the set W can be covered by a controlled number N δ of δ-boxes. In other words, N δ δ p should be bounded for all δ below certain threshold. Choosing n such that δ n+1 < δ ≤ δ n gives N δ δ p ≤ N δ n+1 δ p n = N δ n+1 δ p n+1 K p n , and the factor K p n → ∞ makes it impossible to have a constant upper estimate for N δ δ p . The idea here is that we need to leave a "gap" for an unbounded sequence in the estimate for N δn δ p n and to make sure that K p n fits inside that gap. The realisation of that gap is the inequality (3.32).
To finish the introduction, let us briefly explain why we should be concerned about porosity points. A point x ∈ W is called its porosity point if there exists λ > 0 such that for any r > 0 there is a point y ∈ B(x, r) such that B(y, λ y − x ) ∩ W = ∅. If x is a porosity point of W then the distance to W , f (·) = dist(·, W ), is a 1-Lipschitz function not differentiable at x. Since our aim is to construct a universal differentiability set, we try to avoid as much as possible constructions that lead to many porosity points inside the set we construct. More information about porous and σ-porous sets can be found in the survey [10], and further discussion of relations between problems about differentiability of Lipschitz functions and the theory of porous and σ-porous sets is presented in the recent book [6].
Structure of the paper
In Section 2 we establish a criterion for the universal differentiability property based on the results of [2] and [4]. Section 3 is devoted to the construction of the new set of upper Minkowski dimension one. Finally, in Section 4, we apply the result established in Section 2 to prove that our set has the universal differentiability property.
Differentiability
In this section, we prove a criterion for the universal differentiability property. We begin by defining the two key notions of differentiability of a real valued function f on a Banach space X, according to [1, p. 83].
Definition 2.1. A function f : X → R is said to be Fréchet differen- tiable at a point x ∈ X if the limit f ′ (x, e) = lim t→0 f (x + te) − f (x) t
exists uniformly in e ∈ B(0, 1) and is a bounded linear map.
Next, we formalise what it means for a function f : X → R to be Lipschitz [3, p. 2].
Definition 2.2. A function f : X → R is called Lipschitz if there exists L > 0 such that |f (y) − f (x)| ≤ L y − x X for all x, y ∈ X with y =
x. If f : X → R is a Lipschitz function then the number
Lip(f ) = sup |f (y) − f (x)| y − x X : x, y ∈ X, y = x
is finite and is called the Lipschitz constant of f . Theorem 2.3 is an amalgamation of the results in [2] and [3]. Roughly speaking, it says that a closed set S has the universal differentiability property if every point in S can be approximated, in a special way, by line segments contained in S. We use this theorem in Section 3 to construct a universal differentiability set with upper Minkowski dimension one.
Theorem 2.3. Let d ≥ 2. Suppose that (U λ ) λ∈[0,1] is a family of closed subsets of R d satisfying U λ 1 ⊆ U λ 2 whenever 0 ≤ λ 1 ≤ λ 2 ≤ 1. Suppose further that for every λ ∈ [0, 1), ψ ∈ (0, 1 − λ) and η ∈ (0, 1), there exists δ 1 = δ 1 (λ, ψ, η) > 0 such that whenever x ∈ U λ , δ ∈ (0, δ 1 ) and v 1 , v 2 , v 3 are in the closed unit ball in R d , there exists v 1 ′ , v 2 ′ , v 3 ′ ∈ R d such that v i ′ − v i ≤ η and [x + δv 1 ′ , x + δv 3 ′ ] ∪ [x + δv 3 ′ , x + δv 2 ′ ] ⊆ U λ+ψ . Then the set (2.1) S = q<1 U q
is a universal differentiability set. Moreover, S has the property that whenever y ∈ S, ρ > 0 and g : R d → R is a Lipschitz function, there exists x ∈ S such that x − y < ρ and g is differentiable at y.
Proof of Theorem 2.3. Let y ∈ S, ρ > 0 and g : R d → R be a Lipschitz function. We show that there exists a point x ∈ S such that g is differentiable at x and x − y < ρ. We may assume Lip(g) > 0. Let H be the Hilbert space R d . Set λ 1 = 1 and fix λ 0 ∈ (0, 1) and y ′ ∈ U λ 0 such that y ′ − y < ρ/3. Taking
δ < min{δ 1 , ρ/4} we can find a line segment [a, b] ⊆ R d with [a, b] ⊆ U λ 0 ∩ B ρ/3 (y ′ ), where λ 0 ∈ ( λ 0 , 1). Then, by Lebesgue's theorem, there exists a point x 0 ∈ [a, b] such that the directional derivative g ′ (x 0 , e 0 ) exists, where e 0 = (b − a)/ b − a . Set f 0 = g, (S, ) = ([0, 1], ≤), K = 25 2Lip(g), σ 0 = ρ/3, µ = Lip(g), and apply [3, Theorem 2.7] where H = R d .
This theorem provides us with the Lipschitz function f , the number λ ∈ (λ 0 , λ 1 ), the point x ∈ U λ ∩ B σ 0 (x 0 ), the direction e ∈ S d−1 and, for each ǫ > 0, the numbers σ ǫ > 0 and λ ǫ ∈ (λ, 1) such that f − f 0 is linear with operator norm less than or equal to µ and the directional derivative f ′ (x, e) > 0 is almost locally maximal in the following sense: Whenever
(i) x ′ ∈ F ǫ = U λǫ ∩ B σǫ (x), f ′ (x ′ , e ′ ) ≥ f ′ (x, e) and (ii) for any t ∈ R (2.2) |(f (x ′ + te) − f (x ′ )) − (f (x + te) − f (x))| ≤ K f ′ (x ′ , e ′ ) − f ′ (x, e)|t|,then we have f ′ (x ′ , e ′ ) < f ′ (x, e) + ǫ.
We then verify that the conditions of [3, Lemma 2.8] hold for the function f : R d → R, the pair (x, e) and the family of sets F ǫ ⊆ R d defined in (i). This would imply that the function f is differentiable at x.
We already have that the derivative f ′ (x, e) exists and is non-negative. We now verify condition (1) of [3, Lemma 2.8]. Given ǫ > 0 and η ∈ (0, 1) we put ψ ǫ = λ ǫ − λ and define
δ ⋆ = δ ⋆ (ǫ, η) = 1 2 min {δ 1 (λ, η, ψ ǫ ), σ ǫ } , where δ 1 (λ, η, ψ ǫ ) is defined in the hypothesis of Theorem 2.3. Let v 1 , v 2 , v 3 ∈ B 1 (0) ⊆ R d and δ ∈ (0, δ ⋆ ). Since 0 < δ < δ 1 (λ, η, ψ ǫ ), there exist, by the hypothesis of Theorem 2.3, points v 1 ′ , v 2 ′ , v 3 ′ ∈ R d with v i ′ − v i ≤ η and [x + δv 1 ′ , x + δv 3 ′ ] ∪ [x + δv 3 ′ , x + δv 2 ′ ] ⊆ U λ+ψǫ = U λǫ .
Moreover, given that δ < σ ǫ , we have
[x + δv 1 ′ , x + δv 3 ′ ] ∪ [x + δv 3 ′ , x + δv 2 ′ ] ⊆ U λǫ ∩ B σǫ (x) = F ǫ .
Thus, condition (1) Since g − f is linear, we conclude that g is also differentiable at x. Moreover, we have that x ∈ U λǫ ⊆ S and
x − y ≤ x − x 0 + x 0 − y ′ + y ′ − y < ρ/3 + ρ/3 + ρ/3 = ρ.
The Set
We let d ≥ 2 and construct a universal differentiability set of upper Minkowski dimension one in R d . There are many equivalent ways of defining the (upper and lower) Minkowski dimension of a bounded subset of R d . In addition to Definition 1.1 in Section 1, several examples can be found in [7, p. 41-45]. The equivalent definition given below will be most convenient for our use. By an ǫ-cube, with centre x ∈ R d , parallel to e ∈ S d−1 , we mean any subset of R d of the form
(3.1) C(x, ǫ, e) = x + d i=1 t i e i : e 1 = e, t i ∈ [−ǫ, ǫ] .
where e 2 , . . . , e d ∈ S d−1 and e i , e j = 0 whenever i = j.
Definition 3.1. Given a bounded subset A of R d and ǫ > 0, we denote by N ǫ (A) the minimum number of (closed) ǫ-cubes required to cover A. That is, N ǫ (A) is the smallest integer n for which there exist ǫ-cubes C 1 , C 2 , . . . , C n such that For a point x ∈ R d and w > 0, we shall write B w (x) for the closed ball with centre x and radius w with respect to the Euclidean norm. For a subset S of R d , we let B w (S) = x∈S B w (x). The cardinality of a finite set F shall be denoted by |F |. Given a real number α, we write [α] for the integer part of α.
A ⊆ n i=1 C i .
Fix two sequences of positive integers (s k ) and (M k ) such that the following conditions are satisfied:
(3.2) 3 ≤ M k ≤ s k ; M k , s k → ∞; M k log s k s k → 0
and there exists a sequences k ≥ s k such that
(3.3)s k −s k−1 s k → 0.
Note that if the sequence (s k −s k−1 ) k≥2 is bounded and s k → ∞, then (3.3) is satisfied. Hence an example of sequences (s k ), (s k ) satisfying (3.2) and (3.3) iss k = ak + b with a > 0 and any integer sequence
s k → ∞ such that 3 ≤ s k ≤s k .
We also remark that if s k → ∞ is such that
(3.4) s k s k+1 → 1, then (3.3) is satisfied withs k = s k . Indeed,s k −s k−1 s k = 1 − s k−1 s k → 0. An example of an integer sequence s k → ∞ satisfying (3.4) is s k = max{3, [F (k)]}, where F (x)
is a linear combination of powers of x such that the highest power of x is positive and has a positive coefficient. Also, whenever s k → ∞ satisfies the condition (3.4), the sequence s ′ k = [log s k ] also satisfies this condition and tends to infinity.
Once (s k ) is defined there is much freedom to choose (M k ). For example, we may take M k = [s α k ] with α ∈ (0, 1) or M k = [log s k ] etc. Let now E k be a maximal 1 s k -separated subset of S d−1 .
We therefore get a collection of finite subsets E k ⊆ S d−1 such that
(3.5) |E k | ≤ s 2d k and ∀e ∈ S d−1 ∃ e ′ ∈ E k s.t. e − e ′ ≤ 1 s k .
Definition 3.2. Given a line segment l = x + [a, b]e ⊆ R d and 0 < w < length(l)/2, define F w (l) to be a finite collection of cubes of the form C(x i , w, e), defined by (3.1), with x i ∈ l, such that
(3.6) B w (l) ⊆ C∈Fw(l) C and |F w (l)| < length(l)/w.
Fix an arbitrary number Q ∈ (1, 2). Let l 1 be a line segment in R d of length 1, set
w 1 = Q −s 1 , L 1 = {l 1 }, C 1 = F w 1 (l 1 ) and T 1 = B w 1 (l 1 ) .
We refer to the collection L 1 as 'the lines of level 1', the collection C 1 as 'the cubes of level 1' and the collection T 1 as 'the tubes of level 1'. Note that C 1 is a cover of the union of tubes in T 1 . Suppose that k ≥ 2, and that we have defined real numbers w r > 0 and the collections L r of lines, T r of tubes and C r of cubes of level r = 1, 2, . . . , k − 1 in such a way that
T r = B wr (l) : l ∈ L r ; C r = l∈Lr F wr (l) is a cover of {T : T ∈ T r }.
We now describe how to construct the lines, cubes and tubes of the kth level. We start with the definition of the new width. Set
(3.7) w k = Q −s k w k−1 .
The collections L k of lines, T k of tubes and C k of cubes will be partitioned into exactly M k + 1 classes and each class will be further partitioned into categories according to the length of the lines. We first define the collections of lines, tubes and cubes of level k, class 0 respectively by (3.8)
L (k,0) = L k−1 , T (k,0) = B w k (l) : l ∈ L (k,0) and C (k,0) = l∈L (k,0) F w k (l).
We will say that all lines, tubes and cubes of level k, class 0 have the empty category. From (3.8) and Definition 3.2, we have that C (k,0) is a cover of the union of tubes in T (k,0) . Using (3.6), we also have that
(3.9) C (k,0) = l∈L (k,0) |F w k (l)| ≤ 1 w k l∈L (k,0) length(l).
The intersection of each line l in L (k,0) = L k−1 with a cube C ∈ C k−1 is a line segment of length at most 2w k−1 . Moreover, the collection of cubes C k−1 is a cover of the union of lines in L (k,0) . It follows that (3.10)
l∈L (k,0) length(l) ≤ 2 |C k−1 | w k−1 .
Combining (3.9), (3.10) and (3.7) yields
(3.11) C (k,0) ≤ 2 |C k−1 | Q s k .
Definition 3.3. Given a bounded line segment l ⊆ R d , an integer j ≥ 1 with length(l) ≥ Q j w k /s k and a direction e ∈ S d−1 , we define a collection of line segments R l (j, e) as follows: Let Φ ⊆ l be a maximal
Q j w k s k -separated set and set R l (j, e) = {φ x : x ∈ Φ} ,
where φ x is the line defined by
(3.12) φ x = x + [−1, 1]Q j w k e.
We note for future reference that
(3.13) |R l (j, e)| ≤ 2s k length(l) Q j w k .
For j ∈ {1, 2, . . . , s k }, we define the collection of lines of level k, class 1, category (j) by
(3.14) L (j) (k,1) = l∈L (k,0) e∈E k R l (j, e).
We emphasise that all the lines in L (j) (k,1) have the same length. Indeed, from Definition 3.3, we get (3.15) length(l) = 2Q j w k for all lines l ∈ L (j) (k,1) . From (3.13) and (3.14), it follows that
(3.16) L (j) (k,1) ≤ 2w −1 k s k |E k | Q −j l∈L (k,0) length(l).
Together, (3.16), (3.10) and (3.7) imply
(3.17) L (j) (k,1) ≤ |C k−1 | (4s k |E k |)Q s k −j . Let (3.18) C (j) (k,1) = l∈L (j) (k,1) F w k (l).
Then, using (3.6), (3.15) and (3.17) we obtain (k,1) is a cover of the union of tubes in T (j) (k,1) . We can also use (3.20) and (3.17) to conclude that
C (j) (k,1) ≤ l∈L (j) (k,1) |F w k (l)| ≤ 1 w k l∈L (j) (k,1) length(l) ≤ L (j) (k,1) × 2 × Q j w k w k ≤ 8 |C k−1 | s k |E k | Q s k .(3.21) T (j) (k,1) = L (j) (k,1) ≤ |C k−1 | (4s k |E k |)Q s k −j .
The collections of lines, cubes and tubes of level k, class 1 are now defined by
# (k,1) = s k j=1 # (j) (k,1) ,
where # stands for L, C or T . Note that C (k,1) is a cover of the union of tubes in T (k,1) . Moreover, in view of (3.19), we get
(3.22) C (k,1) ≤ s k j=1 C (j) (k,1) ≤ 2 |C k−1 | (4s 2 k |E k |)Q s k .
Suppose that 1 ≤ m < M k and that we have defined the collections ,
L (k,(I m ) L (j 1 ,...,jm) (k,m) ≤ |C k−1 | (4s k |E k |) m Q s k −jm , (II m ) T (j 1 ,...,jm) (k,m) = B w k (l) : l ∈ L (j 1 ,...,jm) (k,m) , (III m ) C (j 1 ,...,jm) (k,m) = l∈L (j 1 ,...,jm) (k,m) F w k (l), (IV m ) C (j 1 ,...,jm) (k,m) ≤ 2 |C k−1 | (4s k |E k |) m Q s k . (V m )
For an integer sequence (j 1 , . . . , j m , j m+1 ) satisfying (3.23) we define the collection of lines of level k, class (m + 1), category (j 1 , . . . , j m+1 ) by Note that every line in the collection L (j 1 ,...,j m+1 ) (k,m+1)
has the same length. In fact, by Definition 3.3 we have that (I m+1 ) is satisfied. Combining (3.13), (I m ) and (II m ) we deduce the following:
L (j 1 ,...,j m+1 ) (k,m+1) ≤ 2s k |E k | w −1 k Q −j m+1 l∈L (j 1 ,...,jm) (k,m) length(l) ≤ |C k−1 | (4s k |E k |) m+1 × Q s k −j m+1 . (3.25)
Thus, (II m+1 ) is satisfied. We define the collection of tubes and cubes of level k and class m + 1, category (j 1 , . . . , j m+1 ) by (III m+1 ) and (IV m+1 ). Using (3.6), (I m+1 ) and (3.25) we obtain
C (j 1 ,...,j m+1 ) (k,m+1) ≤ l∈L (j 1 ,...,j m+1 ) (k,m+1) |F w k (l)| = 1 w k l∈L (j 1 ,...,j m+1 ) (k,m+1) length(l) ≤ 1 w k × L (j 1 ,...,j m+1 ) (k,m+1) × 2 × Q j m+1 w k ≤ 2 |C k−1 | (4s k |E k |) m+1 Q s k ,
and this verifies (V m+1 ). The collections of lines, tubes and cubes of level k, class m + 1 are given by
# (k,m+1) = s k ≥j 1 ≥···≥j m+1 ≥1 # (j 1 ,...,j m+1 ) (k,m+1) ,
where # stands for L, T or C. Note that C (k,m+1) is a cover of the union of tubes in T (k,m+1) . Moreover, in view of (V m+1 ) and (3.23) we have
(3.26) C (k,m+1) ≤ (j 1 ,...,j m+1 ) C (j 1 ,...,j m+1 ) (k,m+1) ≤ 2 |C k−1 | (4s k |E k |) m+1 s m+1 k Q s k = 2 |C k−1 | (4s 2
k |E k |) m+1 Q s k , and this generalises (3.22) for arbitrary 0 ≤ m < M k . Finally, the collections of lines, tubes and cubes of level k are given by
(3.27) L k = M k m=0 L (k,m) , T k = M k m=0 T (k,m) and C k = M k m=0 C (k,m) .
Note that C k is a cover of the union of tubes in T k . Moreover, using (3.11) and (3.26) we get
(3.28) |C k | ≤ 2(M k + 1) |C k−1 | (4s 2 k |E k |) M k Q s k .
The construction of the lines, tubes and cubes of all levels is now complete.
We now define a collection of closed sets (U λ ) λ∈[0,1] . Eventually, we will use these sets to form a compact universal differentiability set S with upper Minkowski dimension one, defined by (2.1). To do this, we follow a method invented by Doré and Maleva and used in [2], [3] and [4]. The sets U λ are defined similarly to the sets (T λ ) in [
(3.29) U λ = ∞ k=1 0≤m k ≤λM k l∈L (k,m k ) B λw k (l) .
We emphasise that the single line segment l 1 of level 1 is contained in the set U λ for every λ ∈ [0, 1]. Hence, every U λ is non-empty. Note also that U λ 1 ⊆ U λ 2 whenever 0 ≤ λ 1 ≤ λ 2 ≤ 1. Finally, since the unions in (3.29) are finite, it is clear that the sets U λ are closed. Proof. For any λ ∈ [0, 1] we have that U λ contains a line segment. Hence, each of the sets U λ has upper Mikowski dimension at least one. We also have U λ ⊆ U 1 for all λ ∈ [0, 1]. Therefore, to complete the proof, it suffices to show that the set U 1 has upper Minkowski dimension one. From (3.27), it is clear that for all k ≥ 1 and 0 ≤ m ≤ M k
l∈L (k,m) B w k (l) ⊆ l∈L k B w k (l) = T ∈T k T.
We conclude, using Definition 3.4, that for all k ≥ 1
(3.30) U 1 ⊆ T ∈T k T.
Let k ≥ 1. Recall that C k is a finite collection of w k -cubes which cover the union of tubes T in T k . Therefore, in view of (3.30), we have that C k is also a cover of U 1 . By Definition 3.1, this means
(3.31) N w k (U 1 ) ≤ |C k | for all k ≥ 1.
Fix an arbitrary p ∈ (1, 2). We complete the proof of this lemma by showing that dim M (U 1 ) ≤ p. For this fixed 1 < p < 2 we claim that the sequence |C k | w k p Q ps k is bounded, i.e. there exist H > 0 such that
(3.32) |C k |w k p Q ps k ≤ H ∀k ≥ 1.
Assume that the claim is valid. Fix an arbitrary w ∈ (0, w 1 ). There exists an integer k ≥ 1 such that w k+1 ≤ w < w k . This implies
N w (U 1 ) ≤ N w k+1 (U 1 ) so that (3.33) N w (U 1 )w p ≤ N w k+1 (U 1 )w p k = N w k+1 (U 1 )w p k+1 Q ps k+1 ≤ H.
Hence, the sequence N w (U 1 )w p is uniformly bounded from above by a fixed constant H. Since this is true for any arbitrarily small w ∈ (0, w 1 ), we conclude that dim M (U 1 ) ≤ p.
It only remains to establish the claim (3.32). We prove a more general statement, namely, that the sequence |C k |w p k Q ps k tends to zero. From (3.2), it follows that s k ≥ M k + 1 ≥ 4 for sufficiently large k. Using this, together with (3.28), (3.5) and (3.7) we obtain
|C k |w p k Q ps k |C k−1 |w p k−1 Q ps k−1 ≤ 2(M k + 1)(4s 2 k |E k |) M k Q −(p−1)s k Q p(s k −s k−1 ) ≤ s 2 k s (3+2d)M k k Q −(p−1)s k × Q p(s k −s k−1 ) ≤ Q (p−1)s k /2 × Q −(p−1)s k × Q p(s k −s k−1 ) (3.34)
for sufficiently large k. The latter inequality follows from
M k log s k s k < (p − 1) log Q 2(5 + 2d) ,
which is true for sufficiently large k. We then see that the product of the three terms in (3.34) tends to zero as k → ∞, since (3.3) implies that p(s k −s k−1 ) < (p − 1)s k /4 for k sufficiently large.
Main Result
The objective of this section is to prove Theorem 4.6 which guarantees, in every finite dimensional space, the existence of a compact universal differentiability set of upper and lower Minkowski dimension one. We should note that one cannot achieve a better Minkowski dimension as any universal differentiability set has Hausdorff dimension at least one [3, Lemma 1.5], hence Minkowski dimension of a universal differentiability set should be at least one. We also note that we will always assume d ≥ 2 as the case d = 1 is trivial, we can simply take S = [0, 1].
We will first need to establish several lemmas. The statements we prove typically concern a line l of level k, class m, category (j 1 , . . . , j m ) where 0 ≤ m ≤ M k . When m = 0, we interpret the category (j 1 , . . . , j m ) as the empty category and assume j ≤ j m for all integers j. . Let e ∈ E k and 1 ≤ j m+1 ≤ j m ≤ s k . If x ∈ l, then there exists x ′ ∈ l such that x ′ − x ≤ Q j m+1 w k /s k and
l ′ = x ′ + [−1, 1]Q j m+1 w k e ∈ L (j 1 ,...,jm,j m+1 ) (k,m+1)
Proof. We observe that by definition, the collection R l (j m+1 , e) has an element l ′ satisfying the conclusions of this lemma. , such that l ′ is parallel to l and there exists a point
x ′ ∈ l ′ with x ′ − x ≤ m×Q im w k s k .
Proof. Suppose that either (i) n = 1, or (ii) 2 ≤ n ≤ M k and the statement of Lemma 4.2 holds for integers m = 1, . . . , n − 1. We prove that in both cases, the statement of Lemma 4.2 holds for m = n. The proof will then be complete, by induction.
Let the line l, integers j 1 , . . . , j n , i n and point x ∈ l be given by the hypothesis of Lemma 4.2 when we set m = n. Let e ∈ E k be the direction of l. By (3.14) in case (i), or (3.24) in case (ii), there exists a line l (n−1) of level k, class n − 1, category (j 1 , . . . , j n−1 ) such that the line l belongs to the collection R l (n−1) (j n , e).
Let the line segment l (n−1) be parallel to f (n−1) ∈ S 1 . By Definition 3.3, the line l has the form
l = z + [−1, 1]Q jn w k e
where z ∈ l (n−1) . Therefore, we may write
(4.1) z = x + βe, where (4.2) |β| ≤ Q jn w k .
We now distinguish between two cases. First suppose that i n ≤ j n−1 . Note that this is certainly the case if n = 1. Setting i a = j a for a = 1, . . . , n − 1, we get that s k ≥ i 1 ≥ . . . ≥ i n−2 ≥ i n−1 ≥ i n . The line l (n−1) ∈ L (i 1 ,...,i n−1 ) (k,n−1)
, the direction e ∈ E k , the integer i n and the point z ∈ l (n−1) now satisfy the conditions of Lemma 4.1. Hence there is a line l ′ of level k, class n, category (i 1 , . . . , i n ) and a point z ′ with
(4.3) z ′ − z ≤ Q in w k s k ,
such that the line segment l ′ is given by
l ′ = z ′ + [−1, 1]Q in w k e.
Finally, set x ′ = z ′ − βe, so that x ′ ∈ l ′ , using (4.2). We deduce, using (4.3) and (4.1) that
x ′ − x ≤ Q in w k s k ≤ n×Q in w k s k .
This completes the proof for the case i n ≤ j n−1 . Now suppose that i n > j n−1 . In this situation, we must be in case (ii). We set i n−1 = i n > j n−1 . The conditions of Lemma 4.2 are now readily verified for z ∈ l (n−1) ∈ L (j 1 ,...,j n−1 ) (k,n−1)
, and the integer i n−1 .
Therefore, by (ii) and Lemma 4.2, there exists an integer sequence s k ≥ i 1 ≥ . . . ≥ i n−2 ≥ i n−1 and a line l ′′ ∈ L (i 1 ,...,i n−1 ) (k,n−1) such that l ′′ is parallel to l (n−1) and there exists a point y ′′ ∈ l ′′ such that
(4.4) y ′′ − z ≤ (n − 1) × Q i n−1 w k s k .
The conditions of Lemma 4.1 are now readily verified for the line l ′′ ∈ L (i 1 ,...,i n−1 ) (k,n−1)
, the direction e ∈ E k , the integer i n and the point y ′′ ∈ l ′′ .
Hence there exists a line l ′ ∈ L (i 1 ,...,in) (k,n) and a point y ′ ∈ l ′ such that
(4.5) y ′ − y ′′ ≤ Q in w k /s k .
and the line l ′ is given by
l ′ = y ′ + [−1, 1]Q in w k e.
We set x ′ = y ′ − βe.
Using (4.2) and i n > j n we get that x ′ ∈ l ′ . Moreover, using (4.1), (4.4) and (4.5), we obtain
x ′ − x ≤ n×Q in w k s k . Lemma 4.3. Let λ ∈ [0, 1), ψ ∈ 0, 1 − λ and suppose that x ∈ U λ .
Suppose that the integer n and number δ > 0 satisfy (4.6) ψQ t−1 w n < δ ≤ ψQ t w n and 1 s n ≤ ψ
where t ∈ {0, 1, . . . , s n − 1}. Let f ∈ E n and suppose that y ∈ l ∈ L (h 1 ,...,hr) (n,r)
, where (4.7) r ≤ (λ + ψ)M n − 2, h r = t + 1.
Then there exists a line l ′ ∈ L (h 1 ,...,hr,t+1) (n,1+r) and a point y ′ ∈ l such that
y ′ − y ≤ Q ψs n × δ, (4.8) l ′ = y ′ + [−1, 1]Q t+1 w n f,(4.9)
and
y ′ + [−1, 1] τ f ⊆ U λ+ψ ∩ l ′ , whenever 0 ≤ τ ≤ (Q − Q ψs n )δ − y − x . (4.10)
Proof. Choose a sequence of integers (m k ) k≥1 with 0 ≤ m k ≤ λM k , and a sequence (l k ) k≥1 of line segments such that l k ∈ L (k,m k ) is a line of level k, class m k and (4.11)
x ∈ ∞ k=1 B λw k (l k ).
Note that Qδ < ψw n Q t+1 ≤ ψw n Q sn = ψw n−1 ≤ ψw k for all k ≤ n − 1. This, together with (4.11), implies that
(4.12) B Qδ (x) ⊆ B (λ+ψ)w k (l k ) for 1 ≤ k ≤ n − 1.
Now, the line l ∈ L (h 1 ,...,hr) (n,r)
, the direction f ∈ E n , the integer t + 1 and the point y ∈ l satisfy the conditions of Lemma 4.1. Therefore, there exists a line l ′ of level n, class 1 + r, category (h 1 , . . . , h r , t + 1) and a point y ′ ∈ l ′ such that (4.9) holds and
(4.13) y ′ − y ≤ Q t w n s n = Q ψs n × ψQ t−1 w n ≤ Q ψs n × δ.
Recall that l ′ is a line of level n. Hence, from (3.8) we have that l ′ is a line of level k, class 0 for all k ≥ n + 1. We now set (4.14) l k ′ = l ′ for all k ≥ n and l k ′ = l k for 1 ≤ k ≤ n − 1.
Then for each k ≥ 1, we have that l k ′ is a line of level k class m k ′ where
m k ′ = m k if 1 ≤ k ≤ n − 1, 1 + r if k = n 0 if k ≥ n + 1.
From m k ≤ λM k and (4.7) we have that 0 ≤ m k ′ ≤ (λ + ψ)M k for all k. Hence, by Definition 3.4,
(4.15) ∞ k=1 B (λ+ψ)w k (l ′ k ) ⊆ U λ+ψ .
Suppose τ is a real number satisfying (4.10) (by (4.6) we have that ψ − 1/s n is non-negative). As ψ < 1,
0 ≤ τ ≤ Q t+1 ψ − 1 s n w n ≤ Q t+1 w n .
Hence y ′ + [−1, 1]τ f ⊆ l ′ by (4.9). From (4.12), (4.13) and (4.10) we have that for all 1 ≤ k ≤ n − 1
y ′ + [−1, 1]τ f ⊆ l ′ ∩ B Qδ (x) ⊆ l ′ ∩ B (λ+ψ)w k (l k ).
Putting this together with (4.14), we conclude that
y ′ + [−1, 1]τ f ⊆ U λ+ψ ∩ l ′ as l ′ ⊆ B (λ+ψ)w k (l ′ ) = B (λ+ψ)w k (l k ) for all k ≥ n.
The next Lemma represents the crucial step towards our main result Theorem 4.6.
Lemma 4.4. Let λ ∈ (0, 1), ψ ∈ 0, 1 − λ and η ∈ (0, 1/4). Then there exists a real number (4.16) δ 0 = δ 0 (λ, ψ, η) > 0 such that for any x ∈ U λ , e ∈ S d−1 and δ ∈ (0, δ 0 ), there exists e ′ ∈ S d−1 , integers n, t and a pair (x ′ , l ′ ), consisting of a point and a straight line segment, with x ′ ∈ l ′ ∈ L (h 1 ,...,hr) n,r , satisfying the following properties.
(i) Condition (4.6) of Lemma 4.3 is satisfied;
(ii) Condition
(4.17) r ≤ (λ + ψ)M n − 4, h r = t + 1
is satisfied (a stronger version of (4.7));
(iii) x ′ − x ≤ ηδ, e ′ − e ≤ η and (4.18) x ′ + [−1, 1]δe ′ ⊆ U λ+ψ ∩ l ′ .
Moreover, δ 0 can be chosen to be independent of Q ∈ (1, 2).
Proof. We will find δ ′ 0 = δ ′ 0 (λ, ψ, η) such that for any x ∈ U λ , e ∈ S d−1 and δ ∈ (0, δ ′ 0 ), conclusions (i), (ii) and (iii) of Lemma 4.4 are valid when (4.18) is replaced by a weaker statement (4.19) x
′ + [−1, 1] δ 2 e ′ ⊆ U λ+ψ ∩ l ′ .
Then, defining δ 0 = 1 2 δ ′ 0 (λ, ψ, η/2), we will get that the conclusion of this lemma including (4.18) is satisfied.
Since (w k ) k≥1 is strictly decreasing, and the sequences (s k ) and (M k ) satisfy s k , M k → ∞, s k /M k → 0 by (3.2), we may choose δ ′ 0 ∈ (0,
ψ 2 w 1 ) small enough so that whenever ψw k ≤ 2δ ′ 0 we have 1 s k ≤ min {η, ψ} , M k + 4 s k ≤ ηψ 4 , ψM k ≥ 6.
As Q ∈ (1, 2), this implies that whenever ψw k ≤ Qδ ′ 0 we have
(4.20) 1 s k ≤ min {η, ψ} , M k + 4 s k ≤ ηψ Q 2 , ψM k ≥ 6.
Let x ∈ U λ and fix δ ∈ (0, δ ′ 0 ). Choose a sequence of integers (m k ) k≥1 with 0 ≤ m k ≤ λM k and a sequence (l k ) k≥1 of line segments such that l k ∈ L (k,m k ) is a line of level k, class m k and
x ∈ ∞ k=1 B λw k (l k ).
Note that Qδ < ψw 1 as Q < 2. Since w k → 0, there is a unique natural number n ≥ 2 satisfying (4.21) ψw n ≤ Qδ < ψw n−1 .
Using (4.21) and w n−1 = Q sn w n , we can find t ∈ {0, 1, . . . , s n − 1} satisfying ψQ t w n ≤ Qδ < ψQ t+1 w n .
Further, from (4.20), δ ∈ (0, δ 0 ) and (4.21) we have that 1/s n ≤ ψ. Hence δ, n and t satisfy (4.6). By (3.5) there exists a direction e ′ ∈ E n such that e ′ − e ≤ 1/s n , whilst 1/s n ≤ η follows from (4.20), δ ∈ (0, δ ′ 0 ) and (4.21). Hence, we have e ′ − e ≤ η as required. Note that B wn (l n ) is a tube of level n, class m n , containing the point x. Let the line l n have category (j 1 , . . . , j mn ). We can write x = z + αg where z ∈ l n , g ∈ S d−1 and α ∈ [0, λw n ]. Next, using (3.5), pick g ′ ∈ E n such that g ′ − g ≤ 1/s n . Apply now Lemma 4.1 to z ∈ l n to find a line l ′′′ = z ′ + [−1, 1]Qw n g ′ ∈ L (j 1 ,...,jm n ,1) (k,1+mn) , where z ′ ∈ l n and z ′ − z ≤ Qw n /s n . Let x ′′′ = z ′ + αg ′ ; then
(4.22) x ′′′ − x ≤ z ′ − z + α g ′ − g ≤ 2Qw n s n ≤ 2Q ψs n × ψQ t w n ≤ 2Q 2 ψs n × δ.
From (4.20), δ ∈ (0, δ ′ 0 ) and (4.21), we have ψM n ≥ 6. In particular, m n + 2 ≤ λM n + 2 ≤ (λ + ψ)M n − 4, and (4.17) is satisfied when r = m n + 2 and h r = t + 1.
We will now show that there exists a line l ′ of level n, class 2 + m n category (j 1 , . . . , j 1+mn , t + 1), and a point x ′ ∈ l ′ such that
(4.23) x ′ − x ′′′ ≤ (m n + 2)Q 2 ψs n × δ and x ′ + [−1, 1] δ 2 e ′ ⊆ U λ+ψ ∩ l ′ .
Once (4.23) is established, the proof is completed by combining (4.23) and (4.22) to get
(4.24) x ′ − x ≤ (m n + 4)Q 2 ψs n × δ ≤ ηδ,
where the final inequality follows from (4.20), δ ∈ (0, δ ′ 0 ) and (4.21). Thus, it only remains to verify (4.23). We distinguish two cases; namely the case t = 0 and the case t ≥ 1.
If t = 0 then the conditions of Lemma 4.3 are satisfied for λ, ψ, x, δ, t, n, f = e ′ , l = l ′′′ , r = 1 + m n , (h 1 , . . . , h r ) = (j 1 , . . . , j 1+mn ) and y = x ′′′ ∈ l ′′′ . Therefore, by Lemma 4.3, there exists a line l ′ of level n, class 2 + m n category (j 1 , . . . , j 1+mn , 1) and point x ′ ∈ l ′ such that
x ′ − x ′′′ ≤ Q ψs n × δ ≤ (m n + 2)Q 2 ψs n × δ and,
(4.25)
x ′ + [−1, 1]τ e ′ ⊆ U λ+ψ ∩ l ′ whenever 0 ≤ τ ≤ (Q − Q ψs n )δ − x ′′′ − x .
(4.26) From (4.20), δ ∈ (0, δ ′ 0 ) and (4.21) we can deduce that (4.27) Q 2 ψs n ≤ η 3 and Q 2 (m n + 4) ψs n ≤ η.
Therefore, using η < 1/4, (4.27) and (4.22) we get
(Q − Q ψs n )δ − x ′′′ − x ≥ Q − η 3 − η δ ≥ δ 2 .
Hence, by (4.26) we have x ′ + [−1, 1] δ 2 e ′ ⊆ U λ+ψ ∩ l ′ , and we obtain (4.23).
Now assume that we are in the remaining case, t ≥ 1. Set i 1+mn = t + 1, so that j 1+mn = 1 < i 1+mn ≤ s n . Observe that the line l ′′′ ∈ L (j 1 ,...,j 1+mn ) (n,1+mn)
, the integer i 1+mn > j 1+mn and the point x ′′′ ∈ l ′′′ satisfy the conditions of Lemma 4.2. Therefore, by Lemma 4.2, there exists an integer sequence s k ≥ i 1 ≥ . . . ≥ i 1+mn ≥ 1 together with a line l ′′ of level n, class 1 + m n , category (i 1 , . . . , i 1+mn ) such that l ′′ is parallel to l ′′′ and there exists a point x ′′ ∈ l ′′ with (4.28)
x
′′ − x ′′′ ≤ (1 + m n ) × Q t+1 w n s n .
Set i 2+mn = t + 1, so that i 2+mn = i 1+mn . Note that the conditions of Lemma 4.3 are satisfied for λ, ψ, x, δ, t, n, f = e ′ , l = l ′′ , r = 1 + m n , (h 1 , . . . , h r ) = (i 1 , . . . , i 1+mn ) and y = x ′′ ∈ l ′′ . Hence, by Lemma 4.3, there exists a line segment l ′ of level n, class 2 + m n , category (i 1 , . . . , i 1+mn , t + 1) and a point x ′ ∈ l ′ with
x ′ − x ′′ ≤ Q ψs n × δ and, (4.29)
x ′ + [−1, 1]τ e ′ ⊆ U λ+ψ ∩ l ′ whenever 0 ≤ τ ≤ Q − Q ψs n − x ′′ − x .
(4.30)
We observe that
x ′ − x ′′′ ≤ (m n + 2)Q t+1 w n s n ≤ (m n + 2)Q 2 ψs n × δ,
using (4.29), (4.28) and (4.21). Moreover, combining (4.28) with (4.22) yields
x ′′ − x ≤ (3 + m n )Q t+1 w n s n ≤ (m n + 3)Q 2 ψs n × δ.
Therefore, by (4.27) and η < 1/4,
Q − Q ψs n δ − x ′′ − x ≥ Q − (m n + 4)Q 2 ψs n δ ≥ (Q − η)δ ≥ δ 2 .
We conclude, using (4.30) that x ′ + [−1, 1] δ 2 e ′ ⊆ U λ+ψ ∩ l ′ . We have now verified (4.23). Lemma 4.5. Let λ ∈ [0, 1), ψ ∈ (0, 1 − λ) and η ∈ (0, 1). Then there exists a number δ 1 = δ 1 (λ, ψ, η) > 0 such that whenever x ∈ U λ , δ ∈ (0, δ 1 ) and v 1 , v 2 , v 3 are in the closed unit ball in R d , there exist v 1
′ , v 2 ′ , v 3 ′ ∈ R d such that v i ′ − v i ≤ η and (4.31) [x + δv 1 ′ , x + δv 3 ′ ] ∪ [x + δv 3 ′ , x + δv 2 ′ ] ⊆ U λ+ψ . (4.32)
Moreover, δ 1 can be chosen to be independent on Q ∈ (1, 2).
Proof. Fix positive numbers a, b, c such that (4.33) a + 2b + 3c < 1 2 .
Using the notation of Lemma 4.4, choose 0 < δ 1 ≤ δ 0 (λ, ψ, aη) such that 2 ψs k ≤ bη whenever ψw k < 2δ 1
implying that (4.34) Q ψs k ≤ bη whenever ψw k < Qδ 1 as Q ∈ (1, 2). Fix x ∈ U λ , δ ∈ (0, δ 1 ) and v 1 , v 2 , v 3 in the closed unit ball in R d . We may assume that (4.35) 0 < v i ≤ c for each i = 1, 2, 3
and v 1 , v 2 , v 3 are distinct vectors. Set e 1 = v 1 / v 1 . Since δ < δ 0 (λ, ψ, aη), Lemma 4.4 asserts that there exists e ′ 1 ∈ S d−1 , integers n, t and x ′ ∈ l ′ ∈ L (h 1 ,...,hr) (n,r)
such that (4.6) and (4.17) are satisfied, together with (4.36)
x ′ − x ≤ aηδ, e 1 ′ − e 1 ≤ aη and x ′ + [−1, 1]δe ′ 1 ⊆ U λ+ψ ∩ l ′ . Denote l 1 := l ′ and set (4.37)
x 1 = x ′ + δ v 1 e 1 ′ and e 3 = (v 3 − v 1 )/ v 3 − v 1 .
Let e ′ 3 ∈ E n be such that e ′ 3 − e 3 ≤ 1/s n . We note that (4.34) implies 1/s n ≤ Q/(ψs n ) ≤ bη, as by (4.6) we have ψw n ≤ ψQ t w n < Qδ < Qδ 1 . We now apply Lemma 4.3 to point x ∈ U λ , integers n, t found above, δ satisfying (4.6), f := e ′ 3 , y := x 1 ∈ [x ′ , x ′ + δe ′ 1 ] ⊆ l 1 ∈ L (h 1 ,...,hr) (n,r)
.
Let the point y ′ ∈ l 1 and the line l ′ 1 ∈ L (h 1 ,...,hr,hr) (n,1+r) be given by the conclusion of Lemma 4.3.
We now define x ′ 1 = y ′ and note that (4.8) and (4.34) imply x ′ 1 − x 1 = y ′ − x 1 ≤ bηδ, so that using (4.35) we get x ′ 1 − x ′ ≤ (bη + c)δ.
Next,
(x ′ 3 −x)−δv 3 ≤ x ′ 3 −x 3 + (x 3 −x ′ 1 )−δ(v 3 −v 1 ) + (x ′ 1 −x)−δv 1 ≤ bηδ + δ v 3 − v 1 × e ′
3 − e 3 + (a + b + ac)ηδ ≤ (a + 2b + ac + 2bc)ηδ < ηδ
using v 3 − v 1 ≤ 2c and e ′ 3 − e 3 ≤ bη. Finally, using the definition of x ′ 2 , we get
(x ′ 2 − x) − δv 2 = (x ′ 3 − x) + δ v 2 − v 3 e ′ 2 − δv 2 ≤ (x ′ 3 − x) + δ(v 2 − v 3 ) − δv 2 + δ v 2 − v 3 × e ′ 2 − e 2 = (x ′ 3 −x)−δv 3 +δ v 2 −v 3 × e ′
2 −e 2 ≤ (a+2b+ac+4bc)ηδ < ηδ as a + 2b + ac + 4bc < 2(a + 2b + 3c) < 1.
We are now ready to prove our main result.
Theorem 4.6. For every d ≥ 1, there exists a compact subset S ⊆ R d of upper Minkowski dimension one with the universal differentiability property. Moreover if g : R d → R is Lipschitz, the set of points x ∈ S such that g is Fréchet differentiable at x is a dense subset of S.
Proof. From Lemma 4.5, we have that the family of closed sets (U λ ), λ ∈ [0, 1] satisfy the conditions of Theorem 2.3. Therefore, by Theorem 2.3, the set S = q<1 U q is a universal differentiability set. Moreover, Theorem 2.3 asserts that whenever g is a Lipschitz function on R d , the set of points x ∈ S such that g is differentiable at x is a dense subset of S. All that remains is to show that S has upper Minkowski dimension one. This follows from the observation that U 1/2 ⊆ S ⊆ U 1 , together with Lemma 3.5.
of tubes of level k, class 1, category (j) is defined by (3.20) T (j) (k,1) = B w k (l) : l ∈ L
m) , C (k,m) and T (k,m) of lines, cubes and tubes of level k, class m. Assume that these collections are partitioned into categories L (j 1 ,...,jm) (k,m) , C (j 1 ,...,jm) (k,m) and T (j 1 ,...,jm) (k,m)where the j i are integers satisfying(3.23) 1 ≤ j i+1 ≤ j i ≤ s k for all i.Suppose that the following conditions hold.length(l) = 2Q jm w k for all lines l in L (j 1 ,...,jm) (k,m)
( 3 .
324) L (j 1 ,...,j m+1 ) (k,m+1) = l∈L (j 1 ,...,jm) (k,m) e∈E k R l (j m+1 , e) .
Lemma 3 . 5 .
35For λ ∈ [0, 1], the set U λ has upper Minkowski dimension one.
Lemma 4. 1 .
1Let k ≥ 2, 0 ≤ m < M k and and l ∈ L (j 1 ,...,jm) (k,m)
Lemma 4 . 2 .
42Let k ≥ 2 and suppose 1 ≤ m ≤ M k . Let x ∈ l ∈ L (j 1 ,...,jm) (k,m) and i m be an integer with j m < i m ≤ s k . Then there exists an integer sequence s k ≥ i 1 ≥ . . . ≥ i m−1 ≥ i m and a line
As in Definition 1.1, we define the lower Minkowski dimension of A by (1.1) and the upper Minkowski dimension of A by (1.2).
We claim that the straight line segment [x ′ 1 − 1 2 δe ′ 3 , x ′ 1 + 1 2 δe ′ 3 ] is inside U λ+ψ . Indeed, we verify that τ = δ/2 satisfies (4.10). Using(4.38)xand Q > 1, together with (4.33) and 0 < η < 1, we getLet2 − e 2 ≤ 1 sn and apply Lemma 4.3 to point x ∈ U λ , n, t and δ satisfying (4.6) and found earlier,. We note that the condition (4.7) of Lemma 4.3 is satisfied for r + 1 instead of r because of (4.17). Let the point y ′ ∈ l 3 and the line l 2 ∈ L (h 1 ,...,hr,hr,hr) (n,2+r) be given by the conclusion of Lemma 4.3.We again show that τ = δ/2 satisfies (4.10). Indeed using (4.38) we getHence, using (4.33) and 0 < η < 1, we concludeFinally, defineWe are now left to see that v ′ i , i = 1, 2, 3 defined according to(4.39)xFirst we see that x ′ 3 ∈ l 3 and by (4.33)ηδ for all i = 1, 2, 3. We note first that using (4.33)1 − e 1 + (a + b)ηδ ≤ (a + b + ac)ηδ < ηδ.
Geometric Nonlinear Functional Analysis. Y Benyamini, J Lindenstrauss, American Mathematical Society48Y. Benyamini and J. Lindenstrauss. Geometric Nonlinear Functional Analysis, volume 48. American Mathematical Society, 2000.
A compact null set containing a differentiability point of every Lipschitz function. M Doré, O Maleva, Mathematische Annalen. 3513M. Doré and O. Maleva. A compact null set containing a differentiability point of every Lipschitz function. Mathematische Annalen, 351(3):633-663, 2009.
A compact universal differentiability set with Hausdorff dimension one. M Doré, O Maleva, Israel Journal of Mathematics. M. Doré and O. Maleva. A compact universal differentiability set with Haus- dorff dimension one. Israel Journal of Mathematics, pages 1-12, 2010.
A universal differentiability set in Banach spaces with separable dual. M Doré, O Maleva, Journal of Functional Analysis. 2616M. Doré and O. Maleva. A universal differentiability set in Banach spaces with separable dual. Journal of Functional Analysis, 261(6):1674-1710, 2011.
K Falconer, Fractal Geometry: Mathematical Foundations and Applications. WileyK. Falconer. Fractal Geometry: Mathematical Foundations and Applications. Wiley, 2003.
Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces. D Lindenstrauss, J Preiss, Tišer, Princeton University PressJ Lindenstrauss, D. Preiss, and J. Tišer. Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces. Princeton University Press, 2012.
Geometry of sets and measures in Euclidean spaces: Fractals and Rectifiability. P Mattila, Cambridge University Press44P. Mattila. Geometry of sets and measures in Euclidean spaces: Fractals and Rectifiability, volume 44. Cambridge University Press, 1999.
Differentiability of Lipschitz functions on Banach spaces. D Preiss, Journal of Functional Analysis. 912D. Preiss. Differentiability of Lipschitz functions on Banach spaces. Journal of Functional Analysis, 91(2):312-345, 1990.
Sur l'ensemble des points de non-derivabilite d'une fonction continue. Z Zahorski, Bull. Soc. Math. France1946Z. Zahorski. Sur l'ensemble des points de non-derivabilite d'une fonction con- tinue. Bull. Soc. Math. France, (74), 1946.
Sets of σ-porosity and sets of σ-porosity (q). Z Zajíček, Časopis pro pěstováni matematikyZ. Zajíček. Sets of σ-porosity and sets of σ-porosity (q).Časopis pro pěstováni matematiky, 1976.
Birmingham B15 2TT UK E-mail address: [email protected] School of Mathematics. [email protected] B15 2TT UK E-mailSchool of Mathematics, University of Birmingham ; University of BirminghamSchool of Mathematics, University of Birmingham, Birmingham B15 2TT UK E-mail address: [email protected] School of Mathematics, University of Birmingham, Birmingham B15 2TT UK E-mail address: [email protected]
| [] |
[
"ON THE NUMBER OF K-SKIP-N-GRAMS",
"ON THE NUMBER OF K-SKIP-N-GRAMS"
] | [
"Dmytro Krasnoshtan [email protected] "
] | [] | [] | The paper proves that the number of k-skip-n-grams for a corpus of size L iswhere k ′ = min(L − n + 1, k). | null | [
"https://arxiv.org/pdf/1905.05407v1.pdf"
] | 153,312,620 | 1905.05407 | ff156cfff059005318bcfabe6d4d74814d062893 |
ON THE NUMBER OF K-SKIP-N-GRAMS
14 May 2019
Dmytro Krasnoshtan [email protected]
ON THE NUMBER OF K-SKIP-N-GRAMS
14 May 2019NLP, skip-grams
The paper proves that the number of k-skip-n-grams for a corpus of size L iswhere k ′ = min(L − n + 1, k).
Introduction
Skip-gram [1] is a popular technique used in natural language processing, where in addition to sequences of words, we allow to substitute a word with a skip token. The model is used to overcome the data sparsity problem and provides an efficient method for learning high-quality vector representations for phrases.
Guthrie et al. further investigated the use of skip-grams by introducing k-skip-n-grams [2] and empirically shown that they can be more effective than increasing the size of the training corpus. In their paper, they also provided the following formula for calculating the number of k-skip-trigrams (n = 3) for a corpus of size L:
(k + 1)(k + 2) 6 (3L − 2k − 6)
The purpose of this paper is to derive the general case formula for arbitrary L, n, and k.
Proof
The proof of the general formula can be derived from the algorithm of constructing the k-skip-n-grams. There are a few recursive algorithms to construct them, but the one that makes the counting easier relies on the following intuition:
The number of k-skip-n-grams is equal to the sum of the number of n-grams with 0 skips plus the number of n-grams with exactly 1 skip plus the number of n-grams with exactly 2 skips plus so on till the number of n-grams with exactly k skips. So if we number of n-grams with exactly k skips is f (L, n, k), then the total number of all k-skip-n-grams is k i=0 f (L, n, i).
To derive the formula for f , let's see how we can generate an n-gram with exactly k skips. One can notice that generating n-grams with k skips is equivalent of selecting a sequence of length n + k and substituting any k element with skips. It is important to realize is that you can't substitute the first or the last element, as this n-gram will be equivalent to
• (k-1)-skip-n-gram if you substitute only one (first or last) element with a skip • (k-2)-skip-n-gram if you substitute both (first and last) elements with a skip So we need to choose k substitutions from n + k − 2 positions which can be done in n+k−2 k different ways. Because we can generate L − n − k + 1 (should be > 0) different substrings of length n + k from the corpus of size L, the total number of n-grams with exactly k skips is
f (L, n, k) = n + k − 2 k · (L − n − k + 1) = n + k − 2 n − 2 · (L − n − k + 1)
Therefore the total formula for k-skip-n-grams is
A = k i=0 n + i − 2 n − 2 · (L − n − i + 1)
This expression can be simplified using the following identities:
A = k i=0 n − 2 + i n − 2 · (L − n + 1) − k i=0 i n − 2 + i n − 2 = = n − 1 + k n − 1 · (L − n + 1) − k(k + 1) n n − 1 + k n − 2 = = n − 1 + k n − 1 · (L − n + 1) − k(n − 1) n · n − 1 + k n − 1 = = Ln + n + k − n 2 − kn n · n − 1 + k n − 1
The formula is almost complete apart of a few corner cases. If n = 0, we do not select any n-grams and the result should be zero. Previously it was also mentioned that L − n − k + 1 > 0, which is the same as k = min(L − n + 1, k)
Additional materials
The code and verification for the formula are available at https://github.com/salvador-dali/k-skip-n-gram
Efficient estimation of word representations in vector space. Tomas Mikolov, Kai Chen, Greg Corrado, Jeffrey Dean, ICLR Workshop. Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. ICLR Workshop, 2013.
A Closer Look at Skip-gram Modelling. David Guthrie, Ben Allison, Wei Liu, Louise Guthrie, Yorick Wilk, Proceedings of the Fifth International Conference on Language Resources and Evaluation (LREC). the Fifth International Conference on Language Resources and Evaluation (LREC)David Guthrie, Ben Allison, Wei Liu, Louise Guthrie and Yorick Wilk. A Closer Look at Skip-gram Modelling. Proceedings of the Fifth International Conference on Language Resources and Evaluation (LREC), 2016.
| [
"https://github.com/salvador-dali/k-skip-n-gram"
] |
[
"Managing Volatility Risk: An Application of Karhunen-Loève Decomposition and Filtered Historical Simulation",
"Managing Volatility Risk: An Application of Karhunen-Loève Decomposition and Filtered Historical Simulation"
] | [
"Jinglun Yao ",
"Sabine Laurent ",
"Brice Bénaben "
] | [] | [] | Implied volatilities form a well-known structure of smile or surface which accommodates the Bachelier model and observed market prices of interest rate options. For the swaptions that we study, three parameters are taken into account for indexing the implied volatilities and form a "volatility cube": strike (or moneyness), time to maturity of the option contract, duration of the underlying swap contract. It should be noted that the implied volatility structure changes across time, which makes it important to study its dynamics in order to well manage the volatility risk. As volatilities are correlated across the cube, it is preferable to decompose the dynamics on orthogonal principal components, which is the idea of Karhunen-Loève decomposition that we have adopted in the article. The projections on principal components are investigated by Filtered Historical Simulation in order to predict the Value at Risk (VaR), which is then examined by standard tests and non-arbitrage condition to ensure its appropriateness. | null | null | 55,873,266 | 1710.00859 | 1cbc7a8e6307a5b7e2eb545545f48cde65ecef6f |
Managing Volatility Risk: An Application of Karhunen-Loève Decomposition and Filtered Historical Simulation
October 4, 2017 2 Oct 2017
Jinglun Yao
Sabine Laurent
Brice Bénaben
Managing Volatility Risk: An Application of Karhunen-Loève Decomposition and Filtered Historical Simulation
October 4, 2017 2 Oct 2017Bachelier modelimplied volatility smiles and surfacesswaptionKarhunen-Loève decompositionfiltered historical simulationvalue at risk (VaR)volatility risk 1
Implied volatilities form a well-known structure of smile or surface which accommodates the Bachelier model and observed market prices of interest rate options. For the swaptions that we study, three parameters are taken into account for indexing the implied volatilities and form a "volatility cube": strike (or moneyness), time to maturity of the option contract, duration of the underlying swap contract. It should be noted that the implied volatility structure changes across time, which makes it important to study its dynamics in order to well manage the volatility risk. As volatilities are correlated across the cube, it is preferable to decompose the dynamics on orthogonal principal components, which is the idea of Karhunen-Loève decomposition that we have adopted in the article. The projections on principal components are investigated by Filtered Historical Simulation in order to predict the Value at Risk (VaR), which is then examined by standard tests and non-arbitrage condition to ensure its appropriateness.
Introduction
Since the liquidity trap of the eurozone and negative interest rates associated with it, Bachelier model (Bachelier [1900]) has become more appropriate than Black-Scholes model (Black and Scholes [1973]) for interest rate products because of its ability to handle negative interest rates. While volatility of the underlying asset in the most basic Bachelier model is constant, in practice we should calibrate volatilities for different derivative products (e.g. call options with different strikes) at time t, leading to what we call "volatility smile" or "volatility surface". It is preferable and heuristic to assume the smile to be static and invariant across time t, which is the case when practitioners adopt the so-called "sticky moneyness" or "sticky strike" rule. Unfortunately, however, according to Rosenberg [2000] and Cont et al. [2002], we observe in reality rather continuous change in a smile. This dynamics of volatility smiles or surfaces is important for risk management since it would affect the volatility parameter in the Bachelier pricing function, resulting in the change of values of interest rate derivatives. The bank's portfolio is thus influenced and suffers from Value at Risk (VaR). More particularly, this is what we call "Vega Risk" in the terminology of risk management since the risk factor is volatility. In our study, we are particularly interested in swaptions, which have 3 parameters for Bachelier implied volatility σ B t : strike (or moneyness), time to maturity of the option contract (hereafter "expiry"), duration of the underlying interest rate swap contract (hereafter "tenor").
Despite the obvious importance of modeling the dynamics, it is not easy to accomplish this task. The dynamics of smiles concerns not only at-the-money (ATM) volatilities but also non-ATM ones. For the series of ATM volatilities, we might well model it using traditional time series models. Yet it is not appropriate to model non-ATM volatilities separately from the ATM ones since there exists clearly a correlation between ATM and non-ATM volatilities. The smile, though not a rigid body with only three degrees of freedom, is not a "soft body" with infinitely many degrees of freedom, either. The stickiness of volatilities across different strikes requires the study of smile dynamics in a holistic fashion. Some major moves are more dominant than others and it is reasonable to use them as a concise description of the dynamics. For example, parallel shift of the smile is a common phenomenon. Nonetheless, these major moves are not known a priori, which should be estimated from data instead.
It should be noted that we have resisted the temptation of using a local volatility model (Dupire [1997]) or a stochastic volatility model (e.g. SABR model in Hagan et al. [2002]). These models, by introducing an infinitesimal description of volatility, adds more degrees of freedom for the volatility and explains the empirical derivation from Bachelier model with constant volatility. Implied volatility, by contrast, is a state variable which accommodates Bachelier model and market prices. However, similar to the dynamics of implied volatility smiles, we cannot assume the parameters in local or stochastic volatility models to be constant. In fact, as Hagan et al. [2002] remarked, the dynamics of the market smile predicted by local volatility models is opposite of observed market behavior: when the price of the underlying decreases, local volatility models predict that the smile shifts to higher prices; when the price decreases, these models predict that the smile shifts to lower prices. In reality, asset prices and market smiles move in the same direction. In consequence, frequent recalibration is needed in order to ensure the correctness of the local volatility model and model parameters bear a dynamics. This is also the case for SABR model in practice, even if asset prices and market smiles move in the same direction. The dynamics of model parameters in a local or stochastic volatility model is of course a reflection of the smile dynamics. But it should be noted that the relationship between them is not clear-cut: a calibration method is needed for estimating model parameters since there are less parameters in say, SABR model, than the degree of freedom of volatility smile. This implies that we should establish an objective function of model parameters to optimize. Yet it is difficult to tell which calibration method is "best " or "better" since none of these methods would reproduce the empirically observed implied volatilities. What's more, even if we admit a calibration method, whether it is appropriate to use the time series of calibrated parameters for risk management remains to be a question. Because "real" realized values are needed in order to evaluate the performance of VaR prediction, yet the "real" values of model parameters are not directly (or indirectly) observable in the markets.
A second and deeper reason for using a model-free approach (in the sense that we do not use a local or stochastic volatility model, but we have of course used the Bachelier model) is explained by Cont et al. [2002]: option markets have become increasingly autonomous and option prices are driven, in addition to movements in the underlying asset, also by internal supply and demand in the options market. For example, Bakshi et al. [2000] has documented evidence of violations of quantitative dynamical relations between options and their underlying. In fact, practitioners would resort to supply-demand equilibrium and implied volatilities instead of stochastic volatility models for liquid assets. This is the case for vanilla swaptions in our studies. Despite these arguments favoring a model-free approach, it should be remarked that stochastic volatility models provide an essential way of evaluating price and managing risk for less liquid assets which are out of the scope of this article.
Facing the challenges of modeling smile dynamics, we have adopted the Karhunen-Loève decomposition which can be seen as a generalized version of Principal Component Analysis conceived for functions. The method was proposed in Loeve [1978] and adopted by Cont et al. [2002] for studying volatility surface dynamics of stock index options. We have extended the usage of this method to swaptions and explored the implications for risk management. Since swaptions have three parameters (moneyness, expiry, tenor) for implied volatilities, it would be interesting to explore the dynamics for each dimension. Moreover, Karhunen-Loève decomposition can be applied to multivariate functions, making it possible to explore the dynamics of surfaces. In this way, the dynamics of implied volatility smiles or surfaces can be characterized by several principal components and the time series of projections on these components. The time series of projected values can then be studied by Historical Simulation or Filtered Historical Simulation in order to evaluate VaR. Historical Simulation has been overwhelmingly popular in assessing VaR in recent years because of its non-parametric approach. And Filtered Historical Simulation, based on Historical Simulation and proposed by Barone-Adesi et al. [1999], has overcome the problem of volatility clustering illustrated in Cont [2001].
The article is organized in the following way: Section 2 gives an introduction to the theory of Karhunen-Loève decomposition, which is then applied to implied volatility smiles and surfaces of swaptions in section 3. Empirical evidence and intuitions for different dimensions are also explored. Section 4 studies the properties of projected time series and uses Filtered Historical Simulation for evaluating VaR of volatility increments. Note that the most important (and perhaps the only) hypothesis of derivative pricing is non-arbitrage. Sections 5 checks if the estimated VaR violates this fundamental assumption. Moreover, Section 6 checks if the estimated VaR meets some standard backtesting criteria. Section 7 concludes the article.
Karhunen-Loève Decomposition: Theoretical Background
The Karhunen-Loève decomposition is a generalization of Principal Component Analysis conceived for functions. Let D ⊂ R n be a bounded domain. For example, in the case where we want to study the dynamics of smiles (as a function of moneyness), D could be [m min , m max ], where m min and m max are respectively the smallest and largest moneyness. Then we can define the integral operator K on L 2 (D), K : u → Ku for u ∈ L 2 (D), by:
[Ku](x) = D k(x, y)u(y) dy(1)
It can be shown that if k :
D × D → R is a Hilbert-Schmidt kernel, i.e. D D |k(x, y)| 2 dx dy < ∞,(2)
then K is a compact operator. Furthermore, if k(x, y) = k(y, x) ∀x, y ∈ D, then K is a self-adjoint operator. More particularly, K and k are defined in the way presented below for Karhunen-Loève decomposition so that they satisfy these properties.
In order to get a series of smiles, we add another parameter ω ∈ Ω to u, i.e., u : Ω × D → R. We shall not distinguish between ω and t for notation since in data, each t represents a realization of ω. Intuitively, for each ω fixed, u(ω, ·) is for example a volatility smile (more precisely the log-return of the smile, for reason explained below), while for each x fixed, u(·, x) can be seen as a random variable. So we can calculate the covariance between u(·, x) and u(·, y) and define k(x, y) as follows:
k(x, y) = cov(u(·, x), u(·, y))
Then K is a compact and self-adjoint operator. Remark that the kernel function k is analogous to the covariance matrix of a random vector in a conventional Principal Component Analysis. It can also be shown that K is positive, so according to Mercer's theorem (Mercer [1909]), k(x, y) can be represented by:
k(x, y) = i λ i e i (x)e i (y)(4)
where {λ i } i and {e i } i are eigenvalues and eigenvectors of K, i.e.
K(e i ) = D k(·, y)e i (y) dy = λ i e i(5)
Without loss of generality, assume that λ 1 ≥ λ 2 ≥ · · · ≥ 0. Let u i (ω) = u(ω, ·), e i be the projection of a smile or surface u(ω, ·) on the eigenfunction e i . Then
u(ω, ·) = i u i (ω)e i (·) and u i (ω) = D u(ω, x)e i (x) dx(6)
Equation 6 is what we call Karhunen-Loève decomposition. It is important to note that if u(·, x) is a centered variable for each x, then u i (·) is also centered. To see this, it suffices to take expectation on Equation 6. Moreover, {u i (·)} i are mutually uncorrelated, since
E(u i u j ) = E[ D u(·, x)e i (x) dx D u(·, y)e j (y) dy] (7) = E[ D D u(·, x)u(·, y)e i (x)e j (y) dx dy] (8) = D D E[u(·, x)u(·, y)]e i (x)e j (y) dx dy (9) = D D k(x, y)e i (x)e j (y) dx dy (10) = D [Ke i ](y)e j (y) dy (11) = Ke i , e j (12) = λ i e i , e j (13) = λ i δ ij(14)
where δ ij is Kronecker's symbol. This can help us study separately the properties of projection times series without worrying about the correlations between them. This also explains the following notation often used in practice which is equivalent to Equation 6:
u(ω, x) = i λ i ξ i (ω)e i (x)(15)
where ξ i are centered mutually uncorrelated random variables with unit variance and are given by:
ξ i (w) = 1 √ λ i D u(ω, x)e i (x) dx(16)
We shall stick with this notation in the following sections.
How to solve the problem numerically? Suppose we have observed a random field {u(t, ·)} t , we can first of all use the empirical covariance to estimate k(x, y), i.e.
k(x, y) = 1 T T t=1 [u(t, x) −ū(x)][u(t, y) −ū(y)](17)= 1 T T t=1 [u(t, x) − 1 T T t=1 u(t, x)][u(t, y) − 1 T T t=1 u(t, y)](18)
For solving the problem of eigenvalues and eigenfunctions in Equation 5, we can reduce it to a finite-dimensional problem by using a Galerkin scheme, i.e. using a linear combination of basis functions for approximating each eigenfunction. For example, we can choose Legendre functions for the basis functions. Approximately, we have:
e i (x) = N n=1 d (i) n φ n (x) = Φ T (x)D (i)(19)
where {φ n } n are basis functions, Φ(x) is the vector of basis functions, D (i) is the vector of coefficients of shape N × 1 to be estimated. Combining Equation 19 and Equation 5, we get:
N n=1 d (i) n D k(x, y)φ n (y) dy = λ i N n=1 d (i) n φ n (x)(20)
Multiplying both sides by φ m (x) and integrating on D, we get:
N n=1 d (i) n D D k(x, y)φ m (x)φ n (y) dx dy = λ i N n=1 d (i) n D φ m (x)φ n (x) dx(21)
Or
AD (i) = λ i BD (i)(22)
where A and B are positive symmetric and defined in the following way:
A mn = D D k(x, y)φ m (x)φ n (y) dx dy (23) B mn = D φ m (x)φ n (x) dx(24)
Numerically solving the problem 22 will yield the eigenvalues and eigenfunctions of operator K.
Karhunen-Loève Decomposition: Application on Swaption Implied Volatilities
The structure of implied volatilities is important for accommodating Bachelier model and market price of options. For vanilla swaptions, in addition to the two parameters (strike or moneyness, time to maturity) of any European option, the duration of the underlying swap contract ("tenor") is also a crucial parameter. In the article, moneyness is defined as
moneyness = strike − f orward rate(25)
Keeping expiry and tenor fixed, we can study the dynamics of moneyness-indexed smile using Karhunen-Loève decomposition. Studying the dynamics of the whole smile is important since it helps us to manage volatility risk related to skewness and convexity of the smile, in addition to the evolution of ATM volatilities which is the most important but not sufficient to fully characterize the smile dynamics. As the bank has not only ATM options but also non-ATM options, managing volatility risk related to non-ATM volatilities is crucial and requires the study of dynamics of the whole smile. The same is true for expiry-indexed smiles or tenor-indexed smiles, keeping the other two parameters fixed. It should be noted that implied volatilities {I(ω, x)} ω∈Ω is obviously not centered, so we take the log-return of volatilities in order to apply the Karhunen-Loève decomposition, i.e.
u(t, x) = log(I(t, x)) − log(I(t − 1, x))(26)
The data we use are implied volatilities for USD dollars, from 2007 to 2017. Eigenfunctions λ 1 = 3. 63e − 04, 89.62% explained λ 2 = 3. 69e − 05, 9.10% explained λ 3 = 4. 69e − 06, 1.16% explained Figure 1: First three eigenfunctions and eigenvalues of Karhunen-Loève decomposition for moneyness-indexed smile log-return. expiry=10Y, tenor=10Y, currency=USD. The x-axis is moneyness and has been multiplied by 100, i.e. x = 2 means strike = f orward rate + 2% Figure 1 illustrates the result of Karhunen-Loève decomposition for moneyness-indexed smile of swaption with expiry=10Y and tenor=10Y (hereafter "10Y 10Y swaption"). Remark that the first eigenfunction, which can be interpreted as parallel shift of the smile, explains 89.62% of the dynamics. The second eigenfunction explains most of the remaining dynamics and can be interpreted as skew change (rotation). The third eigenfunction, whose influence is more debatable, can be understood as the change of convexity. In fact, the same understanding can be gained in Figure 2 for tenor-indexed smile.
These intuitive interpretations are, however, no longer valid for expiry-indexed smile. Figure 3 illustrates the eigenfunctions and eigenvalues for log-return of expiry-indexed smile. While the first eigenfunction can still be interpreted as parallel shift, the second one, which accounts for nearly 10% of the dynamics, has hardly any interpretation. This difference between expiry-indexed smile and tenor-indexed smile is consistent with practitioners' experience that tenor-indexed smile is more "rigid" than expiry-indexed smile. Because when tenor is fixed, different expiries would incorporate different levels of uncertainty in the option contract. The implied volatilities across different expiries are loosely related and exhibit less regularities.
Karhunen-Loève decomposition can also be applied on series of multivariate functions. Figure 4 illustrates the first three eigenfunctions for log-return of expiry-and-tenor-indexed volatility surfaces. The first eigenfunction, even though not quite flat, is always positive and can be interpreted as parallel shift. The second eigenfunction reflects the rotation across the expiry since it monotonously crosses 0, but parallel shift across the tenor. One important remark is that marginal functions of Eigenfunctions λ 1 = 3. 08e − 03, 98.26% explained λ 2 = 3. 53e − 05, 1.13% explained λ 3 = 8. 23e − 06, 0.26% explained the two-dimensional eigenfunctions in Figure 4 do not correspond exactly to the one-dimensional eigenfunctions in Figure 2 and 3. This should be due to the fact that some volatilities on the volatility surface are significantly correlated. For example, the implied volatility of 10Y 10Y swaption should be intimately linked to that of 9Y 11Y swaption because of the existence of Bermudan swaptions, which are the most actively traded exotic swaptions. For a Bermudan swaption contract which will end in 20 years, the holder of the contract may be able to exercise the contract in 9 years or 10 years. So the implied volatilities of 10Y 10Y swaption is in this way linked to the implied volatilities of 9Y 11Y swaption. It should also be noted that in Figure 3 and 2, eigenvalues for expiry-indexed smiles are larger than those for tenor-indexed smiles, given the same domain of definition [0,30]. So the movements related to expiry should be more significant than those related to tenor in Figure 4. This might explain the second eigenfunction in Figure 4 in which the marginal function is approximately invariant for a fixed expiry. We shall focus on moneyness-indexed smiles in the following sections for studying the implications on risk management. The projections of their log-returns on the first three components in Figure 1 are shown in Figure 5. These are the times series of ξ i defined in Equation 15. Table 1 shows that they are in fact mutually uncorrelated.
Filtered Historical Simulation for Projected Time Series
Value at Risk (VaR), originally an internal risk management measure of JPMorgan, has become the benchmark of the banking system for evaluating market risk. Despite the critics of not providing the expected shortfall (Dowd [1998]) or not being subadditive (Artzner et al. [1999]), VaR is an intuitive and easy-to-calculate measure summarizing important risk informations. Market risk comes from different risk factors and the risk factor that we are here concerned with is volatility. But implied volatility is a smile, a surface or in the case of swaption, a cube, which renders the study of its dynamics difficult. As we have shown in the previous sections, Karhunen-Loève decomposition can be adopted to have a concise and accurate description of the dynamics. At most three eigenvectors and the projections on these eigenvectors are needed to characterize the dynamics. Hence the volatility risk factor can be resumed to three mutually uncorrelated process which can be studied separately to evaluate the VaR.
The most widely used approach for computing VaR is historical simulation. For a given time series {r t } t (for example {ξ i (t)} t ), we want to calculate the VaR(α), i.e. the α-th quantile of the distribution F of the random process. Unlike a parametric approach which assumes a priori a parametric distribution F and estimates the distribution parameters, historical simulation uses directly the Eigenfunction #2 (λ 2 = 1. 59e − 02, 11.65% explained) empirical distribution based on a rolling window. For example, for calculating VaR(α) at time t, we generate first of all the empirical cumulative distribution functionF by using L most recent historical observations r t−L , r t−L+1 , · · · , r t−1 , where L is the length of the rolling window. Then V aR t (α) =F −1 (α). Interpolations are often needed sinceF is a step function.
Yet as Cont [2001] points out, financial return time series often exhibits volatility clustering phenomenon, i.e. conditional heteroscedasticity. For a more general time series, conditional mean structure is also very common, for example in an autoregressive model. It should be noted that both conditional mean structure and conditional heteroscedasticity structure are compatible with the stationarity of the time series. The later one is usually assumed for time series under study, except for special cases where a unit root can be detected. Despite this compatibility, the existence of conditional mean or conditional heteroscedasticity would render a Historical Simulation less credible, since even though the observations r t−L , r t−L+1 , · · · , r t−1 have the same distribution in the unconditional sense, they come from conditionally different distributions. For example, if conditional volatility at time t − 1 is larger than that at time t − L, then using them simultaneously for estimatingF t is not quite reasonable. This is the reason why Filtered Historical Simulation has been proposed.
Let's first of all consider the conditional mean structure. For ξ 1 , whose time series is shown in the first panel of Figure 5, its autocorrelation function (ACF) and partial autocorrelation function (PACF) are shown in Figure 6. Looking at the PACF, we can model ξ 1 by AR(1), because of the following theorem proved in Roueff [2016].
Theorem 1 Let X be a centered weakly stationary process with partial autocorrelation function κ. Then X is an AR(p) process if and only if κ(m) = 0 for all m > p.
Fitting ξ 1 with AR(1), we get the following model:
ξ 1 (t) = β 1 ξ 1 (t − 1) + 1 (t) with β 1 = 0.179634(27)
The time series, as well as ACF and PACF of residues { 1 (t)} t are shown in Figure 7. While the ACF and PACF resemble those of a white noise, the time series clearly have a volatility clustering effect. In fact, taking absolute value of the residues, and recalculating ACF and PACF, we get Figure 8. Thus 1 can be modeled as a weakly white noise but not an IID one. It exhibits conditional heteroscedasticity structure which should be removed before estimating VaR.
How to estimate conditional volatilities of 1 ? The simplest approach would be the standard deviation based on a rolling window, which unfortunately works very badly. More complex considerations such as GARCH(p, q) model can surely accomplish this task. Here we have adopted a parsimonious approach called Exponentially Weighted Moving Average (EWMA). More specifically, for a centered process X, we can estimated its conditional volatility by using the following recursive formula:
σ 2 (t) = θσ 2 (t − 1) + (1 − θ)X 2 (t − 1)(28)
where θ ∈ (0, 1). The method is called "exponential" because Equation 28 can be applied recursively to get the following equation:
σ 2 (t) = (1 − θ) W i=1 θ i−1 X 2 (t − i) + θ W σ 2 (t − W )(29)
where W is the window length for estimating conditional volatilities. In our study, W = 60, θ = 0.9 so θ W σ 2 (t − W ) can be neglected. Calculating EWMA volatilities σ 1 of residues 1 and devolatising 1 by σ 1 , we get the times series { 1 (t) σ 1 (t) } t illustrated in Figure 9. The ACF and PACF of | 1 σ 1 | are shown in Figure 10. The volatility clustering phenomenon has been completely removed! In summary, we have firstly removed the conditional mean structure by applying AR(1) on projected time series ξ 1 to get the residues 1 , then the conditional heteroscedasticity structure by descaling the residues 1 by conditional volatilities σ 1 . Now Historical Simulation can be safely applied to 1 σ 1 . In our study, the 1st and the 99th rolling quantiles are estimated, which are then "revolatised" by multiplying σ 1 to get the 1st and 99th rolling quantiles of residues 1 . Denote the 1st and 99th quantile respectively
1 (t) = σ 1 (t)F −1 1,t (α)(30)
where σ 1 is estimated conditional volatility by using EWMA method,F 1,t is the rolling empirical cumulative distribution defined by:F
1,t (x) = 1 L L l=1 1 x≤ 1 (t−l) σ 1 (t−l)(31)
The comparison ofˆ Figure 11. It can be seen thatˆ
ξ (α) 1 (t) = β 1 ξ 1 (t − 1) +ˆ (α) 1 (t)(32)
where β 1 = 0.179634, and an extreme parallel shift of smile log-return is √ λ 1ξ (α) 1 (t)e 1 (x). Consequently, the predicted extreme smile iŝ
I (α) 1 (t, x) = I(t − 1, x) + exp( λ 1ξ (α) 1 (t)e 1 (x))(33)
Note we have used the subscript 1 to emphasize the fact that the predicted smile is only related to the move along the first principal component. Denote C(f t , κ, σ) the pricing function of a payer swaption at time t whose option contract expires at t + T . f t is the forward rate of the fixed leg of the underlying swap contract. Then VaR of the swaption at time t resulting from parallel smile shift would be:
C(f t , κ,Î (α) 1 (t, κ − f t )) − C(f t−1 , κ, I(t, κ − f t−1 ))(34)
What if we want to calculate the VaR resulting from the combined move of the first several principal components? Repeating the procedure described in this section,ξ
(α) 2 andξ (α) 3
can be estimated in a similar way 1 Hence according to Equation 15, the quantile of smile log-return can be calculated by:
u (α) (t, x) = 3 i=1 λ iξ (α) i (t)e i (x)(35)
Here we have used only the first 3 eigenvectors as they explain nearly 100% of the dynamics. But there is a caveat here: the VaR of a 3-dimensional vector, which is the case ofû (α) (t, x), is not well defined. For example, in the case of moneyness-indexed smile, if we take α = 0.01 and look at
I (α) (t, x) = I(t − 1, x) + exp(û (α) (t, x))(36)
Consequently the VaR of the swaption at time t resulting from smile move would be:
C(f t , κ,Î (α) (t, κ − f t )) − C(f t−1 , κ, I(t, κ − f t−1 ))(37)
Do Reconstructed Volatility Smiles Violate Non-Arbitrage Condition?
As it is well-known, non-arbitrage condition is the most basic assumption in derivative pricing. It has direct implications on the prices of options, even in model-free conditions. In the case of a payer swaption, whose option contract is a call option, the price of the swaption should be decreasing and convex as a function of the strike. To demonstrate the idea, it suffices to investigate a simple European call option Call (S 0 , κ, σ). It should satisfy the following conditions:
∂Call(S 0 , κ, σ) ∂K ≤ 0 (38) ∂ 2 Call(S 0 , κ, σ) ∂K 2 ≥ 0(39)
The first equation is straightforward if we look at the original pricing function Call(S 0 , κ, σ) = E(S T − κ) + . For the second one, imagine a portfolio X consisting of a long position of λ calls with strike κ 1 , a long position of 1 − λ calls with strike κ 2 , and a short position of a call with strike λκ 1 + (1 − λ)κ 2 . According to the convexity of the function x + and using Jensen's inequality, we have:
λ(S T − κ 1 ) + + (1 − λ)(S T − κ 2 ) + − (S T − λκ 1 − (1 − λ)κ 2 ) + ≥ 0(40)
This means that the terminal value of the portfolio at maturity T is non-negative. Hence taking the expectation, the price of the portfolio at time 0 should also be non-negative. Because λ can be arbitrary, we get the convexity of Call with respect to κ.
DoesÎ (α) (t, x) constructed in the previous section violate the non-arbitrage condition? We should resort to the price function in order to answer to this question. Using the notations from previous sections, the predicted extreme price of the swaption would be C(f t , κ,Î (α) (t, κ − f t )). Note that we have used κ − f t to replace x because x represents moneyness defined in 25. An example of the price function with respect to strike is shown in Figure 12 which observes conditions in Equations 39. In fact, for the historical period that we have studied (2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017), none of the days violates the non-arbitrage condition. This shows that the Karhunen-Loève decomposition, together with the Filtered Historical Simulation that we have adopted, is well compatible with the pricing framework.
Backtesting VaR
Is the VaR of 1 (or of ξ 1 ) well estimated? From Figure 11 we see thatˆ form a good contour of 1 , but what is exactly the "goodness" of a VaR estimation? Campbell [2006] summarizes backtesting approaches for VaR. In order to proceed with the backtesting, we should first of all define "hit sequences" as follows:
h (0.01) (t) = 1 if 1 (t) ≤ˆ (0.01) 1 (t) 0 if 1 (t) >ˆ (0.01) 1 (t)(41)h (0.99) (t) = 1 if 1 (t) ≥ˆ (0.99) 1 (t) 0 if 1 (t) <ˆ (0.99) 1 (t)(42)
so that the hit sequence tallies the history of whether or not a move in excess of the reported VaR has been realized. Figure 12: An example of pricing function with respect to strike for a payer swaption, when extreme predicted implied volatilities are used. t=2008-06-12, expiry T=10Y, tenor=10Y, currency: USD, α = 0.01.
Christopherson et al. [1998] points out that the problem of determining the accuracy of a VaR model can be reduced to the problem of determining whether the hit sequence, h (α) (t), satisfies the following two properties:
1. Unconditional Coverage Property: The probability of realizing a move in excess of the reported VaR must be precisely α×100% ((1−α)×100%) for α < 0.5 (α > 0.5), i.e. P(h (α) (t) = 1) = α (P(h (α) (t) = 1) = 1 − α). Otherwise, we have either overestimated or underestimated VaR.
To test this property, Kupiec [1995] constructed the following test statistic P OF (proportion of failures) for α < 0.5. The P OF for α > 0.5 can be defined analogously. Under null hypothesis:
H 0 :α = α,(43)
P OF asymptotically follows χ 2 1 , i.e. chi-square distribution with one degree of freedom.
P OF = −2ln( (1 − α) T 0 α T 1 (1 −α) T 0α T 1 ) (44) whereα = T 1 T 0 + T 1 , T 1 = T t=1 h (α) (t), T 0 = T − T 1(45)
2. Independence Property: Any two elements of the hit sequence, (h (α) (t + j), h (α) (t + k)) must be independent from each other. Intuitively, this condition requires that the previous history of VaR violations, {. . . , h (α) (t − 1), h (α) (t)}, must not convey any information about whether or not an additional VaR violation, h (α) (t + 1), will occur. If not, previous VaR violations presage a future VaR violation, which further suggests a lack of responsiveness in the reported VaR measure as changing market risks fail to be fully incorporated into the reported VaR.
Christopherson et al. [1998] proposed the following test statistic IN D for α < 0.5. The test statistic for α > 0.5 can be defined analogously. Under null hypothesis:
H 0 :α 01 =α 11 =ᾱ(46)
IN D asymptotically follows χ 2 1 .
IN D = −2ln( (1 −ᾱ) T 00 +T 10ᾱ T 01 +T 11 (1 −α 01 ) T 00α T 01 01 (1 −α 11 ) T 10α T 11 11 ) (47) whereα ij = T ij T i0 + T i1 ,ᾱ = T 01 + T 11 T 00 + T 01 + T 10 + T 11(48)
and T ij is the frequency of a i value followed by a j value in the hit sequence.
These two properties of the "hit sequence" {h (α) (t)} T t=1 are often combined into the single statement:
h (α) (t) i.i.d. ∼ Bernoulli(α) if α < 0.5 Bernoulli(1 − α) if α > 0.5(49)
Now we can use these these two tests for 1 . The results are shown in Table 2 and all tests have been successfully passed, which suggests that the estimated VaR well corresponds to the unconditional probability and adjusts sufficiently fast to incorporate conditional information. Note that we can also conduct these tests for ξ 1 , which would yield similar results.
Conclusion
Managing market risk has always been crucial for ensuring the soundness and stability of financial institutions. Among all kinds of risk factors, volatility risk is one of the most important and has been the central concern of this article. More specifically, we are concerned with volatilities of swaptions which are one of the most liquid interest rate derivatives. Instead of using a local volatility or a stochastic volatility model, which gives an infinitesimal description of the volatility dynamics, we have adopted Bachelier model together with implied volatilities. This relatively simple approach is appropriate because of the liquid characteristic of swaptions, and the advantage of being observable in the markets. The dynamics of volatility smiles or surfaces is studied by Karhunen-Loève decomposition, a generalized version of Principal Component Analysis conceived for functions. The decomposition gives a concise and precise description of the dynamics, using at most 3 principal components. As swaption has three important parameters (moneyness, expiry, tenor) for indexing the volatility smile, the decomposition has been separately conducted for these 3 cases and different behaviors have been observed along these three dimensions. For moneyness-or tenor-indexed smile, the first three components can generally be explained as parallel shift, rotation and convexity change. While for expiry-indexed smile, the interpretation is more difficult. What's more, volatilities along different dimensions have inter-connections, which explains the difference between a marginal of a 2-dimensional principal component and the corresponding 1-dimensional principal component.
Thanks to null correlations, projections on different principal components can be investigated separately for evaluating VaR. We have particularly paid attention to the conditional mean structure and the conditional heteroscedasticity structure of the time series, in order to have a more "stable" time series which is favorable for Historical Simulation. This approach is called Filtered Historical Simulation in the general sense. Extreme moves of volatility smile can thus be estimated in a holistic way and are compatible with non-arbitrage hypothesis. Moreover, estimated VaR well passes Kupiec's unconditional coverage test and Christofferson's independence test, which are the most widely accepted criteria for VaR backtesting. Consequently, volatility risk of a swaption portfolio can be consistently evaluated.
It should be noted, nonetheless, that special attention should be paid if we want to extend the method to other financial products. For less liquid products, modeling the dynamics of the underlying using local or stochastic volatility model remains a crucial step for pricing the products and managing market risk. Further investigations should be made in order to appropriately calibrate the parameters and to manage volatility risk.
Figure 2 :
2First three eigenfunctions and eigenvalues of Karhunen-Loève decomposition for tenorindexed smile log-return. moneyness=0, expiry=10Y, currency=USD. The unit for x-axis is year. 13e − 03, 79.21% explained λ 2 = 5. 09e − 04, 9.76% explained λ 3 = 4. 29e − 04, 8.24% explained
Figure 3 :
3First three eigenfunctions and eigenvalues of Karhunen-Loève decomposition for expiryindexed smile log-return. moneyness=0, tenor=10Y, currency=USD. The unit for x-axis is year.
Figure 4 :Figure 5 :
45First three eigenfunctions and eigenvalues of Karhunen-Loève decomposition for expiryand-tenor-indexed surface log-return. moneyness=0, currency=USD. x 1 is expiry and x 2 is tenor. Projections (ξ i ) of moneyness-index smile log-return on first three components. ex-piry=10Y, tenor=10Y, currency=USD.
Figure 6 :
6ACF and PACF of ξ 1 , obtained from the Karhunen-Loève decomposition of moneynessindexed smile.
realized time series 1 are illustrated in
Figure 7 :Figure 8 :Figure 9 :Figure 10 :
78910Time series, ACF and PACF of 1 , which is obtained from the AR(1) model in Equation27. ACF and PACF of | 1 |. Time series of 1 /σ 1 , where σ 1 is EWMA volatility of 1 with decay factor θ = 0.9 and volatility estimation rolling window W = 60. ACF and PACF of | 1 /σ 1 |.
Figure 11 :
11Predicted
Figure 1
1e 3 (x) means an extreme flattening of the smile. Thusû (0.01) (t, x) defined in Equation 35 incorporates these three aspects. Yet we might as well define the VaR as the combination of extreme positive parallel shift, extreme counter-clockwise rotation and extreme flattening of the smile, i.e 3 (x). The definition of smile VaR would thus be based on the concerns in practice and we shall stick with this definition for the simplicity of notation. Based on Equations 26 and 35, if we are at time t − 1, we can estimate the extreme cases of volatility smile (or surface etc.) at time t by:
Table 1 :
1Correlations between {ξ i } i for Karhunen-Loève decomposition of moneyness-indexed smile
log-return. expiry=10Y, tenor=10Y, currency=USD
combinations of projections {ξ i } i
1st and 2nd 1st and 3rd 2nd and 3rd
Pearson's correlation
0.00001%
0.00016%
-0.00007%
p-value
99.99975% 99.99346%
99.99715%
Table 2 :
2Backtesting results for 1 Kupiec test Christofferson test p-value T 00 T 01 T 10 T 11 p-valueq1
29.36% 2226 28 28
0 40.42%
q99
65.30% 2233 24 24
1 27.71%
For some case, the autoregressive structure may not be necessary. This is the case for ξ 2 and ξ 3 since their PACF look like a white noise.
Coherent measures of risk. Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, David Heath, Mathematical finance. 93Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath. Coherent measures of risk. Mathematical finance, 9(3):203-228, 1999.
Théorie de la spéculation. Louis Bachelier, Gauthier-VillarsLouis Bachelier. Théorie de la spéculation. Gauthier-Villars, 1900.
Do call prices and the underlying stock always move in the same direction?. Gurdip Bakshi, Charles Cao, Zhiwu Chen, The Review of Financial Studies. 133Gurdip Bakshi, Charles Cao, and Zhiwu Chen. Do call prices and the underlying stock always move in the same direction? The Review of Financial Studies, 13(3):549-584, 2000.
Var without correlations for portfolio of derivative securities. Giovanni Barone-Adesi, Kostas Giannopoulos, Les Vosper, Università della Svizzera italianaTechnical reportGiovanni Barone-Adesi, Kostas Giannopoulos, and Les Vosper. Var without correlations for portfolio of derivative securities. Technical report, Università della Svizzera italiana, 1999.
The pricing of options and corporate liabilities. Fischer Black, Myron Scholes, Journal of political economy. 813Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of political economy, 81(3):637-654, 1973.
A review of backtesting and backtesting procedures. D Sean, Campbell, The Journal of Risk. 92Sean D Campbell. A review of backtesting and backtesting procedures. The Journal of Risk, 9(2): 1, 2006.
Conditioning manager alphas on economic information: Another look at the persistence of performance. Wayne E Jon A Christopherson, Debra A Ferson, Glassman, The Review of Financial Studies. 111Jon A Christopherson, Wayne E Ferson, and Debra A Glassman. Conditioning manager alphas on economic information: Another look at the persistence of performance. The Review of Financial Studies, 11(1):111-142, 1998.
Empirical properties of asset returns: stylized facts and statistical issues. Rama Cont, Rama Cont. Empirical properties of asset returns: stylized facts and statistical issues. 2001.
Dynamics of implied volatility surfaces. Rama Cont, José Da Fonseca, Quantitative finance. 21Rama Cont, José Da Fonseca, et al. Dynamics of implied volatility surfaces. Quantitative finance, 2(1):45-60, 2002.
Beyond value at risk: the new science of risk management. Kewin Dowd, Kewin Dowd. Beyond value at risk: the new science of risk management. 1998.
Pricing and hedging with smiles. Mathematics of derivative securities. Bruno Dupire, 1Bruno Dupire. Pricing and hedging with smiles. Mathematics of derivative securities, 1(1):103-111, 1997.
Managing smile risk. Deep Patrick S Hagan, Kumar, S Andrew, Diana E Lesniewski, Woodward, The Best of Wilmott. 1Patrick S Hagan, Deep Kumar, Andrew S Lesniewski, and Diana E Woodward. Managing smile risk. The Best of Wilmott, 1:249-296, 2002.
Techniques for verifying the accuracy of risk measurement models. H Paul, Kupiec, The journal of Derivatives. 32Paul H Kupiec. Techniques for verifying the accuracy of risk measurement models. The journal of Derivatives, 3(2):73-84, 1995.
Probability theory. Michel Loeve, Graduate texts in mathematics. 46Michel Loeve. Probability theory, vol. ii. Graduate texts in mathematics, 46:0-387, 1978.
Functions of positive and negative type, and their connection with the theory of integral equations. Philosophical transactions of the royal society of London. Series A, containing papers of a mathematical or physical character. James Mercer, 209James Mercer. Functions of positive and negative type, and their connection with the theory of integral equations. Philosophical transactions of the royal society of London. Series A, containing papers of a mathematical or physical character, 209:415-446, 1909.
Implied volatility functions: A reprise. V Joshua, Rosenberg, The Journal of Derivatives. 73Joshua V Rosenberg. Implied volatility functions: A reprise. The Journal of Derivatives, 7(3): 51-64, 2000.
Time Series Analysis. François Roueff, François Roueff. Time Series Analysis. 2016. Book can be found at http://perso. telecom-paristech.fr/~blanchet/FCFD/Scilab/docs/polyTimeSeries.pdf.
| [] |
[
"Classical dimers and dimerized superstructure in orbitally degenerate honeycomb antiferromagnet",
"Classical dimers and dimerized superstructure in orbitally degenerate honeycomb antiferromagnet"
] | [
"G Jackeli \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany\n",
"D I Khomskii \nII. Physikalisches Institut\nUniversität zu Köln\nZülpicher Strasse 7750937KölnGermany\n"
] | [
"Max-Planck-Institut für Festkörperforschung\nHeisenbergstrasse 1D-70569StuttgartGermany",
"II. Physikalisches Institut\nUniversität zu Köln\nZülpicher Strasse 7750937KölnGermany"
] | [] | We discuss the ground state of the spin-orbital model for spin-one ions with partially filled t2g levels on a honeycomb lattice. We find that the orbital degrees of freedom induce a spontaneous dimerization of spins and drive them into nonmagnetic manifold spanned by hard-core dimer (spinsinglet) coverings of the lattice. The cooperative "dimer Jahn-Teller" effect is introduced through a magnetoelastic coupling and is shown to lift the orientational degeneracy of dimers leading to a peculiar valence bond crystal pattern. The present theory provides a theoretical explanation of nonmagnetic dimerized superstructure experimentally seen in Li2RuO3 compound at low temperatures. PACS numbers: 75.10.Jm, 75.30.EtThe nearest-neighbor Heisenberg antiferromagnet on a bipartite lattice has a Néel-type magnetically long-range ordered ground state. However, such a classical order of spins can be destabilized by introducing a frustration into the system through the competing interactions that may lead to the extensively degenerate classical ground states[1]. In such systems exotic quantum phases without longrange order can emerge as the true ground states. In this Letter we want to point out and discuss another scenario, that can appear when magnetic ions on a bipartite lattice possess also an orbital degeneracy. The physics of such systems may be drastically different from that of pure spin models, as the occurrence of an orbital ordering can modulate the spin exchange and preclude the formation of magnetically ordered state on a bipartite lattice. In the following we focus on a system with threefold-orbitally-degenerate S = 1 magnetic ions on a honeycomb lattice. This model is suitable to describe d 2 and d 4 -type transition-metal compounds with partially filled t 2g levels, like the layered compound Li 2 RuO 3 [2]. Here the layers are formed by edge-sharing network of RuO 6 and LiO 6 octahedra. The Ru ions make a honeycomb lattice and Li ions reside in the centers of hexagons. These layers are well separated by the remaining Li ions. The magnetically active Ru 4+ -ions are characterized by four electrons in the threefold degenerate t 2g -manifold coupled into a S = 1 state. Li 2 RuO 3 undergoes a metalto-insulator transition on cooling below 540 K [2]. At the transition the magnetic susceptibility shows a steep decrease and its low temperature value can be considered to be due almost entirely to the Van Vleck paramagnetism. The structural analyses have revealed the formation of dimerized superstructure of Ru-Ru bonds in the low temperature phase. These observations indicate that Ruthenium spin-one degrees of freedom are mysteriously missing at low temperatures and suggest the formation of an unusual spin-singlet dimer phase in the ground state of the system.Here we describe the microscopic theory behind the stabilization of such a spin-singlet dimer state. We argue that, remarkably, such a novel phase can be realized on a honeycomb lattice because of orbital degeneracy, without invoking any exotic spin-only interactions. A possibility of formation of orbitally driven magnetically disordered states has been suggested within various coupled spinorbital models[3,4,5,6,7,8,9]. The orbital induced frustration in t 2g based systems on a bipartite (cubic) lattice has also been considered[10,11]. The emergence of new phases due to the p-orbital degeneracy of cold atoms in optical lattices has been recently discussed within a spinless fermion model on a honeycomb and other twodimensional lattices[12,13]. Yet, the peculiar case of partially filled t 2g levels on a honeycomb lattice leads to new results: the onset of an orbitally driven spin-singlet dimer phase in a spin-one system. The model.-We assume that the low-temperature insulating phase of Li 2 RuO 3 is of Mott-Hubbard type and describe the low energy physics within the Kugel-Khomskii type spin-orbital Hamiltonian[14]. We consider undistorted honeycomb lattice of Ru ions and look for possible instabilities towards symmetry reductions. In the Li 2 RuO 3 crystal structure three distinct bonds of honeycomb lattice are in xy, xz, and yz planes (in cubic notations). We consider the leading part of the nearestneighbor (NN) hopping integral of t 2g orbitals (d xy , d yz , and d xz ) due to the direct σ-type overlap. The ddσ overlap in αβ plane (α, β = x, y, z) connects only the orbitals of same αβ type. The effective spin-orbital Hamiltonian for such a system been reported in Refs.[15,16]. It has the following form:where the sum is taken over pairs of NN sites, S i are spin-one operators, and the orbital contribution are described byŌ ij and O ij operators. The second-order virtual processes locally conserve orbital index. The orbital | 10.1103/physrevlett.100.147203 | [
"https://arxiv.org/pdf/0802.2692v2.pdf"
] | 11,493,824 | 0802.2692 | c8f3b1faf6744aebb0a77c02740534af69631b4f |
Classical dimers and dimerized superstructure in orbitally degenerate honeycomb antiferromagnet
18 Mar 2008 (Dated: March 18, 2008)
G Jackeli
Max-Planck-Institut für Festkörperforschung
Heisenbergstrasse 1D-70569StuttgartGermany
D I Khomskii
II. Physikalisches Institut
Universität zu Köln
Zülpicher Strasse 7750937KölnGermany
Classical dimers and dimerized superstructure in orbitally degenerate honeycomb antiferromagnet
18 Mar 2008 (Dated: March 18, 2008)arXiv:0802.2692v2 [cond-mat.str-el]
We discuss the ground state of the spin-orbital model for spin-one ions with partially filled t2g levels on a honeycomb lattice. We find that the orbital degrees of freedom induce a spontaneous dimerization of spins and drive them into nonmagnetic manifold spanned by hard-core dimer (spinsinglet) coverings of the lattice. The cooperative "dimer Jahn-Teller" effect is introduced through a magnetoelastic coupling and is shown to lift the orientational degeneracy of dimers leading to a peculiar valence bond crystal pattern. The present theory provides a theoretical explanation of nonmagnetic dimerized superstructure experimentally seen in Li2RuO3 compound at low temperatures. PACS numbers: 75.10.Jm, 75.30.EtThe nearest-neighbor Heisenberg antiferromagnet on a bipartite lattice has a Néel-type magnetically long-range ordered ground state. However, such a classical order of spins can be destabilized by introducing a frustration into the system through the competing interactions that may lead to the extensively degenerate classical ground states[1]. In such systems exotic quantum phases without longrange order can emerge as the true ground states. In this Letter we want to point out and discuss another scenario, that can appear when magnetic ions on a bipartite lattice possess also an orbital degeneracy. The physics of such systems may be drastically different from that of pure spin models, as the occurrence of an orbital ordering can modulate the spin exchange and preclude the formation of magnetically ordered state on a bipartite lattice. In the following we focus on a system with threefold-orbitally-degenerate S = 1 magnetic ions on a honeycomb lattice. This model is suitable to describe d 2 and d 4 -type transition-metal compounds with partially filled t 2g levels, like the layered compound Li 2 RuO 3 [2]. Here the layers are formed by edge-sharing network of RuO 6 and LiO 6 octahedra. The Ru ions make a honeycomb lattice and Li ions reside in the centers of hexagons. These layers are well separated by the remaining Li ions. The magnetically active Ru 4+ -ions are characterized by four electrons in the threefold degenerate t 2g -manifold coupled into a S = 1 state. Li 2 RuO 3 undergoes a metalto-insulator transition on cooling below 540 K [2]. At the transition the magnetic susceptibility shows a steep decrease and its low temperature value can be considered to be due almost entirely to the Van Vleck paramagnetism. The structural analyses have revealed the formation of dimerized superstructure of Ru-Ru bonds in the low temperature phase. These observations indicate that Ruthenium spin-one degrees of freedom are mysteriously missing at low temperatures and suggest the formation of an unusual spin-singlet dimer phase in the ground state of the system.Here we describe the microscopic theory behind the stabilization of such a spin-singlet dimer state. We argue that, remarkably, such a novel phase can be realized on a honeycomb lattice because of orbital degeneracy, without invoking any exotic spin-only interactions. A possibility of formation of orbitally driven magnetically disordered states has been suggested within various coupled spinorbital models[3,4,5,6,7,8,9]. The orbital induced frustration in t 2g based systems on a bipartite (cubic) lattice has also been considered[10,11]. The emergence of new phases due to the p-orbital degeneracy of cold atoms in optical lattices has been recently discussed within a spinless fermion model on a honeycomb and other twodimensional lattices[12,13]. Yet, the peculiar case of partially filled t 2g levels on a honeycomb lattice leads to new results: the onset of an orbitally driven spin-singlet dimer phase in a spin-one system. The model.-We assume that the low-temperature insulating phase of Li 2 RuO 3 is of Mott-Hubbard type and describe the low energy physics within the Kugel-Khomskii type spin-orbital Hamiltonian[14]. We consider undistorted honeycomb lattice of Ru ions and look for possible instabilities towards symmetry reductions. In the Li 2 RuO 3 crystal structure three distinct bonds of honeycomb lattice are in xy, xz, and yz planes (in cubic notations). We consider the leading part of the nearestneighbor (NN) hopping integral of t 2g orbitals (d xy , d yz , and d xz ) due to the direct σ-type overlap. The ddσ overlap in αβ plane (α, β = x, y, z) connects only the orbitals of same αβ type. The effective spin-orbital Hamiltonian for such a system been reported in Refs.[15,16]. It has the following form:where the sum is taken over pairs of NN sites, S i are spin-one operators, and the orbital contribution are described byŌ ij and O ij operators. The second-order virtual processes locally conserve orbital index. The orbital
We discuss the ground state of the spin-orbital model for spin-one ions with partially filled t2g levels on a honeycomb lattice. We find that the orbital degrees of freedom induce a spontaneous dimerization of spins and drive them into nonmagnetic manifold spanned by hard-core dimer (spinsinglet) coverings of the lattice. The cooperative "dimer Jahn-Teller" effect is introduced through a magnetoelastic coupling and is shown to lift the orientational degeneracy of dimers leading to a peculiar valence bond crystal pattern. The present theory provides a theoretical explanation of nonmagnetic dimerized superstructure experimentally seen in Li2RuO3 compound at low temperatures. The nearest-neighbor Heisenberg antiferromagnet on a bipartite lattice has a Néel-type magnetically long-range ordered ground state. However, such a classical order of spins can be destabilized by introducing a frustration into the system through the competing interactions that may lead to the extensively degenerate classical ground states [1]. In such systems exotic quantum phases without longrange order can emerge as the true ground states. In this Letter we want to point out and discuss another scenario, that can appear when magnetic ions on a bipartite lattice possess also an orbital degeneracy. The physics of such systems may be drastically different from that of pure spin models, as the occurrence of an orbital ordering can modulate the spin exchange and preclude the formation of magnetically ordered state on a bipartite lattice. In the following we focus on a system with threefold-orbitally-degenerate S = 1 magnetic ions on a honeycomb lattice. This model is suitable to describe d 2 and d 4 -type transition-metal compounds with partially filled t 2g levels, like the layered compound Li 2 RuO 3 [2]. Here the layers are formed by edge-sharing network of RuO 6 and LiO 6 octahedra. The Ru ions make a honeycomb lattice and Li ions reside in the centers of hexagons. These layers are well separated by the remaining Li ions. The magnetically active Ru 4+ -ions are characterized by four electrons in the threefold degenerate t 2g -manifold coupled into a S = 1 state. Li 2 RuO 3 undergoes a metalto-insulator transition on cooling below 540 K [2]. At the transition the magnetic susceptibility shows a steep decrease and its low temperature value can be considered to be due almost entirely to the Van Vleck paramagnetism. The structural analyses have revealed the formation of dimerized superstructure of Ru-Ru bonds in the low temperature phase. These observations indicate that Ruthenium spin-one degrees of freedom are mysteriously missing at low temperatures and suggest the formation of an unusual spin-singlet dimer phase in the ground state of the system.
Here we describe the microscopic theory behind the stabilization of such a spin-singlet dimer state. We argue that, remarkably, such a novel phase can be realized on a honeycomb lattice because of orbital degeneracy, without invoking any exotic spin-only interactions. A possibility of formation of orbitally driven magnetically disordered states has been suggested within various coupled spinorbital models [3,4,5,6,7,8,9]. The orbital induced frustration in t 2g based systems on a bipartite (cubic) lattice has also been considered [10,11]. The emergence of new phases due to the p-orbital degeneracy of cold atoms in optical lattices has been recently discussed within a spinless fermion model on a honeycomb and other twodimensional lattices [12,13]. Yet, the peculiar case of partially filled t 2g levels on a honeycomb lattice leads to new results: the onset of an orbitally driven spin-singlet dimer phase in a spin-one system. The model.-We assume that the low-temperature insulating phase of Li 2 RuO 3 is of Mott-Hubbard type and describe the low energy physics within the Kugel-Khomskii type spin-orbital Hamiltonian [14]. We consider undistorted honeycomb lattice of Ru ions and look for possible instabilities towards symmetry reductions. In the Li 2 RuO 3 crystal structure three distinct bonds of honeycomb lattice are in xy, xz, and yz planes (in cubic notations). We consider the leading part of the nearestneighbor (NN) hopping integral of t 2g orbitals (d xy , d yz , and d xz ) due to the direct σ-type overlap. The ddσ overlap in αβ plane (α, β = x, y, z) connects only the orbitals of same αβ type. The effective spin-orbital Hamiltonian for such a system been reported in Refs. [15,16]. It has the following form:
H = ij J S i · S j − 1 Ō ij − J 0 S i · S j + J 1 O ij , (1)
where the sum is taken over pairs of NN sites, S i are spin-one operators, and the orbital contribution are described byŌ ij and O ij operators. The second-order virtual processes locally conserve orbital index. The orbital degrees are thus static Potts-like variables and their contribution can be expressed simply in terms of projectors P i,αβ onto the singly occupied orbital state αβ at site i. With this definition of the projectors the orbital part of the Hamiltonian can be written in the form equally valid for t 2 2g and its particle-hole symmetry related t 4 2g configurations. The orbital operatorsŌ ij and O ij along the bond ij in αβ-plane are given by:Ō ij = P i,αβ P j,αβ and O ij = P i,αβ (1 − P j,αβ ) + P j,αβ (1 − P i,αβ ). For further analysis it is convenient to rewrite the Hamiltonian as the sum of three terms: [17]. For the full expressions of the exchange constants in terms of hopping integral t, on-site Coulomb repulsion U and Hund's coupling J H we refer reader to Refs. [15,16]. To the leading order in small parameter η = J H /U , the coupling constants are given by
H = E 0 + H AF + H FM , H AF = ij J S i · S j + ζ Ō ij , H FM =−J 0 ij S i · S j O ij ,(2)E 0 = −2J 1 N , N is number of lattice sites, and ζ = 2J 1 /J − 1J ≈ (1 − η) t 2 U , J 0 ≈ η t 2 U , J 1 ≈ (1 + 2η) t 2 U
, and ζ ≈ 1 + 6η. For our further analysis, it will be important that ζ ≥ 1 for all values of the Hund's coupling. The inequality implies that any magnetically ordered ground state has a positive classical energy on antiferromagnetic (AF) bonds. It suggests that the formation of a low dimensional network of AF spin-coupling patterns are energetically favorable due to a larger gain of quantum spinenergy per bond. This picture is substantiated by the exact solution of the ground state problem at zero Hund's coupling described in the next section.
Zero Hund's coupling.-The spin-orbital model defined in Eq. (2) has a small parameter η ≪ 1. We start our analysis from the limit of zero Hund's coupling η = 0 (H FM =0) and look for the possible spin-coupling patterns generated by H AF term of the Hamiltonian.
The antiferromagnetic term H AF is active only on the bonds with corresponding orbital being singly occupied at both ends of a bond (e.g on a bond ij in xy plane both sites have xy orbital singly occupied). The AF bonds can only form non-intersecting linear open and/or closed chains (see Fig. 1A). The longer is the chain, the more energy can be gained from the AF spin interaction. However, for each AF bond we pay a positive energy ζ ≥ 1 (the second term in H AF ). Under these condi- energies E 2 = −2 and E 3 = −3. The dimer states have a lower energy than the closed chains ifĒ M > − 3M 2 , whereĒ M is the ground-state energy of the M -site spinone AF Heisenberg ring. The last inequality is satisfied for any AF Heisenberg ring with M > 4 (see for example Ref. [18]). Note that on a honeycomb lattice only the closed chains with M ≥ 6 can be formed. Therefore, at zero Hund's coupling η = 0, in the ground state manifold the pattern of AF bonds corresponds to a hardcore dimer covering of the lattice and on each inter-dimer bond only one site has active orbital singly occupied. The H AF term is inactive on such inter-dimer bonds, and spins of different dimers are decoupled. Spins are coupled into quantum spin-singlet state on dimer bonds and are thus gapped. Such a spin-singlet dimer states form the exact ground state manifold of spin-orbital Hamiltonian Eq. (2) in the limit of zero Hund's coupling.
Ground state manifold.-The ground state manifold is extensively degenerate: there are infinitely many ways of covering a honeycomb lattice with hard-core dimers, and each dimer covering has its own Ising-type degeneracy connected with the orientation of an "inactive" orbital. Thus, in contrast to the case of one d-electron (S=1/2) where the state is determined by the covering of the lattice with spin-singlet dimers [9], here we have an extra degeneracy. At each site the second orbital points along one of two remaining bonds (the arrows in Fig.1B), with the rule that at each of such "empty" bonds there should be only one orbital (one arrow). For any dimer covering the inter-dimer ("empty") bonds form non-intersecting linear chains. Along an inter-dimer path all arrows must be directing along the same direction [see as an example the zig-zag like linear chain in Fig. 1B]. However, on each inter-dimer path we can inverse all the arrows and still remain in the ground state manifold. Thus each inter-dimer path has a two-fold degeneracy of Ising-type. The degeneracy of a state with given dimer covering is thus equal to 2 Nc , where N c is a number of decoupled inter-dimer chains. One finds that N c ∼ √ N for all dimer coverings except one case when all inter-dimer chains are hexagonal loops. In this case the honeycomb lattice is divided into non-overlapping hexagons with no dimers, and the remaining hexagons are occupied by three dimers. This gives N c = N/6 leading to a finite contribution to a bulk entropy from Ising-type degeneracy. The above degeneracy is, however, easily lifted by any interaction which induces a non-zero coupling of non-active orbitals on dimer bonds and thus correlates the Ising-like variables (direction of arrows) of neighboring inter-dimer chains. For example, the coupling of Jahn-Teller distortions on NN edge sharing oxygen octahedra favors the ferro-type orbital order of singly occupied non-active orbitals on dimer bonds (see Fig. 1B) [19]. The π-type contribution to the hopping integral instead stabilizes an antiferro-type orbital order by increasing the AF coupling on a dimer bond [20]. In real materials the two mechanisms will have different energy scales and the dominant one will dictate the resulting orbital pattern. In both cases one completely lifts the degeneracy of a given dimer state.
The degeneracy of second type originates from the orientational degeneracy of dimers and is exactly equal to the number of hard-core dimer coverings of the honeycomb lattice. The latter is well known to be an extensive quantity. The classical dimers on a honeycomb lattice exhibit a power-law decay of dimer-dimer correlation function and are thus in a critical state [21]. Therefore, one expects that any perturbation which may favor one or another type of dimer orientation may lift this extensive degeneracy, resulting in a long-range ordered pattern of dimers. We have considered a possibility of perturbing the exact dimer ground state manifold by small but finite Hund's coupling 0 < η ≪ 1. The Hund's coupling produces a ferromagnetic (FM) term H FM active on interdimer bonds (see Fig. 1B). At small η, the dimer state is stable against the weak FM inter-dimer interaction. In this case the magnetic contribution along the FM bond is zero ( S i · S j = 0 for i and j belonging to different dimers). However, the weak inter-dimer coupling may, in principle, lift the orientational degeneracy through order out of disorder by triplet fluctuations [9]. We find that on a honeycomb lattice these quantum fluctuations do not fully lift the dimer degeneracy. They only disfavor the configurations with three dimers on a hexagon because such a hexagonal loop of alternating AF and FM bonds is frustrated in the classical limit. In the next section we introduce a magnetoelastic coupling and show that it fully lifts the orientational degeneracy of dimers.
Degeneracy breaking by magnetoelastic coupling.-The magnetoelastic mechanism of lifting the extensive degeneracy of frustrated systems has been successful in explaining the experimentally observed structures [5,22,23,24]. The physics behind this mechanism is that the correlated nature of structural distortions that appear due to the modulation of magnetic energies on bonds may select a particular pattern of distortions and thus lift the extensive degeneracy of the ground state manifold.
In the ground state manifold the spin degrees of freedom are described by the product of spin-singlet dimer states. A shortening of a bond where the singlet is located enlarges the magnetic energy gain, because of the increase in the exchange coupling J on that bond. The magnetic energy gain, being linear in distortion, outweights the increase in elastic energy and would always lead to such a contraction of singlet bonds for any dimer covering. But due to elastic coupling of these distortions different distorted patterns will lead to different elastic energy and hence to the lifting of dimer degeneracy. We, first, formalize this picture within the simplest model and later discuss its extension. The schematic structure of Li 2 RuO 3 is shown in Fig.2D. In the minimal model the exchange coupling is assumed to depend solely on a distance between Ru ions, and the lattice degrees of freedom are described within Einstein phonon model for Ru ions:
E ME = −γǫ ij u i − v j l ij + k 2 i u 2 i(3)
where γ = − 1 J ∂J(r) ∂r is a magnetoelastic coupling constant, ǫ is magnetic energy gain on a dimer bond, u i is a displacement of Ru ions at site i, and k is Einstein phonon constant. The dimer positions are described by l ij : the latter is an unit vector along a bond ij occupied by dimer and is zero otherwise. We minimize the energy functional E ME with respect to set of u i and eliminate them in favor of l ij . We find E ME = − k 2 i ū 2 i , and ū i = γǫ k j l ij . The set of ū i define the distortion pattern in which Ru ions on dimer bonds are moved towards each other. However, as the dimer bonds do not share a common site, E ME is a constant. It is independent of dimer variables l ij . In order to introduce the coupling between the dimers we extend the model by including a finite force induced on Li ions by distorted pattern of Ru ions. Consider an isolated complex of a Ru-Ru bond together with neighboring Li ions, shown in Fig. 2A. When this bond is occupied by spin-singlet dimer it gets contracted, and the displacements of Ru ions induce the corresponding distortion of Li complex as seen in Fig. 2A. On a lattice the distorted dimer bonds will now be coupled through the force induced on a common Li ion, an effect similar to a cooperative Jahn-Teller physics. The dimer configurations for which the induced forces on Li sites interfere in a non-destructive manner result in a larger distortion and hence more gain in energy. The two orientations of neighboring dimers shown in Fig. 2B and C are energetically unfavorable, as in both cases the induced forces exactly cancel each other. Thus the ground state dimer pattern should satisfy the constraint of no-B and no-C type configurations. Note that a dimer covering satisfying the former constraint automatically satisfies the latter. It is possible to check that the only possibility to fulfill such a no-B constraint is realized for the dimer pattern shown in Fig. 2D. This dimerization pattern exactly reproduces the one observed in the insulating phase of Li 2 RuO 3 [2]. We note that above described mechanism of the selection of spin-singlet dimer pattern equally applies to the case when singlets are formed by spin-one-half degrees of freedom. The spin-singlet dimer nature of the ground state manifold for d 1 systems on a honeycomb lattice has been proven in Ref. [9]. We therefore predict the same dimerized superstructure also for d 1 systems, such as V 4+ and Ti 3+ based compounds with such structure (if orbital degeneracy will not be lifted by some external mechanism like trigonal distortion, often present in such structure).
Summary.-To summarize, we have studied the ground state of spin-one honeycomb antiferromagnet with partially filled t 2g levels. We have demonstrated that the orbital degeneracy induces spontaneous dimerization of spins and drives them into extensively degenerate manifold of spin-singlet dimer states. The orientational degeneracy of dimers is then lifted through the magnetoelastic interaction that stabilizes a peculiar valence bond crystal state. Our theory provides an explanation for the observed nonmagnetic dimerized superstructure in Li 2 RuO 3 compound.
We are grateful to G. Khaliullin and H. Takagi for useful discussions. We acknowledge kind hospitality at KITP, UCSB where the part of this work has been done.
PACS numbers: 75.10.Jm, 75.30.Et
FIG. 1 :
1tions the minimal possible AF energy is achieved when all AF chains are dimers. The proof of the stability of dimer states against the formation of longer open Heisenberg chains is based on the variational estimate on the ground-state energy of the M -site spin-one AF Heisenberg chain with open ends: E M ≥ 1 − 3M 2 , with the equality attained only at M = 2. This estimate can be obtained by dividing the chain into shorter overlapping sub-chains of lengths two and three with exactly known (Color online) Examples of orbital and spin-coupling patterns on the honeycomb lattice of Ru ions. A) Decoupled AF chain and ring with corresponding orbital pattern. B) An example of the spin-singlet dimer covering minimizing the energy at zero Hund's coupling. Thick (thin) lines denote AF (FM) intra-(inter-)dimer bonds, respectively. Dashed lines stand for the noninteracting bonds.
online) Lifting dimer degeneracy by magnetoelastic coupling. The small light (large dark) circles denote Ru (Li) ions. The arrows show the displacements of corresponding ions. A) The sketch of Li displacements induced by dimerized Ru-Ru bond. B) and C) Two examples of destructive interference of Li displacements induced by neighboring dimers. D)The ground state dimer pattern selected by magnetoelastic coupling. This pattern exactly corresponds to the one found in Li2RuO3 in Ref.[2].
. Georgia † Tbilisi, UKat the Loughborough UniversityTbilisi, Georgia † Also at the Loughborough University, UK
For reviews, see Magnetic Systems with Competing Interactions. H. T. DiepWorld ScientificSingaporeFor reviews, see Magnetic Systems with Competing In- teractions, edited by H. T. Diep (World Scientific, Singa- pore, 1994).
. Y Miura, Y Yasui, M Sato, N Igawa, K Kakurai, J. Phys. Soc. Jpn. 7633705Y. Miura, Y. Yasui, M. Sato, N. Igawa, and K. Kakurai, J. Phys. Soc. Jpn. 76, 033705 (2007).
. H F Pen, J Van Den, D I Brink, G A Khomskii, Sawatzky, Phys. Rev. Lett. 781323H. F. Pen, J. van den Brink, D. I. Khomskii, and G. A. Sawatzky, Phys. Rev. Lett. 78, 1323 (1997).
. L F Feiner, A M Oles, J Zaanen, Phys. Rev. Lett. 782799L. F. Feiner, A. M. Oles, and J. Zaanen, Phys. Rev. Lett. 78, 2799 (1997).
. S Di Matteo, G Jackeli, C Lacroix, N B Perkins, Phys. Rev. Lett. 9377208S. Di Matteo, G. Jackeli, C. Lacroix, and N. B. Perkins, Phys. Rev. Lett. 93, 077208 (2004);
. S Di Matteo, G Jackeli, N B Perkins, Phys. Rev. B. 7224431S. Di Matteo, G. Jackeli, and N. B. Perkins, Phys. Rev. B 72, 024431 (2005).
. F Vernay, K Penc, P Fazekas, F Mila, Phys. Rev. B. 7014428F. Vernay, K. Penc, P. Fazekas, and F. Mila, Phys. Rev. B 70, 014428 (2004).
. D I Khomskii, T Mizokawa, Phys. Rev. Lett. 94156402D. I. Khomskii and T. Mizokawa, Phys. Rev. Lett. 94, 156402 (2005).
. D I For, Khomskii, Progr. Theor. Phys. Suppl. 159319For review, see D. I. Khomskii, Progr. Theor. Phys. Suppl. 159, 319 (2005).
. G Jackeli, D A Ivanov, Phys. Rev. B. 76132407G. Jackeli and D. A. Ivanov, Phys. Rev. B 76, 132407 (2007).
. See G For Review, Khaliullin, Prog. Theor. Phys. Suppl. 160155For review, see G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).
. A M Oles, P Horsch, L F Feiner, G Khaliullin, Phys. Rev. Lett. 96147205A. M. Oles, P. Horsch, L. F. Feiner, and G. Khaliullin, Phys. Rev. Lett. 96, 147205 (2006).
. C Wu, D Bergman, L Balents, S. Das Sarma, Phys. Rev. Lett. 9970401C. Wu, D. Bergman, L. Balents, and S. Das Sarma, Phys. Rev. Lett. 99, 070401 (2007);
. C Wu, arXiv:0801.0888C. Wu, arXiv:0801.0888.
. E Zhao, W V Liu, arXiv:0801.0589E. Zhao and W. V. Liu, arXiv:0801.0589.
. K I Kugel, D I Khomskii, Sov. Phys. Usp. 136Usp. Fiz. NaukK. I. Kugel and D. I. Khomskii, Usp. Fiz. Nauk 136, 621 (1982) [Sov. Phys. Usp. 231, 25 (1982)].
. Y Motome, H Tsunetsugu, Phys. Rev. B. 70184427Y. Motome and H. Tsunetsugu, Phys. Rev. B 70, 184427 (2004)
. S Di Matteo, G Jackeli, N B Perkins, Phys. Rev. B. 7220408S. Di Matteo, G. Jackeli, and N. B. Perkins, Phys. Rev. B 72, 020408(R) (2005).
) ij. In deriving Eq. 2Ōij + Oij ] = 2N was usedIn deriving Eq. (2) ij [2Ōij + Oij ] = 2N was used.
. J P Neirotti, M J De Oliveira, Phys. Rev. B. 53668J. P. Neirotti and M. J. de Oliveira, Phys. Rev. B 53, 668 (1996).
This conclusion is drawn based on the similar analysis as done by D. I. Khomskii and K. I. Kugel. Phys. Rev. B. 67134401This conclusion is drawn based on the similar analysis as done by D. I. Khomskii and K. I. Kugel, Phys. Rev. B 67, 134401 (2003).
along xy bond generates a finite hopping amplitude for xz and yz orbitals. The strongest is a non-diagonal hybridization between xz and yz orbitals. Therefore the largest AF exchange on a xy bond is achieved when xz orbital is singly occupied at the one site of the bond. while yz is at the other sideThe π-type hybridization e.g. along xy bond generates a finite hopping amplitude for xz and yz orbitals. The strongest is a non-diagonal hybridization between xz and yz orbitals. Therefore the largest AF exchange on a xy bond is achieved when xz orbital is singly occupied at the one site of the bond while yz is at the other side.
. See, R Example, S L Moessner, Sondhi, Phys. Rev. B. 6864411See, for example, R. Moessner and S. L. Sondhi, Phys. Rev. B 68, 064411 (2003).
. O Tchernyshyov, R Moessner, S L Sondhi, Phys. Rev. Lett. 8867203O. Tchernyshyov, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 88, 067203 (2002);
. Phys. Rev. B. 6664403Phys. Rev. B 66, 064403 (2002).
. K Penc, N Shannon, H Shiba, Phys. Rev. Lett. 93197203K. Penc, N. Shannon, and H. Shiba, Phys. Rev. Lett. 93, 197203 (2004).
. D L Bergman, R Shindou, G A Fiete, L Balents, Phys. Rev. B. 74134409D. L. Bergman, R. Shindou, G. A. Fiete, and L. Balents, Phys. Rev. B 74, 134409 (2006).
| [] |
[
"Asymptotically Optimal Multiple-access Communication via Distributed Rate Splitting",
"Asymptotically Optimal Multiple-access Communication via Distributed Rate Splitting"
] | [
"Student Member, IEEEJian Cao ",
"Member, IEEEEdmund M Yeh "
] | [] | [] | We consider the multiple-access communication problem in a distributed setting for both the additive white Gaussian noise channel and the discrete memoryless channel. We propose a scheme called Distributed Rate Splitting to achieve the optimal rates allowed by information theory in a distributed manner. In this scheme, each real user creates a number of virtual users via a power/rate splitting mechanism in the M -user Gaussian channel or via a random switching mechanism in the M -user discrete memoryless channel. At the receiver, all virtual users are successively decoded. Compared with other multipleaccess techniques, Distributed Rate Splitting can be implemented with lower complexity and less coordination. Furthermore, in a symmetric setting, we show that the rate tuple achieved by this scheme converges to the maximum equal rate point allowed by the information-theoretic bound as the number of virtual users per real user tends to infinity. When the capacity regions are asymmetric, we show that a point on the dominant face can be achieved asymptotically. Finally, when there is an unequal number of virtual users per real user, we show that differential user rate requirements can be accommodated in a distributed fashion.Index Terms-Multiple access, rate splitting, successive decoding, stripping, interference cancellation, ALOHA.PLACE PHOTO HERE | 10.1109/tit.2006.887497 | [
"https://arxiv.org/pdf/cs/0605041v2.pdf"
] | 11,519,115 | cs/0605041 | 5f8192b76263512862d46e68d0178a66bd51711f |
Asymptotically Optimal Multiple-access Communication via Distributed Rate Splitting
3 Oct 2006
Student Member, IEEEJian Cao
Member, IEEEEdmund M Yeh
Asymptotically Optimal Multiple-access Communication via Distributed Rate Splitting
3 Oct 2006arXiv:cs/0605041v2 [cs.IT] 1Index Terms-Multiple accessrate splittingsuccessive decod- ingstrippinginterference cancellationALOHA
We consider the multiple-access communication problem in a distributed setting for both the additive white Gaussian noise channel and the discrete memoryless channel. We propose a scheme called Distributed Rate Splitting to achieve the optimal rates allowed by information theory in a distributed manner. In this scheme, each real user creates a number of virtual users via a power/rate splitting mechanism in the M -user Gaussian channel or via a random switching mechanism in the M -user discrete memoryless channel. At the receiver, all virtual users are successively decoded. Compared with other multipleaccess techniques, Distributed Rate Splitting can be implemented with lower complexity and less coordination. Furthermore, in a symmetric setting, we show that the rate tuple achieved by this scheme converges to the maximum equal rate point allowed by the information-theoretic bound as the number of virtual users per real user tends to infinity. When the capacity regions are asymmetric, we show that a point on the dominant face can be achieved asymptotically. Finally, when there is an unequal number of virtual users per real user, we show that differential user rate requirements can be accommodated in a distributed fashion.Index Terms-Multiple access, rate splitting, successive decoding, stripping, interference cancellation, ALOHA.PLACE PHOTO HERE
I. INTRODUCTION
We consider the basic multiple-access communication problem in a distributed setting. In Gallager's survey paper [1], it is pointed out that the multiple-access problem has been studied from a number of different perspectives, each having its own advantages and shortcomings. In the data networking community, a well-known distributed multiple access scheme is ALOHA [2], [3]. In ALOHA, it is assumed that a "collision" happens whenever more than one user transmit simultaneously. Those packets involved in a collision are discarded and retransmitted according to some retransmission probability. The collision channel model, however, does not accurately describe the underlying physical multiple-access channel. It is well known that there exist coding techniques which can decode multiple users' messages when simultaneous transmissions occur. Indeed, more sophisticated models such as signal capture [4], [5], spread ALOHA [6], and multi-packet reception [7], [8], [9] have been developed to enhance ALOHA. Even these improved schemes, however, are not optimal from the viewpoint of information theory. In the information theory literature, the capacity regions of various multiple-access channel (MAC) models have been characterized (see [10], [11], [12], [13], [14], [15]). Rate splitting multiple-access techniques (or generalized time-sharing) are presented in [16], [17], [18], [19] to achieve every point in the Gaussian or the discrete memoryless MAC capacity region using only singleuser codes. These schemes, however, require a pre-defined decoding order, which makes distributed implementation difficult. Finally, in the spread spectrum community, CDMA techniques are adopted. Here, users are decoded regarding all other users' signals as interference. This, however, is not optimal from the information theoretic viewpoint.
To address some of the shortcomings mentioned above, Medard et al. [20] use information-theoretic techniques to analyze different notions of capacity for time-slotted ALOHA systems. A coding/decoding scheme which combines rate splitting and superposition coding is constructed. This scheme allows some bits to be reliably received even when collision occurs, and more bits to be reliably received in the absence of collisions. Shamai [21] proposes a similar scheme to apply a broadcast strategy to multiple-access channel under static fading where the fading coefficients are not available to the transmitters or the receiver. To implement the scheme in [20], however, a pre-defined decoding order is required, as in [16], [17], [18], [19]. In [22], Cheng proposes a distributed scheme called "stripping CDMA" for the L out of K Gaussian MAC. Here, no pre-defined decoding order is required. It is shown in [22] that stripping CDMA is asymptotically optimal, although the optimal operating parameters are not specified.
In this paper, we investigate distributed multiple-access schemes based on the idea of rate splitting for both the M -user additive white Gaussian noise MAC and the Muser discrete memoryless MAC. We characterize the optimal operating parameters as well as the asymptotic optimality of these schemes from the viewpoint of information theory. Assume that every user has an infinite backlog of bits to send, and that every user knows the total number of users M . We propose a distributed scheme, called Distributed Rate Splitting (DRS), to achieve the optimal communication rates allowed by information theory. In this scheme, each real user creates a number of virtual users via a power/rate splitting mechanism in the M -user Gaussian channel or via a random switching mechanism in the M -user discrete memoryless channel. At the receiver, all virtual users are successively decoded. A possible advantage of the DRS scheme is that it can be implemented with lower complexity when compared with multiple-access schemes such as joint coding 1 and less coordination when compared with time-sharing and rate splitting.
In Sections II and III, we focus first on symmetric situations where the channel capacity regions are symmetric and every real user creates the same number of virtual users. In this case, the DRS scheme entails the following. Each user i creates L virtual users indexed by i k , k = 1, 2, ..., L. The virtual user class V k consists of users {1 k , ..., M k } (i.e. we have altogether L virtual user classes and there are M virtual users in each class). In the M -user Gaussian MAC, virtual users are created via a power/rate splitting mechanism. The signal transmitted by a real user is the superposition of all its virtual users' signals. The receiver receives the sum of the virtual users' signals plus noise. All virtual users are then successively decoded in increasing order of their class. That is, all virtual users in class V k , k = 1, ..., L, are decoded before any virtual user in V j , where j > k, is decoded. In contrast to [22], the optimal operating parameters, such as power and rate, are explicitly specified for any finite L. In the M -user discrete memoryless MAC, virtual users with the same input distribution as the real users are created, and the transmitted signal of a real user is determined by a random switch. The receiver successively decodes all virtual users in increasing order of their class given the side information of already decoded virtual users. The optimal switch is found for any finite L for the 2-user case. Finally, it is shown that for both channel models, the rate tuple achieved by the DRS scheme converges to the maximum equal rate point allowed by the information-theoretic bound as the number of virtual users per real user tends to infinity.
Next, in Section IV, we consider more general situations where the capacity regions can be asymmetric and real users may generate different numbers of virtual users. For the case of asymmetric capacity regions, new operating parameters are specified for any finite number of virtual users per real user. We show that the DRS scheme still can achieve a point on the dominant face as the number of virtual users per real user tends to infinity. For the case of unequal number of virtual users per real user, we present a variation of DRS which supports differential user rate requirements in a distributed manner. In this new scheme, each user i, independently from other users, generates L i virtual users according to its own rate requirement. All virtual users are then decoded reliably at the receiver. Furthermore, as each real user generates more virtual users, the rate tuple achieved under this variation of DRS converges the maximum equal rate point on the dominant face.
II. M -USER GAUSSIAN MULTIPLE-ACCESS CHANNEL
We first examine a Gaussian MAC with a symmetric capacity region. Later in Section IV, we consider the asymmetric case. Consider an M -user Gaussian MAC where each transmitter has transmission power P and the receiver has 1 The lower complexity comes from the fact that the DRS scheme uses single-user codes instead of multi-user codes. As we show later, the DRS scheme with a reasonable number of virtual users per real user allows us to get close to the optimal operating rates.
R = (R 1 , ..., R M ) ∈ R M + satisfying 2 i∈S R i ≤ 1 2 log 1 + |S|P N ∀S ⊆ {1, ..., M } ,(1)
where |S| is the cardinality of the set S. The dominant face D is the subset of rate tuples which gives equality in (1) for S = {1, ..., M }. For this symmetric setting, it is easy to see that the maximum common rate that every user can achieve 3 is R * = 1 2M log(1 + MP N ). It is well-known that rate tuples on the dominant face other than the vertices cannot be achieved via standard successive decoding [11]. Note that the optimal rate tuple R * ≡ (R * , ..., R * ), called the maximum equal rate point, is such a point. For the two-user Gaussian MAC, the maximum equal rate point is shown in Fig. 1. Currently, three methods are known to achieve general points on the dominant face: joint encoding/decoding, time-sharing, and rate-splitting. Joint encoding/decoding is not practical because of its high complexity [1]. In time-sharing, all M users need to coordinate their transmissions. Therefore, some communication overhead is required. The rate-splitting method in [16] achieves every point in C via a generalized successive decoding scheme. For the two-user case, user 1 creates two virtual users, say 1a and 1b, by splitting its power P into δ and P − δ and setting
r 1a = 1 2 log(1+ δ 2P −δ+N ), r 1b = 1 2 log(1+ P −δ N )
. User 2 does not split its power and sets its rate to R 2 = 1 2 log(1+ P P −δ+N ). The decoding order is (1a, 2, 1b). In order to achieve the maximum equal rate point, we solve R 2 = r 1a + r 1b , yielding δ = 1 2 (N + 2P − N (N + 2P )). Thus, both time-sharing and rate splitting require some coordination among users.
In this paper, we focus on distributed multiple-access communication schemes. In particular, we introduce the Distributed Rate Splitting (DRS) scheme. The DRS scheme offers the possibility of multiple-access communication with lower complexity when compared with joint coding, and communi- cation with less coordination when compared with the timesharing or rate splitting method. Moreover, we show that DRS can achieve the maximum equal rate point of the MAC capacity region asymptotically. We now formally present the DRS scheme. In this scheme, each user creates L virtual users by splitting its power P into (p 1 , p 2 , ..., p L ), where p k is the power allocated to the kth virtual user and L k=1 p k = P . Each user then assigns transmission rate r k to virtual user k. Note that the proposed DRS scheme is symmetric, i.e. all M users split their powers and set their rates in the same way. The signal transmitted by a user is the superposition of its virtual users' signals. As defined in Section I, virtual user class V k consists of all virtual users indexed by k. The receiver receives the sum of all virtual users' signals plus noise. All virtual users are then successively decoded in increasing order of their class.
To illustrate the DRS scheme, consider the case L = 2. Each real user splits its power P into δ and P − δ. Notice there are two major differences between our scheme and the traditional rate splitting scheme in [16]. First, in our scheme, all real users split in the same way, whereas there is at least one user who does not split in the traditional rate splitting scheme. Second, virtual users in the same class, (i.e. with the same index k), are allocated the same rate in our scheme, whereas all virtual users have different rates according to the pre-defined decoding order in the traditional rate splitting scheme. 4 These differences are illustrated in Fig. 2.
Since we assume the receiver uses successive decoding method, some virtual user must be decoded first. Without loss of generality we assume one of the δ virtual users is decoded first. For the case L = 2, we show that there is a unique way for a real user to split its power in order to maximize its total throughput.
Lemma 1: For L = 2 and for a fixed δ, each real user's throughput is maximized by setting r 1 = 1 2 log(1 + δ MP −δ+N ) and r 2 = 1 2 log(1 + P −δ (M−1)(P −δ)+N ). 4 In terms of achievable rate, the DRS scheme with L = 2 is not optimal. Later in this section, we demonstrate the asymptotic optimality of DRS by taking L to infinity.
Proof: The δ virtual user who is decoded first must have r 1 = 1 2 log(1 + δ MP −δ+N ) (i.e. the virtual user regards all other virtual users as interference) in order to be decoded successfully. Due to symmetry, all other δ virtual users must have the same r 1 . Then the problem of maximizing each real user's throughput reduces to max r 2 , subject to (i) r 1 = 1 2 log(1 + δ MP −δ+N ), (ii) one of the δ virtual users is decoded first and (iii) (r 1 , r 2 , r 1 , r 2 , ..., r 1 , r 2 ) must be decodable.
Note that r 2 is maximized when the interference plus noise faced by all the (P − δ) virtual users is minimized, and the only way to minimize the interference plus noise faced by all the (P − δ) virtual users is to decode all the δ virtual users before decoding any (P − δ) virtual user. 5 Therefore, the minimum interference plus noise faced by any
(P − δ) virtual user is M P − M δ − (P − δ) + N = (M − 1) (P − δ) + N .
Hence, the maximum rate associated with a (P − δ) virtual user is
r 2 = 1 2 log(1 + P −δ (M−1)(p−δ)+N ).
Using the DRS scheme with L = 2, each user can strictly increase its throughput relative to the case where users do not split their powers and decode against each other as noise. This is easily verified by observing that for any δ < P ,
log 1 + δ M P − δ + N +log 1 + P − δ (M − 1) (P − δ) + N > log 1 + P (M − 1) P + N .(2)
Now consider the case where each user creates more than two virtual users (L > 2). Here, we show that each user's throughput increases further.
r L ′ = 1 2 log 1 + p ′ L M p L − p ′ L + N r L ′′ = 1 2 log 1 + p ′′ L (M − 1) p ′′ L + N .
Now each real user has L + 1 virtual users. Notice that we do not change the power and decoding order of any of the other virtual users (i.e. virtual users 1, . . . , L − 1). From a real user's view point, the virtual user with r L ′ is decoded second to last among all virtual users generated by this real user and the virtual user with r L ′′ is decoded last. Thus, all virtual users can be decoded and from (2), r L ′ +r L ′′ > r L . Therefore, every real user with L virtual users can strictly increase its throughput by splitting its power among L+1 virtual users.
Before we examine the asymptotic behavior of DRS, we solve the problem of how to split a user's power optimally among a fixed number of virtual users. The main difficulty here is that the objective function is not concave. In order to find the optimal splitting method, we prove the following lemma.
Lemma 3: Consider the following optimization problem:
max p k ,pj 1 2 log 1 + p k A − p k + 1 2 log 1 + p j A − M p k − p j (3) subject to p k +p j = c and p k , p j ≥ 0, where A, M and c are positive constants and A ≥ M c. The unique solution to (3) is also the unique solution to p k A−p k = pj A−Mp k −pj , subject to p k + p j = c and p k , p j ≥ 0, where A ≥ M c. Proof: Substitute p j = c − p k into the objective function, we have f (p k ) = 1 2 log 1 + p k A − p k + 1 2 log 1 + c − p k A − M p k − (c − p k )
.
Setting df (p k ) dp k = 0 subject to 0 ≤ p k ≤ c, the unique solution is p * k = 1 M (A − A(A − cM )
). Thus, p * k is the unique stationary point of f (p k ). We can also verify that
f (p * k ) > f (0) and f (p * k ) > f (c). So (p * k , c − p * k )
is the unique solution to our maximization problem. We can
directly solve p k A−p k = pj A−Mp k −pj subject to p k + p j = c and p k , p j ≥ 0. The unique solution is also (p * k , c − p * k ).
We now present the optimal splitting method. Theorem 1 states a necessary condition for the optimal splitting method, and Theorem 2 implies there is a unique optimal splitting method. In Corollary 1, we formally present the optimal splitting method and the required power levels.
Theorem 1: Let each real user split its power into L virtual users. Let p k be the power allocated to the kth virtual user and r k = 1 2 log (1 +
p k MP −M j<k pj −p k +N ). If (p * 1 , ...p * L ) maximizes L k=1 r k and satisfies L k=1 p * k = P , p * k ≥ 0 for k = 1, 2..., L, then r k (p * 1 , ...p * L ) = r * , for all k.
That is, the optimal power split must lead to equal transmission rates for all virtual users.
Proof: We use a perturbation argument. Suppose (p 1 , ...,p L ) maximizes L k=1 r k and satisfies L k=1p k = P ,p k ≥ 0 ∀k and the resulting r k (p 1 , ...,p L ) is not the same for all k. Then we can find a pair of virtual users (k, k + 1), where virtual user k and k + 1 are decoded at the kth and (k + 1)th places respectively, and r k (p 1 , ...,p L ) = r k+1 (p 1 , ...,p L ).
Without loss of generality, let us consider the case where r k (p 1 , ...,p L ) > r k+1 (p 1 , ...,p L ). By the definition of r k , we havep
k M P − M j<kp j −p k + N >p k+1 M P − M j<kp j − Mp k −p k+1 + N .(4)
We can verify that if we changep k top k − ε andp k+1 tõ p k+1 + ε, where ε is a small positive number, then the first term of (4) decreases and the second term of (4) increases. Let ε * be the solution tõ
p k − ε M P − M j<kp j − (p k − ε) + N =p k+1 + ε M P − M j<kp j − M (p k − ε) − (p k+1 + ε) + N . (The existence of ε * can be demonstrated). Let p k =p k − ε * andp k+1 =p k+1 + ε * . Notice that p k +p k+1 =p k +p k+1 .
Since the maximization considered in Lemma 3 has a unique solution,
r k (p 1 , ...,p k ,p k+1 , ...,p L ) + r k+1 (p 1 , ...,p k ,p k+1 , ...,p L ) > r k (p 1 , ...,p k ,p k+1 , ...,p L ) + r k+1 (p 1 , ...,p k ,p k+1 , ...,p L ).
This contradicts our assumption that (p 1 , ...,p L ) maximizes L k=1 r k . Therefore, the theorem follows.
By Theorem 1, if (p * 1 , ..., p * L ) maximizes L k=1 r k and satisfies L k=1 p * k = P , p * k ≥ 0 for k = 1, 2..., L, then we must have SIR 1 (p * 1 , ..., p * L ) = ... = SIR L (p * 1 , ..., p * L ) where SIR k (p 1 , ..., p L ) = p k MP −M j<k pj −p k +N .
Therefore, if we show that SIR k (p 1 , ..., p L ) = SIR j (p 1 , ..., p L ) for all k = j, has a unique solution, then there is at most one feasible solution to the maximization problem.
Theorem 2: The set of equations: SIR 1 (p 1 , ..., p L ) = ... = SIR L (p 1 , ..., p L ), subject to L k=1 p k = P and p k ≥ 0 ∀k, has a unique solution.
Proof: See Appendix I.
Corollary 1: If a real user splits its power P into L virtual users, then the unique way to maximize this user's throughput is to set
p k = N M (1+ MP N ) L−k L [(1+ MP N ) 1 L −1] for k = 1, ..., L.
Proof: Since the constraint region {p : L k=1 p k = P and p k ≥ 0 for k = 1, 2, .., L} is a simplex and L k=1 r k is continuous, there exists at least one solution. We denote one solution by (p 1 , ...,p L ). By the necessary condition stated in Theorem 1, (p 1 , ...,p L ) must satisfy r 1 (p 1 , ...,p L ) = ... = r L (p 1 , ...,p L ). Moreover, by the uniqueness property stated in Theorem 2, (p 1 , ...,p L ) is the unique solution to max p1,..,pL L k=1 r k subject to L k=1 p k = P and p k ≥ 0 ∀k. Next, we plug p k into the expression of SIR k (p 1 , ..., p L ).
Let A = 1 + MP N , we are able to verify that SIR k (p 1 , ..., p L ) = A 1 L −1 (M−1)A 1 L +1
, which is independent of k. Hence, the corollary follows.
We now examine the asymptotic behavior of the DRS scheme. We first demonstrate the interesting fact that the rate tuple converges to the maximum equal rate point for a general power split as long as all virtual users' powers go to 0 as L → ∞. This implies a convergence result for the optimal power split. We then analyze the rate of convergence under the optimal power split.
Theorem 3: Given any power split (p 1 , ..., p L ), a sufficient condition for lim L→∞
L k=1 r k = 1 2M log(1 + MP N ) is max k∈{1,...,L} p k → 0 as L → ∞. Proof: lim L→∞ L k=1 r k = lim L→∞ L k=1 1 2 log 1 + p k M P − M j<k p j − p k + N = lim L→∞ L k=1 1 2 p k M P − M j<k p j − p k + N (5) ≥ lim L→∞ L k=1 1 2 p k M P − M j<k p j + N (6) = lim L→∞ L k=1 1 2 β k − β k−1 M P + N − M β k−1 (7) = 1 2 P 0 1 M P + N − M x dx (8) = 1 2M log 1 + M P N(9)
where β 0 = 0, β k = k j=1 p j . The equality in (5) is justified as follows. Note that, lim x→0
log(1+x)−x x 2 = − 1 2 . Hence, for sufficiently small posi- tive δ, |x| < δ ⇒ | log(1+x)−x x 2 | < 1 ⇒ | log(1 + x) − x| ≤ x 2 . Now, we examine the error term L k=1 log 1 + p k M P − M j<k p j − p k + N − p k M P − M j<k p j − p k + N ≤ L k=1 log 1 + p k M P − M j<k p j − p k + N − p k M P − M j<k p j − p k + N ≤ L k=1 p k M P − M j<k p j − p k + N 2 (10) ≤ L k=1 1 N 2 p 2 k (11) ≤ 1 N 2 max k p k L k=1 p k (12) = P 1 N 2 max k p k(13)
where inequality in (10) holds because p k MP −M j<k pj −p k +N < δ when L → ∞, and the inequality in (11) follows from the fact that
p k MP −M j<k pj −p k +N ≤ p k N .
Since max k p k → 0 as L → ∞, the error term goes to zero in the limit. This justifies the equality in (5). Using the capacity bound, we also have lim L→∞
L k=1 r k ≤ 1 2M log(1 + MP N ). Therefore, lim L→∞ L k=1 r k = 1 2M log(1 + MP N ).
Note that our optimal power split satisfies the sufficient condition in Theorem 3. Therefore, its convergence is assured.
Corollary 2:
If each real user adopts the optimal splitting method specified in Corollary 1, then
lim L→∞ L k=1 r k = 1 2M log 1 + M P N ≡ R * .
Next, we examine the rate of convergence to the maximum equal rate point under the optimal power split. Define the error term e[L] ≡ R * − L k=1 r k , we analyze how fast this error term tends to 0 as L → ∞. We prove the following: 6
Theorem 4: e [L] = Θ 1 L . Proof: Let A 1 + MP N . lim L→∞ Le [L] = lim L→∞ L 1 2M log (A) − L 2 log M M − 1 + A − 1 L = lim y→0 y log(A) 2M − 1 2 log M M−1+A −y y 2 (14) = lim y→0 log(A) 2M − 1 2 A −y log(A) M−1+A −y 2y (15) = (M − 1) (log (A)) 2 4M 2 .(16)
Note that equalities in (15) and (16) can be verified by L'Hospital's rule. Consequently, given any ε > 0, there exists a positive integer n 0 such that for all L > n 0 , we have
Le [L] − (M−1)(log(A)) 2 4M 2 < ε. This implies (M−1)(log(A)) 2 4M 2 − ε < Le [L] < (M−1)(log(A)) 2 4M 2 + ε. Therefore, we can choose a small enough ε such that (M−1)(log(A)) 2 4M 2 − ε > 0. Let c 1 = (M−1)(log(A)) 2 4M 2 − ε, c 2 = (M−1)(log(A)) 2 4M 2 + ε, we have c 1 ≤ Le [L] ≤ c 2 .
This implies that there exists a positive Finally, we note that all virtual users in one virtual user class can be decoded in parallel. Thus, the decoding delay of DRS is proportional to the number of virtual users L and independent of the number of real users. Since L is controlled by the designer, DRS offers a tradeoff between the throughput of a real user and the decoding delay. In Fig. 3, we present some numerical simulations illustrating the tradeoff between the number of virtual users and the throughput for both the high and low SNR regimes. A system with 100 real users is used in the simulations.
III. M-USER DISCRETE MEMORYLESS MAC
An M -user discrete memoryless MAC is defined in terms of M discrete input alphabets X i , i ∈ {1, ..., M }, an output alphabet Y and a stochastic matrix W : X 1 × X 2 × · · · × X M → Y with entries W (y | x 1 , ..., x M ). For any product input distribution P X1 · · · P XM , let the achievable region R[W ; P X1 · · · P XM ] be the set of
R ∈ R M + satisfying i∈S R i ≤ I (X S ; Y | X S c ) , ∀S ⊆ {1, ..., M } where X S (X i ) i∈S and S c {1, ..., M } \ S.
The capacity region of the asynchronous MAC is [14] [15]
C = PX 1 ···PX M R[W ; P X1 · · · P XM ].(17)
We fix the input product distribution P X1 · · · P XM and focus on achieving the desired operating point in R[W ; P X1 · · · P XM ]. In this section, we consider only discrete memoryless channels satisfying the following symmetry condition: In the Gaussian MAC, virtual users are created via a power/rate splitting mechanism. For the discrete memoryless MAC, we adopt the random switching mechanism of [19] where virtual users with the same input distribution as the real users are created and the transmitted signal of a real user is determined by a random switch. We first consider the two-user discrete memoryless MAC (M = 2), and illustrate the random switching mechanism. The optimal random switches and the asymptotic behavior of DRS under the optimal switching are presented. We then examine the M -user case (M > 2), and present a sufficient condition for the random switching mechanism to converge to the information theoretic upper bound. Finally, we investigate the rate of convergence for a simple suboptimal random switch.
I (X S ; Y | X S c ) = I (X T ; Y | X T c ) ,(18)
A. Two-user Case (M = 2)
Consider a two-user MAC, W : X 1 × X 2 → Y. For a fixed input product distribution P X1 P X2 , the achievable region is given by:
R 1 ≤ I (X 1 ; Y |X 2 ) = I (X 1 ; Y, X 2 ) R 2 ≤ I (X 2 ; Y |X 1 ) = I (X 2 ; Y, X 1 ) R 1 + R 2 ≤ I (X 1 , X 2 ; Y ) .
Under our symmetry assumption (cf (18)), we have I(X 1 ; Y ) = I(X 2 ; Y ), I(X 1 ; Y, X 2 ) = I(X 2 ; Y, X 1 ), I(X 1 ; Y, X 2 ) > I(X 1 ; Y ), and the optimal rate tuple is
(R * , R * ) ≡ 1 2 I (X 1 , X 2 ; Y ) , 1 2 I (X 1 , X 2 ; Y ) .(19)
Let us consider the random switching mechanism for this channel. We first consider the case where each real user generates two virtual users. Later, we consider the case where the number of virtual users per real user goes to infinity. We split by means of two switches, as shown in Fig. 4. Each switch has two inputs, X i1 ∈ X i and X i2 ∈ X i and one output X i ∈ X i . Switch i is controlled by a random variable
S i ∈ {1, 2} with P (S i = 1) = λ.
The output is given by
X i = X i1 if S i = 1, and X i = X i2 if S i = 2.
The switching random variables {S 1 , S 2 } are independent of the channel inputs. We also assume that {S 1 , S 2 } are available at the receiver. In practice, one would generate S 1 and S 2 at the transmitters and at the receiver, e.g. by means of a pseudorandom sequence generator. Assign to the channel inputs X 11 , X 12 , X 21 and X 22 the probability mass function P X11,X12,X21,X22 (x 11 , x 12 , x 21 , x 22 ) = P X1 (x 11 )P X1 (x 12 )P X2 (x 21 )P X2 (x 22 ). Notice that X i1 and X i2 are independent and each has the same probability mass function as the random variable X i for i = 1, 2.
In successive decoding for the discrete memoryless MAC, the signals of decoded virtual users are used as side information to aid the decoding process of subsequent virtual users. The first constituent decoder observes the output {Y, S 1 , S 2 } and tries to decode X 11 and X 21 . The second constituent decoder is informed of the decision about {X 11 , X 21 } made by the previous constituent decoder and tries to decode X 12 and X 22 . Without loss of generality, let us focus on real user 1.
r X11 = I(X 11 ; Y, S 1 , S 2 ) = I(X 11 ; Y, S 2 |S 1 ) = λI(X 11 ; Y, S 2 |S 1 = 1) + (1 − λ) I(X 11 ; Y, S 2 |S 1 = 2) = λI(X 1 ; Y )
where the second equality follows from the independence between X 11 and S 1 , and the last equality follows from the fact that when S 1 = 2, X 11 is independently of the output Y and S 2 . Similarly, we have
r X12 = I(X 12 ; Y, S 1 , S 2 , X 11 , X 21 ) = (1 − λ) [λI(X 1 ; Y, X 2 ) + (1 − λ)I(X 1 ; Y )] .
It can be verified that in the two-user discrete memoryless MAC, both real users' throughput can be strictly increased by splitting their inputs via a random switch, relative to the case where they do not split, (i.e. r X11 + r X12 > I(X 1 ; Y ) for λ ∈ (0, 1)). Next, we show that by generating more virtual users, the throughput of each real user increases further.
Lemma 4: For M = 2, consider a distributed rate splitting scheme with L virtual users per real user. The random switch for user i is controlled by S i , where P(S i = k) = λ k for k = 1, ..., L. It is possible to strictly increase the throughput via an (L + 1) virtual user system by splitting the Lth virtual user into two virtual users.
Proof: Without loss of generality, we consider user 1. For the kth virtual user, we have r X 1k = I X 1k ; Y, S 1 , S 2 , X 11 , X 21 , ..., X 1(k−1) , X 2(k−1) = I (X 1k ; Y, S 2 , X 11 , X 21 , ...,
X 1(k−1) , X 2(k−1) | S 1 (20) = λ k I (X 1k ; Y, S 2 , X 11 , X 21 , ..., X 1(k−1) , X 2(k−1) | S 1 = k + (1 − λ k ) I (X 1k ; Y, S 2 , X 11 , X 21 , ..., X 1(k−1) , X 2(k−1) | S 1 = k = λ k I (X 1k ; Y, S 2 , X 11 , X 21 , ..., X 1(k−1) , X 2(k−1) | S 1 = k (21) = λ k j<k λ j I (X 1k ; Y, X 11 , X 21 , ..., X 1(k−1) , X 2(k−1) | S 1 = k, S 2 < k + 1 − j<k λ j I (X 1k ; Y, X 11 , X 21 , ..., X 1(k−1) , X 2(k−1) | S 1 = k, S 2 ≥ k = λ k j<k λ j I (X 1 ; Y, X 2 ) + 1 − j<k λ j I (X 1 ; Y ) ,(22)
where equality in (20) is due to the independence between X 1k and S 1 , and equality in (21) follows from the fact that when S 1 = k, X 1k is independent of the output Y and all the other random variables. Finally, equality in (22) holds because when S 2 < k, one of the random variables X 21 , ..., X 2(k−1) is the switch output, and when S 2 ≥ k, none of them is the switch output. Therefore,
r X1L = λ L [(1 − λ L ) I (X 1 ; Y, X 2 ) + λ L I (X 1 ; Y )] .
Now let us split r X1L into r X 1 1L and r X 2 1L by using a switch controlled by a binary random variable S ′ with P(S ′ = 0) = α. We have
r X 1 1L = αλ L ((1 − λ L ) I (X 1 ; Y, X 2 ) + λ L I (X 1 ; Y )) , r X 2 1L = αλ L ((1 − αλ L ) I (X 1 ; Y, X 2 ) + αλ L I (X 1 ; Y )) , whereᾱ = 1 − α. Hence, r X 1 1L + r X 2 1L − r X1L = αλ L ((1 − λ L ) I (X 1 ; Y, X 2 ) + λ L I (X 1 ; Y )) +αλ L ((1 − αλ L ) I (X 1 ; Y, X 2 ) + αλ L I (X 1 ; Y )) −λ L [(1 − λ L ) I (X 1 ; Y, X 2 ) + λ L I (X 1 ; Y )] = ααλ 2 L [I (X 1 ; Y, X 2 ) − I (X 1 ; Y )] ≥ 0
with strict inequality if α ∈ (0, 1).
Before we examine the asymptotic behavior of the DRS scheme for M = 2, we first solve the problem of how to find the optimal switches for a fixed number of virtual users per real user.
Theorem 5: For M = 2, if a real user has L virtual users, then the optimal random variable to control the switch for user i is S i ∈ {1, ..., L} with P(S i = k) = 1 L for k = 1, ..., L and i = 1, 2.
Proof: We use a perturbation argument. Suppose the random variables S 1 , S 2 ∈ {1, ..., L} with P(S 1 = k) = P(S 2 = k) = λ k maximize R X1 , where λ k ≥ 0 and L k=1 λ k = 1. Moreover, suppose there exists λ k such that λ k = 1 L . Let λ k be the first element which is not equal to 1 L . We consider the pair (λ k , λ k+1 ). We have
r 1k = λ k k − 1 L I(X 1 ; Y, X 2 ) + 1 − k − 1 L I(X 1 ; Y ) , r 1(k+1) = λ k+1 k − 1 L + λ k I(X 1 ; Y, X 2 ) + 1 − k − 1 L − λ k I(X 1 ; Y ) .
Therefore,
r 1k + r 1(k+1) = (λ k + λ k+1 ) k − 1 L + λ k λ k+1 I(X 1 ; Y, X 2 ) + (λ k + λ k+1 ) L − k + 1 L − λ k λ k+1 I(X 1 ; Y ).
First, we consider the case where λ k > λ k+1 . We letλ k = λ k − ε andλ k+1 = λ k+1 + ε for ε < λ k −λ k+1 2 . We havê r 1k +r 1(k+1)
= (λ k + λ k+1 ) k − 1 L + (λ k − ε) (λ k+1 + ε) I(X 1 ; Y, X 2 ) + (λ k + λ k+1 ) L − k + 1 L − (λ k − ε) (λ k+1 + ε) I(X 1 ; Y ).
Thus,r 1k +r 1(k+1) -(r 1k + r 1(k+1) ) = [(λ k + λ k+1 ) ε -ε 2 ] (I(X 1 ; Y, X 2 )-I(X 1 ; Y )). Notice that the second term of the R.H.S. expression is positive and there exists ε such that ((λ k + λ k+1 ) ε-ε 2 ) > 0. This, however, contradicts our assumption that the random variables S 1 , S 2 ∈ {1, ..., L} with P(S 1 = k) = P(S 2 = k) = λ k , maximize R X1 . We obtain similar contradictions for the case where λ k ≤ λ k+1 .
We now examine the asymptotic behavior of the DRS scheme in the two-user discrete memoryless MAC. Proof: Without loss of generality, we consider real user 1. Since
r X 1k = 1 L k − 1 L I(X 1 ; Y, X 2 ) + 1 − k − 1 L I(X 1 ; Y ) , we have lim L→∞ R X1 = lim L→∞ 1 L 1 + 1 L + ... + L − 1 L I(X 1 ; Y ) + 1 L L − 1 L + ... + 1 L I(X 1 ; Y, X 2 ) = lim L→∞ L + 1 2L I(X 1 ; Y ) + L − 1 2L I(X 1 ; Y X 2 ) = 1 2 [I(X 1 ; Y ) + I(X 1 ; Y, X 2 )] = 1 2 I(X 1 , X 2 ; Y ),
where the last equality follows from the fact that I(X 1 ; Y, X 2 ) = I(X 2 ; Y, X 1 ) in the symmetric setting. Next, we examine the rate of convergence. We have the error term e
[L] = 1 2 I(X 1 ; Y ) + 1 2 I(X 1 ; Y, X 2 ) − ( L+1 2L I(X 1 ; Y ) + L−1 2L I(X 1 ; Y, X 2 )). Thus, e[L] = 1 2L (I(X 1 ; Y, X 2 ) − I(X 1 ; Y )) , which implies lim L→∞ Le[L] = 1 2 (I(X 1 ; Y, X 2 ) − I(X 1 ; Y )) c 2 .
For any ε > 0, there exists an n 0 such that for all L > n 0 , |Le[L] − c 2 | < ε. Hence, c 2 − ε < Le[L] < c 2 + ε. We can choose ε small enough such that c 2 − ε > 0. This implies e[L] = Θ( 1 L ).
B. M -user Case (M > 2)
There are M real users and each real user creates L virtual users. We have M switches (S 1 , ..., S M ) to do the splitting with probabilities P(S i = k) = λ k for i = 1, ..., M and k = 1, ..., L. We assume the receiver also knows (S 1 , ..., S M ). This may require common randomness to exist between all transmitters and the receiver. Due to symmetry, we focus on one user, say user 1.
| S 1 = k) + (1 − λ k ) I(X 1k ; Y, S M 2 , X 1(k−1) 11
, ...,
X M(k−1) M1 | S 1 = k) = λ k I(X 1k ; Y, S M 2 , X 1(k−1) 11 , ..., X M(k−1) M1 | S 1 = k) (25) = λ k I(X 1 ; Y, X 1(k−1) 11 , ..., X M(k−1) M1 |S 1 = k, S M 2 ) = s2,...,sM P (S 2 = s 2 , ..., S M = s M ) λ k I(X 1 ; Y, X 1(k−1) 11 , ..., X M(k−1) M1 |S 1 = k, S 2 = s 2 , ..., S M = s M ) = λ k 1 − j<k λ j M−1 I(X 1 ; Y ) + M−1 i=1 M − 1 i j<k λ j i · 1 − j<k λ j M−1−i I(X 1 ; Y, X 2 , ..., X i+1 ) , (26) where S j i {S i , ..., S j } and X ik i1
{X i1 , ..., X ik }. Equality in (24) is due to the independence between X 1k and S 1 . Equality in (25) holds because when S 1 = k, X 1k is independent of the output Y and all the other random variables. The first term in (26) follows from P(S 2 ≥ k, ..., S M ≥ k) = (1 − j<k λ j ) M−1 . The second summation term in (26) follows from the fact that the probability of i switching random variables among (S 2 , ..., S M ) having values less than k is
M−1 i ( j<k λ j ) i (1− j<k λ j ) M−1−i .
It can be verified that real user 1 with L virtual users can strictly increase its total throughput via an L + 1 virtual user system.
In order to maximize the total throughput of real user 1 for fixed L, we need to find the optimal (λ * 1 , ..., λ * L ) to maximize L k=1 r X 1k . This is a non-convex optimization problem and appears to be difficult. We are able to verify that for the general M -user case (unlike the two-user case), random switches with a uniform distribution are in general suboptimal. Nevertheless, it is possible to generalize the asymptotic result of Theorem 6. We first demonstrate the fact that the convergence result holds for a general switch controlled by S i , where P (S i = k) = λ k for k = 1, ..., L, as long as max k∈{1,...,L} λ k → 0 as L → ∞. We then analyze the rate of convergence for a particular suboptimal switch, the uniform switch. Proof: Without loss of generality, let us examine real user 1.
lim L→∞ R X1 = lim L→∞ L k=1 r X 1k = lim L→∞ L k=1 λ k 1 − j<k λ j M−1 I(X 1 ; Y ) + M−1 i=1 M − 1 i j<k λ j i · 1 − j<k λ j M−1−i I(X 1 ; Y, X 2 , ..., X i+1 ) = lim L→∞ M−1 i=0 c i I i , where I 0 = I(X 1 ; Y ) I i = I(X 1 ; Y, X 2 , ..., X i+1 ) c 0 = L k=1 λ k 1 − j<k λ j M−1 c i = L k=1 λ k M − 1 i j<k λ j i 1 − j<k λ j M−1−i , for i ≥ 1. It is sufficient to prove lim L→∞ c i = 1 M for all i. For i ≥ 1, lim L→∞ c i = lim L→∞ L k=1 λ k M − 1 i j<k λ j i 1 − j<k λ j M−1−i = lim L→∞ L k=1 (β k − β k−1 ) M − 1 i β i k−1 (1 − β k−1 ) M−1−i = M − 1 i 1 0 x i (1 − x) M−1−i dx = M − 1 i B (i + 1, M − i) = (M − 1)! i!(M − i − 1)! i!(M − i − 1)! M ! = 1 M , where β 0 = 0, β k = k j=1 λ k and B(m, n) = (m−1)!(n−1)! (m+n−1)!
is the beta function. The term c 0 can be shown to converge to 1 M as L → ∞ by a similar argument.
Next, we analyze a particular suboptimal switch, the uniform switch. Since the uniform switch satisfies the sufficient condition in Theorem 7, the convergence result holds. The next lemma presents its rate of convergence.
R X1 = L k=1 r X 1k = 1 L I(X 1 ; Y ) + L k=2 1 L M−1 i=0 M − 1 i k − 1 L i · 1 − k − 1 L M−1−i I(X 1 ; Y, X 2 , ..., X i+1 ) , where I(X 1 ; Y, X 2 , ..., X i+1 ) = I(X 1 ; Y ) for i = 0. We denote R X1 c 0 I 0 + ... + c M−1 I M−1 ,
where I i is defined in the proof of Theorem 7 for i ≥ 0, and
c 0 = 1 L L k=1 1 − k − 1 L M−1 c i = L k=1 1 L M − 1 i k − 1 L i 1 − k − 1 L M−1−i .
for i ≥ 1. Therefore, the error term can be calculated as follows,
e[L] = 1 M I(X 1 , X 2 , ..., X M ; Y ) − M−1 i=0 c i I i = M−1 i=0 1 M − c i I i ≤ M−1 i=0 1 M − c i I i ≤ M max i 1 M − c i I i .
Note that
x i (1 − x) M−1−i is maximized at x = i M−1 . For i ≥ 1, it can be verified that 1 L L−1 l=0 l L i 1 − l L M−1−i + i M − 1 i 1 − i M − 1 M−1−i 1 L ≥ 1 0 x i (1 − x) M−1−i dx, 1 L L−1 l=0 l L i 1 − l L M−1−i − i M − 1 i 1 − i M − 1 M−1−i 1 L ≤ 1 0 x i (1 − x) M−1−i dx.
Multiplying both sides of the above two inequalities by M−1
i , we have 1 M − c i ≤ M − 1 i i M − 1 i 1 − i M − 1 M−1−i 1 L .
Therefore,
max i 1 M − c i I i ≤ max i M − 1 i i M − 1 i 1 − i M − 1 M−1−i I i 1 L ≡ α 1 L , so e[L] ≤ M α 1 L , where α > 0.
For the term c 0 , a similar argument can be used to show that |c 0 − 1
M | ≤ 1 L . Therefore, e[L] = O 1 L .
IV. VARIATIONS OF DISTRIBUTED RATE SPLITTING
In Section II and Section III, we imposed two symmetry constraints. The first is that the capacity region for the Gaussian MAC and the achievable rate region for the discrete memoryless MAC are symmetric. The second is that users generate the same number of virtual users. In this section, we describe two variations of DRS. The first variation is presented in Section IV-A, where we relax the symmetric region constraint. In this case, we show that as the number of virtual user per real user tends to infinity, the rate tuple achieved under DRS approaches a point on the dominant face. The second variation is presented in Section IV-B, where each real user may generate a different number of virtual users. The main advantage of this variation is that it can accommodate different user rate requirements in a distributed fashion.
A. Asymmetric Capacity/Achievable Rate Region 1) M -user Gaussian MAC: In this section, we consider the case where real users in a Gaussian MAC may have different transmission powers (i.e. the capacity region may not be symmetric). We assume that user i has transmission power P i and the power vector (P 1 , ..., P M ) is known to all users. We also assume that all real users split their powers into L virtual users according to the common power splitting rule defined by the vector (γ 1 , γ 2 , ..., γ L ), where γ k > 0 ∀k and L k=1 γ k = 1. The power vector for the virtual users generated by user i is (γ 1 P i , ..., γ L P i ) for i = 1, ..., M . Proof: By replacing p k by γ k P i for k = 1, ..., L, we can use arguments similar to those in Section II to prove the following: Fig. 5. Achievable point for a two-user Gaussian MAC where user 1 has higher transmission power. The transmission powers and the achievable rates satisfy
R 2 R 1 Achievable Point R 2 * R 1 *R * 1 R * 2 = P 1 P 2 .
1) Given a DRS scheme with L virtual users per real user,
it is possible to strictly increase the throughput via an (L + 1) virtual user system. 2) Under the optimal power split, all virtual users generated by real user i must have the same rate for i = 1, ..., M . (Virtual users generated by different real users may have different rates.) 3) For any real user with L virtual users, the unique way to maximize this user's throughput is to set We illustrate this achievable point on the dominant face for a two-user Gaussian MAC in Fig. 5.
γ k = N M i=1 Pi 1 + M i=1 Pi N L−k L 1 + M i=1 Pi
2) M -user Discrete Memoryless MAC: In Section III, we considered the symmetric setting (cf (18)): Proof: We can replace M−1 i I (X 1 ; Y, X 2 , ..., X i+1 ) by ∅⊂S⊆−i,|S|=j I (X i ; Y, X S ) and use arguments similar to those for Theorem 7 to prove the above lemma.
I (X S ; Y | X S c ) = I (X T ; Y | X T c ) ,
B. Unequal Number of Virtual Users
1) M -user Gaussian MAC: In this section, we retain the assumption that every user has the same transmission power P , but we do not require all real users to create the same number of virtual users. That is, user i and user j create L i and L j virtual users independently, where L i may not be equal to L j . The signal transmitted by a real user is the superposition of all its virtual users' signals. We also assume in this section that user i transmits the number L i in a header message to the receiver. The receiver receives the sum of M i=1 L i signals plus noise. We now describe a protocol which allows each user to split its power and set its rates independently, and allows the receiver to decode all virtual users one by one via a generalized successive decoding mechanism. Recall that for the Gaussian MAC, successive decoding works as follows. Users are decoded one after another regarding all other users that have not been decoded as interference, and the signals of decoded users are subtracted from the overall received signal.
PROTOCOL 1: For user i, the power split and rate allocation rule are defined as follows: for k = 1, ..., L i ,
p ik = N M 1 + M P N L i −k L i 1 + M P N 1 L i − 1 , r ik = 1 2 log 1 + p ik M P − j<k p ij − p ik + N .
Note that the power split and rate allocation rule in PROTO-COL 1 are the same as that discussed in Section II. The generalized successive decoding algorithm is given by the following pseudo-program. Note that after a virtual user is decoded, its signal is subtracted from the overall received signal. To illustrate the decoding algorithm, let us carefully examine a three-user example shown in Fig. 6. The shaded regions correspond to the virtual users (11,21,31), which are decoded in any order. Suppose we decode "11" first. By the rate allocation rule, r 11 = 1 2 log(1 + p11 3P −p11+N ), which means the maximum amount interference plus noise that virtual user "11" can tolerate is 3P − p 11 + N . This is exactly the amount of interference plus noise it faces. Therefore, "11" can be decoded reliably and we can subtract the signal of virtual user "11" from the overall received signal. Similarly, (21, 31) can be decoded reliably and subtracted from the overall received signal. Now p = (p 11 , p 21 , p 31 ). The subsequent decoding order is illustrated by the numbers in Fig. 6. In the first run of the while loop, since p 31 is the minimum in p, the receiver decodes virtual user "32". By the rate allocation rule, r 32 = 1 2 log(1 + p32 3(P −p31)−p32+N ), which implies the maximum amount interference plus noise it can tolerate is 3(P −p 31 )−p 32 +N . However, the real interference plus noise it faces is (P − p 11 ) + (P − p 21 ) + (P − p 31 − p 32 ) + N , which is smaller than what it can tolerate since 2p 31 < p 11 + p 21 . So virtual user "32" can be decoded reliably and subtracted from the received signal. By searching for the minimum entry in p search P 31 P 11 P 21 ( in each run, we always decode a virtual user that can tolerate more interference than what it really faces. This assures the validity of our decoding algorithm. All of the other virtual users can be decoded in a similar fashion. By Lemma 2 and Lemma 8, user i can choose any L i ∈ Z + , independently from other users, and have all virtual users decoded reliably at the receiver. Therefore, user i can choose L i according to its own service requirement. For example, if user i wants to send low rate voice communication packets, it can set L i = 1, which corresponds to the basic CDMA scheme. If user i wants to send high rate stream video, it can set L i equal to a large value in order to get higher throughput at the expense of higher coding complexity. Thus, this variation of DRS provides an explicit way for end users to trade off throughput and coding complexity, making differential rate requirements achievable in a distributed manner. Finally, Corollary 2 and Theorem 4 demonstrate the asymptotic optimality of this scheme and its rate of convergence.
2) M -user Discrete Memoryless MAC: In this section, we describe a variation of the DRS scheme for the discrete memoryless MAC which supports differential rate requirements to end users in a distributed manner. In this scheme, we adopt the uniform switch, but we do not require every user to have the same number of virtual users. We split by means of M independent switches. Without loss of generality, let us consider user i. If user i has L i virtual users, then switch i has L i inputs, X ik ∈ X i for k = 1, ..., L i , and one output X i ∈ X i . Switch i is controlled by a uniform random variable
S i ∈ {1, ..., L i } with P(S i = k) = 1
Li for k = 1, ..., L i . The output is given by:
X i = X ik , if S i = k.
We now describe the protocol for the discrete memoryless MAC which allows user i to choose L i independently. We show the asymptotic optimality of this variation of DRS under the protocol.
PROTOCOL 2: For user i with L i virtual users, the switch i is controlled by a uniform random variable S i where P(S i = k) = 1
Li for k = 1, ..., L i . The rate allocation rule is defined as follows:
r X ik = 1 L i 1 − k − 1 L i M−1 I(X 1 ; Y ) + M−1 l=1 M − 1 l k − 1 L i l · 1 − k − 1 L i M−1−l I(X 1 ; Y, X 2 , .., X l+1 ) .
The decoding algorithm is given by the following pseudoprogram. Note that after a virtual user is decoded, its signal is used as side information to aid the decoding process of subsequent virtual users. decode virtual users (11, 21, ..., M 1) in any order or in parallel.
set s = ( 1 L1 , 1 L2 , ..., 1 LM ) while (some virtual users are not decoded),
Find the minimal element in s, say the ith entry; Decode the subsequent virtual user of user i; Update the ith entry of s: s(i) = s(i) + 1 Li ; end Lemma 9: If all M users adopt the rate allocation rule in PROTOCOL 2, then for any L i ∈ Z + ∀i, the decoder can decode all virtual users one by one following the decoding algorithm.
Proof : Virtual users (11, 21, ..., M 1) can be decoded in any order if and only if r Xi1 ≤ I(X i1 ; Y, S M 1 ) for all i. This is true because under our rate allocation rule, r Xi1 = 1
Li I(X 1 ; Y ) = 1 Li I(X i ; Y ) = I(X i1 ; Y, S M 1 ). We set s = ( 1 L1 , ..., 1 LM ).
In the first run of the while loop, if 1
Li is the minimum entry of s, the receiver decodes virtual user i2. By the rate allocation rule
r Xi2 = 1 L i 1 − 1 L i M−1 I(X 1 ; Y ) + M−1 l=1 M − 1 l 1 L i l · 1 − 1 L i M−1−l I(X 1 ; Y, X 2 , .., X l+1 ) (27)
Virtual user i2 can be decoded reliably if r Xi2 ≤ I(X i2 ; Y, S M 1 , X 11 , ..., X M1 ). This mutual information can be simplified in the same way as described in equations (23)
I X i2 ; Y, S M 1 , X 11 , ..., X M1 = 1 L i I (X i2 ; Y, X 11 , ...X M1 |S −i )(28)= 1 L i s−i P (S −i = s −i ) I (X 1 ; Y, X 11 , ..., X M1 |S −i = s −i ) .(29)
We can verify that the I(X 1 ; Y ) term in (27) is less than or equal to the corresponding I(X 1 ; Y ) term in Equation (29). This follows from
P (S 1 > 1, ..., S i−1 > 1, S i+1 > 1, ...S M > 1) = j =i 1 − 1 L j ≥ 1 − 1 L i M−1 , since 1 Li ≤ 1
Lj for all j = i. Similarly, it can be verified that the second term in (27) is less than or equal to the corresponding term in Equation (29). Therefore, virtual user i2 can be decoded reliably.
Suppose the decoding process succeeds in the tth run of the while loop. Now, s (s t 1 , ..., s t M ). Let us consider the (t + 1)th run of the while loop. Suppose the ith entry, s t i , is the minimum entry in s. The receiver decodes the subsequent virtual user of user i, denoted by ij. By the rate allocation rule
r Xij = 1 L i 1 − j − 1 L i M−1 I(X 1 ; Y ) + M−1 l=1 M − 1 l j − 1 L i l · 1 − j − 1 L i M−1−l I(X 1 ; Y, X 2 , .., X l+1 ) .
Again, we can simplify the mutual information I(X ij ; Y, S M 1 , X j−1 1 , ..., X j−1 M ) and show that it is great than or equal to r Xij , which implies that virtual user ij can be decoded reliably. Hence, the lemma follows by induction.
Let us illustrate the decoding algorithm by the following example. We consider a two-user discrete memoryless MAC where user 1 creates 2 virtual users and user 2 creates 3 virtual users. Random switch 1 is controlled by S 1 where P(S 1 = 1) = P(S 1 = 2) = 1 2 , and random switch 2 is controlled by S 2 where P(S 2 = k) = 1 3 for k = 1, 2, 3. By the rate allocation rule, the virtual users' rates can be simplified as follows
r X11 = 1 2 I(X 1 ; Y ) r X12 = 1 2 1 2 I(X 1 ; Y ) + 1 2 I(X 1 ; Y, X 2 ) r X21 = 1 3 I(X 1 ; Y ) r X22 = 1 3 2 3 I(X 1 ; Y ) + 1 3 I(X 1 ; Y, X 2 ) r X23 = 1 3 1 3 I(X 1 ; Y ) + 2 3 I(X 1 ; Y, X 2 ) .
We first decode (11,21) in any order. Suppose we decode "21" first. Virtual user "21" can be decoded reliably if r X21 ≤ I (X 21 ; Y, S 1 , S 2 ). The condition holds because In the first run of the while loop, the receiver decodes "22" since 1 3 < 1 2 . Let us calculate the mutual information between X 22 and Y given S 1 , S 2 and previously decoded X 11 , X 21 .
I (X 22 ; Y, S 1 , S 2 , X 11 , X 21 ) = 1 3 I (X 22 ; Y |S 1 , X 11 , X 21 , S 2 = 2) = 1 3 1 2 I(X 2 ; Y, X 1 ) + 1 2 I(X 2 ; Y ) = 1 3
1 2 I(X 1 ; Y, X 2 ) + 1 2 I(X 1 ; Y ) ≥ 1 3 1 3 I(X 1 ; Y, X 2 ) + 2 3 I(X 1 ; Y ) = r X22 .
The third equality is due to our symmetric assumption. Therefore, virtual user "22" can be decoded. By searching for the minimum entry in s in each run, we always decode a virtual user whose rate is smaller than or equal to the corresponding mutual information. This guarantees the correctness of our decoding algorithm. Virtual users (12, 23) can be decoded reliably at the receiver in a similar fashion. The asymptotic optimality of this scheme in the discrete memoryless MAC can be demonstrated by Theorem 7.
V. CONCLUDING REMARKS
In this paper, we take an information-theoretic approach to the problem of distributed multiple-access communication. We present a Distributed Rate Splitting scheme whereby each real user creates a number of virtual users and all virtual users are successively decoded at the receiver. One possible advantage of Distributed Rate Splitting is that it can be implemented with lower complexity when compared with joint coding schemes, and less coordination among users when compared with either time-sharing or rate splitting. For the symmetric M -user Gaussian MAC, each real user creates the same number of virtual users via a power/rate splitting mechanism. The transmitted signal of a real user is the superposition of all its virtual users' signals. For the symmetric M -user discrete memoryless MAC, each real user creates the same number of virtual users via a random switching mechanism, and the transmitted signal of a real user is determined by a random switch. All virtual users are successively decoded at the receiver. It is shown that DRS can achieve the maximum equal rate point for both channel models as the number of virtual users per real user tends to infinity. Finally, we present two variations of the DRS scheme. For the case of asymmetric capacity regions, we show that a point on the dominant face can be achieved asymptotically. For the case of an unequal number of virtual users, we show that different user rates requirements can be accommodated independently in a distributed manner.
APPENDIX I PROOF OF THEOREM 2
Proof: We use induction on the number of virtual users. For L = 2, the original problem reduces to: SIR 1 (p 1 , p 2 ) = SIR 2 (p 1 , p 2 ) subject to p 1 + p 2 = P and p 1 , p 2 ≥ 0. The unique solution is For L = j − 1, suppose (p * 1 (P ), ..., p * j−1 (P )) uniquely solves SIR 1 (p 1 , ..., p j−1 ) = ... = SIR j−1 (p 1 , ..., p j−1 ) for any P > 0, subject to j−1 k=1 p k = P and p k ≥ 0 for k = 1, 2, .., j−1. Let us consider the j virtual users case. Given any tuple (p 1 , ..., p j ) such that j k=1 p k = P and p k ≥ 0 for k = 1, 2, .., j, we can fix p j , so p 1 + ... + p j−1 = P − p j . We now solve SIR 1 (p 1 , ..., p j−1 ) = ... = SIR j−1 (p 1 , ..., p j−1 )
subject to j−1 k=1 p k = P −p j , p k ≥ 0 for k=1, 2, .., j. For fixed p j , by the induction hypothesis, we have a unique solution (p * 1 (P − p j ), ...,p * j−1 (P − p j )) which solves equation (30). Let s * (p j ) SIR j−1 p * 1 (P − p j ) , ..., p * j−1 (P − p j ) = p * j−1 (P −pj ) (M−1)p * j−1 (P −pj )+N . We are able to verify that s * (p j ) is a strictly decreasing function of p j , and SIR j (p j ) = pj (M−1)pj +N is a strictly increasing function of p j . Moreover, the function values at boundary points satisfy s * (0) > SIR j (0) and s * (P ) < SIR j (P ). So there exists a unique p * j such that SIR j p * j = s * p * j . Hence, we conclude p * 1 P − p * j , ..., p * j−1 P − p * j , p * j is the unique solution to the j virtual users case. The theorem follows by induction.
Manuscript received April 19, 2005; revised September 21, 2006. This work was supported in part by Army Research Office (ARO) Young Investigator Program (YIP) grant DAAD19-03-1-0229 and by National Science Foundation (NSF) grant CCR-0313183. The material in this paper was presented in part at the Allerton Conference on Communication, Control, and Computing, Monticello, IL, September, 2004, the IEEE International Symposium on Information Theory, Adelaide, Australia, September, 2005, and the Global Telecommunications Conference, St. Louis, MO, November, 2005. The authors are with the Department of Electrical Engineering, Yale University, New Haven, CT 06520, U.S.A. (Email:[email protected]; [email protected]). Communicated by Y. Steinberg, Associate Editor for Shannon Theory.
Fig. 2 .
2Comparison of rate splitting with distributed rate splitting. In the right-hand figure, the virtual user class V 1 = {11, 21} is decoded before virtual user class V 2 = {12, 22}.
Lemma 2 :L
2Given a DRS scheme with L virtual users per real user, where (p 1 , ..., p L ) are the virtual users' powers, it is possible to strictly increase the throughput via an (L + 1) virtual user system with powers (p 1 , ..., p = p L .Proof: Suppose that every user splits its power into L virtual users: (p 1 , p 2 , ..., p L−1 , p L ) subject to L k=1 p k = P , where L is an arbitrary integer and p k is the power of kth virtual user. Since virtual user L is decoded last, following the reasoning in the proof of Lemma 1, we have r L = 1 2 log(1 + pL (M−1)pL+N ). We now split the virtual user with power p L into two new virtual users with powers p
Fig. 3 .
3Throughput per real user v.s. the number of virtual users per real user for both high and low SNR regimes. Note that the scales of the vertical axes in both figures are different. integer n 0 and for all L > n 0 , we have c1 L ≤ e [L] ≤ c2 L .
∀S, T ⊆ {1, ..., M } such that |S| = |T |. Later, in Section IV, we consider the more general asymmetric case. We further assume that for ∀S, T ⊆ {1, 2, ..., M }, if S ∩ T = ∅, then I (X S ; Y ) < I (X S ; Y |X T ). Under our symmetric setting, the maximum common rate that every user can achieve is R * = 1 M I (X 1 , ..., X M ; Y ).
Theorem 6 :
6For M = 2, if both real users adopt P (S i = k) = 1 L for k = 1, ..., L and i
For k = 1, ..., L, we setr X 1k = I(X 1k ; Y, S M 1 , X 1(k−1) 11 , ..., X M(k−1) M1 ) (23) = I(X 1k ; Y, S M 2 , X 1(k−1) 11 , ..., X M(k−1) M1 | S 1 ) (24) = λ k I(X 1k ; Y, S M 2 , X 1(k−1) 11, ..., X M(k−1) M1
Theorem 7 :
7For a general random switch controlled byS i , where P(S i = k) = λ k for k = 1, ..., L, a sufficient condition for lim L→∞ L k=1 r X ik = 1 M I (X 1 , ..., X M ; Y ) is max k∈{1,...,L} λ k → 0 as L → ∞ for i = 1, ..., M .
Lemma 5 :
5Consider an M -user discrete memoryless MAC. Let each real user have L virtual users and each switch be controlled by an i.i.d. random variable S i ∈ {1, 2, ..., L} with P(S i = k) = 1 L , k = 1, ..., L. Define the error term e[L] ≡ 1 M I(X 1 , X 2 , ..., X M ; Y )− L k=1 r X ik . Then e[L] = O( 1 L ) for all i. Proof: In the uniform switch setting, P(S i = k) = 1 L for i = 1, ..., M and k = 1, ..., L. Without loss of generality, we examine the total throughput of real user 1.
Lemma 6 :
6For any real user with L virtual users, the unique way to maximize this user's throughput is to set i = 1, ..., M .
N 1 L
1− 1 for k = 1, ..., L. 4) If all real users adopt this power allocation rulei = 1, ..., M
I
∀S, T ⊆ {1, ..., M } such that |S| = |T |. In this section, we relax this constraint and consider an asymmetric achievable region. We require only that for ∀S, T ⊆ {1, 2, ..., M }, if S ∩ T = ∅, then I (X S ; Y ) < I (X S ; Y |X T ). The M switches (S 1 , ..., S M ) have probabilities P (S i = k) = λ k for i = 1, ..., M and k = 1, ..., L. Lemma 7: Consider a general random switch controlled by S i , where P (S i = k) = λ k , k = 1, ..., L. (X i ; Y, X S ) , for i = 1, ..., M .
decode virtual users(11, 21, ..., M 1) in any order or in parallel.set p = (p 11 , p 21 , ..., p M1 ) while (some virtual users are not decoded), Find the minimal element in p, say the ith entry; Decode the subsequent virtual user of user i; Update the ith entry of p: p(i) = p(i) +p, wherep is the power of the virtual user being decoded in the previous step;endLemma 8: If all M users adopt the power split and the rate allocation rule described in PROTOCOL 1, then for any L i ∈ Z + , i = 1, 2, ..., M , the decoder can decode all virtual users one by one following the decoding algorithm.Proof: By the rate allocation rule,r i1 = 1 2 log(1 + pi1 MP −pi1+N ) for i = 1, ..., M .Thus, each of them can tolerate the maximum amount of interference plus noise, M P − p i1 + N . It is then easy to see that virtual users (11, ..., M 1) can be decoded reliably in any order or in parallel. Now, we set p = (p 11 , ..., p M1 ).In the first run of the while loop, if p i1 is the minimum entry in p, the receiver decodes virtual user i2. By the rate allocation rule, r i2 = 1 2 log(1 + pi2 M(P −pi1)−pi2+N ). This implies that the maximum amount of interference plus noise that virtual user i2 can tolerate is M (P − p i1 )−p i2 +N . However, the real amount of interference plus noise it faces is M P − M j=1 p j1 − p i2 + N , which is smaller than or equal to what it can tolerate because M p i1 ≤ M j=1 p j1 . Therefore, virtual user i2 can be decoded reliably at the receiver.Suppose the decoding process succeeds in the tth run of the while loop. Now, p (p t 1 , ..., p t M ). Let us consider the (t + 1)th run of the while loop. Suppose the ith entry, p t i , is the minimum entry in p. The receiver decodes the subsequent virtual user of user i, denoted by il. By the rate allocation rule,r il = 1 2 log(1 + p il M(P − j<l pij )−p il +N ). This implies the maximum amount of interference plus noise that virtual user il can tolerate is M (P − j<l p ij ) − p il + N = M (P −p t i )−p il +N . However, the real amount of interference plus noise it faces is M P − M j=1 p t j − p il + N , which is smaller than or equal to what virtual user il can tolerate because M p t i ≤ M j=1 p t j . Therefore, virtual user il can be decoded reliably. Hence, the lemma follows by induction.
-(26). Recall the definition −i ≡ {1, ..., M } \ {i}.
I
(X 21 ; Y, S 1 , S 2 ) = I (X 21 ; Y, S 1 |S 2 (X 1 ; Y ) = r X21Virtual user "11" can be decoded similarly. Now, we set s =
noise variance N . The capacity region C is the set ofR 2
R 1
A
R max
Maximum Equal
Rate Point
B
R min
R min
R max
Fig. 1.
Two-user Gaussian multiple-access capacity region, where the
dominant face has been highlighted.
Fig. 4. Switches for two-user discrete memoryless MACS 1 =1,2
S 1
Switch 1
Switch 2
X 11
X 12
X 21
X 22
MAC
X 1
X 2
Y
S 2
S 2 =1,2
We use R to denote (R 1 , ..., R M ) throughout.3 R is achievable if for any ε > 0, there exists an (n, R 1 − ε, ..., R M − ε) multiple-access code with overall error probability Pe < ε, where n is the block length.
We assume a genie-aided[16] decoding scheme where the previously decoded messages have been decoded correctly. In practice, errors can be made in previous decodings. However, for purposes of analyzing the overall error probability, the genie-aided model is sufficient.
Recall that f [n] = Ω (g [n]) if there exist positive constants c 1 and n 0 such that f [n] ≥ c 1 g [n] for all n ≥ n 0 , and f [n] = O(g[n]) if there are positive constants c 2 and n 0 , such that f [n] ≤ c 2 g[n] for all n ≥ n 0 . Finally, f [n] = Θ (g[n]) if f [n] = Ω (g[n]) and f [n] = O (g[n]).
ACKNOWLEDGMENT Edmund Yeh gratefully acknowledges useful discussions with Prof. Shlomo Shamai on this topic.
A perspective on multiacces channels. R Gallager, IEEE Trans. Inform. Theory. 31R. Gallager, "A perspective on multiacces channels," IEEE Trans. Inform. Theory, vol. 31, pp. 124-142, Mar. 1985.
The aloha system-another alternative for computer communications. N Abramson, Fall Joint Computer Conference. N. Abramson, "The aloha system-another alternative for computer communications," in Fall Joint Computer Conference, 1970.
. D Bertsekas, R Gallager, Data Networks, Prentice HallD. Bertsekas and R. Gallager, Data Networks. Prentice Hall, 1992.
The throughput of packet broadcasting channels. N Abramson, IEEE Trans. Commun. 25N. Abramson, "The throughput of packet broadcasting channels," IEEE Trans. Commun., vol. 25, pp. 117-128, 1977.
Analysis of mobile radio slotted aloha networks. C Namislo, IEEE J. Select. Areas Commun. 2C. Namislo, "Analysis of mobile radio slotted aloha networks," IEEE J. Select. Areas Commun., vol. 2, pp. 583 -588, July 1984.
Spread slotted aloha techniques for mobile and personal satellite communication systems. D Makrakis, K Murthy, IEEE J. Select. Areas Commun. 10D. Makrakis and K. Murthy, "Spread slotted aloha techniques for mobile and personal satellite communication systems," IEEE J. Select. Areas Commun., vol. 10, pp. 985 -1002, Aug. 1992.
Optimal decentralized control in the random access multipacket channel. S Ghez, S Verdu, S Schwartz, IEEE Trans. Automat. Contr. 34S. Ghez, S. Verdu, and S. Schwartz, "Optimal decentralized control in the random access multipacket channel," IEEE Trans. Automat. Contr., vol. 34, pp. 1153-1163, Nov. 1989.
Stability and delay of slotted aloha with multipacket reception. V Naware, G Mergen, L Tong, IEEE Trans. Inform. Theory. submitted for publicationV. Naware, G. Mergen, and L. Tong, "Stability and delay of slotted aloha with multipacket reception," IEEE Trans. Inform. Theory, submitted for publication.
Performance analysis of random access packetswitched code division multiple access systems. D Raychaudhuri, IEEE Trans. Commun. 29D. Raychaudhuri, "Performance analysis of random access packet- switched code division multiple access systems," IEEE Trans. Commun., vol. 29, pp. 895-901, 1981.
Recent results in Shannon Theory. A D Wyner, IEEE Transactions on Information Theory. 20A. D. Wyner, "Recent results in Shannon Theory," IEEE Transactions on Information Theory, vol. 20, pp. 2-10, 1974.
Elements of Information Theory. T Cover, J Thomas, Wiley-InterscienceT. Cover and J. Thomas, Elements of Information Theory. Wiley- Interscience, 1991.
Multi-way communication channels. R Ahlswede, Proc. IEEE International Symposium on Information Theory. IEEE International Symposium on Information TheoryR. Ahlswede, "Multi-way communication channels," in Proc. IEEE International Symposium on Information Theory, 1971, pp. 23-52.
Multiple access channels. H Liao, University of HawaiiPh.D. dissertationH. Liao, "Multiple access channels," Ph.D. dissertation, University of Hawaii, 1972.
Coding for channel with asynchronous multiple access. G Poltyrev, Probl. Pered. Inform. 19G. Poltyrev, "Coding for channel with asynchronous multiple access," Probl. Pered. Inform., vol. 19, 1983.
The capacity region of the totally asynchronous multiple-access channel. J Hui, P Humblet, IEEE Trans. Inform. Theory. 31J. Hui and P. Humblet, "The capacity region of the totally asynchronous multiple-access channel," IEEE Trans. Inform. Theory, vol. 31, Mar. 1985.
A rate-splitting approach to the gaussian multiple-access channel. B Rimoldi, R Urbanke, IEEE Trans. Inform. Theory. 42B. Rimoldi and R. Urbanke, "A rate-splitting approach to the gaussian multiple-access channel," IEEE Trans. Inform. Theory, vol. 42, pp. 364- 375, Mar. 1996.
Achieving the multiple access capacity region via projective time sharing. E Yeh, R Gallager, Proc. IEEE International Symposium on Information Theory. IEEE International Symposium on Information TheoryCambridge, MA213E. Yeh and R. Gallager, "Achieving the multiple access capacity region via projective time sharing," in Proc. IEEE International Symposium on Information Theory, Cambridge, MA, Aug. 1998, p. 213.
Rate-splitting multiple-access for discrete memoryless channels. A Grant, B Rimoldi, R Urbanke, P Whiting, IEEE Trans. Inform. Theory. 47A. Grant, B. Rimoldi, R. Urbanke, and P. Whiting, "Rate-splitting multiple-access for discrete memoryless channels," IEEE Trans. Inform. Theory, vol. 47, pp. 873 -890, Mar. 2001.
Generalized time sharing: a low-complexity capacity achieving multiple-access technique. B Rimoldi, IEEE Trans. Inform. Theory. 47B. Rimoldi, "Generalized time sharing: a low-complexity capacity achieving multiple-access technique," IEEE Trans. Inform. Theory, vol. 47, pp. 2432 -2442, Sept. 2001.
Capacity of time-slotted aloha packetized multiple-access systems over the awgn channel. M Medard, J Huang, A Goldsmith, S Meyn, T Coleman, IEEE Trans. Wireless Commun. 3M. Medard, J. Huang, A. Goldsmith, S. Meyn, and T. Coleman, "Capacity of time-slotted aloha packetized multiple-access systems over the awgn channel," IEEE Trans. Wireless Commun., vol. 3, pp. 486-499, Mar. 2004.
A broadcast approach for the multiple-access slow fading channel. S Shamai, Proc. IEEE International Symposium on Information Theory. IEEE International Symposium on Information TheorySorrento, Italy128S. Shamai, "A broadcast approach for the multiple-access slow fading channel," in Proc. IEEE International Symposium on Information The- ory, Sorrento, Italy, June 2000, p. 128.
PLACE PHOTO HERE Jian Cao Jian Cao received the B.S. degree in Electrical and Electronics Engineering with first class honor from the Hong Kong University of Science and Technology. R Cheng, Proc. IEEE Global Telecommunications Conference. IEEE Global Telecommunications ConferenceSorrento, Italy; Hong Kong; New HavenElectrical Engineering from Yale UniversityStripping cdma -an asymptotically optimal coding scheme for l-out-of-k white gaussian channels. in 2002, the M.S. degree in. and is currently working toward the Ph.D. degree in Electrical Engineering at Yale. His research interests include information theory and queueing theoryR. Cheng, "Stripping cdma -an asymptotically optimal coding scheme for l-out-of-k white gaussian channels," in Proc. IEEE Global Telecom- munications Conference, Sorrento, Italy, Nov. 1996, pp. 142-146. PLACE PHOTO HERE Jian Cao Jian Cao received the B.S. degree in Elec- trical and Electronics Engineering with first class honor from the Hong Kong University of Science and Technology, Hong Kong, in 2002, the M.S. degree in Electrical Engineering from Yale Univer- sity, New Haven, in 2003, and is currently working toward the Ph.D. degree in Electrical Engineering at Yale. His research interests include information theory and queueing theory.
| [] |
[
"On Markov Decision Processes with Borel Spaces and an Average Cost Criterion *",
"On Markov Decision Processes with Borel Spaces and an Average Cost Criterion *"
] | [
"Huizhen Yu [email protected] \nDepartment of Computing Science\nRLAI Lab\nUniversity of Alberta\nCanada\n"
] | [
"Department of Computing Science\nRLAI Lab\nUniversity of Alberta\nCanada"
] | [] | We consider average-cost Markov decision processes (MDPs) with Borel state and action spaces and universally measurable policies. For the nonnegative cost model and an unbounded cost model, we introduce a set of conditions under which we prove the average cost optimality inequality (ACOI) via the vanishing discount factor approach. Unlike most existing results on the ACOI, which require compactness/continuity conditions on the MDP, our result does not and can be applied to problems with discontinuous dynamics and one-stage costs. The key idea here is to replace the compactness/continuity conditions used in the prior work by what we call majorization type conditions. In particular, among others, we require that for each state, on selected subsets of actions at that state, the state transition stochastic kernel is majorized by finite measures, and we use this majorization property together with Egoroff's theorem to prove the ACOI.We also consider the minimum pair approach for average-cost MDPs and apply the majorization idea. For the case of a discrete action space and strictly unbounded costs, we prove the existence of a minimum pair that consists of a stationary policy and an invariant probability measure induced by the policy. This result is derived by combining Lusin's theorem with another majorization condition we introduce, and it can be applied to a class of countable action space MDPs in which, with respect to the state variable, the dynamics and one-stage costs are discontinuous. | null | [
"https://arxiv.org/pdf/1901.03374v1.pdf"
] | 119,132,742 | 1901.03374 | 342804a2d9bd773c1f62b83c43fb3a88e0bb47fe |
On Markov Decision Processes with Borel Spaces and an Average Cost Criterion *
10 Jan 2019
Huizhen Yu [email protected]
Department of Computing Science
RLAI Lab
University of Alberta
Canada
On Markov Decision Processes with Borel Spaces and an Average Cost Criterion *
10 Jan 20191 2 On Average-Cost Borel-Space MDPsMarkov decision processesBorel spacesuniversally measurable policiesaverage costoptimality inequalityminimum pairmajorization conditions *
We consider average-cost Markov decision processes (MDPs) with Borel state and action spaces and universally measurable policies. For the nonnegative cost model and an unbounded cost model, we introduce a set of conditions under which we prove the average cost optimality inequality (ACOI) via the vanishing discount factor approach. Unlike most existing results on the ACOI, which require compactness/continuity conditions on the MDP, our result does not and can be applied to problems with discontinuous dynamics and one-stage costs. The key idea here is to replace the compactness/continuity conditions used in the prior work by what we call majorization type conditions. In particular, among others, we require that for each state, on selected subsets of actions at that state, the state transition stochastic kernel is majorized by finite measures, and we use this majorization property together with Egoroff's theorem to prove the ACOI.We also consider the minimum pair approach for average-cost MDPs and apply the majorization idea. For the case of a discrete action space and strictly unbounded costs, we prove the existence of a minimum pair that consists of a stationary policy and an invariant probability measure induced by the policy. This result is derived by combining Lusin's theorem with another majorization condition we introduce, and it can be applied to a class of countable action space MDPs in which, with respect to the state variable, the dynamics and one-stage costs are discontinuous.
Introduction
We consider discrete-time Markov decision processes (MDPs) with Borel state and action spaces, for the average cost criterion where the objective is to minimize the (limsup) expected long-run average cost per unit time. Specifically, we are interested in the universal measurability framework, which involves lower semi-analytic one-stage cost functions and universally measurable policies. It is a mathematical formulation of MDPs developed to resolve measurability difficulties in dynamic programming on Borel spaces (Strauch [39]; Blackwell [4]; Blackwell, Freedman, and Orkin [5]; Shreve [35]; Shreve and Bertsekas [2,36,37]). An in-depth study of this theoretical framework is given in the monograph [2,Part II], and optimality properties of finite-and infinite-horizon problems with discounted and undiscounted total cost criteria have been analyzed [2,22,36,37]. The average cost problem has not been thoroughly studied in this framework, however, and the primary purpose of this paper is to investigate the subject further.
To progress toward our goal, we will draw heavily from the rich literature on a subclass of Borel-space MDPs that have certain compactness and continuity properties-these properties help remove major measurability-related issues and also lead to strong optimality results. Most notably, there has been extensive research on lower semicontinuous models and sophisticated theories for the average cost criterion have been developed. (The literature on this subject is too vast to list in full; see the early work [31,32,33], an early survey paper [1], the books [17,18,27,34], and the recent work [13,19,42] and the references therein.)
In order to derive analogous average-cost optimality results for general Borel-space MDPs, our main idea is to replace the compactness/continuity conditions used in the prior work for lower semicontinuous models by what we call majorization type conditions. These conditions will have different forms when we employ different methods of analysis. But roughly speaking, we want to have finite measures that majorize the state transition stochastic kernel of the MDP or some substochastic kernel created from that kernel, at certain action sets for each state or at the admissible state-action pairs, depending on the context. Our idea is to use those majorizing finite measures in combination with Egoroff's or Lusin's theorem, which would then allow us to extract arbitrarily large sets (large as measured by a given finite measure) on which certain functions involved in our analyses have desired uniform convergence or continuity properties. We use this technique, along with other analysis techniques developed in the prior work, to obtain two main results in this paper that can be applied to certain classes of MDPs with discontinuous dynamics and one-stage costs.
Our first result based on this majorization idea is a proof of the average cost optimality inequality (ACOI) for two types of MDPs, the nonnegative cost model and an unbounded cost model with a Lyapunov-type condition, without using compactness/continuity conditions. The study of ACOI was initiated by Sennott [33], who proved it for countable-space MDPs; prior to [33], the ACOE (average cost optimality equation) was the research focus. Cavazos-Cadena's counterexample [8] showed that the ACOI is more general: in a countable-space MDP in the example, the ACOI has a solution and yet the ACOE does not. For Borel-space MDPs under various compactness/continuity conditions, the ACOI was first established by Schäl [32], whose results have been further extended since then (see e.g., Hernández-Lerma and Lasserre [17,18] and more recently, Vega-Amaya [41]; Jaśkiewicz and Nowak [19]; Feinberg, Kasyanov, and Zadoianchuk [13]).
The two MDP models we consider have been studied in some of the references just mentioned. As in those studies, to prove the ACOI, we also use the vanishing discount factor approach, which treats the average cost problem as the limiting case of the discounted problems, and we adopt some of the conditions formalized in [13,18,32] regarding the value functions of the discounted problems. In place of the compactness/continuity conditions used in the prior work, we introduce a set of new conditions of the majorization type: among others, we require that for each state, on selected subsets of actions at that state, the state transition stochastic kernel is majorized by finite measures (see Assumptions 3.2 and 3.4). We use this majorization property together with Egoroff's theorem (which shows pointwise convergence of functions is "almost" uniform convergence as measured by a given finite measure) to prove the ACOI (see Theorems 3.1 and 3.2).
For comparison, let us mention a few early results that are either about or applicable to averagecost MDPs with universally measurable policies. In particular, Gubenko and Shtatland also introduced a majorization condition to prove the ACOE for Borel-space MDPs without compactness/continuity conditions [15, Theorem 2 ′ ] (measurability issues are assumed away in this theorem). However, they pursued a contraction-based fixed point approach and their majorization condition, formed to make the contraction argument work, not only differs in essential ways from ours but is also too stringent to be practical (see Remark 3.1 for details). Gubenko and Shtatland [15] studied the ACOE also under an alternative, minorization condition using the same fixed point approach, and that type of sufficient condition for the ACOE has been generalized by Kurano [20] to one of a multistep-contraction type. Dynkin and Yushkevich [10,Chap. 7.9] and Piunovski [26] studied the characteristic properties of canonical systems-a general form of the ACOE together with stationary policies that solve or almost solve the ACOE. These early researches on general Borel-space MDPs differ significantly from ours in both the approaches taken and the results obtained.
Our second result based on the majorization idea is about the existence of a minimum pair in average-cost MDPs. A minimum pair refers to a policy together with an initial state distribution that attains the minimal average cost over all policies and initial state distributions. Of interest is the existence of such a pair with special structures, in particular, a stationary policy with an associated invariant probability measure, for the stationary policy is then not only average-cost optimal for that initial distribution but it is also pathwise optimal under additional recurrence conditions [17,Chap. 5.7]. The minimum pair approach was proposed by Kurano [21], motivated by the methods of occupancy measures from Borkar [6,7]. Unlike the vanishing discount factor approach, it is a direct method. Kurano [21] considered bounded costs and compact spaces, and Hernández-Lerma ( [16]; see also the book [17,Chap. 5.7]) analyzed the case of strictly unbounded costs, both working with lower semicontinuous MDP models.
Our result is for a discrete action space and strictly unbounded costs. We prove the existence of a minimum pair that consists of a stationary policy and an invariant probability measure of the Markov chain it induces, under another majorization condition we introduce (see Assumption 4.1 and Theorem 4.1). The result is derived by combining the majorization property with Lusin's theorem (which is about the continuity of Borel measurable functions when restricted to some arbitrarily "large" closed sets, with largeness measured by a given finite measure). It applies to a class of discrete action space MDPs where the dynamics and one-stage costs are discontinuous with respect to (w.r.t.) the state variable. It can be compared with the minimum pair results for lower semicontinuous models in [16,17,21], although its scope is limited because with our current proof arguments, we can only handle discrete action spaces. Future work is to extend this result to Borel action spaces and universally measurable policies.
We remark that Lusin's theorem has been used earlier in a similar way by the author to tackle measurability-related issues in policy iteration for a lower semicontinuous, Borel-space MDP model [45,Sec. 6]. The minimum pair problem we address in this paper and the other arguments involved in our analysis are entirely different from those in [45], however.
Besides the results mentioned above, in this paper, we also derive a basic average-cost optimality theorem for the nonnegative cost and unbounded cost MDP models mentioned earlier, without extra ACOI-or majorization-related conditions. It shows that the optimal average cost function is lower semi-analytic and there always exists a universally measurable, ǫ-optimal semi-Markov policy (see Theorem 2.1). Based on known counterexamples from Dynkin and Yushkevich [10,Chap. 7] and Feinberg [11], without additional assumptions on the MDP, this is the strongest conclusion possible (see Remark 2.1 for details).
The rest of the paper is organized as follows. In Section 2, we introduce the universal measurability framework for Borel-space MDPs, and to prepare for subsequent analyses, we derive several basic optimality results for the two models we consider, under the average and discounted cost criteria. In Section 3, we consider the vanishing discount factor approach, propose new majorization type conditions, and prove the ACOI for the aforementioned models. In Section 4, we consider the minimum pair approach in the case of strictly unbounded costs, and we present our results for discrete action spaces under a new majorization condition we introduce. Some background material and proof details are given in Appendix A.
Background and Preliminary Analysis
To study general Borel-space MDPs, we need to go beyond Borel measurable functions and policies because there are measurability difficulties otherwise [4,39]. The universal measurability framework for MDPs is quite involved, however, so before describing it, we need to first introduce several basic definitions and terminologies. We present these introductory materials in Section 2.1. They are largely based on the monograph [2, Part II] and are similar to the background overview the author gave in [45].
We then present a preliminary analysis of average-cost MDPs in Section 2.2, where we define two model classes and derive some basic optimality results for them (Theorems 2.1 and 2.2). In the subsequent section, we will impose further conditions on the two models in order to derive more special average-cost optimality results.
Borel-space MDPs in the Universal Measurability Framework
Definitions for some Sets and Functions
A Borel space is a topological space that is homeomorphic to a Borel subset of some Polish space (i.e., a separable and completely metrizable topological space) [2,Def. 7.7]. For a Borel space X, let B(X) denote the Borel σ-algebra and P(X) the set of probability measures on B(X). We shall refer to these probability measures as Borel probability measures. We endow the space P(X) with the topology of weak convergence; then P(X) is also a Borel space [2,Chap. 7.4]. Each Borel probability measure p has a unique extension on a larger σ-algebra B p (X), which is the σ-algebra generated by B(X) and all the subsets of X with p-outer measure 0. This extension is called the completion of p (cf. [9,Chap. 3.3]). The universal σ-algebra on X is defined as U(X) := ∩ p∈P(X) B p (X).
If a function is U(X)-measurable, we say it is universally measurable. Since B(X) ⊂ U(X), a Borel measurable function is universally measurable. Conversely, a universally measurable function f is measurable w.r.t. the completion of any Borel probability measure p since U(X) ⊂ B p (X), and one implication of this is that the integral f dp for a nonnegative f can be defined w.r.t. the completion of p. This is the definition for integration that will be used for Borel-space MDPs.
Let X and Y be Borel spaces. A Borel or universally measurable stochastic kernel on Y given X is a function q : X → P(Y ), denoted q(dy | x), such that for each B ∈ B(Y ), the function q(B | ·) : X → [0, 1] is Borel or universally measurable, respectively. The definition is equivalent to that q is a measurable function from the space (X, B(X)) or (X, U(X)), respectively, to the space (P(Y ), B(P(Y ))); see [2,Def. 7.12,Prop. 7.26 and Lemma 7.28]. If q is a continuous function, we say that the stochastic kernel q(dy | x) is continuous (also known as weak Feller in the literature).
We now introduce analytic sets and lower semi-analytic functions. Analytic sets in a Polish space have several equivalent definitions, one of which is that they are the images of Borel subsets of some Polish space under continuous or Borel measurable functions (see e.g., [2,Prop. 7.41], [9,Sec. 13.2]). More precisely, in a Polish space Y , the empty set is analytic by definition, and a nonempty set D is analytic if D = f (B) for some Borel set B in a Polish space and Borel measurable function f : B → Y [9, Thm. 13.2.1(c ′ )]. In a Polish space every Borel set is analytic and every analytic set is universally measurable ([2, Cor. 7.42.1], [9,Thm. 13.2.6]). The σ-algebra generated by the analytic sets is called the analytic σ-algebra and lies in between the Borel and universal σ-algebras. Thus functions that are analytically measurable (i.e., measurable w.r.t. the analytic σ-algebra) are also universally measurable.
Lower semi-analytic functions are extended real-valued functions whose lower level sets are analytic. Specifically, a function f : D → [−∞, ∞] is called lower semi-analytic if D is an analytic set and for every a ∈ R, the level set {x ∈ D | f (x) ≤ a} of f is analytic [2,Def. 7.21]. An equivalent definition is that the epigraph of f , {(x, a) | x ∈ D, f (x) ≤ a, a ∈ R}, is analytic (cf. [2, p. 186]).
For comparison, f is lower semicontinuous if its epigraph is closed. Since Borel sets are analytic, every extended real-valued, Borel measurable function on a Borel space is lower semi-analytic; since analytic sets are universally measurable, every lower semi-analytic function is universally measurable.
Some Properties of Analytic Sets and Lower Semi-analytic Functions
Analytic sets and lower semi-analytic functions play instrumental roles in the universal measurability framework for Borel-space MDPs. These sets and functions were chosen to be the foundation for a theoretical MDP model, because they possess many properties that are relevant to and important for stochastic dynamic programming. A full account of these properties is beyond our scope, however. For that, we refer the reader to the papers [5,22,36] and the monograph [2, Chap. 7] (for general properties of analytic sets, see also the books [10, Appendix 2] and [25,38]). Below we will only mention some properties that will be used frequently in this paper. They concern measurable selection theorems and the preservation of analyticity or lower semi-analyticity under various operations.
The class of analytic sets in a Polish space is closed under countable unions and countable intersections, and moreover, Borel preimages of analytic sets are also analytic ( [2,Cor. 7.35.2,Prop. 7.40], [38,Chap. 4]). These properties of analytic sets are reflected in the properties of lower semianalytic functions, whose lower level sets are analytic. Specifically, in the statements below, let D be an analytic set, and let X and Y be Borel spaces. Throughout the paper, for arithmetic operations involving extended real numbers, we define
∞ − ∞ = −∞ + ∞ = ∞, 0 · ±∞ = ±∞ · 0 = 0.
The following operations on lower semi-analytic functions result in lower semi-analytic functions (see [2,Lemma 7.30]):
(i) For a sequence of lower semi-analytic functions f n : D → [−∞, ∞], n ≥ 1, the functions inf n f n , sup n f n , lim inf n→∞ f n , and lim sup n→∞ f n are also lower semi-analytic.
(ii) If g : X → Y is Borel measurable and f : g(X) → [−∞, ∞] is lower semi-analytic, then the composition f • g is lower semi-analytic.
(iii) If f, g : D → [−∞, ∞] are lower semi-analytic functions, then f + g is lower semi-analytic. In addition, if f, g ≥ 0 or if g is Borel measurable and g ≥ 0, then f g is lower semi-analytic.
Another operation on lower semi-analytic functions is integration w.r.t. a stochastic kernel. If f : X × Y → [0, ∞] is lower semi-analytic and q(dy | x) is a Borel measurable stochastic kernel on Y given X, then the integral
φ(x) = Y f (x, y) q(dy | x)
is a lower semi-analytic function on X [2, Prop. 7.48]. (If q(dy | x) is analytically or universally measurable instead, then φ is universally measurable [2,Prop. 7.46 and Sec. 11.2] but not necessarily lower semi-analytic.) The preceding properties are closely related to the structure of the optimal cost functions and the selection of measurable policies in the MDP context. The next two properties concern analytic sets in product spaces or lower semi-analytic functions involving two variables. The first property is closely related to the validity of value iteration as well as the structure of the optimal cost function in the MDP context. If D is an analytic set in X × Y , then the projection of D on X, proj X (D) = {x | (x, y) ∈ D for some y}, is analytic [2,Prop. 7.39]. When applied to level sets of functions, an implication of this is that if D ⊂ X × Y is analytic and f : D → [−∞, ∞] is lower semi-analytic, then after partial minimization of f over the vertical sections D x of D for each x, the resulting function f * : proj X (D) → [−∞, ∞] given by
f * (x) = inf y∈Dx f (x, y), where D x = {y | (x, y) ∈ D}, (2.1)
is also lower semi-analytic [2,Prop. 7.47]. The Jankov-von Neumann measurable selection theorem asserts that if D is an analytic set in X × Y , then there exists an analytically measurable function φ : proj X (D) → Y such that the graph of φ lies in D, i.e., (x, φ(x)) ∈ D for all x ∈ proj X (D) [2,Prop. 7.49]. For minimization problems of the form (2.1), the theorem is applied to the level sets or epigraphs of lower semi-analytic functions. Together with other properties, it yields, for each ǫ > 0, the existence of an analytically measurable ǫ-minimizer, as well as the existence of a universally measurable ǫ-minimizer φ(·) that attains the minimum f * (x) at every x ∈ proj X (D) where this is possible: 1
φ(x) ∈ arg min y∈Dx f (x, y), if arg min y∈Dx f (x, y) = ∅,(2.2)
and for x ∈ proj X (D) with arg min y∈Dx f (x, y) = ∅,
φ(x) ∈ D x and f (x, φ(x)) ≤ f * (x) + ǫ, if f * (x) > −∞; −1/ǫ, if f * (x) = −∞. (2.3)
Further details about these measurable selection theorems can be found in [2,Prop. 7.50]. For MDPs, this is closely related to the existence of optimal or nearly optimal policies and their structures.
Definitions for Borel-space MDPs
In the universal measurability framework, a Borel-space MDP has the following elements and model assumptions (cf. [2, Chap. 8.1]):
• The state space X and the action space A are Borel spaces.
• The control constraint is specified by a set-valued map A : x → A(x), where for each state x ∈ X, A(x) ⊂ A is a nonempty set of admissible actions at that state, and the graph of A(·),
Γ = {(x, a) | x ∈ X, a ∈ A(x)} ⊂ X × A,
is analytic.
• The one-stage cost function c : Γ → [−∞, +∞] is lower semi-analytic.
• State transitions are governed by q(dy | x, a), a Borel measurable stochastic kernel on X given X × A.
We consider infinite horizon control problems. A policy consists of a sequence of stochastic kernels on A that specify for each stage, which admissible actions to apply, given the history up to that stage. In particular, a universally measurable policy is a sequence π = (µ 0 , µ 1 , . . .), where for each k ≥ 0, µ k da k | x 0 , a 0 , . . . , a k−1 , x k is a universally measurable stochastic kernel on A given (X × A) k × X and obeys the control constraint of the MDP: 2
µ k A(x k ) | x 0 , a 0 , . . . , a k−1 , x k = 1, ∀ (x 0 , a 0 , . . . , a k−1 , x k ) ∈ (X × A) k × X. (2.4)
A policy π is Borel measurable if each component µ k is a Borel measurable stochastic kernel; π is then also universally measurable by definition. (A Borel measurable policy, however, may not exist [4].) We define the policy space Π of the MDP to be the set of universally measurable policies. We shall simply refer to these policies as policies, dropping the term "universally measurable," if there is no confusion or no need to emphasize their measurability. We define several subclasses of policies in the standard way: A policy π is nonrandomized if for every k ≥ 0 and every (x 0 , a 0 , . . . , a k−1 , x k ), µ k da k | x 0 , a 0 , . . . , a k−1 , x k is a Dirac measure that assigns probability one to a single action in A(x k ). A policy π is semi-Markov if for every k ≥ 0, the function (x 0 , a 0 , . . . , a k−1 , x k ) → µ k (da k | x 0 , a 0 , . . . , a k−1 , x k ) depends only on (x 0 , x k ); Markov if for every k ≥ 0, that function depends only on x k ; stationary if π is Markov and µ k = µ for all k ≥ 0. For the stationary case, we simply write µ for π = (µ, µ, . . .). A nonrandomized stationary policy µ can also be viewed as a function that maps each x ∈ X to an action in A(x). We denote this mapping also by µ and we will use both notations µ(x), µ(da | x) in the paper.
Because the graph Γ of the control constraint A(·) is analytic, by the Jankov-von Neumann selection theorem [2,Prop. 7.49], there exists at least one universally measurable, nonrandomized stationary policy. Thus the policy space Π is non-empty. Given a policy π ∈ Π and an initial state distribution p 0 ∈ P(X), the collection of stochastic kernels
µ 0 (da 0 | x 0 ), q(dx 1 | x 0 , a 0 ), µ 1 (da 1 , | x 0 , a 0 , x 1 ), q(dx 2 | x 1 , a 1 ), . . . ,
. . . , µ k da k | x 0 , a 0 , . . . , a k−1 , x k , q(dx k+1 | x k , a k ), . . . , determines uniquely a probability measure r(π, p 0 ) on the universal σ-algebra on (X × A) ∞
(X × A) k+1 → [0, ∞] equals the iterated integral X A · · · X A f (x 0 , a 0 , . . . , x k , a k )µ k (da k | x 0 , a 0 , . . . , x k ) q(dx k | x k−1 , a k−1 ) · · · µ 0 (da 0 | x 0 ) p 0 (dx 0 ).
(Recall that whenever a Borel probability measure appears in the integral of a universally measurable function, the integration is defined w.r.t. the completion of the Borel probability measure.) In general, for a universally measurable function f :
(X × A) ∞ → [−∞, +∞], define Ef := Ef + − Ef − where f + = max{0, f } and f − = − min{0, f }; if Ef + = Ef − = +∞, define Ef = +∞ by following the convention ∞ − ∞ = −∞ + ∞ = ∞.
In the control problems that we will study, however, we will not encounter such summations.
The Expected Average Cost and Discounted Cost Criteria
We consider the average cost criterion and the discounted cost criterion. The n-stage value function of a policy π is given by
J n (π, x) := E π x n−1 k=0 c(x k , a k ) , x ∈ X,
where E π x denotes expectation w.r.t. the probability measure induced by π and the initial state x 0 = x (cf. the explanation given in Section 2.1.3). By [2,Prop. 7.46], the function J n (π, ·) is universally measurable. We define the average cost function of π by J(π, x) := lim sup n→∞ J n (π, x)/n,
x ∈ X, and the optimal average cost function by
g * (x) := inf π∈Π J(π, x) = inf π∈Π lim sup n→∞ J n (π, x)/n, x ∈ X.
For the discounted cost criterion, with a discount factor 0 < α < 1, we define the α-discounted value function of a policy π by v π α (x) := lim sup
n→∞ E π x n−1 k=0 α k c(x k , a k ) ,
x ∈ X, and the optimal α-discounted value function by
v α (x) := inf π∈Π v π α (x), x ∈ X.
Both J(π, ·) and v π α (·) are universally measurable (the latter by [2,Prop. 7.46]). However, whether the optimal cost functions g * and v α are universally measurable cannot be deduced immediately from their definitions. It will be shown in the next subsection, for two classes of MDP models, that g * and v α are indeed lower semi-analytic functions. This analysis relies on various properties of lower semi-analytic functions and a deep connection between a certain subset of the policy space and an analytic set, which we will explain more in the next subsection and in Appendix A.1.
We now introduce several classes of functions and the dynamic programming operators for an MDP, which will be needed in the subsequent analysis. Let M(X) denote the set of extended real-valued, universally measurable functions on X, and M b (X) the subset of bounded functions in M(X). We shall also consider certain subsets of unbounded functions in M(X). For a universally measurable function w : X → (0, +∞), which we shall refer to as a weight function, let
M w (X) := f | f w < ∞, f ∈ M(X) , where f w := sup x∈X f (x) /w(x).
The space M w (X) endowed with the weighted norm · w and the space M b (X) with the supreme norm · ∞ are both Banach spaces. Let A(X) denote the set of extended real-valued, lower semi-analytic functions on X. Note that A(X) ∩ M b (X) and A(X) ∩ M w (X) are closed subsets of M b (X) and M w (X), respectively. 4 For 0 < α < 1, define an operator T α that maps v ∈ M(X) to a function on X according to
(T α v)(x) := inf a∈A(x) c(x, a) + α X v(y) q(dy | x, a) ,
x ∈ X.
For α = 1, define an operator T likewise. We shall refer to them as dynamic programming operators.
Lemma 2.1 (cf. [2, Chap. 7]). The operators T and T α , 0 < α < 1, map A(X) into A(X).
This lemma follows from the model assumptions in the universal measurability framework for MDPs and the properties of analytic sets and lower semi-analytic functions given in Section 2.1.2. 5
Two Model Classes and some Basic Optimality Properties
We consider two model classes which we designate as (PC) and (UC):
• (PC) is simply the nonnegative model where c ≥ 0. For the average-cost or discounted problem, it is equivalent to the case where c is bounded from below.
• In (UC), the one-stage cost function c can be unbounded below or above, but it needs to satisfy a growth condition and moreover, there is a Lyapunov-type condition on the dynamics of the MDP. The precise definition is as follows.
Definition 2.1 (the model (UC)). There exist a universally measurable weight function w(·) ≥ 1 and constants b,ĉ ≥ 0 and λ ∈ [0, 1) such that for all x ∈ X,
(a) sup a∈A(x) |c(x, a)| ≤ĉ w(x); (b) sup a∈A(x) X w(y) q(dy | x, a) ≤ λw(x) + b.
For (UC), the conditions (a)-(b) in its definition ensure that the average cost function of any policy π satisfies J(π, ·) w ≤ ℓ for the constant ℓ =ĉ b/(1 − λ), and hence the optimal average cost function also satisfies g * w ≤ ℓ and in particular, g * is finite everywhere. For (PC), g * ≥ 0 and it is possible that at some state x, g * (x) = +∞. This possibility will be eliminated later under further assumptions on the model.
The nonnegative model (PC) has been analyzed in [2, Part II] under the expected total cost criterion. This book also discusses the expected discounted cost criterion and analyzes a model with bounded costs. It does not address the average cost criterion; nonetheless, some part of its analysis can be applied to the average cost case. In particular, the relations between a Borel-space MDP and a corresponding deterministic control model (DM) defined on spaces of probability measures ( [37] and [2, Chaps. 9.2-9.3]), by which many optimality results for the total or discounted cost criterion are derived in the book, give us a starting point to study the average cost case in the universal measurability framework.
The optimality properties stated in Theorem 2.1 below follow from those arguments from the book [2, Part II]. In this theorem as well as in what follows, by an ǫ-optimal or optimal policy, we mean a policy that is ǫ-optimal or optimal for all initial states. If a policy is only ǫ-optimal or optimal for a certain initial state or initial state distribution, we will state that explicitly. (ii) For each ǫ > 0, there exists a universally measurable, ǫ-optimal, randomized semi-Markov policy. If there exists an optimal policy for each state x ∈ X, then there exists a universally measurable, optimal, randomized semi-Markov policy.
We give the proof details in Appendix A. Specifically, we explain the corresponding deterministic control model (DM) in Section A.1, which will also be needed later in two other proofs, and we then prove Theorem 2.1 in Section A.2. Here let us make a few remarks about this theorem and its proof.
Remark 2.1 (comparison with some prior results). It is known that even for MDPs with a countable state space, a finite action space, and bounded one-stage costs, there need not exist an ǫ-optimal nonrandomized semi-Markov policy [10, Example 3, Chap. 7] nor an ǫ-optimal randomized Markov policy [11,Sec. 5]. In both of these counterexamples, there exists an optimal policy for each state. So without extra conditions on the MDP, Theorem 2.1(ii) is the strongest possible.
It is pointed out by Feinberg [11] that Strauch's results [39, Lemma 4.1 and the proof of Theorem 8.1] can be applied to the average cost case where the one-stage costs are bounded below, and they yield, for any p ∈ P(X) and ǫ > 0, the existence of a randomized semi-Markov policy that is ǫoptimal p-almost everywhere. The p-almost-everywhere optimality here is due to the restriction to only Borel measurable policies. With universally measurable policies, there exist policies that are optimal or nearly optimal everywhere. This is true for the finite-horizon and infinite-horizon total cost problems [36,37] and also true for the average cost problem, as reflected by Theorem 2.1(ii).
The ǫ-optimality mentioned above involves a constant ǫ. It can be generalized to a strictly positive function ǫ(·); such notions of optimality have been considered by Feinberg [12, Sec. 2.2]. Theorem 2.1(ii) holds as well with ǫ(·) in place of a constant ǫ. This can be shown by using the above version of Theorem 2.1(ii) to construct another policy with the desired ǫ(·)-optimality (the construction is similar to that given in Footnote 1). Alternatively, one can make a slight change in the proof of Theorem 2.1 to handle the function ǫ(·) directly, by using the implication/extension of a measurable selection theorem mentioned in Footnote 1.
Remark 2.2 (about the proof of Theorem 2.1 and the role of (DM)). To prove the second part of the theorem, we construct the desired ǫ-optimal (or optimal) policy directly from a universally measurable ǫ-optimal (or optimal) solution of (DM). This differs from the proofs of similar existence results for the total and discounted cost criteria given in [2,Chap. 9.6], where the existence of nearly optimal policies is analyzed by relating the dynamic programming operator of the original problem to that in (DM) and by transferring the optimality equations from (DM) to the original problem. In the average cost case, neither (DM) nor the original problem need to admit optimality equations or possess other dynamic-programming type of properties, so our proof cannot rely on such properties. The deterministic control model (DM) facilitates greatly the analysis. This is not because the average cost problem in (DM) could somehow be solved by dynamic programming, nor is it because a deterministic model can help us evade measurability issues in the original problem. (DM) is useful because the structure of its optimization problem permits more readily the application of the theory for analytic sets and lower semi-analytic functions. The optimality properties of (DM) thus obtained can then be transferred to the original problem via their correspondence relations.
Another comment is that the proofs based on (DM) share similarities with but differ from Strauch's proof of [39, Theorem 8.1] mentioned in the preceding remark. The major difference is that in [39] one deals directly with the set of probability measures on the trajectory space induced by all policies, whereas with (DM), one deals with only the set of sequences of marginal probability measures induced by the policies and this set turns out to have nicer properties with regard to measurable selection, as mentioned above. (See [2, Chap. 9.2] and Appendix A for more details).
In the next section, we will use the vanishing discount factor approach to prove the ACOI for (PC) and (UC) under additional conditions. That analysis starts with the optimality equations for the α-discounted cost criteria (α-DCOE), which are given below: Theorem 2.2 (the α-DCOE and existence of ǫ-optimal policies). (PC)(UC) For 0 < α < 1, the optimal value function v α is lower semi-analytic and satisfies the α-
DCOE v α = T α v α , i.e., v α (x) = inf a∈A(x) c(x, a) + α X v α (y) q(dy | x, a) ,
x ∈ X.
For (PC), v α is the smallest solution of the α-DCOE in A(X), whereas for (UC), v α is the unique solution of the α-DCOE in the space A(X) ∩ M w (X)
. Furthermore, in both cases, for each ǫ > 0, there exists a universally measurable, ǫ-optimal, nonrandomized stationary policy.
Under additional compactness and continuity conditions, proofs of the α-DCOE for (PC) and (UC) can be found in e.g., the papers [13,30] and the books [17,Chap. 5], [18,Chap. 8]. In the case here, we will use the results of [2, Part II] for general Borel-space MDPs to prove the above theorem. Specifically, for (PC), this theorem is implied by the optimality results for nonnegative models [2, Props. 9.8, 9.10, and 9.19].
For (UC), it can be shown (see Lemma A.2 in Appendix A.4) that for some universally measurable weight functionw ≥ w, the operator T α is a contraction on the closed subset A(X) ∩ Mw(X) of the Banach space (Mw(X), · w ). More precisely, for some β ∈ (α, 1),
T α v ∈ A(X) ∩ Mw(X) and T α v − T α v ′ w ≤ β v − v ′ w , ∀ v, v ′ ∈ A(X) ∩ Mw(X). (2.5)
We use this contraction property of T α together with the correspondence between the original problem and the deterministic control model (DM) [2, Chap. 9] to prove Theorem 2.2 for (UC). The proof is given in Appendix A.3. It is similar to, but does not follow exactly the one given in [2,Chap. 9] for bounded one-stage costs; see Remark A.1 at the end of Appendix A.3 for further explanations. As another preparation for the subsequent analysis, let us state a lemma about an implication of the ACOI on the existence and structure of average-cost optimal or nearly optimal policies. For comparison, recall that what Theorem 2.1 just showed is the existence of an ǫ-optimal, randomized semi-Markov policy, in the general case where g * need not be constant. The proof of this lemma uses mostly standard arguments and is given in Appendix A.4. Lemma 2.2 (a consequence of ACOI). Consider the models (PC) and (UC) with the average cost criterion. Suppose that the optimal average cost function g * is constant and finite. Suppose also that for some finite-valued h ∈ A(X), with h ≥ 0 for (PC) and h w < ∞ for (UC), the ACOI holds:
g * + h ≥ T h, i.e., g * + h(x) ≥ inf a∈A(x) c(x, a) + X h(y) q(dy | x, a) , x ∈ X. (2.6)
Then there exist an optimal nonrandomized Markov policy and, for each ǫ > 0, an ǫ-optimal nonrandomized stationary policy. If, in addition, the infimum in the right-hand side of the ACOI is attained for every x ∈ X, then there exists an optimal nonrandomized stationary policy.
Main Results: The ACOI
In this section, we place additional conditions on the models (PC) and (UC), under which we study the ACOI via the vanishing discount factor approach. Some of these conditions are standard and from prior work, and some are new conditions that we introduce to replace the compactness and continuity conditions used in the prior work to prove the ACOI for lower semicontinuous models. The arguments for the two models (PC) and (UC) are similar but differ in details, so we will discuss (PC) and (UC) in two separate subsections.
The Case of Unbounded Costs (UC)
We consider the model (UC) first. Letx be some fixed state and consider the relative value functions of the α-discounted problems:
h α (x) := v α (x) − v α (x), x ∈ X. (3.1)
Assumptions
The first assumption is extracted from the prior work on ACOI:
Assumption 3.1. For the model (UC), the set of functions {h α | α ∈ (0, 1)} as defined above is bounded in M w (X), i.e., sup α∈(0,1) h α w < ∞.
In the prior work, a w-geometric ergodicity condition has been used to ensure that the functions h α have the above boundedness property. Specifically, it is assumed, or ensured through other conditions, that every stationary nonrandomized policy induces a w-geometric ergodic Markov chain on the state space (see e.g., [18,Lemma 10.4.2] and [19, p. 498]; see [23,Chap. 15] for the definition of such Markov chains). This together with the existence of ǫ-optimal nonrandomized stationary policies for each ǫ > 0 (cf. Theorem 2.2) then guarantees the boundedness of the family {h α | α ∈ (0, 1)}. In addition to the w-geometric ergodicity condition, compactness and continuity conditions are also involved in deriving the ACOI or ACOE in the prior work [18,19].
To start the analysis of the ACOI, we will need an implication of Assumption 3.1 given in the following lemma. Take a sequence α n ↑ 1 such that for some finite number ρ * ,
(1 − α n ) v αn (x) → ρ * as n → ∞. (3.2)
This is possible because the model conditions of (UC) imply that for each x ∈ X, We now introduce new conditions, which we use to replace compactness and continuity conditions used in the prior work on the ACOI for (UC):
(1 − α) v α (x) is bounded over α ∈ (0,Assumption 3.2.
In the model (UC), for each x ∈ X and ǫ > 0, the following hold:
(i) There exist a compact set K ⊂ A and 0 <ᾱ < 1 such that for all α ∈ [ᾱ, 1),
inf a∈K∩A(x) c(x, a) + α X v α (y) q(dy | x, a) ≤ v α (x) + ǫ. (3.5) (ii) There exists a (nonnegative) finite measure ν on B(X) that majorizes every q(dy | x, a), a ∈ K ∩ A(x): sup a∈K∩A(x) q(B | x, a) ≤ ν(B), ∀ B ∈ B(X). (3.6) (iii) The weight function w(·) is uniformly integrable w.r.t. {q(dy | x, a) | a ∈ K ∩A(x)} in the sense that lim ℓ→∞ sup a∈K∩A(x) X w(y) 1 w(y) ≥ ℓ q(dy | x, a) = 0, (3.7)
where 1(·) denotes the indicator function.
Let us give here a preliminary discussion about these conditions. We will discuss them further and also give an illustrative example after we prove the ACOI, since the roles of some of these conditions can be better seen then (see Section 3.1.3).
First, there are cases where all or some of the conditions in Assumption 3.2 hold obviously. For example, if for each x, A(x) is finite, then all three conditions are satisfied by letting K = A(x). (This simple setting is not of primary interest to us, however, because optimality results can be derived directly in this case, without using the proof approach that we are going to take.) More generally, if A(x) or A is compact, then (i) holds trivially for K = A(x) or A. If w(·) is bounded from above on the union of the supports of the probability measures q(dy | x, a), a ∈ K ∩ A(x), such as in the case where the union is contained in a compact set and w(·) is continuous, then (iii) is clearly satisfied. If c(·) is bounded, then w(·) can be chosen to be constant and (iii) then holds trivially. Note that Assumption 3.2 does not require the one-stage cost function c(·) to have any special properties.
Regarding Assumption 3.2(iii), it is implied by the slightly stronger, yet much simpler-looking condition w dν < ∞, where ν is the majorizing finite measure in Assumption 3.2(ii). The condition w dν < ∞, however, can be inconvenient to verify when the measure ν is too complicated and a direct evaluation of the integral w dν is impractical. In comparison, verifying the condition (iii) can be straightforward when the weight function w(·) has a simple analytical expression (e.g., when x ∈ R and w(x) = e x or x 2 ). This is why we have Assumption 3.2(iii) as is. Note that the situation is different for the condition (ii) in which ν appears, because to verify (ii), we do not need the exact expression of ν. It suffices that some finite measure with the desired majorization property exists. This can be inferred qualitatively, in some cases, from the properties of the state transition stochastic kernel q(dy | x, a), without the need for exact calculation.
Assumption 3.2(ii) is the key condition. Our purpose is to use this majorization condition to handle a certain class of discontinuous models. Although lower semicontinuous models cover a large class of problems and are mathematically elegant, discontinuity naturally occurs in physical systems. The behavior of such systems can vary gradually within certain regions of the state space but change abruptly across the boundaries of these regions, depending on which physical mechanisms come into effect. The class of discontinuous models for which Assumption 3.2(ii) can hold naturally are those where q(dy | x, a) is not continuous in a or (x, a), but for all a ∈ K ∩ A(x), q(dy | x, a) has a density f x,a w.r.t. a common (σ-finite) reference measure ϕ. The pointwise supremum of the density functions, f x := sup a∈K∩A(x) f x,a (or a measurable function that upper-bounds it), when it belongs to L 1 (X, B(X), ϕ), defines a finite measure ν, with dν = f x dϕ, that has the desired majorization property (3.6).
Note that Assumption 3.2(ii) need not be satisfied by continuous state transition stochastic kernels. For example, if A(x) = [0, 1] and q(dy | x, a) = δ a (the Dirac measure at a), there is no finite measure with the desired majorization property.
Regarding Assumption 3.2(i), in some circumstances, the existence of a compact set K with the property (3.5) is close to being a necessary condition for the ACOI to hold. We shall discuss this condition further after we analyze the ACOI. There we will explain where this condition comes from, in particular, its connection with the theory on epi-convergence of functions (see Section 3.1.3). Here let us remark that first, our proof of the ACOI will not use the compactness property of K, so it suffices that for some subset K of actions, (3.5) and the rest of the assumptions hold. Second, the subset K introduced in this condition is important in the subsequent Assumption 3.2(ii): It would be too stringent to require a finite measure ν to majorize q(dy | x, a) for all a ∈ A(x) instead of a ∈ K ∩ A(x). For example, if A(x) = R and q(dy | x, a) is the normal distribution N (a, 1) on R, no finite measure can majorize these distributions for all a ∈ R.
Remark 3.1 (about the majorization condition in [15]). Gubenko and Shtatland used a minorization condition and a majorization condition, alternatively, to convert the dynamic programming operator T into a contraction (roughly speaking), thereby proving the ACOE via a contraction-based fixed point approach [15, Theorem 2 and 2 ′ ]. Their majorization condition [15, Sec. 3, Condition (II)] is like a symmetric counterpart of their minorization condition, and it requires that there exists a finite measure ν on B(X) such that
q(B | x, a) ≤ ν(B), ∀ B ∈ B(X), (x, a) ∈ Γ, and ν(X) < 2. (3.8)
Note that here the same measure ν needs to majorize q(dy | x, a) for all states and admissible actions, whereas in our Assumption 3.2, ν can be different for each state. The requirement ν(X) < 2 (needed for converting T into a contraction) is too stringent and renders their condition (3.8) impractical.
Optimality Results
We now prove the ACOI for (UC) under the assumptions introduced in the preceding subsection. The result and its proof involve the relative value functions {h n }, the functions h, {h n },h, {h n }, and also the scalar ρ * = lim n→∞ (1 − α n ) v αn (x) that we defined earlier (cf. (3.1)-(3.4)).
Theorem 3.1 (the ACOI for (UC)). For the (UC) model, under Assumptions 3.1-3.2, the optimal average cost function g * (·) = ρ * , and with h ∈ A(X) ∩ M w (X) as given in (3.3), the pair (ρ * , h) satisfies the ACOI:
ρ * + h(x) ≥ inf a∈A(x) c(x, a) + X h(y) q(dy | x, a) , x ∈ X. (3.9)
Hence there exist an optimal nonrandomized Markov policy and for each ǫ > 0, an ǫ-optimal nonrandomized stationary policy.
This theorem also implies that ρ * does not depend on our choice of the sequence {α n } or the statex, and lim
α→1 (1 − α) v α (x) = g * , ∀ x ∈ X.
To prove the theorem, we first prove a lemma.
lim n→∞ inf a∈K∩A(x) c(x, a) + α n X h n (y) q(dy | x, a) = inf a∈K∩A(x) c(x, a) + X h(y) q(dy | x, a) .
Proof. For the state x, ǫ > 0, and the set K given in the lemma, let ν be the corresponding finite measure on B(X) in Assumption 3.2(ii). Recall that h n ↑ h and these functions are finite-valued and universally measurable (Lemma 3.1). Therefore, by Egoroff's Theorem [9, Theorem 7.5.1], for any δ > 0, there exists a universally measurable set D δ ⊂ X with ν(X \ D δ ) < δ such that on the set D δ , h n converges to h uniformly as n → ∞. Consequently, for any η > 0, it holds for all n sufficiently large that
D δ h(y) − h n (y) q(dy | x, a) ≤ η, ∀ a ∈ A(x). (3.10)
We now bound the integral of h − h n on the complement set X \ D δ . By Lemma 3.1, for all n ≥ 0, h − h n w ≤ ℓ for some constant ℓ. So for all a ∈ A(x),
X\D δ h(y) − h n (y) q(dy | x, a) ≤ ℓ X\D δ w(y) q(dy | x, a).
By the choice of D δ and the majorization property of ν in Assumption 3.2(ii), we have
sup a∈K∩A(x) q X \ D δ | x, a ≤ ν X \ D δ < δ. (3.11)
By an alternative characterization of uniform integrability [9, Theorem 10.3.5], (3.11) together with the uniform integrability assumption in Assumption 3.2(iii) implies that for any given η > 0, it holds for all δ sufficiently small that X\D δ w(y) q(dy | x, a) ≤ η for all a ∈ K ∩ A(x). Consequently, given η > 0, by choosing δ sufficiently small, we can make
sup a∈K∩A(x) X\D δ h(y) − h n (y) q(dy | x, a) ≤ η.(3.12)
Combining this with (3.10), we obtain that for all n sufficiently large,
sup a∈K∩A(x) X h(y) − h n (y) q(dy | x, a) ≤ 2η and hence inf a∈K∩A(x) c(x, a) + α n X h n (y) q(dy | x, a) ≥ inf a∈K∩A(x) c(x, a) + α n X h(y) q(dy | x, a) − 2η.
Since η is arbitrary and h n ≤ h, the lemma follows by letting n → ∞ on both sides of the preceding inequality (and using also the fact that since sup a∈K∩A(x) X |h(y)| q(dy | x, a) < ∞ under Assumption 3.1 and the model condition of (UC), (1 − α n ) sup a∈K∩A(x) X |h(y)| q(dy | x, a) → 0).
Proof of Theorem 3.1.
For each x ∈ X and ǫ > 0, by the α-DCOE (Theorem 2.2) and Assumption 3.2(i), for all n sufficiently large
(1 − α n ) v αn (x) + h n (x) = inf a∈A(x) c(x, a) + α n X h n (y) q(dy | x, a) (3.13) ≥ inf a∈K∩A(x) c(x, a) + α n X h n (y) q(dy | x, a) − ǫ ≥ inf a∈K∩A(x) c(x, a) + α n X h n (y) q(dy | x, a) − ǫ,(3.14)
where the last inequality used the fact h n ≤ h n and that h n is universally measurable (Lemma 3.1).
Letting n → ∞ in both sides of (3.14), we have
ρ * + h(x) + ǫ ≥ lim inf n→∞ inf a∈K∩A(x) c(x, a) + α n X h n (y) q(dy | x, a) = inf a∈K∩A(x) c(x, a) + X h(y) q(dy | x, a) ≥ inf a∈A(x) c(x, a) + X h(y) q(dy | x, a) ,
where the equality follows from Lemma 3.2. Since this holds for every x ∈ X and ǫ is arbitrary, the desired inequality (3.9) is proved.
To show g * (·) = ρ * , as in the analysis in [19], it suffices to show that for the pair (ρ * ,h), wherē h = lim sup n→∞ h n as we recall, the opposite inequality holds:
ρ * +h(x) ≤ inf a∈A(x) c(x, a) + Xh (y) q(dy | x, a) ,
x ∈ X. (3.15) This is because the inequality (3.15) implies that for all policies π and state x, the average cost J(π, x) ≥ ρ * , whereas the inequality (3.9) just proved implies the opposite relation J(π, x) ≤ ρ * .
(The proof of the former is standard and similar to that of Lemma 2.2 given in Appendix A.4, and the proof of the latter is the same as that of Lemma 2.2.) From the α-DCOE (3.13), we have that for all a ∈ A(x),
(1 − α n ) v αn (x) + h n (x) ≤ c(x, a) + α n X h n (y) q(dy | x, a),(3.16)
so by taking limit supremum of both sides as n → ∞ and using also the facth n ≥ h n by definition, we have
ρ * +h(x) ≤ c(x, a) + lim sup n→∞ α n Xh n (y) q(dy | x, a) = c(x, a) + Xh (y) q(dy | x, a),
where the last inequality follows from the dominated convergence theorem, in view of Lemma 3.1 and the model condition of (UC). This proves (3.15) and hence g * (·) = ρ * as discussed earlier.
Finally, that h ∈ A(X) ∩ M w (X) follows from Lemma 3.1, and the existence of an optimal Markov policy and ǫ-optimal stationary policy follows from the ACOI proved above and Lemma 2.2.
Further Discussion on Assumption 3.2 and an Illustrative Example
In this discussion, to simplify notation, for pointwise limits of functions, we shall abbreviate the expression "n → ∞" and write, for instance, "lim n " or "lim inf n " instead. We shall assume c(x, a) = +∞ if a ∈ A(x), so that we can write "inf a f (a)" instead of "inf a∈A(x) f (a)" when f is of the form f (·) = c(x, ·) + ψ(x, ·) for a given state x and some function ψ.
Let us first explain the origin of Assumption 3.2(i). In introducing this condition, we have been influenced by the theory on epi-convergence of functions on R d [29,Chap. 7]. A sequence of extended real-valued functions {f n } on R d converges epigraphically to a function f , denoted f n e → f , if as n → ∞, epi(f n ) (the epigraph of f n ) converges to epi(f ) in the sense of set convergence. 6 We denote this limit by e-lim n f n . It is, by definition, lower semicontinuous and lies below lim inf n f n . For a nondecreasing sequence {f n }, e-lim n f n always exists and equals sup n (cl f n ) [29,Prop. 7.4(d)], where cl f n is the closure of f n (i.e., the function whose epigraph equals the closure of epi(f n ) or in other words, the largest lower semicontinuous function majorized by f n ). To relate such f n and their epi-limits to the functions involved in our problem, suppose that for every x ∈ X, A(x) ⊂ ℜ d . Assume also that h n = h n ≥ 0 for all n, for simplicity. Let us investigate when the ACOI is impossible. This will show us what kind of condition is needed for the desired ACOI to hold.
For a given state x, let f n (a) = c(x, a) + α n X h n (y) q(dy | x, a). Since the nonnegative functions h n ↑ h, the sequence {f n } is nondecreasing and hence e-lim n f n exists, as discussed earlier. By the monotone convergence theorem, the pointwise limit lim n f n also exists, and it is the function c(x, ·) + X h(y) q(dy | x, ·) and lies above the epi-limit. Hence for all m ≥ 0, Recall that for (UC), we have −∞ < inf a (e-lim n f n )(a) < +∞. In such a case, by [29,Theorem 7.31], (3.19) holds if and only if for every ǫ > 0, there exists a compact set K ⊂ R d such that inf a∈K f n (a) ≤ inf a f n (a) + ǫ, for all n sufficiently large. (3.20) In the preceding discussion we have assumed A ⊂ R d so that we can use the epi-convergence results in [29,Chap. 7] to shorten the discussion. When A is a general Borel space in the above setup, and also when h n = h n are not assumed to be nonnegative, one can directly verify that the same conclusion is reached for a compact set K ⊂ A. The inequality (3.20) is unwieldy to verify for a given problem, since, among others, f n depends on the particular choice of the sequence {α n }. Therefore, we consider a similar condition instead: For all α sufficiently close to 1,
cl f m ≤ e-lim n f n ≤ lim n f n ,(3.inf a∈K c(x, a) + α X h α (y) q(dy | x, a) ≤ inf a c(x, a) + α X h α (y) q(dy | x, a) + ǫ.
Since h α differs from v α by a constant and with v α in place of h α , the right-hand side (r.h.s.) above is v α (x) + ǫ by the α-DCOE, the above condition is equivalent to Assumption 3.2(i).
Note that this condition alone does not guarantee the equality (3.19) in general when h n = h n . Even when (3.19) holds, the second inequality in (3.18) can still be strict, ruling out the ACOI. So Assumption 3.2(i) alone is insufficient. It is Assumption 3.2(ii)-(iii) that give us the rest of the help needed in establishing the ACOI.
We now use a simple example to illustrate when Assumption 3.2 holds and can be verified relatively straightforwardly. For the model (UC), because the one-stage cost c(x, a) is bounded over the action set A(x) for a given state x, we have not yet found an easy way to identify the compact set K in Assumption 3.2(i) when A(x) is unbounded. (For comparison, in the case of the model (PC) that we will discuss next, an unbounded c(x, ·) can help identify the set K.) It is for this reason that the example below deals only with a compact action space A. x n+1 = x n + η n (x n , a n ) − ξ n (x n , a n ) + , n ≥ 0.
Here we assume that given (x n , a n ) = (x, a) ∈ X × A, η n (x, a) and ξ n (x, a) are random variables that are independent of the history x k , a k , η k (x k , a k ), ξ k (x k , a k ) k<n and also mutually independent. Furthermore, we assume that their conditional probability distributions are parametrized by (x, a) only and independent of n, and we denote these distributions by F x,a and G x,a for η n (x, a) and ξ n (x, a), respectively. This specifies indirectly q(dy | x, a) for the problem. The admissible action sets A(x), x ∈ X, are just subsets of A; we do not need them to be compact. This type of model appears in several applications, e.g., random-release dams, queueing and inventory-production systems (see the aforementioned examples in [18]). In a random-release dam model, for example, x n could be the amount of water in the reservoir at the beginning of the nth stage, η n (x n , a n ) and ξ n (x n , a n ) could be the amount of inflow and outflow, respectively, during the nth stage. In an inventory-production system, x n could be the stock level and a n the amount of product ordered at the beginning of the nth stage; η n (x n , a n ) could be the amount of product received and ξ n (x n , a n ) the demand during the nth stage.
X w(y) q(dy | x, a) ≤ λ w(x) + 1, ∀ x ∈ X.
Suppose that the one-stage cost function also satisfies the bound sup a∈A(x) |c(x, a)| ≤ĉ e κx , x ∈ X, for someĉ > 0, so that the problem under consideration belongs to the (UC) class. Let us discuss now some cases where Assumption 3.2 holds. Let us choose the compact set K = A = [0, L] for every state x and ǫ > 0; then Assumption 3.2(i) is satisfied trivially. Next, consider Assumption 3.2(iii) in two different situations:
(a) Suppose that for every (x, a) ∈ Γ, we have η 0 (x, a), ξ 0 (x, a) ≥ 0 (which is the case in the aforementioned applications). In addition, suppose that for all a ∈ A(x), η 0 (x, a) ∈ [0, ℓ x ] for some state-dependent constant ℓ x . Then given x 0 = x, x 1 lies in some bounded interval D x .
Since the exponential weight function w(y) = e κy is continuous, it is bounded above on D x . Then Assumption 3.2(iii) is satisfied trivially, as discussed earlier in Section 3.1.1.
(b) Alternatively, suppose that for each state x, η 0 (x, a) and ξ 0 (x, a) are normally distributed random variables that have means and variances bounded uniformly in a over A(x). Or more generally, suppose that they have light-tailed distributions with bounded means and tail probabilities that decrease faster than e −βx|y| as |y| → ∞, for some β x > 2κ.
In this situation, it can be verified straightforwardly that the weight function w(y) = e κy is uniformly integrable w.r.t. {q(dy | x, a) | a ∈ A(x)}, as required by Assumption 3.2(iii).
Finally, consider Assumption 3.2(ii) for each state x. Suppose w.r.t. the Lebesgue measure, F x,a , G x,a , a ∈ A(x), have density functions that are bounded above uniformly. Then, in the situation (a) above, since with x 0 = x, x 1 takes values in the bounded set D x , we can simply let the majorizing finite measure ν in Assumption 3.2(ii) be a multiple of the Lebesgue measure on D x . In the situation (b) above, there exists a function that not only majorizes all those density functions of F x,a , G x,a , a ∈ A(x), but also decays faster than exponentially; subsequently, from this function, one can define a finite measure ν to satisfy Assumption 3.2(ii).
Note that we do not need the continuity of F x,a and G x,a in a or in (x, a). Nor do we need the continuity of their density functions.
The Case of Nonnegative Costs (PC)
We now consider the nonnegative cost model (PC) and introduce two additional assumptions for the study of the ACOI. The first assumption includes two conditions from the prior work on the ACOI for MDP models satisfying certain continuity conditions. Let
m α := inf x∈X v α (x),
α ∈ (0, 1).
Assumption 3.3. For the model (PC):
(G) For some policy π and state x, the average cost J(π, x) < ∞.
(B) For every x ∈ X, lim inf α↑1 (v α (x) − m α ) < ∞.
Precursors to these conditions were introduced in the early work of Sennott [33] and Schäl [32] and then evolved as the research progressed. In particular, the condition (G) is the same as that in [32] and by [32, Lemma 1.2(b)], it implies that
lim sup α→1 (1 − α) m α ≤ inf x∈X g * (x) < ∞.
The condition (B) is introduced more recently by Feinberg, Kasyanov, and Zadoianchuk [13] to weaken one condition in [32] (cf. (3.25) in Example 3.2). That one needs only a pointwise (instead of uniform) upperbound on the relative value functions was first shown by Sennott [33] for countablespace MDPs. Besides (G) and (B), most existing prior work studies the ACOI under additional compactness/continuity conditions; see [13,32] for the details of these conditions and their earlier forms (see also [14,42] and the references therein for related work).
Let
h α := v α − m α , h := lim inf α→1 h α . (3.22)
For each α ∈ [0, 1), define a function h α as
h α (x) := inf β∈[α,1) h β (x), x ∈ X. (3.23)
Note that h α ≤ h β ∀ α ≤ β < 1, and h α ↑ h as α ↑ 1. Proof. That the functions h and h α are finite-valued is clear from Assumption 3.3(B). To prove that they are lower semi-analytic, we consider them as functions of (α, x), and we show first that v α is a lower semi-analytic function of (α, x) on (0, 1) × X. This proof uses the deterministic control model (DM) corresponding to the MDP and is given in Appendix A.4 (see Lemma A.1). Next, consider m α as a function of α. Since the one-stage cost c ≥ 0 in (PC), for each policy π and initial state x, the α-discounted value is non-decreasing as α increases. Therefore, m α = inf x∈X v α (x) is also monotonically non-decreasing as α increases. It follows that m α is a Borel-measurable function of α on (0, 1) and so is −m α . The latter together with the first part of the proof implies, by [2, Lemma 7.30(4)], that h α (x) = v α (x) − m α is lower semi-analytic in (α, x). Now, since for each α, h α is the partial minimization of h β (x) over β ∈ [α, 1), h α is a lower semi-analytic function of x by [2, Prop. 7.47]. As h is the pointwise limit of h α as α ↑ 1, h is lower semi-analytic by [2, Lemma 7.30 (2)].
We now impose three additional conditions on the model (PC). They are similar to Assumption 3.2 in the previous (UC) case. As before, we use these conditions in place of the compactness/continuity conditions used in the aforementioned prior work, to prove the ACOI for (PC).
Assumption 3.4. In the model (PC), for each x ∈ X and ǫ > 0, (i)-(ii) Assumption 3.2(i)-(ii) hold; (iii) Assumption 3.2(iii) holds with the function h in place of the weight function w.
We have already discussed extensively the roles of the conditions (i)-(ii) and how they can be verified, in Sections 3.1.1, 3.1.3. An additional observation is that for the (PC) model, verifying the condition (i) can be straightforward when A is non-compact but c(x, ·) is "coercive": (3.25) (This is the condition (B) in [32]. Under the condition (G), it is equivalent to lim sup α↑1 h α (x) < ∞ for x ∈ X [13, Lemma 5].) Then Assumption 3.4(i) is also satisfied. To see this, note that Regarding the condition (iii), it is difficult to verify directly in practice. Instead, a viable way is to first find an upper boundĥ ≥ h that has a simple analytical expression like the weight function w in the previous (UC) case, and then verify the inequality (3.7) withĥ in place of h. If such a boundĥ is available (e.g., obtained in the process of verifying Assumption 3.3(B)), then the discussions given in Sections 3. 1.1 and 3.1.3 about verifying (3.7) using the properties of w and the state transition stochastic kernel also apply here withĥ in place of w.
sup α∈(0,1) v α (x) − m α = sup α∈(0,1) h α (x) < ∞, ∀ x ∈ X.inf a∈A(x) c(x, a) + α X h α (y) q(dy | x, a) = (1 − α) m α + h α (x), ∀ α < 1,
Our ACOI result is as follows. Its conclusions are almost the same as those of Theorem 3.1 for the (UC) model and the proof is also similar.
Theorem 3.2 (the ACOI for (PC)). For the (PC) model, suppose Assumptions 3.3-3.4 hold. Let
ρ * = lim sup α↑1 (1 − α) m α and let the finite-valued function h ∈ A(X) be as given in (3.22). Then the optimal average cost function g * (·) = ρ * , and the pair (ρ * , h) satisfies the ACOI:
ρ * + h(x) ≥ inf a∈A(x) c(x, a) + X h(y) q(dy | x, a) ,
x ∈ X. (3.26) Hence there exist an optimal nonrandomized Markov policy and for each ǫ > 0, an ǫ-optimal nonrandomized stationary policy.
Proof. Consider an arbitrary x ∈ X and fix it in the proof below. Choose a sequence {α n } such that
lim n→∞ h αn (x) = lim inf α→1 h α (x) = h(x).
Note that ρ * ≥ lim sup n→∞ (1 − α n ) m αn . Recall also that ρ * ≤ inf y∈X g * (y) by [
(1 − α n ) m αn + h n (x) = inf a∈A(x) c(x, a) + α n X h n (y) q(dy | x, a) .
(3.27)
Then similarly to the derivation of (3.14), we apply Assumption 3.4(i) and replace h n by h n in the r.h.s. above before letting n → ∞ in both sides of (3.27). This results in the inequality
ρ * + h(x) ≥ lim inf n→∞ inf a∈K∩A(x) c(x, a) + α n X h n (y) q(dy | x, a) − ǫ (3.28)
where K is the compact set in Assumption 3.4(i) for the fixed state x and ǫ > 0, and the integration operation h n (y) q(dy | x, a) is valid since h n is universally measurable by Lemma 3.3. What remains to show is that the conclusion of Lemma 3.2 holds so that
lim inf n→∞ inf a∈K∩A(x) c(x, a) + α n X h n (y) q(dy | x, a) = inf a∈K∩A(x)
c(x, a) + X h(y) q(dy | x, a) .
(3.29) For this, we can use the same proof of Lemma 3.2 except for two small changes. The first change is in the proof of the bound (3.12), sup a∈K∩A(x) X\D δ h(y) − h n (y) q(dy | x, a) ≤ η for a given η > 0 and for all n sufficiently large. Here we use the fact that h − h n ≤ h in this case, and we also use the uniform integrability condition in Assumption 3.4(iii) for the nonnegative function h, instead of the weight function w. The second change is that at the end of the proof of Lemma 3.2, to have (1 − α n ) sup a∈K∩A(x) X h(y) q(dy | x, a) → 0, we need sup a∈K∩A(x) X h(y) q(dy | x, a) < ∞, and this is implied by the uniform integrability condition on h in Assumption 3.4(iii).
We now combine the relations (3.28) and (3.29) to obtain, as before,
ρ * + h(x) + ǫ ≥ inf a∈K∩A(x) c(x, a) + X h(y) q(dy | x, a) ≥ inf a∈A(x) c(x, a) + X h(y) q(dy | x, a) .
Since this holds for an arbitrary x ∈ X and an arbitrary ǫ > 0, we obtain the desired ACOI. Since ρ * ≤ g * (·) [32, Lemma 1.2(b)], the ACOI just established implies that we must have g * (·) = ρ * [32, Prop. 1.3] (the proof is essentially the same as that of Lemma 2.2 given in Appendix A.4). Finally, the existence of an average-cost optimal Markov policy and ǫ-optimal stationary policy follows from Lemma 3.3, the ACOI proved above, and Lemma 2.2. This completes the proof.
Other Results: The Existence of a Minimum Pair
In this section, we consider the minimum pair approach ( [16,21]; [17,Chap. 5.7]) for the case of nonnegative, strictly unbounded costs. Compared with the vanishing discount factor approach, this is a direct approach to analyzing average-cost MDPs. Let J(π, p) denote the average cost of a policy π for an initial state distribution p ∈ P(X). A pair (π * , p * ) ∈ Π × P(X) is called a minimum pair if
J(π * , p * ) = inf p∈P(X) inf π∈Π J(π, p).
The question we shall focus on is whether there exists a minimum pair with π * being a stationary policy. When this is the case, under an additional recurrence condition on the Markov chain induced by π * , one can ensure that π * is not only optimal but also pathwise optimal for the average cost criterion (see [17,Theorem 5.7.9(b)]), and furthermore, it is also possible to construct from π * a nonrandomized stationary policy with the same property (see [21, Theorem 2.2 and its proof]). The minimum pair approach for lower semicontinuous models with strictly unbounded costs has been discussed in detail in the books [17,Chap. 5.7], [18,Chap. 11.4]. Our interest is again to replace the continuity conditions with majorization type conditions similar to the ones in Section 3. In the present case, however, our results are less general: they apply only to a discrete action space. With a discrete A, one can work with Borel measurable policies without encountering measurability issues, so in this section, we let the policy space Π be the set of Borel measurable policies. The state space X is still assumed to be a Borel space in our results. (For comparison, for a discrete A, a measure space X with measurable singleton subsets has been considered [28]. ) We state the conditions below. The condition (G) is to exclude vacuous problems, and the condition (SU) defines what we mean by strictly unbounded costs. These are standard conditions. The condition (M) is the new majorization condition that we introduce.
q (O \ D) ∩ B | x, a ≤ ν(B), ∀ B ∈ B(X), (x, a) ∈ Γ, (4.1)
where D ⊂ X is some closed set (possibly empty) such that restricted to D × A, the state transition stochastic kernel q(dy | x, a) is continuous and the one-stage cost function c is lower semicontinuous.
In the majorization condition (M), roughly speaking, we divide the state space into two parts, a closed set D on which the model has the appealing continuity properties, and the complement set of D on which we impose a majorization condition (4.1). The condition (M) is satisfied trivially by letting D = X, if the entire model is lower semicontinuous (this means, in the discrete action setting considered here, that for each a ∈ A, q(dy | x, a) is continuous in x and c(x, a) is lower semicontinuous in x). For discontinuous models in general, the condition (M) seems natural in cases where the probability measures {q(· | x, a) | (x, a) ∈ Γ} have densities on the complement set D c := X \ D, w.r.t. a common (σ-finite) reference measure. In such cases, under practical conditions on those density functions, the condition (M) holds; see Section 4.2 for an illustrative example.
Our main results are as follows. They are analogous to the prior results for lower semicontinuous models (see [ 1, for any pair (π, p) ∈ Π × P(X) with J(π, p) < ∞, there exists a stationary policyμ andp ∈ P(X) such thatp is an invariant probability measure of the Markov chain on X induced byμ and J(μ,p) ≤ J(π, p). Assumption 4.1, there exists a minimum pair (μ,p), whereμ is a stationary policy andp is an invariant probability measure of the Markov chain on X induced byμ.
Proofs
We prove Prop. 4.1 and Theorem 4.1 in this subsection. The major proof steps are the same as those for [17, Theorem 5.7.9 and Lemma 5.7.10], but since the model here is not a lower semicontinuous model, the details in some steps are different. We will focus on those steps.
Let C b (X) denote the set of bounded continuous functions on X. Recall that a set of probability measures on a topological space is called tight if for any ǫ > 0, there exists a compact set whose measure is greater than 1 − ǫ w.r.t. every probability measure in that set [9, p. 293].
Consider the pair (π, p) in Prop. 4.1. Let γ n ∈ P(X × A) be the marginal distribution of (x n , a n ) under the policy π with initial distribution p. For n ≥ 1, defineγ n ∈ P(X × A) to be the averagē γ n := 1 n n k=1 γ k .
By assumption, the average cost J(π, p) = lim sup n→∞ c dγ n < ∞. Together with Assumption 4.1(SU), this implies that {γ n } is tight. So by Prohorov's Theorem [3, Theorem 6.1], there exists a subsequence {γ n k } k≥0 converging weakly to someγ ∈ P(X × A).
By [2, Cor. 7.27.2], we can decomposeγ into its marginalp on X and a Borel measurable stochastic kernelμ(da | x) on A given X, and by modifyingμ(da | x) at a set ofp-measure zero if necessary, we can make it obey the control constraint of the MDP:
γ(d(x, a)) =μ(da | x)p(dx), andμ(A(x) | x) = 1, ∀ x ∈ X.
In order to prove Prop. 4.1, we need to show the following:
(i)p is an invariant probability measure of the Markov chain on X induced by the stationary policyμ, namely,p
(B) = X A q(B | x, a)μ(da | x)p(dx), ∀ B ∈ B(X).
By [9,Prop. 11.3.2], this is equivalent to that for every v ∈ C b (X), To prove (i)-(ii), we will make use of the following implication of Assumption 4.1(M). Letp n denote the marginal ofγ n on X. Recall thatp is the marginal ofγ on X.
X×A X v(y) q(dy | x, a)γ(d(x, a)) = X v(x)p(dx).
On Average-Cost Borel-Space MDPs
p (O \ D) ∩ B ≤ ν(B),p n (O \ D) ∩ B ≤ ν(B), ∀ n ≥ 1.
Proof. For n ≥ 1, consider the marginal distribution γ n of (x n , a n ). For any E ∈ B(X), γ n (E × A) = q(E | x, a) γ n−1 (d(x, a) We now proceed to prove (4.3) and then (4.2). To compare the integrals in (4.4), consider an arbitrary ǫ > 0. There exists a sufficiently largē n such that for the corresponding compact sets K := Kn and F := An in Assumption 4.1(SU), the complement set (K × F ) c satisfies that
γ n (K × F ) c ≤ ǫ, ∀ n ≥ 0, andγ (K × F ) c ≤ ǫ. (4.5)
In the above, the existence of suchn and the first inequality of (4.5) follow from Assumption 4.1(SU) and the fact lim sup n→∞ c dγ n < ∞. The second inequality of (4.5) follows from the fact that We now compare the integrals of c m with those ofc m and bound their differences: Since δ and ǫ are both arbitrary, we obtain lim inf k→∞ c m dγ n k ≥ c m dγ, which is (4.4) and implies the desired inequality (4.3) as discussed earlier. Proof. Recall thatp n andp are the marginals ofγ n andγ, respectively, on X. For any v ∈ C b (X), since {γ n k } converges weakly toγ, it is clear that the r.h.s. of (4.2) satisfies v dp = lim k→∞ v dp n k .
X×A c m −c m dγ n k = (D∪B)×F c c m −c m dγ n k ≤ K\(D∪B) ×F c m −c m dγ n k + (K×F ) c c m −c m dγ n k ≤ m O\(D∪B) ×A dγ n k + m ǫ (4.7) ≤ m ν(B c ) + m ǫ (4.8) ≤ m (δ + ǫ),
The same proof given in [17, p. 119] (which is based on a martingale argument) establishes that
lim n→∞ X×A X v(y) q(dy | x, a)γ n (d(x, a)) − X v(x)p n (dx) = 0.
Therefore, to prove (4.2), it suffices to show that for any v ∈ C b (X), x, a)).
lim k→∞ X×A X v(y) q(dy | x, a)γ n k (d(x, a)) = X×A X v(y) q(dy | x, a)γ(d(X × A with φ ∞ ≤ φ ∞ ≤ v ∞ .
Since the functionφ is bounded and continuous and {γ n k } converges weakly toγ, we have lim k→∞ φ dγ n k = φ dγ. (4.12)
We now compare the integrals of φ with those ofφ and bound their differences, similarly to the derivation of (4.9)-(4.10):
X×A φ −φ dγ n k = (D∪B)×F c φ −φ dγ n k ≤ K\(D∪B) ×F φ −φ dγ n k + (K×F ) c φ −φ dγ n k ≤ 2 v ∞ · O\(D∪B) ×A dγ n k + 2 v ∞ · ǫ ≤ 2 v ∞ · ν(B c ) + 2 v ∞ · ǫ ≤ 2 v ∞ · (δ + ǫ),
where we used (4.5), Lemma 4.1, and the fact ν(X \ B) ≤ δ by the choice of B, to derive the last three inequalities, respectively. By the same arguments, the same conclusion holds for the integrals w.r.t.γ:
X×A φ −φ dγ ≤ 2 v ∞ · (δ + ǫ).
Combining these two relations with (4.12), we have lim sup k→∞ φ dγ n k − φ dγ ≤ 4 v ∞ · (δ + ǫ).
Since δ and ǫ are arbitrary, we obtain the desired inequality (4.11), which implies (4.2), as discussed earlier. Proof of Theorem 4.1. By Prop. 4.1, we can construct a sequence of pairs (μ n ,p n ), n ≥ 1, such that µ n is a stationary policy,p n an invariant probability measure of the Markov chain on X induced bȳ µ n , and lim n→∞ J(μ n ,p n ) = inf p∈P(X) inf π∈Π J(π, p) < ∞. Letγ n (d(x, a)) =μ n (da | x)p n (dx). By the invariance property ofp n , we have J(μ n ,p n ) = c dγ n , so c dγ n is bounded by some constant for all n ≥ 1. In view of Assumption 4.1(SU), this implies, as in the preceding proofs, that {γ n } is tight and hence there is a subsequence {γ n k } converging weakly to someγ ∈ P(X × A). The rest of the proof now parallels that of Prop. 4.1. Decomposeγ into the marginalp on X and a Borel measurable stochastic kernelμ(da | x) on A given X that obeys the control constraint. To prove the theorem, we need to show that (4.2) and (4.3) hold for {γ n } andγ in this case. For all n ≥ 1, we havep n (E) = q(E | x, a)γ n (d(x, a)) for all E ∈ B(X), by the invariance property ofp n . It follows from this relation and Assumption 4.1(M) that the conclusion of Lemma 4.1 holds forp n , and then the second half of the proof of that lemma shows that its conclusion also holds forp in this case. We then use Lemma 4.1 to prove that for any v ∈ C b (X), (4.2) holds. Clearly, the proof amounts to showing that (4.11) holds, and the arguments are the same as those given in the proof of Lemma 4.3. This establishes thatp is an invariant probability measure of the Markov chain on X under the stationary policyμ. Finally, the proof for (4.3) in this case is exactly the same as the proof of Lemma 4.2, and this establishes that c dγ = lim k→∞ c dγ n k = inf p∈P(X) inf π∈Π J(π, p). Hence (μ,p) is the desired minimum pair.
Proof of
An Illustrative Example
For simplicity, we consider a problem similar to a one-dimensional linear-quadratic (LQ) control problem but with a discretized action space and piecewise quadratic costs. The same reasoning can be applied to higher dimensional problems with nonlinear dynamics and additive noise. Let X = R, A ⊂ R, and c(x, a) = β(x) (x 2 + a 2 ), where β(·) is some piecewise constant function such that lim inf |x|→∞ β(x) > 0, sup x∈R β(x) < ∞. Let x n+1 = x n + a n + ζ n (x n , a n ), n ≥ 0, where ζ n (x n , a n ) is a random disturbance whose distribution, given the history {(x k , a k )} k≤n , is assumed to depend only on (x n , a n ) and is given by F x,a ∈ P(R) for (x n , a n ) = (x, a). Since our results only apply to discrete action spaces, as an illustrative example, consider a small number δ > 0 and let A = kδ | k = 0, ±1, ±2, . . . . The one-stage costs in this problem are strictly unbounded, and we can let the compact subsets of states and actions in Assumption 4.1(SU) be K n = [−n, n] and A n = 0, ±δ, ±2δ, . . . , ±nδ for n ≥ 1, for example.
Now similar to what we discussed in Section 3.1.1, suppose that w.r.t. the Lebesgue measure, the distributions F x,a have densities f x,a that are bounded uniformly from above by ℓ. Regarding Assumption 4.1(G), suppose that F x,a , (x, a) ∈ Γ, have zero means and variances bounded uniformly by σ 2 . Let A(x) = A for all x ∈ X, for simplicity. Then a policy π that satisfies Assumption 4.1(G) for the initial state x 0 = 0 is the one that chooses the action a n = arg min a∈A,|a|≤|xn| |x n + a|, since J(π, 0) ≤ 2(δ 2 + σ 2 ) · sup x∈R β(x) < ∞, as can be verified. In this simple example, we can see immediately a solution to the condition (G). For more complicated problems, one can use Markov chain theory in finding a stationary policy π with a finite average cost; see [ (a) Given p 0 ∈ P(X) and a policy π ∈ Π of (SM), we can define a policyπ of (DM) that corresponds to π at p 0 in the following sense. Let p k ∈ P(X) and γ k ∈ P(X × A), k ≥ 0, be the marginal distributions of the state x k and the state-action pairs (x k , a k ) in (SM), respectively, at time k, under the policy π and with the initial state distribution being p 0 . Then we can define a policyπ = (μ 0 ,μ 1 , . . .) for (DM) such that those marginal distributions γ k , p k+1 , k ≥ 0, are exactly the actions and states generated by the policyπ in (DM) according to Def. A.1(iii), when the initial state is p 0 . Furthermore, if π is Markov, there exists a policyπ of (DM) that corresponds to π at every p 0 ∈ P(X) in the above sense.
(b) Conversely, given p 0 ∈ P(X) and a policyπ of (DM), we can define a Markov policy π of (SM) such that the two policies correspond at p 0 in the sense explained above. 9
Let us now consider the average cost problem in (DM) and relate it to the average-cost MDP. In (DM), for an initial state p 0 ∈ P(X), we denote the n-stage cost and the average cost of a policȳ π byJ n (π, p 0 ) andJ(π, p 0 ), respectively. Since the problem is deterministic, the former is given bȳ J n (π, p 0 ) = 1 n n−1 k=0c (γ k ) (recall the convention −∞ + ∞ = ∞ − ∞ = ∞), where {γ k } is the action sequence generated byπ according to Def. A.1(iii). As in (SM), the optimal average cost at p 0 is defined asḡ
* (p 0 ) := inf π∈ΠJ (π, p 0 ) = inf π∈Π lim sup n→∞J n (π, p 0 )/n.
Consider the valueḡ * (δ x ) for the Dirac measure δ x at x ∈ X. Note first that by the correspondence between (DM) and (SM) described in (b) above, if (SM) is in the model class (UC), then in its corresponding (DM),J n (π, δ x ) is finite for all x ∈ X,π ∈Π and n ≥ 0, whereas if (SM) is in the model class (PC),J n (π, δ x ) is nonnegative and possibly infinite. So for both (PC) and (UC),
g * (δ x ) > −∞, ∀ x ∈ X. (A.2)
More importantly, by the relations described in (a)-(b) above, the average costs of the corresponding policies in (DM) and (SM) are equal as well:
(a ′ ) Given x ∈ X and a policy π of (SM), there exists a policyπ of (DM) such that J(π, x) = J(π, δ x ).
(b ′ ) Conversely, given x ∈ X and a policyπ of (DM), there exists a Markov policy π of (SM) such that J(π, x) =J(π, δ x ).
Thus we must have g * (x) =ḡ * (δ x ) for all x ∈ X.
A.2 Proof of Theorem 2.1
We prove Theorem 2.1 in this subsection. We first prove several average cost optimality properties for (DM), and we then transfer the results to (SM) via the correspondence relations between the two models given in the preceding subsection.
Deriving several desired optimality properties for (DM): For (DM), we can write the optimal average cost functionḡ * as the result of partial minimization of a lower semi-analytic function as 9 The policy π is constructed by decomposing the action sequence {γ k } generated from the initial state p 0 by the policyπ in (DM) according to Def. A.1(iii). Specifically, each γ k is decomposed into the marginal p k on X and a stochastic kernel µ k on A given X that obeys the original control constraint. Then π is defined to be the collection of the stochastic kernels (µ 0 , µ 1 , . . .). See [2, Prop. 9.2] and its proof for further details.
follows. Recall the set ∆, which consists of all admissible sequences (p 0 , γ 0 , γ 1 , . . .) (cf. [2,Def. 9.7] and Def. A.1(iii)). For each p 0 ∈ P(X), denote the vertical section of ∆ at p 0 by ∆ p0 , i.e., ∆ p0 := (γ 0 , γ 1 , . . .) | (p 0 , γ 0 , γ 1 , . . .) ∈ ∆ ; ∆ p0 is the set of all action sequences (γ 0 , γ 1 , . . .) that can be generated by some policy of (DM) for the initial state Moreover, by a measurable selection theorem [2,Prop. 7.50], for any ǫ > 0, we can select a measurable ǫ-minimizer for the optimization problem in (A.3). More precisely, there exists a universally measurable mapping ψ : P(X) → P(X × A) ∞ such that for all p 0 ∈ P(X), where E = p 0 ∈ P(X) ∃ (γ 0 , γ 1 , . . .) ∈ ∆ p0 withḡ * (p 0 ) = G p 0 , γ 0 , γ 1 , . . . .
ψ(p 0 ) ∈ ∆ p0 and G p 0 , ψ(p 0 ) ≤ ḡ * (p 0 ) + ǫ, ifḡ * (p 0 ) > −∞; −1/ǫ, ifḡ * (p 0 ) = −∞.
As noted earlier in (A.2),ḡ * (δ x ) > −∞ for all x ∈ X if the original problem (SM) is in the class (PC) or (UC). Therefore,
ψ(δ x ) ∈ ∆ δx and G δ x , ψ(δ x ) ≤ḡ * (δ x ) + ǫ, ∀ x ∈ X. (A.5)
By (A.4), in the case δ x ∈ E for all x ∈ X, ψ can be chosen to attain the minimal values at those Dirac measures:
ψ(δ x ) ∈ ∆ δx and G δ x , ψ(δ x ) =ḡ * (δ x ), ∀ x ∈ X. (A.6)
We now transfer these results to (SM).
Proof of the first part of the theorem: The argument is the same as that in [2, Chap. 9.3]. Sinceḡ * is lower semi-analytic and the mapping x → δ x is a homeomorphism [2, Cor. 7.21.1], the function g * (x) =ḡ * (δ x ) is lower semi-analytic by [2,Lemma 7.30(3)]. This proves Theorem 2.1(i).
Proof of the second part of the theorem: The second part of the theorem concerns the existence of universally measurable, ǫ-optimal or optimal semi-Markov policies in (SM), and to prove it, we start from (A.5) or (A.6) and construct a policy for (SM) with desired properties. We prove the existence of an ǫ-optimal policy first; the proof for the existence of an optimal policy is largely the same and will be given at the end.
For the average cost problem (A.3) in (DM), (A.5) gives an ǫ-optimal solution σ(x) = ψ(δ x ) for x ∈ X, which is universally measurable in x by the universal measurability of ψ(·) together with [2, Cor. 7.21.1 and Lemma 7.30 (3)]. Express this solution as
σ(x) = (γ 0 (x), γ 1 (x), . . .)
where each γ k : X → P(X × A) is a universally measurable stochastic kernel on X × A given X. We then apply [2,Prop. 7.27] to decompose each γ k , k ≥ 0, into two universally measurable stochastic kernels µ k (da k | x, x k ) and p k (dx k | x) that satisfy
γ k (x)(B) = X A 1 B (x k , a k ) µ k (da k | x, x k ) p k (dx k | x), ∀ B ∈ B(X × A), (A.7)
where 1 B (·) denotes the indicator function for the set B. For k = 0, by the control constraint in (DM) (cf. Def. A.1(i)), we have p 0 (dx 0 | x) = δ x and γ 0 (x)(Γ) = 1, so µ 0 (da 0 | x, x) must satisfy the control constraint of (SM):
µ 0 A(x) | x, x = 1, ∀ x ∈ X. (A.8)
For k ≥ 1, the stochastic kernels µ k need not satisfy the control constraint of (SM). However, for each x, since γ k (x)(Γ) = 1 by the control constraint in (DM) (cf. Def. A.1(i)), we have, by (A.7),
X µ k (A(x k ) | x, x k ) p k (dx k | x) = 1,
which implies that for each x,
µ k (A(x k ) | x, x k ) = 1, p k (dx k | x)-almost surely. (A.9)
We now alter each µ k at those points (x, x k ) where the control constraint in (SM) is violated, in order to make the collection of these stochastic kernels a valid policy for (SM). To this end, for each k ≥ 1, define a set
D k := (x, x k ) | µ k (A(x k ) | x, x k ) < 1 . (A.10) Since µ k A(x k ) | x, x k = A 1 Γ (x k , a k ) µ k (da k | x, x k )
and the indicator function 1 Γ (x k , a k ) is universally measurable (because Γ is analytic), by [2,Prop. 7.46], the function µ k (A(x k ) | x, x k ) is universally measurable in (x, x k ). Consequently, the set D k is universally measurable. Then for each x, as the vertical section of D k , D k,x := {x k | (x, x k ) ∈ D k } is also universally measurable [2,Lemma 7.29], and by (A.9),
p k D k,x | x = 0, ∀x ∈ X. (A.11)
Recall that since Γ is analytic, by the Jankov-von Neumann selection theorem [2,Prop. 7.49], there exists a universally (in fact, analytically) measurable stationary policy µ o . We now define stochastic kernelsμ k , k ≥ 1, by altering each µ k as follows:
µ k (da k | x, x k ) = µ k (da k | x, x k ), if (x, x k ) ∈ D k ; µ o (da k | x k ), if (x, x k ) ∈ D k . (A.12)
By definition eachμ k above is a universally measurable stochastic kernel that obeys the control constraint in (SM). For k = 0, letμ By [2,Prop. 7.44],μ 0 is universally measurable, and by (A.8), it satisfies the control constraint in (SM). Then, identifying x with x 0 , we have that
π = μ 0 (da 0 | x),μ 1 (da 1 | x, x 1 ),μ 2 (da 2 | x, x 2 ), . . .
is a universally measurable, semi-Markov policy for (SM). What remains to prove is that π is an ǫ-optimal policy. This follows from (A.8)-(A.9), (A.11)-(A.12) and the correspondence relations between π and a policy in (DM) that generates the admissible sequence δ x , σ(x) = δ x , γ 0 (x), γ 1 (x), . . . for each x ∈ X. Specifically, in (SM), under π, if the initial state x 0 = x, the marginal distributionγ k (x) of (x k , a k ) for k ≥ 1 is given bỹ
γ k (x)(B) := X A 1 B (x k , a k )μ k (da k | x, x k )p k (dx k | x), ∀ B ∈ B(X × A), wherep k (dx k | x)
is the marginal distribution of x k , and for k = 0, the marginal distributions of x 0 and (x 0 , a 0 ) satisfyp
0 (dx 0 | x) = δ x = p 0 (dx 0 | x),γ 0 (x) = γ 0 (x)
, in view of the definitions ofμ 0 and µ 0 . We now use induction on k and the relation (A.9) to verify thatp k (dx k | x) = p k (dx k | x) andγ k (x) = γ k (x) for all k ≥ 0. Suppose that for some k ≥ 1, we have shown that they hold for 0, 1, . . . k−1; let us prove that they also hold for k. Sinceγ k−1 (x) = γ k−1 (x) by assumption, the marginal distributions of x k are also equal:p k (dx k | x) = p k (dx k | x), in view of the definition of the transition function in (DM) (cf. Def. A.1(i)). Together with (A.11) and the definition (A.12) ofμ k , this implies that for any B ∈ B(X × A),
X A 1 B (x k , a k )μ k (da k | x, x k )p k (dx k | x) = X A 1 B (x k , a k )μ k (da k | x, x k ) p k (dx k | x) = X\D k,x A 1 B (x k , a k )μ k (da k | x, x k ) p k (dx k | x) = X\D k,x A 1 B (x k , a k ) µ k (da k | x, x k ) p k (dx k | x) = X A 1 B (x k , a k ) µ k (da k | x, x k ) p k (dx k | x).
This showsγ k (x) = γ k (x). So by the induction argument, we haveγ k (x) = γ k (x) for all k ≥ 0.
The result just proved implies that for each x ∈ X, the average cost of the policy π equals G δ x , ψ(δ x ) ≤ḡ * (δ x ) + ǫ (cf. (A.5)). As proved earlier, g * (x) =ḡ * (δ x ) for x ∈ X, so π is ǫ-optimal. This proves the existence of a universally measurable, ǫ-optimal, randomized semi-Markov policy.
Finally, consider the last statement of the theorem. By assumption, for each x ∈ X, there exists a policy in (SM) attaining the optimal average cost g * (x). By the correspondence relations between (SM) and (DM) discussed at the end of Section A.1, this means that for each x ∈ X, there exists a policy in (DM) attaining the optimal average costḡ * (δ x ). Thus we can choose a solution ψ(·) of the average cost problem (A.3) in (DM) to satisfy (A.6). Then, letting σ(x) = ψ(δ x ) for x ∈ X in the preceding proof, we obtain, with exactly the same arguments, that the semi-Markov policy π constructed above for (SM) has average costs J(π, x) = G δ x , ψ(δ x ) =ḡ * (δ x ) = g * (x) for all x ∈ X. So π is a universally measurable, optimal, randomized semi-Markov policy. This completes the proof of Theorem 2.1(ii).
A.3 Proof of Theorem 2.2 for (UC)
Note first that the two conditions (a)-(b) in Definition 2.1 of the model (UC) ensure that for all policies π, starting from x ∈ X, the expected α-discounted total cost v π α (x) is bounded in absolute value byĉ
· ∞ n=0 α n E π x w(x n ) ≤ĉ w(x) 1 − αλ +ĉ b (1 − α)(1 − λ) .
Therefore, the optimal value function v α is not only finite everywhere but also satisfies v α w < ∞. We now consider the corresponding deterministic control model (DM) described in Section A.1. For a policyπ ∈Π and initial state p 0 ∈ P(X), define its α-discounted total cost to bē Similarly to the average cost case, the correspondence between (DM) and (SM) implies the following:
• v α (x) =v α (δ x ) for all x ∈ X.
•v α can be expressed as the result of a partial minimization problem: Then, with arguments almost identical to those given in the preceding subsection for proving Theorem 2.1, we can draw the following conclusions similar to those derived in the average cost case:
(i) The functionv α is lower semi-analytic. This implies (as in the proof of Theorem 2.1(i)) that the optimal value function of (SM), v α (x) =v α (δ x ), is a (finite-valued) lower semi-analytic function.
(ii) Similarly to the proof of Theorem 2.1(ii), for each ǫ > 0, we can construct, from an ǫ-optimal solution to (A.13), a universally measurable, ǫ-optimal, semi-Markov policy π ǫ for the original problem (SM). The value function v πǫ α of this policy then satisfies v πǫ α ≤ v α + ǫ. (A.15) Note that v πǫ α is universally measurable.
We now focus on the original problem (SM) and use the above results together with the contraction property of the dynamic programming operator T α to prove Theorem 2.2 for (UC). By Lemma A.2 (see Appendix A.4) and the Banach fixed point theorem, T α has a unique fixed point in A(X) ∩ Mw(X) for some weight functionw ≥ w. Since v α ∈ A(X) ∩ M w (X) by the preceding proof and M w (X) ⊂ Mw(X), we have v α ∈ A(X) ∩ Mw(X). So to prove the theorem, we only need to show v α = T α v α . To this end, we prove first v α ≤ T α v α and then v α ≥ T α v α .
Since v α ∈ A(X) ∩ M w (X), T α v α ∈ A(X) ∩ M w (X) by Lemma 2.1 and the model condition of (UC). So by a measurable selection theorem [2,Prop. 7.50], there exists a universally measurable nonrandomized stationary policy µ such that T µ α v α ≤ T α v α + ǫ, (A. 16) where T µ α : M(X) → M(X) and it is given by (T µ α v)(x) = c(x, µ(x)) + α X v(y) q(dy | x, a) for v ∈ M(X). Consider the policy π that applies µ at the first stage and then applies π ǫ afterwards. By the monotonicity of T µ α and the inequalities (A.15) and (A. 16),
v π α = T µ α v πǫ α ≤ T µ α v α + αǫ ≤ T α v α + 2ǫ.
Since v π α ≥ v α and ǫ is arbitrary, we obtain v α ≤ T α v α . For the reverse inequality, consider again the ǫ-optimal semi-Markov policy π ǫ and express it as µ 0 (da 0 | x 0 ), µ 1 (da 1 | x 0 , x 1 ), µ 2 (da 2 | x 0 , x 2 ), . . . . For x ∈ X, since E πǫ x ∞ n=1 α n c(x n , a n ) = E πǫ x E πǫ x ∞ n=1 α n c(x n , a n ) x 0 ,
x 1 ≥ E πǫ x [α v α (x 1 )] ,v α + ǫ ≥ v πǫ α ≥ T α v α , =⇒ v α ≥ T α v α .
Hence v α = T α v α . Finally, given ǫ ′ > 0, the nonrandomized stationary policy µ in (A.16) with ǫ = (1 − α)ǫ ′ is an ǫ ′ -optimal policy. This follows from combining (A.16) with the monotonicity of T µ α and with the observation that T µ α is a contraction on Mw(X) by the same proof of Lemma A.2 and has the value function v µ α as its unique fixed point in Mw(X). (We omit the details of the arguments since they are standard and straightforward.) This completes the proof of Theorem 2.2 for (UC).
Remark A.1 (about the proof). We mentioned in the discussion after Theorem 2.2 that our proof of its (UC) part is similar to, but does not follow exactly the one given in [2,Chap. 9] for the case of a bounded one-stage cost function. Let us explain this more here.
First, there are special cases of the weight function w(·) for which a well-known technique can be applied to convert the problem to one with bounded one-stage costs, and then Theorem 2.2(UC) follows immediately from the result of [2,Chap. 9]. In particular, suppose that the weight function w(·) is Borel measurable, instead of universally measurable, so that for each α < 1, the weight functionw(·) constructed in the proof of Lemma A.2 is also Borel measurable. Then using the contraction property of T α w.r.t. the · w norm (Lemma A.2), one can apply Veinott's similarity transformation ( [43]; see also [40,Chap. 5.2,) to convert the α-discounted problem into a β-discounted problem with bounded costs:
c(x, a) → c(x, a)/w(x), α q(dy | x, a) → α w(y)/w(x) q(dy | x, a). When w(·) is universally measurable, the conversion just mentioned no longer works. This is because after the similarity transformation (A.18), the resulting problem has c(x, a)/w(x) as the one-stage cost function and α w(y)/w(x) q(dy | x, a) as the state transition (sub)-stochastic kernel. The former function is universally measurable and not necessarily lower semi-analytic, and the latter stochastic kernel is universally measurable instead of Borel measurable. So the resulting MDP no longer satisfies the model assumptions in the universal measurability framework for MDPs.
Note also that even when the conversion can be done for each α-discounted problem, the model condition of (UC) does not imply that the average cost problem can also be converted to one with bounded one-stage costs.
The proof we gave in this appendix differs from the counterpart in [2,Chap. 9] in the following way. Like in the average cost case, we used (DM) to establish v α ∈ A(X) and the existence of a universally measurable, ǫ-optimal semi-Markov policy in the original MDP. We then combined these facts with the contraction property of T α to derive the theorem. In [2,Chap. 9], the α-DCOE for the case of bounded one-stage costs is derived first for (DM) and then transferred to the original problem. This route seems inconvenient for (UC) because with unbounded one-stage costs, the optimal value function in (DM) can take ±∞ values for some initial distributions p 0 ∈ P(X), although its value at any Dirac measure on X is finite due to the (UC) model conditions. Thus the dynamic programming operator in (DM) works on the space of extended real-valued, lower semi-analytic functions on P(X) and is not a contraction on that space. It is more convenient to work directly with the operator T α and the original problem, after obtaining the needed optimality properties from (DM).
A.4 Three Technical Lemmas
The first lemma is about the lower semi-analyticity of v α (x) as a function of (α, x). We use its (PC) part to prove Lemma 3.3 in Section 3.2 for establishing the ACOI for the (PC) model. Proof. We use the correspondence between the deterministic control model (DM) and the original problem (SM) to prove this lemma. Recall that in the preceding Section A.3, in the course of proving Theorem 2.2 for (UC), we have shown the following results regarding the optimal value functionv α in (DM), which hold for (PC) as well by essentially the same arguments: For α ∈ (0, 1) and x ∈ X, v α (δ x ) = v α (x); andv α can be expressed as the result of partial minimization: (A.19) Since α ≥ 0 andc(·) is lower semi-analytic (cf. Def. A.1(i)), the product α kc (γ k ) is a lower semianalytic function of (α, z) on (0, 1) × ∆ by [2,Lemma 7.30(4)]. Therefore, G α (z) is a lower semianalytic function of (α, z) on (0, 1) × ∆ by [2,Lemma 7.30 (2) and (4)]. Then, as the result of the partial minimization (A. 19),v α (p 0 ) is a lower semi-analytic function of (α, p 0 ) by [2,Prop. 7.47]. Since f (α, x) = v α (x) =v α (δ x ) and the mapping (α, x) → (α, δ x ) is a homeomorphism [2, Cor. 7.21.1], this implies, by [2,Lemma 7.30(3)], that f is a lower semi-analytic function of (α, x).
The next lemma is about the contraction property of the dynamic programming operator T α stated in (2.5) for the (UC) model. We use it in the proof of Theorem 2.2 for (UC).
Lemma A.2. (UC) For some universally measurable wight functionw ≥ w, the operator T α is a contraction on the closed subset A(X) ∩ Mw(X) of the Banach space (Mw(X), · w ).
Proof.
We have already shown that T α maps A(X) into A(X) (Lemma 2.1). To prove the lemma, we need to show that for some β ∈ (0, 1) andw ≥ w, T α v − T α v ′ w ≤ β v − v ′ w for all v, v ′ ∈ A(X) ∩ Mw(X). The proof is essentially the same as the analysis given in [18, p. 45-46] (cf. the proof of Prop. 8.3.4 and Remark 8.3.5 therein). To constructw with the desired property, we proceed as follows. Define universally measurable functions c n , n ≥ 0, by c 0 = w, c n = λ n w + (1 + λ + · · · + λ n−1 ) b, n ≥ 1.
Theorem 2.1 (some average-cost optimality properties). (PC)(UC) (i) The optimal average cost function g * is lower semi-analytic.
1). Corresponding to the sequence {α n }, consider the sequence of functions h n := h αn . Define as n → ∞, h n ↑ h andh n ↓h. The next lemma about these functions follows directly from Theorem 2.2, [2, Lemma 7.30(2)], and Assumption 3.1.Lemma 3.1. (UC) Under Assumption 3.1, all the functions h,h, h n ,h n , n ≥ 0, are lower semianalytic and lie in a bounded subset of M w (X).
Lemma 3. 2 .
2(UC) Under the assumptions of Theorem 3.1, let K be the compact set in Assumption 3.2 for a given x ∈ X and ǫ > 0. Then
17) and since inf a (cl f m )(a) = inf a f m (a), it follows thatlim n inf a f n (a) ≤ inf a (e-lim n f n )(a) ≤ inf a c(x, a) + X h(y) q(dy | x, a) . (3.18)For the ACOI to hold, we need equality to hold throughout in(3.18), so if the first inequality in (3.18) is strict, it becomes impossible to obtain the desired ACOI. Thus we need the equality lim n inf a f n (a) = inf a (e-lim n f n )(a).(3.19)
X = [0, ∞), A = [0, L] for some L > 0, and with ℓ + := max{ℓ, 0}, let the states evolve according to
Define Z(x, a) := η 0 (x, a) − ξ 0 (x, a). Similarly to [18, Examples 8.6.4 and 10.9.3], suppose that the collection of probability distributions F x,a and G x,a are such that for all state and admissible action pairs (x, a) ∈ Γ, the random variables Z(x, a) have negative means and furthermore, for some κ > 0, λ := sup (x,a)∈Γ E e κZ(x,a) < 1. (3.21) This constraint relates to system stability and can be met, for instance, by choosing the action sets A(x) at each state accordingly. Then, similarly to the derivations in [18, Examples 8.6.4, p. 71-73], it can be shown that the weight function w(x) := e κx satisfies the (UC) model condition (b) (cf. Def. 2.1) for the constants b = 1 and λ < 1 given in (3.21): sup a∈A(x)
( 3 . 24 )
324Lemma 3.3. (PC) Under Assumption 3.3, the functions h and h α , α ∈ (0, 1), are finite-valued and lower semi-analytic.
Example 3 . 2 .
32Suppose that for each x ∈ X, there exists a sequence of compact sets A n ⊂ A such that inf a ∈An c(x, a) → +∞ as n → ∞. (For instance, A = R d and for each state x, c(x, ·) is coercive: c(x, a) → +∞ as a → +∞.) Let Assumption 3.3(G) hold and in addition, suppose that a stronger condition than Assumption 3.3(B) holds:
by the α-DCOE (Theorem 2.2). The r.h.s. is bounded over α, in view of (3.25) and an implication of the condition (G), sup α∈(0,1) (1 − α)m α < ∞ [32, Lemma 1.2(b)]. So, for any ǫ > 0, there exists n sufficiently large such that for any a ∈ A n , c(x, a) > (1 − α)m α + h α (x) + ǫ for all α ∈ (0, 1). Consequently, Assumption 3.4(i) is satisfied by letting K = A n .
32 ,
32Lemma 1.2(b)]. Define h n := h αn and h n := h αn . Then h n ≤ h n and h n ↑ h as n → ∞.The rest of the proof is similar to the first part of the proof for Theorem 3.1. Let ǫ > 0. For the fixed state x, by subtracting αm α from both sides of the α-DCOE (Theorem 2.2), we have that for all n ≥ 0,
Assumption 4 . 1 .
41For the model (PC), the action space A is a countable space with the discrete topology, the one-stage cost function c and the graph Γ of the control constraint are Borel measurable, and furthermore, the following hold:(G) For some policy π and state x ∈ X, the average cost J(π, x) < ∞.(SU) There exist two increasing sequences of compact sets {K n } and {A n } with K n ⊂ X and A n ⊂ A, such that lim n→∞ inf (x,a) ∈Kn×An c(x, a) = +∞. (M) For each set K ∈ {K n }, there exist an open set O ⊃ K and a finite measure ν on B(X) such that
) If (i) is established, then J(μ,p) = c dγ by the invariance property ofp, so to prove the desired relation J(π, p) ≥ J(μ,p), we also need to show that lim sup n→∞ c dγ n ≥ c dγ.
Lemma 4. 1 .
1Let the open set O, the closed set D, and the finite measure ν on B(X) be as in Assumption 4.1(M) for some K ∈ {K n }. Then for all B ∈ B(X),
), so by Assumption 4.1(M), γ n ({(O\D)∩B}×A) ≤ ν(B) for any B ∈ B(X).Sincep n (·) =γ n (· × A) k (· × A), the desired inequality forp n follows.We now prove the first inequality forp.As the set O \ D is open, for any open set B ⊂ X, the set (O \ D) ∩ B is also open. Since {γ n k } converges weakly toγ, by [9, Theorem 11.1.1], for any open set E,γ(E) ≤ lim inf k→∞γn k (E). Then, letting E = {(O \ D) ∩ B} × A for any open set B, we havep((O \ D) ∩ B) ≤ lim inf k→∞pn k ((O \ D) ∩ B) ≤ ν(B). This inequality must also hold for any B ∈ B(X). To see this, first, definep ′ (·) =p((O \ D) ∩ ·), to simplify notation. By [9, Theorem 7.1.3], on a metric space, finite Borel measures are closed regular, which means, in our case, that for any Borel set B,p ′ (B) = sup{p ′ (F ) | F ⊂ B, F closed} and the same is true for ν(B). This in turn implies thatp ′ (B) = inf{p ′ (F ) | F ⊃ B, F open} and the same for ν(B). Now given B ∈ B(X), for any open set F ⊃ B, we havep ′ (F ) ≤ ν(F ) as proved earlier, and thereforep ′ (B) ≤ ν(B).
Lemma 4 . 2 .
42The inequality (4.3) holds. Proof. For m ≥ 0, define c m : X×A → R by c m (x, a) = min{c(x, a), m}. Since lim inf k→∞ c dγ n k ≥ lim inf k→∞ c m dγ n k and c m dγ ↑ c dγ as m → ∞ by the monotone convergence theorem, to prove (4.3), it suffices to prove lim inf k→∞ c m dγ n k ≥ c m dγ.
(K × F ) c is an open set and hence, as the weak limit of {γ n k },γ satisfies thatγ((K × F ) c ) ≤ lim inf k→∞γn k (K×F ) c by [9, Theorem 11.1.1]. We use (4.5) to bound the integrals (K×F ) c c m dγ n k and (K×F ) c c m dγ by m ǫ. We now compare the integrals of c m on K × F . By Assumption 4.1(M), c is lower semicontinuous on the closed set D × A; therefore, so is c m on D × A. Since the set F is finite, applying Lusin's Theorem [9, Theorem 7.5.2], for any δ > 0, we can choose a closed set B ⊂ X with ν(X \ B) ≤ δ such that c m is continuous on B × F . 7 Then c m is lower semicontinuous on the closed set (D ∪ B) × F . By the Tietze-Urysohn extension theorem [9, Theorem 2.6.4] and an approximation property for lower semicontinuous functions [2, Lemma 7.14], the restriction of c m to (D ∪ B) × F can be extended to a nonnegative lower semicontinuous functionc m on X × A with the extension also bounded above by m. 8 For the functionc m , since {γ n k } converges weakly toγ, by [17, Prop. E.2], lim inf k→∞ c m dγ n k ≥ c m dγ.
set O in (4.7) is the set O ⊃ K appearing in Assumption 4.1(M), and we used (4.5) to derive this inequality, and we used Lemma 4.1 and the fact ν(X \ B) ≤ δ to derive (4.8) and (4.9), respectively. By the same arguments, X×A c m −c m dγ ≤ m (δ + ǫ).
dγ n k ≥ c m dγ − 2m (δ + ǫ).
Lemma 4. 3 .
3The equality (4.2) holds.
> 0 and let the sets K ⊂ O, F and D, and the finite measure ν be as in the proof of Lemma 4.2. Recall that by Assumption 4.1(M), q(dy | x, a) is continuous on D × A. Since the space P(X) is separable and metrizable [2, Prop. 7.20] and the set F is finite, applying Lusin's Theorem [9, Theorem 7.5.2] (cf. Footnote 7), for any δ > 0, we can choose a closed set B ⊂ X with ν(X \ B) ≤ δ such that q(dy | x, a) is continuous on B × F . Then q(dy | x, a) is continuous on the closed set (D ∪ B) × F , so by [2, Prop. 7.30], φ(x, a) := X v(y) q(dy | x, a) is a bounded continuous function on the closed set (D ∪ B) × F , and by the Tietze-Urysohn extension theorem [9, Theorem 2.6.4], this restriction of φ to (D ∪ B) × F can be extended to a continuous functionφ on
Prop. 4 . 1 .
41The proposition follows from Lemmas 4.2-4.3 and the discussion given immediately before Lemma 4.1.
For a closed interval K = [−n, n], consider the open interval O = (−n − 1, n + 1) ⊃ K. A finite measure ν satisfying Assumption 4.1(M) is simply given by ℓ times the Lebesgue measure on the open interval O. (We let D = ∅ in Assumption 4.1(M) in this case.)
24, Sec. IV.A]. Thus, Theorem 4.1 holds in this example. Like in the previous Section 3, there is no need here for the continuity of q(dy | x, a) or c(x, a) in x. Such a sequence (p 0 , γ 0 , γ 1 , . . .) is called an admissible sequence. The set of all admissible sequences that can be generated by the policies in (DM) is denoted by ∆ (which is an analytic set [2, Lemma 9.1]). The policies of the original problem (SM) and (DM) are related (cf. [2, Def. 9.9 and Prop. 9.2]):
p 0 .
0Define a function G : ∆ → [−∞, +∞] by G p 0 , γ 0 , γ 1 , . . . can be written equivalently as the result of partial minimization of G: g * (p 0 ) = inf (γ0,γ1,...)∈∆p 0 G p 0 , γ 0 , γ 1 , . . . . (A.3) A crucial result is that the set ∆ is an analytic subset of the space P(X) × P(X × A) ∞ (endowed with the product topology) [2, Lemma 9.1]. Then, since the sum of a finite number of lower semianalytic functions are lower semi-analytic [2, Lemma 7.30(4)] and lim sup n→∞ f n is lower semianalytic for a sequence {f n } of lower semi-analytic functions [2, Lemma 7.30(2)], the function G is lower semi-analytic. Consequently, being the result of partial minimization of G (cf. (A.3)), the functionḡ * is lower semi-analytic by [2, Prop. 7.47].
Furthermore, by [ 2 ,
2Prop. 7.50(b)], ψ can be chosen to attain the infimum in (A.3) for all p 0 at which this is possible; i.e., G p 0 , ψ(p 0 ) =ḡ * (p 0 ), ∀ p 0 ∈ E, (A.4)
kc (γ k )where {γ k } is the action sequence generated byπ according to Def. A.1(iii). Define the optimal α-discounted value function in (DM) as v α (p 0 ) := inf π∈Πvπ α (p 0 ), p 0 ∈ P(X).
,γ1,...)∈∆p 0 G α p 0 , γ 0 , γ 1 , . . . (A.13) for the function G α p 0 , γ 0 , γ 1 , . . . := lim sup n→∞ n k=0 α kc (γ k ). (A.14)
(y) q(dy | x, a) = (T α v α )(x). (A.17)Combining (A.15) and (A.17) and taking ǫ to be arbitrarily small, we have
Lemma A.1. (PC)(UC) On (0, 1) × X, the function f (α, x) = v α (x) is lower semi-analytic.
,γ1,...)∈∆p 0 G α p 0 , γ 0 , γ 1 , . . . , where G α p 0 , γ 0 , γ 1 , . . . = lim sup n→∞ n k=0 α kc (γ k ).
Furthermore, by[2, Prop. 7.45], w.r.t. this probability measure, the expectation Ef for any nonnegative, universally measurable function f :[2, Prop.
7.45]. 3
Here ǫ is a constant. The result[2, Prop. 7.50], however, extends to a more general case where the required degree of optimality is different for each x. Specifically, given a pair of strictly positive, real-valued functions ǫ(·) and ℓ(·), both assumed to be analytically or universally measurable, there exists a universally measurable function φ(·) that satisfies (2.2)-(2.3) with ǫ(x) and ℓ(x) in place of ǫ and 1/ǫ, respectively, in (2.3). Such a function φ(·) can be constructed as follows: For n ≥ 1, let φn denote the function that satisfies (2.2)-(2.3) with ǫ = 1/n. LetEn = x ∈ proj X (D) | ǫ(x) > 1/n, f * (x) > −∞, arg min y∈Dx f (x, y) = ∅ and let Fn = x ∈ proj X (D) | ℓ(x) < n, f * (x) = −∞, arg min y∈Dx f (x, y) = ∅ . Clearly, ∪ n≥1 (En ∪ Fn) = x ∈ proj X (D) | arg min y∈Dx f (x, y) = ∅ ,which is a universally measurable set[2, Prop. 7.50(b)]. The sets En, Fn are also universally measurable. Then define φ(x) = φn(x) on the set (En∪Fn)\∪ k<n (E k ∪F k ) for n ≥ 1; and on the set x ∈ proj X (D) | arg min y∈Dx f (x, y) = ∅ , let φ(x) = φ 1 (x). This function φ(·) is universally measurable and satisfies (2.2)-(2.3) with ǫ(x) and ℓ(x) in place of ǫ and 1/ǫ, respectively, as required.
In(2.4), the probability of the set A(x k ) is measured w.r.t. the completion of the Borel probability measure µ k (da k | x 0 , a 0 , . . . , a k−1 , x k ). This is valid because for each x ∈ X, the vertical section A(x) of the analytic set Γ is universally measurable by[2, Lemma 7.29].3 Because the universal σ-algebra on (X × A) ∞ is not a product σ-algebra, the existence of a unique probability measure r(π, p 0 ) here does not follow immediately from the Ionescu Tulcea theorem.
This is because convergence in the · ∞ or · w norm implies pointwise convergence, and the pointwise limit of a sequence of lower semi-analytic functions is lower semi-analytic [2, Lemma 7.30(2)].5 Details: Since the state transition stochastic kernel q(dy | x, a) is Borel measurable, by[2, Prop. 7.48], the integral v(y) q(dy | x, a) for a function v ∈ A(X) is a lower semi-analytic function in (x, a). Since α ≥ 0, by[2, Lemma 7.30(4)], the integral multiplied by α remains to be lower semi-analytic in (x, a). Then, as the one-stage cost function c(·) is lower semi-analytic, the sum c(x, a) + α v(y) q(dy | x, a) is lower semi-analytic in (x, a) by[2, Lemma 7.30(4)]. This shows that Tαv and T v are the result of partial minimization of a lower semi-analytic function on the analytic set Γ (the graph of the control constraint). So by[2, Prop. 7.47], Tαv and T v are lower semi-analytic.
For E, En ⊂ R d , n ≥ 0, we say {En} converges to E if the following two conditions are met: (i) For every subsequence {n k } and convergent sequence {yn k } with yn k ∈ En k and yn k →ȳ as k → ∞, the limitȳ ∈ E. (ii) For everyȳ ∈ E, there exist yn ∈ En for all n sufficiently large, such that yn →ȳ as n → ∞. (See[29, Definition 4.1].)
Details: We apply Lusin's Theorem for each a ∈ F to obtain a closed set Ba ⊂ X such that ν(X \ Ba) ≤ δ/|F | and c m (·, a) is continuous on Ba. We then take B = ∩ a∈F Ba.8 Details: Denote the restriction of c m to (D ∪ B) × F by f . Since it is lower semicontinuous and bounded above by m, by [2, Lemma 7.14], there exists a sequence of nonnegative continuous functions {fn} on (D ∪ B) × F with fn ↑ f . We apply the Tietze-Urysohn extension theorem to extend each fn to a nonnegative continuous functionfn on X × A that is also bounded above by m. We then letc m = sup nf n.
(da 0 | x) = µ 0 (da 0 | x, x).
AcknowledgmentThe author would like to thank Professor Eugene Feinberg, who gave helpful comments on a preliminary draft of this manuscript and pointed her to several related prior results on Borel-space MDPs, and Dr. Martha Steenstrup, whose suggestions helped her improve the presentation. This research was supported by a grant from Alberta Innovates-Technology Futures.Appendix A Some Basic Optimality Properties for the Average and Discounted Cost CriteriaIn this appendix, we first define a deterministic control model (DM) that corresponds to a given Borel-space MDP, which will be referred to below as (SM) for short (this is the term used in[2], standing for the stochastic control model). This background material, given in Section A.1, is based on the book [2, Chap. 9.2], and it is an important part of the theory for Borel-space MDPs with universally measurable policies, as we explained in Section 2.2. The text of Section A.1 is similar to an online appendix the author wrote earlier for a paper on total cost Borel-space MDPs[44]. The rest of this appendix (Sections A.2-A.4) contains the proof details for the results given in Section 2.2, as well as a technical lemma used in proving Lemma 3.3 in Section 3.2. These proofs make use of the correspondence relations between (DM) and (SM) explained in Section A.1, inA(p) := γ ∈ P(X × A) γ(B × A) = p(B), γ(Γ) = 1, ∀ B ∈ B(X) .I.e., the actions at state p are those Borel probability measures on X × A that have p as the marginal on X and assign probability one to the graph Γ of the original control constraint A(·). (Like A(·), the graph ofĀ(·) is an analytic set in the state-action space P(X) × P(X × A); see the proof of [2, Lemma 9.1].) • System functionf : P(X × A) → P(X), which maps each action γ ∈ P(X × A) to a statē f (γ) ∈ P(X), defined according to the state transition stochastic kernel of the original problem asf (γ)(B) :This functionf specifies how the states evolve in (DM). (ii) A policy for (DM) is a sequence of mappingsπ = (μ 0 ,μ 1 , . . .) such that for each k ≥ 0, µ k : P(X) → P(X × A) andμ k (p) ∈Ā(p) for every p ∈ P(X). In other words,μ k maps each state p to an admissible action at p. (Note that every policy here is Markov and there are no measurability conditions on them in this deterministic control problem.) The set of all policies for (DM) is denoted byΠ.(iii) Given an initial state p 0 ∈ P(X), applying a policyπ in (DM) generates recursively a sequence of state and action pairs (p 0 , γ 0 ), (p 1 , γ 1 ), . . ., according to the system functionf asNow chooseα ∈ (α, 1) and definew = ∞ n=0α n c n . Thenw ∈ M(X) and it is finite-valued becausẽFurthermore, for any x ∈ X and a ∈ A(x), by (A.20),Recall also that sup a∈A(x) |c(x, a)| ≤ĉ w(x) ≤ĉw(x) for all x ∈ X. This and the relation (A.21)together imply that T α maps A(X) ∩ Mw(X) into A(X) ∩ Mw(X) and for β = α/α < 1, T α has the desired contraction property:Finally, we prove Lemma 2.2. Recall that it is about the existence of optimal and nearly optimal, nonrandomized Markov or stationary policies in (PC) and (UC), as a consequence of the ACOI (2.6):x ∈ X where it is assumed that g * is constant and h ∈ A(X), both being finite and with h ≥ 0 for (PC), h w < ∞ for (UC). The proof arguments are mostly standard.Proof of Lemma 2.2.For all x ∈ X, the r.h.s. of the ACOI (2.6) is greater than −∞ under the assumption on h and the model conditions for (PC) and (UC). Then, since h is lower semi-analytic and Γ is analytic, by a measurable selection theorem[2,Prop. 7.50], for each ǫ > 0, there exists a universally measurable function f : X → A such that for all x ∈ X, f (x) ∈ A(x) andFor k ≥ 0, let f k be the function satisfying the above for ǫ = ǫ k = 2 −k . Then by the ACOI,Consider the nonrandomized Markov policy π = (f 0 , f 1 , . . .), and let us show that it is averagecost optimal. First, note that for (PC), by iterating the inequality (A.23) and using the fact c, h ≥ 0, we have 0 ≤ E π x h(x n ) < ∞,x ∈ X, n ≥ 0. (A.24)For (UC), the assumption h w < ∞ and the model condition of (UC) ensure that E π x |h(x n )| ≤ h w · E π x w(x n ) ≤ h w · λ n w(x) + b/(1 − λ) , x ∈ X, n ≥ 0. (A.25)
. A Arapostathis, V S Borkar, E Fernández-Gaucherand, M K Ghosh, S I Marcus, Arapostathis, A., Borkar, V. S., Fernández-Gaucherand, E., Ghosh, M. K., and Marcus, S. I. (1993).
Discrete-time controlled Markov processes with average cost criterion: A survey. SIAM J. Control Optim. 312Discrete-time controlled Markov processes with average cost criterion: A survey. SIAM J. Control Optim., 31(2):282-344.
Stochastic Optimal Control: The Discrete Time Case. D P Bertsekas, S E Shreve, Academic PressNew YorkBertsekas, D. P. and Shreve, S. E. (1978). Stochastic Optimal Control: The Discrete Time Case. Aca- demic Press, New York.
Convergence of Probability Measures. P Billingsley, John Wiley & SonsNew YorkBillingsley, P. (1968). Convergence of Probability Measures. John Wiley & Sons, New York.
A Borel set not containing a graph. D Blackwell, Ann. Math. Statist. 39Blackwell, D. (1968). A Borel set not containing a graph. Ann. Math. Statist., 39:1345-1347.
The optimal reward operator in dynamic programming. D Blackwell, D Freedman, M Orkin, Ann. Probability. 25Blackwell, D., Freedman, D., and Orkin, M. (1974). The optimal reward operator in dynamic program- ming. Ann. Probability, 2(5):926-941.
Controlled Markov chains and stochastic networks. V S Borkar, SIAM J. Control Optim. 214Borkar, V. S. (1983). Controlled Markov chains and stochastic networks. SIAM J. Control Optim., 21(4):652-666.
On minimum cost per unit time control of Markov chains. V S Borkar, SIAM J. Control Optim. 226Borkar, V. S. (1984). On minimum cost per unit time control of Markov chains. SIAM J. Control Optim., 22(6):965-978.
A counterexample on the optimality equation in Markov decision chains with the average cost criterion. R Cavazos-Cadena, System and Control Lett. 165Cavazos-Cadena, R. (1991). A counterexample on the optimality equation in Markov decision chains with the average cost criterion. System and Control Lett., 16(5):387-392.
Real Analysis and Probability. R M Dudley, Cambridge University PressCambridgeDudley, R. M. (2002). Real Analysis and Probability. Cambridge University Press, Cambridge.
E B Dynkin, A A Yushkevich, Controlled Markov Processes. New YorkSpringerDynkin, E. B. and Yushkevich, A. A. (1979). Controlled Markov Processes. Springer, New York.
An ǫ-optimality control of a finite Markov chain with an average reward criterion. E A Feinberg, Theory Probab. Appl. 251Feinberg, E. A. (1980). An ǫ-optimality control of a finite Markov chain with an average reward criterion. Theory Probab. Appl., 25(1):70-81.
Controlled Markov processes with arbitrary numerical criteria. E A Feinberg, Theory Probab. Appl. 273Feinberg, E. A. (1982). Controlled Markov processes with arbitrary numerical criteria. Theory Probab. Appl., 27(3):486-503.
Average cost Markov decision processes with weakly continuous transition probabilities. E A Feinberg, P O Kasyanov, N V Zadoianchuk, Math. Oper. Res. 374Feinberg, E. A., Kasyanov, P. O., and Zadoianchuk, N. V. (2012). Average cost Markov decision processes with weakly continuous transition probabilities. Math. Oper. Res., 37(4):591-607.
Optimality inequalities for average cost Markov decision processes and the stochastic cash balance problem. E A Feinberg, M E Lewis, Math. Oper. Res. 324Feinberg, E. A. and Lewis, M. E. (2007). Optimality inequalities for average cost Markov decision processes and the stochastic cash balance problem. Math. Oper. Res., 32(4):769-783.
On controlled, discrete-time Markov decision processes. L G Gubenko, E S Shtatland, Theory Probab. Math. Statist. 7Gubenko, L. G. and Shtatland, E. S. (1975). On controlled, discrete-time Markov decision processes. Theory Probab. Math. Statist., 7:47-61.
Existence of average optimal policies in MCPs with strictly unbounded costs. O Hernández-Lerma, Kybernetika. 291Hernández-Lerma, O. (1993). Existence of average optimal policies in MCPs with strictly unbounded costs. Kybernetika, 29(1):1-17.
O Hernández-Lerma, J B Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria. New YorkSpringerHernández-Lerma, O. and Lasserre, J. B. (1996). Discrete-Time Markov Control Processes: Basic Optimality Criteria. Springer, New York.
O Hernández-Lerma, J B Lasserre, Further Topics on Discrete-Time Markov Control Processes. New YorkSpringerHernández-Lerma, O. and Lasserre, J. B. (1999). Further Topics on Discrete-Time Markov Control Processes. Springer, New York.
On the optimality equation for average cost Markov control processes with Feller transition probabilities. A Jaśkiewicz, A S Nowak, J. Math. Anal. Appl. 316Jaśkiewicz, A. and Nowak, A. S. (2006). On the optimality equation for average cost Markov control processes with Feller transition probabilities. J. Math. Anal. Appl., 316:495-509.
Markov decision processes with a Borel measurable cost function-the average case. M Kurano, Math. Oper. Res. 112Kurano, M. (1986). Markov decision processes with a Borel measurable cost function-the average case. Math. Oper. Res., 11(2):309-320.
The existence of a minimum pair of state and policy for Markov decision processes under the hypothesis of Doeblin. M Kurano, SIAM J. Control Optim. 272Kurano, M. (1989). The existence of a minimum pair of state and policy for Markov decision processes under the hypothesis of Doeblin. SIAM J. Control Optim., 27(2):296-307.
The optimal reward operator in negative dynamic programming. A Maitra, W Sudderth, Math. Oper. Res. 174Maitra, A. and Sudderth, W. (1992). The optimal reward operator in negative dynamic programming. Math. Oper. Res., 17(4):921-931.
S Meyn, R L Tweedie, Markov Chains and Stochastic Stability. CambridgeCambridge University Press2nd editionMeyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability. Cambridge University Press, Cambridge, 2nd edition.
The policy iteration algorithm for average reward Markov decision processes with general state space. S P Meyn, IEEE Trans. Automat. Contr. 4212Meyn, S. P. (1997). The policy iteration algorithm for average reward Markov decision processes with general state space. IEEE Trans. Automat. Contr., 42(12):1663-1680.
Probability Measures on Metric Spaces. K R Parthasarathy, Academic PressNew YorkParthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York.
General Markov models with the infinite horizon. Problems of Control and Information Theory. A B Piunovski, 18Piunovski, A. B. (1989). General Markov models with the infinite horizon. Problems of Control and Information Theory, 18(3):169-182.
Markov Decision Processes: Discrete Stochastic Dynamic Programming. M L Puterman, John Wiley & SonsNew YorkPuterman, M. L. (1994). Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley & Sons, New York.
Optimal stationary policies in general state space Markov decision chains with finite action sets. Math. R K Ritt, L I Sennott, Oper. Res. 174Ritt, R. K. and Sennott, L. I. (1992). Optimal stationary policies in general state space Markov decision chains with finite action sets. Math. Oper. Res., 17(4):901-909.
R T Rockafellar, R J Wets, .-B , Variational Analysis. BerlinSpringer1st editionRockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis. Springer, Berlin, 1st edition.
Conditions for optimality in dynamic programming and for the limit of n-stage optimal policies to be optimal. M Schäl, Z. Wahrscheinlichkeitstheorie verw. Gebiete. 32Schäl, M. (1975). Conditions for optimality in dynamic programming and for the limit of n-stage optimal policies to be optimal. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 32:179-196.
On the second optimality equation for semi-Markov decision models. M Schäl, Math. Oper. Res. 172Schäl, M. (1992). On the second optimality equation for semi-Markov decision models. Math. Oper. Res., 17(2):470-486.
Average optimality in dynamic programming with general state space. M Schäl, Math. Oper. Res. 181Schäl, M. (1993). Average optimality in dynamic programming with general state space. Math. Oper. Res., 18(1):163-172.
Average cost optimal stationary policies in infinite state Markov decision processes with unbounded costs. L I Sennott, Oper. Res. 374Sennott, L. I. (1989). Average cost optimal stationary policies in infinite state Markov decision processes with unbounded costs. Oper. Res., 37(4):626-633.
Stochastic Dynamic Programming and the Control of Queueing Systems. L I Sennott, John Wiley & SonsNew YorkSennott, L. I. (1999). Stochastic Dynamic Programming and the Control of Queueing Systems. John Wiley & Sons, New York.
Resolution of measurability problems in discrete-time stochastic control. S E Shreve, Stochastic Control Theory and Stochastic Differential Systems. BerlinSpringerShreve, S. E. (1979). Resolution of measurability problems in discrete-time stochastic control. In Stochastic Control Theory and Stochastic Differential Systems, pages 580-587. Springer, Berlin.
Alternative theoretical frameworks for finite horizon discretetime stochastic optimal control. S E Shreve, D P Bertsekas, SIAM J. Control Optim. 166Shreve, S. E. and Bertsekas, D. P. (1978). Alternative theoretical frameworks for finite horizon discrete- time stochastic optimal control. SIAM J. Control Optim., 16(6):953-978.
Universally measurable policies in dynamic programming. S E Shreve, D P Bertsekas, Math. Oper. Res. 41Shreve, S. E. and Bertsekas, D. P. (1979). Universally measurable policies in dynamic programming. Math. Oper. Res., 4(1):15-30.
A Course on Borel Sets. S M Srivastava, SpringerNew YorkSrivastava, S. M. (1998). A Course on Borel Sets. Springer, New York.
Negative dynamic programming. R E Strauch, Ann. Math. Statist. 37Strauch, R. E. (1966). Negative dynamic programming. Ann. Math. Statist., 37:871-890.
Stochastic Dynamic Programming. The Mathematical Centre. J Van Der Wal, Amsterdam2nd editionvan der Wal, J. (1984). Stochastic Dynamic Programming. The Mathematical Centre, Amsterdam, 2nd edition.
The average cost optimality equation: A fixed point approach. O Vega-Amaya, Bol. Soc. Mat. Mexicana. 91Vega-Amaya, O. (2003). The average cost optimality equation: A fixed point approach. Bol. Soc. Mat. Mexicana, 9(1):185-195.
On the vanishing discount factor approach for Markov decision processes with weakly continuous transition probabilities. O Vega-Amaya, J. Math. Anal. Appl. 426Vega-Amaya, O. (2015). On the vanishing discount factor approach for Markov decision processes with weakly continuous transition probabilities. J. Math. Anal. Appl., 426:978-985.
On discrete dynamic programming with sensitive discount optimality criteria. A F Veinott, Ann. Math. Statist. 40Veinott, A. F. (1969). On discrete dynamic programming with sensitive discount optimality criteria. Ann. Math. Statist., 40:1635-1660.
On convergence of value iteration for a class of total cost Markov decision processes. H Yu, SIAM J. Control Optim. 534Yu, H. (2015). On convergence of value iteration for a class of total cost Markov decision processes. SIAM J. Control Optim., 53(4):1982-2016.
A mixed value and policy iteration method for stochastic control with universally measurable policies. H Yu, D P Bertsekas, Math. Oper. Res. 404Yu, H. and Bertsekas, D. P. (2015). A mixed value and policy iteration method for stochastic control with universally measurable policies. Math. Oper. Res., 40(4):926-968.
| [] |
[
"Distance distribution in configuration model networks",
"Distance distribution in configuration model networks"
] | [
"Mor Nitzan \nRacah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael\n\nDepartment of Microbiology and Molecular Genetics\nFaculty of Medicine\nThe Hebrew University\n91120JerusalemIsrael\n",
"Eytan Katzav \nRacah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael\n",
"Reimer Kühn \nDepartment of Mathematics\nKing's College London\nWC2R 2LSStrand, LondonUK\n",
"Ofer Biham \nRacah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael\n"
] | [
"Racah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael",
"Department of Microbiology and Molecular Genetics\nFaculty of Medicine\nThe Hebrew University\n91120JerusalemIsrael",
"Racah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael",
"Department of Mathematics\nKing's College London\nWC2R 2LSStrand, LondonUK",
"Racah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael"
] | [] | We present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks. The results, which are based on recursion equations, are shown to be in good agreement with numerical simulations for networks with degenerate, binomial and power-law degree distributions. The mean, mode and variance of the distribution of shortest path lengths are also evaluated. These results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution, illuminating the connection between the two distributions. | 10.1103/physreve.93.062309 | [
"https://arxiv.org/pdf/1603.04473v2.pdf"
] | 22,550,115 | 1603.04473 | 48c9196966cf3414172340f9a6f3ccd823fa36ed |
Distance distribution in configuration model networks
5 Jun 2016
Mor Nitzan
Racah Institute of Physics
The Hebrew University
91904JerusalemIsrael
Department of Microbiology and Molecular Genetics
Faculty of Medicine
The Hebrew University
91120JerusalemIsrael
Eytan Katzav
Racah Institute of Physics
The Hebrew University
91904JerusalemIsrael
Reimer Kühn
Department of Mathematics
King's College London
WC2R 2LSStrand, LondonUK
Ofer Biham
Racah Institute of Physics
The Hebrew University
91904JerusalemIsrael
Distance distribution in configuration model networks
5 Jun 2016arXiv:1603.04473v2 [cond-mat.dis-nn]
We present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks. The results, which are based on recursion equations, are shown to be in good agreement with numerical simulations for networks with degenerate, binomial and power-law degree distributions. The mean, mode and variance of the distribution of shortest path lengths are also evaluated. These results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution, illuminating the connection between the two distributions.
I. INTRODUCTION
The study of complex networks has attracted much attention in recent years. It was found that network models provide a useful description of a large number of processes which involve interacting objects [1][2][3][4][5]. In these models, the objects are represented by nodes and the interactions are expressed by edges. Pairs of adjacent nodes can affect each other directly. However, the interactions between most pairs of nodes are indirect, mediated by intermediate nodes and edges.
A pair of nodes, i and j, may be connected by a large number of paths. The shortest among these paths are of particular importance because they are likely to provide the fastest and strongest interaction. Therefore, it is of interest to study the distribution of shortest path lengths (DSPL) between pairs of nodes in different types of networks. Such distributions, which are also referred to as distance distributions, are expected to depend on the network structure and size. They are of great importance for the temporal evolution of dynamical processes on networks, such as signal propagation [6], navigation [7][8][9] and epidemic spreading [10,11]. Central measures of the DSPL such as the average distance between pairs of nodes, and extremal measures such as the diameter were studied [12][13][14].
However, apart from a few studies [15][16][17][18][19][20], the entire DSPL has attracted little attention.
Recently, an analytical approach was developed for calculating the DSPL [21] in the Erdős-Rényi (ER) network, which is the simplest mathematical model of a random network [22][23][24]. Using recursion equations, analytical results for the DSPL were obtained in different regimes, including sparse and dense networks of small as well as asymptotically large sizes.
The resulting distributions were found to be in good agreement with numerical simulations.
ER networks are random graphs in which the degrees follow a Poisson distribution and there are no degree-degree correlations between connected pairs of nodes. In fact, ER networks can be considered as a maximum entropy ensemble under the constraint that the mean degree is fixed. Moreover, there is a much broader class of networks, named the configuration model, which generates maximum entropy ensembles when the entire degree distribution is constrained [4,14,15,25]. The ER ensemble is equivalent to a configuration model in which the degree distribution is constrained to be a Poisson distribution. For any given degree distribution, one can produce an ensemble of configuration model networks and perform a statistical analysis of its properties. Therefore, the configuration model provides a general and highly powerful platform for the analysis of networks. It is the ideal model to use as a null model when one tries to analyze an empirical network of which the degree distribution is known. For a given empirical network, one constructs a configuration model network of the same size and the same degree distribution. Properties of interest such as the DSPL [26], the betweenness centrality [27] and the abundance of network motifs [28] are compared between the two networks. The differences provide a rigorous test of the systematic features of the empirical network vs. the random network.
A theoretical framework for the study of the shell structure in configuration model networks was developed in a series of papers [29][30][31]. The shell structure around the largest hub in a scale free network was analyzed in Ref. [29]. This approach was later extended into a general theory of the shell structure arond a random node in a configuration model network [30,31]. This formulation is based on recursion equations for the number of nodes in each shell and for the degree distributions in the shells. In the special case of the ER network, the results of Refs. [30,31] for the number of nodes in each shell coincide with those of Ref. [16].
The shell structure around a random node in the configuration model was recently utilized for the study of epidemic spreading [32]. In a study of biological networks, the DSPL in a protein-protein interaction network was analyzed and compared to a corresponding configuration model network [26]. It was found that the distances in the configuration model are shorter than in the original empirical network. This highlights the features of the biological network which tend to increase the distances. These studies demonstrate the applicability of the configuration model in the analysis of the structure and dynamics in empirical networks.
In this paper we develop a theoretical framework, based on the cavity approach [33][34][35][36], for the calculation of the DSPL in networks which belong to the configuration model class.
Using this framework we derive recursion equations for the calculation of the DSPL in configuration model networks. We apply these equations to networks with degenerate, binomial and power-law degree distributions, and show that the results are in good agreement with numerical simulations. Using the tail-sum formula we calculate the mean and the variance of the DSPL. Evaluating the discrete derivative of the tail distribution, we also obtain the mode of the DSPL. These results provide closed form expressions for the central measures and dispersion measures of the DSPL in terms of the moments of the degree distribution and the size of the network, illuminating the connection between the two distributions.
The paper is organized as follows. In Sec. II we present the class of configuration model networks. In Sec. III we use the cavity approach to derive the recursion equations for the calculation of the DSPL in these networks. In Sec. IV we consider properties of the DSPL such as the mean, mode and variance. In Sec. V we present the results obtained from the recursion equations for different network models and compare them to numerical simulations. In Sec. VI we present a summary of the results.
II. THE CONFIGURATION MODEL
The configuration model is a maximum entropy ensemble of networks under the condition that the degree distribution is imposed [4,15]. Here we focus on the case of undirected networks, in which all the edges are bidirectional. To construct such a network of N nodes, one can draw the degrees of all nodes from a desired degree distribution p(k), k = 0, 1, . . . , N − 1, producing the degree sequence k i , i = 1, . . . , N (where k i must be even). The degree distribution p(k) satisfies k p(k) = 1. The mean degree over the ensemble of networks is c = k = k kp(k), while the average degree for a single instance of the network isk = i k i /N. Here we consider networks which do not include isolated nodes, namely p(0) = 0. This does not affect the applicability of the results, since the distribution of shortest path lengths is evaluated only for pairs of nodes which reside on the same cluster, for which the distance is finite. Actually, if a network includes isolated nodes, one can discard them by considering a renormalized degree distribution of the form p(k)/[1 − p(0)], for k = 1, . . . , N − 1.
A convenient way to construct a configuration model network is to prepare the N nodes such that each node, i, is connected to k i half edges [4]. Pairs of half edges from different nodes are then chosen randomly and are connected to each other in order to form the network. The result is a network with the desired degree sequence but no correlations. Note that towards the end of the construction the process may get stuck. This may happen in case that the only remaining pairs of half edges are in the same node or in nodes which are already connected to each other. In such cases one may perform some random reconnections in order to enable completion of the construction.
III. DERIVATION OF THE RECURSION EQUATIONS
Consider a random pair of nodes, i and j, in a network of N nodes. Assuming that the two nodes reside on the same connected cluster, they are likely to be connected by a large number of paths. Here we focus on the shortest among these paths (possibly more than one). More specifically, we derive recursion equations for the length distribution of these shortest paths. To this end we introduce the indicator function
χ N (d ij > ℓ) = 1 d ij > ℓ 0 d ij ≤ ℓ,(1)
where d ij is the length of the shortest path between nodes i and j, and ℓ is an integer. We also introduce the conditional indicator function
χ N (d ij > ℓ|d ij > ℓ − 1) = χ N (d ij > ℓ ∩ d ij > ℓ − 1) χ N (d ij > ℓ − 1) .(2)
Under the condition that the length d ij is larger than ℓ − 1, this function indicates whether d ij is also larger than ℓ. If it is, the conditional indicator function χ = 1, otherwise (namely if d ij = ℓ) χ = 0. In case the condition d ij > ℓ − 1 is not satisfied, the value of the conditional indicator function is undetermined. In order to extend this definition we adopt the convention that in case the condition is not satisfied the conditional indicator function takes the value χ N (d ij > ℓ|d ij > ℓ − 1) = 1. We note that all the subsequent results are independent of the value adopted here. The indicator function χ N (d ij > ℓ) can be expressed as a product of the conditional indicator functions in the form
χ N (d ij > ℓ) = χ N (d ij > 0) ℓ ℓ ′ =1 χ N (d ij > ℓ ′ |d ij > ℓ ′ − 1),(3)
where χ N (d ij > 0) = 1, since i and j are assumed to be two different nodes.
In the analysis below we calculate the mean of the indicator function over an ensemble of networks to obtain the distribution of shortest path lengths P N (d > ℓ). To this end we define the mean conditional indicator function m i (ℓ) ∈ [0, 1], obtained by averaging the conditional indicator function χ N (d ij > ℓ|d ij > ℓ − 1) over all suitable choices of the final node, j:
m i (ℓ) = χ N (d ij > ℓ|d ij > ℓ − 1) j .(4)
The averaging is done only over nodes j which reside on the same cluster as node i and for which the condition d ij > ℓ − 1 is satisfied.
A path of length ℓ from node i to node j can be decomposed into a single edge connecting node i and node r ∈ ∂ i (where ∂ i is the set of all nodes directly connected to i), and a shorter path of length ℓ − 1 connecting r and j. Thus, the existence of a path of length ℓ between nodes i and j can be ruled out if there is no path of length ℓ − 1 between any of the nodes r ∈ ∂ i , and j ( Fig. 1). The conditional indicator functions for these paths of length ℓ − 1 are χ
(i) N −1 (d rj > ℓ − 1|d rj > ℓ − 2)
, since they are embedded in a smaller network of N − 1 nodes, which does not include node i. The superscript (i) stands for the fact that the node r is reached by a link from node i. This is often referred to as the cavity indicator function [33][34][35][36]. Similarly, we define the mean cavity indicator function as
m (i) r (ℓ) = χ (i) N (d rj > ℓ|d rj > ℓ − 1) j .(5)
This reasoning enables us to express the conditional indicator function
χ N (d ij > ℓ|d ij > ℓ−1)
as a product of conditional indicator functions for shorter paths between nodes r ∈ ∂ i and j
χ N (d ij > ℓ|d ij > ℓ − 1) = r∈∂ i \{j} χ (i) N −1 (d rj > ℓ − 1|d rj > ℓ − 2).(6)
Under the assumption that the local structure of the network is tree-like, one can approximate the average of the product in Eq. (6) by the product of the averages. This assumption is fulfilled in the limit of large networks. In the analysis below we assume that N → ∞ and thus obtain recursion equations of the form
m i (ℓ) = r∈∂ i \{j} m (i) r (ℓ − 1).(7)
The mean cavity indicator function m
m (i) r (ℓ) = s∈∂r\{i,j} m (r) s (ℓ − 1).(8)
The number of neighbors r ∈ ∂ i is given by the degree, k i , of node i, while the number of neighbors s ∈ ∂ r is given by the degree, k r , of node r. Node i is a randomly chosen node and thus its degree, k i , is drawn from p(k). Node r is an intermediate node along the path and its probability to be encountered is proportional to its degree. Thus, its degree, k r , is drawn from the distribution (k/c)p(k), where c takes care of the normalization.
Considering an ensemble of networks, the variables m i (ℓ) and m (i) r (ℓ), which were defined for a specific node, i, on a given instance of the network, turn into the random variables m(ℓ) andm(ℓ), respectively. These random variables are drawn from suitable probability distributions, which respect the recursion equations (7) and (8). We denote these distribu-
tions by π ℓ (m) = P r[m(ℓ) = m] andπ ℓ (m) = P r[m(ℓ) = m]. These distributions obey the equations π ℓ (m) = ∞ k=1 p(k) 1 0 1 0 . . . 1 0 k ν=1π ℓ−1 (m ν )dm ν δ m − k ν=1 m ν (9) andπ ℓ (m) = ∞ k=1 k c p(k) 1 0 1 0 . . . 1 0 k−1 ν=1π ℓ−1 (m ν )dm ν δ m − k−1 ν=1 m ν .(10)
Eq. (9) refers to the random node, i, thus its degree is drawn from p(k). Eq. (10) The expectation values of m(ℓ) andm(ℓ) over the graph ensemble yield the conditional probabilities
m ℓ = P (d > ℓ|d > ℓ − 1) = 1 0 mπ ℓ (m)dm (11) andm ℓ =P (d > ℓ|d > ℓ − 1) = 1 0 mπ ℓ (m)dm.(12)
Plugging Eqs. (9) and (10) into Eqs. (11) and (12), respectively, we obtain the recursion equations
m ℓ = ∞ k=1 p(k)(m ℓ−1 ) k(13)
and
m ℓ = ∞ k=1 k c p(k)(m ℓ−1 ) k−1 ,(14)
which are valid for ℓ ≥ 2. Recalling that p(0) = 0, Eqs. (13) and (14) can be written using the degree generating functions [15]
m ℓ = G 0 (m ℓ−1 )(15)
andm
ℓ = G 1 (m ℓ−1 ) ,(16)
where
G 0 (x) = ∞ k=0 p(k)x k(17)
and
G 1 (x) = ∞ k=0 k c p(k)x k−1 .(18)
Eq. (13) can be understood intuitively as follows. Consider the simplified scenario in which node i is known to have a degree k. In this case, excluding a path of length ℓ from i to j is equivalent to excluding a path of length ℓ − 1 from all k neighbors of i to j, namely m ℓ = (m ℓ−1 ) k . Such reasoning was applied in Ref. [21], to obtain the DSPL from a node with a given degree to all other nodes in the network. In practice, the degree of a random node is unknown, and is distributed according to p(k). Therefore, Eq. (13) averages over all possible degrees with suitable weights, provided by p(k). Eq. (14) can be understood using a similar reasoning.
In the case of finite networks, we obtain
m N,ℓ = N −2 k=1 p(k)(m N −1,ℓ−1 ) k(19)
andm
N,ℓ = N −2 k=1 k c p(k)(m N −1,ℓ−1 ) k−1 ,(20)
for ℓ ≥ 2. For ℓ = 1 we can directly obtain the results
m N,1 = N −1 k=1 p(k) 1 − 1 N − 1 k (21) andm N,1 = N −1 k=1 k c p(k) 1 − 1 N − 1 k−1 .(22)
The tail distribution of the shortest path lengths can be expressed as a product of the form
P N (d > ℓ) = P N (d > 0) ℓ ℓ ′ =1 P N (d > ℓ ′ |d > ℓ ′ − 1) ≡ P N (d > 0) ℓ ℓ ′ =1 m N,ℓ ′ .(23)
Actually, since we choose two different nodes as the initial and final nodes, P N (d > 0) = 1, which further simplifies Eq. (23).
In Fig. 2 The probability distribution function, namely, the probability P N (ℓ) = P N (d = ℓ) that the shortest path length between a random pair of nodes is equal to ℓ can be obtained from the tail distribution by
P N (ℓ) = P N (d > ℓ − 1) − P N (d > ℓ),(24)
for ℓ = 1, 2, . . . , N − 1.
It should be noted that Eqs. (9) and (10), presenting the distributions π ℓ (m) andπ ℓ (m) enable the analysis of fluctuations of the conditional probabilities within an ensemble of networks with a given degree distribution in the large N limit.
IV. PROPERTIES OF THE DSPL
The distribution of shortest path lengths, P N (ℓ), can be characterized by its moments.
The nth moment, ℓ n , can be obtained using the tail-sum formula [37]
ℓ n = N −2 ℓ=0 [(ℓ + 1) n − ℓ n ]P N (d > ℓ).(25)
Note that the sum in Eq. (25) does not extend to ∞ because the longest possible shortest path in a network of size N is N − 1. The average distance between pairs of nodes in the network is given by the first moment
ℓ = N −2 ℓ=0 P N (d > ℓ).(26)
The average distance between nodes in configuration model networks has been studied extensively [15,18,20,[38][39][40][41][42]. It was found that
ℓ ≃ ln N ln k 2 − k k + O(1).(27)
The width of the distribution can be characterized by the variance σ 2 ℓ = ℓ 2 − ℓ 2 , where
ℓ 2 = N −2 ℓ=0 (2ℓ + 1)P N (d > ℓ).(28)
In addition to the average distance ℓ , another common measure of the typical distance between nodes in the network is the mode. Here we present a way to extract the mode of P N (ℓ) directly from the recursion equations, in the limit of a large network. It is based on the following observations: (a) The tail-distribution, P N (d > ℓ), is a sigmoid function,
i.e. it starts at 1 at the origin and drops to 0 at infinity. The transition between the two levels occurs over a relatively narrow interval; (b) Actually, P N (d > ℓ) can be expressed as a product of conditional probabilities of the form m N,ℓ ′ , where each term has the form of a sigmoid function [Eq. (23)]. Therefore, the product becomes an even sharper sigmoid function, and to a good approximation its maximal slope is determined by the the last term in the product. Therefore, in the analysis below we focus on the conditional probability m N,ℓ .
Considering the large N limit we can use the recursion equations (15) and (16). The generating functions satisfy G 0 (1) = G 1 (1) = 1, thus both equations exhibit a (repelling) fixed point at m ℓ =m ℓ = 1. Note that in this formulation, the network size N does not appear explicitly in the recursion equations, but only enters through the initial conditions, given by Eqs. (21) and (22). For simplicity, we approximate Eqs. (21) and (22) by
m 1 ≃ 1 − c N − 1 + O 1 N 2 ,(29)andm 1 ≃ 1 − k 2 − k k (N − 1) + O 1 N 2 ,(30)
respectively. For networks which are not too dense, these values are only slightly smaller than 1. Therefore, the linearized versions of Eqs. (15) and (16) hold as long as m ℓ andm ℓ are sufficiently close to 1. Note that these expressions require that the second moment k 2 would be finite. This condition may limit the validity of the derivation presented below to networks for which k 2 is bounded. Thus, networks for which k 2 diverges require special attention.
The location of the maximum value of the probability distribution function (namely the mode) is obtained at the point where the tail distribution falls most sharply. Up to that point the linear approximation holds quite well. This motivates us to perform the analysis in terms of the deviations
ǫ ℓ = 1 − m ℓ ,(31)andǫ ℓ = 1 −m ℓ .(32)
Linearizing Eqs. (15) and (16) in terms of ǫ ℓ andǫ ℓ , respectively, we obtain
ǫ ℓ = k ǫ ℓ−1 ,(33)andǫ ℓ = k 2 − k k ℓ−1ǫ 1 ,(34)
for any ℓ ≥ 2, whereǫ 1 = (
k 2 − k )/[ k (N − 1)].
Our aim is to determine the value of ℓ at which the reduction in m ℓ is maximal. We denote the discrete derivative
∆P = m ℓ−1 − m ℓ .(35)
Using the recursion equations (15) and (16), we can express this as
∆P = G 0 (m ℓ−2 ) − G 0 [G 1 (m ℓ−2 )],(36)
and we are therefore interested in the value of x, denoted by x max , at which the function
∆P (x) = G 0 (x)−G 0 [G 1 (x)]
is maximal. This is determined by the solution of the extremum
condition d∆P dx = G ′ 0 (x) − G ′ 0 [G 1 (x)]G ′ 1 (x) = 0.(37)
As long as x max is close to 1 we can use the linear approximation leading to Eq.
ℓ mode = ln [(N − 1)(1 − x max )] ln k 2 − k k + 2 + O(1).(38)
It is interesting to note that the mode exhibits the same scaling with the network size as the average distance shown in Eq. (27). This analysis is in the spirit of the renormalization group approach, where the flow of an initial small deviation from the critical temperature (here from the fixed point m = 1), under the linearized renormalization transformation determines the scaling behaviour of the system.
V. ANALYSIS OF NETWORK MODELS
To examine the recursion equations we apply them to the calculation of the DSPL in configuration model networks with different choices of the degree distribution. The results are compared to numerical simulations. In these simulations we generate instances of the configuration model networks with the required degree distribution. We then calculate the distances between all pairs of nodes in each network and generate a histogram. The process is repeated over a large number network instances. In case that the network includes more than one connected cluster we take into account only the distances between pairs of nodes which reside on the same cluster. The DSPL obtained from the numerical simulations is normalized accordingly.
To cover a broad class of networks, we consider configuration models which exhibit narrow as well as broad degree distributions. For networks with narrow degree distributions we study the the regular network (degenerate distribution) and networks with a binomial distribution.
For networks with broad degree distributions we study configuration models with power-law degree distributions (scale-free networks). A detailed analysis of the distributions of shortest path lengths in these configuration models is presented below.
A. Regular Networks
The simplest case of the configuration model is the regular graph, in which the degree distribution is p(k) = δ k,c , namely all N nodes have the same degree, (where c ≥ 2 and Nc is even). For c = 2 the network consists only of loops, while for c ≥ 3 more complex network structures appear. The random regular graph ensemble has been studied extensively and enjoys many analytical results [43]. In particular, there is an interesting phase transition at c = 3 above which the network becomes connected with probability 1 in the asymptotic limit.
In case of the regular graph the recursion equations (19) and (21)
and
m N,1 = 1 − 1 N − 1 c ,(40)
respectively. The subsequent equations, derived from Eqs. (20) and (22) take the form
m N,ℓ = (m N −1,ℓ−1 ) c−1(41)
andm
N,1 = 1 − 1 N − 1 c−1 .(42)
probabilities
P N d > ℓ|d > ℓ − 1) = m N,ℓ = 1 − 1 N − ℓ c(c−1) (ℓ−1) .(43)
Inserting the conditional probabilities into Eq. (23), and using the approximation N −ℓ ≃ N,
we obtain the tail distribution
P N d > ℓ) = exp − c(c − 1) ℓ N(c − 2) ,(44)
in agreement with Eq. (1.10) in Ref. [40].
Actually, in this case, Eqs. (9) and (10), describing the fluctuations in the ensemble in the large N limit, can be solved analytically yielding
π ℓ (m) = δ m − 1 − 1 N c(c−1) (ℓ−1) .(45)
This means that in regular networks, for sufficiently large N, the fluctuations are negligible.
The mean distance, ℓ , for the regular graph thus takes the form
ℓ = N −2 ℓ=0 e − c(c−1) ℓ N(c−2) .(46)
It is useful to define
s = ln N ln(c − 1) ,(47)
where ⌊x⌋ is the integer part of x. It is easy to see that for ℓ = 0, 1, . . . , s, the exponents on the right hand side of Eq. (46) are very close to 1, while for ℓ > s these exponents are quickly reduced. Therefore, to a very good approximation ℓ = ln N/ ln(c − 1). In order to obtain a more systematic approximation of ℓ we take into account explicitly a few terms around ℓ = s in Eq. (46). For example, taking three terms explicitly we obtain
ℓ = (s − 1) + s+1 ℓ=s−1 e − c(c−1) ℓ N(c−2) .(48)
One can easily improve the approximation by including additional explicit terms to the right and left of ℓ = s. Higher order moments can be evaluated in a similar fashion, yielding
ℓ n = (s − r) n + s+r ℓ=s−r [(ℓ + 1) n − ℓ n ]e − c(c−1) ℓ N(c−2) ,(49)
where r is the number of terms taken into account explicitly on the right and on the left.
The variance of P N (ℓ) is thus
σ 2 ℓ = r ℓ ′ =−r (2ℓ ′ + 2r + 1)e − c(c−1) s+ℓ ′ N(c−2) − r ℓ ′ =−r e − c(c−1) s+ℓ ′ N(c−2) 2 .(50)
In Fig. 3 we present the DSPL for regular networks of N = 1000 nodes, with c = 5, 20 and 50, obtained from Eq. (44). The probability distribution function P (d = ℓ) is shown in Fig. 3(a) and the tail distribution P (d > ℓ) is shown in Fig. 3(b). The results are compared with computer simulations showing excellent agreement.
In Fig. 4 we present the mean distance in regular graphs of N = 1000 nodes vs. the degree c, obtained from the recursion equations (⋄). The results are in excellent agreement with numerical simulations (+). As expected, the average distance decreases logarithmically as c is increased, in very good agreement with the exact result ℓ = ln N/ ln(c − 1).
For the regular graph, k = c and k 2 = c 2 . Plugging the degenerate degree distribution p(k) = δ k,c into Eqs. (17) and (18) we obtain that for the regular network G 0 (x) = x c and G 1 (x) = x c−1 . Since the distribution P N (ℓ) for the regular network is narrow, one expects the mode ℓ mode of this distribution to follow closely the mean value ℓ and to increase logarithmically as a function of N. Here we evaluate ℓ mode using Eq. (38). Inserting
x max = (c − 1) −1/(c−1) into Eq. (38) we obtain ℓ mode = ln N ln(c − 1) + O(1).(51)
Unlike ℓ the mode takes only integer values. Therefore, it must take the form of a step function vs. N. In Fig. 5 we present ℓ max vs. N on a semi-logarithmic scale. The general trend indeed satisfies ℓ max ∼ ln N, but the graph is decorated by steps at integer values of ℓ max .
B. Networks with Binomial Degree Distributions
To further examine the recursion equations, we extend the analysis to networks which exhibit a narrow or bounded degree distribution, with an average k = c and variance σ 2 k .
Since the degree distribution, p(k), is a discrete distribution, the binomial distribution
p(k) = n k p k (1 − p) n−k ,(52)
where n is an integer and 0 < p < 1, is particularly convenient. Its mean is given by k = np and its variance is given by σ 2 k = np(1 − p). In order to obtain desired values of k and σ 2 k , we choose the parameters n and p according to
n = Round k 2 k − σ 2 k ,(53)
where Round(x) is the nearest integer to x, and
p = k − σ 2 k k .(54)
It is important to note that the parameter, n, is not related to the network size, N, and can be either larger or smaller than N. However, one should choose a combination of n and p for which the probability, p(k), for k > N − 1 is vanishingly small, otherwise a truncation will be needed, which will deform the distribution. In Fig. 6(a) These deviations are due to the fact that in sparse networks the weight of the small, isolated clusters may be non-negligible even above the percolation threshold. This gives rise to some discrepancy between the theoretical and the numerical results for P (d > ℓ) for small values of c.
Plugging the binomial degree distribution of Eq. (52) into Eqs. (17) and (18) we obtain
that G 0 (x) = [1 − p(1 − x)] n and G 1 (x) = [1 − p(1 − x)] n−1 .
In the asymptotic limit, where
n ≫ 1, this expression converges to G 0 (x) ≃ G 1 (x) ≃ e −c(1−x) .
Here we evaluate ℓ mode for a network with a binomial degree distribution using Eq. (38).
For such networks x max = 1 − ln c/c. Inserting the results above into Eq. (38) we obtain
ℓ mode = ln N ln c + O(1).(55)
Note that Eqs. (51) and (55) differ in their denominators, where the former is ln(c − 1)
while the latter is ln c. The reason for this difference comes from the fact that in the regular network each node has exactly c neighbours, and so only c − 1 of them actually connect inner to outer shells. However, in the binomial case (as in the ER case), each neighbour of the initial node has on average an extra edge, and thus c edges connect an inner shell to an outer shell.
C. Networks with Power-Law Degree Distributions
Studies of empirical networks revealed that many of them exhibit power-law degree distributions of the form p(k) ∼ k −γ , where 2 < γ < 3. This is the range of values of γ for which the average degree is bounded but its variance diverges in the infinite system limit.
To construct a configuration model network with a power-law distribution p(k), we first choose a lower cutoff k min ≥ 1 and an upper cutoff k max ≤ N − 1. We then draw the degree sequence k i , i = 1, . . . , N from the distribution
p(k) = Ak −γ ,(56)
where the normalization coefficient is
A = [ζ(γ, k min ) − ζ(γ, k max + 1)] −1 ,(57)
and ζ(s, a) is the Hurwitz zeta function [44]. In the analytical calculations we insert p(k)
from Eq. (56) into the recursion equations in order to obtain the distribution of shortest path lengths for the ensemble of networks produced using this degree distribution. In the numerical simulation we repeatedly draw degree sequences from this distribution, produce instances of configuration model networks, calculate the distribution of shortest path lengths in these networks and average over a large number of instances.
In Fig. 7(a) we present the degree distributions of three scale-free network ensembles with N = 1000 nodes and γ = 2.5. The lower cutoffs of the degree distributions of these networks are given by k min = 2, 5 and 8, respectively. In each one of these three ensembles, the upper cutoff, k max was chosen such that p(k max ) ≃ 0.01, which means that in a network of 1000 nodes there will be on average about 10 nodes with degree k max . In Fig. 7(b) we present the tail distribution P (d > ℓ) for a scale free network with the degree distributions shown in Fig. 7(a). The analytical results are in very good agreement with the numerical simulations.
In the asymptotic limit, where k max → ∞, the power-law distribution satisfies k = ζ(γ − 1, k min )/ζ(γ, k min ) and k 2 = ζ(γ − 2, k min )/ζ(γ, k min ). Plugging the power-law degree distribution (56) into Eqs. (17) and (18) we obtain that
G 0 (x) = Φ(x, γ, k min ) ζ(γ, k min ) x k min(58)
and
G 1 (x) = Φ(x, γ − 1, k min ) ζ(γ − 1, k min ) x k min −1 ,(59)
where Φ(x, γ, k) is the Lerch transcendent [45]. Evaluating ℓ mode for a network with a power-law degree distribution using Eq. (38) we obtain
ℓ mode = ln N ln k 2 − k k + O(1).(60)
Note that in scale free networks characterized by 2 < γ < 3, the value of the second moment k 2 is dominated by the upper cutoff, k max . As long as k max is kept finite, ℓ mode will depend on this upper cutoff. On the other hand, in case that k max = N −1, then for γ = 3 one obtains that ( k 2 − k )/ k diverges logarithmically with N. As a result, ℓ mode ∼ ln N/ ln ln N for large N. For 2 < γ < 3 one obtains that (
k 2 − k )/ k ∼ (N − 1) 3−γ , entailing that ℓ mode = O(1).
The mean distance between nodes in scale free networks was studied in Ref. [46]. Using an analytical argument it was shown that scale free networks with degree distribution of the form p(k) ∼ k −γ are ultrasmall, namely exhibit a mean distance which scales like ℓ ∼ ln ln N for 2 < γ < 3. For γ = 3 it was shown that the mean distance scales like ℓ ∼ ln N/ ln ln N, while for γ > 3 it coincides with the common scaling of small world networks, namely ℓ ∼ ln N. As of now, our approach does not yield a closed form expression for the mean and thus we cannot provide a conclusive result for its scaling with N. We do see that the scaling of the mode of the DSPL coincides with the scaling predicted for the mean of the DSPL in Ref. [46] for γ > 3. In the range 2 < γ < 3 we find that the mode is of order 1, namely independent of N, which is even shorter than ln ln N. This is consistent with the ultrasmall scaling of the mean, reported in Ref. [46], since the mode is expected to be smaller than the mean and less sensitive to extreme values.
VI. SUMMARY AND DISCUSSION
We presented a theoretical framework for the calculation of the distributions of shortest path lengths between random pairs of nodes in configuration model networks. This framework, which is based on recursion equations derived using the cavity approach, provides analytical results for the distribution of shortest path lengths. We used the recursion equations to study a broad class of configuration model networks, with degree distributions that follow the degenerate, binomial and power-law distributions. The results were shown to be in good agreement with numerical simulations. The mean, mode and variance of the distribution of shortest path lengths were also evaluated and expressed in terms of moments of the degree distribution, illuminating the important connection between the two distributions.
The DSPL is of great relevance to transport processes on networks such as information flow and epidemic spreading. For example, an epidemic tends to spread outwards from the node where it was initiated. As time proceeds, it may reach nodes in shells farther away from the initial node and increases the fraction of infected nodes in the inner shells. Therefore, the number of nodes in each shell and their connectivity affect the rate and efficiency in which the epidemic progresses in the population [32].
The approach presented in this paper is aimed at the calculation of the entire distribution of distances between pairs of nodes in configuration model networks. In general, it does not provide a closed form expression for the DSPL but a set of recursion equations which can be evaluated for a given network size and a given degree distribution. As a result, it is difficult to obtain a closed form expression for the mean distance, except for special cases such as the regular graph. In fact, for the regular graph, our result for the mean distance coincides with the exact result presented in Ref. [40]. Regarding the mode of the DSPL, we do manage to obtain an analytical expression in the general case. The mode turns out to be more amenable to analysis than the mean because it can be determined by a local criterion.
For degree distributions with a finite second moment, the mean and the mode tend to scale in a similar fashion. However, in the case of scale free networks, the mean and the mode may scale differently. This is related to the fact that in scale free networks with 2 < γ < 3, the second moment of the degree distribution, k 2 , diverges in the infinite system limit.
The second moment appears in the equations for the mean distance and for the mode, thus calling for a special care in scale free networks. The mode is less sensitive to extreme values and therefore is expected to be smaller. We find that for 2 < γ < 3 the mode is of order 1, namely does not scale with the network size. Lacking a closed form expression for the mean, we cannot provide a conclusive result for the scaling of the mean with the system size. This is an important issue which deserves further research.
M.N. is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship. respectively), and from numerical simulations (+, × and * , respectively), for the three networks described above. It is observed that as the mean degree is increased, the average distance decreases. distributions P (d > ℓ), obtained from the recursion equations (♦, and , respectively), and from numerical simulations (+, × and * , respectively), for the three networks described above. It is observed that as the lower cutoff, k min , is increased, the mean distance decreases.
r
(ℓ) obeys a similar equation of the form
refers to intermediate nodes along the path, thus the degrees are drawn from the distribution (k/c)p(k). An additional feature of the intermediate nodes is that one of their edges is consumed by the incoming link, leaving only k − 1 links for the outgoing paths.
we illustrate the way the recursion equations are iterated ℓ ′ − 1 times along the diagonal in order to obtain m N,ℓ ′ . Starting fromm N −ℓ ′ ,1 (squares), Eq. (20) is iterated ℓ ′ − 2 times (empty circles), followed by a single iteration (full circles) of Eq. (19). The desired value of P N (d > ℓ) is obtained from Eq. (23). This product runs from bottom to top along the rightmost column of Fig. 2.
we can equateǫ ℓ mode −2+O(1) = 1 − x max , where the O(1) term comes from the fact that we are using a linearized equation while potentially higher order corrections should have been considered. This term is small and could be omitted when x max is close to 1, which is the situation in various known cases. Combining this result with Eq. (34) we obtain
take the form m N,ℓ = (m N −1,ℓ−1 ) c
we present the binomial degree distributions of three ensembles of networks of N = 1000 nodes, c = 5 (+), 20 (×) and 50 ( * ) and σ k = 4. In Fig. 6(b) we present the tail distributions P (d > ℓ) for these three network ensembles, obtained from the recursion equations for c = 5 (⋄), 20 ( ) and 50 (•). The results are found to be in very good agreement with numerical simulations, (+, × and * , respectively), except for the case of c = 5, where some small deviations are observed.
FIG. 1 :FIG. 2 :FIG. 3 :
123(Color online) Illustration of the possible paths of length ℓ between two random nodes, i and j, in a network of N nodes. The first edge of such a path connects node i to some other node, r, which may be any one of the k neighbors of node i. The rest of the path, from node r to node j is of length ℓ − 1 and it resides on a smaller network of N − 1 nodes, from which node i is excluded. (Color online) Illustration of the iteration process of the recursion equations(19), and(20), which carry over along the diagonals (empty circles). Starting fromm N −ℓ ′ ,1 (squares), given by Eq.(22), the iteration gives rise to m N,ℓ ′ (full circles). Eventually, P N (d > ℓ) is obtained as a product of the results in the right-most column [Eq. (23)]. (Color online) Distribution of shortest path lengths in a regular graph. The results of the recursion equations for P (ℓ) (a) and P (d > ℓ) (b), for c = 5, 20 and 50 (♦, and , respectively), fit well the numerical results (+, × and * , respectively). The numerical results were averaged over 50 graph instances in a graph of size N = 1000.
online) Mean shortest path length, ℓ , vs. the degree, c, in a regular graph of size N = 1000. The results of the recursion equations (♦) are in very good agreement with the numerical results (+). The numerical results were averaged over 50 graph instances.
FIG
. 5: (Color online) The mode of the distribution P N (ℓ) as a function of the network size, N , for a regular network of degree c = 3. Overall, the mode scales logarithmically with the network size. However, on a finer scale it forms steps due to the discreteness of the distance ℓ.
FIG. 6 :
6(Color online) (a) The degree distributions of three networks of size N = 1000, where p(k) was drawn from binomial distributions with means c = 5 , 20 and 50 (+, × and * , respectively), for which the standard deviation is σ k = 4. The results were obtained from numerical simulations, averaging over 50 graph instances. These results verify the construction of the configuration model network. (b) The tail distribution P (d > ℓ), obtained from the recursion equations (♦, and ,
FIG. 7 :
7(Color online) (a) The degree distributions of three networks of size N = 1000, where p(k) was drawn from power-law distributions with γ = 2.5 and lower cutoffs at k min = 2, 5 and 8 (+, × and * , respectively). The upper cutoffs, k max were set such that p(k max ) = 10/N . The results were obtained from numerical simulations, averaging over 50 graph instances. (b) The tail
. R Albert, A L Barabási, Rev. Mod. Phys. 7447R. Albert and A.L. Barabási, Rev. Mod. Phys. 74, 47 (2002).
Scale free networks: complex webs in nature and technology. G Caldarelli, Oxford University PressOxfordG. Caldarelli, Scale free networks: complex webs in nature and technology (Oxford University Press, Oxford, 2007).
S Havlin, R Cohen, Complex Networks: Structure, Robustness and Function. New YorkCambridge University PressS. Havlin and R. Cohen, Complex Networks: Structure, Robustness and Function (Cambridge University Press, New York, 2010).
M E J Newman, Networks: an Introduction. OxfordOxford University PressM.E.J. Newman, Networks: an Introduction (Oxford University Press, Oxford, 2010).
E Estrada, The structure of complex networks: Theory and applications. OxfordOxford University PressE. Estrada, The structure of complex networks: Theory and applications (Oxford University Press, Oxford, 2011).
. A Maáyan, S L Jenkins, S Neves, A Hasseldine, E Grace, B Dubin-Thaler, N J Eungdamrong, G Weng, P T Ram, J J Rice, A Kershenbaum, G A Stolovitzky, R D Blitzer, R Iyengar1, Science. 3091078A. Maáyan, S.L. Jenkins, S. Neves, A. Hasseldine, E. Grace, B. Dubin-Thaler, N.J. Eung- damrong, G. Weng, P.T. Ram, J.J. Rice, A. Kershenbaum, G.A. Stolovitzky, R.D. Blitzer, R. Iyengar1, Science 309, 1078 (2005).
. E W Dijkstra, Numer. Math. l. 269E.W. Dijkstra, Numer. Math. l, 269 (1959).
Engineering Route Planning Algorithms, in Algorithmics of Large and Complex Networks: Design, Analysis, and Simulation. D Delling, P Sanders, D Schultes, D Wagner, J. Lerner, D. Wagner, and K.A. Zweig117D. Delling, P. Sanders, D. Schultes and D. Wagner, Engineering Route Planning Algorithms, in Algorithmics of Large and Complex Networks: Design, Analysis, and Simulation, J. Lerner, D. Wagner, and K.A. Zweig (Eds.), p. 117 (2009).
. I Abraham, D Delling, A V Goldberg, R F Werneck, J. Experimental Algorithmics. 183I. Abraham, D. Delling, A.V. Goldberg and R.F. Werneck, J. Experimental Algorithmics 18, Article 1.3 (2013).
. R Pastor-Satorras, A Vespignani, Phys. Rev. Lett. 863200R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001).
. R Pastor-Satorras, C Castellano, P Van Mieghem, A Vespignani, Rev. Mod. Phys. 87925R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Rev. Mod. Phys. 87, 925 (2015).
Random Graphs. B Bollobas, Academic PressLondonSecond EditionB. Bollobas, Random Graphs, Second Edition (Academic Press, London, 2001).
. D J Watts, S H Strogatz, Nature. 393440D.J. Watts and S.H. Strogatz, Nature 393, 440 (1998).
. A Fronczak, P Fronczak, J A Holyst, Phys. Rev. E. 7056110A. Fronczak, P. Fronczak, and J.A. Holyst, Phys. Rev. E 70, 056110 (2004).
. M E J Newman, S H Strogatz, D J Watts, Phys. Rev. E. 6426118M.E.J. Newman, S.H. Strogatz, and D.J. Watts, Phys. Rev. E 64, 026118 (2001).
. V D Blondel, J.-L Guillaume, J M Hendrickx, R M Jungers, Phys. Rev. E. 7666101V.D. Blondel, J.-L. Guillaume, J.M. Hendrickx and R.M. Jungers, Phys. Rev. E 76, 066101 (2007).
. S N Dorogotsev, J F F Mendes, A N Samukhin, Nuclear Physics B. 653307S.N. Dorogotsev, J.F.F. Mendes and A.N. Samukhin, Nuclear Physics B 653, 307 (2003).
. R Van Der Hofstad, G Hooghiemstra, D Znamenski, Electronic Journal of Probability. 12703R. van der Hofstad, G. Hooghiemstra and D. Znamenski, Electronic Journal of Probability 12, 703 (2007).
. R Van Der Hofstad, G Hooghiemstra, J. Math. Phys. 49125209R. van der Hofstad and G. Hooghiemstra, J. Math. Phys. 49, 125209 (2008).
. H Van Der Esker, R Van Der Hofstad, G Hooghiemstra, J. Stat. Phys. 133169H. van der Esker, R. van der Hofstad and G. Hooghiemstra, J. Stat. Phys. 133, 169 (2008).
. E Katzav, M Nitzan, D Ben-Avraham, P L Krapivsky, R Kühn, N Ross, O Biham, EPL. 11126006E. Katzav, M. Nitzan, D. ben-Avraham, P.L. Krapivsky, R. Kühn, N. Ross and O. Biham, EPL 111, 26006 (2015).
. P Erdős, Rényi , Publ. Math. 6290P. Erdős and Rényi, Publ. Math. 6, 290 (1959).
. P Erdős, Rényi , Publ. Math. Inst. Hung. Acad. Sci. 517P. Erdős and Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960).
. P Erdős, Rényi , Bull. Inst. Int. Stat. 38343P. Erdős and Rényi, Bull. Inst. Int. Stat. 38, 343 (1961).
. M Molloy, B Reed, Random Struct. Algorithms. 6161M. Molloy and B. Reed, Random Struct. Algorithms 6, 161 (1995).
. Giot, Science. 3021727Giot et al., Science 302 1727 (2003).
. K.-I Goh, E Oh, B Kahng, D Kim, Phys. Rev. E. 6717101K.-I. Goh, E. Oh, B. Kahng and D. Kim, Phys. Rev. E 67, 017101 (2003).
. R Milo, S Shen-Orr, S Itzkovitz, N Kashtan, D Chklovskii, U Alon, Science. 298824R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii and U. Alon, Science 298, 824 (2002).
. T Kalisky, R Cohen, O Mokryn, D Dolev, Y Shavitt, S Havlin, Phys. Rev. E. 7466108T. Kalisky, R. Cohen, O. Mokryn, D. Dolev, Y. Shavitt and S. Havlin, Phys. Rev. E 74, 066108 (2006).
. J Shao, S V Buldyrev, R Cohen, M Kitsak, S Havlin, H E Stanley, EPL. 8448004J. Shao, S.V. Buldyrev, R. Cohen, M. Kitsak, S. Havlin and H.E. Stanley, EPL 84, 48004 (2008).
. J Shao, S V Buldyrev, L A Braunstein, S Havlin, H E Stanley, Phys. Rev. E. 8036105J. Shao, S.V. Buldyrev, L.A. Braunstein, S. Havlin and H.E. Stanley Phys. Rev. E 80, 036105 (2009).
. S Shao, X Huang, H E Stanley, S Havlin, New J. Phys. 1723049S. Shao, X. Huang, H.E. Stanley and S. Havlin, New J. Phys. 17, 023049 (2015).
. M Mézard, G Parisi, M A Virasoro, J. Physique Lett. 46217M. Mézard, G. Parisi and M.A. Virasoro, J. Physique Lett. 46, 217 (1985).
. M Mézard, G Parisi, J. Stat. Phys. 1111M. Mézard and G. Parisi, J. Stat. Phys. 111, 1 (2003).
M Mézard, A Montanari, Information, Physics and Computation. Oxford University PressM. Mézard and A. Montanari, Information, Physics and Computation (Oxford University Press, 2009).
Cavity Method -Message Passing from a Physics Perspective, Statistical Physics, Optimization, Inference and Message-Passing Algorithms. G Ferraro, C Wang, D Martí, M Mézard, Lecture Notes of the Les Houches School of Physics. F. Krzakala, F. Ricci-Tersenghi, L. Zdeborova, R. Zecchina, E.W. Tramel, and L.F. CugliandoloOxford University PressG. Del Ferraro, C. Wang, D. Martí and M. Mézard, Cavity Method -Message Passing from a Physics Perspective, Statistical Physics, Optimization, Inference and Message-Passing Al- gorithms, Lecture Notes of the Les Houches School of Physics, Eds. F. Krzakala, F. Ricci- Tersenghi, L. Zdeborova, R. Zecchina, E.W. Tramel, and L.F. Cugliandolo (Oxford University Press, 2015).
. J Pitman, Probability. SpringerJ. Pitman, Probability (Springer, New York, 1993).
F Chung, L Lu, Proc. Nat. Acad. Sci. USA. Nat. Acad. Sci. USA9915879F. Chung and L. Lu, Proc. Nat. Acad. Sci. USA 99, 15879 (2002)
. F Chung, L Lu, Internet Mathematics. 191F. Chung and L. Lu, Internet Mathematics 1, 91 (2004).
. R Van Der Hofstad, G Hooghiemstra, P Van Mieghem, Random Structures Algorithms. 2776R. van der Hofstad, G. Hooghiemstra and P. Van Mieghem, Random Structures Algorithms 27, 76 (2005).
. H Van Den Esker, R Van Der Hofstad, G Hooghiemstra, D Znamenski, Extremes. 8111H. van den Esker, R. van der Hofstad, G. Hooghiemstra and D. Znamenski, Extremes 8, 111 (2006).
. B Bollobas, S Janson, O Riordan, Random Structures and Algorithms. 313B. Bollobas, S. Janson and O. Riordan, Random Structures and Algorithms 31, 3 (2007).
Models of random regular graphs. N C Wormald, LMS Lecture Note Series. J.D. Lamb and D.A. PreeceCambridge University PressN.C. Wormald, Models of random regular graphs, in LMS Lecture Note Series, Surveys in Combinatorics Eds. J.D. Lamb and D.A. Preece, pages 239-298 (Cambridge University Press, 1999).
F W J Olver, D M Lozier, R F Boisvert, C W Clark, NIST Handbook of Mathematical Functions. Cambridge University PressF.W.J. Olver, D.M. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, 2010).
I S Gradshteyn, I M Ryzhik, Tables of Integrals, Series, and Products. San DiegoAcademic Press6th editionI.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series, and Products, 6th edition (Aca- demic Press, San Diego, 2000).
. R Cohen, S Havlin, Phys. Rev. Lett. 9058701R. Cohen and S. Havlin, Phys. Rev. Lett. 90, 058701 (2003).
| [] |
[
"ADM-like Hamiltonian formulation of gravity in the teleparallel geometry: derivation of constraint algebra",
"ADM-like Hamiltonian formulation of gravity in the teleparallel geometry: derivation of constraint algebra"
] | [
"Andrzej Okołów [email protected] \nInstitute of Theoretical Physics\nWarsaw University ul\nHoża 6900-681WarsawPoland\n"
] | [
"Institute of Theoretical Physics\nWarsaw University ul\nHoża 6900-681WarsawPoland"
] | [] | We derive a new constraint algebra for a Hamiltonian formulation of the Teleparallel Equivalent of General Relativity treated as a theory of cotetrad fields on a spacetime. The algebra turns out to be closed. | 10.1007/s10714-013-1636-4 | [
"https://arxiv.org/pdf/1309.4685v1.pdf"
] | 118,556,304 | 1309.4685 | 4927778abd8916844d5f95af87cd80d7f2e02939 |
ADM-like Hamiltonian formulation of gravity in the teleparallel geometry: derivation of constraint algebra
18 Sep 2013 September 18, 2013
Andrzej Okołów [email protected]
Institute of Theoretical Physics
Warsaw University ul
Hoża 6900-681WarsawPoland
ADM-like Hamiltonian formulation of gravity in the teleparallel geometry: derivation of constraint algebra
18 Sep 2013 September 18, 2013arXiv:1309.4685v1 [gr-qc]
We derive a new constraint algebra for a Hamiltonian formulation of the Teleparallel Equivalent of General Relativity treated as a theory of cotetrad fields on a spacetime. The algebra turns out to be closed.
Introduction
In our previous paper [1] we presented a Hamiltonian formulation of the Teleparallel Equivalent of General Relativity (TEGR) regarded as a theory of cotetrad fields on a spacetime-the formulation is meant to serve as a point of departure for canonical quantization à la Dirac of the theory (preliminary stages of the quantization are described in [2,3,4]). In [1] we found a phase space, a set of (primary and secondary) constraints on the phase space and a Hamiltonian. We also presented an algebra of the constraints. An important fact is that this algebra is closed i.e. a Poisson bracket of every pair of the constraints is a sum of all the constraints multiplied by some factors. This property of the constraint algebra together with a fact that the Hamiltonian is a sum of the constraints allowed us to conclude that (i) the set of constraints is complete and (ii) all the constraints are of the first class.
Let us emphasize that knowledge of a complete set of constraints and their properties as well as knowledge of an explicite form of a constraint algebra is very important from the point of view the Dirac's approach to canonical quantization of constrained systems since the knowledge enables a right treatment of constraints in the procedure of quantization.
However, the derivation of the constraint algebra turned out to be too long to be included in [1]. To fill this gap, that is, to prove that the constraint algebra is correct we carry out the derivation in the present paper 1 . Moreover, to the best of our knowledge a derivation of constraint algebra of TEGR treated as a theory of cotetrad fields has never been presented before-in papers [5,6] describing a distinct Hamiltonian formulation of this version 2 of TEGR one can find a constraint algebra but its derivation is not shown.
The paper is organized as follows: in Section 2 we recall the description of the phase space and the constraints on it derived in [1]. In Section 3 we derive the constraint algebra. Section 4 contains a short summary.
Let us finally emphasize that since the present paper plays a role of an appendix to [1] we do not discuss here the results nor compare them to results of previous works-all these can be found in [1].
Preliminaries
Let M be a four-dimensional oriented vector space equipped with a scalar product η of signature (−, +, +, +). We fix an orthonormal basis (v A ) (A = 0, 1, 2, 3) such that the components (η AB ) of η given by the basis form a matrix diag(−1, 1, 1, 1). The matrix (η AB ) and its inverse (η AB ) will be used to, respectively, lower and raise capital Latin letter indeces.
Let Σ be a three-dimensional oriented manifold. We assume moreover that it is a compact manifold without boundary.
In [1] we obtained the phase space of TEGR as a Cartesian product of 1. a set of all quadruplets of one-forms (θ A ) (A = 0, 1, 2, 3) on Σ such that for each quadruplet a metric q := η AB θ A ⊗ θ B (2.1) on Σ is Riemannian (i.e. positive definite);
2. a set of all quadruplets (p B ) (B = 0, 1, 2, 3) of two-forms on Σ-a two-form p A is the momentum conjugate to θ A .
The metric q defines a volume form ǫ on Σ and a Hodge operator * acting on differential forms on the manifold. Throughout the paper we will often use functions on Σ defined as follows [9]:
ξ A := − 1 3! ε A BCD * (θ B ∧ θ C ∧ θ D ),(2.
2) 1 The derivation is also an example of an application of differential form calculus to a derivation of a constraint algebra which usually is done by means of tensor calculus. 2 There is another version of TEGR configuration variables of which are cotetrad fields and flat Lorentz connections of non-zero torsion. For a complete Hamiltonian analysis of this version of TEGR see [7].
where ε ABCD are components of a volume form on M given by the scalar product η.
Components of q in a local coordinate frame (x i ) (i = 1, 2, 3) on Σ will be denoted by q ij . Obviously
q ij = η AB θ A i θ B j ,(2.3)
where θ A i are components of θ A . The metric q and its inverse q −1 ,
q −1 := q ij ∂ i ⊗ ∂ j , q ij q jk = δ i k ,(2.4)
will be used to, respectively, lower and raise, indeces (here: lower case Latin letters) of components of tensor fields defined on Σ. In particular we will often map one-forms to vector fields on Σ-a vector field corresponding to a one form α will be denoted by α i.e. if α = α i dx i then α := q ij α i ∂ j .
Let us emphasize that all object defined by q (as ǫ, * , ξ A and q −1 ) are functions of (θ A ) which means that they are functions on the phase space.
In [1] we found some constraints on the phase space of TEGR. Smeared versions of the constraints read
B(a) := Σ a ∧ (θ A ∧ * dθ A + ξ A p A ), (2.5) R(b) := Σ b ∧ (θ A ∧ * p A − ξ A dθ A ), (2.6) S(M ) := Σ M 1 2 (p A ∧ θ B ) ∧ * (p B ∧ θ A ) − 1 4 (p A ∧ θ A ) ∧ * (p B ∧ θ B ) − ξ A ∧ dp A + + 1 2 (dθ A ∧ θ B ) ∧ * (dθ B ∧ θ A ) − 1 4 (dθ A ∧ θ A ) ∧ * (dθ B ∧ θ B ) ,(2.
Derivation of the constraint algebra
In this section we will calculate Poisson brackets of all pairs of the constraints presented above and show that each Poisson bracket is a sum of the constraints smeared with some fields.
Calculations needed to achieve the goal will be long, laborious and complicated. We assume that the reader is familiar with tensor calculus, differential form calculus (including the contraction X α of a vector field X with a differential form α) and properties of a Hodge operator on three-dimensional manifold defined by a Riemannian metric.
Preliminaries
Poisson bracket
If F and G are functionals on the phase space then their Poisson bracket [8]
{F, G} = Σ δF δθ A ∧ δG δp A − δG δθ A ∧ δF δp A ,
where the functional derivatives with respect to θ A and p A are defined as follows [9]: δF/δθ A is a differential two-form on Σ and δF/δp A is a differential one-form on Σ such that
δF = Σ δθ A ∧ δF δθ A + δp A ∧ δF δp A
for every δθ A and δp A . Calculating functional derivatives of the smeared constraints would be straightforward if (i) the Hodge operator * did not depend on θ A and (ii) the constraints did not depend on ξ A being a complicated function of θ A . Thus explicite formulae describing these derivatives are needed.
Given k-forms α and β, denote
α ∧ * ′ A β ≡ θ B [η AB α ∧ * β − ( θ A α) ∧ * ( θ B β) − ( θ B α) ∧ * ( θ A β)]. (3.1)
If the forms α, β do not depend on the canonical variables then [9] δ
δθ A Σ α ∧ * β = α ∧ * ′ A β. (3.2)
An important property of every two-form α ∧ * ′ A β is that it vanishes once contracted with the function ξ A [9]:
ξ A (α ∧ * ′ A β) = 0. (3.3)
Consider now a three-form κ A on Σ which does not depend on θ A and p A and a functional
F = Σ ξ D κ D = Σ ( * ξ D ) ∧ * (κ D ) = Σ − 1 3! ε D BCA θ B ∧ θ C ∧ θ A ∧ * κ D .
Using (3.2) we obtain
δF = Σ − 1 3! ε D BCA (δθ B ∧ θ C ∧ θ A + θ B ∧ δθ C ∧ θ A + θ B ∧ θ C ∧ δθ A ) ∧ * κ D + + δθ A ∧ ( * ξ D ) ∧ * ′ A κ D and therefore δF δθ A = − 1 2 ε D BCA θ B ∧ θ C ∧ * κ D + ( * ξ D ) ∧ * ′ A κ D (3.4)
Auxiliary formulae
Auxiliary formulae presented below will be used throughout the calculations. Except them we will need many other formulae which will be derived in the subsequent subsections.
The functions (ξ A ) satisfy the following important conditions [10]:
ξ A ξ A = −1, ξ A θ A = 0.
These two equations imply
ξ A dξ A = 0, dξ A ∧ θ A + ξ A dθ A = 0.
These formulae will be used very often and therefore it would be troublesome to refer to them each time. Therefore we kindly ask the reader to keep the formulae in mind since they will be used without any reference. For any one-form α and any k-form β [9] * ( * β ∧ α) = α β. (3.5) Setting β = * γ and taking into account that * * = id we obtain an identity * (γ ∧ α) = α ( * γ) (3.6) valid for every l-form γ. It was shown in [1] that
θ B θ A = η AB + ξ A ξ B , (3.7) θ A ∧ ( θ A α) = kα (3.8)
where α is a k-form on Σ.
Tensor calculus
Although our original wish was to carry out all necessary calculations using differential form calculus only we were forced in some cases to use tensor calculus. Below we gathered some expressions which will be applied repeatedly in the sequel. Let ∇ denote a covariant derivative on Σ defined by the Levi-Civita connection given by the metric q. Consequently, by virtue of (2.3)
0 = ∇ i q jk = ∇ i (θ Aj θ A k ) = (∇ i θ Aj )θ A k + θ Aj (∇ i θ A k ) (3.9)
and
(∇ a θ Bb )θ Bb = 0. (3.10)
Note also that ∇ a ǫ ijk = 0 (3.11) because ǫ is defined by q.
For any one-forms α and β
dα = ∇ a α b dx a ∧ dx b , * dα = (∇ a α b )ǫ ab c dx c , dα ∧ β = (∇ a α b )β c dx a ∧ dx b ∧ dx c , * (dα ∧ β) = (∇ a α b )β c ǫ abc , (3.12)
If α is a one-form, β a two-form and γ a three-form then
* d * α = q ab ∇ a α b = ∇ a α a , * d * β = q ab ∇ a β cb dx c = ∇ b β cb dx c , * d * γ = q ab ∇ a γ dcb dx d ⊗ dx c = ∇ b γ dcb dx d ⊗ dx c = 1 2 ∇ b γ dcb dx d ∧ dx c ,(3.13)
We will also apply the following identities (for a proof see e.g. [9]):
ǫ ibc ǫ abc = 2δ a i , ǫ ijc ǫ abc = 2δ [a i δ b] j , ǫ ijk ǫ abc = 3!δ [a i δ b j δ c] k .
(3.14)
and a formula
α ∧ * β = 1 k! α a 1 ...a k β a 1 ...a k ǫ,(3.15)
where α and β are k-forms.
Poisson brackets of B(a) and R(b)
In this subsection we will calculate Poisson brackets {B(a),
B(a ′ )}, {R(b), R(b ′ )} and {B(a), R(b)}.
Auxiliary formulae
The following formulae will be used while calculating the brackets:
b ∧ α ∧ * (b ′ ∧ β) − (b ↔ b ′ ) = * (b ∧ b ′ ) ∧ α ∧ β, (3.16) (α ∧ * ′ A β) ∧ * (b ∧ θ A ) = 0, (3.17) θ A ∧ * (α ∧ θ A ) = (3 − k) * α, (3.18) ǫ DBCA θ B ∧ θ C ∧ * (b ∧ θ A ) = 0, (3.19) − 1 2 ǫ D BCA θ B ∧ θ C ξ A = * θ D . (3.20)
In (3.16) α, β, b, b ′ are one-forms, in (3.17) α and β are k-forms, and b is a one-form, in (3.18) α is a k-form, and in (3.19) b is a one-form.
Proof of (3.16).
b ∧ α ∧ * (b ′ ∧ β) − (b ↔ b ′ ) = b ∧ * * [α ∧ * (b ′ ∧ β)] − (b ↔ b ′ ) = = −b ∧ * [( α b ′ )β − b ′ α β] − (b ↔ b ′ ) = α (b ∧ b ′ ) ∧ * β = −(b ∧ b ′ ) α * β = = (b ∧ b ′ ) ∧ * (α ∧ β) = * (b ∧ b ′ ) ∧ (α ∧ β), (3.21)
where in the second step we used (3.5), in the third the fact that b ∧ * b ′ is symmetric in b and b ′ and finally in the fifth step we applied (3.6).
Proof of (3.17). To prove (3.17) note that the two-form α ∧ * ′ A β given by (3.1) is of the form θ B γ AB , where the three-form γ AB is symmetric in A and B: γ AB = γ BA . Thus
(α∧ * ′ A β)∧ * (b∧θ A ) = ( θ B γ AB )∧ * (b∧θ A ) = γ AB ∧ θ B * (b∧θ A ) = γ AB ∧ * (b∧θ A ∧θ B )
, where in the last step we used (3.6). But (b ∧ θ A ∧ θ B ) is antisymmetric in A and B, hence (3.17) follows.
Proof of (3.18).
θ A ∧ * (α ∧ θ A ) = θ A ∧ θ A * α = (3 − k) * α,
where in the first step we used (3.6) and in the second one we applied (3.8).
Proof of (3.19). Let us transform the following expression by means of (3.6):
θ B ∧ θ C ∧ * (b ∧ θ A ) = θ B ∧ θ C ∧ θ A * b = − θ A (θ B ∧ θ C ) ∧ * b = = −( θ A θ B ) ∧ θ C ∧ * b + θ B ( θ A θ C ) ∧ * b. (3.22)
Note that the first term at the r.h.s. of the equation above is symmetric in A and B, while the second one-in A and C. This means that both terms vanish once contracted with ǫ DBCA .
The last formula (3.20) is proven in [9].
3.2.2 Poisson bracket {B(a), B(a ′ )} Let B 1 (a) := Σ a ∧ θ A ∧ * dθ A , B 2 (a) := Σ a ∧ ξ A p A = Σ ( * ξ A ) ∧ * (a ∧ p A ). (3.23)
Taking into account (2.5) we see that B(a) = B 1 (a) + B 2 (a) and consequently
{B(a), B(a ′ )} = {B 1 (a), B 1 (a ′ )}+ {B 1 (a), B 2 (a ′ )}−{B 1 (a ′ ), B 2 (a)} +{B 2 (a), B 2 (a ′ )}.
(3.24) Corresponding variational derivatives read
δB 1 (a) δθ A = −a ∧ * dθ A + (a ∧ θ B ) ∧ * ′ A dθ B + d * (a ∧ θ A ), δB 1 (a) δp A = 0, δB 2 (a) δθ A = − 1 2 ǫ D BCA θ B ∧ θ C * (a ∧ p D ) + ( * ξ B ) ∧ * ′ A (a ∧ p B ), δB 2 (a) δp A = aξ A .
(3.25)
Obviously, {B 1 (a), B 1 (a ′ )} = 0. The next term in (3.24)
{B 1 (a), B 2 (a ′ )}−{B 1 (a ′ ), B 2 (a)} = Σ −a∧ * dθ A ∧a ′ ξ A +d * (a∧θ A )∧a ′ ξ A −(a ↔ a ′ ) = = Σ 2 * (a ∧ a ′ )ξ A dθ A − a ∧ θ A ∧ * (a ′ ∧ dξ A ) − (a ↔ a ′ ) = = Σ 2 * (a ∧ a ′ )ξ A dθ A − * (a ∧ a ′ )θ A ∧ dξ A = Σ * (a ∧ a ′ )ξ A dθ A ,
where in the first step we used (3.3) and in the third one (3.16). The last term in (3.24) due to (3.3) reads
{B 2 (a), B 2 (a ′ )} = Σ − 1 2 ǫ D BCA θ B ∧ θ C ∧ * (a ∧ p D ) ∧ a ′ ξ A − (a ↔ a ′ ).
By virtue of (3.20) and (3.5)
− 1 2 ǫ D BCA θ B ∧ θ C ∧ * (a ∧ p D ) ∧ a ′ ξ A = * θ D ∧ a ′ * (a ∧ p D ) = a ∧ p D θ D a ′ .
Thus
{B 2 (a), B 2 (a ′ )} = − Σ θ D (a∧a ′ )p D = Σ a∧a ′ ∧ θ D p D = Σ −a∧a ′ ∧ * (θ D ∧ * p D ) = = Σ − * (a ∧ a ′ )θ A ∧ * p A ,
where in the third step we used (3.5).
We conclude that (see (2.6))
{B(a), B(a ′ )} = − Σ * (a ∧ a ′ ) ∧ (θ A ∧ * p A − ξ A dθ A ) = −R( * (a ∧ a ′ )). (3.26) 3.2.3 Poisson bracket {R(b), R(b ′ )} Let us define R 1 (b) := Σ b ∧ θ A ∧ * p A , R 2 (b) := Σ b ∧ ξ A dθ A . (3.27)
Then by virtue of (2.6)
R(b) = R 1 (b) − R 2 (b) and {R(b), R(b ′ )} = {R 1 (b), R 1 (b ′ )} − {R 1 (b), R 2 (b ′ )} − {R 1 (b ′ ), R 2 (b)} + + {R 2 (b), R 2 (b ′ )}. (3.28)
Corresponding variational derivatives read
δR 1 (b) δθ A = −b ∧ * p A + (b ∧ θ B ) ∧ * ′ A p B δR 1 (b) δp A = * (b ∧ θ A ), δR 2 (b) δθ A = d(bξ A ) + (b ∧ dθ B ) ∧ * ′ A ( * ξ B ) − 1 2 ǫ DBCA θ B ∧ θ C * (b ∧ dθ D ) δR 2 (b) δp A = 0.
(3.29) Due to (3.17) the first bracket at the r.h.s. of (3.28) reduces to
{R 1 (b), R 1 (b ′ )} = Σ −b ∧ * p A ∧ * (b ′ ∧ θ A ) − (b ↔ b ′ ) = Σ * (b ∧ b ′ ) ∧ θ A ∧ * p A ,
where in the last step we used (3.16). Similarly, by virtue of (3.17) the next two brackets in (3.28) reduce to
{R 1 (b), R 2 (b ′ )} − {R 1 (b ′ ), R 2 (b)} = = Σ [−d(b ′ ξ A ) + 1 2 ǫ DBCA θ B ∧ θ C * (b ′ ∧ dθ D )] ∧ * (b ∧ θ A ) − (b ↔ b ′ ) = = Σ [b ′ ∧ dξ A ∧ * (b ∧ θ A ) + 1 2 ǫ DBCA θ B ∧ θ C ∧ * (b ∧ θ A ) * (b ′ ∧ dθ D )] − (b ↔ b ′ ).
Note now that due to (3.19) the term containing ǫ DBCA vanishes. Therefore
{R 1 (b), R 2 (b ′ )} − {R 1 (b ′ ), R 2 (b)} = Σ * (b ∧ b ′ ) ∧ ξ A dθ A , where we applied (3.16). Because {R 2 (b), R 2 (b ′ )} = 0 we finally obtain {R(b), R(b ′ )} = Σ * (b ∧ b ′ ) ∧ (θ A ∧ * p A − ξ A dθ A ) = R( * (b ∧ b ′ )).{B(a), R(b)} = {B 1 (a), R 1 (b)} + {B 2 (a), R 1 (b)} − {B 1 (a), R 2 (b)}− − {B 2 (a), R 2 (b)}. (3.31)
Using (3.17) we immediately obtain
{B 1 (a), R 1 (b)} = Σ −a ∧ * dθ A ∧ * (b ∧ θ A ) + d * (a ∧ θ A ) ∧ * (b ∧ θ A ) = = Σ * (a ∧ b) ∧ θ A ∧ * dθ A − b ∧ * dθ A ∧ * (a ∧ θ A ) + d * (a ∧ θ A ) ∧ * (b ∧ θ A ),
where we transformed the first term by means of (3.16). Three terms constituting the next bracket in (3.31) vanish by virtue of (3.19), (3.17) and (3.3) and consequently
{B 2 (a), R 1 (b)} = Σ b ∧ * p A ∧ aξ A = Σ * (a ∧ b) ∧ ξ A p A .
The bracket {B 1 (a), R 2 (b)} is obviously zero and
− {B 2 (a), R 2 (b)} = Σ [(d(bξ A )) − 1 2 ǫ DBCA θ B ∧ θ C * (b ∧ dθ D )] ∧ aξ A = = Σ −db∧a+ * θ D ∧a∧ * (b∧dθ D ) = Σ −db∧a+ * a∧θ D ∧ * (db∧θ D )−θ D ∧ * a∧ * d(b∧θ D ) = = Σ −db ∧ a + * a ∧ * db + d * (θ D ∧ * a) ∧ b ∧ θ D = Σ * d * (θ D ∧ * a) ∧ * (b ∧ θ D ),
where in the first step we used (3.3), in the second one we applied (3.20) and in the fourth step (3.18). Gathering the partial results we obtain
{B(a), R(b)} = Σ * (a ∧ b)(θ A ∧ * dθ A + ξ A p A ) − b ∧ * dθ A ∧ * (a ∧ θ A )+ + [d * (a ∧ θ A ) + * d * (θ A ∧ * a)] ∧ * (b ∧ θ A ) = B( * (a ∧ b))+ + Σ −b ∧ * dθ A ∧ * (a ∧ θ A ) + [ * * d * (a ∧ θ A ) + * d * (θ A ∧ * a)] ∧ * (b ∧ θ A ). (3.32)
Let us show now that the integrand in the last line of (3.32) is zero. The first term in the integrand can be expressed as follows:
− b ∧ * dθ A ∧ * (a ∧ θ A ) = −[b ∧ * d * ( * θ A )] ∧ * [a ∧ θ A ] = − 1 2 [b ∧ * d * ( * θ A )] lj [a ∧ θ A ] lj ǫ = b l (∇ k θ i A )ǫ ijk (a j θ Al − a l θ Aj )ǫ = δ l n ǫ ijk b n (∇ k θ i A )(a j θ A l − a l θ Aj )ǫ
-the second equality holds by virtue of (3.15) and the third one due to (3.13) and (3.11). Using (3.14) we rewrite δ l n by means of "epsilons" and continue transformations
− b ∧ * dθ A ∧ * (a ∧ θ A ) = 1 2 ǫ nab ǫ lab ǫ ijk b n (∇ k θ i A )(a j θ A l − a l θ Aj )ǫ = = ǫ nab (δ l i δ a j δ b k + δ a i δ b j δ l k + δ b i δ l j δ a k )b n (∇ k θ i A )(a j θ A l − a l θ Aj )ǫ = = −a i b n ǫ njk (∇ k θ i A )θ Aj ǫ + a j b n ǫ nij (∇ k θ i A )θ A k ǫ − a k b n ǫ nij (∇ k θ i A )θ Aj ǫ = = a j b n ǫ jni (∇ k θ i A )θ A k ǫ + a k b n ǫ nij θ Aj (∇ i θ k A − ∇ k θ i A )ǫ
-here in the second step we applied (3.14) to express ǫ lab ǫ ijk by means of "deltas". On the other hand the second term in the integrand
[ * * d * (a ∧ θ A ) + * d * (θ A ∧ * a)] ∧ * (b ∧ θ A ) = = [∇ j (a i θ Aj − a j θ Ai ) + ∇ i (θ Aj a j )]ǫ ikl b k θ l A ǫ = = a i b k ǫ ikl θ l A (∇ j θ A j )ǫ + a j b k ǫ ikl θ Al (∇ i θ j A − ∇ j θ i A )ǫ
-the second equality holds by virtue of (3.15) and (3.11). Thus the sum of the last two terms in (3.32) is equal to
a i b k ǫ ikl [θ A j (∇ j θ l A ) + θ l A (∇ j θ A j )]ǫ = 0,
where we used (3.9). Finally Following [9] we split the constraint S(M ) given by (2.7) into three functionals
S 1 (M ) := Σ M [ 1 2 (p A ∧ θ B ) ∧ * (p B ∧ θ A ) − 1 4 (p A ∧ θ A ) ∧ * (p B ∧ θ B )] S 2 (M ) := − Σ M ξ A dp A , S 3 (M ) := Σ M [ 1 2 (dθ A ∧ θ B ) ∧ * (dθ B ∧ θ A ) − 1 4 (dθ A ∧ θ A ) ∧ * (dθ B ∧ θ B )].
(3.34)
Then
{S(M ), S(M ′ )} = {S 1 (M ), S 1 (M ′ )} + {S 2 (M ), S 2 (M ′ )} + {S 3 (M ), S 3 (M ′ )}+ {S 1 (M ), S 2 (M ′ )} + {S 2 (M ), S 3 (M ′ )} + {S 3 (M ), S 1 (M ′ )} − (M ↔ M ′ ) . (3.35) 3.3.1 Poisson brackets {S i (M ), S j (M ′ )}
Functional derivatives of the functionals read:
δS 1 (M ) δθ A =M p B * (p A ∧ θ B ) − 1 2 p A * (p B ∧ θ B )+ + 1 2 (p C ∧ θ B ) ∧ * ′ A (p B ∧ θ C ) − 1 4 (p B ∧ θ B ) ∧ * ′ A (p C ∧ θ C ) , (3.36) δS 1 (M ) δp A =M θ B * (p B ∧ θ A ) − 1 2 θ A * (p B ∧ θ B ) , (3.37) δS 2 (M ) δθ A =M 1 2 ǫ D BCA θ B ∧ θ C * dp D − ( * ξ B ) ∧ * ′ A dp B , (3.38) δS 2 (M ) δp A =d(M ξ A ) = ξ A dM + M dξ A , (3.39) δS 3 (M ) δθ A =d M θ B * (dθ B ∧ θ A ) − M 2 θ A * (dθ B ∧ θ B ) + M dθ B * (dθ A ∧ θ B )− − 1 2 dθ A * (dθ B ∧ θ B ) + 1 2 (dθ C ∧ θ B ) ∧ * ′ A (dθ B ∧ θ C )− − 1 4 (dθ B ∧ θ B ) ∧ * ′ A (dθ C ∧ θ C ) , (3.40) δS 3 (M ) δp A =0.{S 1 (M ), S 1 (M ′ )} = Σ δS 1 (M ) δθ A ∧ δS 1 (M ′ ) δp A − (M ↔ M ′ ) = 0
because the first term under the integral is symmetric in M and M ′ . It was shown in [9] that
{S 2 (M ), S 2 (M ′ )} = − Σ ( m θ A ) ∧ dp A , (3.42) where m := M dM ′ − M ′ dM.
Because S 3 (M ) does not depend on the momenta
{S 3 (M ), S 3 (M ′ )} = 0.
Next,
{S 1 (M ), S 2 (M ′ )} − (M ↔ M ′ ) =
Thus we obtain an explicite expression for the r.h.s. of (3.35):
{S(M ), S(M ′ )} = Σ −( m θ A ) ∧ dp A + m ∧ p B * (ξ A p A ∧ θ B ) − 1 2 m ∧ ξ A p A * (p B ∧ θ B )+ + m ∧ dθ B * (ξ A dθ A ∧ θ B ) − 1 2 m ∧ ξ A dθ A * (dθ B ∧ θ B ) − m ∧ θ B ∧ θ A * (dθ B ∧ * p A )− − 1 2 m ∧ θ A ∧ * p A * (dθ B ∧ θ B ) + 1 2 m ∧ θ B ∧ * dθ B * (p A ∧ θ A ). (3.46)
Isolating constraints
Our goal now is to isolate constraints at the r.h.s. of (3.46), that is, to show that the r.h.s. of (3.46) is a sum of the constraints (2.5)-(2.8) smeared with appropriately chosen fields.
It is clear (see (2.8)) that the first term at the r.h.s. of (3.46)
Σ −( m θ A ) ∧ dp A = V ( m) + Σ dθ A ∧ m p A .
The second and the third terms are equal to
Σ [θ B * (m ∧ p B ) − 1 2 m * (p B ∧ θ B )] ∧ ξ A p A = B θ B * (m ∧ p B ) − 1 2 m * (p B ∧ θ B ) + + Σ − * (m ∧ p B )θ B ∧ θ A ∧ * dθ A + 1 2 * (p B ∧ θ B )m ∧ θ A ∧ * dθ A .
Similarly, the fourth and the fifth ones are equal to
Σ [θ B * (m∧dθ B )− 1 2 m * (dθ B ∧θ B )]∧ξ A dθ A = −R θ B * (m∧dθ B )− 1 2 m * (dθ B ∧θ B ) + + Σ * (m ∧ dθ B )θ B ∧ θ A ∧ * p A − 1 2 * (dθ B ∧ θ B )m ∧ θ A ∧ * p A .
Thus (3.46) can be rewritten as follows:
{S(M ), S(M ′ )} = V ( m) + B θ B * (m ∧ p B ) − 1 2 m * (p B ∧ θ B ) − − R θ B * (m ∧ dθ B ) − 1 2 m * (dθ B ∧ θ B ) + remaining terms, (3.47)
where the remaining terms read
Σ − * (m ∧ p B )θ B ∧ θ A ∧ * dθ A + * (m ∧ dθ B )θ B ∧ θ A ∧ * p A + + * (p B ∧ θ B )m ∧ θ A ∧ * dθ A − * (dθ B ∧ θ B )m ∧ θ A ∧ * p A − − m ∧ θ B ∧ θ A * (dθ B ∧ * p A ) + dθ A ∧ m p A . (3.48)
Now we will transform the remaining terms (3.48) to a form which will be a convenient starting point for isolating constraints. The first term in (3.48)
− * (m ∧ p B )θ B ∧ θ A ∧ * dθ A = −m ∧ p B * (θ B ∧ θ A ∧ * dθ A ) = = −m ∧ p B θ B * (θ A ∧ * dθ A ) = − θ B (m ∧ p B ) ∧ * (θ A ∧ * dθ A ) = = −( θ B m)p B ∧ * (θ A ∧ * dθ A ) + m ∧ * ( * p B ∧ θ B ) ∧ * (θ A ∧ * dθ A ) = = −( θ B m)θ A ∧ * dθ A ∧ * p B + m ∧ * (θ A ∧ * dθ A ) ∧ * (θ B ∧ * p B ), (3.49)
where we used (3.6) in the second step and (3.5) in the fourth one. Similarly, the second term in (3.48
) * (m ∧ dθ B )θ B ∧ θ A ∧ * p A = = ( θ B m) ∧ θ A ∧ * p A ∧ * dθ B − m ∧ * (θ A ∧ * p A ) ∧ * (θ B ∧ * dθ B ). (3.50)
Next, we transform the third term in (3.48):
m ∧ θ A ∧ * dθ A * (p B ∧ θ B ) = m ∧ θ A ∧ * dθ A θ B * p B = θ B (m ∧ θ A ∧ * dθ A ) ∧ * p B = = ( θ B m)θ A ∧ * dθ A ∧ * p B − m ∧ ( θ B θ A ) * dθ A ∧ * p B + + m ∧ θ A ( θ B * dθ A ) ∧ * p B = ( θ B m) ∧ θ A ∧ * dθ A ∧ * p B − − m ∧ * dθ A ∧ * p A − m ∧ * (ξ A dθ A ) ∧ * (ξ B p B ) + m ∧ θ A ∧ * p B * (dθ A ∧ θ B ), (3.51)
where in the first step we used (3.6), in the fourth one (3.7) and in the last one (3.6) again. Similarly, the fourth term in (3.48)
− m ∧ θ A ∧ * p A * (dθ B ∧ θ B ) = −( θ B m)θ A ∧ * p A ∧ * dθ B + + m ∧ ( θ B θ A ) * p A ∧ * dθ B − m ∧ θ A ∧ * dθ B * (p A ∧ θ B ). (3.52)
By virtue of (3.5) the last term in (3.48) can be expresses as follows [9]:
dθ A ∧ m p A = dθ A ∧ * ( * p A ∧ m) = * p A ∧ m ∧ * dθ A = m ∧ * dθ A ∧ * p A . (3.53)
Now it is easy to see that the following pairs 1. the first term at the r.h.s. of (3.49) and the first term at the r.h.s. of (3.51), 2. the first term at the r.h.s. of (3.50) and the first term at the r.h.s. of (3.52),
3. the second term at the r.h.s. of (3.51) and (3.53) sum up to zero. Moreover, the second term at the r.h.s. of (3.49) is equal to the second term at the r.h.s. of (3.50). Consequently, the terms (3.48) can be expresses as
Σ ( θ B θ A )m ∧ * p A ∧ * dθ B − m ∧ θ A ∧ * dθ B * (p A ∧ θ B ) + m ∧ θ A ∧ * p B * (dθ A ∧ θ B )+ +m∧ * (ξ A p A )∧ * (ξ B dθ B )−m∧θ B ∧θ A * (dθ B ∧ * p A )+2m∧ * (θ A ∧ * dθ A )∧ * (θ B ∧ * p B ). (3.54)
Note now that the term above containing ξ A can be transformed as follows:
Σ m∧ * (ξ A p A )∧ * (ξ B dθ B ) = Σ − * [m∧ * (ξ B dθ B )]∧(ξ A p A ) = −B * (m∧ξ B * dθ B ) + + Σ * [m ∧ * (ξ B dθ B )] ∧ θ A ∧ * dθ A = −B * (m ∧ ξ B * dθ B ) − − Σ * [m ∧ * (θ A ∧ * dθ A )] ∧ ξ B dθ B = −B * (m ∧ ξ B * dθ B ) + R * [m ∧ * (θ A ∧ * dθ A )] − − Σ m ∧ * (θ A ∧ * dθ A ) ∧ * (θ B ∧ * p B ). (3.55)
Gathering the result above, (3.54) and (3.47) we obtain
{S(M ), S(M ′ )} = V ( m) + B θ B * (m ∧ p B ) − 1 2 m * (p B ∧ θ B ) − − R θ B * (m ∧ dθ B ) − 1 2 m * (dθ B ∧ θ B ) − − B * (m ∧ ξ B * dθ B ) + R * [m ∧ * (θ A ∧ * dθ A )] + remaining terms, (3.56)
where the remaining terms read now
Σ ( θ B θ A )m ∧ * p A ∧ * dθ B − m ∧ θ A ∧ * dθ B * (p A ∧ θ B ) + m ∧ θ A ∧ * p B * (dθ A ∧ θ B )− − m ∧ θ B ∧ θ A * (dθ B ∧ * p A ) + m ∧ * (θ A ∧ * dθ A ) ∧ * (θ B ∧ * p B ). (3.57)
Now let us show that the remaining terms (3.57) can be expressed as R(b) with the one-form b being a complicated function of the canonical variables and m. By shifting the contraction θ B in the first term above and using (3.6) one can easily show that the sum of the first and the second terms in (3.57) reads
Σ ( θ B θ A )m ∧ * p A ∧ * dθ B − m ∧ θ A ∧ * dθ B * (p A ∧ θ B ) = = Σ θ B (m ∧ * dθ B ) ∧ θ A ∧ * p A = R θ B (m ∧ * dθ B ) + Σ θ B (m ∧ * dθ B ) ∧ ξ A dθ A . (3.58)
Let us now express the third term in (3.57) by means of the components of the canonical variables and the covariant derivative ∇ a (see (3.12)):
m ∧ θ A ∧ * p B * (dθ A ∧ θ B ) = m ∧ 1 2 θ A i p Bjk ǫ jk l dx i ∧ dx l (∇ a θ Ab )θ B c ǫ abc = = m ∧ θ A i p B ab dx i ∧ dx c (∇ a θ Ab )θ B c + m ∧ θ A i p B bc dx i ∧ dx a (∇ a θ Ab )θ B c + + m ∧ θ A i p B ca dx i ∧ dx b (∇ a θ Ab )θ B c ,
where in the last step we used (3.14). Similarly, the fourth term in (3.57)
−m ∧ θ B ∧ θ A * (dθ B ∧ * p A ) = −m ∧ θ B i θ A c dx i ∧ dx c (∇ a θ Bb )p A ab
and consequently the sum of the third and the fourth terms in (3.57) is of the following form:
m ∧ θ A ∧ * p B * (dθ A ∧ θ B ) − m ∧ θ B ∧ θ A * (dθ B ∧ * p A ) = = m ∧ θ A i p B bc dx i ∧ dx a (∇ a θ Ab )θ B c + m ∧ θ A i p B ca dx i ∧ dx b (∇ a θ Ab )θ B c = = −m ∧ (θ Bc p Bc b )(dθ A ) ba dx a ∧ θ A = −m ∧ [ − −−−− → ( θ B p B ) dθ A ] ∧ θ A = = −m ∧ * [ * dθ A ∧ * ( * p B ∧ θ B )] ∧ θ A = − * [ * (m ∧ θ A ) ∧ * dθ A ] ∧ θ B ∧ * p B
-here in the fourth step we used (3.5). Integrating the equation above over Σ we obtain:
Σ m ∧ θ A ∧ * p B * (dθ A ∧ θ B ) − m ∧ θ B ∧ θ A * (dθ B ∧ * p A ) = = R − * [ * (m ∧ θ B ) ∧ * dθ B ] + Σ − * [ * (m ∧ θ B ) ∧ * dθ B ] ∧ ξ A dθ A . (3.59)
Finally, the last term in (3.57)
Σ m ∧ * (θ B ∧ * dθ B ) ∧ * (θ A ∧ * p A ) = Σ * [m ∧ * (θ B ∧ * dθ B )] ∧ θ A ∧ * p A = = R * [m ∧ * (θ B ∧ * dθ B )] + Σ * [m ∧ * (θ B ∧ * dθ B )] ∧ ξ A dθ A . (3.60)
Gathering the three results (3.58), (3.59) and (3.60) we conclude that the terms (3.57) can be expressed as
R θ B (m ∧ * dθ B ) − * [ * (m ∧ θ B ) ∧ * dθ B ] + * [m ∧ * (θ B ∧ * dθ B )] + + terms independent of p A , (3.61) where the terms independent of p A read Σ θ B (m∧ * dθ B )∧ξ A dθ A − * [ * (m∧θ B )∧ * dθ B ]∧ξ A dθ A + * [m∧ * (θ B ∧ * dθ B )]∧ξ A dθ A . (3.62)
Now it is enough to show that the terms (3.62) sums up to zero for all m and θ A . To this end let us isolate the factor mξ A in each term of (3.62)-using (3.6) and (3.5) we obtain
Σ −mξ A ∧ * dθ B ∧ θ B dθ A − ( θ B * m) ∧ * dθ B ∧ ξ A * dθ A − m ∧ θ B dθ B ∧ ξ A * dθ A = = Σ mξ A ∧ − * dθ B ∧ θ B dθ A + * ( θ B ( * dθ B ∧ * dθ A )) − θ B dθ B ∧ * dθ A .
Consider now the terms in the square bracket above: Setting this result to (3.56) we obtain
− * dθ B ∧ θ B dθ A + * ( θ B ( * dθ B ∧ * dθ A )) − θ B dθ B ∧ * dθ A = = − * dθ B ∧ θ B dθ A + ( θ B * dθ B )dθ A − dθ B θ B * dθ A − θ B dθ B ∧ * dθ A = = θ B ( * dθ B ∧ dθ A ) − θ B (dθ B ∧ * dθ A ) = θ B ( * dθ B ∧ dθ A ) − θ B ( * dθ B ∧ dθ A ) = 0.{S(M ), S(M ′ )} = V ( m)+ + B θ B * (m ∧ p B ) − 1 2 m * (p B ∧ θ B ) − R θ B * (m ∧ dθ B ) − 1 2 m * (dθ B ∧ θ B ) − − B * (m ∧ ξ B * dθ B ) + R * [m ∧ * (θ A ∧ * dθ A )] + + R θ B (m ∧ * dθ B ) − * [ * (m ∧ θ B ) ∧ * dθ B ] + * [m ∧ * (θ B ∧ * dθ B )] (3.64) 3.3.3 Another form of {S(M ), S(M ′ )}
Let us now transform the result to a form in which both constraints B given by (2.5) and R defined by (2.6) appear on an equal footing. Consider the following transformation:
p A → −dθ A , dθ A → p A . (3.65)
It is easy to see that under this transformation
B(a(p A , dθ B )) → R(a(−dθ A , p B )), R(b(p A , dθ B )) → −B(b(−dθ A , p B ))
Let us now express the formula (3.55) in the following form:
Σ m ∧ * (ξ A p A ) ∧ * (ξ B dθ B ) + m ∧ * (θ A ∧ * dθ A ) ∧ * (θ B ∧ * p B ) = = −B * (m ∧ ξ B * dθ B ) + R * [m ∧ * (θ A ∧ * dθ A )] .
Note now that the l.h.s. of the identity above is invariant with respect to the transformation (3.65). Consequently, the r.h.s. has to be invariant too. Thus the fourth and the fifth terms in (3.64)
− B * (m ∧ ξ B * dθ B ) + R * [m ∧ * (θ A ∧ * dθ A )] = −R * (m ∧ ξ B * p B ) − − B * [m ∧ * (θ A ∧ * p A )] = − 1 2 B * (m ∧ ξ B * dθ B ) + * [m ∧ * (θ A ∧ * p A )] + + 1 2 R − * (m ∧ ξ B * p B ) + * [m ∧ * (θ A ∧ * dθ A )] , (3.66)
where the last equation holds by virtue of the following trivial fact: if x = y then x = 1 2 (x + y). Similarly, the expression (3.57) is also invariant with respect to (3.65). Thus by virtue of the identity (3.63) the last term in (3.64)
R θ B (m ∧ * dθ B ) − * [ * (m ∧ θ B ) ∧ * dθ B ] + * [m ∧ * (θ B ∧ * dθ B )] = = −B θ B (m ∧ * p B ) − * [ * (m ∧ θ B ) ∧ * p B ] + * [m ∧ * (θ B ∧ * p B )] = = 1 2 R θ B (m ∧ * dθ B ) − * [ * (m ∧ θ B ) ∧ * dθ B ] + * [m ∧ * (θ B ∧ * dθ B )] − − 1 2 B θ B (m ∧ * p B ) − * [ * (m ∧ θ B ) ∧ * p B ] + * [m ∧ * (θ B ∧ * p B )] .′ )} = V ( m) + B θ B * (m ∧ p B ) − 1 2 * (m ∧ ξ B * dθ B )− − * [m ∧ * (θ B ∧ * p B )] − 1 2 * ( * m ∧ θ B ) * p B + 1 2 * [ * (m ∧ θ B ) ∧ * p B ] + + R − θ B * (m ∧ dθ B ) − 1 2 * (m ∧ ξ B * p B )+ + * [m ∧ * (θ B ∧ * dθ B )] + 1 2 * ( * m ∧ θ B ) * dθ B − 1 2 * [ * (m ∧ θ B ) ∧ * dθ B ] . (3.68)
Note that the sum of the constraints B and R at the r.h.s. of (3.68) is explicitely invariant with respect to the transformation (3.65).
Poisson bracket of R(b) and S(M)
Recall that the constraints R(b) and S(M ) are defined by, respectively, (2.6) and (2.7).
To show that the bracket {R(b), S(M )} is a sum of the constraints (2.5)-(2.8) smeared with some fields we will proceed according to the following prescription. The bracket under consideration can be expressed as
{R(b), S(M )} = 2 i=1 3 j=1 (−1) i+1 {R i (b), S j (M )},
where the functionals at the r.h.s. are given by (3.27) and (3.34). It is not difficult to see that each bracket in the sum is either quadratic in p A , linear in p A or independent of p A . So we will first calculate the brackets and then we will gather similar terms according to the classification. Next, it will turn out that the terms quadratic in p A can be re-expressed as a constraint plus a term linear in p A . Then it will turn out that all the linear terms can be re-expressed as some constraints plus a term independent of p A . Finally we will show that all the term independent of p A sum up to zero. Except the prescription we will need some formulae and identities which will make easier the calculations.
Auxiliary formulae
The following formulae will be used in the sequel while calculating both {R(b), S(M ))} and {B(a), S(M )}:
θ A * ξ B = − 1 2 ǫ B CDA θ C ∧ θ D + ξ A * θ B , (3.69) − 1 2 ǫ A BCD θ B ∧ θ C ∧ α = −ξ D α ∧ * θ A + ( * ξ A ) θ D α, (3.70) (α ∧ * ′ A β) ∧ γ = α ∧ * β θ A γ − [( θ A α) ∧ * β + (α ↔ β)] ∧ γ = = [(−1) k α ∧ ( θ A * β) − ( θ A β) ∧ * α] ∧ γ, (3.71) (α ∧ * ′ A β) ∧ δS 1 (M ) δp A = M (k − 5 2 )α ∧ * β * (p A ∧ θ A )+ + (−1) 4−k M [ * (α ∧ θ A ) ∧ * p A ∧ β + (α ↔ β)], (3.72) α A ∧ β A = θ A ∧ θ B α A ∧ β B + (−1) k θ A ∧ α A ∧ θ B β B − − ξ A α A ∧ ξ B β B , (3.73) d( * ( * θ A ∧ θ B )) = ξ B dξ A + ξ A dξ B , (3.74)
where in (3.70) α is a one-form, and in (3.71) α and β are k-forms and γ a one-form, in (3.72) α and β are k-forms and finally in (
3.73) α A is a k-form, while β A is a (3−k)-form.
Moreover, we will apply the following two identities:
α ∧ * ( * γ ∧ β) − * α * (β ∧ γ) − β ∧ * ( * γ ∧ α) + * β * (α ∧ γ) = 0, (3.75) * [α ∧ * (κ ∧ β)] − α * (β ∧ * κ) − * [β ∧ * (κ ∧ α)] + β * (α ∧ * κ) = 0, (3.76)
where α, β and κ are one-forms, and γ is a two-form. Note that if in (3.71) α and β are three-forms then the formula can be simplified further. Indeed, in this case * α and * β are zero-forms thus
( θ A α) ∧ * β ∧ γ = α ∧ * β θ A γ
and setting this equality to the r.h.s. of the first line of (3.71) we obtain
(α ∧ * ′ A β) ∧ γ = −α ∧ * β θ A γ. (3.77)
Using this result we can also simplify (3.72)-if α and β are three-forms then setting to
(3.77) γ = (δS 1 (M ))/(δp A ) we obtain (α ∧ * ′ A β) ∧ δS 1 (M ) δp A = M 2 α ∧ * β * (p A ∧ θ A ). (3.78)
Proof of (3.69). Recall that the functions ξ B are given by the formula (2.2). Using it we obtain
θ A * ξ B = − 1 2 ǫ B CDE ( θ A θ C )θ D ∧ θ E = − 1 2 ǫ B ADE θ D ∧ θ E − 1 2 ǫ B CDE ξ A ξ C θ D ∧ θ E = = − 1 2 ǫ B CDA θ C ∧ θ D + ξ A * θ B ,
where in the second step we used (3.7), and in the last one (3.20).
Proof of (3.70). By virtue of (3.7) and (3.20) the l.h.s. of (3.70) can be transformed as follows:
− 1 2 ǫ A BCD θ B ∧ θ C ∧ α = − 1 2 ǫ A BCE θ B ∧ θ C ∧ ( θ D θ E − ξ D ξ E )α = = − 1 2 ǫ A BCE θ B ∧ θ C ∧ ( θ D θ E )α − ( * θ A ) ∧ ξ D α.
Shifting the contraction θ D in the first of the two resulting terms and applying once again (3.7) and (3.20) we obtain
− 1 2 ǫ A BCD θ B ∧ θ C ∧ α = ǫ A DCE θ C ∧ θ E ∧ α − 1 2 ǫ A BCE θ B ∧ θ C ∧ θ E ( θ D α)− − 3( * θ A ) ∧ ξ D α.
Now to justify (3.70) it is enough to note that (i) the first term on the r.h.s of the equation above is proportional to the term on the l.h.s. and (ii) the second term on the r.h.s. by virtue of (2.2) is equal to 3( * ξ A ) θ D α.
Proof of (3.71). Let us now consider the l.h.s. of (3.71):
(α * ′ A β) ∧ γ = θ B η AB α ∧ * β − [( θ A α) ∧ * ( θ B β) + (α ↔ β)] ∧ γ = = α ∧ * β( θ A γ) − [( θ A α) ∧ * ( θ B β) + (α ↔ β)] θ B γ.
By virtue of (3.5) and (3.
8) * ( θ B β) θ B γ = * β ∧ θ B ∧ θ B γ = * β ∧ γ.
Setting this result to the r.h.s. of the previous equation we obtain the r.h.s. of the first line of (3.71). To obtain the result at the second line it is enough to shift the contraction θ A in the term −( θ A α) ∧ * β ∧ γ.
Proof of (3.72). By virtue of the first line of Equation (3.71) just proven
(α ∧ * ′ A β) ∧ δS 1 (M ) δp A = α ∧ * β θ A δS 1 (M ) δp A − [( θ A α) ∧ * β + (α ↔ β)] ∧ δS 1 (M ) δp A = = − M 2 α∧ * β * (p A ∧θ A )−M [( θ A α)∧ * β+(α ↔ β)]∧ θ B * (p B ∧θ A )− 1 2 θ A * (p B ∧θ B ) . (3.79) Because * (p B ∧ θ A ) is a zero-form (i.e. a function) ( θ A α) * (p B ∧ θ A ) = * ( * α ∧ θ A ) * (p B ∧ θ A ) = * ( * α ∧ θ A ∧ * (p B ∧ θ A )) = * ( * α ∧ * p B ),
where the first equality holds true by virtue of (3.5) and the last one-due to (3.18). On the other hand due to (3.8)
( θ A α) ∧ * β ∧ θ A = θ A ∧ ( θ A α) ∧ * β = kα ∧ * β.
Setting the two results above to (3.79) we get
(α ∧ * ′ A β) ∧ δS 1 (M ) δp A = M (k − 1 2 )α ∧ * β * (p A ∧ θ A )− − M [ * ( * α ∧ * p A ) ∧ * β ∧ θ A + (α ↔ β)]. (3.80)
The last step of the proof aims at simplifying the last term in the equation above:
* ( * α ∧ * p A ) ∧ * β ∧ θ A = * α ∧ * p A ∧ * ( * β ∧ θ A ) = * α ∧ * p A ∧ θ A β = = (−1) 3−k θ A ( * α ∧ * p A ) ∧ β = (−1) 3−k * (α ∧ θ A ) ∧ * p A ∧ β + * α ∧ β * (p A ∧ θ A )
(here we used (3.5) and (3.6)). Taking into account that * α ∧ β = α ∧ * β we set the result above to (3.80) obtaining thereby (3.72).
Proof of (3.73). By virtue of (3.7)
α A ∧ β A = ( θ B θ A − ξ B ξ A )α A ∧ β B = θ B θ A α A ∧ β B − ξ A α A ∧ ξ B β B .
Now to get (3.73) it is enough to shift the contraction θ B in the first term at the r.h.s. of the equation above.
Proof of (3.74). First transform the l.h.s. of (3.7) by means of (3.5) then act on the both sides of the resulting formula by d.
Proof of (3.75). Note that β ∧ γ is a three form. Therefore * α *
(β ∧ γ) = * [ * (β ∧ γ) ∧ α] = α (β ∧ γ) = α β ∧ γ − β ∧ α γ,
where in the second step we used (3.5). Transforming similarly the term * β * (α ∧ γ) we obtain
α ∧ * ( * γ ∧ β) − * α * (β ∧ γ) − β ∧ * ( * γ ∧ α) + * β * (α ∧ γ) = = α ∧ β γ − α β ∧ γ + β ∧ α γ − β ∧ α γ + β α ∧ γ − α ∧ β γ = 0.
Proof of (3.76). Act by the Hodge operator * on both sides of (3.75) and set κ = * γ.
Terms quadratic in p A
Terms quadratic in the momenta come form the Poisson bracket
{R 1 (b), S 1 (M )} = Σ [−b ∧ * p A + (b ∧ θ B ) ∧ * ′ A p B ] ∧ δS 1 (M ) δp A − − M p B * (p A ∧ θ B ) − 1 2 p A * (p B ∧ θ B ) ∧ * (b ∧ θ A ), (3.81)
where we used (3.17) to simplify the r.h.s. It is not difficult to see that
−b ∧ * p A ∧ δS 1 (M ) δp A − M p B * (p A ∧ θ B ) − 1 2 p A * (p B ∧ θ B ) ∧ * (b ∧ θ A ) = 0.
Let us now transform the remaining term in (3.81)-by virtue of (3.72)
[(b ∧ θ B ) ∧ * ′ A p B ] ∧ δS 1 (M ) δp A = − M 2 b ∧ θ B ∧ * p B * (p A ∧ θ A )+ + M [ * (b ∧ θ B ∧ θ A ) ∧ * p A ∧ p B + * (p B ∧ θ A ) ∧ * p A ∧ b ∧ θ B ] = = − M 2 b ∧ θ B ∧ * p B * (p A ∧ θ A ) + M * (p B ∧ θ A ) * p A ∧ b ∧ θ B .
To justify the last step let us note that (b ∧ θ B ∧ θ A ) is antisymmetric in A and B while * p A ∧ p B is symmetric. Consequently,
{R 1 (b), S 1 (M )} = Σ − M 2 b ∧ θ B ∧ * p B * (p A ∧ θ A ) + M * (p B ∧ θ A )b ∧ θ B ∧ * p A .δS 2 (M ) δθ A ∧ δR 1 (b) δp A = 0, thus {R 1 (b), S 2 (M )} = Σ [−b ∧ * p A + (b ∧ θ B ) ∧ * ′ A p B ] ∧ d(M ξ A ) = Σ −dM ∧ b ∧ * (ξ A p A )− − M b ∧ * p A ∧ dξ A + M b ∧ θ B ∧ dξ A θ A * p B − M θ A p B ∧ * (b ∧ θ B ) ∧ dξ A , (3.83)
where we have used the second line of (3.71) and (3.3). The other bracket,
{R 2 (b), S 1 (M )} = Σ [d(bξ A )+(b∧dθ B )∧ * ′ A ( * ξ B )− 1 2 ǫ DBCA θ B ∧θ C * (b∧dθ D )]∧ δS 1 (M ) δp A .
(3.84) The last two terms in the square bracket above give together zero once multiplied by (δS 1 (M ))/(δp A ). Indeed, due to (3.78)
[(b ∧ dθ B ) ∧ * ′ A ( * ξ B )] ∧ δS 1 (M ) δp A = M 2 (b ∧ ξ B dθ B ) * (p A ∧ θ A ).
On the other hand,
− 1 2 ǫ DBCA θ B ∧ θ C ∧ δS 1 (M ) δp A * (b ∧ dθ D ) = * ξ D * (b ∧ dθ D ) θ A δS 1 (M ) δp A = = − M 2 (b ∧ ξ B dθ B ) * (p A ∧ θ A )
-the first equality holds by virtue of (3.70) and due to a fact that ξ A (δS 1 (M ))/(δp A ) = 0 ((δS 1 (M ))/(δp A ) is of the form θ A i γ i j dx j form some tensor field γ i j ), in the last step we used (3.7). Using the fact just mentioned and (3.6) we transform the only remaining term in (3.84) as follows:
d(bξ A ) ∧ δS 1 (M ) δp A = −M b ∧ dξ A ∧ θ B * (p B ∧ θ A ) + M 2 b ∧ dξ A ∧ θ A * (p B ∧ θ B ) = = M b ∧ θ B ∧ dξ A θ A * p B − M 2 b ∧ θ A ∧ dξ A * (p B ∧ θ B ). (3.85)
Gathering all the terms linear in p A , that is, (3.83) and (3.85) we obtain being an exact three-form-the integral of this term over Σ is zero hence
{R 1 (b), S 2 (M )} − {R 2 (b), S 1 (M )} = Σ −dM ∧ b ∧ * (ξ A p A ) − M b ∧ * p A ∧ dξ A − − M θ A p B ∧ * (b ∧ θ B ) ∧ dξ A + M 2 b ∧ θ A ∧ dξ A * (p B ∧ θ B ).{R 2 (b), S 2 (M )} = Σ [(b ∧ dθ B ) ∧ * ′ A ( * ξ B ) − 1 2 ǫ DBCA θ B ∧ θ C * (b ∧ dθ D )] ∧ d(M ξ A ).
Applying (3.77), (3.20) and (3.70) it is not difficult to show that
{R 2 (b), S 2 (M )} = Σ dM ∧ * θ D * (b ∧ dθ D ).
Consider now the last bracket {R 1 (b), S 3 (M )}. Due to (3.17)
{R 1 (b), S 3 (M )} = − Σ d[M θ B * (dθ B ∧θ A )− M 2 θ A * (dθ B ∧θ B )]+M dθ B * (dθ A ∧θ B )− − M 2 dθ A * (dθ B ∧ θ B ) ∧ * (b ∧ θ A ).
Our goal now is to express the r.h.s. of the equation above as a sum of a term containing M b and a one containing dM . To this end we first act by the operator d on the factors constituting terms in the square brackets. Next, in those cases when it is possible, we use (3.18) to simplify θ A ∧ * (b ∧ θ A ) to 2 * b and θ A ∧ * (dθ B ∧ θ A ) to * dθ B , finally in all the terms containing M and * b we shift the Hodge operator * to get b. Thus we obtain
{R 1 (b), S 3 (M )} = − Σ M b ∧ [− * dθ A ∧ * dθ A + θ A ∧ * dθ B * (dθ A ∧ θ B )− − θ A ∧ * dθ A * (dθ B ∧ θ B ) − θ A ∧ * (θ B ∧ d * (dθ B ∧ θ A )) − * d * (dθ A ∧ θ A )]+ + dM ∧ [θ A ∧ * (b ∧ * dθ A ) − * b * (dθ A ∧ θ A )].
Note that since * dθ A is a one-form the first term at the r.h.s. above vanishes. Thus the terms independent of p A read
{R 1 (b), S 3 (M )} − {R 2 (b), S 2 (M )} = Σ the term containing M b − − Σ dM ∧ [θ B ∧ * (b ∧ * dθ B ) − * b * (dθ B ∧ θ B ) + * θ D * (b ∧ dθ D )].
The term containing dM can be expressed as
− Σ dM ∧ [− * b * (θ B ∧ dθ B ) − θ B ∧ * ( * dθ B ∧ b) + * θ B * (b ∧ dθ B )].
Now by setting in (3.75) α = b, β = θ B and γ = dθ B the term under consideration can be simplified to
− Σ dM ∧ [−b ∧ * ( * dθ B ∧ θ B )] = − Σ * (dM ∧ b) ∧ θ B ∧ * dθ B .
This means that the terms independent of p A read
{R 1 (b), S 3 (M )} − {R 2 (b), S 2 (M )} = − Σ M b ∧ [θ A ∧ * dθ B * (dθ A ∧ θ B )− − θ A ∧ * dθ A * (dθ B ∧ θ B ) − θ A ∧ * (θ B ∧ d * (dθ B ∧ θ A )) − * d * (dθ A ∧ θ A )]− − Σ * (dM ∧ b) ∧ θ A ∧ * dθ A . (3.87)
Isolating constraints
Our goal now is to express the bracket Terms quadratic in p A We are going to transform the last term of (3.82) to a form containing the factor θ A ∧ * p A being a part of the constraint R(b): applying (3.73) to the term with α A = * p A we obtain
M * (p B ∧ θ A ) ∧ b ∧ θ B ∧ * p A = * p A ∧ M * (p C ∧ θ A ) ∧ b ∧ θ C = = θ A ∧ θ B * p A ∧ M * (p C ∧ θ B ) ∧ b ∧ θ C − θ A ∧ * p A ∧ θ B M * (p C ∧ θ B ) ∧ b ∧ θ C . (3.88)
The first term in the last line is zero-indeed, using (3.6) we get
θ A ∧ θ B * p A ∧ M * (p C ∧ θ B ) ∧ b ∧ θ C = −M * (p A ∧ θ B ) * (p C ∧ θ B )θ A ∧ θ C ∧ b.
Note now that * (p A ∧θ B ) * (p C ∧θ B ) is symmetric in A and C, while θ A ∧θ C antisymmetric. Transforming the remaining term in (3.88) we obtain
M * (p B ∧ θ A ) ∧ b ∧ θ B ∧ * p A = = −M θ A ∧ * p A ∧ * (p C ∧ θ B θ B b) ∧ θ C − * (p C ∧ θ B )b θ B θ C = = −M θ A ∧ * p A ∧ * (p C ∧ b)θ C − * (p B ∧ θ B )b .
where in the last step we used (3.8) and (3.7). Setting this result to (3.82) we arrive at the terms (3.82) quadratic in where now the phrase "terms linear in p A " means the terms (3.86) and the last term in (3.89). Now we are going to isolate constraints from the linear terms.
p A = − Σ M [θ B * (p B ∧ b) − 1 2 b * (p B ∧ θ B )] ∧ θ A ∧ * p A = = −R M [θ B * (p B ∧b)− 1 2 b * (p B ∧θ B )] − Σ M [θ B * (p B ∧b)− 1 2 b * (p B ∧θ B )]∧θ A ∧dξ A .
Terms linear in p A The terms read
Σ −dM ∧ b ∧ * (ξ A p A ) + M b ∧ dξ A ∧ * p A − M θ A p B ∧ * (b ∧ θ B ) ∧ dξ A + + M b ∧ θ A ∧ dξ A * (p B ∧ θ B ) − M θ B ∧ θ A ∧ dξ A * (p B ∧ b). (3.91)
The first term can be written as
Σ −dM ∧b∧ * (ξ A p A ) = Σ − * (dM ∧b)∧ξ A p A = −B( * (dM ∧b))+ Σ * (dM ∧b)∧θ A ∧ * dθ A .
(3.92)
Transformation of the remaining terms in (3.91) (i.e. those which do not contain dM ) to an appropriate form takes more effort. Applying (3.73) to the first of them by
setting α A = b ∧ dξ A and β A = * p A we obtain M b ∧ dξ A ∧ * p A = M θ A ∧ θ B (b ∧ dξ A ) ∧ * p B + M θ A ∧ b ∧ dξ A θ B * p B = = M θ A ∧ dξ A ∧ * p B θ B b − M θ A ∧ b ∧ * p B θ B dξ A − M b ∧ θ A ∧ dξ A * (p B ∧ θ B )
-here in the last step we used (3.6). The last term of (3.91) can be transformed as follows
− M θ B ∧ θ A ∧ dξ A * (p B ∧ b) = −M θ A ∧ dξ A ∧ * * [ * (p B ∧ b) ∧ θ B ] = = −M θ A ∧dξ A ∧ * [ θ B (p B ∧b)] = −M θ A ∧dξ A ∧ * [( θ B p B )∧b]−M θ A ∧dξ A ∧ * p B θ B b,
where in the second step we used (3.5). The last two results allow us to express in a more simpler form the sum of the terms in (3.91) which do not contain dM :
−M θ A ∧b∧ * p B θ B dξ A −M θ A p B ∧ * (b∧θ B )∧dξ A −M θ A ∧dξ A ∧ * [( θ B p B )∧b]. (3.93)
Our goal now is to rewrite the sum above in a form of a single term containing the factor θ A ∧ * p A . Let us begin with the first term in (3.93):
−M θ A ∧ b ∧ * p B θ B dξ A = M * (b ∧ θ C ) θ A dξ C ∧ p A . Setting in (3.73) α A = * (b ∧ θ C ) θ A dξ C and β A = p A we obtain − M θ A ∧ b ∧ * p B θ B dξ A = = M θ A ∧ θ B * (b ∧ θ C ) θ A dξ C ∧ p B − M θ A * (b ∧ θ C ) θ A dξ C ∧ θ B p B = = M θ A ( θ A dξ C ) ∧ * (b ∧ θ C ∧ θ B ) ∧ p B − M θ A ( θ A dξ C ) ∧ * (b ∧ θ C ) ∧ * ( * p B ∧ θ B ) = = M dξ C ∧ * (b ∧ θ C ∧ θ B ) ∧ p B + M * [dξ C ∧ * (b ∧ θ C )] ∧ θ B ∧ * p B , (3.94)
where we applied (3.6) and (3.5) in the third step and (3.8) in the last step. The second term in (3.93)
− M θ A p B ∧ * (b ∧ θ B ) ∧ dξ A = M p B [ θ A * (b ∧ θ B )] ∧ dξ A − M p B ∧ * (b ∧ θ B ) θ A dξ A = = M p B * (b ∧ θ B ∧ θ A ) ∧ dξ A − M θ C dξ C * (b ∧ θ A ) ∧ p A = = −M dξ C * (b ∧ θ C ∧ θ B ) ∧ p B − M θ C dξ C b ∧ θ B ∧ * p B (3.95)
(in the second step we applied (3.6)). Finally, the last term in (3.93) by virtue of (3.5) can be written as
− M θ A ∧ dξ A ∧ * [( θ B p B ) ∧ b] = −M * ( * p B ∧ θ B ) ∧ b ∧ * (θ A ∧ dξ A ) = M * [b ∧ * (θ A ∧ dξ A )] ∧ θ B ∧ * p B .
Gathering (3.94), (3.95) and the equation above we obtain the desired expression for these terms in (3.91) which do not contain dM :
Σ M * [dξ C ∧ * (b ∧ θ C )] − θ C dξ C b + * [b ∧ * (θ A ∧ dξ A )] ∧ θ B ∧ * p B = = Σ M * [b ∧ * (θ A ∧ dξ A )] − b * (dξ A ∧ * θ A ) − * [dξ A ∧ * (θ A ∧ b)] ∧ θ B ∧ * p B ,
where again we used (3.5). We can now simplify the term in the big parenthesis -it is enough to use (3.76) setting α = b, β = dξ A and κ = θ A to get where the phrase "terms independent of p A " means here the terms given by (3.87), the last term in (3.92) and the last one in (3.96).
Σ M − dξ A * (b ∧ * θ A ) ∧ θ B ∧ * p B = −R M dξ A * (b ∧ * θ A ) − − Σ M * (b ∧ * θ A )dξ A ∧ ξ B dθ B ,
Terms independent of p A Our goal now is to show that the terms independent of p A sum up to zero. Gathering all the terms under consideration which appear in (3.97) we see that the sum of the last term of (3.87) and the last term of (3.92) is zero. Note now that the last term in (3.96) contains ξ A which does not appear in the others term.
To get rid of ξ A let us use (3.74):
− Σ M * (b ∧ * θ A )dξ A ∧ ξ B dθ B = − Σ M * (b ∧ * θ A )d * ( * θ A ∧ θ B ) ∧ dθ B = = − Σ M b ∧ * θ A * [d * ( * θ A ∧ θ B ) ∧ dθ B ].
Consequently, the terms in (3.97) independent of p A read
− Σ M b∧ θ A ∧ * dθ B * (dθ A ∧θ B )−θ A ∧ * dθ A * (dθ B ∧θ B )−θ A ∧ * (θ B ∧d * (dθ B ∧θ A ))− − * d * (dθ A ∧ θ A ) + * θ A * (d * ( * θ A ∧ θ B ) ∧ dθ B ) . (3.98)
Now we are going to show that the expression above is zero for every M, b and θ A . This will be achieved by proving that the terms in the big parenthesis sum up to zero for every θ A . The proof will be carried out with application of tensor calculus (see formulae in Section 3.1.3).
The first term in (3.98)
θ A ∧ * dθ B * (dθ A ∧ θ B ) = (∇ i θ Aj )(∇ a θ Bb )θ Bk θ A d ǫ abc ǫ ijk dx d ∧ dx c .
By virtue of the third equation in (3.14) we can express the r.h.s. above as a sum of six terms. Four of them vanish: two of them contain the vanishing factor (∇ a θ Bb )θ Bb (see (3.10)), the remaining two vanish because they are of the form
γ dc dx d ∧ dx c , γ dc = γ cd . (3.99) Thus θ A ∧ * dθ B * (dθ A ∧ θ B ) = (∇ b θ Ac )(∇ a θ Bb )θ Ba θ A d dx d ∧ dx c − − (∇ c θ Ab )(∇ a θ Bb )θ Ba θ A d dx d ∧ dx c . (3.100)
In the case of the second term in (3.98) we proceed similarly-applying (3.14) we obtain six terms and again four of them vanish: two of them are of the form (3.99), the other two can be transformed to this form by means of (3.9) hence
− θ A ∧ * dθ A * (dθ B ∧ θ B ) = −(∇ a θ b A )(∇ c θ B a )θ Bb θ A d dx d ∧ dx c + + (∇ a θ b A )(∇ c θ B b )θ Ba θ A d dx d ∧ dx c . (3.101)
Let us consider now the third term in (3.98):
− θ A ∧ * (θ B ∧ d * (dθ B ∧ θ A )) = −∇ b [(∇ i θ Bj )θ Ak ]θ Ba θ A d ǫ abc ǫ ijk dx d ∧ dx c = = −(∇ b ∇ i θ Bj )θ Ba q kd ǫ abc ǫ ijk dx d ∧ dx c − (∇ i θ Bj )(∇ b θ Ak )θ Ba θ A d ǫ abc ǫ ijk dx d ∧ dx c ,
where in the second step we used (2.3). Applying (3.14) we obtain twelve terms: six of them contain a second covariant derivative of components of θ B , while the remaining ones are quadratic in covariant derivatives of the components. Three terms of those containing second covariant derivatives vanish: two terms turn out to be of the form (3.99), the third one can be transformed to this form by means of (3.10):
− (∇ d ∇ c θ Ba )θ Ba dx d ∧ dx c = −∇ d [(∇ c θ Ba )θ Ba ] + (∇ c θ Ba )(∇ d θ Ba ) dx d ∧ dx c = = (∇ c θ Ba )(∇ d θ Ba ) dx d ∧ dx c .
Four terms of those quadratic in covariant derivatives are zero: two of them contain the factor (3.10), the other two can be transformed to the form (3.99) by means of (3.9). Finally (3.13) to express the fourth term in (3.98) as follows
− θ A ∧ * (θ B ∧ d * (dθ B ∧ θ A )) = −(∇ b ∇ b θ Bc )θ B d dx d ∧ dx c + (∇ d ∇ a θ Bc )θ Ba dx d ∧ dx c + + (∇ b ∇ c θ Bb )θ B d dx d ∧ dx c − (∇ a θ Bb )(∇ b θ Ac )θ Ba θ A d dx d ∧ dx c + + (∇ c θ Bb )(∇ b θ Aa )θ Ba θ A d dx d ∧ dx c . (3.102) Since dθ A ∧ θ A = 3!(∇ [d θ Ac )θ A b] dx d ⊗ dx c ⊗ dx b we can use− * d * (dθ A ∧ θ A ) = − 3! 2 ∇ b [(∇ [d θ Ac )θ A b] ] dx d ∧ dx c = −∇ b [(∇ d θ Ac )θ A b ] dx d ∧ dx c − − ∇ b [(∇ c θ Ab )θ A d ] dx d ∧ dx c − ∇ b [(∇ b θ Ad )θ A c ] dx d ∧ dx c .
Acting by ∇ b on the factors in the square brackets we obtain
− * d * (dθ A ∧ θ A ) = −(∇ b ∇ d θ Ac )θ A b dx d ∧ dx c − (∇ b ∇ c θ Ab )θ A d dx d ∧ dx c − − (∇ b ∇ b θ Ad )θ A c dx d ∧ dx c − (∇ d θ Ac )(∇ b θ A b ) dx d ∧ dx c − (∇ c θ Ab )(∇ b θ A d ) dx d ∧ dx c , (3.103) where we omitted the term −(∇ b θ Ad )(∇ b θ A c ) dx d ∧ dx c which being of the form (3.99) is zero.
The last term in (3.98) by virtue of (3.5) and (3.14) can be expressed as follows
* θ A * (d * ( * θ A ∧ θ B ) ∧ dθ B ) = 1 2 θ Ab ∇ i (θ Aa θ Ba )(∇ j θ Bk )ǫ ijk ǫ bdc dx d ∧ dx c = = θ Ab ∇ b (θ Aa θ Ba )(∇ d θ Bc ) dx d ∧ dx c + θ Ab ∇ d (θ Aa θ Ba )(∇ c θ Bb ) dx d ∧ dx c + + θ Ab ∇ c (θ Aa θ Ba )(∇ b θ Bd ) dx d ∧ dx c .
The last term in the second line above vanishes-indeed, due to (2.3) the term is equal to
θ Ab (∇ d θ Aa )θ Ba (∇ c θ Bb ) dx d ∧ dx c + (∇ d θ Bb )(∇ c θ Bb ) dx d ∧ dx c
and each term in this sum is of the form (3.99). Using (2.3) again we obtain
* θ A * (d * ( * θ A ∧ θ B ) ∧ dθ B ) = (∇ b θ Bb )(∇ d θ Bc ) dx d ∧ dx c + (∇ c θ Bb )(∇ b θ Bd ) dx d ∧ dx c + + (∇ b θ Aa )(∇ d θ Bc )θ Ab θ Ba dx d ∧ dx c + (∇ c θ Aa )(∇ b θ Bd )θ Ab θ Ba dx d ∧ dx c . (3.104)
Now we are ready to gather all the results (3.100)-(3.104) to show that (3.98) is zero. To make the task easier let us note that we obtained three kinds of terms: (i) ones containing second covariant derivatives of θ A a , (ii) ones quadratic in covariant derivatives of θ A a and (iii) ones quadratic both in the covariant derivatives and in θ A a . Expressions containing second covariant derivatives of θ A a appear in (3.102) and (3.103) and they read
− (∇ b ∇ b θ Bc )θ B d dx d ∧ dx c + (∇ d ∇ a θ Bc )θ Ba dx d ∧ dx c + (∇ b ∇ c θ Bb )θ B d dx d ∧ dx c − − (∇ b ∇ d θ Ac )θ A b dx d ∧ dx c − (∇ b ∇ c θ Ab )θ A d dx d ∧ dx c − (∇ b ∇ b θ Ad )θ A c dx d ∧ dx c
We see now that the first and the last term sum up to zero, similarly do the third and the fifth ones. The sum of the remaining second an fourth terms can be expressed as
[(∇ d ∇ a − ∇ a ∇ d )θ b B ]θ Ba q bc dx d ∧ dx c = R b eda θ e B θ Ba q bc dx d ∧ dx c = = R b eda q ea q bc dx d ∧ dx c = R ceda q ea dx d ∧ dx c = R cd dx d ∧ dx c = 0,
where (i) in the first step we used the Riemann tensor R b eda of the Levi-Civita connection compatible with q to express the commutator (∇ d ∇ a −∇ a ∇ d ) acting on θ b B and (ii) in the second step we applied (2.3). Note that the last equality holds by virtue of symmetricity of the Ricci tensor R cd .
Terms quadratic in covariant derivatives of θ A a appear in (3.103) and (3.104):
− (∇ d θ Ac )(∇ b θ A b ) dx d ∧ dx c − (∇ c θ Ab )(∇ b θ A d ) dx d ∧ dx c + (∇ b θ Bb )(∇ d θ Bc ) dx d ∧ dx c + + (∇ c θ Bb )(∇ b θ Bd ) dx d ∧ dx c
It is easy to see that the first and the third terms sum up to zero, similarly do the second and fourth ones. Let us finally consider the terms quadratic in covariant derivatives of θ A a and quadratic in θ A a -there are eight of them and they can be grouped into pairs such that each pair sums up to zero. These pairs are:
1. the first term at the r.h.s. of (3.100) and the fourth one at the r.h.s. of (3.102), 2. the second term at the r.h.s. of (3.100) and the third one at the r.h.s. of (3.104) (apply (3.9) to the latter term),
3. the first term at the r.h.s. of (3.101) and the fifth one at the r.h.s. of (3.102), 4. the second term at the r.h.s. of (3.101) and the fourth one at the r.h.s. of (3.104) (apply (3.9) to the latter term).
In this way we demonstrated that (3.98) is zero for every M , b and θ A . Thus the formula (3.97) turns into the final expression of the Poisson bracket of R(b) and S(M ):
{R(b), S(M )} = −R M [θ B * (p B ∧ b)− 1 2 b * (p B ∧ θ B )+ dξ A * (b∧ * θ A )] − B * (dM ∧ b) .
Terms quadratic in p A
The only term in (3.106) quadratic in p A is
{B 2 (a), S 1 (M )} = Σ δB 2 (a) δθ A ∧ δS 1 (M ) δp A − δS 1 (M ) δθ A δB 2 (a) δp A . (3.107)
The first term of the r.h.s. of this equation turns out to be zero. To show this let us express the term as follows
δB 2 (a) δθ A ∧ δS 1 (M ) δp A = − M 2 ǫ D BCA θ B ∧ θ C ∧ θ E * (a ∧ p D ) * (p E ∧ θ A )+ + M 4 ǫ D BCA θ B ∧ θ C ∧ θ A * (a ∧ p D ) * (p E ∧ θ E ) + [( * ξ D ) ∧ * ′ A (a ∧ p D )] ∧ δS 1 (M ) δp A .
(3.108)
Our strategy now is to restore in each term above the function ξ A which originally appears in B 2 (a). Thus by virtue of (3.70)
− M 2 ǫ D BCA θ B ∧θ C ∧θ E * (a∧p D ) * (p E ∧θ A ) = M ( * ξ D )( θ A θ E ) * (a∧p D ) * (p E ∧θ A ) = = M a ∧ ξ D p D * (p A ∧ θ A ). (3.109)
Due to (2.2) the second term at the r.h.s of (3.108)
M 4 ǫ D BCA θ B ∧ θ C ∧ θ A ∧ * (a ∧ p D ) * (p E ∧ θ E ) = − 3M 2 a ∧ ξ D p D * (p A ∧ θ A ). (3.110) By virtue (3.78) [( * ξ D ) ∧ * ′ A (a ∧ p D )] ∧ δS 1 (M ) δp A = M 2 a ∧ ξ D p D * (p A ∧ θ A ). (3.111)
Gathering the three results (3.109), (3.110) and (3.111) we see that, indeed, the first term on the r.h.s. of (3.107) is zero.
The second term at the r.h.s. of (3.107) requires (3.3) to be applied, then some simple transformations give us an expression for terms in (3.106) quadratic in the momenta:
{B 2 (a), S 1 (M )} = − Σ M [θ B * (p B ∧ a) − 1 2 a * (p B ∧ θ B )] ∧ ξ A p A . (3.112)
Terms linear in p A
It turns out that {B 1 (a), S 1 (M )} and {B 2 (a), S 2 (M )} give terms linear in p A . The first of the two brackets can be calculated as follows
{B 1 (a), S 1 (M )} = Σ [−a ∧ * dθ A + d * (a ∧ θ A )] ∧ M [θ B * (p B ∧ θ A ) − 1 2 θ A * (p B ∧ θ B )]+ + [(a ∧ θ B ) ∧ * ′ A dθ B ] ∧ δS 1 (M ) δp A
Using (3.72) after some simple algebra we obtain
{B 1 (a), S 1 (M )} = Σ M a ∧ θ B ∧ * dθ A * (p B ∧ θ A ) + M d * (a ∧ θ A ) ∧ θ B * (p B ∧ θ A )− − M a ∧ θ B ∧ * dθ B * (p A ∧ θ A ) − M 2 d * (a ∧ θ A ) ∧ θ A * (p B ∧ θ B )+ + M * (a ∧ θ B ∧ θ A ) * p A ∧ dθ B − M a ∧ * p A ∧ θ B * (dθ B ∧ θ A ).
The next bracket reads
{B 2 (a), S 2 (M )} = Σ − 1 2 ǫ D BCA θ B ∧ θ C ∧ (ξ A dM + M dξ A ) * (a ∧ p D )+ + M [( * ξ B ) ∧ * ′ A (a ∧ p B )] ∧ dξ A − 1 2 ǫ D BCA θ B ∧ θ C ξ A ∧ M a * dp D (3.113)
-here we omitted two terms which are zero by virtue of (3.3). To simplify the resulting expression let us first consider the two terms above containing dξ A -the first of the terms can be transformed by means of (3.70):
− 1 2 ǫ D BCA θ B ∧ θ C ∧ M dξ A * (a ∧ p D ) = M ( * ξ D ) θ A dξ A * (a ∧ p D ) = M ( θ A dξ A )a ∧ ξ D p D .
On the other hand by virtue of (3.77) the other term
M [( * ξ B ) ∧ * ′ A (a ∧ p B )] ∧ dξ A = −M ( * ξ B ) ∧ * (a ∧ p B )( θ A dξ A ) = −M ( θ A dξ A )a ∧ ξ B p B .
Thus the sum of the two terms in (3.113) containing dξ A is zero. Consequently,
{B 2 (a), S 2 (M )} = Σ − 1 2 ǫ D BCA θ B ∧ θ C ξ A ∧ (dM * (a ∧ p D ) + M a * dp D ) = = Σ * θ D ∧ (dM * (a ∧ p D ) + M a * dp D ), (3.114)
where we applied (3.20). Finally, the terms in (3.106) linear in p A read
{B 1 (a), S 1 (M )} + {B 2 (a), S 2 (M )} = Σ * θ D ∧ (dM * (a ∧ p D ) + M a * dp D )+ + M a ∧ θ B ∧ * dθ A * (p B ∧ θ A ) + M d * (a ∧ θ A ) ∧ θ B * (p B ∧ θ A )− − M a ∧ θ B ∧ * dθ B * (p A ∧ θ A ) − M 2 d * (a ∧ θ A ) ∧ θ A * (p B ∧ θ B )+ + M * (a ∧ θ B ∧ θ A ) * p A ∧ dθ B − M a ∧ * p A ∧ θ B * (dθ B ∧ θ A ).{B 1 (a), S 2 (M )} = Σ [−a ∧ * dθ A + (a ∧ θ B ) ∧ * ′ A dθ B ] ∧ (dM ξ A + M dξ A ) = = Σ −dM ∧ a ∧ * (ξ A dθ A ) − M a ∧ * dθ A ∧ dξ A + M a ∧ θ B ∧ dξ A * (dθ B ∧ θ A )− − M * ( * dθ B ∧ θ A ) ∧ * (a ∧ θ B ) ∧ dξ A ,
where in the second step we used the second line of (3.3), (3.71), (3.6) and (3.5). The third bracket
{B 2 (a), S 3 (M )} = − Σ d M θ B * (dθ B ∧ θ A ) − M 2 θ A * (dθ B ∧ θ B ) ∧ aξ A + + M dθ B * (dθ A ∧ θ B ) − 1 2 dθ A * (dθ B ∧ θ B ) ∧ aξ A = − Σ −M a ∧ θ B ∧ ξ A d * (dθ B ∧ θ A )+ + M a ∧ ξ A dθ B * (dθ A ∧ θ B ) − M a ∧ ξ A dθ A * (dθ B ∧ θ B ),
-here in the first step we applied (3.3) and in the second one we carried out the exterior differentiation at the r.h.s. of the first line. Thus the terms in (3.106) independent of p A read Terms quadratic in p A We immediately see that the formula (3.112) can be expressed as where now the phrase "terms linear in p A " means (3.115) and the last term in (3.117).
{B 1 (a), S 2 (M )} + {B 2 (a), S 3 (M )} = Σ −dM ∧ a ∧ * (ξ A dθ A ) − M a ∧ * dθ A ∧ dξ A + + M a ∧ θ B ∧ dξ A * (dθ B ∧ θ A ) − M * ( * dθ B ∧ θ A ) ∧ * (a ∧ θ B ) ∧ dξ A + + M a ∧ θ B ∧ ξ A d * (dθ B ∧ θ A ) − M a ∧ ξ A dθ B * (dθ A ∧ θ B ) + M a ∧ ξ A dθ A * (dθ B ∧ θ B ).−B M [θ B * (p B ∧a)− 1 2 a * (p B ∧θ B )] + Σ M [θ B * (p B ∧a)− 1 2 a * (p B ∧θ B )]∧θ A ∧ * dθ A ,(3.
Terms linear in p A According to the last statement of the previous paragraph the remaining terms linear in
p A read Σ M [θ B * (p B ∧ a) − 1 2 a * (p B ∧ θ B )] ∧ θ A ∧ * dθ A + * θ D ∧ (dM * (a ∧ p D ) + M a * dp D )+ + M a ∧ θ B ∧ * dθ A * (p B ∧ θ A ) + M d * (a ∧ θ A ) ∧ θ B * (p B ∧ θ A )− − M a ∧ θ B ∧ * dθ B * (p A ∧ θ A ) − M 2 d * (a ∧ θ A ) ∧ θ A * (p B ∧ θ B )+ + M * (a ∧ θ B ∧ θ A ) * p A ∧ dθ B − M a ∧ * p A ∧ θ B * (dθ B ∧ θ A ). (3.119)
Let us now transform the terms containing dM and dp D appearing in the first line of (3.119):
Σ * θ D ∧ dM * (a ∧ p D ) = Σ * ( * dM ∧ θ D )a ∧ p D Σ = ( θ D dM )a ∧ p D = = Σ dM ( θ D a) ∧ p D − dM ∧ a θ D p D = Σ dM ( θ D a) ∧ p D + * (dM ∧ a)θ A ∧ * p A = = R * (dM ∧ a) + Σ dM ( θ D a) ∧ p D + * (dM ∧ a) ∧ ξ A dθ A ,
where we used (3.5). On the other hand the term with dp D
Σ * θ D ∧ M a * dp D = Σ M * ( * a ∧ θ D )dp D = − Σ d[M * ( * a ∧ θ D )] ∧ p D = = − Σ dM ( θ D a) ∧ p D + M d * ( * a ∧ θ D ) ∧ p D .
Consequently, the sum of the two terms
Σ * θ D ∧ (dM * (a ∧ p D ) + M a * dp D ) = R * (dM ∧ a) + + Σ −M d * ( * a ∧ θ A ) ∧ p A + dM ∧ a ∧ * (ξ A dθ A ). (3.120)
The result just obtained means that (3.118) can be re-expressed as
{B(a), S(M )} = −B M [θ B * (p B ∧ a) − 1 2 a * (p B ∧ θ B )] + R * (dM ∧ a) + terms linear in p A + terms independent of p A , (3.121)
where now (i) the phrase "terms linear in p A " means the second term at the r.h.s. of (3.120) and (3.119) except the terms containing dM and dp D and (ii) "terms independent of p A " means the last term in (3.120) and the terms (3.116).
Note now that the form of constraints at the r.h.s. of (3.121) we managed to isolate so far resemble closely the form of the constraints at the r.h.s. of (3.105). Let us then assume that
{B(a), S(M )} = −B M [θ B * (p B ∧a)− 1 2 a * (p B ∧θ B )+dξ B * (a∧ * θ B )] +R * (dM ∧a) .
(3.122) To justify the assumption we will proceed as follows: we will add to the r.h.s. of (3.121) zero expressed as
0 = −B M dξ B * (a ∧ * θ B ) + B M dξ B * (a ∧ * θ B ) = −B M dξ B * (a ∧ * θ B ) + + Σ M dξ B * (a ∧ * θ B ) ∧ (θ A ∧ * dθ A + ξ A p A ) = −B M dξ B * (a ∧ * θ B ) + + Σ M dξ B * (a ∧ * θ B )θ A ∧ * dθ A + M d * ( * θ B ∧ θ A ) ∧ p A * (a ∧ * θ B ) (3.123)
(here in the last step we used (3.74)). Next we will show that all the remaining terms linear in p A sum up to zero, and that similarly do all the remaining terms independent of the momenta. Note that now the description of the terms linear in and independent of the momenta given just below Equation (3.121) has to be completed by taking into account the two last term in (3.123).
To demonstrate that all the remaining terms linear in p A , that is,
Σ M θ B * (p B ∧ a) ∧ θ A ∧ * dθ A − 1 2 a * (p B ∧ θ B ) ∧ θ A ∧ * dθ A − d * ( * a ∧ θ A ) ∧ p A + + a ∧ θ B ∧ * dθ A * (p B ∧ θ A ) + d * (a ∧ θ A ) ∧ θ B * (p B ∧ θ A )− − a ∧ θ B ∧ * dθ B * (p A ∧ θ A ) − 1 2 d * (a ∧ θ A ) ∧ θ A * (p B ∧ θ B )+ + * (a ∧ θ B ∧ θ A ) * p A ∧ dθ B − a ∧ * p A ∧ θ B * (dθ B ∧ θ A ) + d * ( * θ B ∧ θ A ) ∧ p A * (a ∧ * θ B ) (3.124)
sum up to zero let us first perform some transformations. First we are going to show that the first, fourth, sixth and eighth terms above give together zero. To this end let us transform the eighth one as follows
* (a ∧ θ B ∧ θ A ) * p A ∧ dθ B = −[ θ B * (a ∧ θ A )] * p A ∧ dθ B = = − * (a ∧ θ A ) * (p A ∧ θ B ) ∧ dθ B + * (a ∧ θ A ) ∧ * p A ∧ * ( * dθ B ∧ θ B ),
where in the first step we applied (3.6), and in the second one we shifted the contraction θ B and used (3.6) and (3.5). Let us now transform the last term above in an analogous
way: * (a ∧ θ A ) ∧ * p A ∧ * ( * dθ B ∧ θ B ) = θ A * a ∧ * p A ∧ * ( * dθ B ∧ θ B ) = = − * a ∧ * (p A ∧ θ A ) * ( * dθ B ∧ θ B ) + * a ∧ * p A * ( * dθ B ∧ θ B ∧ θ A ) = = a ∧ θ B ∧ * dθ B * (p A ∧ θ A ) − θ A * (p A ∧ a) ∧ θ B ∧ * dθ B . Thus the eighth term * (a ∧ θ B ∧ θ A ) * p A ∧ dθ B = −a ∧ θ A ∧ * dθ B * (p A ∧ θ B )+ + a ∧ θ B ∧ * dθ B * (p A ∧ θ A ) − θ A * (p A ∧ a) ∧ θ B ∧ * dθ B
and indeed the first, fourth, sixth and eighth terms disappear from (3.124). Let us consider now the second and the seventh terms in (3.124). After a slight transformation of the second one their sum can be expressed as
− 1 2 [ * (a ∧ θ A ) ∧ dθ A + d * (a ∧ θ A ) ∧ θ A ] * (p B ∧ θ B ) = = − 1 2 [2 * (a ∧ θ A ) ∧ dθ A − d(θ A * (a ∧ θ A ))] * (p B ∧ θ B ) = − [ * (a ∧ θ A ) ∧ dθ A − d * a] * (p B ∧ θ B ),
where in the last step we used (3.18). Now the terms (3.124) can be re-expressed in a simpler form as
Σ M − * (a ∧ θ A ) ∧ dθ A * (p B ∧ θ B ) + (d * a) * (p B ∧ θ B ) − d * ( * a ∧ θ A ) ∧ p A + + d * (a∧ θ A )∧ θ B * (p B ∧ θ A )− a∧ * p A ∧ θ B * (dθ B ∧ θ A )+ d * ( * θ B ∧ θ A )∧ p A * (a∧ * θ B ) .
Note that in each term above one can isolate the factor p B obtaining thereby
Σ M p B ∧ − θ B * [ * (a ∧ θ A ) ∧ dθ A ] + θ B * d * a − d * ( * a ∧ θ B )+ + θ A * [d * (a ∧ θ A ) ∧ θ B ] + * (a ∧ θ A ) * (dθ A ∧ θ B ) + [d * ( * θ A ∧ θ B )] * (a ∧ * θ A ) . (3.125)
Now using tensor calculus (see Section 3.1.3) we will show that the terms in the big parenthesis above sum up to zero for every θ A and a. The first term in (3.125) can be expressed as
− θ B * [ * (a ∧ θ A ) ∧ dθ A ] = − 1 2 θ B c (a ∧ θ A ) ab (dθ A ) ab dx c = = −θ B c a a θ Ab (∇ a θ Ab − ∇ b θ Aa )dx c = a a θ Ab (∇ b θ Aa )θ B c dx c , (3.126)
where in the first step we applied (3.15) and * ǫ = 1, and in the last one (3.10). The second term in (3.125)
θ B * d * a = (∇ a a a )θ B c dx c (3.127)
by virtue of (3.13). Due to (3.5) the third one
− d * ( * a ∧ θ B ) = −∇ c (θ Ba a a )dx c = −(∇ c θ Ba )a a dx c − θ Ba (∇ c a a )dx c . (3.128)
The fourth term in (3.125) by means of the last formula in (3.12) (set α = * (a ∧ θ A ) and β = θ B ), (3.14) and (2.3) can be expressed as
θ A * [d * (a ∧ θ A ) ∧ θ B ] = ∇ d (a a θ b A )θ B e θ A c ǫ abf ǫ df e dx c = = −(∇ a a a )θ B c dx c − a a (∇ a θ b A )θ B b θ A c dx c + (∇ c a a )θ B a dx c + a a (∇ b θ b A )θ B a θ A c dx c . (3.129) The fifth one * (a ∧ θ A ) * (dθ A ∧ θ B ) = a a θ Ab (∇ d θ Ae )θ B f ǫ abc ǫ def dx c .
Using (3.14) we obtain six terms and two of them vanish by virtue of (3.10). Thus
* (a ∧ θ A ) * (dθ A ∧ θ B ) = a a θ Ab (∇ b θ Ac )θ B a dx c + a a θ Ab (∇ c θ Aa )θ B b dx c − − a a θ Ab (∇ b θ Aa )θ B c dx c − a a θ Ab (∇ a θ Ac )θ B b dx c .[d * ( * θ A ∧ θ B )] * (a ∧ * θ A ) = ∇ c (θ b A θ B b )a a θ Aa dx c = (∇ c θ b A )θ B b a a θ Aa dx c + + (∇ c θ Ba )a a dx c . (3.131)
Collecting all the results (3.126)-(3.131) we note that we obtain two kinds of terms: ones containing a covariant derivative of a a and ones containing a covariant derivative of θ A a . The terms containing ∇ b a a appear in (3.127), (3.128) and (3.129) and sum up to zero:
(∇ a a a )θ B c dx c − θ Ba (∇ c a a )dx c − (∇ a a a )θ B c dx c + (∇ c a a )θ B a dx c = 0.
Regarding the terms containing ∇ a θ A b , there are ten of them and they can be grouped into pairs such that each pair sums up to zero. These pairs are:
3. the second term at the r.h.s. of (3.129) and the last term at the r.h.s. of (3.130) (shift the derivative by means of (3.9) in the latter term), 4. the last term at the r.h.s. of (3.129) and the first term at the r.h.s. of (3.130) (shift the derivative by means of (3.9) in the latter term), 5. the second term at the r.h.s. of (3.130) and the first term at the r.h.s. of (3.131) (again shift the derivative by means of (3.9) in the latter term).
Terms independent of p A Our goal now is to show that all the remaining terms independent of the momenta i.e. the terms (3.116), the last term in (3.120) and the second term at the r.h.s. of (3.123) sum up to zero. Note that the first term in (3.116) cancels the last term in (3.120). Now in all remaining terms there is the factor M a and therefore they can be expressed as
Σ M a ∧ − * dθ A ∧ dξ A + θ B * (dθ B ∧ θ A ) ∧ dξ A + θ B ∧ * [ * ( * dθ B ∧ θ A ) ∧ dξ A ]+ + θ B ∧ ξ A d * (dθ B ∧ θ A ) − ξ A dθ B * (dθ A ∧ θ B ) + ξ A dθ A * (dθ B ∧ θ B )+ + * θ A * (dξ A ∧ θ B ∧ * dθ B ) (3.132)
In the fourth, fifth and sixth terms above there appears the function ξ A while in the remaining ones there is the derivative dξ A . Let us then transform the three terms to obtain ones containing dξ A . To transform the fourth one we note that
0 = −d θ B ∧ ξ A * (dθ B ∧ θ A ) = −dθ B ∧ ξ A * (dθ B ∧ θ A ) + θ B ∧ dξ A * (dθ B ∧ θ A )+ + θ B ∧ ξ A d * (dθ B ∧ θ A ) = θ B * (dθ B ∧ θ A ) ∧ dξ A + θ B ∧ ξ A d * (dθ B ∧ θ A ),
which means that the sum of the second and the fourth term in (3.132) is zero. The fifth term in (3.132)
−ξ A dθ B * (dθ A ∧ θ B ) = −dθ B * (ξ A dθ A ∧ θ B ) = −dθ B * (θ A ∧ dξ A ∧ θ B ).
Transforming similarly the sixth term we can rewrite (3.132) as follows:
Σ M a ∧ − * dθ A ∧ dξ A + θ B ∧ * [ * ( * dθ B ∧ θ A ) ∧ dξ A ] − dθ B * (θ A ∧ dξ A ∧ θ B )+ + θ A ∧ dξ A * (dθ B ∧ θ B ) + * θ A * (dξ A ∧ θ B ∧ * dθ B )
By a direct calculation using tensor calculus we will demonstrate that the terms in the big parenthesis sum up to zero for all θ A . More precisely, we will show that * − * dθ
A ∧ dξ A + θ B ∧ * [ * ( * dθ B ∧ θ A ) ∧ dξ A ] − dθ B * (θ A ∧ dξ A ∧ θ B )+ + θ A ∧ dξ A * (dθ B ∧ θ B ) + * θ A * (dξ A ∧ θ B ∧ * dθ B ) (3.133)
is equal to zero. The first term in the expression above reads by virtue of (3.5)
− * ( * dθ A ∧ dξ A ) = − − − → dξ A dθ A = −(∇ a ξ A )(∇ a θ Ab )dx b + (∇ a ξ A )(∇ b θ Aa )dx b . (3.134)
Using twice (3.5) we express the second term in (3.133) as follows
* θ B ∧ * [ * ( * dθ B ∧ θ A ) ∧ dξ A ] = − θ B [ θ A dθ B ∧ dξ A ] = θ a A θ Bb (∇ b θ Ba )(∇ c ξ A )dx c + + θ a A (∇ a θ Bb )θ Bc (∇ c ξ A )dx b − θ a A (∇ b θ Ba )θ Bc (∇ c ξ A )dx b , (3.135)
where in the second step we applied (3.10).
The third term
− * dθ B * (θ A ∧ dξ A ∧ θ B ) = −θ Aa (∇ b ξ A )θ B c (∇ d θ e B
)ǫ abc ǫ def dx f . Applying (3.14) we again obtain six terms, two of them vanish by virtue of (3.10) and we are left with the following expression
− * dθ B * (θ A ∧ dξ A ∧ θ B ) = −θ Aa (∇ b ξ A )θ B c (∇ a θ b B )dx c − θ Aa (∇ b ξ A )θ B c (∇ c θ a B )dx b + + θ Aa (∇ b ξ A )θ B c (∇ b θ a B )dx c + θ Aa (∇ b ξ A )θ B c (∇ c θ b B )dx a . (3.136) The fourth term in (3.133) reads * (θ A ∧dξ A ) * (dθ B ∧θ B ) = θ a A (∇ b ξ A )(∇ d θ Be )θ B f ǫ def ǫ abc dx c = θ a A (∇ b ξ A )(∇ a θ Bb )θ B c dx c + + θ a A (∇ b ξ A )(∇ b θ Bc )θ B a dx c + θ a A (∇ b ξ A )(∇ c θ Ba )θ B b dx c − θ a A (∇ b ξ A )(∇ b θ Ba )θ B c dx c − − θ a A (∇ b ξ A )(∇ a θ Bc )θ B b dx c − θ a A (∇ b ξ A )(∇ c θ Bb )θ B a dx c . (3.137) Finally, the last term in (3.133) θ A * (dξ A ∧ θ B ∧ * dθ B ) = (∇ d ξ A )θ B e (∇ a θ b B )θ A c ǫ abf ǫ def dx c = −(∇ b ξ A )θ B a (∇ a θ b B )θ A c dx c ,(3.
138) where in the last step we used (3.14) and (3.10).
In this way we managed to express (3.133) in terms of the components θ A a and ξ A and their covariant derivatives obtaining altogether sixteen terms (3.134)-(3.138). As before those terms can be grouped into pairs such that terms in each pair sum up to zero. Let us now enumerate the pairs: 7. the third term at the r.h.s. of (3.136) and the fourth term at the r.h.s. of (3.137), 8. the fourth term at the r.h.s. of (3.136) and the term at the r.h.s. of (3.138).
Thus we managed to demonstrate that all the remaining terms (3.133) independent of p A sum up to zero and thereby proved the assumption (3.122).
Poisson brackets of V ( M)
The functional derivatives of the smeared scalar constraint V ( M ) (see (2.8)) are of the following form [9]:
δV ( M ) δθ A = −L M p A , δV ( M ) δp A = L M θ A . (3.139)
It was shown in [9] that Derivations of brackets of V ( M ) and the other constraints will be based on the following formula [9]:
{V ( M ), V ( M ′ )} = V ([ M , M ′ ]) = V (L M M ′ ),(3.L M (α ∧ * β) = L M α ∧ * β + α ∧ * L M β + L M θ A ∧ (α ∧ * ′ A β).
(3.141)
We will also apply the following well known properties of the Lie derivative: where α ∧ β is a three-form, and γ any k-form on Σ.
0 = Σ L M (α ∧ β) = Σ (L M α) ∧ β + Σ α ∧ L M β,(3.L M p A ∧ M θ B * (p B ∧ θ A )− − M p B * (p A ∧ θ B ) + 1 2 (p C ∧ θ B ) ∧ * ′ A (p B ∧ θ C ) ∧ L M θ A = = − Σ M L M (p A ∧ θ B ) ∧ * (p B ∧ θ A ) + L M θ A ∧ 1 2 [(p C ∧ θ B ) ∧ * ′ A (p B ∧ θ C )]
-functional derivatives of S 11 (M ) used to calculate the bracket can be read off from The first term at the r.h.s. of the formula above
Σ d M θ B * (dθ B ∧ θ A ) ∧ L M θ A = Σ M θ B * (dθ B ∧ θ A ) ∧ L M dθ A ,
where we shifted the derivative d and applied (3.143). Thus the sum of the first two terms at the r.h.s. of (3.147) reads where B 1 (a) and B 2 (a) are given by (3.23). We have
Σ M L M (dθ A ∧ θ B ) * (dθ B ∧ θ A ) = = Σ M 2 L M (dθ A ∧ θ B ) ∧ * (dθ B ∧ θ A ) + dθ A ∧ θ B ∧ * L M (dθ B ∧ θ A ) .{V ( M ), B 1 (a)} = − Σ −a ∧ * dθ A ∧ L M θ A + [(a ∧ θ B ) ∧ * ′ A dθ B ] ∧ L M θ A + + d * (a ∧ θ A ) ∧ L M θ A . (3.151)
The first term at the r.h.s. above The other bracket
{V ( M ), B 2 (a)} = Σ −L M p A ∧ aξ A + 1 2 ǫ D BCA θ B ∧ θ C * (a ∧ p D ) ∧ L M θ A − − [( * ξ B ) ∧ * ′ A (a ∧ p B )] ∧ L M θ A . (3.153)
The first term at the r.h.s. above
− L M p A ∧ aξ A = ξ A L M a ∧ p A − ξ A L M (a ∧ p A ) = L M a ∧ ξ A p A − ( * ξ A ) ∧ * L M (a ∧ p A )
and the second one at the r.h.s. of (3.153)
1 2 ǫ D BCA θ B ∧ θ C ∧ L M θ A * (a ∧ p D ) = 1 3 L M 1 2 ǫ D BCA θ B ∧ θ C ∧ θ A * (a ∧ p D ) = = −L M ( * ξ D ) ∧ * (a ∧ p D ),
where in the last step we used (2.2). Setting these two results to (3.151) and applying (3.141) and (3.142) we obtain In the formulae above L M denotes the Lie derivative on Σ with respect to the vector field M . A discussion of the results can be found in [1], here we restrict ourselves to a statement that a Poisson bracket of every pair of the constraints (2.5)-(2.8) is a sum of the constraints smeared with some fields. In other words, the constraint algebra presented above is closed.
{V ( M ), B 2 (a)} = Σ L M a ∧ ξ A p A − L M [ * ξ A ∧ * (a ∧ p A )] = Σ L M a ∧ ξ A p A = B 2 (L M a).
A ∧ ( M p A ) − ( M θ A ) ∧ dp A ,(2.8) where a, b, M and M are smearing fields on Σ: a and b are one-forms, M is a function and M a vector field on the manifold. The smearing field possess altogether ten degrees of freedom per point of Σ. In [1] we called B(a) boost constraint and R(b) rotation constraint. S(M ) is a scalar constraint and V ( M ) a vector constraint of TEGR.
{B(a), R(b)} = B( * (a ∧ b)).
bracket of S(M) and S(M ′ )
begin the calculations with the bracket {S 1 (M ), S 1 (M ′ )}:
This means that indeed (3.62) is zero for all m and θ A . Consequently, the terms (3.57) = R θ B (m∧ * dθ B )− * [ * (m∧θ B )∧ * dθ B ]+ * [m∧ * (θ B ∧ * dθ B )] . (3.63)
results (3.66) and (3.67) to (3.64) after some simple calculations with application of (3.5) and (3.6) we obtain the final form of the Poisson bracket of the scalar constraints {S(M ), S(M
Now we are ready to begin the calculations of {R(B), S(M )}. Let us recall that variations needed to calculate the bracket are given by (3.29) and (3.36)-(3.41).
linear in p A Here we will calculate the brackets {R 1 (b), S 2 (M )} and {R 2 (b), S 1 (M )}, which give terms linear in the momenta.Considering {R 1 (b), S 2 (M )} we immediately see that by virtue of (3.17) and(3.19)
independent of p A It turns out that the remaining three brackets, {R 2 (b), S 3 (M )}, {R 2 (b), S 2 (M )} and {R 1 (b), S 3 (M )} do not depend on the momenta. The first bracket is zero since both R 2 (b) and S 3 (M ) do not contain p A . The second bracket contains a term d(bξ A ) ∧ d(M ξ A )
{R(b), S(M )} = the terms (3.82) quadratic in p A + + the terms (3.86) linear in p A + the terms (3.87) independent of p A as a sum of the constraints (2.5)-(2.8) smeared with some fields.
the bracket {R(b), S(M )} is of the following form {R(b), S(M )} = −R M [θ B * (p B ∧ b) − 1 2 b * (p B ∧ θ B )] + terms linear in p A + + the terms (3.87) independent of p A , (3.90)
the final form of the terms in (3.91) which do not contain dM . The equations (3.92) and (3.96) allow us to express the terms linear in p A appearing in (3.90) as a sum of constraints and terms independent of the momenta. Consequently, (3.90) can be written in the following form: {R(b), S(M )} = −R M [θ B * (p B ∧b)− 1 2 b * (p B ∧θ B )+dξ A * (b∧ * θ A )] −B * (dM ∧b) + + terms independent of p A , (3.97)
calculated in a similar way to {R(b), S(M )}. Recall that the constraints under consideration are defined by (2.5) and (2.7), the functionals appearing at the r.h.s. of (3.106) are given by (3.23) and (3.34) and that formulae (3.25) and (3.36)-(3.41) describe variations needed to calculate the bracket.
independent of p A The remaining three brackets {B 1 (a), S 3 (M )}, {B 1 (a), S 2 (M )} and {B 2 (a), S 3 (M )} give terms independent of p A . The first bracket is zero since both B 1 (a) and S 3 (M ) do not depend on p A . The second bracket contains a term d * (a ∧ θ A ) ∧ d(M ξ A ) being an exact three-form-the integral of this term over Σ is zero hence
Again, our goal now is to express the bracket {B(a), S(M )} = the terms (3.112) quadratic in p A + + the terms (3.115) linear in p A + the terms (3.116) independent of p A as a sum of the constraints (2.5)-(2.8) smeared with some fields.
a), S(M )} = −B M [θ B * (p B ∧ a) − 1 2 a * (p B ∧ θ B )] + + terms linear in p A + the terms (3.116) independent of p A , (3.118)
140) where [ M , M ′ ] denotes the Lie bracket of the vector fields M , M ′ on Σ.
142) d(L M γ) = L M (dγ), (3.143)
bracket of V ( M ) and S(M ) To calculate the bracket {V ( M ), S(M )} we will use the split of S(M ) into the three functionals (3.34):{V ( M ), S(M )} = 3 i=1 {V ( M ), S i (M )}.To calculate the first term {V ( M ), S 1 (M )} at the r.h.s. of this equation let us split S 1 (M ) into a sum S 1 (M ) = S 11 (M ) + S 12 (M ), where S 11 (M ) := Σ M 2 (p A ∧ θ B ) ∧ * (p B ∧ θ A ), S 12 (M ) := − Σ M 4 (p A ∧ θ A ) ∧ * (p B ∧ θ B ) and consider the bracket {V ( M ), S 11 (M )}: {V ( M ), S 11 (M )} = Σ −
( 3 .
336) and(3.37). It is easy to see that the first term in the last line aboveM L M (p A ∧θ B )∧ * (p B ∧θ A ) = M 2 L M (p A ∧θ B )∧ * (p B ∧θ A )+(p A ∧θ B )∧ * L M (p B ∧θ A ) .
A ∧ θ B ) ∧ * (p B ∧ θ A ) = S 11 (L M M ), (3.144) where the second equality holds by virtue of (3.142). It can be shown in a similar way that {V ( M ), S 12 (M )} = S 12 (L M M ). (3.145) According to calculations carried out in [9] {V ( M ), S 2 (M )} = S 2 (L M M ). (3.146) Let us split S 3 (M ) as follows: S 3 (M ) = S 31 (M ) + S 32 (M ), where S 31 (M ) := Σ M2 (dθ A ∧ θ B ) ∧ * (dθ B ∧ θ A ), S 32 (M ) := − Σ M 4 (dθ A ∧ θ A ) ∧ * (dθ B ∧ θ B ).Reading off from (3.40) and (3.41) functional derivatives of S 31 (M ) we obtain{V ( M ), S 31 (M )} = − Σ d M θ B * (dθ B ∧ θ A ) ∧ L M θ A + M dθ B * (dθ A ∧ θ B ) ∧ L M θ A + 1 2 (dθ C ∧ θ B ) ∧ * ′ A (dθ B ∧ θ C ) ∧ L M θ A (3.147)
Setting this result to (3.147), applying (3.141) and (3.142) we obtain{V ( M ), S 31 (M )} = Σ (L M M )dθ A ∧ θ B ∧ * (dθ B ∧ θ A ) = S 31 (L M M ).(3.148) In analogous way one can show that {V ( M ), S 32 (M )} = S 32 (L M M ) (3.149) Gathering all partial results (3.144)-(3.149) (except (3.147)) we obtain {V ( M ), S(M )} = S(L M M ).(3.150) 3.6.2 Poisson bracket of V ( M ) and the constraints B(a) and R(b) Here we explicitely calculate the bracket {V ( M ), B(a)}. The bracket {V ( M ), R(b)} can be calculated similarly. Obviously, {V ( M ), B(a)} = {V ( M ), B 1 (a)} + {V ( M ), B 2 (a)},
Σ
−a ∧ * dθA ∧ L M θ A = Σ L M (a ∧ θ A ) ∧ * dθ A − L M a ∧ θ A ∧ * dθ A and the last term in (3.151) Σ d * (a ∧ θ A ) ∧ L M θ A = Σ * (a ∧ θ A ) ∧ dL M θ A = Σ * (a ∧ θ A ) ∧ L M dθ A = = Σ a ∧ θ A ∧ * L M dθ A-here in the second step we used (3.143). Setting these two results to (3.151) and applying (3.141) and (3.142) we obtain{V ( M ), B 1 (a)} = − Σ −L M a ∧ θ A ∧ * dθ A + L M (a ∧ θ A ∧ * dθ A ) = = Σ L M a ∧ θ A ∧ * dθ A = B 1 (L M a).(3.152)
The equation above and (3.152) give us the final result {V ( M ), B(a)} = B(L M a). (3.154) Similarly, {V ( M ), R(b)} = R(L M b). summarize the calculation let us list the results. The Poisson brackets of boosts and rotation constrains (Equations (3.26), (3.30) and (3.33)):{B(a), B(a ′ )} = −R( * (a ∧ a ′ )), {R(b), R(b ′ )} = R( * (b ∧ b ′ )), {B(a), R(b)} = B( * (a ∧ b)).The bracket of the scalar constraints (Equation (3.68)):{S(M ), S(M ′ )} = V ( m) + B θ B * (m ∧ p B ) − 1 2 * (m ∧ ξ B * dθ B )− − * [m ∧ * (θ B ∧ * p B )] − 1 2 * ( * m ∧ θ B ) * p B + 1 2 * [ * (m ∧ θ B ) ∧ * p B ] + + R − θ B * (m ∧ dθ B ) − 1 2 * (m ∧ ξ B * p B )+ + * [m ∧ * (θ B ∧ * dθ B )] + 1 2 * ( * m ∧ θ B ) * dθ B − 1 2 * [ * (m ∧ θ B ) ∧ * dθ B ] .In this formulam := M dM ′ − M ′ dM.The brackets of the boost and rotation constraints and the scalar one (Equations (3.122) and (3.105)):{B(a), S(M )} = − B M [θ B * (p B ∧ a) − 1 2 a * (p B ∧ θ B ) + dξ B * (a ∧ * θ B )] + + R * (dM ∧ a) , {R(b), S(M )} = − R M [θ B * (p B ∧ b) − 1 2 b * (p B ∧ θ B ) + dξ A * (b ∧ * θ A )] − − B * (dM ∧ b) .The brackets of the vector constraint (Equations(3.140), (3.150), (3.154) and (3.155)): {V ( M ), V ( M ′ )} =V (L M M ′ ) ≡ V ([ M , M ′ ]), {V ( M ), S(M )} =S(L M M ), {V ( M ), B(a)} =B(L M a), {V ( M ), R(b)} =R(L M b).
Acknowledgments I am grateful to Jędrzej Świeżewski for his cooperation in the research on a Hamiltonian model described in[9]which was for me a preparatory exercise for deriving the results described in this paper. I am also grateful to Jerzy Lewandowski for a valuable discussion.
2. the first term at the r.h.s. of (3.128) and the last term at the r.h.s. of (3.131), 1. the first term at the r.h.s. of (3.134) and the second term at the r.h.s. of (3.137). Here the latter term needs the following transformation: θ a A (∇ b ξ A )(∇ b θ Bc )θ B a dx c = (∇ b ξ A )(∇ b θ Bc )(δ B A + ξ B ξ A ). the term at the r.h.s. of (3.126) and the third term at the r.h.s. of (3.130. dx c = = (∇ b ξ A )(∇ b θ Ac )dx cthe term at the r.h.s. of (3.126) and the third term at the r.h.s. of (3.130), 2. the first term at the r.h.s. of (3.128) and the last term at the r.h.s. of (3.131), 1. the first term at the r.h.s. of (3.134) and the second term at the r.h.s. of (3.137). Here the latter term needs the following transformation: θ a A (∇ b ξ A )(∇ b θ Bc )θ B a dx c = (∇ b ξ A )(∇ b θ Bc )(δ B A + ξ B ξ A )dx c = = (∇ b ξ A )(∇ b θ Ac )dx c ,
) and the sixth term at the r.h.s. of (3.137) (the latter term needs a transformation analogous to that shown above), 3. the first term at the r.h.s. of (3.135) and the second term at the r. the second term at the r.h.s. of (3.134. h.s. of (3.136), 4. the second term at the r.h.s. of (3.135) and the first term at the r.h.s. of (3.137) (apply (3.9) to the latter termthe second term at the r.h.s. of (3.134) and the sixth term at the r.h.s. of (3.137) (the latter term needs a transformation analogous to that shown above), 3. the first term at the r.h.s. of (3.135) and the second term at the r.h.s. of (3.136), 4. the second term at the r.h.s. of (3.135) and the first term at the r.h.s. of (3.137) (apply (3.9) to the latter term),
ADM-like Hamiltonian formulation of gravity in the teleparallel geometry accepted for publication in Gen. A Okołów, arXiv:1111.5498v2Rel. Grav. E-printOkołów A 2011 ADM-like Hamiltonian formulation of gravity in the teleparallel ge- ometry accepted for publication in Gen. Rel. Grav. (E-print arXiv:1111.5498v2)
Variables suitable for constructing quantum states for the Teleparallel Equivalent of General Relativity I. A Okołów, arXiv:1305.4526E-printOkołów A 2013 Variables suitable for constructing quantum states for the Teleparallel Equivalent of General Relativity I E-print arXiv:1305.4526
Variables suitable for constructing quantum states for the Teleparallel Equivalent of General. A Okołów, arXiv:1308.2104Relativity II E-printOkołów A, Variables suitable for constructing quantum states for the Teleparallel Equivalent of General Relativity II E-print arXiv:1308.2104
Kinematic quantum states for the Teleparallel Equivalent of. A Okołów, arXiv:1304.6492General Relativity E-printOkołów A 2013 Kinematic quantum states for the Teleparallel Equivalent of General Relativity E-print arXiv:1304.6492
Hamiltonian formulation of general relativity in the teleparallel geometry. J W Maluf, J Da Rocha-Neto, arXiv:gr-qc/0002059Phys. Rev. D. 6484014Maluf J W, da Rocha-Neto J F 2001 Hamiltonian formulation of general relativity in the teleparallel geometry Phys. Rev. D 64 084014 E-print arXiv:gr-qc/0002059
Hamiltonian formulation of unimodular gravity in the teleparallel geometry. J F Da Rocha-Neto, J W Maluf, S Ulhoa, arXiv:1101.2425Phys. Rev. D. 82124035da Rocha-Neto J F, Maluf J W, Ulhoa S C 2010 Hamiltonian formulation of unimodular gravity in the teleparallel geometry Phys. Rev. D 82 124035 E-print arXiv:1101.2425
Hamiltonian structure of the teleparallel formulation of GR. M Blagojević, I Nikolić, arXiv:hep-th/0002022Phys. Rev. D. 6224021E-printBlagojević M, Nikolić I A 2000 Hamiltonian structure of the teleparallel formulation of GR Phys. Rev. D 62 024021 E-print arXiv:hep-th/0002022
Ashtekar's Complex Variables in General Relativity and Its Teleparallelism Equivalent. E Mielke, Ann. Phys. 219Mielke E W 1992 Ashtekar's Complex Variables in General Relativity and Its Telepar- allelism Equivalent Ann. Phys. 219 78-108
Hamiltonian formulation of a simple theory of the teleparallel geometry. A Okołów, J Świeżewski, arXiv:1111.5490Class. Quant. Grav. 2945008Okołów A, Świeżewski J 2012 Hamiltonian formulation of a simple theory of the teleparallel geometry Class. Quant. Grav. 29 045008 E-print arXiv:1111.5490
Positive energy via the teleparallel Hamiltonian Int. J Nester, J. Mod. Phys. A. 4Nester J M 1989 Positive energy via the teleparallel Hamiltonian Int. J. Mod. Phys. A 4 1755-1772
| [] |
[
"Shaping oscillations via mixed feedback",
"Shaping oscillations via mixed feedback"
] | [
"Weiming Che ",
"Fulvio Forni "
] | [] | [] | We study the problem of controlling oscillations in closed loop by combining positive and negative feedback in a mixed configuration. We develop a complete design procedure to set the relative strength of the two feedback loops to achieve steady oscillations. The proposed design takes advantage of dominance theory and adopts classical harmonic balance and fast/slow analysis to regulate the frequency of oscillations. The design is illustrated on a simple two-mass system, a setting that reveals the potential of the approach for locomotion, mimicking approaches based on central pattern generators. arXiv:2103.13790v2 [eess.SY] 6 Sep 2021 | 10.1109/cdc45484.2021.9683238 | [
"https://arxiv.org/pdf/2103.13790v2.pdf"
] | 232,352,424 | 2103.13790 | 479a2b13d72aecf8d8f6da24ad88e41c99fd9bae |
Shaping oscillations via mixed feedback
Weiming Che
Fulvio Forni
Shaping oscillations via mixed feedback
We study the problem of controlling oscillations in closed loop by combining positive and negative feedback in a mixed configuration. We develop a complete design procedure to set the relative strength of the two feedback loops to achieve steady oscillations. The proposed design takes advantage of dominance theory and adopts classical harmonic balance and fast/slow analysis to regulate the frequency of oscillations. The design is illustrated on a simple two-mass system, a setting that reveals the potential of the approach for locomotion, mimicking approaches based on central pattern generators. arXiv:2103.13790v2 [eess.SY] 6 Sep 2021
I. INTRODUCTION
Oscillations are important system behaviors. There are rich examples in biology, like the rhythmic movements of respiration and locomotion [1], the cardiac rhythm, and several forms of biochemical oscillations [2]. In those examples, oscillations are robust to disturbances, yet flexible to respond to external inputs. The mechanisms of biological oscillations have inspired several attempts in engineering, like in robot locomotion [3], [4], and in neuromorphic circuits design [5]. These examples encourage the question of how to design a feedback controller to enforce oscillations in closed loop which are robust to perturbations yet tunable, that is, flexible enough to adapt their frequency and other features to the needs of specific engineering tasks, [6].
In this paper, we study the generation and tuning of oscillations through feedback. We look into a controller with a mixed feedback structure, identified by two parallel feedback loops with opposite signs. The reason to look into this controller is that the presence of both positive and negative feedback loops is a recurrent structure in biology [7], [8], [9], with sharp examples in neuroscience [10], [11], [12]. Similarly in engineering, various combinations of positive and negative feedback are widespread in the design of electronic oscillators [13], [14], [15], [16]. In fact, the presence of positive and negative feedback is not accidental but motivated by the specific features of robustness and flexibility that this combination guarantees. In this paper we explore the mixed feedback control structure, proposing a design that tunes the balance and strength of positive and negative feedback loops to achieve oscillations of desired frequencies.
Finding oscillations in nonlinear systems is challenging. In contrast to the large body of methods to stabilize system equilibria, we have a few tools in control theory to enforce and stabilize periodic trajectories. The problem of designing controlled oscillations can be divided into two W. Che is supported by CSC Cambridge Scholarship. W. Che and F. Forni are with the Department of Engineering, University of Cambridge, CB2 1PZ, UK wc289|[email protected] main components. The first is to determine the existence of a stable limit cycle given the system's dynamics. The principal tool here is the Poincaré-Bendixon theorem [17], which is constrained to planar systems. This limitation is overcome here using dominance theory [18], [19], rooted in the theory of monotone systems with respect to high rank cones [20], [21], [22], [23]. Dominance theory is able to determine when a high-dimensional system has a low dimensional attractor, possibly captured by planar dynamics. The combination of dominance theory and differential dissipativity also provides a way to characterize robustness and interconnections of oscillating nonlinear systems.
The second component is to shape the limit cycle, to achieve a certain frequency of oscillations in closed loop. In this paper we will take advantage of the harmonic balance method for oscillations in the quasi-harmonic regime [24], [25], [26], [27]. We will also look into the literature of relay feedback systems to tackle relaxation oscillations [28], [29].
In what follows we study the oscillator design problem using the mixed feedback amplifier proposed in our previous work [30]. The mixed feedback controller is tuned by two parameters: the balance β regulates the relative strength between positive and negative feedback, and the gain k determines the collective feedback strength. The objective is to select balance and gain such that the closed loop oscillates at a predefined frequency. We discuss why the mixed feedback structure leads to robust oscillations and we use dominance theory to find the region R osc of balances β and gains k that guarantee steady oscillations. We show that the mixed feedback controller can achieve both quasi-harmonic oscillations and relaxation oscillations, as regulated by the balance parameter β. Thus, we use harmonic balance and fast/slow analysis to find the specific parameter values within R osc that guarantee the desired frequency of oscillations.
The paper is organized as follows. Section II presents the mixed feedback amplifier, where we also show how the combination of positive and negative feedback leads to a robust destabilizing mechanism that enables oscillations. Section III discusses the main design methods, namely dominance theory, harmonic balance, and fast/slow analysis. We also show how to combine these methods for design purposes. The discussion in Sections IV and V focuses on parameter tuning based on harmonic balance and fast/slow analysis, with the goal of achieving a desired oscillation frequency in closed loop. The design is illustrated on a two-mass spring-damper system in Section VI, which provides a simplified, rudimentary model of locomotion. Taking advantage of asymmetric frictions, we show how different controlled oscillations lead to different locomotion regimes.
The paper is concluded in section VII.
II. THE MIXED FEEDBACK AMPLIFIER
We consider the mixed feedback amplifier proposed in [30], which has the structure shown in Fig. 1. The load L(s) is controlled by a positive feedback channel C p (s) combined with a negative feedback channel C n (s). In this paper we restrict the analysis to first order transfer functions
C p (s) = 1 τ p s + 1 C n (s) = 1 τ n s + 1 τ n > τ p > 0,(1)
which guarantee that the phases of these transfer functions each never exceeds 90 degrees, so that the splitting between negative and positive feedback channels is consistent at any frequency. We also assume that the load L(s) has at least relative degree one, with poles and zeros whose real part is to the left of −1/τ n , and that L(0) = 1 (normalized DC gain). The combination of positive and negative feedback is regulated by the parameters k and β. The gain k ≥ 0 controls the overall magnitude of the feedback and the balance 0 ≤ β ≤ 1 controls the relative strength of two feedback channels. We denote by G(s) the transfer function from u to z:
L(s) r x − k C p (s) + − x p x n C n (s) β 1 − β y u + z ϕG(s) = C(s)L(s) = − β(τ n + τ p ) − τ p s + 2β − 1 (τ p s + 1)(τ n s + 1) L(s)
(2) The two feedback channels are mixed and fed into a sigmoid function ϕ (bounded, differentiable, non decreasing, real function with one inflection point) whose slope satisfies 0 ≤ ϕ (y) ≤ 1. This monotone nonlinearity preserves the direction of the control action, in the sense that ϕ(y)y ≥ 0. It also guarantees boundedness of the closed loop trajectories if L(s), C p (s), and C n (s) have poles in the left half of the complex plane (by BIBO stability of G(s)). With this representation, the mixed-feedback closed loop has the structure of a Lure system, as shown in Fig. 2.
For any constant reference input r (in Fig. 1), the closedloop equilibria must be compatible with the equation
ϕ(y) − r = y kG(0) = y k(2β − 1)(3)
where −k(2β − 1) is the DC gain of kG(s). The stability of these equilibria can be verified numerically, [30]. However, by construction, the root locus of the linearized closed-loop system has the form in Fig. 3. The red line crossing the real axis at −λ, λ > 1 τp , separates the poles and the zeros of L(s) from the poles of C(s). The zero z β of C(s) can be placed at any point of the positive real axis by tuning the balance β. The curves in blue represent the motion of the closed-loop poles (for increasing value of the feedback gain k) and show how z β guides the loss of stability in closed loop, for sufficiently large gain k. Indeed, the mixed feedback guarantees bounded closed-loop trajectories, and provides a robust destabilizing mechanism, controlled by balance and gain of the feedback. We use these features to control the closed loop into stable oscillations.
III. DETERMINE THE EXISTENCE OF OSCILLATIONS A. Oscillations via 2-dominance
Determining the existence of stable periodic oscillations is difficult for high dimensional systems. Dominance theory [18], [19] simplifies the study of periodic oscillations.
In the definition below from [18], we use ∂f (x) to denote the Jacobian of f . We also enforce an inertia constraint (p, 0, n − p) on a symmetric matrix P , which means that P has p negative eigenvalues and n − p positive eigenvalues.
Definition 1:
The nonlinear systemẋ = f (x) is pdominant with rate λ ≥ 0 if and only if there exist a symmetric matrix P with inertia (p, 0, n − p) and ε ≥ 0 such that the prolonged system
ẋ = f (x) δẋ = ∂f (x)δx (x, δx) ∈ R n × R n .(4)
satisfies the conic constraint δẋ δx
T 0 P P 2λP + εI δẋ δx ≤ 0(5)
along all its trajectories. The property is strict if ε > 0.
Combining (4) and (5) we get the Lyapunov inequality
(∂f (x) + λI) T P + P (∂f (x) + λI) ≤ −εI .(6)
Given the constraint on the inertia of P , a necessary condition for the feasibility of this inequality is that ∂f (x) + λI has p unstable eigenvalues and n − p stable eigenvalues, uniformly in x. That is, ∂f (x) must have p eigenvalues to the right of −λ and n − p to the left of −λ, a condition that is satisfied by the mixed feedback amplifier for p = 2 (for small k at least), as illustrated in Fig. 3. Dominance theory provides an analytical tool to show the existence of a low dimensional attractor in a high dimensional nonlinear system, as clarified by the following theorem ([18, Corollary 1]). Theorem 1: For a strict p-dominant system with dominant rate λ ≥ 0, every bounded trajectory asymptotically converges to • a unique fixed point if p = 0; • a simple attractor if p = 2, that is, a fixed point, a set of fixed points and connecting arcs, or a limit cycle. Theorem 1 clarifies that the asymptotic behavior of a 2dominant system corresponds to the one of a planar system. This enables the use of Poincaré-Bendixson-like approaches to characterize oscillations.
−1 τ p −1 τ n z β −λ
We will use dominance and Theorem (1) to enforce oscillations in the mixed-feedback closed loop. In fact, the mixed-feedback closed loop is 2-dominant in a specific range of k and β, as clarified by the next theorem (adapted from [30,Theorem 5]).
Theorem 2: Consider a rate λ for which the shifted transfer function G(s − λ) has two unstable poles. Then, for any constant r and any β ∈ [0, 1], the mixed-feedback closed loop system is 2-dominant with rate λ for any gain 0 ≤ k < k 2 , where
k 2 = ∞ if min ω (G(jω − λ)) ≥ 0 − 1 min ω (G(jω−λ)) otherwise.(7)
Proof: For k <k 2 , kG(jω − λ) lies to the right of the vertical line passing through −1, which guarantees that the closed loop is 2-dominant by [19,Corollary 4.5]. Note that k 2 is always greater than zero since G(jω − λ) has finite magnitude for all ω ∈ R ± {∞}.
2-dominance and the root-locus in Fig. 3 suggest a way to guarantee stable oscillations in closed loop:
• we first determine the range of balances β and gains k that guarantees 2-dominance. We denote this region in parameter space of (k, β) by R 2dom ; • within R 2dom , stable oscillations can be precisely determined by excluding the parametric ranges where equilibria are stable. If the equilibria are unstable, Theorem 1 guarantees that the mixed-feedback closed loop has periodic oscillations, since trajectories are bounded. We call the region of guaranteed oscillations R osc ⊆ R 2dom .
For a first order load L(s), an example is provided in Figure 6(a). For a detailed study please refer to [30].
B. Oscillations via harmonic balance
We briefly recap the classical harmonic balance method to predict oscillations, also known as describing function method [31,Chapter 7]. The goal is to use this method to characterize the oscillations of the mixed-feedback closed loop in the quasi-harmonic regime, when possible (typically for small β).
The idea is to look for periodic oscillations of the form y(t) = E sin(ωt). We approximate the nonlinearity ϕ by its describing function, denoted by N (E), which is given by the ratio between the first coefficient of the Fourier series of ϕ(E sin(ωt)) (first harmonic) and the oscillation amplitude E. An oscillation is predicted at the frequency ω that satisfies
kG(jω) = − 1 N (E) .(8)
This corresponds to the intersection between the Nyquist plot of kG(s) and the curve − 1 N (E) . There is no closed form solutions to the Fourier series of smooth odd nonlinearities ϕ, while they can be wellapproximated by piecewise-linear function
ϕ pl (y) = y if |y| ≤ 1 1 otherwise.(9)
For simplicity, in what follows we will restrict the harmonic balance analysis of the mixed feedback closed loop to ϕ pl . This leads to the describing function
N pl (E) = 1 if E ≤ 1 2 π [sin −1 ( 1 E )+ 1 E (1− 1 E 2 ) 1 2 ] if E > 1(10)
which is always real (as usual for odd nonlinearities) and monotonically decreasing from 1 to 0 as E → ∞. Indeed, our analysis will not be general for reasons of simplicity but can be adapted to any sigmoidal nonlinearity.
C. Relaxation oscillations via fast/slow analysis
Oscillations can be also predicted in the time domain. We adapt the method in [28], to determine the oscillations of the mixed-feedback closed loop in the relaxation regime, when possible (typically for large β).
Again, we approximate ϕ(·) with the piecewise linear ϕ pl (·) for simplicity. For |y| < 1, the closed loop is a linear system. For |y| ≥ 1, |y| = 1 defines two switching planes, as shown in Fig. 4. We assume that the traveling time between the two switching planes is negligible (fast unstable linear dynamics). This corresponds to the case of a relaxation oscillation. In this setting, the output y jumps between positive and negative saturated value ±1, producing a nearly square wave ϕ(y). We can thus proceed like in [28].
Consider any minimal state-space realization (A, B, C, 0), of G(s) with state given by (x, x p , x n ), wherex is the state component related to the load L(s). C corresponds to the matrix 0 . . . 0 β β − 1 . The half cycle in Fig. 4 starts at the initial state a, with y = kCa = −1 that triggers the fast switch. Then, by symmetry, this half cycle ends at Figure 5 shows a typical shape of f (h) for the mixed feedback amplifier (this is obtained for a first order load L(s) = 1 τ s+1 ). In general, the initial part of the curve corresponds to fast transients (the load, for example) driving the transition between switching planes. So, short time solutions h 1 should be neglected. Indeed, an oscillation with frequency π/h 2 is predicted if there is a solution that corresponds to a long interval h 2 to (12). This identifies the time of the half cycle illustrated in Fig. 4.
There are two situations where there is no long interval solution h 2 of (12) and the fast/slow analysis predicts no oscillations: for fixed k > 0, 1) min h≥0 kf (h) > −1;
2) lim h→+∞ kf (h) = kG(0) = −k(2β − 1) < −1.
1) represents the case in which kf (h) has no intersections at all with the line −1.
2) represents the case in which the output does not reach the switching plane after initial transients, that is, no half-cycle occurs. It follows that 1) sets a lower bound on k. Furthermore, 2) constraints k only when 2β − 1 > 0, that is, when β > 1 2 . In that case, 2) sets an upper bound on k for oscillations. For illustration, these two bounds on k are represented in Figure 6(c), for the case of first order load. Note that for β ≤ 1 2 there is no upper bound on k.
D. Integration of the three methods for control design
The blue regions in Fig. 6 show that all three methods lead to similar predictions. These regions are obtained for the simple setting of a first order load L(s) = 1 τ s+1 , for time-constants τ = 0.01, τ p = 0.1, and τ n = 1.
The difference among the three methods is that dominance analysis is not an approximated method, therefore it can be used to certify the existence of oscillations, in both harmonic and relaxation regimes. In this sense, dominance analysis responds to the shortcoming of harmonic balance and fast/slow analysis, due to their approximation natures. At the same time, dominance analysis does not provide any information about the oscillation frequency of the mixedfeedback closed loop. Here harmonic balance and fast/slow analysis respond to the shortcoming of dominance analysis, providing guidance on the selection of β and k to achieve a desired oscillation frequency in closed loop.
The idea is thus to integrate these methods to achieve a reliable control design of oscillators. We use dominance theory to determine the parameter range that guarantees oscillations. Within this range, we use either harmonic balance or fast/slow analysis for controlling the oscillation frequency.
IV. FREQUENCY SHAPING VIA HARMONIC BALANCE
A. Parameter range for accurate prediction
The harmonic balance method assumes that the linear subsystem G(s) is a low pass filter such that higher order harmonic signals in ϕ(y) are suppressed. This means that to achieve reliable tuning of the mixed-feedback amplifier using the harmonic balance method, we need to identify gain and balance regimes, k and β, that lead to accurate predictions. Consider a desired frequency of oscillations ω r : i) we need ∠G(jω r ) = −180 • . This is achieved by selecting β, which moves the zero z β = 1−2β β(τp+τn)−τp , shaping the phase of G. For a low-pass G, the magnitude of z β should be at least larger than the magnitude of the smallest pole of G(s). This sets an upper bound
β <β = inf β∈[0,1] 1 − 2β β(τ p + τ n ) − τ p ≥ 1 τ n ;(13)
ii) the gain k modulates the magnitude of G(s). To filter out the higher harmonics, G(s) must have gain less than 1 for frequencies nω r , n ≥ 2. For each β <β, this enforces an upper limitk(β) on k given bȳ
k(β) = 1 |G(2jω r )| .(14)
Prediction from dominance analysis As an example,β andk are illustrated in Fig. 6(b) for the case of a first order load. The dark blue region guarantees accurate predictions. For k and β not in the dark blue region, higher order harmonics are not attenuated; prediction accuracy degrades and oscillations start to transform into relaxation oscillations. For k and β not in the dark blue region, prediction accuracy degrades since higher order harmonics are not attenuated. Oscillations start to transform into relaxation oscillations. Thus for design using harmonic balance method, we search for k <k and β <β.
B. Design procedure
Within the parametric range discussed in the last section, we use harmonic balance to solve the following problem: find (k, β) ∈ R osc , k <k, and β <β such that the mixedfeedback closed loop oscillates with desired frequency ω r .
The key is to find a β <β such that ∠(G(jω)) = −180 • . Denote the phase contribution of the numerator and denominator of −C(jω r ) respectively by θ n and θ d . The phase of G(s) at ω r is ∠G(jω r ) = ∠(−1) + θ d + θ n + ∠L(jω r ) = 180 • + 2mπ (15) Note that both β and k only affect θ n (the phase of the denominator is affected by the selection of the time constants τ n and τ p ). Thus, for any given ω r ,
• find β such that θ n = −θ d − ∠L(jω r ) − 2mπ.
• if β <β, select a suitable k <k • If β ≥β , repeat the design for different time constants τ p and τ n . We close this section by clarifying the relationship between β and θ n . Note that θ n = arctan (β(τp+τn)−τp)ωr 2β−1 , whose range depends on the sign of β(τ p + τ n ) − τ p and 2β − 1. Therefore, θ n falls into three different ranges for β ∈ [0, 1]:
The discussion above suggests that the desired oscillation frequency ω r must decrease as β increases. This is because
β ∈ [0, τp τp+τn ) [ τp τp+τn , 0.5) [0.5, 1] θn ∈ [180 • , 270 • ] [90 • , 180 • ] [0 • , 90 • ]
the range of θ n gets smaller, therefore it is harder to balance the phase contribution of θ d +∠L(jω r ) for large frequencies.
V. FREQUENCY SHAPING VIA FAST/SLOW METHOD
A. Parametric range for accurate prediction
The first step for fast/slow method is also to quantify a sub parametric range (k, β) of R osc which gives accurate predicted frequencies. The main source of the approximation error is the presence of a non-negligible switching transient, that is, the time the system needs to move between switching planes in Fig. 4. This time is reduced for large β, which increases positive feedback and pushes the control signal towards saturation. Likewise, the distance d among the switching planes is also reduced by larger feedback gains
k, since d = 2 k √ 2β 2 −2β+1 .
Unlike the harmonic balance method, there is no quantitative criteria, such as loop shape and gain, to set the range of (k, β). We thus restrict the use of the fast/slow method to those parameters for which harmonic balance is unreliable (k >k, and β >β) with lower boundsk andβ defined in Section IV.
B. Design procedure
The design problem for the fast/slow method is as follows: find (k, β) ∈ R osc , k >k and β >β such that the mixedfeedback closed loop oscillates with desired frequency ω r .
The desired frequency ω r defines the half cycle period h r = π/ω r . For this period, (k, β) are found by solving kf (h r ) = −1. Remember that kf (h r ) = kC(I + e Ahr ) −1 A −1 (e Ahr − I)B = −1 , (16) which shows that k and β appear only in kC = k 0 . . . 0 β β − 1 . This strongly simplifies the search for feasible pairs (k, β).
In contrast to the harmonic balance method where each desired frequency ω r corresponds to a specific balance β, the fast/slow method is more flexible, possibly offering several solutions of (k, β) for the same desired frequency ω r .
VI. EXAMPLE: CONTROLLED OSCILLATIONS OF A TWO-MASS SYSTEM
As an illustration, we present an example of two-mass system in Fig. 7. With asymmetric frictions at contact with ground, oscillations of the two masses lead to a positive average displacement of the center of mass of the system. In this setting, the two-mass system is a basic model for studying locomotion. The dynamics of the two-mass system [32] satisfies
m 2 m 1 x 1 x 2 F F f fẍ 1 = −k m (x 1 − x 2 ) − d m (ẋ 1 −ẋ 2 ) + γF − f (ẋ 1 ) x 2 = k m (x 1 − x 2 ) + d m (ẋ 1 −ẋ 2 ) − γF − f (ẋ 2 )(17)
where k m = 100 and d m = 10 are normalized elastic and damping coefficients, respectively; F is the (internal) force produced by the actuator, scaled by a factor γ = 100 for simplicity; and f models asymmetric friction forces.
The objective is to design a feedback controller that drives the two-mass system into oscillations, by acting on the force F . To keep the design within the linear setting of this paper we design our controller by neglecting the asymmetric friction forces, taking f = 0. Define w = x 1 −x 2 . For f = 0 we haveẅ = −2k m w − 2d mẇ + 2γF , which gives the load transfer function
L(s) = 200 s 2 + 20s + 200 .(18)
L(s) has poles at −10 ± 10j. In the notation of Fig 1, w is the output load L(s) block, which is used by the mixed feedback channel to generate the control signal F = ϕ(y) (simulations use ϕ = tanh). We consider r = 0 and we set the positive and negative feedback time constants as τ p = 1 and τ n = 10. With this load and time constants, we get
G(s) = −200 1.1β − 1 s + 2β − 1 (s 2 + 20s + 200)(s + 1)(10s + 1)(19)
As an illustration, we consider the designing of two oscillation frequencies, ω r = 1 rad/s and ω r = 0.1 rad/s. We use harmonic balance method for ω r = 1 rad/s. From (13), the balance upper bound isβ = 0.3548, under which the linear system is low pass. For ω r = 1, (15) leads to β = 0.1538. Using (14) we get the gain upper boundk = 28.9494. By dominance analysis, oscillations exist for k > 14.5217 for β = 0.1538. We thus choose k = 20.
By using the harmonic balance procedure for ω r = 0.1 rad/s, (15) leads to the selection β = 0.8226 which is much greater thanβ, meaning that the oscillation at such low frequency is of the relaxation type. Hence we switch to the fast/slow method. From Section V, we take h r = π/ω r ≈ 31.4 s. Then, β r = 0.5 and k = 24 is a solution to (16) These parameters are compatible with dominance, which guarantees oscillations for any k > 3.1584 given β. Fig. 8 presents the simulation results for F = ϕ(y), where y is generated by the mixed feedback channels in Fig. 1, for f = 0 (no asymmetric frictions). The frequencies of the simulations agree with the specifications. We now reintroduce the asymmetric friction forces f . A complete analysis of the robustness of the oscillations is beyond the scope of this paper. We just emphasize that the mixed feedback induces a hyperbolic instability of the closed loop equilibria, which in turn guarantees robustness of oscillations for small perturbations. This is illustrated through simulations. For i ∈ {1, 2} we take the forward and backward friction forces as:
f (ẋ i ) = 5 ifẋ i > 0 0 ifẋ i = 0 −20 ifẋ i < 0.(20)
The closed loop maintains its oscillation patterns with mild frequency changes, as shown in Fig. 9, (a) and (b). Fig. 9, (c) and (d) illustrate the forward motion of the system.
VII. CONCLUSIONS
We study the problem of controlling oscillations in closed loop by combining positive and negative feedback in a mixed configuration. This is illustrated by developing a complete design, using dominance theory to set balance β and gain k to achieve reliable oscillations, and harmonic balance and fast/slow analysis to regulate those oscillations towards a desired frequency. The design is illustrated on a simple twomass system, where the mixed feedback regulates oscillations to achieve locomotion, emulating approaches based on central pattern generators. In contrast to classical entrainment due to a driving external source, generating endogenous oscillations through feedback opens the way to questions of sensitivity/robustness of the oscillations to interconnections. Understanding how the frequency of oscillations is shaped by the interaction with an external system is a relevant question to enable adaptive control schemes in applications.
Fig. 1 .
1The Block diagram representation of the mixed feedback amplifier.
Fig. 2 .
2The Lure feedback system.poles and zeros of L(s)
Fig. 3 .
3Root locus of G(s).
Fig. 4 .Fig. 5 .
45An illustration of the switching planes projected on xp − xn plane. The red curve denotes the projected state trajectory of a half cycle and 'a' denotes the initial condition on the switching plane. −a, which satisfies −kCa = 1. Using the explicit solution of linear systems for constant inputs, − a = e Ah a − Γ(h) ⇒ a = (I + e Ah ) −1 Γ(h) Aτ dτ B = A −1 (e Ah − I)B. Definef (h) = Ca. Then, the half period of oscillation h satisfies kf (h) = kCa = kC(I + e Ah ) −1 Γ(h) = −1 . An example of kf (h) for k = 10, β = 0.4, τ = 0.01, τp = 0.1, and τn = 1. τ is the the constant of the first order load L(s) = 1 τ s+1 .
Fig. 6 .
6Predictions of the mixed feedback amplifier with L(s) = 1 τ s+1 and τ = 0.01, τp = 0.1, τn = 1, and r = 0. (a) Dominance analysis. Grey region -2-dominant region, R 2dom (stable and oscillatory regimes). Blue region -oscillations region Rosc (unstable equilibrium). (b) Harmonic balance. Gray region -multiple intersection between Nyquist plot and N (E). Blue region -one intersection between Nyquist plot and N (E). Dark blue region: accurate prediction of oscillations (G(s) is low pass). (c) Fast-slow analysis. Blue region -oscillations with estimated period π/h 2 .
Fig. 7 .
7The double mass system.
Fig. 8 .
8Closed-loop controlled oscillations of the two-mass system. Left: desired frequency ωr = 1 rad/s, achieved frequency 0.9906 rad/s. Right: desired frequency ωr = 0.1 rad/s, achieved frequency 0.1238 rad/s.
Fig. 9 .
9Closed-loop oscillations of the two-mass system with frictions Left: desired frequency ω = 1 rad/s, achieved frequency 0.9374 rad/s. Right: desired frequency ωr = 0.1 rad/s, achieved frequency 0.1185 rad/s.
Central pattern generators and the control of rhythmic movements. E Marder, D Bucher, Current biology. 1123E. Marder and D. Bucher, "Central pattern generators and the control of rhythmic movements," Current biology, vol. 11, no. 23, pp. R986- R996, 2001.
Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour. A Goldbeter, Cambridge University PressA. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge University Press, 1997.
Realization of dynamic walking and running of the quadruped using neural oscillator. H Kimura, S Akiyama, K Sakurama, Autonomous robots. 73H. Kimura, S. Akiyama, and K. Sakurama, "Realization of dynamic walking and running of the quadruped using neural oscillator," Au- tonomous robots, vol. 7, no. 3, pp. 247-258, 1999.
From swimming to walking with a salamander robot driven by a spinal cord model. A J Ijspeert, A Crespi, D Ryczko, J.-M Cabelguen, science. 3155817A. J. Ijspeert, A. Crespi, D. Ryczko, and J.-M. Cabelguen, "From swimming to walking with a salamander robot driven by a spinal cord model," science, vol. 315, no. 5817, pp. 1416-1420, 2007.
Neuromodulation of neuromorphic circuits. L Ribar, R Sepulchre, IEEE Transactions on Circuits and Systems I: Regular Papers. L. Ribar and R. Sepulchre, "Neuromodulation of neuromorphic cir- cuits," IEEE Transactions on Circuits and Systems I: Regular Papers, 2019.
Central pattern generators for locomotion control in animals and robots: a review. A J Ijspeert, Neural networks. 214A. J. Ijspeert, "Central pattern generators for locomotion control in animals and robots: a review," Neural networks, vol. 21, no. 4, pp. 642- 653, 2008.
Modeling circadian oscillations with interlocking positive and negative feedback loops. P Smolen, D A Baxter, J H Byrne, Journal of Neuroscience. 2117P. Smolen, D. A. Baxter, and J. H. Byrne, "Modeling circadian oscillations with interlocking positive and negative feedback loops," Journal of Neuroscience, vol. 21, no. 17, pp. 6644-6656, 2001.
Positive feedback in cellular control systems. A Y Mitrophanov, E A Groisman, Bioessays. 306A. Y. Mitrophanov and E. A. Groisman, "Positive feedback in cellular control systems," Bioessays, vol. 30, no. 6, pp. 542-555, 2008.
Robust, tunable biological oscillations from interlinked positive and negative feedback loops. T Y Tsai, Y S Choi, W Ma, J R Pomerening, C Tang, J E Ferrell, Science. 3215885T. Y.-C. Tsai, Y. S. Choi, W. Ma, J. R. Pomerening, C. Tang, and J. E. Ferrell, "Robust, tunable biological oscillations from interlinked positive and negative feedback loops," Science, vol. 321, no. 5885, pp. 126-129, 2008.
Neuromodulation of circuits with variable parameters: Single neurons and small circuits reveal principles of state-dependent and robust neuromodulation. E Marder, T O'leary, S Shruti, Annual Review of Neuroscience. 371E. Marder, T. O'Leary, and S. Shruti, "Neuromodulation of circuits with variable parameters: Single neurons and small circuits reveal principles of state-dependent and robust neuromodulation," Annual Review of Neuroscience, vol. 37, no. 1, pp. 329-346, 2014.
Neuronal behaviors: A control perspective. G Drion, T O'leary, J Dethier, A Franci, R Sepulchre, 54th IEEE Conference on Decision and Control. G. Drion, T. O'Leary, J. Dethier, A. Franci, and R. Sepulchre, "Neuronal behaviors: A control perspective," in 54th IEEE Conference on Decision and Control, pp. 1923-1944, 2015.
Control across scales by positive and negative feedback. R Sepulchre, G Drion, A Franci, Robotics, and Autonomous Systems. 2Annual Review of ControlR. Sepulchre, G. Drion, and A. Franci, "Control across scales by positive and negative feedback," Annual Review of Control, Robotics, and Autonomous Systems, vol. 2, pp. 89-113, 2019.
The history of positive feedback: The oscillating audion, the regenerative receiver, and other applications up to around 1923. D Tucker, Radio and Electronic Engineer. 422D. Tucker, "The history of positive feedback: The oscillating audion, the regenerative receiver, and other applications up to around 1923," Radio and Electronic Engineer, vol. 42, no. 2, pp. 69-80, 1972.
Feedback control: an invisible thread in the history of technology. D S Bernstein, IEEE Control Systems Magazine. 222D. S. Bernstein, "Feedback control: an invisible thread in the history of technology," IEEE Control Systems Magazine, vol. 22, no. 2, pp. 53- 68, 2002.
Van der pol and the history of relaxation oscillations: Toward the emergence of a concept. J Ginoux, C Letellier, Chaos: An Interdisciplinary Journal of Nonlinear Science. 22223120J. Ginoux and C. Letellier, "Van der pol and the history of relaxation oscillations: Toward the emergence of a concept," Chaos: An Inter- disciplinary Journal of Nonlinear Science, vol. 22, no. 2, p. 023120, 2012.
Linear and Nonlinear Circuits. C Chua, L Desoer, E Kuh, Mc Graw-Hill Book CompanyC. Chua, L.O a d Desoer and E. Kuh, Linear and Nonlinear Circuits. Mc Graw-Hill Book Company, 1987.
M Hirsch, S Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press60M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Pure and Applied Mathematics, Vol. 60). Aca- demic Press, 1974.
Differential dissipativity theory for dominance analysis. F Forni, R Sepulchre, IEEE Transactions on Automatic Control. 646F. Forni and R. Sepulchre, "Differential dissipativity theory for dom- inance analysis," IEEE Transactions on Automatic Control, vol. 64, no. 6, pp. 2340-2351, 2018.
Analysis of lur'e dominant systems in the frequency domain. F A Miranda-Villatoro, F Forni, R J Sepulchre, Automatica. 98F. A. Miranda-Villatoro, F. Forni, and R. J. Sepulchre, "Analysis of lur'e dominant systems in the frequency domain," Automatica, vol. 98, pp. 76-85, 2018.
Existence of period orbits of autonomous ordinary differential equations. R Smith, Proceedings of the Royal Society of Edinburgh. 85R. Smith, "Existence of period orbits of autonomous ordinary differ- ential equations," in Proceedings of the Royal Society of Edinburgh, vol. 85A, pp. 153-172, 1980.
Orbital stability for ordinary differential equations. R Smith, Journal of Differential Equations. 692R. Smith, "Orbital stability for ordinary differential equations," Journal of Differential Equations, vol. 69, no. 2, pp. 265 -287, 1987.
Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems. L Sanchez, Journal of Differential Equations. 2465L. Sanchez, "Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems," Journal of Differential Equations, vol. 246, no. 5, pp. 1978 -1990, 2009.
Existence of periodic orbits for high-dimensional autonomous systems. L Sanchez, Journal of Mathematical Analysis and Applications. 3632L. Sanchez, "Existence of periodic orbits for high-dimensional au- tonomous systems," Journal of Mathematical Analysis and Applica- tions, vol. 363, no. 2, pp. 409 -418, 2010.
Multiple-input describing functions and nonlinear system design. A Gelb, W Vander Velde, McGraw HillA. Gelb and W. Vander Velde, Multiple-input describing functions and nonlinear system design. McGraw Hill, 1968.
Describing functions revisited. A Mees, A Bergen, IEEE Transactions on Automatic Control. 20A. Mees and A. Bergen, "Describing functions revisited," IEEE Transactions on Automatic Control, vol. 20, pp. 473-478, Aug 1975.
Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics. A Tesi, E H Abed, R Genesio, H O Wang, Automatica. 329A. Tesi, E. H. Abed, R. Genesio, and H. O. Wang, "Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics," Automatica, vol. 32, no. 9, pp. 1255-1271, 1996.
Multivariable harmonic balance for central pattern generators. T Iwasaki, Automatica. 4412T. Iwasaki, "Multivariable harmonic balance for central pattern gen- erators," Automatica, vol. 44, no. 12, pp. 3061-3069, 2008.
Oscillations in systems with relay feedback. K J Åström, Adaptive control, filtering, and signal processing. SpringerK. J.Åström, "Oscillations in systems with relay feedback," in Adaptive control, filtering, and signal processing, pp. 1-25, Springer, 1995.
Lecture notes on nonlinear vibrations. R H Rand, R. H. Rand, "Lecture notes on nonlinear vibrations," 2012.
A tunable mixed feedback oscillator. W Che, F Forni, European Control ConferenceW. Che and F. Forni, "A tunable mixed feedback oscillator," European Control Conference, 2021.
H Khalil, Nonlinear Systems. Pearson Education. Prentice HallH. Khalil, Nonlinear Systems. Pearson Education, Prentice Hall, 2002.
Oscillatory locomotion. T Majewski, R , CONIELE-COMP 2011, 21st International Conference on Electrical Communications and Computers. IEEET. Majewski and R. Alejos, "Oscillatory locomotion," in CONIELE- COMP 2011, 21st International Conference on Electrical Communi- cations and Computers, pp. 322-327, IEEE, 2011.
| [] |
[
"ON C 1+α REGULARITY OF SOLUTIONS OF ISAACS PARABOLIC EQUATIONS WITH VMO COEFFICIENTS",
"ON C 1+α REGULARITY OF SOLUTIONS OF ISAACS PARABOLIC EQUATIONS WITH VMO COEFFICIENTS"
] | [
"N V Krylov "
] | [] | [] | We prove that boundary value problems for fully nonlinear second-order parabolic equations admit Lp-viscosity solutions, which are in C 1+α for an α ∈ (0, 1). The equations have a special structure that the "main" part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.2010 Mathematics Subject Classification. 35B65, 35D40. | null | [
"https://arxiv.org/pdf/1211.4882v1.pdf"
] | 119,326,403 | 1211.4882 | d362aa3a54cd30f060eb2a1c3a75c8bbd97525a2 |
ON C 1+α REGULARITY OF SOLUTIONS OF ISAACS PARABOLIC EQUATIONS WITH VMO COEFFICIENTS
20 Nov 2012
N V Krylov
ON C 1+α REGULARITY OF SOLUTIONS OF ISAACS PARABOLIC EQUATIONS WITH VMO COEFFICIENTS
20 Nov 2012
We prove that boundary value problems for fully nonlinear second-order parabolic equations admit Lp-viscosity solutions, which are in C 1+α for an α ∈ (0, 1). The equations have a special structure that the "main" part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.2010 Mathematics Subject Classification. 35B65, 35D40.
Introduction
In this article we take a function H(u, t, x),
u = (u ′ , u ′′ ), u ′ = (u ′ 0 , u ′ 1 , ..., u ′ d ) ∈ R d+1 , u ′′ ∈ S, (t, x) ∈ R d+1 , where S is∂ t = ∂ ∂t , D 2 u = (D ij u), Du = (D i u), D i = ∂ ∂x i , D ij = D i D j .
Our main goal is to establish the existence of L p -viscosity solutions of boundary value problems associated with (1.1), solutions, which are in C 1+α for an α ∈ (0, 1).
Let us briefly discuss what the author was able to find in the literature concerning this kind of regularity. The articles cited below contain a very large amount of information concerning all kinds of issues in the theory of fully nonlinear elliptic and parabolic equations, but we will focus only on one of them. Trudinger [12], [13] and Caffarelli [1] were the first authors who proved C 1+α regularity for fully nonlinear elliptic equations of type F (u, Du, D 2 u, x) = f without convexity assumptions on F . The assumptions in these papers are different. In [1] the function F is independent of u ′ and, for each u ′′ uniformly sufficiently close to a function which is continuous with respect to x. In [12] and [13] the function F depends on all arguments but is Hölder continuous in (u ′ 0 , x). Next step in what concerns C 1+α -estimates for the elliptic case was done byŚwiȩch [11], who considered general F , imposed the same condition as in [1] on the x-dependence, which is much weaker than in [12] and [13], but also imposed the Lipschitz condition on the dependence of F on u ′ 1 , ..., u ′ d . In [12] and [13] only continuity with respect to u ′ 1 , ..., u ′ d is assumed.
In case of parabolic equations interior C 1+α -regularity was established by Wang [14] under the same kind of assumption on the dependence of H on (t, x) as in [1] and assuming that H is almost independent of u ′ 1 , ..., u ′ d . Then Crandall, Kocan, andŚwiȩch [2] generalized the result of [14] to the case of full equation again as in [11] assuming that H is uniformly sufficiently close to a function which is continuous with respect to (t, x) and assuming the Lipschitz continuity of H with respect to u ′ 1 , ..., u ′ d and the continuity with respect to u ′ 0 . On the one hand, our class of equations is more narrow than the one in [2] because we require the "main" part of H, called F , be positive homogeneous of degree one. On the other hand, we do not require H to be Lipschitz with respect to u ′ 1 , ..., u ′ d , the continuity with respect to u ′ suffices. Also we only need F to be measurable in t and VMO in x, say, independent of x and measurable in t.
Our methods are absolutely different from the methods of above cited articles. We do not use any ideas or facts from the theory of viscosity solutions. Instead we rely on the methodology brought into the theory of fully nonlinear equations by Safonov [9], [10] and on an idea behind the proof of the main Lemma 4.3 inspired by a probabilistic interpretation of solutions of (1.1). We only focus on interior estimates of solutions in smooth domains leaving to the interested reader investigation of the same issues in nonsmooth domains or near the boundary of sufficiently regular ones.
The article is organized as follows. Section 2 contains main results and some comments on them. In Section 3 we use a theorem from [8] to approximate the equations with H and with its main part F by those for which the solvability is known. We also leave to the interested reader carrying our results over to elliptic equations.
In Section 4 we show that the approximate principal equation with F admits solutions locally well approximated in the sup norm by affine functions. This is the most important part of the article. Section 5 contains estimates of C 1+α -norms of approximate equation with full H and in Section 6 we give the proof of our main Theorem 2.1. The last Section 7 is actually an appendix, which we need in order to be able to represent positive homogeneous of order one functions depending on parameters, such as F , as supinf's of affine functions whose coefficients inherit the regularity properties of the original function with respect to the parameters.
Main results
To state our main results, we introduce a few notation and assumptions. Fix a constant δ ∈ (0, 1], and set
S δ = {a ∈ S : δ|ξ| 2 ≤ a ij ξ i ξ j ≤ δ −1 |ξ| 2 , ∀ ξ ∈ R d },
where and everywhere in the article the summation convention is enforced.
Assumption 2.1. (i)
The function H(u, t, x) is measurable with respect to (t, x) for any u and Lipschitz continuous in u ′′ for every u ′ , (t, x) ∈ R d+1 .
(ii) For any (t, x), at all points of differentiability of H(u, t, x) with respect to u ′′ , we have (H u ′′ ij ) ∈ S δ . (iii) There is a functionH(t, x) and a constant K 0 ≥ 0 such that
|H(u ′ , 0, t, x)| ≤ K 0 |u ′ | +H(t, x).
(iv) There is an increasing continuous function ω(r), r ≥ 0, such that ω(0) = 0 and
|H(u ′ , u ′′ , t, x) − H(v ′ , u ′′ , t, x)| ≤ ω(|u ′ − v ′ |)
for all u, v, t, and x.
For R ∈ (0, ∞) and (t, x) ∈ R d+1 introduce B R = {x ∈ R d : |x| < R}, B R (x) = x + B R , C R = (0, R 2 ) × B R , C R (t, x) = (t, x) + C R .
For a Borel set Γ in R d+1 by |Γ| we denote its Lebesgue measure. Also for a function f on Γ we set
- Γ f (t, x) dxdt = 1 |Γ| Γ f (t, x) dxdt
in case Γ has a nonzero Lebesgue measure in R d+1 . Similar notation is used in case of functions f (x) on R d . We fix a constant R 0 ∈ (0, 1] and for κ ∈ (0, 2] and measurable f (t, x) introduce
f κ = sup R≤R 0 ,t,x R 2−κ - C R (t,x) |f (s, y)| d+1 dyds 1/(d+1) .
Remark 2.1. By Hölder's inequality for p ≥ d + 1
- C R (t,x) |f (s, y)| d+1 dyds 1/(d+1) ≤ N R −(d+2)/p R d+1 |f (s, y)| p dyds 1/p ,
which shows that f κ < ∞ if f ∈ L p (R d+1 ) and κ ≤ 2 − (d + 2)/p. It is useful to observe that one can take κ > 1 if f ∈ L p (R d+1 ) for p > d + 2.
In the following assumption there are three objects κ 1 = κ(d, δ) ∈ (1, 2), any κ ∈ (1, κ 1 ], and θ = θ(κ, d, δ) ∈ (0, 1]. The values of κ 1 and θ are specified later in the proof of Lemma 5.3.
|G(u, t, x)| ≤ K 0 |u ′ | +H(t, x)
and there exists a κ ∈ (1, κ 1 ] such thatH κ < ∞.
(iii) The function F is positive homogeneous of degree one with respect to u ′′ , is Lipschitz continuous with respect to u ′′ , and at all points of differentiability of F with respect to u ′′ we have F u ′′ ∈ S δ .
(iv) For any R ∈ (0, R 0 ], (t, x) ∈ R d+1 , and u ′′ ∈ S with |u ′′ | = 1 (|u ′′ | := (tr u ′′ u ′′ ) 1/2 ), we have
θ R,t,x := - C R (t,x) |F (u ′′ , s, y) −F R,x (u ′′ , s)| dsdy ≤ θ, whereF R,x (u ′′ , s) = - B R (x) F (u ′′ , s, y) dy.
Remark 2.2. Assumption 2.2 (ii) is stronger than Assumption 2.1 (iii) which is singled out for methodological purposes.
Also observe that one can take θ = 0 in Assumption 2.2 (iv) if F is independent of x.
Fix a T ∈ (0, ∞) and for domains Ω ∈ R d define
Ω T = (0, T ) × Ω, ∂ ′ Ω T =Ω T \ ({0} × Ω).
For κ ∈ (0, 1] and functions φ(t, x) onΩ T set
[φ] C κ (Ω T ) = sup (t,x),(s,y)∈Ω T |φ(t, x) − φ(s, y)| |t − s| κ/2 + |x − y| κ , φ C(Ω T ) = sup Ω T |φ|, φ C κ (Ω T ) = φ C(Ω T ) + [φ] C κ (Ω T ) .
For κ ∈ (1, 2] and sufficiently regular φ set
[φ] C κ (Ω T ) = sup t,s∈[0,t],x∈R d |φ(t, x) − φ(s, x)| |t − s| κ/2 + sup x,y∈Ω,t∈[0,T ] |Dφ(t, x) − Dφ(t, y)| |x − y| κ−1 , φ C κ (Ω T ) = φ C 1 (Ω T ) + [φ] C κ (Ω T ) .
The set of functions with finite norm · C κ (Ω T ) is denoted by C κ (Ω T ).
Remark 2.3. According to the above notation C 2 (Ω T ) is not what is usually meant. Therefore, we are going to use the symbol W 1,2 ∞ (Ω T )∩C(Ω T ) instead for the space provided with norm · C 2 (Ω T ) . One should keep this in mind when we consider all κ ∈ (0, 2] at once.
For sufficiently regular functions φ(t, x) we set
H[φ](t, x) = H(φ(t, x), Dφ(t, x), D 2 φ(t, x), t, x).
(2.1)
Similarly we introduce F [φ] and other operators if we are given functions of u, t, x.
Everywhere below Ω is a bounded C 2 domain in R d and T ∈ (0, ∞). The following is the main result of the paper. We refer the reader to [2] for the definition of L p -viscosity solutions and their numerous properties.
Theorem 2.1. Let g ∈ W 1,2 ∞ (Ω T ) ∩ C(Ω T ). Then there is a function v ∈ C κ loc (Ω T ) ∩ C(Ω T ) which, for any p > d + 2, is an L p -viscosity solution of the equation ∂ t v + H[v] = 0 (2.2) in Ω T (a.e.) with boundary condition v = g on ∂ ′ Ω T . Furthermore, for any r, R ∈ (0, R 0 ] satisfying r < R and (t, x) ∈ Ω T such that C R (t, x) ⊂ Ω T we have [v] C κ (Cr (t,x)) ≤ N (R − r) −κ sup C R (t,x) |v| + NH κ , (2.3)
where N depend only on d, δ, K 0 , and κ (in particular, independent of ω).
Remark 2.4. A typical example of applications of Theorem 2.1 arises in connection with the theory of stochastic differential games where the so-called Isaacs equations play a major role. To describe a particular case of these equations, assume that we are given countable sets A and B and, for each α ∈ A and β ∈ B, we have an S δ -valued function a αβ (t, x) defined on R d+1 and a real-valued function G αβ (u ′ , t, x) defined for u ′ , (t, x) ∈ R d+1 . Suppose that these functions are measurable and Assumption 2.2 (ii) is satisfied with G αβ in place of G for any α ∈ A and β ∈ B (andH independent of α ∈ A and β ∈ B). Also suppose that Assumption 2.1 (iv) is satisfied with the same function ω and with G αβ in place of H for any α ∈ A and β ∈ B. Finally, suppose that for any R ∈ (0, R 0 ] and (t,
x) ∈ R d+1 - C R (t,x) sup sup α∈A β∈B |a αβ (s, y) −ā αβ (s)| dsdy ≤ θ, whereā αβ (s) = - B R a αβ (s, y) dy.
Upon introducing
F (u ′′ , t, x) = sup inf α∈A β∈B a αβ ij (t, x)u ′′ ij , G(u, t, x) = sup inf α∈A β∈B a αβ ij (t, x)u ′′ ij + G αβ (u ′ , t, x) − F (u ′′ , t, x)
one easily sees that Theorem 2.1 is applicable to the equation
∂ t v + sup inf α∈A β∈B a αβ ij (t, x)D 2 ij v + G αβ (v, Dv, t, x) = 0.
This example is close to the one from the introduction in [2] and is more general, because G αβ are not assumed to be linear in u ′ . On the other hand, we suppose that Assumption 2.2 (ii) is satisfied with G αβ in place of G uniformly in α, β. In the situation of [2] only
sup inf α∈A β∈B G αβ (0, ·, ·) κ < ∞ is required.
Remark 2.5. Assumption 2.2 (iii), (iv) can be replaced with the following which turns out to be basically weaker (cf. (3.3)): There exist countable sets A and B and functions a αβ (t, x) satisfying the conditions of Remark 2.4 and there are numbers f αβ (independent of (t, x)) such that
F (u ′′ , t, x) = sup inf α∈A β∈B a αβ ij (t, x)u ′′ ij + f αβ and F (0, t, x) ≡ 0.
Auxiliary equations
In the first result of this section only Assumptions 2.1 is used. By Theorem 2.1 of [8] there exists a convex positive homogeneous of degree one function P (u ′′ ) such that at all points of differentiability of P with respect to u ′′ we have P u ′′ (u ′′ ) ∈ Sδ, whereδ =δ(d, δ) ∈ (0, δ/4) and such that the following fact holds in which by P [v] we mean a differential operator constructed as in (2.1).
Theorem 3.1. Let K ≥ 0 be a fixed constant, g ∈ W 1,2 ∞ (Ω T ) ∩ C(Ω T )
. Assume thatH is bounded. Then the equation
∂ t v + max(H[v], P [v] − K) = 0 (3.1)
in
Ω T with boundary condition v = g on ∂ ′ Ω T has a solution v ∈ C(Ω T ) ∩ W 1,2 ∞,loc (Ω T ). In addition, |v|, |Dv|, ρ|D 2 v|, |∂ t v| ≤ N (sup Ω TH + K + g C 1,2 (Ω T ) ) in Ω T (a.e.), where ρ = ρ(x) = dist (x, R d \ Ω)
and N is a constant depending only on Ω, T , K 0 , and δ (in particular, independent of ω).
Theorem 3.1 is applicable to the equation
∂ t u + max(F [u], P [u] − K) = 0,(3.2)
which we want to rewrite in a different form.
First we observe that if in Section 7 we take
B = {0} × S δ , take a strictly convex open set B ′ 0 in S such that S δ ⊂ B ′ 0 ⊂ S δ/2 , and set B 0 = {0} ×B ′ 0 , then by Theorem 7.2 we have F (u ′′ , t, x) = sup inf α∈A 1 β∈B a αβ ij (t, x)u ′′ ij ,F (u ′′ , t) = sup inf α∈A 1 β∈Bā αβ ij (t)u ′′ ij , where A 1 = S, for α ∈ A 1 and β = (0, β ′ ) ∈ B, a αβ (t, x) = λ αβ (t, x)β ′ + (1 − λ αβ (t, x))G u ′′ (α) a αβ (t) =λ αβ (t, x)β ′ + (1 −λ αβ (t, x))G u ′′ (α), G(u ′′ ) = sup β ′ ∈B ′ 0 β ′ ij u ′′ ij , λ αβ (t, x) = 1 ∧ G(α) − F (α, t, x) G(α) − β ′ ij α ij 0 0 = 1 , andλ αβ (t) is defined similarly. From Section 7 we also know that, for a constant µ > 0, we have G(α) − β ′ ij α ij ≥ µ|α| if β = (0, β ′ ) ∈ B and α ∈ A 1 . Next, since P (u ′′ ) is positive homogeneous, convex, and P u ′′ ∈ Sδ, there exists a closed set A 2 ⊂ Sδ such that P (u ′′ ) = sup α∈A 2 α ij u ′′ ij .
For uniformity of notation introduce as a disjoint union of A 1 and A 2 and for β ∈ B and α ∈ A 2 set
a αβ (t, x) =ā αβ (t) = α, f αβ = 0. Also for α ∈Â and β ∈ B introduce σ αβ (t, x) = [a αβ (t, x)] 1/2 ,σ αβ (t) = [ā αβ (t)] 1/2 , L αβ v(t, x) = a αβ ij (t, x)D ij v(t, x),L αβ v(t, x) =ā αβ ij (t)D ij v(t,
x). Next we have the following which is essentially Remark 3.1 of [3] with the proof based on the positive homogeneity and Lipschitz continuity of F with respect to u ′′ .
Lemma 3.2. There is a function θ = θ(µ) = θ(µ, d, δ) > 0 defined for µ > 0 such that Assumptions 2.2 (i), (iii), (iv) being satisfied with this θ(µ) implies that for any R ∈ (0, R 0 ] and (t, x) ∈ R d+1 - C R (t,x) sup u ′′ =0 |F (u ′′ , s, y) −F R,x (u ′′ , s)| |u ′′ | dsdy ≤ µ.
Note that by Lemma 3.2 and Theorem 7.3 for any R ∈ (0, R 0 ] and (t,
x) ∈ R d+1 µ R,t,x := - C R(t,x) sup α∈Â,β∈B |a αβ (s, y) −ā αβ (s)| dsdy ≤ N µ, (3.3)
where the constant N depends only on d and δ. On Sδ the function a 1/2 is Lipschitz continuous and therefore (3.3) also holds if we replace a with σ.
Finally, observe that equation (3.2) is easily rewritten as
∂ t u + sup inf α∈Â β∈B L αβ u(t, x) + f αβ K ] = 0, (3.4) where f αβ K = −KI α∈A 2 .
4. Main estimate for solutions of (3.2)
Take R, K ∈ (0, ∞) and g ∈ W 1,2 ∞ (C R ) ∩ C(C R ). By Theorem 3.1 there exists u ∈ W 1,2 ∞,loc (C R ) ∩ C(C R ) such that u = g on ∂ ′ C R and equation (3.
2) holds (a.e.) in C R . By the maximum principle such u is unique.
Here is the main result of this section.
Theorem 4.1. There exist constants κ 0 ∈ (1, 2] and N ∈ (0, ∞) depending only on d and δ such that for each r ∈ (0, R] one can find an affine function u =û(x) such that
|u −û| ≤ N (µ κ/(6d+6) R ∨ µ 1/(6d+6) R )[g] C κ (C R ) R κ + N r κ 0 (R − r) −κ 0 osc C R (g −ĝ)
inC r for any κ ∈ (0, 2], where µ R = µ R,0,0 andĝ =ĝ(x) is any affine function of x.
By using parabolic dilations one easily sees that one may take R = 1. In that case we first prove a few auxiliary results. Introduceū as a unique solution of (3.2) (in C 1 ) withF in place of F and the same boundary condition on ∂ ′ C 1 . Below by N with occasional subscripts we denote various constants depending only on d and δ. [g] C κ (C 1 ) = 1.
(4.1)
Then for any ε > 0 there exists an infinitely differentiable function g ε on R d+1 such that inC 1
|g − g ε | ≤ N ε κ , |∂ t g ε | + |D 2 g ε | + ε|D 3 g ε | + ε|D∂ t g ε | ≤ N ε κ−2 , (4.2)
where N depends only on d. Furthermore, for w = u,ū inC 1 we have
|w(t, x) − g ε (t, x)| ≤ N ε κ−2 (1 − |x| 2 ) κ/2 + N ε κ . (4.3)
Proof. The first assertion is well known and is obtained by first continuing g(t, x) as a function of t to R to become an even 2-periodic function, then continuing thus obtained function across |x| = 1 almost preserving (4.1) in the whole space and then taking convolutions with δ-like kernels.
Then, since K ≥ 0, for w ε = u − g ε we have
∂ t w ε + ∂ t g ε + max[F (D 2 w ε + D 2 g ε ), P (D 2 w ε + D 2 g ε )] ≥ 0,
which in light of (4.2) implies that
∂ t w ε + max(F [w ε ], P [w ε ]) ≥ −N 1 ε κ−2 .
Next, it is easily seen that there is a constant N (= N (d, δ)) such that for
φ ε (t, x) = N N 1 ε κ−2 (1 − |x| 2 ) we have ∂ t φ ε + max(F [φ ε ], P [φ ε ]) ≤ −N 1 ε κ−2
in C 1 . It follows by the parabolic Alexandrov maximum principle that in C 1 4) where N depends only on d and δ.
w ε ≤ φ ε + sup ∂ ′ C 1 (w ε − φ ε ), u ≤ g ε + N ε κ−2 (1 − |x| 2 ) + N ε κ ,(4.
On the other hand,
∂ t w ε + ∂ t g ε + F (D 2 w ε + D 2 g ε ) ≤ 0, ∂ t w ε + F [w ε ] ≤ N ε κ−2 , and with perhaps different constant in the formula for φ ε w ε ≥ −φ ε + inf ∂ ′ C 1 (φ ε + w ε ), u ≥ g ε − N ε κ−2 (1 − |x| 2 ) − N ε κ ,
which along with (4.4) yields (4.3) for w = u. The proof of (4.3) for w =ū is identical and the lemma is proved.
Lemma 4.3. For any κ ∈ (0, 2] inC 1 we have |u −ū| ≤ N (µ κ/(6d+6) 1 ∨ µ 1/(d+1) 1 )[g] C κ (C 1 ) . (4.5)
Proof. To simplify some formulas observe that if [g] C κ (C 1 ) = 0, then g is an affine function of x independent of t, so that u =ū = g and we have nothing to prove. However, if [g] C κ (C 1 ) > 0, we can divide equation (3.2) by this quantity, and, since our assertion means, in particular, that N in (4.5) is independent of K, we can reduce the general situation to the one in which (4.1) holds. Therefore, below we assume (4.1).
On sufficiently regular functions u(t, x,x), t ∈ R, x,x ∈ R d , introduce
Φ[u](t, x,x) = sup inf α∈Â β∈B L αβ u(t, x,x) + f αβ K , where L αβ u(t, x,x) = a αβ ij (t, x)D x ij u(t, x,x) +â αβ ij (t, x,x)D xx ij u(t, x,x) +ǎ αβ ij (t, x,x)Dx x ij u(t, x,x) +ā αβ ij (t)Dx ij u(t, x,x), D x ij = ∂ 2 ∂x i ∂x j , D xx ij = ∂ 2 ∂x i ∂x j , Dx x ij = ∂ 2 ∂x i ∂x j , Dx ij = ∂ 2 ∂x i ∂x j , a αβ (t, x,x) = σ αβ (t, x)σ αβ (t),ǎ αβ (t, x,x) =σ αβ (t)σ αβ (t, x).
Observe that for λ,λ ∈ R d we have a αβ ij λ i λ j +â αβ ij λ iλj +ǎ αβ ijλ i λ j +ā αβ ijλ iλj = |σ αβ λ +σ αβλ | 2 ≥ 0, so that Φ is a (degenerate) elliptic operator. Next let w ∈ W 1,2 d+1 (C 1 ) ∩ C(C 1 ) be a solution of the equation ∂ t w + sup sup (4.6) One of reasons we need the function w is that, as is easy to see, there is a λ > 0 depending only on d and δ such that for all α, β on C 1 we have
∂ t (λw(t, x) + |x −x| 2 ) + L αβ (t, x,x)(λw(t, x) + |x −x| 2 ) = λ(∂ t w + L αβ w)(t, x) + 2|σ αβ (t, x) − σ αβ (t)| 2 ≤ 0,
where the inequality follows from the fact that a 1/2 is a Lipschitz continuous function on Sδ, so that |σ αβ −σ αβ | 2 ≤ N |a αβ −ā αβ | 2 ≤ N |a αβ −ā αβ |.
After that we proceed in two steps.
Step 1. Estimate of u−ū from above. According to Lemma 4.2 for |x| = 1 and |x| ≤ 1 we have
u(t, x) ≤ g ε (t, x) + N ε κ−2 (1 − |x| 2 ) + N ε κ ≤ g ε (t, x) + N ε κ−2 |x −x| + N ε κ , where ε κ−2 |x −x| ≤ ε κ−4 |x −x| 2 + ε κ , so that u(t, x) ≤ g ε (t, x) + N ε κ−4 |x −x| 2 + N ε κ .
This inequality also obviously holds if |x| = 1, |x| ≤ 1 or if t = 1 and |x|, |x| ≤ 1. This shows that for ε ∈ (0, 1)
u ε (t, x,x) := u(t, x)−[g ε (t, x)−g ε (t,x)+N ε κ−6 e 1−t |x−x| 2 +N ε κ ] ≤ g ε (t,x) on ∂ ′ [(0, 1) × C 2 1
]. Actually, above we could have replaced ε κ−6 with ε κ−4 but later on we will need to deal with terms of order ε κ−6 |x −x| 2 anyway. Also observe that for ε ∈ (0, 1),
I ε (t, x,x) := ∂ t u ε (t, x,x) + Φ[u ε ](t, x,x) = ∂ t u(t, x) + ∂ t g ε (t,x) − ∂ t g ε (t, x) +N ε κ−6 e 1−t |x−x| 2 + sup inf α∈Â β∈B a αβ ij D ij u(t, x)+ā αβ ij D ij g ε (t,x)−a αβ ij D ij g ε (t, x) −N ε κ−6 e 1−t |σ αβ (t, x) −σ αβ (t)| 2 + f αβ K , where |ā αβ ij D ij g ε (t,x) − a αβ ij D ij g ε (t, x)| ≤ |ā αβ (t) − a αβ (t, x)| |D 2 g ε (t, x)| +N |D 2 g ε (t, x) − D 2 g ε (t,x)| ≤ N ε κ−2 h(t, x) + N ε κ−3 |x −x|, |∂ t g ε (t,x) − ∂ t g ε (t, x)| ≤ N ε κ−3 |x −x|, −N ε κ−3 |x −x| + N ε κ−6 |x −x| 2 ≥ −N ε κ .
It follows that for ε ∈ (0, 1),
I ε (t, x,x) ≥ ∂ t u(t, x) + sup inf α∈Â β∈B L αβ u(t, x) + f αβ K −N ε κ−6 h − N ε κ = −N 1 ε κ−6 h − N 1 ε κ in (0, 1) × C 2 1 . On the other hand, u(t,x) ≥ g ε (t,x) − N ε κ−2 (1 − |x| 2 ) − N ε κ , which implies that u ε (t, x,x) :=ū(t,x) + N 2 ε κ−6 ((1 + λ)w(t, x) + |x −x| 2 ) +N 2 (2 − t)ε κ ≥ g ε (t,x) on ∂ ′ [(0, 1)×C 2 1 ].
It is also easily seen that by increasing N 2 if needed (which does not violate the above inequality) we may assume that in (0, 1)
× C 2 1 ∂ tū ε (t, x,x) + Φ[ū ε ](t, x,x) ≤ −N 1 ε κ−6 h − N 1 ε κ .
Hence, by the maximum principle (see, for instance, Theorem 2.1 of [5] or Theorem 3.4.2 of [6]) in [0, 1] ×C 2 1 we havē u(t,x) + N ε κ−6 (w(t, x) + |x −x| 2 ) + N ε κ ≥ u(t, x) −[g ε (t, x) − g ε (t,x) + N ε κ−6 |x −x| 2 + N ε κ ], which for x =x in light of (4.6) yields
u(t, x) −ū(t,x) ≤ N (ε κ + ε κ−6 µ 1/(d+1) 1 ). If µ 1 ≤ 1, then for ε = µ 1/(6d+6) 1 (≤ 1) we get u −ū ≤ N µ κ/(6d+6) 1 and if µ 1 ≥ 1, then for ε = 1 we obtain u −ū ≤ N µ 1/(d+1) 1 , so that generally u −ū ≤ N (µ κ/(6d+6) 1 ∨ µ 1/(d+1) 1
).
Step 2. Estimate of u −ū from below. Notice that
v ε (t, x,x) :=ū(t,x) − N ε κ−4 (λw(t, x) + |x −x| 2 ) − N ε κ ≤ g ε (t,x) on ∂ ′ [(0, 1) × C 2 1 ]
. It is also easily seen that in (0, 1)
× C 2 1 ∂ tv ε (t, x,x) + Φ[v ε ](t, x,x) ≥ 0.
On the other hand,
v ε (t, x,x) := u(t, x) − [g ε (t, x) − g ε (t,x)] +N ε κ−6 e 1−t ((1 + λ)w(t, x) + |x −x| 2 ) + N (2 − t)ε κ ≥ g ε (t,x) on ∂ ′ [(0, 1) × C 2
1 ] and the above computations show that (for sufficiently large N )
∂ t v ε (t, x,x) + Φ[v ε ](t,
x,x) ≤ 0 in (0, 1)×C 2 1 . By the maximum principlev ε ≤ v ε , which leads to the desired estimate of u −ū from below and the lemma is proved.
Lemma 4.4. There exist constants κ 0 ∈ (1, 2] and N ∈ (0, ∞) depending only on d and δ such that for any r ∈ (0, 1)
[ū] C κ 0 (Cr ) ≤ N (1 − r) −κ 0 osc C 1 (g −ĝ), (4.7)
whereĝ =ĝ(x) is any affine function of x.
Proof. First observe thatū −ĝ satisfies the same equation asū and the C κ (C r )-seminorms of these functions coincide if κ ∈ (1,2]. It follows that we may concentrate onĝ ≡ 0.
For any ρ ∈ (0, 1) the function δ hū satisfies an equation of type Next observe that for any function f (x) of one variable x ∈ [0, ε], ε > 0, we have
∂ t δ l,hū + a ij D ij δ l,hū = 0 in C ρ with some (a ij )|f ′ (0)| ≤ |f ′ (0) − (f (ε) − f (0)/ε| + ε −1 osc [0,ε) f ≤ ε γ [f ′ ] C γ [0,ε] + ε −1 osc [0,ε] f.
By applying this fact to functions v(x) given in B 1 we obtain that for any r n+1 < r n+2 ≤ 1 and any ε ∈ (0, 1)
|Dv| ≤ ε γ (r n+2 − r n+1 ) γ [Dv] C γ (Br n+2 ) + ε −1 (r n+2 − r n+1 ) −1 osc B 1 v (4.9) inB r n+1 .
Coming back to (4.8) and setting
r 0 = r, r n = r + (1 − r) n k=1 2 −k , n ≥ 1,
we conclude
A n := sup [0,r 2 n ] [Dū(t, ·)] C γ (Br n ) ≤ N (r n+1 − r n ) −γ sup Cr n+1 |Dū| ≤ N 1 ε γ A n+2 + N 2 (1 − r) −(1+γ) ε −1 2 (1+γ)n osc C 1ū , (4.10)
where the constants N i are different from the one in (4.8) but still depend only on δ and d. We first take ε so that
N 1 ε γ = 2 −5 ,
then take n = 2k, k = 0, 1, ..., multiply both parts of (4.10) by 2 −5k and sum up with respect to k. Then upon observing that (1 + γ)2k ≤ 4k we get
∞ k=0 A 2k 2 −5k ≤ ∞ k=1 A 2k 2 −5k + N (1 − r) −(1+γ) ∞ k=0 2 −k osc C 1ū .
By canceling (finite) like terms we find
sup [0,r 2 ] [Dū(t, ·)] C γ (Br) ≤ N (1 − r) −(1+γ) osc C 1ū
.
(4.11)
Next, we use the fact thatū itself satisfies the equation
0 = ∂ tū + max(F [ū],P [ū] − K) − max(0, −K) = ∂ tū + a ij D ijū
with some (a ij ) taking values in Sδ. Furthermore, for any T ∈ (0, r 2 ] and
|x 0 | ≤ r the function v(t, x) :=ū(t, x) − (x i − x 0i )D iū (T, x 0 ) satisfies the same equation and |v(T, x) − v(T, x 0 )| ≤ [Dū(T, ·)] C γ (Bρ) |x − x 0 | 1+γ for |x − x 0 | ≤ ρ − r,|ū(t, x 0 ) −ū(T, x 0 )| ≤ N [Dū(T, ·)] C γ (Bρ) (T − t) (1+γ)/2 ≤ N (1 − r) −(1+γ) (T − t) (1+γ)/2 osc C 1ū .
This provides the necessary estimate of the oscillation ofū in the time variable and along with (4.11) shows that
[ū] C 1+γ (Cr ) ≤ N (1 − r) −(1+γ) osc C 1ū
. Now the assertion of the lemma follows from the fact that
osc C 1ū = osc C 1 g.
The lemma is proved. Proof of Theorem 4.1. Takeû(t, x) =ū(0, 0) + x i D iū (0, 0) and observe that in C r
|u −û| ≤ |u −ū| + |ū −û| ≤ N (µ κ/(6d+6) 1 ∨ µ 1/(6d+6) 1 )[g] C κ (C 1 ) + I, where I = |ū −û| ≤ 2r κ 0 [ū] C κ 0 (Cr ) ≤ N r κ 0 (1 − r) −κ 0 osc C 1 (g −ĝ)
so that the theorem is proved.
5.
Estimating C κ -norm of solutions of (3.1)
In this section we assume thatH is bounded and investigate solutions of (3.1) which exist by Theorem 3.1. We take κ 0 ∈ (1, 2] from Theorem 4.1, take a µ ∈ (0, 1], and suppose that Assumption 2.2 (iv) is satisfied with θ = θ(µ) so that (3.3) holds for any R ∈ (0, R 0 ] and (t, x) ∈ R d+1 .
Lemma 5.1. Let R ∈ (0, R 0 ] and let v ∈ W 1,2 ∞ (C R ) ∩ C(C R )
be a solution of (3.1) inC R . Then for each r ∈ (0, R) one can find an affine function v(x) such that in C r for any κ ∈ [1, 2]
|v −v| ≤ N µ κ/(6d+6) [v] C κ (C R ) R κ + N r κ 0 (R − r) −κ 0 R κ [v] C κ (C R ) +N K 0 R 2 sup C R (|v| + |Dv|) + N R κH κ ,
where the constants N depend only on d and δ.
Proof. Observe that
max(H[v], P [v] − K) = max(F [v], P [v] − K) + h
where h defined by the above equality satisfies
|h| ≤ |H[v] − F [v]| ≤ K 0 (|v| + |Dv|) +H. Next define u ∈ W 1,2 d+1 (C R ) ∩ C(C R ) as a unique solution ∂ t u + max(F [u], P [u] − K) = 0
with boundary data u = v on ∂ ′ C R . Then there exists an Sδ-valued function a such that in C R we have
∂ t (v − u) + a ij D ij (v − u) + h = 0.
By the parabolic Alexandrov estimate (cf. our comment concerning this estimate in a more general situation in the proof of Lemma 6.1)
|v − u| ≤ N R d/d+1 h L d+1 (C R ) = N R 2 - C R |h| d+1 dxdt 1/(d+1) ≤ N K 0 R 2 sup C R (|v| + |Dv|) + N R κH κ .
After that our assertion follows from Theorem 4.1 and the lemma is proved.
Here is a result, which can be easily extracted from the proof of Theorem 2.1 of [10].
Lemma 5.2. Let r 0 ∈ (0, ∞), κ ∈ (1, 2), φ ∈ C κ (C r 0 ) and assume that there is a constant N 0 such that for any (t, x) ∈ C r 0 and r ∈ (0, 2r 0 ] there exists an affine functionφ =φ(x) such that sup Cr(t,x)∩Cr 0 |φ −φ| ≤ N 0 r κ .
Then
[
φ] C κ (Cr 0 ) ≤ N N 0 ,
where N depends only on d and κ.
Lemma 5.3. Take r 1 ∈ (0, R 0 ], r 0 ∈ (0, r 1 ), and define
κ 1 = 1 + κ 0 2 .
Let v ∈ W 1,2 ∞ (C r 1 ) ∩ C(C r 1 ) be a solution of (3.1) in C r 1 and let κ ∈ (1, κ 1 ]. Then there exists θ = θ(κ, d, δ) ∈ (0, 1] such that, if Assumption 2.2 (iv) is satisfied with this θ, then
[v] C κ (Cr 0 ) ≤ (1/2)[v] C κ (Cr 1 ) + N (K 0 + 1)(r 1 − r 0 ) −κ sup Cr 1 |v| +N (K 0 + 1)(r 1 − r 0 ) −(κ−1) sup Cr 1 |Dv| + NH κ , (5.1) where N = N (d, δ, κ).
Proof. To specify θ we first take a µ ∈ (0, 1] and suppose that Assumption 2.2 (iv) is satisfied with θ = θ(µ) so that (3.3) holds for any R ∈ (0, R 0 ] and (t, x) ∈ R d+1 .
Then take (t 0 , x 0 ) ∈ C r 0 , ε ∈ (0, 1), define
r ′ 0 = ε 3 (r 1 − r 0 ),
and notice that for any (t, x) ∈ C r ′ 0 (t 0 , x 0 ), r ∈ (0, 2r ′ 0 ], and R = ε −1 r, we have C R (t, x) ⊂ C r 1 . Therefore, by Lemma 5.1 we can find an affine functionv(x) such that sup
Cr(t,x)∩C r ′ 0 (t 0 ,x 0 ) |v −v| ≤ sup Cr(t,x) |v −v| ≤ N µ κ/(6d+6) [v] C κ (C R (t,x)) ε −κ r κ + N ε κ 0 −κ (1 − ε) −κ 0 r κ [v] C κ (C R (t,x)) +N K 0 ε −2 r 2 sup C R (t,x) (|v| + |Dv|) + N ε −κ r κH κ ≤ N r κ I(ε, r 1 ),
where the constants N depend only on d and δ and
I(ε, r 1 ) := µ κ/(6d+6) ε −κ + ε κ 0 −κ (1 − ε) −κ 0 [v] C κ (Cr 1 ) +ε −2 K 0 sup Cr 1 (|v| + |Dv|) + ε −κH κ .
It follows by Lemma 5.2 that
[v] C κ (C r ′ 0 (t 0 ,x 0 )) ≤ N 1 I(ε, r 1 ),
where N 1 depends only on d, κ, and δ. We can now specify θ and ε. First we chose ε ∈ (0, 1) so that
N 1 ε κ 0 −κ (1 − ε) −κ 0 = 1/4.
Since κ 0 − κ ≥ (κ 0 − 1)/2 > 0 and κ 0 depends only on d and δ and N 1 depends only on d, κ, and δ, ε also depends only on d, κ, and δ. After that we take µ = µ(d, κ, δ) ∈ (0, 1] so that
N 1 µ 1/(6d+6) ε −2 ≤ 1/4
and set θ = θ(µ(d, κ, δ)). Then 2) where N = N (d, δ, κ) and
[v] C κ (C r ′ 0 (t 0 ,x 0 )) ≤ (1/2)[v] C κ (Cr 1 ) + N J,(5.J = K 0 sup Cr 1 (|v| + |Dv|) +H κ . Now observe that if (t, x), (s, x) ∈ C r 0 and t > s, then either |t−s| ≤ (r ′ 0 ) 2 , in which case (t, x) ∈ C r ′ 0 (s, x) and (t − s) −κ/2 |v(t, x) − v(s, x)| ≤ (1/2)[v] C κ (Cr 1 ) + N J owing to (5.2), or |t − s| ≥ (r ′ 0 ) 2 when |v(t, x) − v(s, x)| ≤ 2(t − s) κ/2 (r ′ 0 ) −κ sup Cr 1 |v| ≤ N (t − s) κ/2 (r 1 − r 0 ) −κ sup Cr 1 |v|.
Next if (t, x), (t, y) ∈ C r 0 and x = y, then either |x − y| < r ′ 0 , in which case (t, y) ∈ C r ′ 0 (t, x) and |x − y| −(κ−1) |Dv(t, x) − Dv(t, y)| ≤ (1/2)[v] C κ (Cr 1 ) + N J,
or else |x − y| ≥ r ′ 0 and |Dv(t, x) − Dv(t, y)| ≤ 2|x − y| κ−1 (r ′ 0 ) −(κ−1) sup Cr 1 |Dv| ≤ N |x − y| κ−1 (r 1 − r 0 ) −(κ−1) sup Cr 1
|Dv|.
This proves (5.1) and the lemma.
Theorem 5.4. Take 0 < r < R ≤ R 0 and take κ 1 , κ ∈ (1, κ 1 ], and θ from
Lemma 5.3. Let v ∈ W 1,2 ∞ (C R ) ∩ C(C R ) be a solution of (3.1) in C R . Then [v] C κ (Cr) ≤ N (R − r) −κ sup C R |v| + NH κ ,(5.3)
where N depends only on d, δ, K 0 , and κ.
Proof. We proceed as in the proof of Lemma 4.4. Fix a number c ∈ (0, 1) such that c 4 > 3/4 and introduce r 0 = r, r n = r + c 0 (R − r) n k=1 c k , n ≥ 1, where c 0 is chosen in such a way that r n → R as n → ∞. Then Lemma 5.3 and (4.9) allow us to find constants N 1 and N depending only on d, δ, K 0 , and κ such that for all n and ε ∈ (0, 1)
A n := [v] C κ (Cr n ) ≤ (2 −1 + N 1 ε κ−1 )A n+2 +N (R − r) −κ c −nκ (1 + ε −1 ) sup C R |v| + NH κ .
we choose ε < 1 so that 2 −1 + N 1 ε κ−1 ≤ 3/4 and then recalling that κ ≤ 2 conclude that
∞ k=0 (3/4) k A 2k ≤ ∞ k=1 (3/4) k A 2k + NH κ +N (R − r) −κ sup C R |v| ∞ k=0 (3/4) k c −4k ,
where the last series converges since 3c −4 /4 < 1. By canceling like terms we come to (5.3) and the theorem is proved.
Proof of Theorem 2.1
First assume thatH is bounded. For K > 0 denote by v K the solution of (3.1) with boundary condition v = g on ∂ ′ Ω T . By Theorem 3.1 such a solution exists is continuous inΩ T and has locally bounded derivatives.
Then the beginning of the proof of Lemma 5.1 shows that for an Sδ-valued function (a ij ) we have
|∂ t v K + a ij D ij v K | ≤ K 0 (|v K | + |Dv K | +H),
and the parabolic Alexandrov estimate shows that |v K | ≤ N ( g C(Ω T ) + H L d+1 (Ω T ) ), (6.1) where N depends only on d, δ, K 0 , and the diameter of Ω.
Also |∂ t (v K − g) + a ij D ij (v K − g)| ≤ K 0 (|v K | + |D(v K − g)|) +H + N (|∂ t g| + |D 2 g| + |Dg|),(6.
2) which, after we continue (v − g)(t, x) for t ≥ T as zero, by Theorem 4.2.6 of [6] yields that there exists an α = α(d, δ) ∈ (0, 1) such that for any domain
Ω ′ ⊂Ω ′ ⊂ Ω |v K | C α (Ω ′ T ) ≤ N,(6.3)
where N depends only on the distance between the boundaries of Ω ′ and Ω and on T , d, δ, K 0 , the diameter of Ω, and the L d+1 (Ω T )-norms ofH and |∂ t g| + |D 2 g| + |Dg|. Now we are going to use one more piece of information available thanks to Theorem 2.1 of [8] which is that v K ∈ W 1,2 p (Ω T ) for any p. Then treating (6.2) near (0, T ) × ∂Ω we can flatten ∂Ω near any given point, then continue v − g (in the new coordinates) across the flat boundary in an odd way. We will then have a function of class W 1,2 d+1 to which Theorem 4.2.6 of [6] is applicable. In this way we estimate the C α -norm of v near the boundary of Ω and in combination with (6.3) obtain that
|v K | C α (Ω T ) ≤ N 0 ,(6.4)
where N 0 depends only on T , d, δ, K 0 , the diameter of Ω, and the L d+1 (Ω T )norms ofH and |∂ t g| + |D 2 g| + |Dg|. It follows that there is a sequence K n → ∞ and a function v such that v n := v Kn → v uniformly inΩ T . Of course, (2.3) holds, owing to Theorem 5.4. Furthermore, (6.4) holds with the same constants and v in place of v K and Dv n → Dv locally uniformly in Ω T .
Next, we need an analog of Lemma 6.1 of [8]. Introduce H 0 (u ′′ , t, x) = H(v(t, x), Dv(t, x), u ′′ , t, x).
Lemma 6.1. There is a constant N depending only on d and δ such that for any C r (t, x) satisfying C r (t, x) ⊂ Ω T and φ ∈ W 1,2 d+1 (C r (t, x)) ∩ C(C r (t, x)) we have on C r (t, (v n − φ) + +N r d/(d+1) (∂ t φ + I n + max(H[φ], P [φ] − K n )) + L d+1 (Cr (t,x)) , (6.7)
x) that v ≤ φ + N r d/(d+1) (∂ t φ + H 0 [φ]) + L d+1 (Cr (t,x)) + max ∂ ′ Cr(t,x) (v − φ) + . (6.5) v ≥ φ − N r d/(d+1) (∂ t φ + H 0 [φ]) − L d+1 (Cr (t,x)) − max ∂ ′ Cr(t,x) (v − φ) − .
where the constant N = N (d, δ). Actually the above references only say that (6.7) holds with N = N (r, d, δ) in place of N r d/(d+1) . However, the way this constant depends on r is easily discovered by using parabolic dilations. We obtain (6.5) from (6.7) by letting n → ∞. In the same way (6.6) is established. The lemma is proved. and the sets {(f, l) ∈ B : f + l, y ≥ H(y)} are nonempty and closed for any y ∈ R d .
Next, let B 0 be a relatively strictly convex closed bounded set in R d 1 +1 such that B 0 ⊃ B and the distance between the relative boundaries of B and B 0 is strictly positive. Then introduce A := R d 1 and for α ∈ A define
Also let (f αβ (t, x), l αβ (t, x)) correspond to H(u, t, x) and (f αβ (t),l αβ (t)) correspond toH(u, t) constructed as before Theorem 7.2. Then Proof. Let λ αβ (t, x) correspond to H(u, t, x) andλ αβ (t) correspond tō H(u, t) constructed as before Theorem 7.2. Then it suffices to prove that (with 0/0 = 0) and our assertion follows. The theorem is proved.
the set of symmetric d × d matrices, and we are dealing with the parabolic equation∂ t v(t, x) + H[v](t, x) := ∂ t v(t, x) + H(v(t, x), Dv(t, x), D 2 v(t, x), t, x) = f (1.1) in subdomains of (0, T ) × R d , where T ∈ (0, ∞), R d = {x = (x 1 , ..., x d ) : x 1 , ..., x d ∈ R},
Assumption 2. 2 .
2We have a representation H(u, t, x) = F (u ′′ , t, x) + G(u, t, x).
(i) The functions F and G are measurable functions of their arguments. (ii) For all values of the arguments
Lemma 4. 2 .
2Let κ ∈ (0, 2] and
L
αβ w = − sup sup α∈A β∈B |a αβ −ā αβ | =: −h in C 1 with zero boundary condition on ∂ ′ C 1 . Such a unique solution exists by Theorem 1.1 of [3] and by the parabolic Alexandrov estimate and (
taking values in Sδ if h is sufficiently small. By Corollary 4.3.6 of[6] for such h and r ∈ (0, ρ) we have[δ l,hū ] C γ (Cr ) ≤ N (ρ − r) −γ sup Cρ |δ l,hū |,where N and γ ∈ (0, 1) depend only on δ and d. By letting h → 0 we conclude[Dū] C γ (Cr ) ≤ N (ρ − r)
where ρ = (1 + r)/2. Therefore, by Lemma 4.4.2 of [6], applied with R = ρ − r = (1 − r)/2 there, for t ∈ [0, T ] we have
−
∂ t φ − max(H 0 [φ], P [φ] − K n ) = −∂ t φ − max(H 0 [φ], P [φ] − K n ) +∂ t v n + max(H 0 [v n ], P [v n ] − K n ) + I n = ∂ t (v n − φ) + a ij D ij (v n − φ) + I n ,where a = (a ij ) is an Sδ-valued function andI n = max(H[v n ], P [v n ] − K n ) − max(H 0 [v n ], P [v n ] − K n ).
− v n | + |Dv − Dv n |) → 0 as n → ∞.It follows by Theorem 3.1 of[5] or Theorem 3.3.9 of [6] that for r ∈ (0, 1] v n ≤ φ + max ∂ ′ Cr(t,x)
αβ (t, x) −f αβ (t)| + |l αβ (t, x) −l αβ (t)| dxdt ≤ N θ,where the constant N depends only on B and B 0 .
αβ (t, x) −λ αβ (t)| dxdt ≤ N θ.
For
β = (f, l) ∈ B we have |λ αβ (t, x) −λ αβ (x)| ≤ G(α) − H(α, t, x) G(α) − [f + l, α ] − G(α) −H(α, t) G(α) − [f + l, α ] ≤ |H(α, t, x) −H(α, t)| γ(α)
After that the proof of Theorem 2.1 in our particular case of bounded H is achieved in the following way. Using(6.5)and repeating the proof of Theorem 2.3 of[8](see Section 6 there), we easily obtain that, if (t 0 , x 0 ) ∈ Ω T and φ ∈ W 1,2 d+1,loc (Ω T ) are such that v −φ attains a local maximum at (t 0 , x 0 ) and v(t 0 , x 0 ) = φ(t 0 , x 0 ), then lim r↓0 ess sup Cr(t 0 ,x 0 ) ∂ t φ(t, x) + H(v(t, x), Dv(t, x), D 2 φ(t, x), t, x) ≥ 0.(6.8)Here v(t, x), Dv(t, x) can be replaced with v(t 0 , x 0 ), Dv(t 0 , x 0 ). Furthermore, if φ ∈ W 1,2 p,loc (Ω T ) with p > d + 2, then by embedding theorems φ ∈ C 1+α loc (Ω T ), where α ∈ (0, 1), and henceIt follows that one can replace v(t, x), Dv(t, x) with φ(t, x), Dφ(t, x) in (6.8) and then, by definition v, is an L p -viscosity subsolution. The fact that it is also an L p -viscosity supersolution is proved similarly on the basis of (6.6).In case of generalH we introduce u n as the solutions found according to Theorem 2.1 of (2.2) in Ω T within place of H(u, t, x) and with the same boundary condition u n = g on ∂ ′ Ω T . From the above we see that the estimates of |u n | C α (Ω T ) and [u n ] C κ (Cr(t,x)) are uniform with respect to n. This allows us to repeat what was said about v n with obvious changes and brings the proof of Theorem 2.1 to an end.A minimax representation of nonlinear functionsHere we complement the results of[7]which originated in[4]by providing a formula better suited for viewing nonlinear PDEs as Isaacs equations.Let d 1 ≥ 1 be an integer. Fix a closed bounded subset B of R d 1 +1 . Let H(u) be a real-valued Lipschitz continuous function given on R d 1 . As a Lipschitz continuous function H is differentiable on a set D ′ H ⊂ R d 1 of full measure. We introducewhere ·, · is the scalar product in R d 1 , and assume thatso that, owing to the boundedness of B, H is boundedly inhomogeneous.Here is Remark 2.1 of[7](in which we correct an obvious misprint).Theorem 7.1. Under the above assumptions we have on R d 1 thatNext, let P be the smallest hyperplane containing B 0 , and, by using the assumption about the boundaries of B and B 0 , define Γ as a closed (in the topology of P) convex subset of P with the origin lying in the relative (in the topology of P) interior of Γ such that (f, l)and observe that since µ(±B 0 ) ⊂ Γ for a constant µ > 0 we have thatFurthermore, since (f, l) + Γ ⊂ B 0 for any (f, l) ∈ B we have that for anywhich shows that is a cone with respect to (ξ, α) which is once continuously differentiable with respect to (λ, α) everywhere apart from the origin due to the strict convexity of B 0 . Since the plane ξ = 1 does not pass through the origin, G(α) is once continuously differentiable. Now for α ∈ A and β = (f, l) ∈ B setFurthermore, if H, F ∈ H(B), then for any α ∈ A and β = (f, l) ∈ B we haveand the first assertion of the theorem follows from Theorem 7.1.To prove the second assertion it suffices to note that, for instance., where the right-hand side is zero if γ(α) = 0, and after that use (7.2). The theorem is proved. Theorem 7.3. Let H also depend on parameters (t, x) ∈ R d+1 and let it satisfy the assumptions in the beginning of the section for each (t, x). Assume that H(u, t, x) is measurable with respect to (t, x) and there is a function H(u, t) also satisfying the assumptions in the beginning of the section for each t and measurable with respect to t. Denote
Interior a priori estimates for solutions of fully nonlinear equations. L Caffarelli, Ann. Math. 130L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. Math., Vol. 130 (1989), 189-213.
L p -theory for fully nonlinear uniformly parabolic equations. M G Crandall, M Kocan, A Świȩch, Comm. Partial Differential Equations. 2511M. G. Crandall, M. Kocan, A.Świȩch, L p -theory for fully nonlinear uniformly par- abolic equations, Comm. Partial Differential Equations, Vol. 25 (2000), No. 11-12, 1997-2053.
On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients. Hongjie Dong, N V Krylov, Xu Li, Algebra i Analiz. 241Hongjie Dong, N.V. Krylov, and Xu Li, On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients, Algebra i Analiz, Vol. 24 (2012), No. 1, 54-95.
Bounded inhomogeneous nonlinear elliptic and parabolic equations in the plane. N V Krylov, English translation in Math. USSR Sbornik. 821Mat. Sb. (N.S.)N.V. Krylov, Bounded inhomogeneous nonlinear elliptic and parabolic equations in the plane, Mat. Sb. (N.S.), Vol. 82 (1970), No. 1, 99-110 in Russian; English trans- lation in Math. USSR Sbornik, Vol. 11 (1970), No. 1, 89-99.
On the maximum principle for nonlinear parabolic and elliptic equations. N V Krylov, Izvestiya Akademii Nauk SSSR, seriya matematicheskaya. 42in Russian; English translation in MathN. V. Krylov, On the maximum principle for nonlinear parabolic and elliptic equa- tions, Izvestiya Akademii Nauk SSSR, seriya matematicheskaya Vol. 42 (1978), No. 5, 1050-1062 in Russian; English translation in Math. USSR Izvestija, Vol. 13 (1979), No. 2, 335-347.
Nonlinear elliptic and parabolic equations of second order. N V Krylov, Nauka, Moscowin Russian; English translation: Reidel, DordrechtN. V. Krylov, Nonlinear elliptic and parabolic equations of second order , Nauka, Moscow, 1985 in Russian; English translation: Reidel, Dordrecht, 1987.
On a representation of fully nonlinear elliptic operators in terms of pure second order derivatives and its applications, Problems of Mathematical Analysis in Russian. N V Krylov, Journal of Mathematical Sciences. 1771English translationN.V. Krylov, On a representation of fully nonlinear elliptic operators in terms of pure second order derivatives and its applications, Problems of Mathematical Anal- ysis in Russian; English translation: Journal of Mathematical Sciences, New York, Vol. 177 (2011), No. 1, 1-26.
An ersatz existence theorem for fully nonlinear parabolic equations without convexity assumptions. N V Krylov, N.V. Krylov, An ersatz existence theorem for fully nonlinear parabolic equations without convexity assumptions, http://arxiv.org/abs/1211.3732.
On the classical solution of Bellman's elliptic equation. M V Safonov, Dokl. Akad. Nauk SSSR. 2784Soviet Math. Dokl.M.V. Safonov, On the classical solution of Bellman's elliptic equation, Dokl. Akad. Nauk SSSR, Vol. 278 (1984), No. 4, 810-813 in Russian; English translation in Soviet Math. Dokl., Vol. 30 (1984), 482-485.
On the classical solution of nonlinear elliptic equations of second order , Izvestija Akad. Nauk SSSR, ser. mat. M V Safonov, Russian. English translation in Math. USSR Izvestija. 1372M.V. Safonov, On the classical solution of nonlinear elliptic equations of second order , Izvestija Akad. Nauk SSSR, ser. mat., Vol. 137, No. 2 (1988), 184-201 in Russian. English translation in Math. USSR Izvestija, Vol. 33, No. 3 (1989), 597- 612.
W 1,p -interior estimates for solutions of fully nonlinear, uniformly elliptic equations. A Świȩch, Adv. Differential Equations. 26A.Świȩch, W 1,p -interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, Vol. 2 (1997), No. 6, 1005-1027.
Hölder gradient estimates for fully nonlinear elliptic equations. N S Trudinger, Proc. Roy. Soc. Edinburgh Sect. A. 1081-2N.S. Trudinger, Hölder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), No. 1-2, 57-65.
On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations. N S Trudinger, in Partial differential equations and the calculus of variations. Boston, Boston, MABirkhäuserIIN.S. Trudinger, On regularity and existence of viscosity solutions of nonlinear sec- ond order, elliptic equations, pp. 939-957 in Partial differential equations and the calculus of variations, Vol. II, Progr. Nonlinear Differential Equations Appl., 2, Birkhäuser, Boston, Boston, MA, 1989.
On the regularity theory of fully nonlinear parabolic equations. L Wang, II. Comm. Pure Appl. Math. 452L. Wang, On the regularity theory of fully nonlinear parabolic equations, II. Comm. Pure Appl. Math., Vol. 45 (1992), No. 2, 141-178.
E-mail address: krylov@math. umn.eduE-mail address: [email protected]
| [] |
[
"ON THE TRIALITY THEORY FOR A QUARTIC POLYNOMIAL OPTIMIZATION PROBLEM",
"ON THE TRIALITY THEORY FOR A QUARTIC POLYNOMIAL OPTIMIZATION PROBLEM"
] | [
"David Yang Gao ",
"Changzhi Wu ",
"\nSchool of Science\nInformation Technology and Engineering\nSchool of Science, Information Technology and Engineering\nUniversity of Ballarat\n3353VictoriaAustralia\n",
"\nSchool of Mathematics\nUniversity of Ballarat\n3353VictoriaAustralia\n",
"\nChongqing Normal University\n400047Shapingba, ChongqingChina\n"
] | [
"School of Science\nInformation Technology and Engineering\nSchool of Science, Information Technology and Engineering\nUniversity of Ballarat\n3353VictoriaAustralia",
"School of Mathematics\nUniversity of Ballarat\n3353VictoriaAustralia",
"Chongqing Normal University\n400047Shapingba, ChongqingChina"
] | [
"AIMS' Journals Volume X, Number 0X, XX 200X pp. X-XX"
] | This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality left in 2003. Results show that the triality theory holds strongly in a tri-duality form if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Four numerical examples are presented to illustrate that this theory can be used to identify not only the global minimum, but also the largest local minimum and local maximum. | 10.3934/jimo.2012.8.229 | [
"https://arxiv.org/pdf/1110.0293v1.pdf"
] | 119,283,220 | 1110.0293 | b594894557d4a525ef54549eb4cb4728f9510ad2 |
ON THE TRIALITY THEORY FOR A QUARTIC POLYNOMIAL OPTIMIZATION PROBLEM
3 Oct 2011
David Yang Gao
Changzhi Wu
School of Science
Information Technology and Engineering
School of Science, Information Technology and Engineering
University of Ballarat
3353VictoriaAustralia
School of Mathematics
University of Ballarat
3353VictoriaAustralia
Chongqing Normal University
400047Shapingba, ChongqingChina
ON THE TRIALITY THEORY FOR A QUARTIC POLYNOMIAL OPTIMIZATION PROBLEM
AIMS' Journals Volume X, Number 0X, XX 200X pp. X-XX
3 Oct 2011Manuscript submitted to(Communicated by Kok Lay Teo)
This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality left in 2003. Results show that the triality theory holds strongly in a tri-duality form if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Four numerical examples are presented to illustrate that this theory can be used to identify not only the global minimum, but also the largest local minimum and local maximum.
1. Introduction and Motivation. The concepts of triality and tri-duality were originally proposed in nonconvex mechanics [4,5]. Mathematical theory of triality in its standard format is composed of three types of dualities: a canonical min-max duality and a pair of double-min and double-max dualities. The canonical min-max duality provides a sufficient condition for global minimum, while the double-min and double max dualities can be used to identify respectively the largest local minimum and local maximum. The tri-duality is a strong form of the triality principle [7]. Together with a canonical dual transformation and a complementary-dual principle, they comprise a versatile canonical duality theory, which can be used not only for solving a large class of challenging problems in nonconvex/nonsmooth analysis and continuous/discrete optimization [7,8], but also for modeling complex systems and understanding multi-scale phenomena within a unified framework [4,5,7] (see also the review articles [10,12,17]).
For example, in the recent work by Gao and Ogden [13] on nonconvex variational/boundary value problems, it was discovered that both the global and local minimizers are usually nonsmooth functions and cannot be determined easily by traditional Newton-type numerical methods. However, by the canonical dual transformation, the nonlinear differential equation is equivalent to an algebraic equation, which can be solved analytically to obtain all solutions. Both global minimizer and local extrema were identified by the triality theory, which revealed some interesting phenomena in phase transitions.
The triality theory has attracted much attention recently in duality ways: Successful applications in multi-disciplinary fields of mathematics, engineering and sciences show that this theory is not only useful and versatile, but also beautiful in its mathematical format and rich in connotation of physics, which reveals a unified intrinsic duality pattern in complex systems; On the other hand, a large number of "counterexamples" have been presented in several papers since 2010. Unfortunately, most of these counterexamples are either fundamentally wrong (see [30,31]), or repeatedly address an open problem left by Gao in 2003 on the double-min duality [9,10].
The main goal of this paper is to solve this open problem left in 2003. The next section will present a brief review and the open problem in the triality theory. In Section 3, the triality theory is proved in its strong form as it was originally discovered. Section 4 shows that both the canonical min-max and the double-max dualities hold strongly in general, but the double-min duality holds weakly in a symmetrical form. Applications are illustrated in Section 5, where a linear perturbation method is used for solving certain critical problems. The paper ended by an Appendix and a section of concluding remarks.
2. Canonical Duality Theory: A Brief Review and an Open Problem. Let us begin with the general global extremum problem
(P) : ext Π (x) = W (x) + 1 2 x, Ax − x, f | x ∈ Xa ,(1)
where Xa ⊂ R n is an open set, x = {x i } ∈ R n is a decision vector, A = {A ij } ∈ R n×n is a given symmetric matrix, f = {f i } ∈ R n is a given vector, and * , * denotes a bilinear form on R n × R n ; the function W : Xa → R is assumed to be nonconvex and differentiable (it is allowed to be nonsmooth and sub-differentiable for constrained problems). The notation ext{ * } stands for finding global extremal of the function given in { * }.
In this paper, we are interested only in three types of global extrema: the global minimum and a pair of the largest local minimum and local maximum. Therefore, the nonconvex term W (x) in (1) is assumed to satisfy the objectivity condition 1 , i.e., there exists a (geometrically) nonlinear mapping Λ : Xa → V ⊂ R m and a canonical function
V : V ⊂ R m → R such that W (x) = V (Λ (x)) ∀x ∈ Xa.
According to [7], a real valued function V : V → R is said to be a canonical function on its effective domain Va ⊂ V if its Legendre conjugate V * :
V * → R V * (ς) = sta{ ξ; ς − V(ξ)| ξ ∈ Va}(2)
is uniquely defined on its effective domain V * a ⊂ V * such that the canonical duality relations
ς = ∇V (ξ) ⇔ ξ = ∇V * (ς) ⇔ V (ξ) + V * (ς) = ξ; ς(3)
hold on Va × V * a , where * ; * represents a bilinear form which puts V and V * in duality. The notation sta{ * } stands for solving the stationary point problem in { * }. By this one-to-one canonical duality, the nonconvex function W (x) = V (Λ(x)) can be replaced by Λ(x); ς − V * (ς) such that the nonconvex function Π(x) in (1) can be written as
Ξ(x, ς) = Λ(x); ς − V * (ς) + 1 2 x, Ax − x, f ,(4)
which is the so-called total complementary (energy) function introduced by Gao and Strang in 1989. By using this total complementary function, the canonical dual function Π d : V * a → R can be formulated as
Π d (ς) = sta{Ξ(x, ς)| ∀x ∈ Xa} = U Λ (ς) − V * (ς),(5)
where
U Λ : V * a → R is called the Λ-conjugate of U (x) = x, f − 1 2 x, Ax , defined by [7] as U Λ (ς) = sta{ Λ(x); ς − U(x)| x ∈ Xa}.(6)
Let Sa ⊂ V * a be the feasible domain of U Λ (ς); then the canonical dual problem is to solve the stationary point problem
(P d ) : ext{ Π d (ς)| ς ∈ Sa}.(7)
Theorem 2.1 (Complementary-duality principle [7]). Problem (P d ) is a canonical dual to (P) in the sense that if (x,ς) is a critical point of Ξ(x, ς), thenx is a critical point of (P),ς is a critical point of (P d ), and
Π(x) = Ξ(x,ς) = Π d (ς).(8)
Theorem 2.1 implies a perfect duality relation (i.e. no duality gap) between the primal problem and its canonical dual 2 . The formulation of Π d (ς) depends on the geometrical operator Λ(x). In many applications, the geometrical operator Λ is usually a quadratic mapping over a given field [7]. In finite dimensional space, this quadratic operator can be written as a vector-valued function (see [8], page 150)
Λ(x) = 1 2 x T B k x m k=1 : Xa ⊂ R n → Va ⊂ R m ,(9)
where B k = {B k ij } ∈ R n×n is a symmetrical matrix for each k = 1, 2, · · · , m, and Va ⊂ R m is defined by
Va = ξ ∈ R m | ξ k = 1 2 x T B k x ∀x ∈ Xa, k = 1, . . . , m .
In this case, the total complementary function has the form
Ξ (x, ς) = 1 2 x, G (ς) x − V * (ς) − x, f ,(10)
where G : R m → R n×n is a matrix-valued function defined by
G(ς) = A + m k=1 ς k B k .(11)
The critical condition ∇xΞ (x, ς) = 0 leads to the canonical equilibrium equation
G (ς) x = f .(12)
Clearly, for any given ς ∈ V * a , if the vector f ∈ C ol (G (ς)), where C ol (G) stands for a space spanned by the columns of G, the canonical equilibrium equation (12) can be solved analytically as 3 x = [G(ς)] −1 f . Therefore, the canonical dual feasible space Sa ⊂ V * a can be defined as Sa = {ς ∈ V * a | f ∈ C ol (G (ς))} , and on Sa the canonical dual Π d (ς) is well-defined as
Π d (ς) = − 1 2 G(ς) −1 f , f − V * (ς).(13)
Theorem 2.2 (Analytic solution [7]). Ifς ∈ Sa is a critical solution of (P d ), then
x = G(ς) −1 f(14)
is a critical solution of (P) and Π(x) = Π d (ς). Conversely, ifx is a critical solution of (P), it must be in the form of (14) for a certain critical solutionς of (P d ).
The canonical dual function Π d (ς) for a general quadratic operator Λ was first formulated in nonconvex analysis, where Theorem 2.2 is called the pure complementary energy principle, [6]. In finite deformation theory, this theorem solved an open problem left by Hellinger (1914) and Reissner (1954) (see [25]). The analytical solution theorem has been successfully applied for solving a class of nonconvex problems in mathematical physics, including Einstein's special relativity theory [7], nonconvex mechanics and phase transitions in solids [13]. In global optimization, the primal solutions to nonconvex minimization and integer programming problems are usually located on the boundary of the feasible space. By Theorem 2.2, these solutions can be analytically determined by critical points of the canonical dual function Π d (ς) (see [3,11,14,16]). In order to identify both global and local extrema of the primal and dual problems, we assume, without losing much generality, that the canonical function V : Ea → R is convex and let
S + a = {ς ∈ Sa | G (ς) 0} ,(15)S − a = {ς ∈ Sa | G (ς) ≺ 0} ,(16)
where G (ς) 0 means that G (ς) is a positive semi-definite matrix and G (ς) ≺ 0 means that G (ς) is negative definite. Theorem 2.3 (Triality Theorem [8]). Let (x,ς) be a critical point of Ξ (x, ς).
If G(ς) 0, thenς is a global maximizer of Problem (P d ), the vectorx is a global minimizer of Problem (P), and the following canonical min-max duality statement holds:
min x∈Xa Π (x) = Ξ (x,ς) = max ς∈S + a Π d (ς) .(17)
If G(ς) ≺ 0, then there exists a neighborhood Xo × So ⊂ Xa × S − a of (x,ς) for which we have either the double-min duality statement
min x∈Xo Π (x) = Ξ (x,ς) = min ς∈So Π d (ς) ,(18)
or the double-max duality statement
max x∈Xo Π (x) = Ξ (x,ς) = max ς∈So Π d (ς) .(19)
The triality theory provides actual global extremum criteria for three types of solutions to the nonconvex problem (P): a global minimizerx(ς) ifς ∈ S + a and a pair of the largest-valued local extrema. In other words,x(ς) is the largest-valued local maximizer ifς ∈ S − a is a local maximizer; x(ς) is the largest-valued local minimizer ifς ∈ S − a is a local minimizer. This pair of largest local extrema plays a critical role in nonconvex analysis of post-bifurcation and phase transitions.
Remark 1 (Relation between Lagrangian Duality and Canonical Duality).
The main difference between the Lagrangian-type dualities (including the equivalent Fenchel-Moreau-Rockfellar dualities) and the canonical duality is the operator Λ : Xa → Va. In fact, if Λ is linear, the primal problem (P) is called geometrically linear in [7] and the total complementary function Ξ(x, ς) is simply the well-known Lagrangian and is denoted as
L(x, ς) = Λx; ς − V * (ς) − F (x).(20)
In convex (static) systems, F (x) = x, f is linear and L(x, ς) is a saddle function. Therefore, the well-known saddle min-max duality links a convex minimization problem (P) to a concave maximization dual problem with linear constraint:
max {Π * (ς) = −V * (ς)| Λ * ς = f , ς ∈ V * a } ,(21)
where Λ * is the conjugate operator of Λ defined via Λx; ς = x, Λ * ς . Using the Lagrange multiplier x ∈ Xa to relax the equality constraint, the Lagrangian L(x, ς) is obtained. By the fact that the (canonical) duality in convex static systems is unique, the saddle min-max duality is refereed as the mono-duality in complex systems (see Chapter 1 in [7]).
Since the linear operator Λ can not change the convexity of W (x) = V (Λx), the Lagrangian duality theory can be used mainly for convex problems. It is known that if W (x) is nonconvex, then the Lagrangian duality as well as the related Fenchel-Moreau-Rockafellar duality will produce the so-called duality gap. Comparing the canonical dual function Π d (ς) in (13) with the Lagrangian dual function Π * (ς) in (21), we know that the duality gap is 1 2 G(ς) −1 f , f . The canonical duality theory is based on the (geometrically) nonlinear mapping Λ : Xa → Va and the canonical transformation W (x) = V (Λ(x)). The total complementary function Ξ(x, ς) is also known as the nonlinear or extended Lagrangian and is denoted by L(x, ς) due to the geometric nonlinearity of Λ(x) (see [7,10]). Relations between the canonical duality and the classical Lagrangian duality are discussed in [16].
Remark 2 (Geometrical Nonlinearity and Complementary Gap Function).
The canonical minmax duality statement (17) was first proposed by Gao and Strang in nonconvex/nonsmooth analysis and mechanics in 1989 [19], where Π(x) = W (x) − F (x) is the so-called total potential energy with W representing the internal (or stored) energy and F the external energy. The geometrical nonlinearity is a standard terminology in finite deformation theory, which implies that the geometrical equation (or the configuration-strain relation) ξ = Λ(x) is nonlinear. By definition in physics, a function F (x) is called the external energy means that its (sub-)differential must be the external force (or input) f . Therefore, in Gao and Strang's work, the external energy should be a linear function(al) F (x) = x, f on its effective domain. In this case, the matrix G(ς) is a Hessian of the so-called complementary gap function (i.e. the Gao-Strang gap function [19])
Gap(x, ς) = −Λc(x); ς ,(22)
where [19]. Actually, in Gao and Strang's original work, the canonical min-max duality statement holds in a general (weak) condition, i.e., Gap(x,ς) ≥ 0, ∀x ∈ Xa in field theory (corresponding to the strong condition G(ς) 0). The related canonical duality theory has been generalized to nonconvex variational analysis of a large deformation (von Karman) plate (where
Λc(x) = − 1 2 x T B k x is called the complementary operator of a Gâteauxdifferential Λt(x)x = x T B k x of Λ(x)A = ∆ 2 [34]), nonconvex (chaotic) dynamical systems (where A = ∆ − ∂ 2 /∂t 2 [10])
, and general nonconvex constrained problems in global optimization. Since F (x) in these general applications is the quadratic function − 1 2 x, Ax + x, f , the Gao-Strang gap function (22) should be replaced by the generalized form Gap(x, ς) = 1 2 x, G(ς)x (see the review article by Gao and Sherali [17]). This gap function recovers the existing duality gap in traditional duality theories and provides a sufficient global optimality condition for general nonconvex problems in both infinite and finite dimensional systems (see review articles [10,17]). By the fact that the geometrical mapping Λ in Gao and Strang's work is a tensor-like operator, it has been realized recently that the popular semi-definite programming method is actually a special application (where W (x) is a quadratic function) of the canonical min-max duality theory proposed in 1989 (see [14,15]).
In a recent paper by Voisei and Zalinescu [31], they unfortunately misunderstood some basic terminologies in continuum physics, such as geometric nonlinearity, internal and external energies, and present "counterexamples" to the Gao-Strang theory based on certain "artificially chosen" operators Λ(x) and quadratic functions F (x). Whereas in the stated contexts, the geometrical operator Λ should be a canonical measure (Cauchy-Reimann type finite deformation operator, see Chapter 6 in [7]) and the external energy F (x) is typically a linear functional on its effective domain; otherwise, its (sub)-differential will not be the external force. Interested readers are refereed by [18] for further discussion.
Remark 3 (Double-Min Duality and Open Problem). The double-min and double-max duality statements were discovered simultaneously in a post-buckling analysis of large deformed beam model [4,5] in 1996, where the finite strain measure Λ is a quadratic differential operator from a 2-D displacement field to a 2-D canonical strain field. Therefore, the triality theory was first proposed in its strong form, i.e. the so-called tri-duality theory (see the next section). Later on when Gao was writing his duality book [7], he realized that this pair of double-min and doublemax dualities holds naturally in convex Hamilton systems. Accordingly, a bi-duality theorem was proposed and proved for geometrically linear systems (where Λ is a linear operator; see Chapter 2 in [7]). Following this, the triality theory was naturally generalized to geometrically nonlinear systems (nonlinear Λ; see Chapter 3 in [7]) with applications to global optimization problems [8]. However, it was discovered in 2003 that if n = m in the quadratic mapping (9), the double-min duality statement needs "certain additional constraints". For the sake of mathematical rigor, the double-min duality was not included in the triality theory and these additional constraints were left as an open problem (see Remark 1 in [9], also Theorem 3 and its Remark in a review article by Gao [10]). By the fact that the double-max duality is always true, the double-min duality was still included in the triality theory in the "either-or" form in many applications (see [12,16]). However, ignoring the open problem related to the "certain additional constraints" on the doublemin duality statement has led to some misleading results.
The goal of this paper is to solve this open problem by providing a simple proof of the triality theory based on linear algebra. To help understanding the intrinsic characteristics of the original problem and its canonical dual, we assume that the nonconvex objective function W (x) is a sum of fourth-order canonical polynomials
W (x) = 1 2 m k=1 β k 1 2 x T B k x − d k 2 ,(23)
where B k = B k ij ∈ R n×n , k = 1, · · · , m, are all symmetric matrices, β k > 0 and d k ∈ R, k = 1, · · · , m are given constants. This polynomial is actually a discretized form of the so-called double-well potential, first proposed by van der Waals in thermodynamics in 1895 (see [26]), which is the mathematical model for natural phenomena of bifurcation and phase transitions in biology, 6 D. GAO AND C.Z. WU chemistry, cosmology, continuum mechanics, material science, and quantum field theory, etc. (see [5,20,23,24]). By using the quadratic geometrical operator Λ(x) given by (9), the canonical function
V (ξ) = 1 2 (ξ − d) T β(ξ − d)(24)
and its Legendre conjugate
V * (ς) = 1 2 ς T β −1 ς + ς T d(25)
are quadratic functions, where β = Diag (β k ) represents the diagonal matrix defined by the nonzero vector {β k }.
In the following discussions, we assume that all the critical points of problem (P) are nonsingular, i.e., if ∇Π(x) = 0, then det ∇ 2 Π(x) = 0.
We will first prove that if n = m, the triality theorem holds in its strong form; otherwise, the theorem holds in its weak form. Three numerical examples are used to illustrate the effectiveness and efficiency of the canonical duality theory.
3. Strong Triality Theory for Quartic Polynomial Optimization: Tri-Duality Theorem. We first consider the case m = n. For simplicity, we assume that β k = 1 in the following discussion (otherwise, B k can be replaced by √ β k B k and d k is replaced by d k / √ β k ). In this case, the problem (1) is denoted as problem (P). Its canonical dual is
(P d ) : ext Π d (ς) = − 1 2 f T [G (ς)] −1 f − 1 2 ς T ς − ς T d | ς ∈ Sa ⊂ R n .(27)
Theorem 3.1 (Tri-Duality Theorem).
Suppose that m = n, that the assumption (26) is satisfied, thatς is a critical point of Problem (P d ) and thatx = [G (ς)] −1 f . Ifς ∈ S + a , thenς is a global maximizer of Problem (P d ) in S + a if and only ifx is a global minimizer of Problem (P), i.e., the following canonical min-max statement holds:
Π(x) = min x∈R n Π (x) ⇐⇒ max ς∈S + a Π d (ς) = Π d (ς).(28)
On the other hand, ifς ∈ S − a , then, there exists a neighborhood Xo × So ⊂ R n × S − a of (x,ς), such that either one of the following two statements holds.
(A) The double-min duality statement
Π(x) = min x∈Xo Π (x) ⇐⇒ min ς∈So Π d (ς) = Π d (ς) ,(29)
or (B) the double-max duality statement
Π(x) = max x∈Xo Π (x) ⇐⇒ max ς∈So Π d (ς) = Π d (ς) .(30)
Proof. Ifς is a critical point of the canonical dual problem (P d ), the criticality condition
∇Π d (ς) = 1 2 f T [G (ς)] −1 B 1 [G (ς)] −1 f · · · f T [G (ς)] −1 B n [G (ς)] −1 f − ς − d = 0 ∈ R n(31)
leads toς = Λ(x). By the fact that
∇Π(x) = G(ς)x − f = 0 ∈ R n , it follows thatx = [G (ς)] −1 f is a critical point of Problem (P).
To prove the validity of the canonical min-max statement (28), letς be a critical point and ς ∈ S + a . Since Π d (ς) is concave on S + a , the critical pointς ∈ S + a must be a global maximizer of Π d (ς) on S + a . On the other hand, by the convexity of V (ξ), we have
V (ξ) − V ξ ≥ ξ −ξ; ∇V ξ = ξ −ξ;ς .(32)
Substituting ξ = Λ (x) andξ = Λ (x) into (32), we obtain
V (Λ (x)) − V (Λ (x)) ≥ Λ (x) − Λ (x) ;ς .
This leads to
Π (x) − Π (x) ≥ Λ (x) − Λ (x) ;ς + 1 2 x, Ax − 1 2 x, Ax − x −x, f , ∀x ∈ R n .(33)
By the fact thatς
= Λ (x) − d,(34)
we have
Π (x) − Π (x) ≥ 1 2 x, G (ς) x − 1 2 x, G (ς)x − x −x, G (ς)x .(35)
For a fixedς ∈ S + a , the convexity of the complementary gap function Gap(x,ς) = 1 2 x, G (ς) x on Xa leads to
Gap(x,ς) − Gap(x,ς) ≥ x −x, ∇xGap(x,ς) = x −x, G (ς)x ∀x ∈ R n .(36)
Therefore, we have
Π (x) − Π (x) ≥ x −x, G (ς)x − x −x, G (ς)x = 0 ∀x ∈ R n .(37)
This shows thatx is a global minimizer of Problem (P). Since it is assumed thatς ∈ S + a , it follows that (28) is satisfied.
We move on to prove the double-min duality statement (29). Letς be a critical point of Π d (ς) andς ∈ S − a . It is easy to verify that
∇Π (x) = n k=1 1 2 x T B k x − d k B k x + Ax − f , ∇ 2 Π (x) = G (ς) + F (x) F (x) T ,(38)
where
F (x) = B 1 x, B 2 x, · · · , B n x .
In light of (31), ∇ 2 Π d (ς) can be expressed in terms ofx = [G (ς)] −1 f as follows:
∇ 2 Π d (ς) = −F (x) T [G (ς)] −1 F (x) − I,
where I is the identity matrix. If the critical pointς ∈ S − a is a local minimizer, we have According to Lemma 6.2 in Appendix, the following matrix inequality is obtained:
∇ 2 Π d (ς) 0. This leads to − F (x) T [G (ς)] −1 F (x) I.(39)∇ 2 Π (x) = G (ς) + F (x) F (x) T 0.
By the assumption (26),x = [G (ς)] −1 f is also a local minimizer of Problem (P) . Therefore, on a neighborhood Xo × So ⊂ R n × S − a of (x,ς) , we have min
x∈Xo Π (x) = Ξ (x,ς) = min ς∈So Π d (ς) .
Similarly, we can show that ifx is a local minimizer of Problem (P) , the correspondingς is also a local minimizer of Problem (P d ).
The next task is to prove the double-max duality statement (30). Letς ∈ S − a be a local maximizer of Problem (P d ). Then, we have ∇ 2 Π d (ς) 0. This gives us
F (x) T [G (ς)] −1 F (x) + I 0.(40)
Now we have two possible cases regarding the invertibility of F (x) . If F (x) is invertible, then by using a similar argument as presented above, we can show that the relations
max x∈Xo Π (x) = Ξ (x,ς) = max ς∈So Π d (ς)
hold on a neighborhood Xo × So ⊂ R n × S − a of (x,ς). If F (x) is not invertible, by Lemma 6.1 in the Appendix, there exists two orthogonal matrices E and K such that
F (x) = EDK,(41)
where E T E = I = K T K and D = Diag (σ 1 , · · · , σr, 0, · · · , 0) with σ 1 ≥ σ 2 ≥ · · · ≥ σr > 0 and r = rank (F (x)). Substituting (41) into (40), we obtain
− K T D T E T [G (ς)] −1 EDK − I 0.(42)− D T E T G (ς) E −1 D − I 0.(43)
Applying Lemma 6.4 in Appendix to (43), it follows that
E T G (ς) E + DD T = E T G (ς) E + DKK T D T 0.
Finally, we have
∇ 2 Π (x) = G (ς) + EDKK T D T E T = G (ς) + F (x) F (x) T 0.
This means thatx is also a local maximizer of Problem (P) under the assumption (26), i.e., there exists a neighborhood X o × So ⊂ R n × S − a of (x,ς) such that max
x∈Xo Π (x) = Ξ (x,ς) = max ς∈So Π d (ς) .
Finally, we can show, in a similar way, that ifx = [G(ς)] −1 f is a local maximizer of Problem (P) andς ∈ S − a , the correspondingς is also a local maximizer of Problem (P d ). Therefore, the tri-duality theorem is proved.
1 2 x T B k x − d k 2 + 1 2 x T Ax − x T f | x ∈ R n ,(44)P d : ext Π d (ς) = − 1 2 f T [G (ς)] −1 f − 1 2 ς T ς − ς T d | ς ∈ Sa ⊂ R m .(45)
Suppose thatx andς are the critical points of Problem (P) and Problem (P d ), respectively, wherē x = [G (ς)] −1 f . It is easy to verify that
∇ 2 Π (x) = G (ς) + F (x) F (x) T ∈ R n×n (46) ∇ 2 Π d (ς) = −F (x) T [G (ς)] −1 F (x) − I ∈ R m×m . (47) In this case, F (x) = B 1 x, B 2 x, · · · , B m x ∈ R n×m .
To continue, we show the following lemmas.
P T ∇ 2 Π (x) P 0.(48)
Proof. By the fact that the critical pointς ∈ S − a is a local minimizer of Π d (ς), we have ∇Π d (ς) = 0 and ∇ 2 Π d (ς) 0. It follows that
− F (x) T [G (ς)] −1 F (x) I ∈ R m×m .
Thus, rank(F(x)) = m. Sinceς ∈ S − a and F (x) F (x) T 0, there exists a non-singular matrix T ∈ R n×n such that T T G (ς) T = Diag (−λ 1 , · · · , −λn) (49) and
T T F (x) F (x) T T = Diag (a 1 , · · · , am, 0, . . . , 0) , (50) where λ i > 0, i = 1, · · · , n, and a j > 0, j = 1, · · · , m.
According to the singular value decomposition theory [22], there exist orthogonal matrices U and E such that
T T F(x) = U √ a 1 . . . √ am 0 · · · 0 · · · 0 · · · 0 E.
Therefore, U is an identity matrix. Let
R = √ a 1 . . . √ am 0 · · · 0 · · · 0 · · · 0 .
Then,
∇ 2 Π d (ς) = −F (x) T [G (ς)] −1 F (x) − I = − F T T T T G (ς) T −1 T T F (x) − I = −E T RU T [Diag (−λ 1 , · · · , −λn)] −1 URE − I ∈ R m×m .
Since ∇ 2 Π d (ς) 0, U is an identity matrix, and E is an orthogonal matrix, we have
−R[Diag (λ 1 , · · · , λn)] −1 R − I m×m = Diag a 1 λ 1 − 1, · · · , am λm − 1 0.
Thus, a i ≥ λ i , i = 1, · · · , m. Note that
T T ∇ 2 Π (x) T = Diag (a 1 − λ 1 , · · · , am − λm, −λ m+1 , · · · , −λn).(51)
Let J = [I m×m , 0 m×(n−m) ] T . Then, we have
J T T T ∇ 2 Π (x) TJ = Diag (a 1 − λ 1 , · · · , am − λm) 0.(52)
Let P = TJ. Clearly, rank(P) = m and P T ∇ 2 Π (x) P = Diag (a 1 − λ 1 , · · · , am − λm) 0. The proof is completed.
In a similar way, we can prove the following lemma.
Lemma 4.2. Suppose that m > n. Letx = [G (ς)] −1 f be a critical point, which is a local minimizer of Problem (P), whereς ∈ S − a . Then, there exists a matrix Q ∈ R m×n with rank(Q) = n such that
Q T ∇ 2 Π d (ς) Q 0.(53)
Let p 1 , · · · , pm be the m column vectors of P and let q 1 , · · · , qn be the n column vectors of Q, respectively. Clearly, p 1 , · · · , pm are m independent vectors and q 1 , · · · , qn are n independent vectors. We introduce the following two subspaces Ifς ∈ S + a , then the canonical min-max duality holds in the strong form:
X ♭ = {x ∈ R n | x =x + θ 1 p 1 + · · · + θmpm, θ i ∈ R, i = 1, · · · , m},(54)S ♭ = {ς ∈ R m | ς =ς + ϑ 1 q 1 + · · · + ϑnqn, ϑ i ∈ R, i = 1, · · · , n}.(55)Π(x) = min x∈R n Π (x) ⇐⇒ max ς∈S + a Π d (ς) = Π d (ς).(56)
Ifς ∈ S − a , then there exists a neighborhood Xo × So ⊂ R n × S − a of (x,ς) such that the double-max duality holds in the strong form
Π(x) = max x∈Xo Π (x) ⇐⇒ max ς∈So Π d (ς) = Π d (ς) .(57)
However, the double-min duality statement holds conditionally in the following super-symmetrical forms.
1. If m < n andς ∈ S − a is a local minimizer of Π d (ς), thenx = [G (ς)] −1 f is a saddle point of Π(x) and the double-min duality holds weakly on Xo ∩ X ♭ × So, i.e.
Π(x) = min x∈Xo∩X ♭ Π (x) = min ς∈So Π d (ς) = Π d (ς);(58)
2. If m > n andx = [G (ς)] −1 f is a local minimizer of Π(x), thenς is a saddle point of Π d (ς) and the double-min duality holds weakly on Xo × So ∩ S ♭ , i.e.
Π(x) = min x∈Xo Π (x) = min ς∈So∩S ♭ Π d (ς) = Π d (ς).(59)
Proof. The proof of the statements (56) and (57) are similar to that given for the proof of Theorem 3.1. Thus, it suffices to prove the double-min duality statements (58) and (59). Firstly, we suppose that m < n andς is a local minimizer of Problem (P d ). Define ϕ(t 1 , · · · , tm) = Π(x + t 1x1 + · · · + tmxm).
From (47), we obtain
−F (x) T [G (ς)] −1 F (x) I ∈ R m×m . Thus, F (x) T [G (ς)] −1 F (x)
is a non-singular matrix and rank (F(x)) = m < n. We claim that x = [G (ς)] −1 f is not a local minimizer of Problem (P) . On a contrary, suppose thatx is also a local minimizer. Then, we have
∇ 2 Π (x) = G (ς) + F (x) F (x) T 0. Thus, F (x) F (x) T −G (ς) .
Sinceς ∈ S − a and rank (F) = m, it is clear that
n = rank (G (ς)) = rank F (x) F (x) T = m.
This is a contradiction. Therefore,x = [G (ς)] −1 f is a saddle point of Problem (P). It is easy to verify that Π(x) = Π d (ς). Thus, to prove (58), it suffices to prove that 0 ∈ R m is a local minimizer of the function ϕ(t 1 , · · · , tn).
It is easy to verify that ∇ϕ(0, · · · , 0) = [(∇Π(x)) T p 1 , · · · , (∇Π(x)) T pm] T = ∇Π(x)) T P = 0 (61) and ∇ 2 ϕ(0, · · · , 0) = P T ∇ 2 Π (x) P.
In light of Lemma 4.1 and the assumption (26), it follows that 0 ∈ R m is, indeed, a local minimizer of the function ϕ(t 1 , · · · , tm).
In a similar way, we can establish the case of m > n. The proof is completed.
Remark 6. Theorem 4.3 shows that both the canonical min-max and double-max duality statements hold strongly for general cases; the double-min duality holds strongly for n = m but weakly for n = m in a symmetrical form. The "certain additional conditions" are simply the intersection Xo X ♭ for m < n and So S ♭ for m > n. Therefore, the open problem left in 2003 [9,10] is solved for the double-well potential function W (x).
The triality theory has been challenged recently in a series of more than seven papers, see, for example, [31,32]. In the first version of [32], Voisei and Zalinescu wrote: "we consider that it is important to point out that the main results of this (triality) theory are false. This is done by providing elementary counter-examples that lead to think that a correction of this theory is impossible without falling into trivia". It turns out that most of these counter-examples simply use the double-well function W (x) with n = m. In fact, these counter-examples address the same type of open problem for the double-min duality left unaddressed in [9,10]. Indeed, by Theorem 4.3, we know that both the canonical min-max duality and the double-max duality hold strongly for the general case n = m. However, based on the weak double-min duality, one can easily construct other V-Z type counterexamples, where the strong double-min duality holds conditionally when n = m. Also, interested readers should find that the references [9,10] never been cited in any one of their papers. ext Π (x) = 1 2
1 2 x T B 1 x − d 1 2 + 1 2 x T B 2 x − d 2 2 + 1 2 x T Ax − x T f | x ∈ R 2 , (63) where A = a 1 0 0 a 2 , B 1 = b 1 0 0 0 , B 2 = 0 0 0 b 2 , f = [f 1 , f 2 ] T .
The canonical dual problem can be expressed as
Π d (ς) = − 1 2 f 2 1 a 1 + ς 1 b 1 + f 2 2 a 2 + ς 2 b 2 − 1 2 ς 2 1 + ς 2 2 − (d 1 ς 1 + d 2 ς 2 ) .
Thus,
∇Π d (ς) = b 1 f 2 1 2(a 1 +ς 1 b 1 ) 2 − ς 1 − 1 b 2 f 2 2 2(a 2 +ς 2 b 2 ) 2 − ς 2 − 1 and ∇ 2 Π d (ς) = −b 2 1 f 2 1 (a 1 + ς 1 b 1 ) −3 − 1 0 0 −b 2 2 f 2 2 (a 2 + ς 2 b 2 ) −3 − 1 . Now, we take b 1 = b 2 = f 1 = f 2 = 1, d 1 = d 2 = 1 and a 1 = −2, a 2 = −3.
It is easy to check that Π d (ς) has only one critical pointς 1 = 1 + √ 2, 3.35991198 in S + a and four critical
pointsς 2 = (3/2, 2.59827880) ,ς 3 = 1 − √ 2, −0.45819078 ,ς 4 = 1 − √ 2, 2.59827880 ,ς 5 = (3/2, −0.45819078) in S − a ,Π d (ς) = − 1 2 0.9 2 −0.2 + ς + 0.3 2 −0.8 + ς − 1 2 ς 2 − 4ς.
We can verify that Π d (ς) has one critical pointς 1 = −0.90489505 in S + a and two critical pointsς 2 = −0.12552589 andς 3 = −3.974788888 in S − a . Furthermore,ς 2 is a local minimizer andς 3 is a local maximizer of Π d (ς). According to θp). Then, it is easy to verify that there exists neighborhoods X 0 ⊂ R and S 0 ⊂ R such that 0 ∈ X 0 ,ς 2 ∈ S 0 and min θ∈X 0
ϕ(θ) = min ς∈S 0 Π d (ς).
This example shows that even if n > m, the canonical min-max duality and the double-max duality still hold strongly. However, the double-min duality statement should be refined into an m− dimensional subspace in this case. Example 3 (n = 1, m = 2). We now consider Problem (P) with n = 1, m = 2, A = −0.2, B 1 = 0.3, B 2 = 0.7, d 1 = 3, d 2 = 2.7 and f = 1.4. Then, its dual problem is
Π d (ς) = − 1 2 f 2 A + ς 1 B 1 + ς 2 B 2 − 1 2 (ς 2 1 + ς 2 2 ) − (d 1 ς 1 + d 2 ς 2 ).
We can verify that Π d (ς) has one critical pointς 1 = (−0.35012607, 3.48303916) T in S + a and two critical pointsς 2 = (−2.98705125, −2.66978626) T andς 3 = (−0.70765026, 2.64881606) T in S − a . Furthermore,ς 2 is a local maximizer andς 3 is a saddle point of Π d (ς). According to Theorem 4.3,x 1 = G(ς 1 ) −1 f = 4.20307342 is the unique global minimizer of Π(x),x 2 = G(ς 2 ) −1 f = −0.29381114 is a local maximizer of Π(x). Let q = (1, 0) T and ψ(ϑ) = Π d (ς 3 + ϑq). Then, it is easy to verify that there exists neighborhoods X 0 ⊂ R and S 0 ⊂ R such thatx 3 ∈ X 0 , 0 ∈ S 0 and min x∈X 0
Π(x) = min ϑ∈S 0 ψ(ϑ).
This example shows that if n < m, the canonical min-max duality and the double-max duality still hold strongly. However, the double-min duality statement should be refined into an n− dimensional subspace in this case. Example 4 Linear Perturbation. Let us consider the following optimization problem without input (f = 0) (P 2 ) : ext Π (x) = 1 2
1 2 (x 1 + x 2 ) 2 − 1 2 2 + 1 2 (x 1 − x 2 ) − 1 2 2 | x ∈ R 2 .
Problem (P 2 ) has four global minimizersx 1 = (1, 0) ,x 2 = (0, −1) ,x 3 = (0, 1) ,x 4 = (−1, 0) and the optimal cost value is 0. Its canonical dual problem is
Π d (ς) = − 1 2 ς 2 1 + ς 2 2 − (ς 1 + ς 2 ) .
Π d (ς) has only one critical pointς = − 1 2 , − 1 2 ∈ S − a . Furthermore, we can check thatx = [G (ς)] −1 f = (0, 0) is a local maximizer of Problem (P 2 ) . Thus, we cannot use the canonical dual transformation method to obtain the global minimizer of Problem (P 2 ) since this problem is in a perfect symmetrical form without input, which allows more than one global minimizer. Now we perturb Problem (P 2 ) as follows.
P b 2 : ext x∈R 2 Π 2 (x) = 1 2 1 2 (x 1 + x 2 ) 2 − 1 2 2 + 1 2 (x 1 − x 2 ) 2 − 1 2 2 − (x 1 f 1 + x 2 f 2 ) .
Its canonical dual function is expressed as
Π d 2 (ς) = − 1 8ς 1 ς 2 (ς 1 + ς 2 ) f 2 1 + f 2 2 + 2 (ς 1 − ς 2 ) f 1 f 2 − 1 2 ς 2 1 + ς 2 2 − 1 2 (ς 1 + ς 2 ) .
Taking f 1 = 0.001, f 2 = 0.005 and solving ∇Π d 2 (ς) = 0, the results obtained are listed in Table 1. We can see thatς = (0.00299107, 0.00199602) ∈ S + a and Π
ς = (ς 1 , ς 2 ) x = (x 1 , x 2 ) G (ς)
The primal problem Lemma 6.1 (Singular value decomposition [22]). For any given G ∈ R n×n with rank(G) = r, there exist U ∈ R n×n , D ∈ R n×n and R ∈ R n×n such that
G = UDR,
where U, R are orthogonal matrices, i.e., U T U = I = R T R, and D = Diag (σ 1 , · · · , σr, 0, · · · , 0), σ 1 ≥ σ 2 ≥ · · · ≥ σr > 0 . Proof. The proof is trivial and is omitted here. Lemma 6.4. Suppose that P ∈ R n×n , U ∈ R m×m , and D ∈ R n×m . Furthermore, D = D 11 0 r×(m−r) 0 (n−r)×r 0 (n−r)×(m−r) ∈ R n×n , where D 11 ∈ R r×r is nonsingular, r = rank(D), and P = P 11 P 12 P 21 P 22 ≺ 0, U = U 11 0 r×(m−r) 0 (m−r)×r U 22 ≻ 0, P ij and U ii , i, j = 1, 2, are of appropriate dimension matrices. Then,
P + DUD T 0 ⇐⇒ −D T P −1 D − U −1 0.(64)
Proof. Suppose that P + DUD T 0. Then, −P − DUD T = −P 11 − D 11 U 11 D T 11 −P 12 −P 21 −P 22 0.
Since P = P 11 P 12 P 21 P 22 ≺ 0, it follows that −P 22 ≻ 0. By Lemma 6.3, we have the following inequality − P 11 − D 11 U 11 D T 11 + P 12 P −1 22 P 21 0 (65) 14 D. GAO AND C.Z. WU which leads to −P 11 + P 12 P −1 22 P 21 D 11 U 11 D T 11 . Since P ≺ 0 and U ≻ 0, it follows from Lemma 6.2 that −P 11 + P 12 P −1 22 P 21 i.e., the right hand side of (64) holds.
In a similar way, we can show that if −D T P −1 D − U −1 0, then P + DUD T 0. The proof is thus completed.
7. Conclusion Remarks. In this paper, we presented a rigorous proof of the double-min duality in the triality theory for a quartic polynomial optimization problem based on elementary linear algebra. Our results show that under some proper assumptions, the triality theory for a class of quartic polynomial optimization problems holds strongly in the tri-duality form if the primal problem and its canonical dual have the same dimension. Otherwise, both the canonical min-max and the double-max still hold strongly, but the double-min duality holds weakly in a symmetric form.
2000
Mathematics Subject Classification. 90C26, 90C30, 90C46, 90C49. Key words and phrases. canonical duality, triality, global optimization, polynomial optimization, counter-examples.
Therefore, − F
−(x) T [G (ς)] −1 F (x) is positive definite and F (x) is invertible. By multiplying F (x) T −1 and F (x) −1 to the left and right sides of (39), respectively, we obtain − [G (ς)] −1 F (x) T −1 (F(x)) −1 .
Remark 4 .
4The strong triality Theorem 3.1 can also be used to identify saddle points of the primal problem, i.e.ς ∈ S − a is a saddle point of Π d (ς) if and only ifx = G(ς) −1 f is a saddle point of Π(x) on Xa. Since the saddle points do not produce computational difficulties in numerical optimization, and do not exist physically in static systems, these points are excluded from the triality theory.Remark 5. By the proof of Theorem 3.1, we know that if there exists a critical pointς ∈ S − a such thatς is a local minimizer of Problem (P d ), then F(x) must be invertible. On the other hand, if the symmetric matrices {B k } are linearly dependent, then F(x) is not invertible for any x ∈ R n . In this case, the corresponding canonical dual problem (P d ) has no local minimizers in S − a , and for any critical pointς ∈ S − a , the analytical solutionx = [G (ς)] −1 f is not a local minimizer of Π(x).
4 .
4Refined Triality Theory for General Quartic Polynomial Optimization. Let us recall the primal problem and its canonical dual problem in the general quartic polynomial case (n = m):
Lemma 4. 1 .
1Suppose that m < n. Let the critical pointς ∈ S − a be a local minimizer of Π d (ς), and letx = [G (ς)] −1 f . Then, there exists a matrix P ∈ R n×m with rank(P) = m such that
Theorem 4. 3 (
3Refined Triality Theorem). Suppose that the assumption (26) is satisfied. Letς be a critical point of Π d (ς) and letx = [G (ς)] −1 f .
5 .
5Numerical Experiments. In this section, some simple numerical examples are presented to illustrate the canonical duality theory. Example 1 (m = n = 2). Let us first consider Problem (P) with n = m = 2.
respectively. Furthermore,ς 2 is a local minimizer andς 3 is a local maximizer; the solutionsς 4 andς 5 are saddle points of Π d (ς) in S − a . Thus, by Theorem 3.1, we know thatx 1 = (2.41421356, 2.77845711) is a global minimizer, whilex 2 = (−2, −2.48928859) is a local minimizer andx 3 = (−0.41421356, −0.28916855) is a local maximizer. The corresponding values of the cost function are Π (x 1 ) = −14.0421 < Π (x 2 ) = −4.3050 < Π (x 3 ) = 0.5971.The graph of Π (x) and its contour are depicted inFigure 1.
Figure 1 .
1Graph of Π(x) (left) and contours of Π(x) (right) for Example 1 Example 2 (n = 2, m = 1). We now consider Problem (P) with n = 2, m = 1, A = Diag (−0.2, −0.8), B = Diag (1, 1), f = (0.9, 0.3) T , d = 4. Then, its dual problem is
Theorem 4.3,x 1 = (A +ς 1 B) −1 f = (1.27678581, 2.86000142) T is the unique global minimizer of Π(x),x 2 = (A +ς 2 B) −1 f = (−2.76475703, −0.32414004) T is a saddle point andx 3 = (A +ς 3 B) −1 f = (−0.21557976, −0.06283003) T is a local maximizer of Π(x). Let p = (1, 0) T and ϕ(θ) = Π(x 2 +
d 2
2(ς) = −0.00500648. Thus, x = [G (ς)] −1 f = (0.000495793, 1.00249) is the global minimizer of Problem P b 2 . Clearly, this x is very close tox 3 . If we take f 1 = 0.001, f 2 = −0.005, the global minimizer of Problem P b 2 isx = [G (ς)] −1 f = (0.000496288, −1.00249) which is close tox 2 = (0, −1). This example shows that if the canonical dual problem has no critical point in S + a , a linear perturbation could be used to solve the primal problem.
.
Appendix. In this Appendix, we present several lemmas which are needed for the proofs of Theorem 3.1 and Theorem 4.3.
Lemma 6. 2 .
2Suppose that G and U are positive definite. Then, G U if and only if U −1 G −1 .
Lemma 6 . 3 (
63Proposition 2.1 in [21]). For any given symmetric matrix M expressed in the form M = M 11 M 12 M 21 M 22 such that M 22 ≻ 0. Then, M 0 if and only if M 11 − M 12 M −1 22 M 21 0. The following lemma plays a key role in the proof of Theorem 3.1 and Theorem 4.3.
Table 1 .
1Numerical results for Example 3
The concept of objectivity in science means that qualitative and quantitative descriptions of physical phenomena remain unchanged when the phenomena are observed under a variety of conditions. That is, the objective function should be independent with the choice of the coordinate systems. In continuum mechanics, the objectivity is also regarded as the principle of frameindifference. See Chapter 6 in[7] for mathematical definitions of the objectivity and geometric nonlinearity in differential geometry and finite deformation field theory. Detailed discussion of objectivity in global optimization will be given in another paper[18].
The complementary-dual in physics means perfect dual in optimization, i.e., the canonical dual in Gao's work, which means no duality gap. Otherwise, any duality gap will violet the energy conservation law. Therefore, each complementary-dual variational statement in continuum mechanics is usually refereed as a principle.3 In this paper G −1 should be understood as the generalized inverse if det G = 0[8].
Vibrations of a nonlinear dynamic beam between two stops. K T Andrews, M F Bengue, M Shillor, Discrete and Continuous Dynamical Systems -Series B. 121K.T. Andrews, M.F. M'Bengue and M. Shillor, Vibrations of a nonlinear dynamic beam between two stops, Discrete and Continuous Dynamical Systems -Series B, 12(1), (2009), 23 -38.
Convex Analysis and Variational Problems. I Ekeland, R Temam, North-HollandI. Ekeland, and R. Temam, "Convex Analysis and Variational Problems", North-Holland, 1976.
Canonical dual approach for solving 0-1 quadratic programming problems. S C Fang, D Y Gao, R L Sheu, S Y Wu, J. Ind. and Manag. Optim. 4S.C. Fang, D.Y. Gao, R.L. Sheu and S.Y. Wu, Canonical dual approach for solving 0-1 quadratic programming problems, J. Ind. and Manag. Optim. 4, (2008), 125-142.
Post-buckling analysis and anomalous dual variational problems in nonlinear beam theory in. D Y Gao, Applied Mechanics in Americans Proc. of the Fifth Pan American Congress of Applied Mechanics. L.A. Godoy, L.E. Suarez4The University of Iowa, Iowa cityD.Y. Gao, Post-buckling analysis and anomalous dual variational problems in nonlinear beam theory in "Applied Mechanics in Americans Proc. of the Fifth Pan American Congress of Applied Mechanics", L.A. Godoy, L.E. Suarez (Eds.), Vol. 4. The University of Iowa, Iowa city, August (1996).
Dual extremum principles in finite deformation theory with applications to postbuckling analysis of extended nonlinear beam theory. D Y Gao, Applied Mechanics Reviews. 5011D.Y. Gao, Dual extremum principles in finite deformation theory with applications to post- buckling analysis of extended nonlinear beam theory, Applied Mechanics Reviews, 50 (11), (1997), S64-S71.
General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics. D Y Gao, Meccanica. 34D.Y. Gao, General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics, Meccanica 34, (1999), 169-198.
D Y Gao, Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic, DordrechtD.Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods and Applications", Kluwer Academic, Dordrecht, 2000.
Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. D Y Gao, J. Glob. Optim. 171D.Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Glob. Optim., 17(1), (2000), 127-160.
Perfect duality theory and complete solutions to a class of global optimization problems. D Y Gao, Optim. 52D.Y. Gao, Perfect duality theory and complete solutions to a class of global optimization problems, Optim., 52(4-5), (2003), 467-493.
Nonconvex semi-linear problems and canonical dual solutions. D Y Gao, Advances in Mechanics and Mathematics. Gao, D.Y., Ogden, R.W.DordrechtKluwer AcademicII261312D.Y. Gao, Nonconvex semi-linear problems and canonical dual solutions, In: Gao, D.Y., Ogden, R.W. (eds.) Advances in Mechanics and Mathematics, vol. II, pp. 261312. Kluwer Academic, Dordrecht (2003).
Solutions and optimality to box constrained nonconvex minimization problems. D Y Gao, J. Ind. Manag. Optim. 32D.Y. Gao, Solutions and optimality to box constrained nonconvex minimization problems, J. Ind. Manag. Optim. 3(2), (2007), 293-304.
Canonical duality theory: theory, method, and applications in global optimization. D Y Gao, Comput. Chem. 33D.Y. Gao, Canonical duality theory: theory, method, and applications in global optimization, Comput. Chem., 33, (2009), 1964-1972.
Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation. D Y Gao, R W Ogden, Quart. J. Mech. Appl. Math. 614D.Y. Gao and R.W. Ogden, Multiple solutions to non-convex variational problems with im- plications for phase transitions and numerical computation, Quart. J. Mech. Appl. Math. 61 (4), (2008), 497-522.
Solutions to quadratic minimization problems with box and integer constraints. D Y Gao, N Ruan, J. Glob. Optim. 473D.Y. Gao and N. Ruan, Solutions to quadratic minimization problems with box and integer constraints, J. Glob. Optim. 47(3), (2010), 463-484.
Canonical dual solutions to sum of fourth-order polynomials minimization problems with applications to sensor network localization. D Y Gao, N Ruan, P M Pardalos, Sensors: Theory, Algorithms and Applications. P.M. Pardalos, Y.Y. Ye, V. Boginski, and C. CommanderSpringerD.Y. Gao, N. Ruan,and P.M. Pardalos, Canonical dual solutions to sum of fourth-order poly- nomials minimization problems with applications to sensor network localization, in Sensors: Theory, Algorithms and Applications, P.M. Pardalos, Y.Y. Ye, V. Boginski, and C. Comman- der (eds). Springer, 2010.
Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality. D Y Gao, N Ruan, H Sherali, J. Glob. Optim. 453D.Y. Gao, N. Ruan, and H. Sherali, Solutions and optimality criteria for nonconvex con- strained global optimization problems with connections between canonical and Lagrangian duality, J. Glob. Optim. 45(3), (2009), 473-497.
Canonical duality: Connection between nonconvex mechanics and global optimization. D Y Gao, H D Sherali, Advances in Appl. Mathematics and Global Optimization. SpringerD.Y. Gao and H.D. Sherali, Canonical duality: Connection between nonconvex mechanics and global optimization, in Advances in Appl. Mathematics and Global Optimization, 249- 316, Springer (2009).
D Y Gao, H D Sherali, G Strang, Canonical duality: Objectivity, Triality, and Gap functions. in preparationD.Y. Gao, H.D. Sherali, and G. Strang, Canonical duality: Objectivity, Triality, and Gap functions, in preparation.
Geometric nonlinearity: Potential energy, complementary energy, and the gap function. D Y Gao, G Strang, Quart. Appl. Math. 473D.Y. Gao and G. Strang, Geometric nonlinearity: Potential energy, complementary energy, and the gap function, Quart. Appl. Math., 47(3), (1989), 487-504.
Multi-scale modelling and canonical dual finite element method in phase transitions of solids. D Y Gao, H F Yu, International Journal of Solids and Structures. 45D.Y. Gao, and H.F. Yu, Multi-scale modelling and canonical dual finite element method in phase transitions of solids, International Journal of Solids and Structures, 45, (2008), 3660- 3673.
The Schur complement and symmetric positive semidefinite (and definite) matrices. J Gallier, J. Gallier, The Schur complement and symmetric positive semidefinite (and definite) matri- ces, www.cis.upenn.edu/˜jean/schurcomp.pdf.
. R A Horn, C R Johnson, Matrix Analysis. Cambridge University PressR.A. Horn, and C.R. Johnson, "Matrix Analysis", Cambridge University Press, 1985.
A Jaffe, Constructive quantum field theory, Mathematical Physics. A. Fokas, A. Grigoryan, T. Kibble, and B. ZegarlinskiLondonImperial College PressA. Jaffe, Constructive quantum field theory, Mathematical Physics 2000, 111-127, Edited by A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski, Imperial College Press, London 2000.
Phase transitions and topological defects in the early universe. T W B Kibble, Aust. J. Phys. 50T.W.B. Kibble, Phase transitions and topological defects in the early universe, Aust. J. Phys., 50, (1997), 697-722.
On dual configuration forces. S F Li, A Gupta, J. of Elasticity. 84S.F. Li and A. Gupta, On dual configuration forces, J. of Elasticity, 84, (2006),13-31.
The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J S Rowlinson, J D Translation, Van Der Waals, J. Statist. Phys. 20J.S. Rowlinson, Translation of J.D. van der Waals' "The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density", J. Statist. Phys., 20, (1979), 197- 244.
Canonical dual least square method for solving general nonlinear systems of equations. N Ruan, D Y Gao, Y Jiao, Comput. Optim. Appl. 472N. Ruan, D.Y. Gao, and Y. Jiao Canonical dual least square method for solving general nonlinear systems of equations, Comput. Optim. Appl. 47 (2), (2010), 335-347.
Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in R n , to appear in. D M M Silva, D Y Gao, J. Math. Analy. Appl. D.M.M. Silva, and D.Y. Gao, Complete solutions and triality theory to a nonconvex opti- mization problem with double-well potential in R n , to appear in J. Math. Analy. Appl.
G Strang, G , Introduction to Applied Mathematics. Wellesley-Cambridge Press758ppG. Strang, G.: Introduction to Applied Mathematics, Wellesley-Cambridge Press, 1986, 758 pp.
R Strugariu, M D Voisei, C Zalinescu, Counter-examples in bi-duality, triality and tri-duality. R. Strugariu, M.D. Voisei, and C. Zalinescu, Counter-examples in bi-duality, triality and tri-duality, http://www.math.uaic.ro/ zalinesc/papers3.php?file=svz.pdf
Some remarks concerning Gao-Strang's complementary gap function. M D Voisei, C Zalinescu, 10.1080/00036811.2010.483427Applicable Analysis. M.D. Voisei, and C. Zalinescu, Some remarks concerning Gao-Strang's complementary gap function, Applicable Analysis, (2010). DOI: 10.1080/00036811.2010.483427.
Counterexamples to some triality tri-duality results. M D Voisei, C Zalinescu, DOI10.1007/s10898-010-9592-yJ. Glob. Optim. M.D. Voisei, and C. Zalinescu, Counterexamples to some triality tri-duality results, J. Glob. Optim., DOI 10.1007/s10898-010-9592-y.
Canonical dual approach to solving the maximum cut problem. Z B Wang, S C Fang, D Y Gao, W X Xing, to appear in J. Glob. OptimZ.B. Wang, S.C. Fang, D.Y. Gao,and W.X. Xing, Canonical dual approach to solving the maximum cut problem, to appear in J. Glob. Optim.
. D Z Gao And C, Wu, D. GAO AND C.Z. WU
Obstacle problem for von Karman equations. S T Yau, D Y Gao, Adv. Appl. Math. 13S.T. Yau, and D.Y. Gao, Obstacle problem for von Karman equations, Adv. Appl. Math., 13, (1992), 123-141.
| [] |
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"Fully on-chip microwave photonics system",
"Fully on-chip microwave photonics system"
] | [
"Yuansheng Tao \nSchool of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina\n",
"Fenghe Yang \nPeking University Yangtze Delta Institute of Optoelectronics\n226010NantongChina\n",
"Zihan Tao \nSchool of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina\n",
"Lin Chang \nSchool of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina\n\nFrontiers Science Center for Nano-optoelectronics\nPeking University\n100871BeijingChina\n",
"Haowen Shu \nSchool of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina\n",
"Ming Jin \nSchool of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina\n",
"Yan Zhou \nPeking University Yangtze Delta Institute of Optoelectronics\n226010NantongChina\n",
"Zhangfeng Ge \nPeking University Yangtze Delta Institute of Optoelectronics\n226010NantongChina\n",
"Xingjun Wang \nSchool of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina\n\nFrontiers Science Center for Nano-optoelectronics\nPeking University\n100871BeijingChina\n\nPeng Cheng Laboratory\n518055ShenzhenChina\n\nPeking University Yangtze Delta Institute of Optoelectronics\n226010NantongChina\n"
] | [
"School of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina",
"Peking University Yangtze Delta Institute of Optoelectronics\n226010NantongChina",
"School of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina",
"School of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina",
"Frontiers Science Center for Nano-optoelectronics\nPeking University\n100871BeijingChina",
"School of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina",
"School of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina",
"Peking University Yangtze Delta Institute of Optoelectronics\n226010NantongChina",
"Peking University Yangtze Delta Institute of Optoelectronics\n226010NantongChina",
"School of Electronics\nState Key Laboratory of Advanced Optical Communications System and Networks\nPeking University\n100871BeijingChina",
"Frontiers Science Center for Nano-optoelectronics\nPeking University\n100871BeijingChina",
"Peng Cheng Laboratory\n518055ShenzhenChina",
"Peking University Yangtze Delta Institute of Optoelectronics\n226010NantongChina"
] | [] | Microwave photonics (MWP), harnessing the tremendous bandwidth of light to generate, process and measure wideband microwave signals, are poised to spark a new revolution for the information and communication fields. Within the past decade, new opportunity for MWP has emerged driven by the advances of integrated photonics. However, despite significant progress made in terms of integration level, a fully on-chip MWP functional system comprising all the necessary photonic and electronic components, is yet to be demonstrated. Here, we break the status quo and provide a complete on-chip solution for MWP system, by exploiting hybrid integration of indium phosphide, silicon photonics and complementary metal-oxide-semiconductor (CMOS) electronics platforms. Applying this hybrid integration methodology, a fully chip-based MWP microwave instantaneous frequency measurement (IFM) system is experimentally demonstrated. The unprecedented integration level brings great promotion to the compactness, reliability, and performances of the overall MWP IFM system, including a wide frequency measurement range (2-34 GHz), ultralow estimation errors (10.85 MHz) and a fast response speed (∼0.3 ns). Furthermore, we deploy the chip-scale MWP IFM system into realistic application tasks, where diverse microwave signals with rapid-varying frequencies at X-band (8-12 GHz) are accurately identified in real-time. This demonstration marks a milestone for the development of integrated MWP, by providing the technology basis for the miniaturization and massive implementations of various MWP functional systems. | null | [
"https://arxiv.org/pdf/2202.11495v1.pdf"
] | 247,058,463 | 2202.11495 | 4a7537a91dc0aa38bcfd1ec894aba2583a339103 |
Fully on-chip microwave photonics system
Yuansheng Tao
School of Electronics
State Key Laboratory of Advanced Optical Communications System and Networks
Peking University
100871BeijingChina
Fenghe Yang
Peking University Yangtze Delta Institute of Optoelectronics
226010NantongChina
Zihan Tao
School of Electronics
State Key Laboratory of Advanced Optical Communications System and Networks
Peking University
100871BeijingChina
Lin Chang
School of Electronics
State Key Laboratory of Advanced Optical Communications System and Networks
Peking University
100871BeijingChina
Frontiers Science Center for Nano-optoelectronics
Peking University
100871BeijingChina
Haowen Shu
School of Electronics
State Key Laboratory of Advanced Optical Communications System and Networks
Peking University
100871BeijingChina
Ming Jin
School of Electronics
State Key Laboratory of Advanced Optical Communications System and Networks
Peking University
100871BeijingChina
Yan Zhou
Peking University Yangtze Delta Institute of Optoelectronics
226010NantongChina
Zhangfeng Ge
Peking University Yangtze Delta Institute of Optoelectronics
226010NantongChina
Xingjun Wang
School of Electronics
State Key Laboratory of Advanced Optical Communications System and Networks
Peking University
100871BeijingChina
Frontiers Science Center for Nano-optoelectronics
Peking University
100871BeijingChina
Peng Cheng Laboratory
518055ShenzhenChina
Peking University Yangtze Delta Institute of Optoelectronics
226010NantongChina
Fully on-chip microwave photonics system
1 *Corresponding author: [email protected] † These authors contributed equally to this manuscript
Microwave photonics (MWP), harnessing the tremendous bandwidth of light to generate, process and measure wideband microwave signals, are poised to spark a new revolution for the information and communication fields. Within the past decade, new opportunity for MWP has emerged driven by the advances of integrated photonics. However, despite significant progress made in terms of integration level, a fully on-chip MWP functional system comprising all the necessary photonic and electronic components, is yet to be demonstrated. Here, we break the status quo and provide a complete on-chip solution for MWP system, by exploiting hybrid integration of indium phosphide, silicon photonics and complementary metal-oxide-semiconductor (CMOS) electronics platforms. Applying this hybrid integration methodology, a fully chip-based MWP microwave instantaneous frequency measurement (IFM) system is experimentally demonstrated. The unprecedented integration level brings great promotion to the compactness, reliability, and performances of the overall MWP IFM system, including a wide frequency measurement range (2-34 GHz), ultralow estimation errors (10.85 MHz) and a fast response speed (∼0.3 ns). Furthermore, we deploy the chip-scale MWP IFM system into realistic application tasks, where diverse microwave signals with rapid-varying frequencies at X-band (8-12 GHz) are accurately identified in real-time. This demonstration marks a milestone for the development of integrated MWP, by providing the technology basis for the miniaturization and massive implementations of various MWP functional systems.
Introduction
With the impending electronic bandwidth bottleneck in our information society for radio-frequency (RF) networks and Internet of Things [1], the use of photonics to generate, process and measure wideband microwave signals has been explored extensively, which nowadays is well known as microwave photonics (MWP) [2,3]. Its large bandwidth and low-loss features enable the realization of key functionalities in microwave systems that are not offered by current RF technology. Recent years, in particular, new opportunity for MWP has emerged with the advances of integrated photonic circuit (PIC) [4,5], which enables a dramatic reduction in the system size, weight, and power-consumption (SWaP). Milestones have been achieved in a wide range of MWP functionalities, including filters [6][7][8], arbitrary waveform generators [9,10], microwave frequency measurement [11][12][13], phase shifters [14], tunable true-time delay lines [15], beamformers [16], and generic programmable processors [17][18][19], exhibiting a various level of system-integration completeness.
However, despite significant progress, a complete on-chip solution to fully incorporate all the required photonic components and supporting electronic hardware of MWP system is still missing: currently, most of MWP systems have only implemented partial photonic components as chip-integrated format [8][9][10][11][12][13][14][15][16][17][18][19], while the rest of the system are still constituted by bulk devices or equipment, thereby will induce issues related to large footprint, poor robustness, and high power-consumption. While indium phosphide (InP) platform allows monolithic integration of all photonic elements [6], it suffers from relatively large passive waveguide loss and elevated amplifier noise [5]. Silicon photonics (SiPh) platform is attractive for its scalable and low-cost implementation of diverse building blocks of MWP systems [20], but the integration of light source requires significant extra efforts. Other integrated photonic materials, such as silicon nitride [21], chalcogenide [22], and lithium niobate [23] also encounter similar obstacles in high-level integration. Therefore, finding a versatile path that combines strength from diverse material platforms is a prior task for the goal of a fully on-chip MWP system.
Aside from the photonic integration, another critical issue is the convergence with electronics integration that has been missing from most of the integrated MWP works. Generally, to offer powerful RF functionalities, many MWP applications should leverage high-performance electronic devices in addition to the photonic parts [4,24]. For instance, in an optoelectronic oscillator (OEO) [25,26], high-gain modulator drivers should be adopted to compensate the optical insertion loss for inspiring the microwave oscillation. In a microwave frequency measurement system [27], lock-in amplifiers are capable to enhance the readout accuracy of weak optical signals. However, the electronic hardware existed in current integrated MWP systems is still being implemented by large-size discrete modules, deeply limiting the scalability, efficiency, and noise performances. Hence, exploiting the integration with modern complementary metal-oxide-semiconductor (CMOS) electronics [28], is also crucial to the miniaturization of MWP systems.
Here, we demonstrated the first complete on-chip solution for MWP system leveraging hybrid integration of different photonic platforms (InP and SiPh) and CMOS electronic circuits. Applying this integration strategy, a fully chip-integrated MWP instantaneous frequency measurement (IFM) system is achieved. The photonic integration is implemented based on directly coupling InP distributed feedback (DFB) laser with SiPh PIC that is compactly united with CMOS transimpedance amplifier (TIA) dies using wire-bonding. Thanks to the unprecedented integration level, the MWP IFM system makes a huge leap compared to the prior studies with similar functionality [11][12][13][29][30][31][32][33][34], in terms of small footprint (tens of mm 2 ), wide frequency measurement range , ultralow estimation error (10.85 MHz) and fast response speed (∼0.3 ns). Also, the key capacity of real-time frequency identification on rapid-varying microwave signals, is experimentally performed at X-band (8)(9)(10)(11)(12) with single-shot detection and high accuracy. Such complete on-chip solution in this work can find immediate practices in numerous MWP operational systems across the microwave signal generation [35], transmission [36], and processing [37] domains.
Results
MWP system architecture and principle
To prove the feasibility of the proposed complete chip-integrated solution, we experimentally implement a fully on-chip MWP functional system for microwave frequency measurement application, as a concrete paradigm. The architecture of the MWP IFM system is illustrated in Fig. 1a, consisting of an InP DFB laser, a monolithic SiPh PIC and electronic CMOS TIAs. The MWP IFM system performs microwave frequency identification based on frequency to optical power mapping approach [38]. Firstly, a continuous-wave (c.w.) optical carrier is generated by the InP DFB laser, and then injected into the SiPh PIC via butt-coupling. The microwave signal of unknown instantaneous frequency (f RF ) is modulated onto the optical carrier by a high-speed Mach-Zehnder (MZ) modulator. The MZ modulator is driven as push-pull scheme and biased at the minimum transmission point to generate a dual-sideband suppressed-carrier (DSB-SC) optical modulated signal. The main reason for selecting DSB-SC modulation is due to its simple generation (only need single bias control) and low power dissipation, thereby is suitable for system-level integration. Noted that the bias-drift of silicon MZ modulator is relatively small [39] compared to those of bulky lithium-niobate modulators used in previous studies, therefore, the produced DSB-SC signal is highly stable during operation. A micro-ring resonator (MRR) with high-Q factor (narrow stopband) is followed to further remove the residual optical carrier. Subsequently, the DSB-SC signal is routed into an asymmetrical MZ interferometer (AMZI) filter, which serves as the key linear-optics frequency discriminator in our MWP IFM system. Once the optical carrier wavelength is finely aligned to the central transmission spectra of the bar/cross ports, thereby, decided by the inherently complimentary responses of AMZI, the DSB-SC signals will undergo f RF -dependent attenuation with opposite slopes at the bar/cross out-ports (as shown in label iv in Fig. 1a). The two channel DSB-SC optical signals are detected in a pair of germanium (Ge)/Si PDs. The resulting weak photocurrents are amplified and converted into voltage-format waveform by dual TIAs, and finally collected by a real-time oscilloscope for post-processing. Assuming the two output electrical powers are severally denoted by P bar (f RF ) and P cross (f RF ), in this case, the power ratio P bar /P cross shall be an unambiguous function of input microwave frequency (f RF ) within the frequency range of half free spectral range (FSR) of the AMZI discriminator, which is generally known as the amplitude comparison function (ACF) [29], given as below:
( ) ( ) ( ) bar RF RF cross RF P f ACF f P f (1)
Based on this frequency-to-power mapping ACF, the unknown frequency of input microwave signal can be estimated. More detailed theoretical analysis is provided in Supplementary Note 1. Fig. 2c). Fig. 2d shows the vertical p-doped-intrinsic-n-doped (PIN) Ge/Si PD, with ultralow dark photocurrents of ∼9 nA, thus supporting a large detection dynamic range. The MRR is designed as multimode waveguide structures to improve the Q-factor (∼1.4×10 5 , shown in Fig. 2e), and the coupling coefficients can be adjusted in virtue of the interferometric scheme (details are provided in Supplementary Note 4) to access critical coupling state. In this sense, the optimized MRR enables nearly ideal carrier suppression of the DSB-SC modulated signals, that is high rejection ratio and slight impact on close-to-carrier sidebands. For the AMZI discriminator, an extra interferometer function as variable beam splitter, is placed at the front (see in Fig. 1a) to form an adaptive double-MZI (DMZI) structure [40]. This particular design is to compensate the imperfections of fabricated 1×2/2×2 multi-mode interference (MMI) splitters and the unbalanced propagation loss of two AMZI arms, further to avoid the degradation of extinction ratio (corresponding to the measurement accuracy). The transmission spectrum of AMZI is characterized (in Fig. 2f), showing an FSR of 0.74 nm (92.5 GHz). TiN thermo-optic (TO) phase shifters are used to tune and maintain the desired operating bias or status for the above-mentioned SiPh devices. The SiPh PIC is directly wire bonded to dual unpackaged commercial TIA dies fabricated in a standard CMOS process (see in Supplementary Note 5). The maximal gain is measured as 34.8 dB at 10 MHz (in Fig. 2b). More details about the performance characterization can be found in Methods.
Design and device characterization
Thanks to the fully chip-scale integration of all required components as noted above, compared with the state-of-the-art benchtop or partially integrated MWP IFM systems, our MWP IFM system only operates at a fraction of the power consumption (884.2 mW, see in Supplementary Note 6) and requires a drastically reduced footprint (Fig. 1c, tens of mm 2 ). These features make it very suitable for mass production and widespread use in some emerging applications (e.g., airborne systems) in near future. after Ge/Si PD. The maximal gain at 10 MHz is 34.8 dB. c, High-speed silicon MZ modulator. The 3-dB operation bandwidth is > 23 GHz. d, Vertical PIN Ge/Si PDs, both have a low dark current of ∼9 nA at -1 V bias. e, High-Q MRR used to perform carrier suppression, with a loaded Q-factor of 1.4×10 5 . f, AMZI discriminator with an FSR of 0.74 nm, function as the optical discriminator in the MWP IFM system.
Static microwave frequency identification
The static microwave frequency identification capability of the MWP IFM system is firstly evaluated, in which the input is c.w. microwave signal with constant frequencies. Owing to the full photonic integration, the experimental setup (shown in Fig. 3a) is quite simple, while the only employed discrete equipment is just as tools for signal generation and characterization, and is not a part of the MWP IFM system. Noted that the CMOS TIAs are not used in this static measurement, since the resulting photocurrents are dc signals.
To initiate the frequency identification operation, we start with calibrating the bias of MZ modulator and the status of high-Q MRR, to realize the generation of DSB-SC modulated signal as shown in Fig. 3b. With the incorporation of the high-Q MRR filtering, the obtained optical carrier suppression ratio is higher than 30 dB, and can be maintained across broad frequency range (Supplementary Note 7). Then, the integrated MWP IFM system is characterized by sweeping the input microwave frequencies across 2-35 GHz and recording the generated photocurrents at the pair of on-chip Ge/Si PDs. The measured results are normalized and shown in the upper panel of Fig. 3c. As expected, the electrical responses of two Ge/Si PDs show opposite trends with frequency, in conformity with the AMZI complementary transmission spectra multiplied by frequency-dependent roll-off of the MZ modulator. It can be noticed that slight ripples exist in the PD frequency responses, which mainly derive from the power fluctuations of the free-running DFB laser. Nonetheless, by making subtraction of these two PD response curves based on Eq. (1), the power ripples can be effectively eliminated for a smooth ACF profile (lower panel of Fig. 3c), because the variations of input power will create exactly same impacts on PD-1 (bar port) and PD-2 (cross port). Following, a polynomial method is adopted to fit the measured data for the realization of continuous ACF curve. The obtained ACF curve, in turn, is used to estimate unknown input microwave frequencies in the whole measurement range, as shown in Fig. 3d. The figure indicates that the MWP IFM system is capable of identifying frequencies with very high precision up to 34 GHz. The estimation errors increased at high frequencies (>34 GHz), seen as inset figure of Fig. 3d, are mainly resulted by the limited EO bandwidth of silicon modulator (∼23 GHz in our system). Meanwhile, the near-dc frequency (<2 GHz) inputs are influenced by the MRR filtering profile and also cannot be accurately estimated. Therefore, the valid high-accuracy frequency measurement range is around 2-34 GHz. Within this measurement range, the frequency estimation errors have an ultralow root mean square (RMS) value of only 10.85 MHz (see in Fig. 3e). Moreover, the long-term stability is also evaluated by monitoring the average estimation errors over 0.5-hour experiment, as shown in Fig. 3f, and the results show acceptable fluctuations (<12.5%) on identification accuracy. More details about the experiment are given in Methods.
An important figure of merit (FOM) for microwave IFM systems is the estimation error as a percentage of the measurement range [11], which is calculated as 0.033% in this work. To the best of our knowledge, the attained FOM is a record-low value among linear-optics MWP IFM systems. This remarkable achievement is attributed to the complete integration of the photonic link, which accordingly leads to great improvement on the robustness for environment perturbations. While in this regards, former partially integrated MWP IFM systems usually require tens of centimeters or longer fibers to connect the on-chip optical discriminator and other off-chip devices, which will be inevitably influenced by operating variations (e.g., the polarization, vibration, temperature) and thereby cause estimation errors. In addition, the full photonic integration avoids the extra light amplification to compensate coupling loss, which itself will add amplified spontaneous emission noise into the system link.
Real-time microwave frequency identification
Besides static microwave frequency measurement, a broad range of practical IFM application scenarios involving radar warning receiver [29], biomedical instrument [41] and cognitive radio [42], commonly demand the identification of dynamic microwave signals with time-varying instantaneous frequencies. To validate that our fully on-chip MWP IFM system also holds the real-time frequency identification capacity, we apply it to recognize several typical time-varying microwave signals including hopping-frequency (HF), linear chirped-frequency (CF) and quadratic CF signals. The experimental setup is shown in Fig. 4a, and details are provided in Methods. Fig. 4b lists the temporal frequency sequences of input microwave signals with μs-level variation rate, generated by an arbitrary waveform generator (AWG, 50 GSa/s). For the HF signal, the frequency sequences hop in the following order: 8.2, 10.6, 9.1, 11.7, 7.5 and 9.8 GHz, each with the duration of 1 μs. For the linear/quadratic CF signals, the chirping frequency range is 7.5-11.5 GHz, with a total chirping duration of 10 μs. When processed through the fully on-chip MWP IFM system, these input time-varying microwave oscillations (∼GHz) are broadcasted onto optical carrier and discriminated based on frequency-to-power mapping in real time, and ultimately the targeted frequency variations with relatively low-speed (∼MHz) will be extracted back to electrical domain. The bandwidth of Ge/Si PDs (∼31 GHz, Supplementary Note 8) and CMOS TIAs (>10 MHz, Fig. 2b) are adequate to recognize the frequency changing rates of these incoming microwave signals. To recover the input frequency sequences, we only need a low-bandwidth oscilloscope to record the temporal output responses of the MWP IFM system, and perform simple off-line processing based on the pre-measured ACF curve shown in Fig. 4e. Fig. 4c displays the real-time readout frequency variation profiles of the HF, linear CF, and quadratic CF microwave signals, respectively. Compared with the original inputs (Fig. 4b), it can be seen that these three distinctly different time-varying microwave signals are all identified with high accuracy. The histograms of estimation errors are given in Fig. 4d. The corresponding RMS is calculated as 60.5 MHz, 59.8 MHz, and 55.0 MHz, respectively. These achieved RMS values are a little larger than that of static frequency identification, which mainly arises from the additional noise of CMOS TIAs, but still better than the measurement accuracy (in static manner) of most previous works. The input frequencies and their variation rates in this experiment are only decided by the off-the-shelf AWG (sampling-rate limited) and TIA dies (bandwidth limited), not the entire capacities of the MWP IFM system. To explore the fastest respond speed for instantaneous frequency-bursts, a HF signal with hopping frequency sequence of 7-12 GHz is chosen as test input (Fig. 4f), and in this case, a discrete low-noise electrical amplifier with higher bandwidth is employed instead of the TIA chips. The corresponding temporal amplitude-stepped waveform is measured and shown in Fig. 4g, indicating an ultra-short transition time of ∼0.3 ns.
It worth emphasized here that the real-time high-accuracy microwave frequency identifications demonstrated in this work are realized by single-shot detection without averaging processing (used in previous work [13]). This is due to the uniting of the SiPh PIC with micro-electronics TIAs, that enables the signal to noise ratio (SNR) of the overall MWP system to be greatly improved. Moreover, our work also avoids the cumbersome frequency-scanning operation [12], thereby reduces the implementation cost and complexity. Therefore, based on its operating simplicity and complete system integration, the fully on-chip MWP IFM system bridges the gap between laboratory test and real-world deployments, and the further efforts are supposed to be focus on the optimizations of element devices and packaging, which will be discussed in later section.
Discussion
A comparison among the representative integrated MWP IFM systems is shown in Table I. We see that the overall frequency identification performance presented in this work is a great advance compared to the pioneering studies, due to the combination of wide measurement range, high accuracy, and fast response speed. This advancement is attributed to the record-high system integration containing all the required photonic and electronic devices, thereby enabling the compactness, measurement latency, reliability, and noise performance of the MWP IFM system to be significantly improved. Noted that though stimulated Brillouin scattering (SBS)-based IFM systems may show lower estimation error [12], they use a large pumping power (>100 mW) and complex frequency-scanning technique (defies real-time frequency identification capacity), which will hinder their practical deployments in many real-world scenarios. In current implementation, the hybrid integration of InP platform and Si platform is realized based on butt-coupling, which can be replaced by photonic wire bonding technique [43] for more efficient optical connection. Heterogeneous integration [44] can further improve the compactness and scalability, but it still needs significant efforts in manufacturing development. Moving forward, the fundamental photonic devices also have the potential for upscaling, toward a more powerful MWP IFM system. For example, the valid frequency measurement range can be enlarged by optimizing the EO bandwidth of silicon modulator (potentially >50 GHz [45]). To promote the identification accuracy, the key is to access a broadband and ultra-sharp ACF profile, which is possible to be satisfied by utilizing Fano resonance structures [46]. On the electronic side, by virtue of the excellent CMOS compatibility of silicon, the low-noise TIAs are capable of being seamlessly integrated with photonics on a monolithic platform [47], providing solutions to drastically reduce the production cost.
Beyond the specific microwave frequency measurement demonstration provided here, our photonic-electronic complete on-chip solution also holds a profound significance to benefit much broader range of MWP applications, due to the universality of this work. For example, in OEO systems [35], the fully chip-scale integration will dramatically improve the power efficiency, stability of oscillation frequency and the key phase noise performance metrics. In the microwave signal processing such as the photonic analog-to-digital converter (ADC) [48] and channelized receiver [49], they usually demand an array of digital electronic circuits for post-processing. Implementing the hybrid photonic-electronic integration will support the reduction in footprint and promotion in the signal-to-noise ratio of the overall system. Therefore, we anticipate our demonstration could serve as a general strategy for the further development of integrated MWP, and paves the way for the massive applications of this technology in next generation information and communication fields.
Methods
Characterizations of the key photonic/electronic devices. To characterize the performances of diverse photonic devices in the MWP system, monitoring ports are preset in several critical positions of the entire on-chip optical link using 1:9 directional couplers. Firstly, the emission spectrum of InP DFB laser is collected and characterized by an ultrahigh-resolution (10 MHz) optical spectrum analyzer (Aragon Photonics, BOSA300 C+L), exhibiting a very narrow spectral linewidth and >40 dB side-mode suppression ratio. The electro-optic (EO) bandwidth of MZ modulator is measured by a vector network analyzer (VNA, Keysight N5247A), with the typical results of >23 GHz (at 2.5 V reverse bias voltage). For the waveguide-coupled Ge/Si PDs, the dark photocurrent is the most concern performance metrics in our IFM application, which is measured by a high-precision source meter (Keysight B2902A) under different bias voltages (using the sweep function mode). For the accurate evaluation of passive MRR and AMZI, a tunable laser (Keysight 81960A) is used to sweep the wavelength, meanwhile the transmission of optical signal is converted to electrical signal by a photodetector and then received by an oscilloscope (Agilent DSO81204B) synchronized with the laser. The CMOS TIA for signal amplification is characterized in terms of its gain value at different input frequency, that is tested by a VNA (Keysight N5247A). Due to the restriction of the measurement equipment, we could only obtain the gain values at >10 MHz range.
Static microwave frequency measurement experiment. The InP-based DFB laser is driven at 300 mA pumping current to generate ∼100 mW c.w. optical carrier, employing a commercial laser diode controller (Thorlabs, ITC 4001). The c.w. optical carrier is launched into the SiPh chip as butt-coupling manner, with a coupling loss of ∼6 dB. Both the InP and SiPh chips are mounted on thermoelectric cooler to stabilize the temperature during operation. An analog signal generator (Keysight E8257D) is employed to produce 16 dBm microwave signals of different frequencies, as input for the integrated IFM system. The microwave signals to be measured are firstly sent into a 180° RF hybrid (Marki); and then the out-of-phase dual outputs are fed to the inputs of silicon MZ modulator using coaxial cables of equal length. Considering the parasitic inductance of Au bonding wires, to support high-bandwidth IFM operation, in this test, the RF connection to the on-chip MZ modulator is implemented by GSGSG high-speed RF probes (Cascade, ACP40). The DC calibrations of TO phase shifters are offered by multi-channel DC power supplies (Keysight E36312A). The output photocurrents of the on-chip Ge/Si PDs are simultaneously collected by a two-channel high-precision source meter unit (Keysight B2902A). The analog signal generator and the source meter are programmatically controlled in MATLAB environment on a desktop via general purpose interface bus (GPIB) interfaces, which enables a fast and accurate measurement.
Real-time microwave frequency measurement experiment. The basic experimental setup is same with that of static IFM demonstration, while the major differences lie in the signal transmitting and receiving parts. In the transmitting part, a RF arbitrary waveform generator (AWG, Tektronix AWG70001) with a sampling rate of 50 GSa/s, is employed to generate the rapidly time-varying HF, linear CF, and quadratic CF microwave signals. The produced signals of the AWG are firstly amplified by a linear electrical amplifier (Mini-circuits), and then are fed into the RF input of the fully on-chip MWP IFM system, or collected by a 39 GHz real-time oscilloscope (Teledyne LeCroy, MCM-Zi-A) for recording original RF temporal waveforms as references (as displayed in Fig. 4b after Fourier transformation). In the receiving part, the generated temporal waveforms that contain the desired time-varying frequency sequences, are collected by a low-speed real-time oscilloscope (RIGOL DS7014, 10 GSa/s). A frequency step sequence 7.5-0.5-12.5 GHz is chosen as input, to obtain the reference ACF curve of the MWP IFM system (Fig. 4e). The data are analyzed offline using MATLAB on a desktop. In addition, to explore the shortest transition time for instantaneous frequency-burst, a HF signal (hopping at 7, 12 GHz) is chosen as test input (Fig. 4f). Since our off-the-shelf TIA dies are bandwidth-limited, in this case, we amplify the photocurrents using a low-noise electrical amplifier (LNA, Mini-circuits) with higher bandwidth and collect time-domain response by a 39 GHz oscilloscope (Teledyne LeCroy, MCM-Zi-A).
Fig. 1
1Photonic-electronic fully on-chip MWP IFM system. a, Illustration and operation principles of the fully chip-integrated MWP IFM system. b, Top-view micrograph of the hybrid-integrated InP DFB laser and SiPh PIC. Noted that the highlighted beam-path of MRR&AMZI is roughly showed, due to their relatively small size. Details can be seen inFig. 2eand f. c, Photograph of the implemented fully on-chip MWP IFM system. The SiPh PIC is directly wire-bonded to the electronic CMOS TIA chips, co-packaged on a PCB.
Fig
. 1b and c show the photographs of the MWP IFM system, that is fully implemented as chip-integrated formats and co-packaged onto a custom-designed print circuit board (PCB) for electrical connections.Fig. 2a-f display the essential photonic and electronic building units of this MWP system and their key performance metrics. For c.w. light generation, a commercially available InP DFB laser diode chip is employed, with a tunable emission wavelength centered at 1552.3 nm (seeFig. 2a). The maximum output power is measured as ∼100 mW under a pumping current of 300 mA (Supplementary Note 2). The monolithic SiPh PIC is designed in-house and fabricated by CompoundTek Foundry Services using its standard 90-nm silicon-on-insulator (SOI) lithography process. The InP and SiPh PICs are butt connected based on a Si inverse-taped edge coupler. The facet-to-facet coupling loss is measured as about 6.05±0.35 dB (Supplementary Note 3). The Si MZ modulator (3-mm long) is implemented based on travelling-wave PN depletion scheme, showing a modulation bandwidth of >23 GHz (see
Fig. 2
2Key photonic/electronic devices and fundamental characteristics. a, InP DFB laser with a tunable emission wavelength around 1552 nm. b, CMOS-based TIA for weak current amplification
Fig. 3
3Static microwave frequency identification using the integrated MWP IFM system. a, Experimental setup to perform the microwave frequency identification as static manner. MS, microwave source; SMU, source/measure unit; MZM, MZ modulator. b, The generated optical DSB-SC modulation signal without and with the aid of high-Q MRR suppression, at a modulation frequency of 10 GHz. c, Upward, electrical responses of the pair of Ge/Si PDs; below, the corresponding ACF curve. d, Frequency estimation measurement over 2-35 GHz. Inset: Enlarged estimation results at the range of 13-15 GHz and 33-35 GHz. e, Frequency estimation errors over 2-34 GHz, showing an RMS of 10.85 MHz. f, Characterization of long-term stability. Measured mean absolute estimation error over 30 minutes.
Fig. 4
4Real-time microwave frequency identification using the fully on-chip MWP IFM system. a, Experimental setup to perform the real-time microwave frequency identification. AWG, arbitrary waveform generator; OSC, oscilloscope. b, Input dynamically time-varying microwave signals, including HF signal, linear-CF (LCF) signal, and quadratic-CF (QCF) signal. c, Real-time frequency readout processed through the integrated MWP IFM system. d, Histogram of the estimation errors. e, The reference ACF employed in these real-time microwave frequency identification tests. f, The time-domain waveform of a HF signal with 7 GHz-12 GHz hopping sequences, used to explore the shortest transition time. g, The temporal response of the integrated MWP IFM system, showing a transition time of ∼0.3 ns.
Table I .
IComparison of state-of-the-art integrated MWP IFM systemsRef
System integration
Measurement
Range (GHz)
Error (MHz)
FOM b
Response
time (ns)
Passive Active a Light source Electronics
[30]
Y
N
N
N
0.5-4
93.6
2.7
N/A
[31]
Y
N
N
N
3-19
500
3.1
N/A
[32]
Y
N
N
N
1-30
237
0.8
~1×10 7
[33]
Y
N
N
N
0-26.62
250
0.9
N/A
[34]
Y
N
N
N
5-20
47.2
0.3
~2×10 8
[11]
Y
N
N
N
0-40
319
0.8
N/A
[12]
Y
N
N
N
9-38
1
0.003
~6×10 5
[13]
Y
N
N
N
0.01-32
755
2.3
1
This work
Y
Y
Y
Y
2-34
10.85
0.03
0.3
a refers to high-speed modulator and PD
b represents the estimation error as a percentage (%) of the measurement range.
Author contributionsThe experiments were conceived by Y.T., with the assistance from F.Y., Z.T., H.S. and M.J. The SiPh PIC was designed by Y.T. The packaging of this on-chip MWP IFM system was conducted by F.Y., Y.T., Y.Z., and Z.G. The results were analyzed by Y.T., F.Y., L.C., and Z.T. All authors participated in writing the manuscript. The project was coordinated by Y.T. and F.Y., under the supervision of X.W.Additional informationSupplementary information is available in the online version of the paper. Reprints and permissions information is available online. Correspondence and requests for materials should be addressed to X.W.Conflict of interestThe authors declare no competing financial interests.
The bandwidth bottleneck that is throttling the internet. J Hecht, Nature. 536139Hecht, J. The bandwidth bottleneck that is throttling the internet. Nature 536, 139 (2016).
Microwave photonics combines two worlds. J Capmany, D Novak, Nat. Photonics. 1319Capmany, J. & Novak, D. Microwave photonics combines two worlds. Nat. Photonics 1, 319 (2007).
Microwave photonics shines. R Won, Nat. Photonics. 5Won, R. Microwave photonics shines. Nat. Photonics 5, 736-736 (2011).
Integrated microwave photonics. D Marpaung, Laser Photonics Rev. 7Marpaung, D. et al. Integrated microwave photonics. Laser Photonics Rev. 7, 506-538 (2013).
Integrated microwave photonics. D Marpaung, J Yao, J Capmany, Nat. Photonics. 13Marpaung, D., Yao, J. & Capmany, J. Integrated microwave photonics. Nat. Photonics 13, 80- 90 (2019).
A monolithic integrated photonic microwave filter. J S Fandiño, P Muñoz, D Dom´enech, J Capmany, Nat. Photonics. 11Fandiño, J. S., Muñoz, P., Dom´enech, D. & Capmany, J. A monolithic integrated photonic microwave filter. Nat. Photonics 11, 124-129 (2017).
Hybrid-integrated high-performance microwave photonic filter with switchable response. Y Tao, Photonics Res. 9Tao, Y. et al. Hybrid-integrated high-performance microwave photonic filter with switchable response. Photonics Res. 9, 1569-1580 (2021).
Integrable microwave filter based on a photonic crystal delay line. J Sancho, Nat. Commun. 3Sancho, J. et al. Integrable microwave filter based on a photonic crystal delay line. Nat. Commun. 3, 1-9 (2012).
Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper. M H Khan, Nat. Photonics. 4Khan, M. H. et al. Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper. Nat. Photonics 4, 117-122 (2010).
Reconfigurable radio-frequency arbitrary waveforms synthesized in a silicon photonic chip. J Wang, Nat. Commun. 6Wang, J. et al. Reconfigurable radio-frequency arbitrary waveforms synthesized in a silicon photonic chip. Nat. Commun. 6, 1-8 (2015).
Low-error and broadband microwave frequency measurement in a silicon chip. M Pagani, Optica. 2Pagani, M. et al. Low-error and broadband microwave frequency measurement in a silicon chip. Optica 2, 751-756 (2015).
Wide-range, high-precision multiple microwave frequency measurement using a chip-based photonic brillouin filter. H Jiang, Optica. 3Jiang, H. et al. Wide-range, high-precision multiple microwave frequency measurement using a chip-based photonic brillouin filter. Optica 3, 30-34 (2016).
Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip. M Burla, X Wang, M Li, L Chrostowski, J Azaña, Nat. Commun. 7Burla, M., Wang, X., Li, M., Chrostowski, L. & Azaña, J. Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip. Nat. Commun. 7, 1-8 (2016).
Brillouin-based phase shifter in a silicon waveguide. L Mckay, Optica. 6McKay, L. et al. Brillouin-based phase shifter in a silicon waveguide. Optica 6, 907-913 (2019).
Continuously tunable ultra-thin silicon waveguide optical delay line. X Wang, Optica. 4Wang, X. et al. Continuously tunable ultra-thin silicon waveguide optical delay line. Optica 4, 507-515 (2017).
Silicon integrated microwave photonic beamformer. C Zhu, Optica. 7Zhu, C. et al. Silicon integrated microwave photonic beamformer. Optica 7, 1162-1170 (2020).
A fully reconfigurable photonic integrated signal processor. W Liu, Nat. Photonics. 10Liu, W. et al. A fully reconfigurable photonic integrated signal processor. Nat. Photonics 10, 190-195 (2016).
Programmable photonic signal processor chip for radiofrequency applications. L Zhuang, C G Roeloffzen, M Hoekman, K.-J Boller, A J Lowery, Optica. 2Zhuang, L., Roeloffzen, C. G., Hoekman, M., Boller, K.-J. & Lowery, A. J. Programmable photonic signal processor chip for radiofrequency applications. Optica 2, 854-859 (2015).
Photonic integrated field programmable disk array signal processor. W Zhang, J Yao, Nat. Commun. 11Zhang, W. & Yao, J. Photonic integrated field programmable disk array signal processor. Nat. Commun. 11, 1-9 (2020).
Silicon-based integrated microwave photonics. W Zhang, J Yao, IEEE J. Quantum Electron. 52Zhang, W. & Yao, J. Silicon-based integrated microwave photonics. IEEE J. Quantum Electron. 52, 1-12 (2015).
Silicon nitride microwave photonic circuits. C G Roeloffzen, Opt. Express. 21Roeloffzen, C. G. et al. Silicon nitride microwave photonic circuits. Opt. Express 21, 22937- 22961 (2013).
On-chip stimulated brillouin scattering for microwave signal processing and generation. R Pant, Laser Photonics Rev. 8Pant, R. et al. On-chip stimulated brillouin scattering for microwave signal processing and generation. Laser Photonics Rev. 8, 653-666 (2014).
Integrated photonics on thin-film lithium niobate. D Zhu, Adv. Opt. Photonics. 13Zhu, D. et al. Integrated photonics on thin-film lithium niobate. Adv. Opt. Photonics 13, 242- 352 (2021).
RF engineering meets optoelectronics: Progress in integrated microwave photonics. S Iezekiel, M Burla, J Klamkin, D Marpaung, J Capmany, IEEE Microw. Mag. 16Iezekiel, S., Burla, M., Klamkin, J., Marpaung, D. & Capmany, J. RF engineering meets optoelectronics: Progress in integrated microwave photonics. IEEE Microw. Mag. 16, 28-45 (2015).
The optoelectronic oscillator. L Maleki, Nat. Photonics. 5Maleki, L. The optoelectronic oscillator. Nat. Photonics 5, 728-730 (2011).
Silicon photonic integrated optoelectronic oscillator for frequency-tunable microwave generation. W Zhang, J Yao, J. Lightwave Technol. 36Zhang, W. & Yao, J. Silicon photonic integrated optoelectronic oscillator for frequency-tunable microwave generation. J. Lightwave Technol. 36, 4655-4663 (2018).
Chip-scale brillouin instantaneous frequency measurement by use of one-shot frequency-to-power mapping based on lock-in amplification. Y Xu, Y Yang, X Li, X Wang, W Zou, Chin. Opt. Lett. 19113902Xu, Y., Yang, Y., Li, X., Wang, X. & Zou, W. Chip-scale brillouin instantaneous frequency measurement by use of one-shot frequency-to-power mapping based on lock-in amplification. Chin. Opt. Lett. 19, 113902 (2021).
Electronic-photonic convergence for silicon photonics transmitters beyond 100 gbps on-off keying. K Li, Optica. 7Li, K. et al. Electronic-photonic convergence for silicon photonics transmitters beyond 100 gbps on-off keying. Optica 7, 1514-1516 (2020).
Towards on-chip photonic-assisted radio frequency spectral measurement and monitoring. L R Cort´es, D Onori, H G De Chatellus, M Burla, J Azaña, Optica. 7Cort´es, L. R., Onori, D., de Chatellus, H. G., Burla, M. & Azaña, J. Towards on-chip photonic-assisted radio frequency spectral measurement and monitoring. Optica 7, 434-447 (2020).
On-chip photonic-assisted instantaneous microwave frequency measurement system. D Marpaung, IEEE Photonics Technol. Lett. 25Marpaung, D. On-chip photonic-assisted instantaneous microwave frequency measurement system. IEEE Photonics Technol. Lett. 25, 837-840 (2013).
High-sensitivity instantaneous microwave frequency measurement based on a silicon photonic integrated fano resonator. B Zhu, W Zhang, S Pan, J Yao, J. Lightwave Technol. 37Zhu, B., Zhang, W., Pan, S. & Yao, J. High-sensitivity instantaneous microwave frequency measurement based on a silicon photonic integrated fano resonator. J. Lightwave Technol. 37, 2527-2533 (2019).
Wideband adaptive microwave frequency identification using an integrated silicon photonic scanning filter. X Wang, Photonics Res. 7Wang, X. et al. Wideband adaptive microwave frequency identification using an integrated silicon photonic scanning filter. Photonics Res. 7, 172-181 (2019).
Demonstration of photonic-assisted microwave frequency measurement using a notch filter on silicon chip. W Jiao, M Cheng, K Wang, J Sun, J. Lightwave Technol. 39Jiao, W., Cheng, M., Wang, K. & Sun, J. Demonstration of photonic-assisted microwave frequency measurement using a notch filter on silicon chip. J. Lightwave Technol. 39, 6786- 6795 (2021).
High-resolution microwave frequency measurement based on dynamic frequency-to-power mapping. S Song, S X Chew, L Nguyen, X Yi, Opt. Express. 29Song, S., Chew, S. X., Nguyen, L. & Yi, X. High-resolution microwave frequency measurement based on dynamic frequency-to-power mapping. Opt. Express 29, 42553-42568 (2021).
Recent advances in optoelectronic oscillators. T Hao, Adv. Photonics. 244001Hao, T. et al. Recent advances in optoelectronic oscillators. Adv. Photonics 2, 044001 (2020).
Microwave photonics: radio-over-fiber links, systems, and applications. K Xu, Photonics Res. 2Xu, K. et al. Microwave photonics: radio-over-fiber links, systems, and applications. Photonics Res. 2, B54-B63 (2014).
Ultra-wideband and adaptive photonic signal processing of microwave signals. R A Minasian, IEEE J. Quantum Electron. 52Minasian, R. A. Ultra-wideband and adaptive photonic signal processing of microwave signals. IEEE J. Quantum Electron. 52, 1-13 (2015).
Recent advances in microwave photonics instantaneous frequency measurements. L A Bui, Prog. Quantum Electron. 69100237Bui, L. A. Recent advances in microwave photonics instantaneous frequency measurements. Prog. Quantum Electron. 69, 100237 (2020).
Bias-drift-free mach-zehnder modulators based on a heterogeneous silicon and lithium niobate platform. S Sun, Photonics Res. 8Sun, S. et al. Bias-drift-free mach-zehnder modulators based on a heterogeneous silicon and lithium niobate platform. Photonics Res. 8, 1958-1963 (2020).
Perfect optics with imperfect components. D A Miller, Optica. 2Miller, D. A. Perfect optics with imperfect components. Optica 2, 747-750 (2015).
Respiratory rate estimation from the ecg using an instantaneous frequency tracking algorithm. L Mirmohamadsadeghi, J.-M Vesin, Biomed. Signal Process. Control. 14Mirmohamadsadeghi, L. & Vesin, J.-M. Respiratory rate estimation from the ecg using an instantaneous frequency tracking algorithm. Biomed. Signal Process. Control 14, 66-72 (2014).
A survey of spectrum sensing algorithms for cognitive radio applications. T Yucek, H Arslan, IEEE Commun. Surv. Tutor. 11Yucek, T. & Arslan, H. A survey of spectrum sensing algorithms for cognitive radio applications. IEEE Commun. Surv. Tutor. 11, 116-130 (2009).
Hybrid integration of silicon photonics circuits and inp lasers by photonic wire bonding. M R Billah, Optica. 5Billah, M. R. et al. Hybrid integration of silicon photonics circuits and inp lasers by photonic wire bonding. Optica 5, 876-883 (2018).
Laser soliton microcombs heterogeneously integrated on silicon. C Xiang, Science. 373Xiang, C. et al. Laser soliton microcombs heterogeneously integrated on silicon. Science 373, 99-103 (2021).
Silicon intensity mach-zehnder modulator for single lane 100 gb/s applications. M Li, L Wang, X Li, X Xiao, S Yu, Photonics Res. 6Li, M., Wang, L., Li, X., Xiao, X. & Yu, S. Silicon intensity mach-zehnder modulator for single lane 100 gb/s applications. Photonics Res. 6, 109-116 (2018).
An actively controlled silicon ring resonator with a fully tunable fano resonance. A Li, W Bogaerts, APL Photonics. 296101Li, A. & Bogaerts, W. An actively controlled silicon ring resonator with a fully tunable fano resonance. APL Photonics 2, 096101 (2017).
Roadmap on silicon photonics. D Thomson, J. Opt. 1873003Thomson, D. et al. Roadmap on silicon photonics. J. Opt. 18, 073003 (2016).
Photonic adc: overcoming the bottleneck of electronic jitter. A Khilo, Opt. Express. 20Khilo, A. et al. Photonic adc: overcoming the bottleneck of electronic jitter. Opt. Express 20, 4454-4469 (2012).
Microwave photonics programs at darpa. R W Ridgway, C L Dohrman, J A Conway, J. Lightwave Technol. 32Ridgway, R. W., Dohrman, C. L. & Conway, J. A. Microwave photonics programs at darpa. J. Lightwave Technol. 32, 3428-3439 (2014).
| [] |
[
"ISOMORPHISM TESTING OF GROUPS OF CUBE-FREE ORDER",
"ISOMORPHISM TESTING OF GROUPS OF CUBE-FREE ORDER"
] | [
"Heiko Dietrich ",
"James B Wilson "
] | [] | [] | A. A group G has cube-free order if no prime to the third power divides |G|. We describe an algorithm that given two cube-free groups G and H of known order, decides whether G ∼ = H, and, if so, constructs an isomorphism G → H. If the groups are input as permutation groups, then our algorithm runs in time polynomial in the input size, improving on the previous super-polynomial bound. An implementation of our algorithm is provided for the computer algebra system GAP.In memory of C.C. Sims.ICapturing the natural concept of symmetry, groups are one the most prominent algebraic structures in science. Yet, it is still a challenge to decide whether two finite groups are isomorphic. Despite abundant knowledge about groups, presently no one has provided an isomorphism test for all finite groups whose complexity improves substantively over brute-force. In the most general form there is no known polynomial-time isomorphism test even for non-deterministic Turing machines, that is, the problem may lie outside the complexity classes NP and co-NP (see [2, Corollary 4.9]). At the time of this writing, the available implementations of algorithms that test isomorphism on broad classes of groups can run out of memory or run for days on examples of orders only a few thousand, see [6, Section 1.1] andTable 1. To isolate the critical difficulties in group isomorphism, it helps to consider special classes of groups as has been done recently in[1,3,6,9,11,31]. is paper is a part of a larger project intended to describe for which orders of groups is group isomorphism tractable: details of this project are given in[11]. In particular, in [11] we have described polynomial-time algorithms for isomorphism testing of abelian and meta-cyclic groups of most orders; the computational framework for these algorithms is built upon type theory and groups of so-called black-box type. By a theorem of Hölder ([26, 10.1.10]), all groups of square-free order are coprime meta-cyclic, that is, they can be decomposed as G = A ⋉ B where A, B G are cyclic subgroups of coprime orders; unfortunately, [11, eorem 1.2] is not guaranteed for all square-free orders. In this paper, we switch to a more restrictive computational model, allowing us to make progress for isomorphism testing of square-free and cube-free groups. Specifically, here we consider groups generated by a set S of permutations on a finite set Ω. at gives us access to a robust family of algorithms by Sims and many others (see[16,28]) that run in time polynomial in |Ω| · |S|. Note that the order of such a group G can be exponential in |Ω| · |S|, even when restricted to groups of square-free order, see Proposition 2.1. e main result of this paper is the following theorem. eorem 1.1. ere is an algorithm that given groups G and H of permutations on finitely many points, decides whether they are of cube-free order, and if so, decides that G ∼ = H or constructs an isomorphism G → H. e algorithm runs in time polynomial in the input size. eorem 1.1 is based on the structure analysis of cube-free groups by Eick & Dietrich [9] and Qiao & Li[25]. A top-level description of our algorithm is given in Section 3.2. Importantly, our algorithm translates to a functioning implementation for the system GAP [14], in the package "Cubefree" [10]. As a side-product, we also discuss algorithms related to the construction of complements of Ω-groups, Sylow towers, socles, and constructive presentations, see Section 4. ese algorithms have applications beyond cube-free groups and might be of general interest in computational group theory. | 10.1016/j.jalgebra.2019.02.008 | [
"https://arxiv.org/pdf/1810.03467v1.pdf"
] | 119,131,312 | 1810.03467 | b2076fe3e114fca30fd76723ebfee7fae366b85e |
ISOMORPHISM TESTING OF GROUPS OF CUBE-FREE ORDER
5 Oct 2018
Heiko Dietrich
James B Wilson
ISOMORPHISM TESTING OF GROUPS OF CUBE-FREE ORDER
5 Oct 2018Date: October 9, 2018.Simon Visiting Professorship in Summer 2016. Both authors thank Alexander Hulpke for advice on conjugacy problems.and phrases finite groupscube-free groupsgroup isomorphisms
A. A group G has cube-free order if no prime to the third power divides |G|. We describe an algorithm that given two cube-free groups G and H of known order, decides whether G ∼ = H, and, if so, constructs an isomorphism G → H. If the groups are input as permutation groups, then our algorithm runs in time polynomial in the input size, improving on the previous super-polynomial bound. An implementation of our algorithm is provided for the computer algebra system GAP.In memory of C.C. Sims.ICapturing the natural concept of symmetry, groups are one the most prominent algebraic structures in science. Yet, it is still a challenge to decide whether two finite groups are isomorphic. Despite abundant knowledge about groups, presently no one has provided an isomorphism test for all finite groups whose complexity improves substantively over brute-force. In the most general form there is no known polynomial-time isomorphism test even for non-deterministic Turing machines, that is, the problem may lie outside the complexity classes NP and co-NP (see [2, Corollary 4.9]). At the time of this writing, the available implementations of algorithms that test isomorphism on broad classes of groups can run out of memory or run for days on examples of orders only a few thousand, see [6, Section 1.1] andTable 1. To isolate the critical difficulties in group isomorphism, it helps to consider special classes of groups as has been done recently in[1,3,6,9,11,31]. is paper is a part of a larger project intended to describe for which orders of groups is group isomorphism tractable: details of this project are given in[11]. In particular, in [11] we have described polynomial-time algorithms for isomorphism testing of abelian and meta-cyclic groups of most orders; the computational framework for these algorithms is built upon type theory and groups of so-called black-box type. By a theorem of Hölder ([26, 10.1.10]), all groups of square-free order are coprime meta-cyclic, that is, they can be decomposed as G = A ⋉ B where A, B G are cyclic subgroups of coprime orders; unfortunately, [11, eorem 1.2] is not guaranteed for all square-free orders. In this paper, we switch to a more restrictive computational model, allowing us to make progress for isomorphism testing of square-free and cube-free groups. Specifically, here we consider groups generated by a set S of permutations on a finite set Ω. at gives us access to a robust family of algorithms by Sims and many others (see[16,28]) that run in time polynomial in |Ω| · |S|. Note that the order of such a group G can be exponential in |Ω| · |S|, even when restricted to groups of square-free order, see Proposition 2.1. e main result of this paper is the following theorem. eorem 1.1. ere is an algorithm that given groups G and H of permutations on finitely many points, decides whether they are of cube-free order, and if so, decides that G ∼ = H or constructs an isomorphism G → H. e algorithm runs in time polynomial in the input size. eorem 1.1 is based on the structure analysis of cube-free groups by Eick & Dietrich [9] and Qiao & Li[25]. A top-level description of our algorithm is given in Section 3.2. Importantly, our algorithm translates to a functioning implementation for the system GAP [14], in the package "Cubefree" [10]. As a side-product, we also discuss algorithms related to the construction of complements of Ω-groups, Sylow towers, socles, and constructive presentations, see Section 4. ese algorithms have applications beyond cube-free groups and might be of general interest in computational group theory.
I
Capturing the natural concept of symmetry, groups are one the most prominent algebraic structures in science. Yet, it is still a challenge to decide whether two finite groups are isomorphic. Despite abundant knowledge about groups, presently no one has provided an isomorphism test for all finite groups whose complexity improves substantively over brute-force. In the most general form there is no known polynomial-time isomorphism test even for non-deterministic Turing machines, that is, the problem may lie outside the complexity classes NP and co-NP (see [2,Corollary 4.9]). At the time of this writing, the available implementations of algorithms that test isomorphism on broad classes of groups can run out of memory or run for days on examples of orders only a few thousand, see [6, Section 1.1] and Table 1. To isolate the critical difficulties in group isomorphism, it helps to consider special classes of groups as has been done recently in [1,3,6,9,11,31]. is paper is a part of a larger project intended to describe for which orders of groups is group isomorphism tractable: details of this project are given in [11]. In particular, in [11] we have described polynomial-time algorithms for isomorphism testing of abelian and meta-cyclic groups of most orders; the computational framework for these algorithms is built upon type theory and groups of so-called black-box type. By a theorem of Hölder ( [26, 10.1.10]), all groups of square-free order are coprime meta-cyclic, that is, they can be decomposed as G = A ⋉ B where A, B
G are cyclic subgroups of coprime orders; unfortunately, [11, eorem 1.2] is not guaranteed for all square-free orders. In this paper, we switch to a more restrictive computational model, allowing us to make progress for isomorphism testing of square-free and cube-free groups. Specifically, here we consider groups generated by a set S of permutations on a finite set Ω. at gives us access to a robust family of algorithms by Sims and many others (see [16,28]) that run in time polynomial in |Ω| · |S|. Note that the order of such a group G can be exponential in |Ω| · |S|, even when restricted to groups of square-free order, see Proposition 2.1. e main result of this paper is the following theorem. eorem 1.1. ere is an algorithm that given groups G and H of permutations on finitely many points, decides whether they are of cube-free order, and if so, decides that G ∼ = H or constructs an isomorphism G → H. e algorithm runs in time polynomial in the input size. eorem 1.1 is based on the structure analysis of cube-free groups by Eick & Dietrich [9] and Qiao & Li [25]. A top-level description of our algorithm is given in Section 3.2. Importantly, our algorithm translates to a functioning implementation for the system GAP [14], in the package "Cubefree" [10]. As a side-product, we also discuss algorithms related to the construction of complements of Ω-groups, Sylow towers, socles, and constructive presentations, see Section 4. ese algorithms have applications beyond cube-free groups and might be of general interest in computational group theory.
1.1. Limitations. In contrast to our work in [11], eorem 1.1 no longer applies to a dense set of orders: the density of positive integers n which are square-free and cube-free tends to 1/ζ(2) ≈ 0.61 and 1/ζ(3) ≈ 0.83, respectively, where ζ(x) is the Riemann ζ-function, see [12, (2)]. It is known that most isomorphism types of groups accumulate at orders with large prime-power divisors. Indeed, Higman, Sims, and Pyber [4] proved that the number of groups of order n, up to isomorphism, tends to n 2µ(n) 2 /27+O(log n) where µ(n) = max{k : n is not k-free}. Specifically, the number of pairwise non-isomorphic groups of a cube-free order n is not more than O(n 8 ), with speculation that the tight bound is o(n 2 ), see [4, p. 236]. e prevailing belief in works like [1,31] is that the difficult instances of group isomorphism are when µ(n) is unbounded, especially when n is a prime power. Isomorphism testing of finite p-groups is indeed a research area that has a racted a lot of a ention.
However, eorem 1.1 completely handles an easily described family of group orders which may make it easier to use in applications. A further point is that groups of cube-free order exhibit many of the fundamental components of finite groups. For instance, groups of cube-free order need not be solvable, to wit the simple alternating group A 5 has cube-free order 60. When decomposed into canonical series, such as the Fi ing series, the associated extensions have nontrivial first and second cohomology groups -a measure of how difficult it is to compare different extensions.
1.2. Structure of the paper. In Section 2 we introduce some notation and comment on the computational model for our algorithm. In Section 3 we recall the structure of cube-free groups and give a top-level description of our isomorphism test. Various preliminary algorithms (for example, related to the construction of Sylow bases and towers, Ω-complements, socles, and constructive presentations) are described in Section 4. e proof of the main theorem is broken up into three progressively more general families: the solvable Fra ini-free case (Section 5), the general solvable case (Section 6), and finally the general case (Section 7). We have implemented many aspects of this algorithm in the computer algebra system GAP and comment on some examples in Section 8.
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2.1. Notation. We reserve p for prime numbers and n for group orders. For a positive integer n we denote by C n a cyclic group of order n, and Z/n for the explicit encoding as integers, in which we are further permi ed to treat the structure as a ring. Let (Z/n) × denote the units of this ring. Direct products of groups are denoted variously by "×" or exponents. roughout, F q is a field of order q and GL d (q) is the group of invertible (d × d)matrices over F q . e group PSL d (q) consists of matrices of determinant 1 modulo scalar matrices.
For a group G and g, h ∈ G, conjugates and commutators are g h = h −1 gh and [g, h] = g −1 g h , respectively. For subsets X, Y ⊂ G let [X, Y ] = [x, y] : x ∈ X, y ∈ Y ; the centralizer and normalizer of X in G are C G (X) = {g ∈ G : [X, g] = 1} and N G (X) = {g ∈ G : [X, g] ⊆ X}, respectively. e derived series of G has terms G (n+1) = [G (n) , G (n) ] for n ≥ 1, with G (1) = G. We read group extensions from the right and use A ⋉ B for split extensions; we also write A ⋉ ϕ B to emphasize the action ϕ :
A → Aut(B). Hence, A ⋉ B ⋉ C ⋉ D stands for ((A ⋉ B) ⋉ C) ⋉ D, etc.
We mostly adhere to protocol set out in standard literature on computational group theory, such as the Handbook of Computation Group eory [16] and the books of Robinson [26] and Seress [28].
Computational model.
roughout we assume that groups are given as finite permutation groups, but it is permissible to include congruences, which are best described as quotients of permutation groups. is allows us to prove that the algorithm of eorem 1.1 runs in polynomial time in the input size. Proving the same for groups given by polycyclic presentations seems difficult, partly because of the challenges involving collection, see [21]. Convention: when we say that an algorithm runs in polynomial time, then this is to be understood to be in time polynomial in the input size, assuming that the groups are input as (quotients of) finite permutation groups.
One simple but critical implication of our computational model is that if a prime p divides the group order |G|, then p divides d!, where d is the size of the permutation domain; so p d, which is less than the input size for G. is shows that all primes dividing the group order are small, allowing for polynomial-time factorization and other relevant number theory. Moreover, many essential group theoretic structures of groups of permutations (and their quotients) can be computed in polynomial time, as outlined in [28, p. 49] and [17,Section 4]. For example, it is possible to compute group orders, to produce constructive presentations, and to test membership constructively. For solvable permutation groups one can also efficiently get a constructive polycyclic presentation (see Lemma 4.7).
Before we begin, we demonstrate that the assumption that our groups are input by permutations is not an automatic improvement in the complexity. In particular, we show that large groups of square-free (and so also cube-free) order can arise as permutation groups in small degrees.
Proposition 2.1. Let G be a square-free group of order n = p 1 · · · p ℓ , with each p i prime. e group G can be faithfully represented in a permutation group of degree p 1 + · · · + p ℓ . For infinitely many squarefree m, there is a faithful permutation representation of the groups of order m on O(log 2 m) points.
P
. Hölder's classification [26, (10.1.10)] shows that G ∼ = C a ⋉ C b with n = ab. Since a is square-free, all subgroups of C a are direct factors, thus C a = C d × C Ca (C b ) for a subgroup C d , and C a ⋉ C b = C d ⋉ C e where the centralizer in C d of C e is trivial. us, we can assume that C a ⋉ C b with C a acting faithfully on C b , and a = p 1 · · · p s and b = p s+1 · · · p ℓ . Using disjoint p i -cycles for each i > s, we faithfully represent C b on p s+1 + · · · + p ℓ points. Since C a acts faithfully on C b , that representation can be given on the disjoint cycles of C b , that is, C a ⋉ C b is faithfully represented on p s+1 + · · · + p ℓ points. e first claim follows. For the last observation, let m = r 1 · · · r ℓ be the product of the first ℓ-primes. ese primorials have asymptotic growth m ∈ exp((1 + Θ(1))ℓ log ℓ), see [27, (3.16)]. Meanwhile, as just shown, the groups of order m can all be represented faithfully on as few as r 1 + · · · + r ℓ points, and r 1 + . . . + r ℓ ∈ Ω(ℓ 2 log ℓ) by [24, eorem C].
S
3.1. Structure of cube-free groups. For a finite group G we denote by Φ(G) and soc(G) its Fra ini subgroup and its socle, respectively; the first is the intersection of all maximal subgroups of G, and the la er is the subgroup generated by all minimal normal subgroups. We write G Φ for the Fra ini quotient G/Φ(G). A group is Fra ini-free if Φ(G) = 1; in particular, G Φ is Fra ini-free. By [9], every group G of cube-free order can be decomposed as
G = A × L
where A is trivial or A = PSL 2 (p) for a prime p > 3 with p ± 1 cube-free, and L is solvable with abelian Fra ini subgroup Φ(L) = Φ(G) whose order is square-free and divides the order of the Fra ini quotient L Φ = L/Φ(L). e la er satisfies
L Φ = K ⋉ (B × C) where soc(L Φ ) = B × C is the socle of L Φ with B = s i=1 Z/p i and C = m j=s+1 (Z/p j ) 2
for distinct primes p 1 , . . . , p m . Let X Y denote a subdirect product, that is, a subgroup of X × Y whose projections to X and Y are surjective. With this notation, we have
K = K 1 . . . K m Aut(B × C) = s i=1 GL 1 (p i ) × m j=s+1 GL 2 (p j );
It follows from work of Gaschütz (see [9,Lemma 9]) that two solvable Fra ini-free groups K⋉(B×C) andK ⋉ (B × C) with K,K Aut(B × C) as above are isomorphic if and only if K andK are conjugate in Aut(B × C); this is one of the reasons why our proposed isomorphism algorithm works so efficiently. Lastly, we recall that L is determined by L Φ : there exists, up to isomorphism, a unique extension M of L Φ by Φ(L) such that Φ(M ) ∼ = Φ(L) and M/Φ(M ) ∼ = L Φ , see [9, eorem 11].
Remark 3.1. Taunt [29] was probably the first who considered the class of cube-free groups. e focus in the work of Dietrich & Eick [9] is on a construction algorithm for all cube-free groups of a fixed order, up to isomorphism; the approach is based on the so-called Fra ini extension method (see [16, §11.4.1]). Complimentary to this work, Qiao & Li [25] also analyzed the structure of cubefree groups. ey proved in [25, eorem 1.1] that for every group G of cube-free order there exist integers a, b, c, d > 0 such that G is isomorphic to
(C c × C 2 d ) ⋉ (C a × C 2 b ) or G 2 ⋉ (C c × C 2 d ) ⋉ (C a × C 2 b ) with G 2 G a Sylow 2-subgroup, or PSL 2 (p) × (C c × C 2 d ) ⋉ (C a × C 2 b )
for some prime p. Le unclassified in this description are the relevant actions of the semidirect products, and a classification up to isomorphism. As we have shown in [11,Section 4], even for meta-cyclic groups, recovering the appropriate actions and comparing them is in general not easy.
Among the implications of these decomposition results is that a solvable group G of cube-free order has a Sylow tower, that is, a normal series such that each section is isomorphic to a Sylow subgroup of G, cf. [25, Corollary 3.4 & eorem 3.9].
3.2.
e algorithm. Let G andG be cube-free groups. We now describe the main steps of our algorithm to construct an isomorphism G →G, which fails if and only if G ∼ =G. Our approach is to determine, for each group, the Fra ini extension structure as described in Section 3. Since our groups are input by permutations, it is possible to decide if |G| = |G| and also to factorize this order. It simplifies our treatment to assume that the groups are of the same order and that the prime factors of this order are known. First, for G (and similarly forG) we do the following:
(i) Decompose G = A × L with A = 1 or A = PSL 2 (p) simple, and L solvable. (ii) Compute the Fra ini subgroup Φ(L) and the Fra ini quotient L Φ = L/Φ(L). (iii) Compute soc(L Φ ) = B × C and K Aut(B × C) such that L Φ = K ⋉ (B × C).
en we proceed as follows; if one of these steps fails, then G ∼ =G is established:
(1) Construct an isomorphism ψ A : A →Ã. (2) Construct an isomorphism ψ Φ : L Φ →L Φ . (3) Extend ψ Φ to an isomorphism ψ L : L →L. (4) Combine ψ A and ψ L to an isomorphism ψ : G →G.
In fact, G andG are isomorphic if and only if we succeed in Steps (1) & (2). us, if we just want to decide whether G ∼ =G, then Steps (3) & (4) need not to be carried out; moreover, it is not necessary to construct ψ A : since A andà are groups of type PSL 2 , we have A ∼ =à if and only if |A| = |Ã|, which can be readily determined in our computational framework.
P
We list a few algorithms which are required later. One important result is the description of an algorithm to construct an abelian Sylow tower for a solvable group, if it exists. is is a key ingredient in [3], but in that work groups are input as multiplication tables; in our se ing multiplication tables might be exponentially larger than the input, so we cannot use this work.
Constructive presentations and Ω-complements.
Let Ω be a set. An Ω-group is a group G on which the set Ω acts via a prescribed map θ : Ω → Aut(G). We first investigate the problem Ω-ComplementAbelian: given an abelian normal Ω-subgroup M G, decide whether G = K ⋉ M for some Ω-subgroup K G, or certify that no such K exists. Variations on this problem have been discussed in several places; the version we describe is based on a proof in [30,Proposition 4.5] which extends independent proofs by Luks and Wright in lectures at the U. Oregon.
We show in Proposition 4.3 that Ω-ComplementAbelian has a polynomial time solution for solvable groups. e proof involves Luks' constructive presentations [23, Section 4.2], which will also be useful later to equip solvable permutation groups with polycyclic presentations, see Lemma 4.7.
Definition 4.1. Let G be a group and N ✂ G. A constructive presentation of a group G/N is a free group F X on a set X, a homomorphism φ : F X → G, a function ψ : G → F X , and a set R ⊂ F X such that g −1 (gψφ) ∈ N for every g ∈ G, and N φ −1 = R F X , the normal closure of R in F X .
is can be interpreted as follows: X | R is a generator-relator presentation of the group G/N , see [23,Lemma 4.1]; the homomorphism φ is defined by assigning the generators X of F X to the generating set S ⊂ G.
e function ψ is in general not a homomorphism, and serves to writes elements of G as a corresponding word in X. e next lemma discusses a constructive presentation for a subgroup of the holomorph Aut(G) ⋉ G of a group G.
Lemma 4.2. Let G be an Ω-group via θ : Ω → Aut(G), and write g w = g wθ for g ∈ G and w ∈ Ω.
Let X | R with φ : F X → G and ψ : G → F X be a constructive presentation of G. Let Ω | S be a presentation for A = Ωθ Aut(G). en Ω ⊔ X | S ⋉ R is a presentation for A ⋉ G where S ⋉ R = S ⊔ R ⊔ {(xφ) w ψ · (x w ) −1 : x ∈ X, w ∈ Ω} ⊂ F Ω⊔X with embedding θ ⊔ φ : Ω ⊔ X → A ⋉ G, z → zθ (z ∈ Ω) zφ (z ∈ X) .
P . Without loss of generality, we can assume that F X = X , F Ω = Ω , and F Ω , F X F Ω⊔X . Let K be the normal closure of S ⋉ R in F Ω⊔X . Recall that, by definition, if x ∈ X, then xφψ and x define the same element in G via φ. It follows that if w ∈ Ω and x ∈ X, then
x w , (xφ) w ψ ∈ F Ω⊔X define the same element in A ⋉ G via θ ⊔ φ: if α : F Ω⊔X → A ⋉ G is the homomorphism defined by θ ⊔ φ, then (x w )α = ((wθ) −1 , 1)(1, xφ)(wθ, 1) = (1, (xφ) w ) = (1, (xφ) w ψφ) = ((xφ) w ψ)α shows that (xφ) w ψ(x w ) −1 ∈ ker α, so K ker α. Now consider N = KF X . From what is said above, if w ∈ Ω and x ∈ X, then Kx w = K(xφ) w ψ N , so N w = K w F w X Kx w : x ∈ X = N . is shows that N ✂ F Ω⊔X ; note that K w = K since K is the normal closure in F Ω⊔X . Now set C = KF Ω . It follows that F Ω⊔X = CN , thus H = F Ω⊔X /K = CN/K = (C/K)(N/K) and N/K is normal in H. Since C/K and N/K satisfy the presentations for A and G respectively, von Dyck's eorem [26, (2.2.1)] implies that H is a quotient of A ⋉ G. To show that H is isomorphic to A ⋉ G it suffices to notice that A ⋉ G satisfies the relations in S ⋉ R with respect to Ω ⊔ X and θ ⊔ φ. As shown above, K ker α. Since H = F Ω⊔X /K is a quotient of the group A ⋉ G = F Ω⊔X α, it follows that K = ker α, and therefore Ω ⊔ X | S ⋉ R is a presentation for A ⋉ G.
We now show that Ω-ComplementAbelian has a polynomial-time solution for solvable groups. Proposition 4.3. Let G be a solvable Ω-group with abelian normal Ω-subgroup M G. ere is a polynomial time algorithm that decides whether G = K ⋉ M for some Ω-subgroup K, or certifies that no such K exists.
P
. Let G be a quotient of a permutation group on n le ers, let θ : Ω → Aut(G) be a function, and let M be an abelian (Ω ∪ G)-subgroup of G. We first describe the algorithm, then prove correctness.
We use the algorithm of [11,Lemma 4.11] to produce a constructive presentation for the solvable quotient G/M with data X | R and maps φ : X → G and ψ : G → F X . For each s ∈ Ω and x ∈ X, define
w s,x = ((xφ) s )ψ · (x s ) −1 ∈ F Ω⊔X .
Let ν : X → M G be a function. Considering each w ∈ F Ω⊔X as a word in Ω ⊔ X, we denote by w(φν) the element in G where each symbol x ∈ Ω ⊔ X in w has been replaced by (xφ)(xν). Use Solve [18, Section 3.2] to decide if there is a a function ν : X → M , where ∀w ∈ R : w(φν) = 1, and (4.1)
∀s ∈ Ω, ∀x ∈ X : w s,x (φν) = 1. (4.2)
If no such ν exists, then report that M has no Ω-complement; otherwise, return the group
K = (xφ)(xν) : x ∈ X .
We show that this is correct. Let A = Ωθ Aut(G) and let Ω | R ′ be a presentation of A with respect to θ. Lemma 4.2 shows that Ω ⊔ X | R ′ ⋉ R is a presentation for A ⋉ (G/M ) with respect to θ ⊔ φ; note that we need not to compute R ′ .
First suppose that the algorithm returns K = (xφ)(xν) :
x ∈ X . As {xφ : x ∈ X} ⊆ KM we get that G = xφ : x ∈ X KM G. Since w(φν) = 1 for all w ∈ R by (4.1), the group K satisfies the defining relations of G/M ∼ = K/(K ∩ M ), which forces K ∩ M = 1, and so G = K ⋉ M . By (4.1) and (4.2), the generator set Ωθ ⊔ {(xφ)(xν) : x ∈ X} of A, K satisfies the defining relations R ′ ⋉ R of (A ⋉ G)/M , and so A, K is isomorphic to a quotient of (A ⋉ G)/M where K is the image of G/M . is shows that K is normal in A, K , in particular, K Ω K.
is proves that if the algorithm returns a subgroup, then the output is correct.
Conversely, suppose G = K ⋉ M such that K Ω ⊂ K and there is an idempotent endomorphism τ : G → G with kernel M and image K. We must show that in this case equations (4.1) and (4.2) have a solution, so that the algorithm returns a complementary Ω-subgroup to M . Define the map ν :
X → M by xν = (xφ) −1 (xφτ ). Now K = Gτ = (xφ)(xν) : x ∈ X is isomorphic to G/M via (xφ)(xν) → xφM , hence {(xφ)(xν) : x ∈ X} satisfies the relations R. Moreover, we have K Ω ⊆ K, so the isomorphism K ∼ = G/M defined by (xφ)(xν) → xφM extends to A ⋉ K → A ⋉ (G/M );
thus, for all s ∈ Ω and x ∈ X we have w s,x (φν) = 1.
e claim on the complexity follows since we only applied polynomial-time algorithms.
We will also need to find direct complements; we follow the algorithm in [30, eorem 4.8]. e analysis has not appeared in print so we include its proof.
Proposition 4.4. Let G be an Ω-group and let U, V G be normal Ω-subgroups with U V . ere is a polynomial time algorithm which decides whether V /U is a direct Ω-factor of G/U and if so, returns a direct complement.
P
. First compute C/U = C G/U (V /U ) via [17, P6], and test whether G = C, V , for example, by computing group orders. If G = C, V , then report that V /U is not a direct Ω-factor of G/U . Otherwise, compute the center Z(V /U ) via [17, P6] and use Ω-ComplementAbelian to compute a G-complement K/U to Z(V /U ) in C/U , or, if none exists, report that V /U is not a direct Ω-factor of G/U . We prove this this is correct. If G/U = K/U × V /U is a direct product of Ω-subgroups with U K G, then K/U C G/U (V /U ) = C/U and K/U complements V /U ∩ C/U = Z(V /U ); the algorithm constructs such an Ω-complement. Conversely, if we find a Ω-complement K/U to Z(V /U ) in C/U , then we have (K/U ) ∩ (V /U ) = U/U , and K/U and V /U centralize each other; therefore so long as G/U = K/U, V /U , the Ω-subgroup K/U is a direct complement to V /U in G/U . We only applied polynomial-time algorithms.
4.2.
Sylow towers and socles. Following [26, Section 9.1], a set of Sylow subgroups, one for each prime dividing the group order, is a Sylow basis if any two such subgroups U and V are permutable, that is, if U V = V U ; every solvable group admits a Sylow basis. A group L has an abelian Sylow tower if there exists a Sylow basis {Y 1 , . . . , Y ℓ } of abelian groups such that L = Y 1 ⋉ · · · ⋉ Y ℓ . Proposition 4.5. Let L be a solvable group which has an abelian Sylow tower. ere is a polynomialtime algorithm that computes a Sylow tower L = Y 1 ⋉ · · · ⋉ Y ℓ .
P
. Compute and factorize |L| = p e 1 1 · · · p e ℓ ℓ . By assumption, L has a normal Sylow subgroup; we run over the prime factors p i and compute a Sylow p i -subgroup P i until [P i , L] is contained in P i ; if so, set Y ℓ = P i . Since all Sylow subgroups are abelian, we use Ω-ComplementAbelian to compute a complement K L to Y ℓ . By construction, L = K ⋉ Y ℓ , and |K| and |Y ℓ | are coprime. Since K ∼ = L/Y ℓ has an abelian Sylow tower, we can recurse with K and compute a Sylow basis for K. We only apply polynomial-time algorithms at most ℓ i=1 i ∈ O((log |G|) 2 ) times.
We also need the ability to compute the socle of a solvable group. Algorithms for that have been given for permutation groups by Luks [17, P15; 22] and for black-box solvable groups by Höfling [15].
Höfling's algorithm reuses the ingredients given above for computing complements, which we will later use to construct Fra ini subgroups. So we pause to note the complexity of Höfling's algorithm. if no such T exists, then we set S i = 1. As T is normal in L, set S i = T . Once this is done for i = 1, . . . , r, return S 1 × · · · × S r . e correctness of this algorithm follows from [15,Proposition 5] where it is shown that soc(L) = S 1 ×· · ·×S r . We only apply algorithms assumed or shown to be polynomial-time.
4.3.
Computing polycyclic constructive presentations. Constructions of polycyclic presentations from solvable permutation groups are done by various means, sometimes invoking steps (such as collection) whose complexities are difficult to analyze; see for instance [28, p. 166]. In that approach, one first chooses a polycyclic generating sequence x 1 , . . . , x s and then uses the constructive membership testing mechanics of permutation groups to si the relations x p i i and x
x j i into words in the x k . at process leaves the resulting words in arbitrary order, rather than in collected order, that is, we need
x p i i = x e i+1
i+1 · · · x es s , but all we can know is that x p i i is a word in x i+1 , . . . , x s in no particular order. Hence, in that approach, a final step of rewriting must be applied to get the words in normalised (collected) form; this comes at a cost, see the discussion in [21]. We present an alternative.
P
. Let L be a solvable group. Use [17, P11] to construct a chief series L = L 0 > . . . > L s = 1. Since L is solvable, each section L i /L i+1 is isomorphic to C f i p i for some prime p i and f i ≥ 1. In the following, set d(i) = f 0 + . . . + f i−1 for i > 0, and denote by F m with m ∈ N the free group on x 1 , . . . , x m . We work with a double recursion through L/L i and within each factor L i /L i+1 .
For the inner recursion we assume L i /L i+1 ∼ = C f i p i and want to create a constructive presentation for this group. Note that every chief series of L i /L i+1 is a composition series, so we use [17, P11] to find generators g 1 , . . . , g f i of a composition series L i0 > L i1 > · · · > L if i = L i+1 such that each L ij = g j+1 , L j+1 and x j | x p i j is a presentation for L ij /L i(j+1) ∼ = C p i . To make this constructive, use ψ j : L ij → F 1 , defined by sending gL i(j+1) ∈ L ij /L i(j+1) to x e 1 where g −1 g e j+1 ∈ L i(j+1) . Since e p 1 is less than the size of the input, ψ j can be evaluated in polynomial time. is yields a constructive polycyclic presentation of L ij /L i(j+1) . Now suppose by induction we have a constructive polycyclic presentation F j → L i0 /L ij . Since we also have a constructive polycyclic presentation of F 1 → L ij /L i(j+1) , we obtain a constructive presentation F j+1 → L i0 /L i(j+1) by Luks' constructive presentation extension lemma [23,Lemma 4.3]. In that new presentation, every polycyclic relation (for example
x p i k = x * k+1 · · · x * j or x x ℓ k = x * ℓ+1 · · · x * j )
is appended with an element of x j+1 , and so the resulting relations are in collected form. us, at the end of this inner recursion we have a polycyclic constructive presentation for the elementary abelian quotients L i /L i+1 . Now consider the outer recursion. In the base case i = 0 we apply the above method to create a constructive polycyclic presentation of L 0 /L 1 . Now suppose by induction we have a polycyclic constructive presentation of L/L i with maps ϕ : F d(i) → L/L i and ψ : L/L i → F d(i) which can be applied in polynomial time. As in the base case, we construct a polycyclic constructive presentation with maps ϕ ′ : F f i → L i /L i+1 and ψ ′ : L i /L i+1 → F f i . Luks' extension lemma now makes a constructive presentation for L/L i+1 with maps ϕ * : F d(i+1) → L/L i+1 and ψ : L/L i+1 → F d(i+1) . In this process, relations of L/L i of the form x p j = x * 1 · · · x * d(i) and x x k j = x * k+1 · · · x * d(i) are appended with normalised words in L i /L i+1 , so these continue to be in collected form. We also add the polycyclic relations for L i /L i+1 , so the extended constructive presentation is polycyclic.
I : F
We now deal with Step (2) of our algorithm as described in Section 3.2. Using the notation of Section 3, throughout the following L andL are finite solvable groups of cube-free order, and we consider their Fra ini-free quotients
L Φ = L/Φ(L) andL Φ =L/Φ(L). Recall that L Φ = K ⋉ soc(L Φ ) with soc(L Φ ) = B × C where |B| = b
and |C| = c 2 with b and c square-free; analogously forL Φ . In the remainder of this section we describe how to construct an isomorphism L Φ →L Φ ; our construction fails if and only if the two groups are not isomorphic.
Proposition 5.1. ere is a polynomial-time algorithm given a solvable Fra ini-free group L Φ of cubefree order, returns generators for the decomposition into subgroups (K, B, C) described above, along with isomorphisms B → s i=1 Z/p i and C → m j=s+1 (Z/p j ) 2 , and a representation
K → Aut(B × C) → s i=1 GL 1 (p i ) × m j=s+1 GL 2 (p j )
induced by conjugation of K on B × C.
P
. Use the algorithms of Propositions 4.6 & 4.3 to compute generators for soc(L Φ ) and for a complement K to soc(L Φ ) in L Φ . en use the algorithm of Proposition 4.5 to decompose soc(L) as a direct product of its Sylow subgroups. Using the decomposition series of each Sylow subgroup, we obtain the decomposition soc(L Φ ) = B × C along with primary decompositions of B = s i=1 Y i and C = m j=s+1 Y j . We can further produce isomorphisms β i : Y i → Z/p i and κ j : Y j → (Z/p j ) 2 , for example, by using our results from [11,Section 3], based on Karagiorgos & Poulakis [19]. Given standard representations for Aut(Z/p i ) ∼ = (Z/p i ) × and Aut((Z/p j ) 2 ) = GL 2 (p j ), compose with β i and κ j respectively to produce an isomorphism
τ : Aut(B × C) → s i=1 GL 1 (p i ) × m j=s+1 GL 2 (p j ).
Finally, define π : K → Aut(B × C) by (bc)(k)π = b k c k , so πτ is the required map from K. e correctness of this algorithm is apparent. e claim on the timing of the first portion follows since we only invoked O(log |L Φ |) many polynomial-time algorithms. We can apply the algorithms of [11,Section 3] to construct an isomorphism in polynomial time since |Y i | = p i and |Y j | = p 2 j , and both p j and p j are bounded by the size of the permutation domain Ω of L. So the complexity of the results used from [11] is sufficient. Our assumption is that all groups here are permutation groups: in the case of the groups m j=s+1 GL 2 (p j ), we can treat the matrices as permutations of pairs m j=s+1 {(a, b)|a, b ∈ Z/p j }; this domain has size O(p s+1 + · · · + p m ) ⊂ O(|Ω| log |L|), so is polynomial in the input size.
To simplify the exposition, we make the following convention and identify
B =B = s i=1 Z/p i and C =C = m i=s+1 (Z/p i ) 2 .
Recall from Section 3 that the conjugation action of K on B × C is faithful. Hence, we also treat K andK as subgroups of
Aut(B × C) = s i=1 GL 1 (p i ) × m i=s+1 GL 2 (p i ).
For j = 1, . . . , m denote by K i andK i the projections of K andK, respectively, into the j-th factor of Aut(B × C); thus K j andK j describe the conjugation action of K andK, respectively, on the Sylow p j -subgroup Y j B × C.
Gaschütz has shown that L Φ ∼ =L Φ if and only if K andK are conjugate in Aut(B × C), see [9,Lemma 9]; hence, the isomorphism problem reduces to finding an element α ∈ Aut(B × C) with α −1 Kα =K. Once such an α is found, the isomorphism ψ Φ can be defined as follows: writing the elements of as (k, b, c) and (k, b, c), respectively, we set Lemma 5.2. Let p be an odd prime and let K GL 2 (p) be a solvable cube-free p ′ -subgroup. a) If K is reducible, then K is conjugate to a subgroup of diagonal matrices. b) If K is irreducible and abelian, then K is conjugate to s (p 2 −1)/r for some r | p 2 − 1, where s is a generator of a Singer cycle in GL 2 (p), that is, s ∼ = C p 2 −1 . c) If K is irreducible and non-abelian, then there are three possibilities. First, K might be conjugate to G 2 ⋉ G 2 ′ where G 2 ′ is an odd order diagonal (but non-scalar) subgroup and G 2 is one of
L Φ = K ⋉ (B × C) andL Φ =K ⋉ (B × C)ψ Φ : L Φ →L Φ , (k, b, c) → (α −1 kα, b α , c α ).( 0 1 1 0 ) , ( 0 z z 0 ) , 0 −1 1 0 , ( 0 1 1 0 ) , −1 0 0 −1 ,
with z ∈ Z/p of order 4 (if it exists). Second, K might be conjugate to S, t where S is a subgroup of a Singer cycle s and t is an involution such that N GL 2 (p) ( s ) = s, t . ird, K might be conjugate to S, ts 2l where S s has even order and p − 1 = 4l with l odd.
In particular,
N GL 2 (p) (K)/C GL 2 (p) (K) is solvable.
We further need an algorithm of Luks & Miyazaki's [20] that demonstrates how to decide conjugacy of subgroups in solvable permutation groups in time polynomial in the input size. eorem 5.3. Let G, K, andK be groups with
K,K G = S = n i=1 GL 2 (p i ),
where K andK are solvable groups of equal cube-free order coprime to p 1 · · · p n . One can decide in polynomial time whether K is conjugate toK and produce a conjugating element, if it exists.
P
. As above, let K i andK i be the projections of K andK, respectively, to the factor GL 2 (p i ). For each i, based on the classification given in Lemma 5.2, we apply basic linear algebra methods to solve for α i ∈ GL 2 (p i ) such that K α i i =K i ; we also construct N i = N GL 2 (p i ) (K i ) based on Lemma 5.2. If we cannot find a particular α i , then K andK are not conjugate and we return that. Once all the α i have been computed, we replace K by K = K α 1 ···αn , so that we can assume that K i =K i for all i. Note that K andK are conjugate if and only if they are conjugate in N = n i=1 N i , which is solvable by Lemma 5.2. Now we apply the algorithm of [20, eorem 1.3(ii)] to solve for β ∈ N such that K β =K, and return α 1 · · · α n β. If we cannot find such a β, then K andK are not conjugate, and we return false. Lastly, we comment on the timing. Note that we can also locate appropriate α i by a polynomial-time brute-force search in GL 2 (p i ): the la er has order at most p 4 i d 4 , where d is the size of the permutation domain of G. We make a total of n log |G| such searches, followed by the polynomial-time algorithm of [20]. e claim follows.
I :
roughout this section L andL are finite solvable groups of cube-free order, given as permutation groups. To decide isomorphism, we first want to use the algorithm of Section 5 to determine whether the Fra ini quotients L Φ andL Φ are isomorphic. For this we need the Fra ini subgroups.
6.1. Frattini subgroups. Since we assume permutation groups as input, we need a polynomialtime algorithm to compute Fra ini subgroups of solvable permutation groups of cube-free order. A candidate algorithm has been provided by Eick [8, Section 2.4] for groups given by a polycyclic (pc) presentation. To adapt to a permutation se ing we have two choices: replace every step of that algorithm with polynomial-time variants for permutation groups, or apply the algorithm in-situ by appealing to a two-way isomorphism between our original permutation group and a constructive pc-presentation as afforded to us by Lemma 4.7. Note that for the efficiency of the inverse isomorphism, elements in a pc-group are straight-line programs (SLPs) in the generators, so evaluation is determined on the generators and computed in polynomial time. us, whenever we take products in the pc-group, we actually carry out permutation multiplications and si these into the polycyclic generators by applying the isomorphism back to the pc-group. is avoids the potential exponential complexity of collection in pc-groups, see the discussion in [21]. at the algorithm in [8,Section 4.2] uses a polynomial number of pc-group operations follows by considering its major steps. It relies on constructing complements of abelian subgroups (shown in Proposition 4.3 to be in polynomial time), and it applies also module decompositions (which can be done in polynomial time see [20, eorem 3.7 & Section 3.5]), and finally computing cores [17, P5]. erefore Eick's algorithm is in fact a polynomial-time algorithm for groups of permutations, and we cite it as such in what follows.
Once Φ(L) and Φ(L) have been constructed, we can compute the quotients L Φ andL Φ , see [17], and use the algorithms of Section 5 to test isomorphism. If we have determined that L Φ ∼ =L Φ , then we can report that L ∼ =L. us, in the following we assume we found an isomorphism ϕ : L Φ →L Φ , so we also know that L ∼ =L by Section 3. In the next sections we describe how to construct an isomorphismφ : L →L such thatφ factors through ϕ in the sense that Φ(L)(gφ) = (Φ(L)g)ϕ for all g ∈ L. is condition is what allows us to not only solve for some isomorphism between L andL, but to also li generators for the automorphism group of L and thus prescribe (generators for) the entire coset of isomorphisms L →L. Our approach to computingφ is to work with each prime divisor of |Φ(L)|. We begin with a key observation about these primes and recall the Fra ini extension structure of groups of cube-free order.
6.2. Frattini extension structure. As above, write A 1 . . . A s for any subdirect product of groups A 1 , . . . , A s . For a group Y and prime p dividing |Y | let Y p be a Sylow p-subgroup of Y . It follows from [26, 9.2] that every finite solvable group has a Sylow basis, and it follows from [7,25] that every solvable cube-free group Y has one of the following abelian Sylow towers
Y = Y r 1 ⋉ Y r 2 ⋉ . . . ⋉ Y r ℓ if |Y | odd Y 2 ⋉ Y r 1 ⋉ Y r 2 ⋉ . . . ⋉ Y r ℓ if |Y | even, Y 2 ✂Y Y r 1 ⋉ Y r 2 ⋉ . . . ⋉ Y r ℓ ⋉ Y 2 (with Y 2 = C 2 2 ) if |Y | even, Y 2 ✂ Y where r 1 < .
. . < r ℓ are the odd prime divisors of |Y | and {(Y 2 ), Y r 1 , . . . , Y r ℓ } forms a Sylow basis of Y . Proposition 4.5 provides an algorithm to construct such a Sylow tower. Lemma 6.1. Let L be Fra ini-free and solvable, and let Y * be a cube-free Fra ini extension of Y , that is,
Y * /Φ(Y * ) ∼ = Y . If p ∤ |Φ(Y * )|, then Y * p ∼ = Y p ; otherwise Y p ∼ = C p and Y * p ∼ = C p 2 . P
. Recall that every prime dividing |Φ(Y * )| must divide |Y |, thus Φ(Y * ) is square-free and the Sylow tower of Y * looks similar to that of Y , where Y * p ∼ = Y p if p ∤ |Φ(Y * )|, and Y p ∼ = C p and Y * p abelian of order p 2 otherwise. We prove that Y * p ∼ = C p 2 . We use the previous notation and consider
M = Φ(Y * ) = C p 1 × . . . × C pm as a Y -module.
It is shown in [9, eorem 12] that Y * is a subdirect product of Fra ini extensions of Y by C p i . us, to prove the lemma, it suffices to consider M = C p for some prime p. First, suppose that p = r i is odd. In this case, Y p ∼ = C p and Y * p is abelian of order p 2 . Suppose, for a contradiction, that Y * p ∼ = C 2 p . It follows from [9,Lemma 5 & eorem 14] that Y * is a non-split extension of Y by M such that N Y (Y p ) acts on M as on Y p . is implies the following:
considering Y * p = Y * r i = (Z/p) 2 as an Z/p-space, there is a basis {m, y} such that M = m and every g ∈ (Y * 2 ⋉)Y * r 1 ⋉ . . . ⋉ Y * r i−1 N Y * (Y * p )
acts on that space as a matrixg = α β 0 α for some α ∈ (Z/p) × and β ∈ Z/p. Since |Y * | is cube-free, g has order coprime to p, and hence β = 0, that is, g acts diagonally on Y * p . Moreover, [16,Proposition 2.44]. is contradiction proves Y * ∼ = C p 2 . Lastly, suppose M = C 2 ; in this case
W = Y * r i+1 ⋉ . . . ⋉ Y * r ℓ (⋉Y * 2 ) centralizes Y * p modulo W . In conclusion, no nontrivial element in Y * p is a non-generator of Y * , contradicting Φ(Y * ) Y * p , seeY 2 ∼ = C 2 and Y * = Y * 2 ⋉ Y * r 1 ⋉ . . . ⋉ Y * r ℓ . If Y * 2 ∼ = C 2 2
, then the same argument shows that no nontrivial element in Y * 2 is a non-generator of Y * , contradicting Φ(Y * ) Y * 2 . us, Y * 2 ∼ = C 4 .
6.3. Constructing the isomorphism. Recall that L ∼ =L if and only if the isomorphism ψ Φ in
Step (2) exists. Suppose ψ Φ has been constructed as described in Section 5, that is, we know that L ∼ =L. As explained in the proof of Lemma 6.1, the groups L andL are iterated Fra ini extensions of L Φ andL Φ , respectively, by cyclic groups of prime order; cf. [9,Definition 4]. Starting with ψ Φ , we iteratively construct isomorphisms of these Fra ini extensions until eventually we obtain an isomorphism L →L. us, we consider the following situation: let Y andỸ be two solvable cube-free groups and let Y * andỸ * be cube-free Fra ini extensions of Y andỸ , respectively, by M = C p . We assume that we have an isomorphism ϕ : Y →Ỹ ; we know that Y * ∼ =Ỹ * , and we aim to construct an isomorphism Y * →Ỹ * . e following preliminary lemma will be handy. Lemma 6.2. Let G be a group and P, Q G such that P is a cube-free p-group and Q = w is cyclic of order q 2 , for distinct primes p and q. Suppose P Q = QP and A = w q is normal in P Q. a) We have P Q = P ⋉ Q or P Q = Q ⋉ P . b) If P Q = Q ⋉ P , then A acts trivially on P . c) If P Q = P ⋉ Q, then the action of P on Q is uniquely determined by its action on Q/A.
P
. Since P Q is cube-free, part a) follows from the structure results mentioned in Section 6.2. For part b), note that Q and Q/A both act on P ; this forces that A acts trivially on P . Now consider part c). Recall that Aut(Q) is cyclic of order q(q − 1), generated by β : Q → Q, w → w k , where k is some primitive root modulo q 2 . Since P Q is cube-free, the element g ∈ P acts on Q via an automorphism α ∈ Aut(Q) of order coprime q. us, α lies in the subgroup T Aut(Q) of order q − 1, and there is a unique e ∈ {1, . . . , q − 1} such that α = (β q ) e . Now (wA)α = (wA) i with i ∈ {0, . . . , q − 1} yields i = k eq mod q. Since k q is a primitive root modulo q, it follows that for any given i ∈ {1, . . . , q − 1} there is a unique e ∈ {1, . . . , q − 1} such that i ≡ k eq mod p, hence for a given i ∈ {0, . . . , q − 1} there is a unique automorphism α ∈ Aut(Q) with (wA)α = (wA) i . Proposition 6.3. Let Y andỸ be two solvable cube-free groups and let Y * andỸ * be cube-free Fra ini extensions of Y andỸ , respectively, by a group isomorphic to C p . Algorithm 1 is a polynomial-time algorithm which, given an isomorphism ϕ : Y →Ỹ , returns an isomorphismφ : Y * →Ỹ * .
P
. We compute the Fra ini subgroups of Y * andỸ * , and the Sylow p-subgroups A Φ(Y * ) andà Φ(Ỹ * ), respectively, see Section 6.1. By assumption, A ∼ =à ∼ = C p , and we can assume that Y = Y * /A andỸ =Ỹ * /Ã. As explained above, the existence of ϕ : Y →Ỹ implies that Y * andỸ * are isomorphic. Use the algorithm of Proposition 4.5 to construct a Sylow tower Y * = Y * 1 ⋉ . . .⋉ Y * n ; for each j let p j be a prime such that Y * j is a Sylow p j -subgroup. Let p = p i , and recall from Lemma 6.1 that Y * i is cyclic; find a generator Y * i = a and note that A = a p Y * i . For every j define Q j = k =j Y * k ; this is a Hall p ′ j -subgroup of Y * . Such a set of Hall r ′ -subgroups (one for each prime divisor r of the group order) is a called a Sylow system in [26, Section 9.2]); in particular, we can recover each Y * j as Y * j = k =j Q k . Since Y * 1 , . . . , Y * n form a Sylow tower of Y * , every x ∈ Y * has a unique factorization x = ha e where h ∈ H = Q i and a e ∈ Y * i with 0 e p 2 − 1; we will use this decomposition later when we define an isomorphismφ : Y * →Ỹ * . We will constructφ via a Sylow basis ofỸ * which is compatible with the above Sylow basis of Y * ; we explain below what this means.
Let Γ : Y * → Y * /A = Y be the natural projection, so that {Q 1 Γϕ, . . . , Q n Γϕ} forms a Sylow system ofỸ * /Ã. For each j we defineQ j Ỹ * to be the full preimage of Q j Γϕ under the natural projectionΓ :Ỹ * →Ỹ * /à =Ỹ . Clearly, if j = i, thenQ j is a Hall p ′ j -subgroup ofỸ * . Moreover, Q i =H ⋉à whereH is some Hall p ′ -subgroup ofQ i and ofỸ * ; we computeH inQ i by first computingà Q i as a Sylow p-subgroup and thenH as a complement toà inQ i . We definẽ
Y * i = k =iQ k andỸ * j =H ∩ k =j,iQ k for each j = i.
It follows from [26, 9.2.1] that {Ỹ * 1 , . . . ,Ỹ * n } is a set of pairwise permutable Sylow subgroups with Y * jΓ = Y * j Γϕ for all j. In particular, we can apply Lemma 6.2 and it follows from our construction that for all u = v we haveỸ *
uỸ * v =Ỹ * u ⋉Ỹ * v if and only if Y * u Y * v = Y * u ⋉ Y * v , andỸ * uỸ * v =Ỹ * v ⋉Ỹ * u if and only if Y * u Y * v = Y * v ⋉ Y * u .
We say that these two Sylow bases are compatible. Let π andπ be the restriction of Γ andΓ to H andH, respectively; note that π : H → HA/A and π :H →HÃ/Ã are isomorphisms, and we define an isomorphism H →H via
H = H/(H ∩ A) π −→ HA/A ϕ −→HÃ/Ãπ −1 −→H/H ∩à =H.
Note that in definingπ : h →Ãh, we identify generators ofH with generators ofHÃ/Ã; as elements ofHÃ/Ã are presumed throughout to be words (or SLPs) in the generators, we can compute preimages ofπ. is affords us an implementation ofπ −1 .
Recall that Y * i = a , and choose a generatorã ∈Ỹ * i such that aΓϕ =ãΓ.
We can now construct an isomorphismφ : Y * →Ỹ * . As mentioned above, every x ∈ Y * has a unique factorization x = ha e where h ∈ H and 0 e p 2 − 1. is shows that ϕ : Y * →Ỹ * , ha e → hπϕπ −1 ·ã e ,
P T 1.1 (I )
We now prove our main result, eorem 1.1, by describing Algorithm 3. Recall from Section 3 that every cube-free group has the form G = A × L, with L solvable and A = 1 or A = PSL 2 (p). If A = 1, then A = G (3) , the third term of the derived series of G, see Remark 3.1. We compute G (3) using the normal closure of commutators [28, p. 23]; since membership testing in permutation groups is in deterministic polynomial time, this can be done efficiently. Furthermore, as G (3) is normal, the algorithm of [17, P6] applies to compute L = C G (A) in polynomial time. us, we may decompose G = A×L, and likewiseG, in polynomial time. For
Step (1) of the general algorithm, the construction of an isomorphism ψ A : A →Ã, we use the next proposition. e correctness of Algorithm 3 now follows from eorem 6.4; together with Proposition 7.1, the runtime is polynomial in the input size.
Proposition 7.1. Let A be isomorphic to a non-abelian simple group of cube-free order. ere is a polynomial-time algorithm that returns an isomorphism A → PSL 2 (p).
P
. By assumption, A ∼ = PSL 2 (p). We can determine p by computing |A|, and then find x, y ∈ A of order p and (p + 1)/2, respectively; note that x, y ∼ = PSL 2 (p) since x generates a Sylow psubgroup, and y generates the image in PSL 2 (p) of the (p − 1)-th power of a Singer cycle in GL 2 (p). Now construct a presentation x, y | R for A from these elements. In PSL 2 (p), list all element pairs (x ′ , y ′ ) of order p and (p + 1)/2, respectively, and search for an identification x → x ′ and y → y ′ that satisfies the relations R. Once found, return the result as the isomorphism. If PSL 2 (p) is represented on n points, then p n and hence |PSL 2 (p)| n 3 . e algorithm searches |PSL 2 (p)| 2 n 6 pairs, so this brute-force test ends in time polynomial in the input. Proposition 7.1 is a shortcut, available because of our focus on a polynomial-time algorithm for permutation groups. Recognizing A ∼ = PSL 2 (p) and constructing an isomorphism has been a subject of intense research; a polynomial time solution for groups of black-box type is discussed in [5].
E
We have implemented the critical features of our algorithm in [10], and we give a few demonstrations of its efficiency in Table 1. For each test, we constructed two (non-)isomorphic groups: we usually started with direct products of groups provided by GAP's SmallGroup Library, and then created isomorphic random copies G and H of these groups (by using random polycyclic generating set). For some of the groups we have used, Table 1 gives their size and code; this data can be used to reconstruct the groups via the GAP function PcGroupCode. We applied our function Isomorphism-CubefreeGroups to find an isomorphism G → H. When comparing the efficiency of our implementation with the GAP function IsomorphismGroups, we have started both calculations with freshly constructed groups G and H, to make sure that previously computed data is not stored. We note that GAP also provides a randomized function (RandomIsomorphismTest) that a empts to decide isomorphism between finite solvable groups (given via their size and code); the current implementation does not return isomorphisms. at algorithm runs exceedingly fast on many examples, see Table 1, but its randomized approach means it cannot be guaranteed to detect all isomorphisms. ere are some practical bo lenecks in our implementation which currently applies available libraries for pcgroups (cf. Section 6.1) and matrix groups (cf. Section 5). e efficiency problems for collection (cf. [21]) become visible when larger primes are involved. ( is is one reason why it takes several minutes to reconstruct some of the groups in Table 1 via PcGroupCode.) Moreover, GAP's functionality for matrix groups is not yet making full use of the promising advances of the matrix group recognition project. ese bo lenecks are responsible for the long runtime of the examples involving the prime 12198421, which is large from the perspective of GAP. Nevertheless, as a proof of concept, these examples demonstrate well the efficiency of our algorithm compared to existing methods.
Proposition 4. 6 .
6Generators for the socle of a solvable group can be computed in polynomial-time.P. Let L be a solvable group, treated as an L-group under conjugation action. Use [17, P11] to compute a chief series 1 = N 0 ✁ N 1 ✁ . . . ✁ N r = L; in particular, N 1 is a minimal normal subgroup of L. We set S 1 = N 1 , and for each i > 1 compute a direct L-complement S i to N i−1 in N i (so S i ✂ L); set S i = 1 if this does not exist. To this end, we proceed as follows: we use the algorithm of Proposition 4.4 to find an L-subgroup T N i such that N i = T × N i−1 ;
Lemma 4. 7 .
7A polycyclic constructive presentation for a solvable group can be computed in polynomialtime.
of α depends very much on the dimension 2 case; in particular, we use a classification of J. Gierster (1881) of the subgroups of GL 2 (p), extracted from [13, eorems 5.1-5.3].
is well-defined; clearly,φ is a bijection, so it remains to show that it is a homomorphism. We use below the important property ofφ that it maps Y * j toỸ * j for each j: this follows from the fact that the Hall subgroups Q 1 , . . . , Q n defining the Sylow basis Y * 1 , . . . , Y * n are mapped underφ to the Hall subgroupsQ 1 , . . . ,Q i−1 ,H,Q i+1 , . . . ,Q n defining the Sylow basisỸ * 1 , . . . ,Ỹ * n . Let x, y ∈ Y * and write x = ha e and y = ka f with h, k ∈ H and e, f ∈ {0, . . . , p 2 − 1}. Write (a e ) k = ma u with m ∈ H and u ∈ {0, . . . , p 2 − 1}, so that xy = hk(a e ) k a f = (hkm)a u+f . is shows that (xy)φ = xφ · yφ ⇐⇒ (ã e ) kπϕπ −1 = mπϕπ −1 ·ã u , and it remains to prove the following: for all k ∈ H and e ∈ {0, . . . ,Recall that every k ∈ H can be wri en as a product of elements in the chosen Sylow tower of Y * , say k = h 1 . . . h l where h u and h v lie in different Sylow subgroups for u = v. We prove the claim by induction on l.and there are two cases to consider.We can write (a e ) k = a i for a uniquely determined i ∈ {0, . . . , p 2 − 1}, which yields (Aa) kπ = (Aa i mod p ) and (Ãã) kπϕ = (Ãã i mod p ).Since k acts on Aa the same way as kπϕ acts on Ãã , it follows from Lemma 6.2 that k acts on A the same way as kπϕπ −1 acts onÃ. us, if (a e ) k = a i , then (ã e ) kπϕπ −1 =ã i , as claimed.∈H is a preimage of [kπϕ,Ãã −e mod p ] ∈HÃ/à under the isomorphismπ :H →HÃ/Ã; recall thatπ is the restriction ofΓ :Ỹ * →Ỹ , andΓ maps kπϕπ −1 and a to kπϕ andÃã, respectively. us, (ã e ) kπϕπ −1 = [k, a −e mod p ]πϕπ −1 ·ã e , as claimed.Second, consider the induction step l ≥ 2 and write k = st such that the induction hypothesis holds for s and t, thatis completes the proof thatφ is an isomorphism between Y * andỸ * . e construction ofφ only employs a finite list of polynomial-time algorithms.As explained in the beginning of this section, if the order of the cube-free group L has k distinct prime divisors, then the algorithm in Proposition 6.3 has to be iterated at most k times to establish an isomorphism from L; note that k log |L|. is proves the following theorem. eorem 6.4. Let L andL be two solvable cube-free groups. Algorithm 2 is a polynomial-time algorithm that constructs an isomorphism L →L, and reports false if and only if L ∼ =L.Algorithm 1 CyclicLiInput: cube-free solvable groups Y * ,Ỹ * with |Y * | = |Ỹ * |, subgroups A Φ(Y * ) andà Φ(Ỹ * ) isomorphic to C p , natural projections Γ : Y * → Y * /A andΓ :Ỹ * →Ỹ * /à with images Y = Y * Γ and Y =Ỹ * Γ , and an isomorphism ϕ : Y →Ỹ Output: an isomorphismφ :
Code equivalence and group isomorphism. L Babai, P Codeno I, J A Grochow, Y Qiao, Proc. of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms. of the 22nd Annual ACM-SIAM Symposium on Discrete AlgorithmsSIAM, PhiladelphiaL. Babai, P. Codeno i, J. A. Grochow, Y. Qiao. Code equivalence and group isomorphism. Proc. of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, 1395-1408, SIAM, Philadelphia, 2011.
On the complexity of matrix group problems I. L Babai, E Szemerédi, Proc. 25th IEEE Sympos. Foundations Comp. Sci. 25th IEEE Sympos. Foundations Comp. SciL. Babai, E. Szemerédi. On the complexity of matrix group problems I. In Proc. 25th IEEE Sympos. Foundations Comp. Sci., 229-240, 1984.
Polynomial-time isomorphism test for groups with abelian Sylow towers. L Babai, Y Qiao, 29th International Symposium on eoretical Aspects of Computer Science. Wadern14LIPIcs. Leibniz Int. Proc. Inform.L. Babai, Y. Qiao. Polynomial-time isomorphism test for groups with abelian Sylow towers. 29th International Sym- posium on eoretical Aspects of Computer Science, 453-464, LIPIcs. Leibniz Int. Proc. Inform., 14, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2012.
Enumeration of finite groups. S R Blackburn, P M Neumann, G Venkataraman, Cambridge PressS. R. Blackburn, P. M. Neumann, G. Venkataraman. Enumeration of finite groups. Cambridge Press 2007.
Adjoint representations of black box groups PSL2(Fq). A Borovik, S Yalcinkaya, doi.org/10.1016/j.jalgebra.2018.02.022J. Algebra. A. Borovik, S. Yalcinkaya. Adjoint representations of black box groups PSL2(Fq) J. Algebra (2018), doi.org/10.1016/j.jalgebra.2018.02.022
A fast isomorphism test for groups whose Lie algebra has genus 2. P A Brooksbank, J Maglione, J B Wilson, J. Algebra. 473P. A. Brooksbank, J. Maglione, J. B. Wilson. A fast isomorphism test for groups whose Lie algebra has genus 2. J. Algebra 473 (2017), 545-590.
Groups of cube-free odd order. M J Curran, Amer. Math. Monthly. 117M. J. Curran. Groups of cube-free odd order. Amer. Math. Monthly 117 (2010) 363-365.
Special presentations for finite soluble groups and computing (pre-)Fra ini subgroups. B Eick, Groups and computation. II (New Brunswick, NJ; Providence, RIAmer. Math. Soc28B. Eick. Special presentations for finite soluble groups and computing (pre-)Fra ini subgroups. In Groups and com- putation, II (New Brunswick, NJ, 1995), 101-112, DIMACS Ser. Discrete Math. eoret. Comput. Sci., 28 , Amer. Math. Soc., Providence, RI, 1997.
On the Groups of Cube-Free Order. H Dietrich, B Eick, with addendum in J. Algebra. 292J. AlgebraH. Dietrich, B. Eick. On the Groups of Cube-Free Order. J. Algebra 292 (2005) 122-137, with addendum in J. Algebra 367 (2012) 247-248.
. H Dietrich, e GAP4 package Cubefree. Available at users.monash.edu/∼heikod/cubefree.htmlH. Dietrich. e GAP4 package Cubefree. Available at users.monash.edu/∼heikod/cubefree.html.
Polynomial time isomorphism tests of black-box type groups of most orders. H Dietrich, J B Wilson, arXiv:1806.08872submi edH. Dietrich, J. B. Wilson. Polynomial time isomorphism tests of black-box type groups of most orders. (submi ed). arXiv:1806.08872.
On the density of the k-free integers. Fibonacci art. R L Duncan, 7R. L. Duncan. On the density of the k-free integers. Fibonacci art. 7 (1969) 140-142.
e linear groups of small degree over finite fields. D L Flannery, E A O'brien, Intern. J. Alg. and Comput. 15D. L. Flannery, E. A. O'Brien. e linear groups of small degree over finite fields. Intern. J. Alg. and Comput. 15 (2005) 467-502.
GAP -Groups, Algorithms and Programming. Available at gap-system.org. GAP -Groups, Algorithms and Programming. Available at gap-system.org.
Computing projectors, injectors, residuals and radicals of finite soluble groups. B Höfling, J. Symb. Comp. 32B. Höfling. Computing projectors, injectors, residuals and radicals of finite soluble groups. J. Symb. Comp. 32 (2001) 499-511.
D F Holt, B Eick, E A O'brien, Handbook of computational group theory. Discrete Mathematics and its Applications. Boca Raton; Boca Raton, FLChapman & Hall/CRCD. F. Holt, B. Eick, E. A. O'Brien. Handbook of computational group theory. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005.
Computing in quotient groups. W M Kantor, E M Luks, Proceedings 22nd ACM Symposium on eory of Computing. 22nd ACM Symposium on eory of ComputingW. M. Kantor, E. M. Luks. Computing in quotient groups. Proceedings 22nd ACM Symposium on eory of Computing (1990) 524-534.
Sylow subgroups in parallel. W M Kantor, E M Luks, P D Mark, J. Algorithms. 31W. M. Kantor, E. M. Luks, P. D. Mark. Sylow subgroups in parallel. J. Algorithms 31 (1999) 132-195.
An algorithm for computing a basis of a finite abelian group. Algebraic informatics. G Karagiorgos, D Poulakis, Lecture Notes in Comput. Sci. 6742SpringerG. Karagiorgos, D. Poulakis. An algorithm for computing a basis of a finite abelian group. Algebraic informatics, 174-184, Lecture Notes in Comput. Sci., 6742, Springer, Heidelberg, 2011.
Polynomial-time normalizers. E M Luks, T Miyazaki, Discrete Math. eor. Comput. Sci. 13E. M. Luks, T. Miyazaki. Polynomial-time normalizers. Discrete Math. eor. Comput. Sci. 13 (2011) 61-96.
On complexity of multiplication in finite soluble groups. M F Newman, A Niemeyer, J. Algebra. 421M. F. Newman, A. Niemeyer. On complexity of multiplication in finite soluble groups. J. Algebra 421 (2015) 425-430.
Computing the composition factors of a permutation group in polynomial time. E M Luks, Combinatorica. 7E. M. Luks. Computing the composition factors of a permutation group in polynomial time. Combinatorica 7 (1987) 87-99.
Computing in Solvable Matrix Groups. E M Luks, Proceedings 33rd Annual Symposium on Foundations of Computer Science. 33rd Annual Symposium on Foundations of Computer ScienceE. M. Luks. Computing in Solvable Matrix Groups. In Proceedings 33rd Annual Symposium on Foundations of Com- puter Science (1992) 111-120.
Bornes effectives pour certaines fonctions concernant les nombres premiers. J-P Massias, G Robin, J. éor. Nombres Bordeaux. 8J-P. Massias, G. Robin, Bornes effectives pour certaines fonctions concernant les nombres premiers. J. éor. Nombres Bordeaux 8 (1996) 215-242.
e finite groups of cube-free order. S Qiao, C H Li, J. Algebra. 334S. Qiao, C. H. Li. e finite groups of cube-free order. J. Algebra 334 (2011) 101-108.
A Course in the eory of Groups. D J S Robinson, Springer-VerlagD. J. S. Robinson, A Course in the eory of Groups. Springer-Verlag, 1982.
Approximate formulas for some functions of prime numbers. J B Rosser, L Schoenfeld, Illinois J. Math. 6J. B. Rosser, L. Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962) 64-94.
Permutation group algorithms. Á Seress, Cambridge University Press152CambridgeÁ. Seress. Permutation group algorithms. Cambridge University Press 152, Cambridge, 2003.
Remarks on the isomorphism problem in theories of construction of finite groups. D Taunt, Proc. Cambridge Philos. Soc. 51D. Taunt. Remarks on the isomorphism problem in theories of construction of finite groups. Proc. Cambridge Philos. Soc. 51 (1955) 16-24.
J B Wilson, arXiv:1005.0548Finding direct product decompositions in polynomial time. J. B. Wilson. Finding direct product decompositions in polynomial time. arXiv:1005.0548.
. J B Wilson, arXiv:1612.01444J. B. Wilson. e threshold for subgroup profiles to agree is log n − 2. arXiv:1612.01444.
| [] |
[
"Visualization of carrier transport in lateral metal-perovskite- metal structures and its influence on device operation",
"Visualization of carrier transport in lateral metal-perovskite- metal structures and its influence on device operation"
] | [
"N Ganesh \nChemistry and Physics of Materials Unit (CPMU)\nJawaharlal Nehru Center for Advanced Scientific Research\nJakkur560064BengaluruIndia\n",
"A Z Ashar \nChemistry and Physics of Materials Unit (CPMU)\nJawaharlal Nehru Center for Advanced Scientific Research\nJakkur560064BengaluruIndia\n",
"Sumukh Purohit \nChemistry and Physics of Materials Unit (CPMU)\nJawaharlal Nehru Center for Advanced Scientific Research\nJakkur560064BengaluruIndia\n",
"K L Narasimhan \nCenter for Nano Science and Engineering (CENSE)\nIndian Institute of Science\n560012BengaluruIndia\n",
"K S Narayan \nChemistry and Physics of Materials Unit (CPMU)\nJawaharlal Nehru Center for Advanced Scientific Research\nJakkur560064BengaluruIndia\n"
] | [
"Chemistry and Physics of Materials Unit (CPMU)\nJawaharlal Nehru Center for Advanced Scientific Research\nJakkur560064BengaluruIndia",
"Chemistry and Physics of Materials Unit (CPMU)\nJawaharlal Nehru Center for Advanced Scientific Research\nJakkur560064BengaluruIndia",
"Chemistry and Physics of Materials Unit (CPMU)\nJawaharlal Nehru Center for Advanced Scientific Research\nJakkur560064BengaluruIndia",
"Center for Nano Science and Engineering (CENSE)\nIndian Institute of Science\n560012BengaluruIndia",
"Chemistry and Physics of Materials Unit (CPMU)\nJawaharlal Nehru Center for Advanced Scientific Research\nJakkur560064BengaluruIndia"
] | [] | The high performance of hybrid perovskite-based devices is attributed to its excellent bulktransport properties. However, carrier dynamics, especially at the metal-perovskite interface, and its influence on device operation are not widely understood. This work presents the dominant transport mechanisms in methylammonium lead iodide (MAPbI3) perovskite-based asymmetric metal-electrode lateral devices. The device operation is studied with inter-electrode lengths varying from 4 μm to 120 μm. Device characteristics indicate distinct ohmic and spacecharge limited current (SCLC) regimes that are controlled by the inter-electrode length and applied bias. The electric-potential mapping using Kelvin-Probe microscopy across the device indicates minimal ion-screening effects and the presence of a transport barrier at the metal-MAPbI3 junction. Further, photocurrent imaging of the channel using near-field excitationscanning microscopy reveals dominant recombination and charge-separation zones. These lateral devices exhibit photodetector characteristics with a responsivity of about 51 mA/W in self-powered mode and 5.2 A/W at 5 V bias, in short-channel devices (4 μm). The low device capacitance enables a fast light-switching response of ~12 ns.2A. IntroductionHybrid organic-inorganic perovskites (HOIP) have evolved as attractive materials for optoelectronic applications. These materials have exhibited high performance in solar cells, photodetectors, LEDs, neuromorphic devices, and lasing applications [1][2][3][4][5][6][7]. The excellent performance of these devices is attributed to their outstanding bulk transport properties such as long diffusion length, extended carrier lifetimes, high absorption coefficient, and mobility [8][9][10][11]. However, interfacial charge transport dynamics, especially across the metal-perovskite interface are not widely understood. Despite numerous reports related to devices based on metal-perovskite junctions, the transport dynamics have not been explored in detail[12][13][14].Moreover, the study of carrier transport in the space charge regime is not conclusive. One of the reasons for this discrepancy is that the experimental studies are carried out on sandwich devices, where ionic motion affects carrier transport, even under equilibrium conditions. The present study demonstrates that lateral structure is a suitable framework to separate the effects of carrier transport and ionic motion. In this work, we study the carrier transport regimes in a HOIP based lateral metal-semiconductor-metal (MSM) device as a function of inter-electrode distance and applied bias. The lateral geometry provides access to spatially probe the transport parameters such as the local potential and photocurrent across the device. These spatially resolved measurements carried out in tandem with the bulk device characteristics provide a more comprehensive picture of the microscopic origins of carrier transport that affect the device response.The transport studies are carried out on lateral back-contact MSM devices, with methylammonium lead iodide, MAPbI3 (MAPI) as the hybrid-perovskite semiconductor.Unlike the widely used sandwich device architecture, lateral devices offer unique advantages such as: (i) Reduced light-absorption losses due to the absence of stack layers, (ii) low dark currents, which allow for small-signal detection, (iii) low device capacitance which reduces the RC lifetime, thereby enabling device operation for high-speed applications[15]. This work 3 primarily investigates the transport characteristics of lateral asymmetric electrode devices with dissimilar work-function metals, i.e., Aluminum (Al) and Gold (Au). The choice of metal electrode work-functions, namely Al (φm = 4.1 eV) and Au (φm = 5.2 eV) in conjunction with MAPI allows for selective extraction of the photogenerated electron and hole, respectively. In the latter part of the work, we demonstrate the utility of lateral devices for photodetector application. Transient measurements on these devices show a high-speed response component of ~ 12 ns. These devices also exhibit a high linear dynamic range of ~ 118 dB, spanning close to six orders of light intensity variation.B. Results and DiscussionDevice OperationThe lateral MSM (metal-semiconductor-metal) devices were fabricated using shadow masks, printed using a 2-photon-polymerization based 3D printer (Nanoscribe GmbH), and maskliftoff techniques developed in the laboratory. After sequential deposition of Au and Al electrodes on the fused silica substrate, MAPI perovskite was deposited (~ 200 nm) to form the back contact lateral device, as shown inFig. 1(a). The absorption and PL spectra of MAPI films shown inFig. S1(a) indicate a bandgap of ~ 1.6 eV. Additionally, the sharp XRD peaks (Fig. S1(b)) and the morphology(Fig. S1(c)) suggest high crystalline order in these films. A detailed explanation of device fabrication is provided in the experimental section of the Supplemental Material. As shown in the device schematic (Fig. 1(a)), l is the inter-electrode channel length and d is the thickness of the metal electrode (~ 80 -100 nm), and w is the width of the electrode.In this work, we study device characteristics as a function of inter-electrode channel lengths (l) in the range of 4 μm ≤ l ≤ 120 μm. In these devices, the electrode width, w >> l, is in the range of ~ 2 -4 mm. The cross-sectional area of these devices is w × d.Fig. S2(a)shows the reflection-microscopy image of the asymmetric electrode device, with l ~ 12 μm. These devices can be implemented to a larger area using inter-digitated electrode patterns, as shown inFig. S2(b). FIG. 1: (a) Schematic of the lateral asymmetric electrode device, where l is the inter-electrode channel length and d is the thickness of the metal electrode. (b) Semi-log plot of J-V characteristics of the asymmetric electrode device for different values of l. (c) The plot of the J-V characteristics, normalized to J at 5 V bias. (d) The plot of the rectification ratio as a function of channel length shows that the rectification ratio increases for short channel devices. | 10.1103/physrevapplied.17.024060 | [
"https://arxiv.org/pdf/2202.12043v1.pdf"
] | 247,084,189 | 2202.12043 | b7d02d3962e43e4987e9e144c8200b32c7f2a648 |
Visualization of carrier transport in lateral metal-perovskite- metal structures and its influence on device operation
N Ganesh
Chemistry and Physics of Materials Unit (CPMU)
Jawaharlal Nehru Center for Advanced Scientific Research
Jakkur560064BengaluruIndia
A Z Ashar
Chemistry and Physics of Materials Unit (CPMU)
Jawaharlal Nehru Center for Advanced Scientific Research
Jakkur560064BengaluruIndia
Sumukh Purohit
Chemistry and Physics of Materials Unit (CPMU)
Jawaharlal Nehru Center for Advanced Scientific Research
Jakkur560064BengaluruIndia
K L Narasimhan
Center for Nano Science and Engineering (CENSE)
Indian Institute of Science
560012BengaluruIndia
K S Narayan
Chemistry and Physics of Materials Unit (CPMU)
Jawaharlal Nehru Center for Advanced Scientific Research
Jakkur560064BengaluruIndia
Visualization of carrier transport in lateral metal-perovskite- metal structures and its influence on device operation
1
The high performance of hybrid perovskite-based devices is attributed to its excellent bulktransport properties. However, carrier dynamics, especially at the metal-perovskite interface, and its influence on device operation are not widely understood. This work presents the dominant transport mechanisms in methylammonium lead iodide (MAPbI3) perovskite-based asymmetric metal-electrode lateral devices. The device operation is studied with inter-electrode lengths varying from 4 μm to 120 μm. Device characteristics indicate distinct ohmic and spacecharge limited current (SCLC) regimes that are controlled by the inter-electrode length and applied bias. The electric-potential mapping using Kelvin-Probe microscopy across the device indicates minimal ion-screening effects and the presence of a transport barrier at the metal-MAPbI3 junction. Further, photocurrent imaging of the channel using near-field excitationscanning microscopy reveals dominant recombination and charge-separation zones. These lateral devices exhibit photodetector characteristics with a responsivity of about 51 mA/W in self-powered mode and 5.2 A/W at 5 V bias, in short-channel devices (4 μm). The low device capacitance enables a fast light-switching response of ~12 ns.2A. IntroductionHybrid organic-inorganic perovskites (HOIP) have evolved as attractive materials for optoelectronic applications. These materials have exhibited high performance in solar cells, photodetectors, LEDs, neuromorphic devices, and lasing applications [1][2][3][4][5][6][7]. The excellent performance of these devices is attributed to their outstanding bulk transport properties such as long diffusion length, extended carrier lifetimes, high absorption coefficient, and mobility [8][9][10][11]. However, interfacial charge transport dynamics, especially across the metal-perovskite interface are not widely understood. Despite numerous reports related to devices based on metal-perovskite junctions, the transport dynamics have not been explored in detail[12][13][14].Moreover, the study of carrier transport in the space charge regime is not conclusive. One of the reasons for this discrepancy is that the experimental studies are carried out on sandwich devices, where ionic motion affects carrier transport, even under equilibrium conditions. The present study demonstrates that lateral structure is a suitable framework to separate the effects of carrier transport and ionic motion. In this work, we study the carrier transport regimes in a HOIP based lateral metal-semiconductor-metal (MSM) device as a function of inter-electrode distance and applied bias. The lateral geometry provides access to spatially probe the transport parameters such as the local potential and photocurrent across the device. These spatially resolved measurements carried out in tandem with the bulk device characteristics provide a more comprehensive picture of the microscopic origins of carrier transport that affect the device response.The transport studies are carried out on lateral back-contact MSM devices, with methylammonium lead iodide, MAPbI3 (MAPI) as the hybrid-perovskite semiconductor.Unlike the widely used sandwich device architecture, lateral devices offer unique advantages such as: (i) Reduced light-absorption losses due to the absence of stack layers, (ii) low dark currents, which allow for small-signal detection, (iii) low device capacitance which reduces the RC lifetime, thereby enabling device operation for high-speed applications[15]. This work 3 primarily investigates the transport characteristics of lateral asymmetric electrode devices with dissimilar work-function metals, i.e., Aluminum (Al) and Gold (Au). The choice of metal electrode work-functions, namely Al (φm = 4.1 eV) and Au (φm = 5.2 eV) in conjunction with MAPI allows for selective extraction of the photogenerated electron and hole, respectively. In the latter part of the work, we demonstrate the utility of lateral devices for photodetector application. Transient measurements on these devices show a high-speed response component of ~ 12 ns. These devices also exhibit a high linear dynamic range of ~ 118 dB, spanning close to six orders of light intensity variation.B. Results and DiscussionDevice OperationThe lateral MSM (metal-semiconductor-metal) devices were fabricated using shadow masks, printed using a 2-photon-polymerization based 3D printer (Nanoscribe GmbH), and maskliftoff techniques developed in the laboratory. After sequential deposition of Au and Al electrodes on the fused silica substrate, MAPI perovskite was deposited (~ 200 nm) to form the back contact lateral device, as shown inFig. 1(a). The absorption and PL spectra of MAPI films shown inFig. S1(a) indicate a bandgap of ~ 1.6 eV. Additionally, the sharp XRD peaks (Fig. S1(b)) and the morphology(Fig. S1(c)) suggest high crystalline order in these films. A detailed explanation of device fabrication is provided in the experimental section of the Supplemental Material. As shown in the device schematic (Fig. 1(a)), l is the inter-electrode channel length and d is the thickness of the metal electrode (~ 80 -100 nm), and w is the width of the electrode.In this work, we study device characteristics as a function of inter-electrode channel lengths (l) in the range of 4 μm ≤ l ≤ 120 μm. In these devices, the electrode width, w >> l, is in the range of ~ 2 -4 mm. The cross-sectional area of these devices is w × d.Fig. S2(a)shows the reflection-microscopy image of the asymmetric electrode device, with l ~ 12 μm. These devices can be implemented to a larger area using inter-digitated electrode patterns, as shown inFig. S2(b). FIG. 1: (a) Schematic of the lateral asymmetric electrode device, where l is the inter-electrode channel length and d is the thickness of the metal electrode. (b) Semi-log plot of J-V characteristics of the asymmetric electrode device for different values of l. (c) The plot of the J-V characteristics, normalized to J at 5 V bias. (d) The plot of the rectification ratio as a function of channel length shows that the rectification ratio increases for short channel devices.
Fig. 1(b)
shows the dark J-V characteristics on asymmetric MSM devices for different values of the inter-electrode distance, l. The magnitude of the dark-J at a given voltage increases for the short-channel devices owing to the higher electric field (V/l). Additionally, the J-V features indicate a rectification-like behavior in the regime of short-channel devices ( Fig. 1(c)).
Likewise, the rectification ratio = (5 ) | (−5 )| , shown in Fig. 1(d) shows a sharp increase for the 5 short channel devices reaching up to ~ 17 for l = 4 μm. To understand the nature of the observed current, we carry out a closer analysis of the non-linear J-V characteristics. Fig. 2(a) and 2(b) show the log-log plots of the J-V characteristics for devices with l = 12 μm and 4 μm respectively, in the positive bias sweep (to the Au electrode). The J-V features are linear (ohmic) at low voltages. Beyond a critical voltage, the current increases as V 2 , suggestive of space charge limited current (SCLC) behavior and is described by [16,17]:
Carrier-transport regimes
= 9 8 0 2 3( 1 )
Where μθ and εr represent the SCLC mobility and the dielectric constant of the semiconductor, respectively. In the expression for the SCLC mobility: μSCLC = μθ, μ corresponds to the free carrier mobility and θ is the reduction factor due to the carrier-trapping effects [18]. The J-V characteristic is linear for longer channel length devices (l > 12 μm), as depicted in Fig. S3.
However, the transition to the SCLC regime is expected to occur at much larger voltages in agreement with the scaling relation (Eq. (1)). is ohmic (J ∝ l -1 ) for long channel devices (l > 24 μm) and scales as l -3 for short channel devices.
If the transport is SCLC, J should satisfy a scaling relation such that: JSCLC = f (V 2 /l 3 ). The SCLC current, JSCLC was estimated considering the device current to be a combination of the ohmic and space charge limited current such that = ℎ + [19] (Details related to the estimation of JSCLC are provided in section 12 of the Supplemental Material). Fig. 2(d) is a plot of JSCLC vs V 2 /l 3 , indicating a linear variation. The current density in the SCLC region is similar for both the l = 4 m and 12 m samples. This confirms that the transport in these samples is SCLC. Using Eq. (1) and εr = 60 [20,21], the SCLC mobility was determined to be in the range, μSCLC ≈ 0.07 -0.15 cm 2 /Vs. These values are consistent with previous observations of μ from FET and contact-based measurements [18,[22][23][24][25]. The low lateral mobility in this system is attributed to scattering at the grain boundaries accompanied by carrier-trapping effects.
Additionally, the plot in Fig. S4 shows the linear J-V characteristics for positive potential sweep to Al electrode (negative bias sweep in Fig. 1(b)) for different values of the channel length. It is noted that this rectification-like characteristic is not observed in the case of symmetric electrode devices (Fig. S5). Therefore, the origin of this trend is attributed to the presence of a charge-selective transport barrier across either of the metal-perovskite interface. To investigate this further, we examine the energy levels and band-bending mechanism across the metalperovskite interface. eV) attributing a p-type character is consistent with previous observations [26,27], and the holecarrier concentration is estimated to be ~ 10 14 cm -3 . Upon forming a contact in short-circuit condition, the Fermi-level equilibrates, and band-bending occurs at the metal-perovskite interface, as shown in Fig. 3(b). At the Al-MAPI interface, this results in an electron injection barrier φ bn = φ m -χ and a hole injection barrier, φ bp = I.E -φ m , where χ and I.E is the electron affinity and ionization energy, respectively. Similar arguments govern the Au-MAPI interface, with an injection barrier for the electrons and holes represented as φ' bn and φ' bp, respectively.
Spatial Potential Mapping
A detailed analysis for estimating barrier heights at the Al/MAPI interface and the Au/MAPI interface is elaborated in Fig. S6. This analysis indicates the presence of a finite injection barrier for both the carriers at the Al-MAPI interface. In contrast, the injection barrier for the holes at Au-HOIP is minimal (~ 0.1 eV) and can be assumed to be negligible. Additionally, a built-in potential is observed in the perovskite (in Fig. 3(b)), represented as Vbi and V'bi at Al-MAPI and Au-MAPI interface, respectively, that arise as a result of band-bending at the interface. In the event of excess carrier generation, this built-in field facilitates electron and hole extraction to the Al and Au electrode, respectively. The lateral device structure offers the advantage to locally probe and map the potential using Kelvin Probe Force Microscopy (KPFM). The KPFM technique, depicted in Fig. 3(c), essentially consists of a conductive tip that is in intermittent contact with the sample. As the tip is scanned across the device, a varied dc-potential with an overriding ac-bias is applied. At the point of Fermi-level equilibration between the tip and the sample, the contact potential difference (CPD), a parameter proportional to the surface potential is recorded. Fig. 3(d) shows the 2D surface plot of the CPD across the device (2D surface plots representing morphology and CPD correlation are presented in Fig. S7). In concurrence with the metal work-function, the CPD depicts a higher value for the Al electrode and a low value for the Au electrode. corresponding to the two electrodes is ~ 0.9 eV, which is in close agreement with the estimated value of 1.1 eV (Fig. S6). As expected, we observe a sharp potential drop across the Al-MAPI interface, which indicates the presence of a high Vbi and an injection barrier at the interface. In contrast, at the Au-HOIP interface, the potential drop is minimal, suggesting a pseudo-ohmic contact for the hole carriers. Interestingly, across the bulk of the device, a gradual reduction of the CPD reveals the presence of a built-in electric field, indicative of negligible screening due to the mobile ions at the metal-perovskite interface. Any degrading process resulting from ionaccumulation at short-circuit operation is minimal [28]. Moreover, in the presence of low bias and low field conditions, degradation due to ionic drifts is not significant. It is to be noted that in our measurements on back contact lateral devices, the CPD corresponding to the metal surface is modified due to the presence of a thin layer (~ 50 nm) of the MAPI perovskite, as depicted in Fig. 3(c). However, the spatial profile is indicative of the potential map across the lateral structure. This has been additionally verified by imaging the potential profile in the presence of an external bias.
The above-illustrated transport barrier at the Al-MAPI interface explains the observed rectification-like behavior in these asymmetric electrode devices. Essentially, in the negative bias condition (positive voltage to Al), an injection barrier is present for the holes and electrons at Al-MAPI and Au-MAPI interface, respectively. However, a high forward bias current is observed in the presence of a positive bias, with an efficient injection of holes across the Au-MAPI interface. Therefore the determined SCLC mobility corresponds to the injected holecarriers. The asymmetry in the J-V (for short channel devices) shown in Fig. 1(c) then arises from the fact that Al does not make an ohmic contact with MAPI.
Previous reports on HOIP-based devices have shown that the classic SCLC formalism is invalid for systems with mobile ions [28][29][30]. The effect of ionic conduction is especially amplified in sandwich devices with an active layer thickness ~ 200-300 nm. This results in a higher electric field (~ 10 5 V/cm) with the dominant voltage drop across the metal-MAPI interface, thereby completely screening the bulk even in short-circuit condition [31,32]. On the other hand, in lateral structures, the low electric field (~ 10 2 V/cm) across the bulk of the lateral device (Fig. 3(e)) suggests the minimal influence of ionic motion.
Spatial Photocurrent Mapping
We study the spatial photocurrent map using a narrowly localized excitation which is scanned across the lateral device. These results are correlated with the KPFM data to gain additional insight regarding photo-carrier transport in the background of the imaged potential landscape.
Near-field Scanning Photocurrent Microscopy (NSPM) technique is employed to carry out these studies. The experimental setup, as depicted in Fig. 4a, consists of a tapered optical fibertip which is integrated with a tuning fork. Using a resonant-frequency-based feedback system, the tuning fork and the fiber-tip is lowered to be in close proximity to the device (experimental details provided in the Supplemental Material). With the sample-to-fiber tip distance in the near-field approximation (≤ λex), beyond Abbe's diffraction limit, spot sizes up to ~ 100 nm can be achieved [33][34][35]. developed. It is to be noted that for local excitation, the magnitude of the Iph is also controlled by the dark resistance of the channel region outside of the illumination zone. The dark interfacial contact resistance is expected to be relatively higher in the short channel devices [36].
Possible origins of the low Iph close to the Al interface points to high recombination, presumably due to the presence of interfacial traps or a complex hetero-junction. The influence on the Iph(x) profile was also studied as a function of l, results of which are presented in a normalized plot shown in Fig. 4(c) (raw data in Fig. S8). For longer l = 56 μm, the non-uniformity of the Iph distribution is skewed towards the MAPI-Au interface. However, for shorter channel l = 13 μm, we see that the Iph(x)-peak is at the center, and the profile tends to a uniform spatial distribution. The dashed line for l = 4 μm, is the Iph(x) estimated using interpolation, also shows a higher degree of spatial uniformity.
Bias-dependence
To understand the microscopic transport features in the entire regime of device operation, we study the potential profiles in the presence of an applied bias. Fig. 5(a) shows the KPFM images of the lateral devices, with 5 V applied alternatively to the Al and Au electrodes (2D surface plots given in Fig. S9). In the presence of 5 V applied to the Al electrode, the measured net potential difference across the electrodes is > 5 V due to the added contribution of the built-in voltage, i.e., 5 V + Vbi. On the other hand, for the 5 V applied to the Au electrode, the net potential difference across the electrodes is < 5 V or 5 V -Vbi. Interestingly, the potential drop across the bulk of the device indicates a gradual reduction, suggesting a constant electric field (Fig. S10). In the case of negative bias (i.e., 5 V to Al), we observe a considerable potential drop across the MAPI-Al interface, indicative of a transport-barrier zone. The reduced dark-J in the reverse bias can be attributed to the inability of hole injection across this barrier. Previous works on lateral devices report dominant ion-migration effects under high electric fields over long time scales [37,38]. In our studies, experimental measurements were performed under conditions of low electric field (< 10 4 V/cm) and shorter timescales to minimize the influence of ionic motion.
In the presence of an applied bias, the band-bending directionality allows for photo-carrier extraction to the respective electrodes. This photo-carrier extraction efficiency also depends on the carrier mobility and position of excitation from the collection electrode.
( ) = 0 exp { − ( √ ( 2 2 + 4 ) − 2 ) }( 2 )
where, δn0, E, D and τ represents the excess carrier density at the point of generation, electric field, diffusion coefficient and recombination lifetime, respectively. To account for unbalanced transport, we simulate carrier density profiles for different values of electron mobility, μe, which is designated to be a fraction of the hole mobility. Fig. 5c shows that the electron distribution profiles exhibit a longer decay length for high μe. Additionally, the directionality of hole and electron decay profiles show that holes drift in the direction of the electric field, and the electrons drift and decay opposite to the electric field direction. Further, to verify the profiles 14 observed in the bias-dependent NSPM experiment (Fig. 5(b)), we simulate the spatial photocurrent density profile, Jph(x), which is given as a combination of diffusion and drift currents:
ℎ ( ) = + = . +(3)
It is additionally noted that the observed current is a result of net eh recombination in the external circuit. Fig. 5(d) presents the simulated normalized photocurrent density profiles for various ℎ ratios. The profiles indicate that, in the case of balanced mobility, i.e., μe = μh, Iph peaks with excitation at the device center. This region of maxima denotes a zone for the optimal collection of electrons and holes at the respective electrodes. On the other hand, for unbalanced mobility (μe < μh), the peak shifts towards the electron-collection electrode. It is additionally noted that the carrier lifetime also affects the distribution profile (as indicated in Eq. (2)).
Overall, the asymmetry of the Iph(x) points to the lower μτ product of the electrons (μeτe < μhτh).
This interpretation is in agreement with earlier reports [11,39].
Photodetector characteristics.
The light-dependent J-V characteristics on lateral MSM devices are shown in Fig. 6(a) under a 532 nm illumination condition (26 mW/cm 2 ). The Iph increases for the short-channel devices due to a higher electric field at any given voltage. It can also be seen that a short-circuit Iph is also observed corresponding to the values at V = 0. The directionality of the Iph confirms the electron extraction to the Al electrode and holes to the Au electrode. Fig. 6(b) shows the variation of the device responsivity with respect to the channel length. We achieved a responsivity, R ≈ 51 mA/W in the self-powered mode and R ≈ 5.2 A/W at 5 V operating voltage for the l = 4 μm device. This corresponds to quantum efficiency, = ℎ = × 1240 ( . / ) of 11.9 % and 1,212 % at 0 V and 5 V, respectively. The gain (Iph/Idark) observed in these devices is in the range of ~ 10 2 (Fig. S11). The performance of these devices compares with lateral photodetectors reported in the literature [13,40,41]. The lateral device structure is characterized by a low dark current in comparison to the sandwich architecture. This reduces the noise level of device operation and allows for weak signal detection, increasing the dynamic range in these devices. At the Al/MAPI interface (Fig. S6(b)), the carrier injection barrier is given as:
Electron= ℎ + (S1)
To account for only the SCLC part of the observed current, at bias beyond the SCLC onset voltage, V', the SCLC current is given as: From the plot in Fig. S12, the extrapolated red line corresponds to the Johmic, which is corrected, using Eq. (S2), to obtain the SCLC current, JSCLC. It must be noted that the voltage represented in Fig. 2(d) corresponds only to the SCLC voltage, i.e., V > V'.
= − ℎ (S2)
Modeling and simulation of carrier drift in MSM devices.
To model the carrier drift in the lateral devices, we solve the transport equation. The transport of excess electrons is given as: The excess carrier concentration profile, obtained as a solution to Eq (S4), is given as:
( ) = 0 exp { − ( √ ( 2 2 + 4 ) − 2 ) } (S5)
The spatial profiles using the above expression for excess electrons and holes are given in Fig.
5(c).
Here we have considered the hole mobility as μh = 100 cm 2 /Vs, recombination lifetime τ = 10 ns, an electric field, E = 1 kV/cm, and δn0 = 10 15 /cm 3 . The electron distribution, shown in
Fig. 5(c)
is simulated for different values of electron mobility, μe, which is designated as a fraction of μh, in accordance with previous reports [4,5].
To simulate the NSPM experiment, the photocurrent due to carrier drift and diffusion is given as:
ℎ ( ) = + = . + (S6)
The net current density is considered to be equivalent to the excess electrons and holes that recombine in the external circuit. This depends on the distribution profile of both the carriers.
It can be seen in Fig. 5(d) that for the case of balanced mobility (μh = μe) the simulated photocurrent density profile exhibits a peak for photo-excitation at the device center, indicating an efficient collection of both the electrons as well as the holes. In the case of unbalanced carrier mobility (μh > μe), the Iph peak shifts towards the electron-extracting electrode.
Experimental Section
Fabrication of asymmetric metal electrodes:
The lateral metal-electrodes were deposited using 3D mask-printing (Nanoscribe GmbH). The masks were printed with a 2-photon-polymerization lithography technique using photo-curable resin (IP Dip) drop cast on fused silica substrates. After the development of the photoresist mask, Au metal was thermally evaporated. The resist was then removed by immersing in liquid N2 for 30 s and then blow dry with air. The second mask corresponding to the channel length (and partially masking the edge of the Au electrode) was then printed using optical markers and then developed. This was followed by thermal evaporation of the Al electrode. Finally, the second mask was also removed to form the asymmetric electrode device. The thickness of the metal coating was in the range of 80 -100 nm.
Fabrication of lateral back contact devices.
The MAPbI3 perovskite films were prepared by spin-coating the precursor solution. The Aldrich) was dynamically dispensed. The films were then annealed at 100 °C for 60 min. All the materials were used as obtained.
J-V measurement on devices
The J-V measurements were carried out on devices using the Keithley Source Meter 2400. The scan on the voltage sweep was carried out at a rate of 0.5 V/s. The light responsivity measurements were carried out using a 532 nm excitation with an optical power of ~ 26 mW/cm 2 .
Several devices (> 20) were fabricated for the studies. The dark I-V studies presented were analyzed from three devices each corresponding to l = 25 μm, 50 μm and 100 μm and from two devices corresponding to channel length, l = 4 μm and 12 μm, and. These devices represented different batches during the fabrication process. We consistently observe that the long channel devices show symmetric I-V characteristics while the short channel devices exhibit rectification-like behavior. These devices were also utilized for the photo-detector studies.
Kelvin Probe Force Microscopy
Kelvin Probe measurements were carried out using the JPK Nanowizard-3 Atomic Force Microscopy instrument. The scans were performed using conductive Cr-Pt cantilever tips (BudgetSensors, Multi 75 EG) with a force calibration of 2 N/m. Calibration of the tip was initially done by measuring the contact potential difference (CPD) on the Au sample (100 nm of Au coated on the glass slide). This was followed by a measurement of the contact potential difference on the lateral devices. The thickness of the perovskite layer was reduced to ~ 150-the active device. In short-circuit condition, both the Al and Au electrodes were grounded with respect to the Kelvin-Probe circuit. In the case of bias-dependent studies, an external bias was applied in parallel to the lateral MSM device. For the spatially resolved studies, we fabricated and measured devices with channel lengths of l = 22 μm and 55 μm. Here a minimum of two devices of each was used, and the measurements were carried out at numerous points across the channel length.
Near-field Scanning Photocurrent Microscopy
JPK Nanowizard-3 Atomic force microscope (AFM) in Near-field scanning optical microscopy (NSOM) mode was used to measure the local photocurrent in the channel region of the devices.
A tapered optical fiber coated with Cr/Au and having aperture ~ 105 nm was raster-scanned using the piezo head of the AFM. The other end of the multi-mode fiber was coupled to a 532 nm laser using a 20x, N.A.= 0.4 microscope objective lens. The input laser power was maintained at 5 mW during the measurement. Additionally, the laser intensity was modulated at 383 Hz using a mechanical chopper. Using a resonant-frequency based feedback system, the assembly consisting of the tuning fork and the fiber-tip was lowered to be in close proximity to the sample. When the fiber tip-to-sample distance is in the near-field approximation (≤ λex), spot sizes up to ~ 100 nm can be achieved beyond Abbe's diffraction limit.
In synchronization with the excitation scanning, the photocurrent signal was measured using Stanford Research Systems SRS 830 lock-in amplifier, and the read-out signal was fed to the JPK scanning probe module (SPM) control for data sampling and storage. A calibrated photodetector was used to measure the transmitted light through the device. The transmission map of the device is used to correlate the local photocurrent value to discern the position of the metal electrodes. The ambient light was completely blocked to avoid any external exposure to the sample. The measurements were performed in ambient air due to the practical limitations of mounting the instrument in a vacuum. During the measurements, desiccants were placed around the setup to reduce the moisture content in the ambient atmosphere.
For the spatially-resolved NSPM measurement, in addition to the measurements on long channel devices, we studied the profiles for a short channel device of l = 13 μm.
Transient Photocurrent
Transient Photocurrent measurements were carried out by using a pulsed supercontinuum laser (YSL Supercontinuum SC PRO 7, λex ~ 400-2300 nm) with a pulse width of ~ 100 ps (with 100 kHz repetition rate) and energy ~ 1 μJ pulse -1 , incident from the glass-substrate side on the lateral device. The device was operated at 5 V bias, and the current was measured in series using a digital oscilloscope (Tektronix MDO3024, 2.5 GS s -1 ) with 50 Ω input coupling.
6 FIG. 2 :
62Log-log plot of the dark J-V data shows distinct ohmic (∝ V 1 ) and SCLC behavior (∝ V 2 ) for (a) l = 12 μm and (b) 4 μm channel respectively. (c) Variation of dark current density at 5 V as a function of channel length shows an ohmic behavior (∝ l -1 ) for the long channel length and an SCLC behavior (∝ l -3 ) for the short channel length. (d) Plot of current density, JSCLC as a function of V 2 /l 3 for short channel devices shows a linear dependence, confirming SCLC behavior.
Fig. 2 (
2c) represents the device current as a function of channel length at 5 V bias. The current
Fig. 3 (
3a) shows the band energy levels and the work-function of the metal electrodes in the absence of circuit formation. The work-function of Al and MAPI were determined to be 4.1 eV and 5.1 eV, respectively, using Kelvin-Probe measurements (with Au as the reference, details in the experimental section, Supplemental Material). The higher work-function of MAPI (5.1
FIG. 3 :
3(a) Band alignment diagram of the asymmetric electrode MSM device. (b) The device in shortcircuit condition shows an injection barrier at the metal-perovskite interface. A built-in potential, Vbi is developed at the metal-MAPI interface as a result of Fermi-level equilibration. (c) Schematic of the KPFM experimental setup used for potential mapping. The measured contact potential difference (CPD) is used to measure the potential profile across the device. (d) A 2D surface plot of the CPD measured across the device. (e) The CPD line-scan profile shows a significant potential drop at the Al-MAPI interface and across the bulk of the device. The potential drop across the Au-MAPI interface is minimal.
Fig. 3 (
3e) shows the line scan profile of CPD across the device. The difference in the CPD value
Fig. 4 (
4b) shows the short-circuit photocurrent (Iph) profile as the excitation is scanned across the asymmetric electrode device with l = 22 μm. The Iph(x) profile indicates regions of low and high Iph for excitation close to the MAPI-Al and MAPI-Au interface, respectively. The higher Iph close to the Au electrode can be attributed to the efficient transport of the holes to the Au electrode. A model for understanding the reduction of the Iph close to the electrodes needs to be
FIG. 4 :
4(a) Schematic of Near-field Scanning Photocurrent Microscopy (NSPM) setup. A tapered fibertip in near-field excitation is scanned across the lateral device. (b) Iph(x) profile obtained for NSPM scan across l = 22 μm MSM device shows the high Iph for excitation close to the Au/MAPI interface. (c)Normalized Iph(x) profile as a function of channel length. For the short channel length devices, Iph(x) tends to a uniform distribution profile.
Fig. 5 (
5b) shows the Iph(x) profile obtained from the NSPM technique on the lateral devices in the presence of both positive and negative bias of 5 V. The profile indicates a spatial asymmetry, where the Iph-peak shifts towards the positively biased (the electron extracting) electrode. This polarity-dependent Iph(x)-asymmetry can be attributed to the unbalanced carrier mobility of the hole and the electron. To understand the microscopic picture of these bias-dependent photocurrent profiles, we simulate the carrier and photocurrent profiles using a simplistic 1D drift-diffusion formalism (details of simulation in section 13 of Supplemental Material). These results emphasize the dominant role of unbalanced mobility and can be obtained from solving the transport equation under conditions of steady-state generation at x = 0, drift condition assuming uniform electricfield ( E = 1 kV/cm) and hole mobility, μh = 100 cm 2 /Vs (this represents free carrier mobility in the ohmic regime of operation)[11].Fig. 5(c)shows the simulated spatial distribution of the excess photogenerated electrons and holes. The spatial dependence of excess photogenerated carriers can be expressed as:
FIG. 5 :
5(a) CPD profiles from KPFM on MSM devices for positive and negative bias of 5 V. Alternatively, this is indicated as 5 V applied to either Au or Al electrodes. The gradual potential drop indicates a uniform electric field across the device. (b) NSPM scans in the presence of external bias show the presence of a photocurrent peak which shifts depending on the polarity of the bias. This Iph(x) asymmetry indicates unbalanced carrier mobility. (c) Simulation using the finite element method shows the decay of excess electrons and holes for excitation at x = 0, under conditions of 1-D drift-diffusion transport for different μe values. (d) The normalized Jph profiles, simulated as a function of excitation position across the device, under 1 kV/cm electric field. The profiles indicate a shift in the Jph(x) peak in the case of unbalanced carrier mobility, which explains the feature observed in (b).
The lateral devices are characterized by low device capacitance owing to the tiny overlap area across the two metal electrodes. This results in a small RC value, which reduces the overall response time of the device[42]. Iph(t) decay measurements on HOIP-based MSM devices have reported operational speeds in the range of ~ 40-90 ns[43]. We have additionally observed a fast decay component of ~ 12 ns (shown inFig. 6(c)), using transient photocurrent measurements on MSM devices with mixed-phase perovskite: (FA0.82MA0.18Pb(I0.82Br0.18)3) (FA = Formamidium) as the active semiconductor. Such fast-response times allow for applications involving high-speed detectors.
FIG. 6 :
6(a) Voltage-dependent responsivity characteristics of the lateral devices for 532 nm illumination at 26 mW/cm 2 . A short circuit Iph is observed at 0 V. (b) Variation of the responsivity as a function of channel length under short circuit conditions and ± 5 V bias conditions. (c) Transient photocurrent measurement on lateral MSM devices with mixed-phase perovskite (FAMA) as the active layer. The biexponential fit reveals a fast response time of 11.8 ns followed by a slow component of ~ 112 ns. (d)Variation of photocurrent with respect to light-intensity (λ = 532nm), shows the LDR ~ 118 dB, spanning over six orders of intensity variation.
Fig. 6 (
6d) shows the linear dynamic range (LDR) of the lateral device with the mixed-phase perovskite in the active layer. The device maintains linear response over a range spanning ~ 6 orders of intensity variation, resulting in a = 20 log ( ) of 118 dB, where Pmax and Pmin correspond to the maximumand minimum light power, respectively, that is resolved with maintained responsivity.C. ConclusionThe lateral asymmetric electrode MSM devices exhibit distinct transport regimes as a function of applied bias and inter-electrode distance. The transport shows ohmic behavior at low voltagesand becomes space charge limited at high voltage in the short channel length devices, consistent with the SCLC scaling relation. The observed rectification-like J-V characteristic is attributed to bias-direction dependent SCLC behavior due to efficient hole injection across the Au-MAPI interface. The KPFM plots across the device indicate an electric field in bulk, both in shortcircuit and bias conditions, suggesting negligible ion-screening effects. In the presence of an applied bias, the spatial photocurrent profiles were understood in the context of drift-diffusion formalism, which indicates unbalanced carrier transport. The lateral devices exhibit photodetection capability and demonstrate a high responsivity of ≈ 5.2 A/W at 5 V operating bias in the short-channel structures. In addition to a higher dynamic range spanning six orders of intensity variation, the low device capacitance allows for high-speed applications in the GHz regime. carrier transport in lateral metal-perovskitemetal structures and its influence on device operation N. Ganesh, A.Z. Ashar, Sumukh Purohit, K.L. Narasimhan and K.S. Narayan* List of Contents: 1. MAPbI3 thin film characterization. FIG. S1: (a) Absorption and PL data for the MAPbI3 thin films. The absorption and Photoluminescence (PL) indicate a band-gap of 1.6 eV. (b) Data shows the thin film XRD of MAPbI3 films. The sharp peaks indicate high crystalline order. c. Shows the morphology image of the MAPbI3 films. 2. Microscopy image of asymmetric electrode device and inter-digitated device. FIG. S2: (a) Microscopy image of the asymmetric electrode lateral device, with l = 12 μm, captured in the reflection geometry. (b) Schematic of an asymmetric interdigitated electrode structure. The enlarged depiction is the microscopic image of the structure with an inter-electrode distance of 10 μm.
FIG. S3 :
S3(a) Schematic of the diagram representing the directionality of the applied bias. Here positive bias is applied to the Au electrode. (b) Log-log J(V) plot for long-channel (l > 20 μm) asymmetric electrode devices.4. Log-log plot of negative bias J(V) characteristics of Al/MAPI/Au devices FIG. S4: (a) Schematic of the diagram representing the directionality of the applied bias. Here positive bias is applied to the Al electrode. (b) Log-log plot of dark J(V) characteristics show single-exponent variation, representing the ohmic regime of operation for devices of different channel lengths, l.5. Dark J(V) characteristics of Au/MAPI/Au symmetric devices. FIG. S5: Dark J(V) characteristics of Au/MAPI/Au symmetric electrode devices with a channel length of (a) 55 μm, (b) 10 μm, and (c) 5 μm. 6. Estimation of barrier height and Vbi in asymmetric electrode device.Fig. S6 shows the band-alignment across the Al/MAPI (left-hand side of Fig. S6) and Au/MAPI (right-hand side of Fig. S6) interface. Fig. S6(a) and S6(c) show the Fermi-levels of the metals and the perovskite. The band-bending results in a barrier at the metal-perovskite interface.
Au/MAPI interface (Fig. S6(d)), the carrier injection barrier is given as: Electron barrier, φ′ = φ − χ = 5.built-in potential is given as: V' bi (Au/MAPI) -V bi (Al/MAPI) = 1.1 V FIG. S6: Schematic shows the band alignment diagram across the Al/MAPI interface (a) before and (b) after the formation of the contact. The right-hand side shows the band-diagram across the Au/MAPI interface (c) before and (d) after the formation of the contact.7. Morphology-KPFM correlation of asymmetric electrode devices. FIG. S7: 2D plots showing the morphology-KPFM correlation for asymmetric electrode device with l = 55 μm.9. 2D surface plots for bias-dependent KPFM FIG. S9: 2D surface plots showing the raw data of the CPD for bias-dependent KPFM measurement with (a) 5 V applied to the Al and (b) 5V applied to Au electrode. (c) shows the data for 5 V applied to Al with additional data processing by subtraction of a zeroth-order polynomial baseline to improve the contrast of the image. (d) shows the data for 5 V to Au electrode, with similar data processing. However, only the raw data was used for all the analysis presented in the main text.10. Electric-filed profile in the presence of bias. FIG. S10: Plot of CPD and lateral electric field for asymmetric electrode devices with l = 56 μm for the case of (a) -5 V and (b) + 5 V. The dominant electric field at the Al-MAPI interface in the negative bias is indicative of the formation of a depletion zone. 12. Representation of the SCLC current correcting for the ohmic contribution. Fig. 2(d) (in the text) shows the plot of JSCLC vs. V 2 /l 3 . JSCLC corresponds to the SCLC dark current corrected for the ohmic contribution. For instance, Fig. S12 shows the log-log J(V) plot for a 4 μm channel device with positive bias. The profile indicates the dominant ohmic and SCLC regimes as a function of the function of applied bias. Here, the ohmic current is given as
FIG. S12 :
S12Log-log plot of the J(V) characteristics for positive bias sweep in a l = 4 μm asymmetric device. The plot shows distinct ohmic and SCLC behavior. The solid red line shows the baseline for the ohmic current.
δn0, E, D, G and τ represents the excess carrier density at the point of generation, electric field, diffusion coefficient, generation function, and recombination lifetime, respectively. In our bias-dependent experiments, we have an electric field in the range of E ~ 1 kV/cm. In the scanning photocurrent experiments, where we have a steady-state point generation of photocarriers, the carrier generation function is given G(x) = G0 δ(x-x0). Solving for 1D transport with carrier collection away from the point of generation (x ≠ x0), the Eq. (S3) reduces to the following second-order homogeneous equation: for the excess carrier concentration by implementing the initial value conditions such that such that δn(x = 0) = δn0 = ɑG0τ, and δn(x = ∞) = 0. Additionally, it is noted that the excess electrons drift in the direction opposite to the electric field, -E. At the charge-selective contacts, the boundary conditions for the excess electrons (δn) and holes (δp) can be assumed as follows:(i)The Al electrode act as electron extraction contacts and reflects the holes such that:
MAPbI3 perovskite precursor was initially prepared using the following by dissolving 357 mg of methylammonium iodide (MAI, Dyesol), 204 mg lead chloride (PbCl2, Sigma Aldrich) and 7 mg of lead iodide (PbI2, 99.999 %, Sigma Aldrich) in 1 ml of DMF (N,N-Dimethylformamide, anhydrous) solution and stirred at high speed for at least 60 min. To render the substrates hydrophilic, they were plasma treated (using ultra-high pure (UHP) nitrogen, 99.9995 %) for 2 min. The substrates were transferred to a glovebox maintained in an inert atmosphere. The perovskite precursor was dispensed onto the substrates and spin-coated at 3000 rpm for 80 s followed by 5000 rpm for 10 s. The films were then annealed in the following sequence: 20 °Cfor 15 min, 70 °C for 15 min, 100 °C for 90 min and 120 °C for 10 min. To encapsulate the devices (for J-V measurements), a layer of PMMA (250 mg ml -1 in chlorobenzene) was spincoated at 2000 rpm for 60 s followed by annealing at 60 °C for 20 min. For the preparation of mixed-phase perovskite precursor solution, 215 mg of FAI (Formamidium Iodide, Sigma Aldrich, > 99 %), 28 mg of MABr (Methylammonium bromide, Dyesol), 634 mg of PbI2 (Lead(II) iodide, Alfa Aesar, 99.9985 %) and 92 mg of PbBr2 (Lead Bromide, Sigma Aldrich, 98 %) was dissolved in 1 ml of 4:1 ratio of (Dimethylformamide : Dimethylsulfoxide) and stirred at 85 °C for about 30 min. 70 μL of the solution was then dispensed onto the substrates with lateral metal electrodes and spin-coated at 1000 rpm for 10 s followed by 6000 rpm for 35 s. In the last 7 s of the spin, 70 μL of Anisole (anhydrous, Sigma
The authors acknowledge the Department of Science and Technology, Government of India, and EPSRC-UKRI Global Challenge Research Fund project, SUNRISE (EP/P032591/1), for the financial assistance. SP acknowledges DST-INSPIRE for the fellowship.
. Q Jiang, Y Zhao, X Zhang, X Yang, Y Chen, Z Chu, Q Ye, X Li, Z Yin, J , Q. Jiang, Y. Zhao, X. Zhang, X. Yang, Y. Chen, Z. Chu, Q. Ye, X. Li, Z. Yin, and J.
. You, Nature Photonics. 13460You, Nature Photonics 13, 460 (2019).
. N R E Laboratory, Golden, CON. R. E. Laboratory, (NREL Golden, CO, accessed June 2021).
. Y.-H Lin, N Sakai, P Da, J Wu, H C Sansom, A J Ramadan, S Mahesh, J Liu, R , Y.-H. Lin, N. Sakai, P. Da, J. Wu, H. C. Sansom, A. J. Ramadan, S. Mahesh, J. Liu, R.
. D Oliver, J Lim, Science. 36996D. Oliver, and J. Lim, Science 369, 96 (2020).
. Y Wang, Z Lv, J Chen, Z Wang, Y Zhou, L Zhou, X Chen, S T Han, Advanced materials. 301802883Y. Wang, Z. Lv, J. Chen, Z. Wang, Y. Zhou, L. Zhou, X. Chen, and S. T. Han, Advanced materials 30, 1802883 (2018).
. Y Wang, R Fullon, M Acerce, C E Petoukhoff, J Yang, C Chen, S Du, S K Lai, S P Lau, D Voiry, Advanced Materials. 291603995Y. Wang, R. Fullon, M. Acerce, C. E. Petoukhoff, J. Yang, C. Chen, S. Du, S. K. Lai, S. P. Lau, and D. Voiry, Advanced Materials 29, 1603995 (2017).
. C Qin, A S Sandanayaka, C Zhao, T Matsushima, D Zhang, T Fujihara, C , C. Qin, A. S. Sandanayaka, C. Zhao, T. Matsushima, D. Zhang, T. Fujihara, and C.
. Adachi, Nature. 58553Adachi, Nature 585, 53 (2020).
. L Zhao, K Roh, S Kacmoli, K Kurdi, S Jhulki, S Barlow, S R Marder, C , L. Zhao, K. Roh, S. Kacmoli, K. Al Kurdi, S. Jhulki, S. Barlow, S. R. Marder, C.
. B P Gmachl, Rand, Advanced Materials. 32Gmachl, and B. P. Rand, Advanced Materials 32, 2000752 (2020).
. Z Chen, Q Dong, Y Liu, C Bao, Y Fang, Y Lin, S Tang, Q Wang, X Xiao, Y , Z. Chen, Q. Dong, Y. Liu, C. Bao, Y. Fang, Y. Lin, S. Tang, Q. Wang, X. Xiao, and Y.
. Bai, Nature communications. 81Bai, Nature communications 8, 1 (2017).
. Y Chen, H Yi, X Wu, R Haroldson, Y Gartstein, Y Rodionov, K Tikhonov, A , Y. Chen, H. Yi, X. Wu, R. Haroldson, Y. Gartstein, Y. Rodionov, K. Tikhonov, A.
. X.-Y Zakhidov, V Zhu, Podzorov, Nature communications. 71Zakhidov, X.-Y. Zhu, and V. Podzorov, Nature communications 7, 1 (2016).
. S D Stranks, G E Eperon, G Grancini, C Menelaou, M J Alcocer, T Leijtens, L , S. D. Stranks, G. E. Eperon, G. Grancini, C. Menelaou, M. J. Alcocer, T. Leijtens, L.
. M Herz, A Petrozza, H J Snaith, Science. 342341M. Herz, A. Petrozza, and H. J. Snaith, Science 342, 341 (2013).
. Q Dong, Y Fang, Y Shao, P Mulligan, J Qiu, L Cao, J Huang, Science. 347967Q. Dong, Y. Fang, Y. Shao, P. Mulligan, J. Qiu, L. Cao, and J. Huang, Science 347, 967 (2015).
. S J Yang, M Kim, H Ko, D H Sin, J H Sung, J Mun, J Rho, M H Jo, K Cho, Advanced Functional Materials. 281804067S. J. Yang, M. Kim, H. Ko, D. H. Sin, J. H. Sung, J. Mun, J. Rho, M. H. Jo, and K. Cho, Advanced Functional Materials 28, 1804067 (2018).
. J Ding, S Du, Y Zhao, X Zhang, Z Zuo, H Cui, X Zhan, Y Gu, H Sun, Journal of Materials Science. 52276J. Ding, S. Du, Y. Zhao, X. Zhang, Z. Zuo, H. Cui, X. Zhan, Y. Gu, and H. Sun, Journal of Materials Science 52, 276 (2017).
. X Lin, A N Jumabekov, N N Lal, A R Pascoe, D E Gómez, N W Duffy, A S , X. Lin, A. N. Jumabekov, N. N. Lal, A. R. Pascoe, D. E. Gómez, N. W. Duffy, A. S.
. K Chesman, M Sears, Y Fournier, Zhang, Nature communications. 81Chesman, K. Sears, M. Fournier, and Y. Zhang, Nature communications 8, 1 (2017).
. D G Georgiadou, J Semple, A A Sagade, H Forstén, P Rantakari, Y.-H Lin, F , D. G. Georgiadou, J. Semple, A. A. Sagade, H. Forstén, P. Rantakari, Y.-H. Lin, F.
. A Alkhalil, K Seitkhan, H Loganathan, Faber, Nature Electronics. 3718Alkhalil, A. Seitkhan, K. Loganathan, and H. Faber, Nature Electronics 3, 718 (2020).
. N Mott, Oxford University PressN. Mott, (Oxford University Press, 1940).
M A Lampert, P Mark, Current injection in solids. Academic pressM. A. Lampert and P. Mark, Current injection in solids (Academic press, 1970).
. M S Alvar, P W Blom, G.-J A Wetzelaer, Nature communications. 111M. S. Alvar, P. W. Blom, and G.-J. A. Wetzelaer, Nature communications 11, 1 (2020).
. J A Röhr, R C Mackenzie, Journal of Applied Physics. 128165701J. A. Röhr and R. C. MacKenzie, Journal of Applied Physics 128, 165701 (2020).
. M Zhang, X Zhang, L.-Y Huang, H.-Q Lin, G Lu, Physical Review B. 96195203M. Zhang, X. Zhang, L.-Y. Huang, H.-Q. Lin, and G. Lu, Physical Review B 96, 195203 (2017).
. N Onoda-Yamamuro, T Matsuo, H Suga, Journal of Physics and Chemistry of Solids. 53935N. Onoda-Yamamuro, T. Matsuo, and H. Suga, Journal of Physics and Chemistry of Solids 53, 935 (1992).
. M Petrovic, T Ye, V Chellappan, S Ramakrishna, ACS applied materials & interfaces. 942769M. Petrovic, T. Ye, V. Chellappan, and S. Ramakrishna, ACS applied materials & interfaces 9, 42769 (2017).
. D Li, G Wang, H.-C Cheng, C.-Y Chen, H Wu, Y Liu, Y Huang, X Duan, Nature communications. 71D. Li, G. Wang, H.-C. Cheng, C.-Y. Chen, H. Wu, Y. Liu, Y. Huang, and X. Duan, Nature communications 7, 1 (2016).
. X Y Chin, D Cortecchia, J Yin, A Bruno, C Soci, Nature communications. 61X. Y. Chin, D. Cortecchia, J. Yin, A. Bruno, and C. Soci, Nature communications 6, 1 (2015).
. S P Senanayak, B Yang, T H Thomas, N Giesbrecht, W Huang, E Gann, B Nair, K Goedel, S Guha, X Moya, Science advances. 31601935S. P. Senanayak, B. Yang, T. H. Thomas, N. Giesbrecht, W. Huang, E. Gann, B. Nair, K. Goedel, S. Guha, and X. Moya, Science advances 3, e1601935 (2017).
. J R Harwell, T Baikie, I Baikie, J L Payne, C Ni, J T S Irvine, G A Turnbull, I D W Samuel, Physical Chemistry Chemical Physics. 1819738J. R. Harwell, T. Baikie, I. Baikie, J. L. Payne, C. Ni, J. T. S. Irvine, G. A. Turnbull, and I. D. W. Samuel, Physical Chemistry Chemical Physics 18, 19738 (2016).
. Y Zhou, G Long, The Journal of Physical Chemistry C. 1211455Y. Zhou and G. Long, The Journal of Physical Chemistry C 121, 1455 (2017).
. E A Duijnstee, J M Ball, V M Le Corre, L J A Koster, H J Snaith, J Lim, ACS Energy Letters. 5376E. A. Duijnstee, J. M. Ball, V. M. Le Corre, L. J. A. Koster, H. J. Snaith, and J. Lim, ACS Energy Letters 5, 376 (2020).
. H J Snaith, A Abate, J M Ball, G E Eperon, T Leijtens, N K Noel, S D Stranks, J T Wang, K Wojciechowski, W Zhang, The journal of physical chemistry letters. 51511H. J. Snaith, A. Abate, J. M. Ball, G. E. Eperon, T. Leijtens, N. K. Noel, S. D. Stranks, J. T.-W. Wang, K. Wojciechowski, and W. Zhang, The journal of physical chemistry letters 5, 1511 (2014).
The journal of physical chemistry letters. W Tress, 83106W. Tress, The journal of physical chemistry letters 8, 3106 (2017).
. S A Weber, I M Hermes, S.-H Turren-Cruz, C Gort, V W Bergmann, L Gilson, A , S. A. Weber, I. M. Hermes, S.-H. Turren-Cruz, C. Gort, V. W. Bergmann, L. Gilson, A.
. M Hagfeldt, W Graetzel, R Tress, Berger, Energy & Environmental Science. 112404Hagfeldt, M. Graetzel, W. Tress, and R. Berger, Energy & Environmental Science 11, 2404 (2018).
. H Wang, A Guerrero, A Bou, A M Al-Mayouf, J Bisquert, Energy & Environmental Science. 122054H. Wang, A. Guerrero, A. Bou, A. M. Al-Mayouf, and J. Bisquert, Energy & Environmental Science 12, 2054 (2019).
. C R Mcneill, H Frohne, J L Holdsworth, J E Furst, B V King, P C Dastoor, Nano Letters. 4219C. R. McNeill, H. Frohne, J. L. Holdsworth, J. E. Furst, B. V. King, and P. C. Dastoor, Nano Letters 4, 219 (2004).
. S Mukhopadhyay, A J Das, K Narayan, The journal of physical chemistry letters. 4161S. Mukhopadhyay, A. J. Das, and K. Narayan, The journal of physical chemistry letters 4, 161 (2013).
. S Mukhopadhyay, S Ramachandra, K Narayan, The Journal of Physical Chemistry C. 11517184S. Mukhopadhyay, S. Ramachandra, and K. Narayan, The Journal of Physical Chemistry C 115, 17184 (2011).
. M Rao, K Narayan, Applied Physics Letters. 92201M. Rao and K. Narayan, Applied Physics Letters 92, 201 (2008).
. F Liu, L Wang, J Wang, F Wang, Y Chen, S Zhang, H Sun, J Liu, G Wang, Y Hu, Advanced Functional Materials. 31F. Liu, L. Wang, J. Wang, F. Wang, Y. Chen, S. Zhang, H. Sun, J. Liu, G. Wang, and Y. Hu, Advanced Functional Materials 31, 2005662 (2021).
. Y Yuan, J Chae, Y Shao, Q Wang, Z Xiao, A Centrone, J Huang, Advanced Energy Materials. 51500615Y. Yuan, J. Chae, Y. Shao, Q. Wang, Z. Xiao, A. Centrone, and J. Huang, Advanced Energy Materials 5, 1500615 (2015).
. O E Semonin, G A Elbaz, D B Straus, T D Hull, D W Paley, A M Van Der Zande, J C Hone, I Kymissis, C R Kagan, X Roy, The journal of physical chemistry letters. 73510O. E. Semonin, G. A. Elbaz, D. B. Straus, T. D. Hull, D. W. Paley, A. M. Van der Zande, J. C. Hone, I. Kymissis, C. R. Kagan, and X. Roy, The journal of physical chemistry letters 7, 3510 (2016).
. Y Liu, Y Zhang, K Zhao, Z Yang, J Feng, X Zhang, K Wang, L Meng, H Ye, M Liu, Advanced Materials. 301707314Y. Liu, Y. Zhang, K. Zhao, Z. Yang, J. Feng, X. Zhang, K. Wang, L. Meng, H. Ye, and M. Liu, Advanced Materials 30, 1707314 (2018).
. B Yang, F Zhang, J Chen, S Yang, X Xia, T Pullerits, W Deng, K Han, Advanced Materials. 291703758B. Yang, F. Zhang, J. Chen, S. Yang, X. Xia, T. Pullerits, W. Deng, and K. Han, Advanced Materials 29, 1703758 (2017).
. A Armin, M Hambsch, I K Kim, P L Burn, P Meredith, E B Namdas, Laser & Photonics Reviews. 8924A. Armin, M. Hambsch, I. K. Kim, P. L. Burn, P. Meredith, and E. B. Namdas, Laser & Photonics Reviews 8, 924 (2014).
. F X Liang, J Z Wang, Z X Zhang, Y Y Wang, Y Gao, L B Luo, Advanced Optical Materials. 51700654F. X. Liang, J. Z. Wang, Z. X. Zhang, Y. Y. Wang, Y. Gao, and L. B. Luo, Advanced Optical Materials 5, 1700654 (2017).
24 2. Microscopy image of asymmetric electrode device and inter-digitated device. MAPbI3 thin film characterization. ................................................................................... 24 2. Microscopy image of asymmetric electrode device and inter-digitated device. ............... 24
Log-log plot of positive bias J(V) characteristics for long channel devices. Log-log plot of negative bias J(V) characteristics of Al/MAPI/Au devicesLog-log plot of positive bias J(V) characteristics for long channel devices. .................... 25 4. Log-log plot of negative bias J(V) characteristics of Al/MAPI/Au devices ..................... 25
. J Dark, MAPI/Au symmetric devices. ....................................... 26Dark J(V) characteristics of Au/MAPI/Au symmetric devices. ....................................... 26
Estimation of barrier height and Vbi in asymmetric electrode device. Morphology-KPFM correlation of asymmetric electrode devicesEstimation of barrier height and Vbi in asymmetric electrode device. .............................. 26 7. Morphology-KPFM correlation of asymmetric electrode devices. .................................. 27
Line profiles of NSPM scan in asymmetric devices for different l. Line profiles of NSPM scan in asymmetric devices for different l. ................................. 28
29 11. Comparison of dark and light I-V characteristics. . . . . . . Kpfm, .......................................................... 29 10 ; . ................................................................ ; ......................................................... 30 12 ; ............. 312D surface plots for bias-dependent. Electric-filed profile in the presence of bias. Representation of the SCLC current correcting for the ohmic contribution. ...2D surface plots for bias-dependent KPFM ...................................................................... 29 10. Electric-filed profile in the presence of bias. ................................................................ 29 11. Comparison of dark and light I-V characteristics ......................................................... 30 12. Representation of the SCLC current correcting for the ohmic contribution. ................ 31
Modeling and simulation of carrier drift in MSM devices. 34 8. Line profiles of NSPM scan in asymmetric devices for different lModeling and simulation of carrier drift in MSM devices. ........................................... 32 14. Experimental Section .................................................................................................... 34 8. Line profiles of NSPM scan in asymmetric devices for different l.
Line scan profiles of the NSPM scans on asymmetric electrodes in short-circuit condition for the inter-electrode channel length of (a) 55 μm (b) 22 μm and (c) 13 μm. 11. Comparison of dark and light I-V characteristics FIG. S11: Comparison of the dark and light (532 nm, 26 mW/cm 2 ) for channel lengths of (a) 4 μm, (b) 12 μm, (c) 25 μm. Fig, S8, d) 55 μm and (e) 122 μm ReferencesFIG. S8: Line scan profiles of the NSPM scans on asymmetric electrodes in short-circuit condition for the inter-electrode channel length of (a) 55 μm (b) 22 μm and (c) 13 μm. 11. Comparison of dark and light I-V characteristics FIG. S11: Comparison of the dark and light (532 nm, 26 mW/cm 2 ) for channel lengths of (a) 4 μm, (b) 12 μm, (c) 25 μm, (d) 55 μm and (e) 122 μm References:
N Mott, GURNEY RW-Electronic processes in ionic crystals. Oxford University PressN. Mott, GURNEY RW-Electronic processes in ionic crystals (Oxford University Press, 1940).
M A Lampert, P Mark, Current injection in solids. Academic pressM. A. Lampert and P. Mark, Current injection in solids (Academic press, 1970).
Analytical description of mixed ohmic and spacecharge-limited conduction in single-carrier devices. J A Röhr, R C Mackenzie, Journal of Applied Physics. 128165701J. A. Röhr and R. C. MacKenzie, Analytical description of mixed ohmic and space- charge-limited conduction in single-carrier devices, Journal of Applied Physics 128, 165701 (2020).
. O E Semonin, G A Elbaz, D B Straus, T D Hull, D W Paley, A M Van Der Zande, O. E. Semonin, G. A. Elbaz, D. B. Straus, T. D. Hull, D. W. Paley, A. M. Van der Zande,
Limits of carrier diffusion in n-type and ptype CH3NH3PbI3 perovskite single crystals. J C Hone, I Kymissis, C R Kagan, X Roy, The journal of physical chemistry letters. 73510J. C. Hone, I. Kymissis, C. R. Kagan, and X. Roy, Limits of carrier diffusion in n-type and p- type CH3NH3PbI3 perovskite single crystals, The journal of physical chemistry letters 7, 3510 (2016).
Electron-hole diffusion lengths > 175 μm in solution-grown CH3NH3PbI3 single crystals. Q Dong, Y Fang, Y Shao, P Mulligan, J Qiu, L Cao, J Huang, Science. 347967Q. Dong, Y. Fang, Y. Shao, P. Mulligan, J. Qiu, L. Cao, and J. Huang, Electron-hole diffusion lengths > 175 μm in solution-grown CH3NH3PbI3 single crystals, Science 347, 967 (2015).
| [] |
[
"Kernel Cross-View Collaborative Representation based Classification for Person Re-Identification",
"Kernel Cross-View Collaborative Representation based Classification for Person Re-Identification"
] | [
"Raphael Prates [email protected] ",
"William Robson Schwartz [email protected] ",
"\nUniversidade Federal de Minas Gerais\n6627Brazil\n",
"\nAv. Pres. Antônio Carlos -Pampulha\nBelo Horizonte -MG\n31270-901\n"
] | [
"Universidade Federal de Minas Gerais\n6627Brazil",
"Av. Pres. Antônio Carlos -Pampulha\nBelo Horizonte -MG\n31270-901"
] | [] | Person re-identification aims at the maintenance of a global identity as a person moves among non-overlapping surveillance cameras. It is a hard task due to different illumination conditions, viewpoints and the small number of annotated individuals from each pair of cameras (smallsample-size problem). Collaborative Representation based Classification (CRC) has been employed successfully to address the small-sample-size problem in computer vision. However, the original CRC formulation is not well-suited for person re-identification since it does not consider that probe and gallery samples are from different cameras. Furthermore, it is a linear model, while appearance changes caused by different camera conditions indicate a strong nonlinear transition between cameras. To overcome such limitations, we propose the Kernel Cross-View Collaborative Representation based Classification (Kernel X-CRC) that represents probe and gallery images by balancing representativeness and similarity nonlinearly. It assumes that a probe and its corresponding gallery image are represented with similar coding vectors using individuals from the training set. Experimental results demonstrate that our assumption is true when using a high-dimensional feature vector and becomes more compelling when dealing with a lowdimensional and discriminative representation computed using a common subspace learning method. We achieve state-of-the-art for rank-1 matching rates in two person reidentification datasets (PRID450S and GRID) and the second best results on VIPeR and CUHK01 datasets. | 10.1016/j.jvcir.2018.12.003 | [
"https://arxiv.org/pdf/1611.06969v1.pdf"
] | 16,603,295 | 1611.06969 | df4525d7d99f7237c864adbcb2dab30d8f7447e0 |
Kernel Cross-View Collaborative Representation based Classification for Person Re-Identification
Raphael Prates [email protected]
William Robson Schwartz [email protected]
Universidade Federal de Minas Gerais
6627Brazil
Av. Pres. Antônio Carlos -Pampulha
Belo Horizonte -MG
31270-901
Kernel Cross-View Collaborative Representation based Classification for Person Re-Identification
Person re-identification aims at the maintenance of a global identity as a person moves among non-overlapping surveillance cameras. It is a hard task due to different illumination conditions, viewpoints and the small number of annotated individuals from each pair of cameras (smallsample-size problem). Collaborative Representation based Classification (CRC) has been employed successfully to address the small-sample-size problem in computer vision. However, the original CRC formulation is not well-suited for person re-identification since it does not consider that probe and gallery samples are from different cameras. Furthermore, it is a linear model, while appearance changes caused by different camera conditions indicate a strong nonlinear transition between cameras. To overcome such limitations, we propose the Kernel Cross-View Collaborative Representation based Classification (Kernel X-CRC) that represents probe and gallery images by balancing representativeness and similarity nonlinearly. It assumes that a probe and its corresponding gallery image are represented with similar coding vectors using individuals from the training set. Experimental results demonstrate that our assumption is true when using a high-dimensional feature vector and becomes more compelling when dealing with a lowdimensional and discriminative representation computed using a common subspace learning method. We achieve state-of-the-art for rank-1 matching rates in two person reidentification datasets (PRID450S and GRID) and the second best results on VIPeR and CUHK01 datasets.
Introduction
Person re-identification (Re-ID) plays a key role in security management applications and has received increasing attention in the past years [3]. Its goal is to identify a person (probe sample), captured by one or more cameras, using a gallery of already known candidates captured from a differ-ent camera. Most of the works consider the single-shot and two surveillance cameras scenario, where a single subject image is available for each camera. The restricted number of samples and cameras makes the problem more challenging due the small-sample-size problem [56]. Despite the efforts from the computer vision community, Re-ID remains an unsolved problem due to appearance changes caused by pose, occlusion, illumination and camera transition.
Due to the lack of available samples or to the high cost for collecting and annotating a large number of images for each subject in the gallery, the small-sample-size problem also appears on the face identification task. To handle this problem, researchers have focused on Sparse Representation based Classification (SRC) [48] and Collaborative Representation based Classification (CRC) [56] by representing each probe image y using all the images in the gallery X with a proper regularization term as min α y − Xα α α 2 2 +λ α α α p ,
where λ is a scalar and α α α is the sparse (p = 1) or the collaborative (p = 2) coding vector [56,48]. These methods assign the probe image to the class that results in the smallest reconstruction error. With this approach, researchers have achieved high performance in applications such as face recognition [56,52,50,54,48], hyperspectral image classification [28] and multimodal biometrics [45]. Despite the accurate results in different computer vision problems, experimental results demonstrate that SRC and CRC classifiers achieve weak matching performance on the person re-identification problem when compared to a baseline obtained by the well-known KISSME [27] approach, as shown in Figure 1 for two widely used Re-ID datasets. While KISSME uses the available image pairs in the training set to learn a discriminative cross-view metric distance, SRC and CRC compare the probe and gallery images directly based on the reconstruction error disregarding the camera transition, which is one of the main challenges of Re-ID. Therefore, the following questions should be ad- and KISSME [27] on two widely used person Re-ID datasets (VIPeR [15] and CUHK01 [29]). The values in parenthesis indicate the rank-1 matching rate.
dressed. How to include discriminative cross-view information in a collaborative representation framework? How to indirectly compare probe and gallery images? An attempt to adapt the collaborative representation to the person Re-ID problem would be to compute the collaborative representation coefficients (coding vectors) α α α x and α α α y by solving the camera-specific optimization problems
where each linear regularized model represents the probe sample, y, or the gallery image, x, using the respective images in the training set (D x or D y ). The feature descriptors extracted from a subject in the training set captured by probe and gallery cameras are assigned to each column of D y and D x , respectively. Therefore, similar coding coefficients would be expected when x and y correspond to a same individuals acquired from different cameras. The matching between probe and gallery images occurs indirectly using the coding vectors (α α α x and α α α y ). Note that D x and D y can be any representation of the training images able to balance discriminative power and robustness to camera transition.
The limitation of Equation 2, however, is that it only considers the representativeness in the camera-specific optimization problems and disregards that elements α α α i x and α α α i y contain information about the i-th subject from training set, which we expect to be similar when x and y represents the same individual captured from different camera-views. Therefore, a better model should consider the balance between the representativeness and similarity when computing α α α x and α α α y in a unique multi-task framework. In addition, previous works demonstrated improved performance when handling the nonlinear behavior of data using kernel functions instead of the linear modeling used in Equation 2 [32,42,41].
In this work, we propose a novel method to address the person Re-ID problem using a supervised CRC framework, the Kernel Cross-View Collaborative Representation based Classification (Kernel X-CRC), inspired on Collaborative Representation based Classification (CRC) in the sense that it has a analytical solution that represents each pair probe y and gallery x images collaboratively using its camera-view specific training samples D y and D x , respectively. Differently from Equation 2, the coding vectors α α α x and α α α y are computed in a unique multi-task learning framework.
A multi-task learning framework provides evidences that related tasks transfer knowledge when learned simultaneously, improving the generalization performance [52,46,4,6]. Considering the computation of the collaborative representation coefficients in each camera-view as a different task, we estimate the coding vectors α α α x and α α α y simultaneously with a similarity term to balance the intra-camera representativeness and the inter-camera discriminative power. Furthermore, we learn α α α x and α α α y in a nonlinear feature space to be able to handle the strong nonlinearity present on the person re-identification data [32,42,41]. Therefore, the Kernel X-CRC computes a multiple regularized linear model with proper regularization of parameters for each pair of probe and gallery images, as min α α αx,α α αy
φ(y) − Φ Φ Φyα α αy 2 2 + φ(x) − Φ Φ Φxα α αx 2 2 +λ α α αy 2 2 +λ α α αx 2 2 + α α αy − α α αx 2 2 ,(3)
where φ(.) is a nonlinear function and, Φ Φ Φ x and Φ Φ Φ y are resulting nonlinear mapping of D x and D y , respectively. Despite its simplicity, the Kernel X-CRC successfully balances the representativeness and similarity to obtain non-linear and discriminative coding vectors for each pair of probe and gallery images. Then, by matching the computed coding vectors using a simple cosine distance, we obtain improved results when compared to state-of-the-art methods. According to experimental results, the proposed Kernel X-CRC outperforms state-of-the-art single-shot based methods in the smaller datasets evaluated (PRID450S and GRID), where the small-sample-size problem is more critical, and is also successful in the larger datasets, holding the second best performance in VIPeR and CUHK01 datasets.
Contributions. The main contributions of this work are the following. We propose a novel approach to the Re-ID problem using a Kernel Cross-View Collaborative Representation based Classification (Kernel X-CRC) that embodies cross-view discriminative information and models the nonlinear behavior of person re-identification data. Furthermore, we present an efficient analytical solutions to Kernel X-CRC with outperforming results when compared with more complex state-of-the-art approaches at four challenging datasets (VIPeR, PRID450S, CUHK01 and GRID).
To the best of our knowledge, this is the first work addressing the person re-identification problem as a multi-task collaborative representation problem. Furthermore, it is important to emphasize that, even though employed to Re-ID problem, the proposed approach provides a general framework that does not consider any extra information at testing stage and could also be employed to other computer vision problems, such as face recognition, hyperspectral image classification and multi-modal biometrics.
Related Work
Different feature descriptors have been proposed for the Re-ID problem by exploiting feature representation and body parts from where they are extracted and matched [37,14,11,5,58,33,36,51,30,44,49,39,7,9]. Regarding the body locations, Farenzena et al. [14] use human symmetry to determine discriminative body locations, while body parts are detected using Pictorial Structures in [11]. Similarly, Cai and Pietikainen [5] extract descriptors from fixed log-polar grids and Liao et al. [30] construct a stable representation using local patches obtained in multiple scales. A different approach constraints the inter-camera matching computing patch saliency information [58] or capturing spatial distribution of patches between cameras [7].
With respect to the feature representation, some works capture information using Fisher Vectors [37], histograms of semantic color names [51], represent local patches with hierarchical Gaussian distribution [39] and design biologically inspired features and covariance descriptors [36]. To handle the camera transition, Chen et al. [9] propose to align feature distribution across disjoint using the Mirror Representation [9], while the feature importance can be com-puted learning a fixed statistical model [44] or adaptively from subsets of similar individuals and random forests [34]. More recently, Wu et al. [49] combine hand-crafted and deep learning-based feature descriptors to obtain a discriminative deep feature representation. In this work, we assume that it is not possible to properly handle the camera transition by directly matching feature descriptors captured by different cameras. Therefore, we use a supervised nonlinear framework to compute coding vectors that capture cross-view discriminative information to perform an indirect matching between probe and gallery images.
In supervised learning-based approaches, feature descriptors are combined with discriminative models learned using labeled images from camera pairs to obtain higher matching performance. For instance, distance metric learning-based approaches use the pairwise constraint to learn a distance function that is smaller between pairs of the same person and larger otherwise [27,59,43,21,20,19,18,17]. As example, WARCA [21] learns a Mahalanobis distance in a low-dimensional subspace computed using orthonormal regularizer, while NLML [20] learns multiple sets of nonlinear transformations using feed-forward neural network and large margin optimization. Differently, Cheng et al. [10] use a triplet loss function that keeps closer instances of the same person in features space learned using multi-channel CNN. Zhang et al. [57] present a distinct approach that relates the model parameters and the feature space using semi-coupled dictionary learning to obtain a model specific for each individual representation. Similarly, in this work we compute models specific for each pair of probe and gallery images. However, we do not assume that it is possible to learn an effective mapping function between feature space and parameters space using a reduced number of training samples.
Subspace learning methods have been widely employed by supervised Re-ID approaches [2,1,32,12,30,42,41,55]. An et al. [1] use Canonical Correlation Analysis (CCA) to learn projections to a latent space where features from different cameras are correlated. Similarly, in [2], the authors address the small-sample-size problem using shrinkage and smoothing techniques to better estimate the covariance matrices and Zhang et al. [55] collapse images of the same person in a single point in a discriminative null space. Prates and Schwartz [12] adapt Partial Least Squares (PLS) to a supervised Re-ID setting using prototypes to indirectly deal with camera transition. To tackle the nonlinearity of the data, Lisanti et al. [32] propose a kernel descriptor to encode person appearance and project the data into common subspace using Kernel Canonical Correlation Analysis (KCCA). Similarly, Kernel PLS [41] and Kernel HPCA [42] have been used to nonlinearly map data into a common subspace. As in the nonlinear subspace learning approaches, this work also represents probe and gallery images nonlin-early by using a set of basis vectors. Differently, our basis vectors are composed of feature representation of each training samples. In fact, this feature representation can be obtained directly from feature descriptor or improved using projections to low-dimensional common subspace.
Some works investigate the person re-identification problem using sparse or collaborative representations [22,31,16,24,25,26,47,53,23]. Lisanti et al. [31] propose an Iterative Sparse Ranking (ISR) method that iteratively applies SRC with adaptive weighting strategies until ranking all the gallery images. In [47,53,23], the authors use CRC in the unsupervised multi-shot Re-ID scenario to compute the distance between probe and gallery images efficiently using both coding residuals and coefficients. Karanam et al. [22] explore the block structure in sparse coefficients to rank gallery images based on the reconstruction error. In [16,26], the authors exploit dictionary learning and sparse coding in a unique framework. Differently, in [24,25], the authors propose a local sparse representation method that uses SRC to represent interest points. Kernel X-CRC has some key advantages when compared to these methods. For instance, Kernel X-CRC is a general method that does not assume a block structure in the coefficients representation as occurs in [22]. Differently from dictionary learning-based approaches [16,26], this work represents probe and gallery images using training samples. More importantly, different from previous works [31,22,16,26], we efficiently model the strong nonlinear transition of features between cameras achieving an analytical solution.
Recently, some methods have employed multi-task learning in person re-identification [46,35,38]. Ma et al. [35] approach the person re-identification by transferring knowledge between the source domain with labeled image pairs and the target domain with unlabeled data using multi-task support vector ranking. In [38], the authors avoid the over-fitting problem learning multiple Mahalanobis distance metrics in a multi-task framework. Su et al. [46] exploit the correlation between low-level features and attributes using a multi-task learning framework with low ranking embedding. The proposed Kernel X-CRC is fundamentally different from previous works because we employ multi-task to learn related collaborative coefficients from multiple regularized linear problems.
The method proposed in this work is related to joint sparse or collaborative representation methods. Such methods have been applied in multi-view face recognition [54,50], hyperspectral image classification [28] and multi-modal biometrics recognition [45]. For instance, the works [28,52,50] use sparsity information to combine complementary features for classification. Differently, we are dealing with a single feature modality and the tasks correspond to the different camera-views. Shekhar et al. [45] consider the correlation between multiple biometric infor-mation using sparse representation, while Zhang et al. [54] employ a joint dynamic sparse representation to exploit the correlation between multiple views of same face image. However, they consider that testing and training images are available for different tasks (e.g., multiple views or modalities). Differently, Re-ID problem aims at predicting the subjects appearance at the target task (i.e., gallery camera) using the image from the source task (i.e., probe camera).
Proposed Approach
In the proposed Kernel X-CRC, we use labeled training images at cameras A and B as columns of matrices D y and D x , respectively (i.e., features belonging to the same subject are in corresponding columns in both matrix representations). Thus, these matrices encode the cross-view discriminative information that reflects in the learned collaborative representation coefficients (α α α x and α α α y ). For instance, when we describe two images of the same person captured by cameras A and B using D y and D x , it is expected the representation coefficients to be more similar than when describing two different subjects. Therefore, we propose to use the similarity between these coefficients to indirectly compute the similarity between probe y and the gallery-set X.
We use the following notation in the description. Bold lower-case letters denote column vectors and bold uppercase letters denote matrices (e.g., a and A, respectively). In this work, we deal with the single-shot scenario (i.e., there exist only one image taken from camera view A and one image taken from camera view B). We represent the ith image from camera A and B, as y i and x i ∈ R m , respectively, where m denotes the dimension of the feature space. Without loss of generality, we assume that l testing images from camera A constitute the probe set Y ∈ R m×l and l testing images from camera B represent the gallery set X ∈ R m×l . Similarly, the set of all n training images from camera A and B compose the matrices D y and D x ∈ R m×n , respectively. The collaborative representation coefficients are denoted by α α α y , α α α x , where α α α ∈ R n , and λ ∈ R is a scalar. We use φ(.) to denote a nonlinear mapping function of input variables to a feature space F, i.e, φ : x i ∈ R m → φ(x i ) ∈ F and, Φ Φ Φ x and Φ Φ Φ y are the resulting matrices after nonlinearly mapping D x and D y , respectively. In the following equations, we use the notation I to indicate the identity matrix.
Considering as related tasks the representation of probe and gallery images using training images from their respective cameras, we propose to simultaneously estimate α α α x and α α α y in a multi-task collaborative representation framework. Thus, we aim at estimating the most similar coding vectors α α α x and α α α y that simultaneously describe probe and gallery subjects. To compute these coding vectors, we introduce a similarity term α α α x −α α α y 2 2 in our multi-task formulation that balances representativeness and similarity resulting in the following optimization problem min α α αx,α α αy
φ(y) − Φ Φ Φyα α αy 2 2 + φ(x) − Φ Φ Φxα α αx 2 2
+λ α α αy 2 2 +λ α α αx 2 2 + α α αy − α α αx 2 2 ,
that we analytically derived with respect to α α α y and α α α x obtaining α α α y = P −1
y α α α x + P −1 y Φ Φ Φ y φ(y)(5)
and
α α α x = P −1 x α α α y + P −1 x Φ Φ Φ x φ(x),(6)
where projections matrices P y and P x are given by
P y = Φ Φ Φ y Φ Φ Φ y + λI and P x = Φ Φ Φ x Φ Φ Φ x + λI.(7)
Note that Equations 5 and 6 are interdependent. Therefore, replacing α α α x for its corresponding equation (Eq. 6) and isolating α α α y , we obtain
α α α y = Q −1 P −1 y P −1 x Φ Φ Φ x φ(x) + Q −1 P −1 y Φ Φ Φ y φ(y) (8)
with projection matrix Q corresponding to
Q = I − P −1 y P −1 x .(9)
Similarly, we can compute the coding vector α α α x as
α α α x = W −1 P −1 x P −1 y Φ Φ Φ y φ(y) + W −1 P −1 x Φ Φ Φ x φ(x) (10)
with W computed as
W = I − P −1 x P −1 y .(11)
To avoid explicitly mapping of data to a highdimensional space, we can use the "kernel trick" substituting cross-product by K = Φ Φ ΦΦ Φ Φ , where K ∈ R n×n is the kernel Gram matrix. Particularly, we define the kernel Gram matrices K x and K y ∈ R n×n to represent the cross-product Φ Φ Φ x Φ Φ Φ x and Φ Φ Φ y Φ Φ Φ y , respectively. Furthermore, we define Φ Φ Φ x φ(x) as the computation of kernel function between x and all vectorsx ∈ Φ Φ Φ x . Identically, Φ Φ Φ y φ(y) denotes the kernel function applied in y and all vectorŝ y ∈ Φ Φ Φ y . Then, the similarity between a pair of probe y and gallery x is computed by the similarity between α α α x and α α α y , as described in Algorithm 1.
Due the multi-task learning framework, a pair of probe (y) and gallery images (x) will compute α α α y and α α α x that balances the representativeness in each camera with the similarity between coding vectors. This balance will only result in a similar coding vector if x corresponds to the respective gallery image of y. For instance, if x is dissimilar when compared to y, similar coding vectors will not be obtained since they should result in poor representativeness (i.e., high reconstruction error) in both cameras.
Algorithm 1: Kernel Cross-View Collaborative Representation based Classification (Kernel X-CRC).
input : Kernel matrices (K x and K y ) output: Ranking list of gallery images R
Compute P x and P y matrices using Equation 7
Compute Q and W using Equations 9 and 11 Pre-compute:
β β β x x ← W −1 P −1 x β β β y x ← W −1 P −1 x P −1 y β β β y y ← Q −1 P −1 y β β β x y ← Q −1 P −1 y P −1 x for y j ∈ Y do for x i ∈ X do α α α x ← β β β x x Φ Φ Φ x φ(x i ) + β β β y x Φ Φ Φ y φ(y j ) α α α y ← β β β x y Φ Φ Φ x φ(x i ) + β β β y y Φ Φ Φ y φ(y j ) sim(i) ← α α α x α α αy α α αx α α αy end R j ← sort(sim, descend) end return R
Experimental Results
In this section, we perform a comprehensive evaluation of the proposed Kernel X-CRC assessing the effect of different strategies in the experimental results (Section 4.1) and providing a broad comparison with other approaches in the state-of-the-art in four datasets (Section 4.2).
Datasets. To perform our experiments, we consider four challenging datasets. The PRID 450S Dataset 1 [43] consists of 450 images pairs of pedestrians captured by two nonoverlapping cameras. The main challenges are related to changes in viewpoint, pose as well as significant differences in background and illumination. The VIPeR Dataset 2 [15] contains 632 labelled image pairs captured by two different outdoor cameras located in an academic environment. Each subject appears once in each camera and most of the image pairs show viewpoint change larger than 90 degrees, making it a very challenging dataset. The CUHK01 Dataset 3 [29] captures two disjoint camera-view images for each person in a campus environment, containing 971 persons, each of which has two images from each camera-view (all the images are normalized to 160×60 pixels for evaluations). The GRID dataset 4 [17] contains 250 image pairs (single-shot) captured by eight disjoint surveillance cameras in a busy underground station generating different poses and poor il-lumination conditions. In addition, different from the other datasets evaluated, it introduces 775 individuals in galleryset without correct matching in the probe-set (distractors) that drastically impacts in the results.
Experimental Setup. As in the majority of the works, we randomly partition the datasets into training and testing subsets with an equal number of individuals. However, due to the odd number of subjects, in CUHK01, we split the 971 individuals into 485 persons for training and the remaining 486 for testing. Furthermore, we also adapt CUHK01 to the single-shot scenario by randomly selecting one image of the same person in each camera, similarly to [39].
To set λ, the only parameter of the proposed Kernel X-CRC, we use a single partition for each dataset. It differs from the multiple partitions commonly used in literature and avoids overfitting the parameter to the data. We also use this single partition to define exponential χ 2 and RBF as kernel functions employed when dealing with the lowdimensional and original descriptors, respectively.
We report the average of results obtained from 10 trials, a common procedure to achieve more stable results. The results are reported using the rank-k matching rate, which consists on the percentage of individuals correctly identified when considering the top-k ranking positions, a widely employed metric to compare Re-ID approaches. We present the evaluated approaches in tables using an ascending order of reported rank-1 matching rate.
Kernel X-CRC Evaluation
In this section, we use the VIPeR dataset to evaluate the performance of Kernel X-CRC according to different aspects: feature descriptors in the original space, performance when operating in a common subspace, contribution of Kernel X-CRC when computed in the common subspace, and the impact of the different choices that resulted in the proposed Kernel X-CRC.
Feature Descriptor Evaluation. This experiment assesses the performance of Kernel X-CRC using feature descriptors widely employed in the literature. We used the descriptors proposed by Zheng et al. [59] and Koestinger et al. [27] that are simple combination of color histograms and texture information extracted from local patches. Due to their limited capability of dealing with the camera transition problem, we also considered descriptors that better handle this problem, such as LOMO [30], WHOS [32], GoG [39] and LOMO+CNN [49]. Table 1 shows that the Kernel X-CRC is able to obtain more accurate results when using a better feature representation. For instance, Kernel X-CRC reached 45.5% of rank-1 using LOMO+CNN [49], which is comparable with state-of-the-art approaches. Due to the superior performance of LOMO [30], WHOS [32], GoG [39] and LOMO+CNN [49], we will focus on these feature descriptors in the following experiments. Subspace Evaluation. Based on the results obtained using different feature descriptors (see Table 1), one might hypothesize that better results can be achieved by improving the feature representation. A straightforward approach for achieving a better feature representation consists in concatenating complementary feature descriptors. However, as we solve regularized linear models for each pair of probe and gallery images, it will result in a prohibitive computational cost. An alternative is to compute a low-dimensional subspace that maintains the computational cost acceptable yet improves the results. Therefore, we project the data onto a common subspace that handles the camera transition problem before matching the probe with gallery images using the proposed Kernel X-CRC. Table 2 presents the experimental results using Kernel X-CRC in a low-dimensional feature representation computed using XQDA [30]. The results were improved for all feature descriptors when compared to those shown in Table 1. It is important to highlight that we employed XQDA due to outperforming results reported in literature yet, other methods to estimate common subspaces could be employed, instead. Metric Function Evaluation. According to the previous experiment, the employment of Kernel X-CRC in a lowdimensional feature space learned using XQDA improved the results. However, it is difficult to define whether the improvement gain are due to Kernel X-CRC or to the better representation learned using XQDA. Therefore, to highlight the contribution of Kernel X-CRC, we compare the proposed method with traditional metric functions to match probe and gallery images in the learned common subspace. Specifically, we compare Kernel X-CRC with cosine and Mahalanobis distances and KISSME metric [27]. According to Table 3, even though reasonable results were achieved in the learned subspace when using the tra-Function Viper (p=316) WHOS [32] LOMO [30] GoG [ ditional metric functions (e.g., cosine and Mahalanobis distance), the Kernel X-CRC achieved greater improvements (5.3 percentage points, on average), when compared to the metric function with the second highest rank-1 matching rate. We attribute this performance gain to the computation of specific coding vectors for each pair of probe and gallery images using a nonlinear model. On the other hand, cosine and Mahalanobis are fixed distances, and the KISSME [27] is a global distance learned using the entire training set. Due the improved results, the remaining experiments will consider the GoG [39] descriptor in the lowdimensional representation computed using XQDA.
Baseline Approaches. This experiment analyzes the impact of different choices that resulted in the Kernel X-CRC model. We evaluate unsupervised methods (SRC [48] and CRC [48]) and supervised methods with and without considering multi-task and kernel extensions. We also compare Kernel X-CRC with the model presented in Equation 2, which we named Cross-View Collaborative Representation classification (C 2 RC), and with a straightforward nonlinear extension of C 2 RC, referred to as Kernel C 2 RC.
According to Table 4, SRC [48] and CRC [48] achieved the lowest results as they directly match probe and gallery images based on the reconstruction error without considering the training samples. Differently, C 2 RC and Kernel C 2 RC improved the results for all ranking positions by using the training samples to relate features with coding vectors that are employed to indirectly match probe and gallery images. Furthermore, due to the nonlinear modeling, the Kernel C 2 RC reached results even better than its linear counterpart, the C 2 RC. The employment of the X-CRC and Kernel X-CRC models achieved improved results when compared to their counterparts C 2 RC and Kernel C 2 RC, respectively. We attribute this gain to the multi-task learning framework that forces the coding vectors to be simultaneously representative in each camera and similar between different cameras.
State-of-the-art Comparisons
In this section, we compare the proposed approach with a large number of state-of-the-art methods on the VIPeR, PRID450S, CUHK01 and GRID datasets.
VIPeR dataset. Table 5 presents the matching rates for different methods, including approaches based on metric learning [20,21,7], common subspace learning [41,32,42,30,55,9,39,13] and deep learning [10,8,49]. According to the results, the proposed method greatly outperforms most of the approaches. For instance, the Kernel X-CRC reached 51.6% of rank-1, while Wu et al. [49] achieved 51.1% combining LOMO and feature descriptors based on deep learning architectures. We believe that our simple approach can outperform more complex methods (e.g., deep learning based approaches) because it learns a coding representation specific for a pair of probe and gallery images, while other methods attempt to learn a matching model considering a small training set, which is prone to overfitting. Similarly to our work, Zhang et al. [57] also employs a specific model for each pair of probe and gallery images. However, their models are obtained using a mapping function to relate feature descriptors to model parameters, which is very challenging for small datasets such as VIPeR.
Method
Viper (p=316) r = 1 r = 5 r = 10 r = 20 r = 30 Prates and Schwartz [13] 32 To the best of our knowledge, SCSP [7] is the unique method with improved results when compared to the proposed Kernel X-CRC. We credit the better results to the combination of global and local matching models. However, learning how to constraint the matching of local regions in images obtained from different camera-views is a very challenge task, mainly in more realistic datasets.
PRID450S dataset. According to the results shown in Table 6 and similarly to the results in VIPeR dataset (Table 5), we achieved improved results when compared to dis-tance metric learning [21], subspace learning [9,30,41,42,39,13], deep learning [49] and other approaches. For instance, using the Kernel X-CRC, we reached higher matching rates for all ranking positions when compared to GoG + XQDA [39]. That can be explained by the use of the same low-dimensional representation in the common subspace to nonlinearly compute coding vectors specific to each pair of probe and gallery images, while they compute a simple cosine distance. To the best of our knowledge, we achieved the best reported results in literature for PRID450S. We attribute the improvement to the better representation obtained using GoG descriptor with XQDA [39] and the nonlinear computation of coding vectors by the Kernel X-CRC. CUHK01 dataset. Table 7 presents the matching rates for different ranking positions of state-of-the-art methods that addressed person ReID on the CUHK01 dataset using the single-shot setting. Notice that results from LOMO + XQDA [30] and Zhang et al. [57] are not included since they focus on the multi-shot scenario. According to the results, the proposed Kernel X-CRC reaches better results than most of the metric learning and subspace learning approaches. As matter of fact, we reached the second best rank-1 matching rate reported in literature, only surpassed by WARCA [21], which is a nonlinear metric model. Regarding the higher ranking positions (e.g., r=10, 20 and 30), the best results are achieved using Multi-CNN [10], which consists in multiple deep learning architectures to combine local and global features. It is important to highlight that both methods WARCA and MultiCNN are very sensitive to the number of training samples available, which justifies the reduced matching rates reported in small datasets. Differently, the Kernel X-CRC remains within the top two best rank-1 approaches for all evaluated datasets.
GRID dataset. Since it is a very challenging dataset that considers real-world problems (e.g., distractors), there are few works with reported experimental results on GRID dataset. representation than the LOMO descriptors. For instance, when comparing both in the subspace learned using XQDA, GoG achieves an improvement of more than 8.0 percentage points for rank-1 matching rate. Furthermore, GoG + XQDA [39] holds higher matching rates than different approaches in literature based on metric learning [20], local and global matching [7] and the sample-specific matching models [57]. More importantly, Kernel X-CRC presents the highest rank-1 matching rate when compared to these approaches, demonstrating the advantage of the nonlinear matching and its robustness to distractors.
Method GRID (p=125) r = 1 r = 5 r = 10 r = 20 r = 30 LOMO + XQDA [30] 16
Conclusions
In this work, we tackled the person re-identification problem using the proposed Kernel Cross-View Collaborative Representation based Classification (Kernel X-CRC) approach. Kernel X-CRC represents probe and gallery images using collaborative representation coefficients that are robust to small-sample-size and balance the intra-camera representativeness with the inter-camera similarity. We performed an extensive experimental evaluation showing that the Kernel X-CRC successfully combines nonlinear modeling and multi-task learning. We also observed that working in a discriminative and low-dimensional subspace, the proposed method reaches outperforming results, obtaining the best rank-1 matching rates for the two smaller datasets evaluated (PRID450S and GRID) and remaining within the two best approaches on the VIPeR and CUHK01 datasets.
Figure 1 :
1Comparison between Sparse Representation based Classification (SRC), Collaborative Representation based Classification (CRC)
Table 2 :
2Subspace evaluation on the VIPeR dataset.
Table 3 :
3Metric functions evaluation on the VIPeR dataset.
Table 4 :
4Results of the baseline approaches on the VIPeR dataset.
Table 5 :
5Top ranked approaches on the VIPer dataset.
Table 6 :
6Top ranked approaches on the PRID450S dataset.
Table 8
8presents these approaches and their respective matching rates for different ranking positions. Based on the results, we observe that GoG captures a better featureMethod
CUHK01 (p=485)
r = 1
r = 5
r = 10
r = 20
r = 30
KPLS ModeA
38.3
66.8
77.7
86.8
90.5
Mirror + KMFA [9]
40.4
64.6
75.3
84.1
-
Kernel HPCA
44.3
73.2
82.7
90.1
93.4
X-KPLS
46.2
74.0
84.3
91.3
94.0
KISSME
49.6
74.7
83.8
91.2
94.3
Sakrapee et al. [40]
53.4
76.4
84.4
90.5
-
MultiCNN [10]
53.7
84.3
91.0
96.3
98.3
GoG + XQDA [39]
57.8
79.1
86.2
92.1
-
WARCA [21]
65.6
85.3
90.5
95.0
-
Kernel X-CRC
61.2
80.9
87.3
93.2
95.6
Table 7 :
7Top ranked approaches on the CUHK01 dataset.
Table 8 :
8Top ranked approaches on the GRID dataset.
Available at: https://lrs.icg.tugraz.at/download.php 2 Available at: https://vision.soe.ucsc.edu/projects 3 Available at:http://www.ee.cuhk.edu.hk/rzhao/ 4 Available at: http://personal.ie.cuhk.edu.hk/˜ccloy
Reference-based person re-identification. L An, M Kafai, S Yang, B Bhanu, Advanced Video and Signal Based Surveillance (AVSS). 10th IEEE International Conference onL. An, M. Kafai, S. Yang, and B. Bhanu. Reference-based person re-identification. In Advanced Video and Signal Based Surveillance (AVSS), 2013 10th IEEE International Conference on, pages 244-249, Aug 2013. 3
Person re-identification by robust canonical correlation analysis. L An, S Yang, B Bhanu, Signal Processing Letters. 228IEEEL. An, S. Yang, and B. Bhanu. Person re-identification by robust canonical correlation analysis. Signal Processing Let- ters, IEEE, 22(8):1103-1107, Aug 2015. 3
A survey of approaches and trends in person re-identification. Image and Vision Computing. B.-G Apurva, S K Shah, 32B.-G. Apurva and S. K. Shah. A survey of approaches and trends in person re-identification. Image and Vision Comput- ing, 32(4):270 -286, 2014. 1
Convex multi-task feature learning. A Argyriou, T Evgeniou, M Pontil, Machine Learning. 73A. Argyriou, T. Evgeniou, and M. Pontil. Convex multi-task feature learning. Machine Learning, 73(3):243-272, 2008. 2
Person re-identification based on global color context. Y Cai, M Pietikinen, Computer Vision ACCV 2010 Workshops. Berlin HeidelbergSpringer6468Y. Cai and M. Pietikinen. Person re-identification based on global color context. In Computer Vision ACCV 2010 Work- shops, volume 6468 of Lecture Notes in Computer Science, pages 205-215. Springer Berlin Heidelberg, 2011. 3
Multitask learning. Machine learning. R Caruana, 28R. Caruana. Multitask learning. Machine learning, 28(1):41-75, 1997. 2
Similarity learning with spatial constraints for person re-identification. D Chen, Z Yuan, B Chen, N Zheng, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition7D. Chen, Z. Yuan, B. Chen, and N. Zheng. Similarity learn- ing with spatial constraints for person re-identification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1268-1277, 2016. 3, 7, 8
Deep ranking for person re-identification via joint representation learning. S.-Z Chen, C.-C Guo, J.-H Lai, IEEE Transactions on Image Processing. 255S.-Z. Chen, C.-C. Guo, and J.-H. Lai. Deep ranking for per- son re-identification via joint representation learning. IEEE Transactions on Image Processing, 25(5):2353-2367, 2016. 7
Mirror representation for modeling view-specific transform in person reidentification. Y.-C Chen, W.-S Zheng, J Lai, International Conference on Artificial Intelligence. AAAI Press7Y.-C. Chen, W.-S. Zheng, and J. Lai. Mirror represen- tation for modeling view-specific transform in person re- identification. In International Conference on Artificial In- telligence, pages 3402-3408. AAAI Press, 2015. 3, 7, 8
Person re-identification by multi-channel parts-based cnn with improved triplet loss function. D Cheng, Y Gong, S Zhou, J Wang, N Zheng, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition7D. Cheng, Y. Gong, S. Zhou, J. Wang, and N. Zheng. Person re-identification by multi-channel parts-based cnn with im- proved triplet loss function. In Proceedings of the IEEE Con- ference on Computer Vision and Pattern Recognition, pages 1335-1344, 2016. 3, 7, 8
Custom pictorial structures for re-identification. D S Cheng, M Cristani, M Stoppa, L Bazzani, V Murino, British Machine Vision Conference (BMVC). D. S. Cheng, M. Cristani, M. Stoppa, L. Bazzani, and V. Murino. Custom pictorial structures for re-identification. In British Machine Vision Conference (BMVC), 2011. 3
Appearancebased person re-identification by intra-camera discriminative models and rank aggregation. R F De Carvalho Prates, W R Schwartz, International Conference on Biometrics. Phuket,ThailandR. F. de Carvalho Prates and W. R. Schwartz. Appearance- based person re-identification by intra-camera discriminative models and rank aggregation. In International Conference on Biometrics, ICB 2015, Phuket,Thailand, 19-22 May, 2015, pages 65-72, 2015. 3
Appearancebased person re-identification by intra-camera discriminative models and rank aggregation. R F De Carvalho Prates, W R Schwartz, International Conference on Biometrics. Phuket, Thailand7R. F. de Carvalho Prates and W. R. Schwartz. Appearance- based person re-identification by intra-camera discriminative models and rank aggregation. In International Conference on Biometrics, ICB 2015, Phuket, Thailand, 19-22 May, 2015, pages 65-72, 2015. 7, 8
Person re-identification by symmetry-driven accumulation of local features. M Farenzena, L Bazzani, A Perina, V Murino, M Cristani, IEEE Conference on Computer Vision and Pattern Recognition (CVPR). 3M. Farenzena, L. Bazzani, A. Perina, V. Murino, and M. Cristani. Person re-identification by symmetry-driven ac- cumulation of local features. In IEEE Conference on Com- puter Vision and Pattern Recognition (CVPR), pages 2360 -2367, June 2010. 3
Evaluating appearance models for recognition, reacquisition, and tracking. D Gray, S Brennan, H Tao, 10th IEEE International Workshop on Performance Evaluation of Tracking and Surveillance (PETS). 25D. Gray, S. Brennan, and H. Tao. Evaluating appearance models for recognition, reacquisition, and tracking. In 10th IEEE International Workshop on Performance Evaluation of Tracking and Surveillance (PETS), 2007. 2, 5
Sparse coding and dictionary learning for symmetric positive definite matrices: A kernel approach. M T Harandi, C Sanderson, R Hartley, B C Lovell, Computer Vision-ECCV 2012. SpringerM. T. Harandi, C. Sanderson, R. Hartley, and B. C. Lovell. Sparse coding and dictionary learning for symmetric positive definite matrices: A kernel approach. In Computer Vision- ECCV 2012, pages 216-229. Springer, 2012. 4
Person re-identification by descriptive and discriminative classification. M Hirzer, C Beleznai, P M Roth, H Bischof, Scandinavian conference on Image analysis. Springer35M. Hirzer, C. Beleznai, P. M. Roth, and H. Bischof. Person re-identification by descriptive and discriminative classifica- tion. In Scandinavian conference on Image analysis, pages 91-102. Springer, 2011. 3, 5
Person reidentification by efficient impostor-based metric learning. M Hirzer, P M Roth, H Bischof, Advanced Video and Signal-Based Surveillance (AVSS). IEEEIEEE Ninth International Conference onM. Hirzer, P. M. Roth, and H. Bischof. Person re- identification by efficient impostor-based metric learning. In Advanced Video and Signal-Based Surveillance (AVSS), 2012 IEEE Ninth International Conference on, pages 203- 208. IEEE, 2012. 3
Relaxed pairwise learned metric for person re-identification. M Hirzer, P M Roth, M Köstinger, H Bischof, European Conference on Computer Vision. SpringerM. Hirzer, P. M. Roth, M. Köstinger, and H. Bischof. Re- laxed pairwise learned metric for person re-identification. In European Conference on Computer Vision, pages 780-793. Springer, 2012. 3
Nonlinear local metric learning for person re-identification. S Huang, J Lu, J Zhou, A K Jain, arXiv:1511.051697arXiv preprintS. Huang, J. Lu, J. Zhou, and A. K. Jain. Nonlinear local metric learning for person re-identification. arXiv preprint arXiv:1511.05169, 2015. 3, 7, 8
Scalable metric learning via weighted approximate rank component analysis. C Jose, F Fleuret, arXiv:1603.003707arXiv preprintC. Jose and F. Fleuret. Scalable metric learning via weighted approximate rank component analysis. arXiv preprint arXiv:1603.00370, 2016. 3, 7, 8
Sparse re-id: Block sparsity for person re-identification. S Karanam, Y Li, R Radke, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops. the IEEE Conference on Computer Vision and Pattern Recognition WorkshopsS. Karanam, Y. Li, and R. Radke. Sparse re-id: Block sparsity for person re-identification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recogni- tion Workshops, pages 33-40, 2015. 4
Person re-identification for real-world surveillance systems. F M Khan, F Brémond, arXiv:1607.05975arXiv preprintF. M. Khan and F. Brémond. Person re-identification for real-world surveillance systems. arXiv preprint arXiv:1607.05975, 2016. 4
Local sparse representation based interest point matching for person reidentification. M I Khedher, M A El Yacoubi, Neural Information Processing. SpringerM. I. Khedher and M. A. El Yacoubi. Local sparse rep- resentation based interest point matching for person re- identification. In Neural Information Processing, pages 241- 250. Springer, 2015. 4
Multi-shot surf-based person re-identification via sparse representation. M I Khedher, M A El Yacoubi, B Dorizzi, Advanced Video and Signal Based Surveillance (AVSS). IEEE10th IEEE International Conference onM. I. Khedher, M. A. El Yacoubi, and B. Dorizzi. Multi-shot surf-based person re-identification via sparse representation. In Advanced Video and Signal Based Surveillance (AVSS), 2013 10th IEEE International Conference on, pages 159- 164. IEEE, 2013. 4
Dictionary learning with iterative laplacian regularisation for unsupervised person reidentification. E Kodirov, T Xiang, S Gong, Proceedings of the British Machine Vision Conference (BMVC). the British Machine Vision Conference (BMVC)BMVA Press12E. Kodirov, T. Xiang, and S. Gong. Dictionary learning with iterative laplacian regularisation for unsupervised person re- identification. In Proceedings of the British Machine Vi- sion Conference (BMVC), pages 44.1-44.12. BMVA Press, September 2015. 4
Large scale metric learning from equivalence constraints. M Koestinger, M Hirzer, P Wohlhart, P M Roth, H Bischof, M. Koestinger, M. Hirzer, P. Wohlhart, P. M. Roth, and H. Bischof. Large scale metric learning from equivalence constraints. 2012. 1, 2, 3, 6, 7
Joint collaborative representation with multitask learning for hyperspectral image classification. Geoscience and Remote Sensing. J Li, H Zhang, L Zhang, X Huang, L Zhang, IEEE Transactions on. 5294J. Li, H. Zhang, L. Zhang, X. Huang, and L. Zhang. Joint col- laborative representation with multitask learning for hyper- spectral image classification. Geoscience and Remote Sens- ing, IEEE Transactions on, 52(9):5923-5936, 2014. 1, 4
Locally aligned feature transforms across views. W Li, X Wang, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition5W. Li and X. Wang. Locally aligned feature transforms across views. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3594- 3601, 2013. 2, 5
Person re-identification by local maximal occurrence representation and metric learning. S Liao, Y Hu, X Zhu, S Z Li, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition7S. Liao, Y. Hu, X. Zhu, and S. Z. Li. Person re-identification by local maximal occurrence representation and metric learning. In Proceedings of the IEEE Conference on Com- puter Vision and Pattern Recognition, pages 2197-2206, 2015. 3, 6, 7, 8
Person re-identification by iterative re-weighted sparse ranking. Pattern Analysis and Machine Intelligence. G Lisanti, I Masi, A Bagdanov, A Del Bimbo, IEEE Transactions on. 378G. Lisanti, I. Masi, A. Bagdanov, and A. Del Bimbo. Person re-identification by iterative re-weighted sparse ranking. Pat- tern Analysis and Machine Intelligence, IEEE Transactions on, 37(8):1629-1642, Aug 2015. 4
Matching people across camera views using kernel canonical correlation analysis. G Lisanti, I Masi, A Del Bimbo, Proceedings of the International Conference on Distributed Smart Cameras, ICDSC '14. the International Conference on Distributed Smart Cameras, ICDSC '14New York, NY, USAACM107G. Lisanti, I. Masi, and A. Del Bimbo. Matching people across camera views using kernel canonical correlation anal- ysis. In Proceedings of the International Conference on Dis- tributed Smart Cameras, ICDSC '14, pages 10:1-10:6, New York, NY, USA, 2014. ACM. 2, 3, 6, 7
On-the-fly feature importance mining for person re-identification. C Liu, S Gong, C C Loy, Pattern Recognition. 474C. Liu, S. Gong, and C. C. Loy. On-the-fly feature impor- tance mining for person re-identification. Pattern Recogni- tion, 47(4):1602 -1615, 2014. 3
Person reidentification: What features are important?. C Liu, S Gong, C C Loy, X Lin, Computer Vision-ECCV 2012. Workshops and Demonstrations. SpringerC. Liu, S. Gong, C. C. Loy, and X. Lin. Person re- identification: What features are important? In Computer Vision-ECCV 2012. Workshops and Demonstrations, pages 391-401. Springer, 2012. 3
Domain transfer support vector ranking for person re-identification without target camera label information. A Ma, P Yuen, J Li, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionA. Ma, P. Yuen, and J. Li. Domain transfer support vector ranking for person re-identification without target camera la- bel information. In Proceedings of the IEEE International Conference on Computer Vision, pages 3567-3574, 2013. 4
Bicov: a novel image representation for person re-identification and face verification. B Ma, Y Su, F Jurie, Proceedings of the British Machine Vision Conference. the British Machine Vision ConferenceBMVA Press11B. Ma, Y. Su, and F. Jurie. Bicov: a novel image represen- tation for person re-identification and face verification. In Proceedings of the British Machine Vision Conference, pages 57.1-57.11. BMVA Press, 2012. 3
Local descriptors encoded by fisher vectors for person re-identification. B Ma, Y Su, F Jurie, Computer Vision ECCV 2012. Workshops and Demonstrations. A. Fusiello, V. Murino, and R. Cucchiara7583B. Ma, Y. Su, and F. Jurie. Local descriptors encoded by fisher vectors for person re-identification. In A. Fusiello, V. Murino, and R. Cucchiara, editors, Computer Vision ECCV 2012. Workshops and Demonstrations, volume 7583 of Lecture Notes in Computer Science, pages 413-422.
Person re-identification over camera networks using multi-task distance metric learning. Image Processing. L Ma, X Yang, D Tao, IEEE Transactions on. 238L. Ma, X. Yang, and D. Tao. Person re-identification over camera networks using multi-task distance metric learning. Image Processing, IEEE Transactions on, 23(8):3656-3670, 2014. 4
Hierarchical gaussian descriptor for person re-identification. T Matsukawa, T Okabe, E Suzuki, Y Sato, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition7T. Matsukawa, T. Okabe, E. Suzuki, and Y. Sato. Hierarchi- cal gaussian descriptor for person re-identification. In Pro- ceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1363-1372, 2016. 3, 6, 7, 8
Learning to rank in person re-identification with metric ensembles. S Paisitkriangkrai, C Shen, A Van Den, Hengel, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition7S. Paisitkriangkrai, C. Shen, and A. van den Hengel. Learn- ing to rank in person re-identification with metric ensembles. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1846-1855, 2015. 7, 8
Kernel partial least squares for person re-identification. R Prates, M Oliveira, W R Schwartz, IEEE International Conference on Advanced Video and Signal-Based Surveillance (AVSS). 7R. Prates, M. Oliveira, and W. R. Schwartz. Kernel par- tial least squares for person re-identification. In IEEE Inter- national Conference on Advanced Video and Signal-Based Surveillance (AVSS), 2016. 2, 3, 7, 8
Kernel hierarchical pca for person re-identification. R Prates, W R Schwartz, 23th International Conference on Pattern Recognition, ICPR 2016. Cancun, MEXICO7R. Prates and W. R. Schwartz. Kernel hierarchical pca for person re-identification. In 23th International Conference on Pattern Recognition, ICPR 2016, Cancun, MEXICO, De- cember 4-8, 2016., 2016. 2, 3, 7, 8
Mahalanobis distance learning for person reidentification. P M Roth, M Hirzer, M Kstinger, C Beleznai, H Bischof, Person Re-Identification. Springer ACVPR35P. M. Roth, M. Hirzer, M. Kstinger, C. Beleznai, and H. Bischof. Mahalanobis distance learning for person re- identification. In Person Re-Identification, Springer ACVPR, pages 247-267. 2014. 3, 5
Learning discriminative appearance-based models using partial least squares. W Schwartz, L Davis, Computer Graphics and Image Processing. XXII Brazilian Symposium onW. Schwartz and L. Davis. Learning discriminative appearance-based models using partial least squares. In Computer Graphics and Image Processing, 2009 XXII Brazilian Symposium on, pages 322-329, Oct 2009. 3
Joint sparse representation for robust multimodal biometrics recognition. Pattern Analysis and Machine Intelligence. S Shekhar, V M Patel, N M Nasrabadi, R Chellappa, IEEE Transactions on. 3614S. Shekhar, V. M. Patel, N. M. Nasrabadi, and R. Chellappa. Joint sparse representation for robust multimodal biomet- rics recognition. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 36(1):113-126, 2014. 1, 4
Multi-task learning with low rank attribute embedding for person re-identification. C Su, F Yang, S Zhang, Q Tian, L S Davis, W Gao, International Conference on Computer Vision. 24C. Su, F. Yang, S. Zhang, Q. Tian, L. S. Davis, and W. Gao. Multi-task learning with low rank attribute embedding for person re-identification. In International Conference on Computer Vision, pages 3739-3747, 2015. 2, 4
Person re-identification based on spatiogram descriptor and collaborative representation. C Tian, M Zeng, Z Wu, IEEE Signal Processing Letters. 2210C. Tian, M. Zeng, and Z. Wu. Person re-identification based on spatiogram descriptor and collaborative representation. IEEE Signal Processing Letters, 22(10):1595-1599, 2015. 4
Sparse representation for computer vision and pattern recognition. J Wright, Y Ma, J Mairal, G Sapiro, T S Huang, S Yan, Proceedings of the IEEE. 9867J. Wright, Y. Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan. Sparse representation for computer vision and pat- tern recognition. Proceedings of the IEEE, 98(6):1031-1044, 2010. 1, 7
An enhanced deep feature representation for person reidentification. S Wu, Y.-C Chen, X Li, J.-J You, W.-S Zheng, WACV2016: IEEE Winter Conference on Applications of Computer Vision. 7S. Wu, Y.-C. Chen, X. Li, J.-J. You, and W.-S. Zheng. An enhanced deep feature representation for person re- identification. In WACV2016: IEEE Winter Conference on Applications of Computer Vision., March 2016. 3, 6, 7, 8
Relaxed collaborative representation for pattern classification. M Yang, L Zhang, D Zhang, S Wang, Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on. IEEE14M. Yang, L. Zhang, D. Zhang, and S. Wang. Relaxed collab- orative representation for pattern classification. In Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Confer- ence on, pages 2224-2231. IEEE, 2012. 1, 4
Salient color names for person re-identification. Y Yang, J Yang, J Yan, S Liao, D Yi, S Z Li, Computer Vision-ECCV 2014. Y. Yang, J. Yang, J. Yan, S. Liao, D. Yi, and S. Z. Li. Salient color names for person re-identification. In Computer Vision-ECCV 2014, pages 536-551, 2014. 3
Visual classification with multitask joint sparse representation. Image Processing. X.-T Yuan, X Liu, S Yan, IEEE Transactions on. 21104X.-T. Yuan, X. Liu, and S. Yan. Visual classification with multitask joint sparse representation. Image Processing, IEEE Transactions on, 21(10):4349-4360, 2012. 1, 2, 4
Efficient person re-identification by hybrid spatiogram and covariance descriptor. M Zeng, Z Wu, C Tian, L Zhang, L Hu, IEEE Conference on Computer Vision and Pattern Recognition Workshops. M. Zeng, Z. Wu, C. Tian, L. Zhang, and L. Hu. Efficient person re-identification by hybrid spatiogram and covariance descriptor. In IEEE Conference on Computer Vision and Pat- tern Recognition Workshops, pages 48-56, 2015. 4
Joint dynamic sparse representation for multi-view face recognition. H Zhang, N M Nasrabadi, Y Zhang, T S Huang, Pattern Recognition. 4544H. Zhang, N. M. Nasrabadi, Y. Zhang, and T. S. Huang. Joint dynamic sparse representation for multi-view face recogni- tion. Pattern Recognition, 45(4):1290-1298, 2012. 1, 4
Learning a discriminative null space for person re-identification. L Zhang, T Xiang, S Gong, arXiv:1603.0213937arXiv preprintL. Zhang, T. Xiang, and S. Gong. Learning a discrimina- tive null space for person re-identification. arXiv preprint arXiv:1603.02139, 2016. 3, 7
Collaborative representation based classification for face recognition. L Zhang, M Yang, X Feng, Y Ma, D Zhang, arXiv:1204.2358arXiv preprintL. Zhang, M. Yang, X. Feng, Y. Ma, and D. Zhang. Collab- orative representation based classification for face recogni- tion. arXiv preprint arXiv:1204.2358, 2012. 1
Samplespecific svm learning for person re-identification. Y Zhang, B Li, H Lu, A Irie, X Ruan, The IEEE Conference on Computer Vision and Pattern Recognition (CVPR). 7Y. Zhang, B. Li, H. Lu, A. Irie, and X. Ruan. Sample- specific svm learning for person re-identification. In The IEEE Conference on Computer Vision and Pattern Recog- nition (CVPR), June 2016. 3, 7, 8
Unsupervised salience learning for person re-identification. R Zhao, W Ouyang, X Wang, Computer Vision and Pattern Recognition (CVPR), 2013 IEEE Conference on. R. Zhao, W. Ouyang, and X. Wang. Unsupervised salience learning for person re-identification. In Computer Vision and Pattern Recognition (CVPR), 2013 IEEE Conference on, pages 3586-3593, June 2013. 3
Person re-identification by probabilistic relative distance comparison. W.-S Zheng, S Gong, T Xiang, Computer vision and pattern recognition (CVPR). IEEE3W.-S. Zheng, S. Gong, and T. Xiang. Person re-identification by probabilistic relative distance comparison. In Computer vision and pattern recognition (CVPR), 2011 IEEE confer- ence on, pages 649-656. IEEE, 2011. 3, 6
| [] |
[
"Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis",
"Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis"
] | [
"Espen Sande \nDepartment of Mathematics\nUniversity of Rome Tor Vergata\nItaly\n",
"Carla Manni \nDepartment of Mathematics\nUniversity of Rome Tor Vergata\nItaly\n",
"Hendrik Speleers \nDepartment of Mathematics\nUniversity of Rome Tor Vergata\nItaly\n"
] | [
"Department of Mathematics\nUniversity of Rome Tor Vergata\nItaly",
"Department of Mathematics\nUniversity of Rome Tor Vergata\nItaly",
"Department of Mathematics\nUniversity of Rome Tor Vergata\nItaly"
] | [] | In this paper, we provide a priori error estimates with explicit constants for both the L 2 -projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends and completes the results recently obtained for spline spaces of maximal smoothness. The presented error estimates indicate that smoother spline spaces exhibit a better approximation behavior per degree of freedom, even for low smoothness of the functions to be approximated. This is in complete agreement with the numerical evidence found in the literature. We begin with presenting results for univariate spline spaces, and then we address multivariate tensor-product spline spaces and isogeometric spline spaces generated by means of a mapped geometry, both in the single-patch and in the multi-patch case. | 10.1007/s00211-019-01097-9 | [
"https://arxiv.org/pdf/1909.03559v3.pdf"
] | 202,539,023 | 1909.03559 | 7eae2f1ba4872643254082e95d3c79cc7de61935 |
Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis
October 2, 2019
Espen Sande
Department of Mathematics
University of Rome Tor Vergata
Italy
Carla Manni
Department of Mathematics
University of Rome Tor Vergata
Italy
Hendrik Speleers
Department of Mathematics
University of Rome Tor Vergata
Italy
Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis
October 2, 2019
In this paper, we provide a priori error estimates with explicit constants for both the L 2 -projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends and completes the results recently obtained for spline spaces of maximal smoothness. The presented error estimates indicate that smoother spline spaces exhibit a better approximation behavior per degree of freedom, even for low smoothness of the functions to be approximated. This is in complete agreement with the numerical evidence found in the literature. We begin with presenting results for univariate spline spaces, and then we address multivariate tensor-product spline spaces and isogeometric spline spaces generated by means of a mapped geometry, both in the single-patch and in the multi-patch case.
Introduction
Spline approximation is a classical topic in approximation theory; we refer the reader to the book [19] for an extended bibliography. Moreover, it has recently received a renewed interest within the emerging field of isogeometric analysis (IGA); see the book [8]. In this context, a priori error estimates in Sobolev (semi-)norms and corresponding projectors for suitably chosen spline spaces are important.
Classical a priori error estimates for spline approximation are explicit in the grid spacing but hide the influence of the smoothness and the degree of the spline space. Such structure, however, is not sufficient for the IGA environment. In particular, IGA allows for a rich assortment of refinement strategies [8], combining grid refinement (h) and/or degree refinement (p) with various interelement smoothness (k). To fully exploit the benefits of the so-called h-p-k refinement, it is necessary to understand how all the parameters involved (i.e., the grid spacing, the degree, and the smoothness) affect the error estimate. Furthermore, it is important to unravel the influence of the geometry map in isogeometric approximation schemes, not only for its effect on the accuracy but also because it helps in defining good mesh quality metrics [10].
Besides their prominent interest for analyzing convergence under different kinds of refinements, error estimates for approximation in suitable reduced spline spaces play a less evident but still pivotal role in other aspects of IGA discretizations, such as the design of fast iterative (multigrid) solvers for the resulting linear systems [15,24]. The convergence rate of fast iterative solvers should ideally be independent of all the parameters involved, and so their explicit impact on the estimates is important to understand.
In the context of IGA, the role of the smoothness and the degree in spline approximation has been theoretically investigated for the first time in [2], providing explicit error estimates for spline spaces of smoothness k and degree p ≥ 2k + 1. The important case of maximal smoothness (k = p − 1) has been recently addressed for uniform grid spacing in [25] and for general grid spacing in [18], where improved error estimates have been achieved as well. The above references all deal with both univariate and multivariate spline spaces.
In order to complete the picture, in this paper we provide a priori error estimates with explicit constants for approximation by spline functions of arbitrary smoothness defined on arbitrary knot sequences. Our results do not only fill the gap of the smoothness that is not yet covered in the literature, but they also improve upon the error estimates in [2,18,25].
The key ingredient to get these results is the representation of the considered Sobolev spaces and the approximating spline spaces in terms of integral operators described by suitable kernels [17]. We consider error estimates for both univariate and multivariate spline spaces, and we also allow for a mapped geometry. After a short description of some preliminary notation, the main theoretical contributions and the structure of the paper are outlined in the next subsections.
Preliminary notation
For k ≥ 0, let C k [a, b] be the classical space of functions with continuous derivatives of order 0, 1, . . . , k on the interval [a, b]. We further let C −1 [a, b] denote the space of bounded, piecewise continuous functions on [a, b] that are discontinuous only at a finite number of points.
Suppose Ξ := (ξ 0 , . . . , ξ N +1 ) is a sequence of (break) points such that a =: ξ 0 < ξ 1 < · · · < ξ N < ξ N +1 := b, and let h := max j=0,. ..,N (ξ j+1 − ξ j ).
Moreover, set I j := [ξ j , ξ j+1 ), j = 0, 1, . . . , N − 1, and I N := [ξ N , ξ N +1 ]. For any p ≥ 0, let P p be the space of polynomials of degree at most p. Then, for −1 ≤ k ≤ p − 1, we define the space S k p,Ξ of splines of degree p and smoothness k by S k p,Ξ := {s ∈ C k [a, b] : s| I j ∈ P p , j = 0, 1, . . . , N }, and we set S p,Ξ := S p−1 p,Ξ . With a slight misuse of terminology, we will refer to Ξ as knot sequence and to its elements as knots.
For real-valued functions f and g we denote the norm and inner product on L 2 (a, b) by
f 2 := (f, f ), (f, g) := b a f (x)g(x)dx,
and we consider the Sobolev spaces H r (a, b) := {u ∈ L 2 (a, b) : ∂ α u ∈ L 2 (a, b), α = 1, . . . , r}.
We use the notation S k p : L 2 (a, b) → S k p,Ξ and S p : L 2 (a, b) → S p,Ξ for the L 2 -projector onto spline spaces, while P p : L 2 (a, b) → P p stands for the L 2 -projector onto the polynomial space P p .
Main results: univariate case
In this paper we focus on general spline spaces of degree p, smoothness k, and arbitrary knot sequence Ξ. We first derive the following (simplified) error estimate:
u − S k p u ≤ e h 4(p − k) r ∂ r u ,(1)
for any u ∈ H r (a, b) and all p ≥ r − 1. We refer the reader to Remark 1, Theorem 3, and Corollary 5 for sharper results. We then show that similar error estimates hold for standard Ritz projections and their derivatives; see Remark 3 and Corollary 6. The inequality in (1) does not only cover the univariate result from [2], but also improves upon it by allowing any smoothness; in particular, the most interesting cases of highly smooth spline spaces are embraced. As already pointed out in [2], a simple error estimate like (1) is not able to give a theoretical explanation for the numerical evidence that smoother spline spaces exhibit a better approximation behavior per degree of freedom. On the other hand, the sharper estimate provided in Theorem 3 seems to be good enough to capture this behavior; see Remark 2 (and Figure 2). For uniform knot sequences, it has been formally shown in [5] that C p−1 spline spaces perform better than C 0 and C −1 spline spaces in almost all cases of practical interest. A similar approximation behavior per degree of freedom is observed for the Ritz projections; see Remark 4 (and Figure 3).
For maximally smooth spline spaces, the best known error estimate for the L 2 -projection is given by
u − S p u ≤ h π r ∂ r u ,(2)
for any u ∈ H r (a, b) and all p ≥ r − 1. This estimate has been recently proved in [18]. Note that the same error estimate also holds for periodic functions/splines [18,21], for which it has been shown to be optimal on uniform knot sequences [13,17,18].
It is easy to see that (2) is sharper than (1) for k = p − 1. Nevertheless, for fixed r, this estimate only ensures convergence in h, and not in p. The role of the grid spacing and the degree is made more clear in the following estimate:
u − S p u ≤ 2eh(b − a) eπ(b − a) + 4h(p + 1) r ∂ r u ,(3)
for any u ∈ H r (a, b) and all p ≥ r − 1; see Remark 6. For small r compared to p, a better estimate is formulated in Remark 7. The general result, covering both (2) and (3), can be found in Corollary 7. Similar estimates hold for Ritz projections and their derivatives; see Remark 9 and Corollary 9. The p-dependence has also been strengthened for the arbitrarily smooth case in Corollary 5. Motivated by their use in the analysis of fast iterative solvers for linear systems arising from spline discretization methods [15], we also provide error estimates for approximation in suitable reduced spline spaces; see Theorems 5 and 6.
Main results: multivariate case
The univariate results can be extended to obtain error estimates for approximation in multivariate isogeometric spline spaces. As common in the related literature [1,3,4], we first address standard tensor-product spline spaces, then investigate the effect of single-patch geometries for isogeometric spline spaces, and finally discuss C 0 multi-patch geometries. In all cases we provide a priori error estimates with explicit constants, highlighting all the actors that play a role in the construction of the considered spline spaces: the knot sequences, the degrees, the smoothness, and the possible geometry map.
For tensor-product spline spaces we provide error estimates for L 2 and Ritz projections in Theorem 7 and Theorem 8, respectively. In case of single-patch geometries, we do not confine ourselves to the plain isoparametric context which is typical in IGA [8], i.e., the same space that generates the geometry is mapped to the physical domain, but we allow for possibly different spaces for the geometry representation and the function approximation. In the first instance, we assume geometric mappings that are sufficiently globally smooth; see Theorem 9 and Example 16. Afterwards, we also provide error estimates for mappings generated by more general geometry function classes that include spline spaces and NURBS spaces of arbitrary smoothness; see Theorem 10 and Example 17. In this perspective, following the literature [3,4], we introduce suitable bent Sobolev spaces, so as to accommodate a less smooth setting for the geometry. We explicitize the role of the (derivatives of the) geometry map in the constants of the error estimates, both for L 2 and Ritz projections. Finally, to deal with the C 0 multi-patch setting, we consider a projector that is local to each of the patches and is closely related to the standard Ritz projector. Indeed, since the global isogeometric space is continuous, we cannot directly use standard L 2 or Ritz projectors as local building blocks on the patches. Instead, we choose each of the projectors to be interpolatory on the patch boundaries [3,24] so that they can be easily combined into a continuous global projector. We provide explicit error estimates for the new local projectors, which immediately give rise to the desired estimates for the global one; see Example 18. Even though the multivariate results emanate from the univariate ones by following arguments similar to those already presented in the literature, see [1,3,4,16,24] and references therein, the novelty of the provided error estimates is twofold:
• they are expressed in terms of explicit constants and cover arbitrary smoothness;
• they hold for a certain (mapped) Ritz projector which is very natural in the context of Galerkin methods.
It is also worthwhile to note that, although the current investigation has been mainly motivated by IGA applications, standard C 0 tensor-product finite elements are included as special cases.
Outline of the paper
The remainder of this paper is organized as follows. In Section 2 we introduce a general framework for dealing with a priori error estimates in standard Sobolev (semi-)norms for L 2 and Ritz projections onto univariate finite dimensional spaces represented in terms of integral operators described by a suitable kernel. Based on these results, error estimates with explicit constants are provided for spline spaces of arbitrary smoothness in Section 3, and further investigated for the salient case of spline spaces of maximal smoothness in Section 4. Section 5 addresses certain reduced spline spaces which can be of interest in several contexts. Then, we extend those univariate results to the multivariate setting. Standard tensor-product spline spaces are considered in Section 6, while isogeometric spline spaces defined on mapped (single-patch) geometries are covered in Section 7; we provide explicit expressions for all the involved constants. In Section 8 we discuss a particular Ritz-type projector and related error estimates for isogeometric spline spaces on C 0 multipatch geometries. Finally, we conclude the paper in Section 9 by summarizing the main theoretical results.
General error estimates
In this section we describe a general framework for error estimates for L 2 -projection and Ritz projection onto spaces defined in terms of integral operators.
General framework
For f ∈ L 2 (a, b), let K be the integral operator
Kf (x) := b a K(x, y)f (y)dy.(4)
As in [17], we use the notation K(x, y) for the kernel of K. We will in this paper only consider kernels that are continuous or piecewise continuous. We denote by K * the adjoint, or dual, of the operator K, defined by
(f, K * g) = (Kf, g).
The kernel of K * is K * (x, y) = K(y, x). Given any finite dimensional subspace Z 0 ⊇ P 0 of L 2 (a, b) and any integral operator K, we let Z t for t ≥ 1 be defined by Z t := P 0 + K(Z t−1 ). We further assume that they satisfy the equality
Z t := P 0 + K(Z t−1 ) = P 0 + K * (Z t−1 ),(5)
where the sums do not need to be orthogonal (or even direct). Moreover, let Z t be the L 2 -projector onto Z t , and define C t,r ∈ R for t, r ≥ 0 to be C t,r := (I − Z t )K r .
Note that C t,0 = 1. In the case t = 0 and r = 1 we further define the constant C ∈ R to be
C := max{ (I − Z 0 )K , (I − Z 0 )K * }.(7)
Lemma 2.1 in [18] then states the following inequality. For completeness we provide a short proof here as well.
Lemma 1. The constants in (6) and (7) satisfy
C t,1 ≤ C, t ≥ 0.
Proof. For t = 0, this is true by the definitions of C 0,1 and C. For t ≥ 1, we see from (5) that KZ t−1 maps into the space Z t . Now, since Z t is the best approximation into Z t we have
(I − Z t )K ≤ K(I − Z t−1 ) = (I − Z t−1 )K * .
Continuing this procedure gives
(I − Z t )K ≤ (I − Z 0 )K , t even, (I − Z 0 )K * , t odd,
and the result again follows from the definitions of C t,1 and C.
Inspired by the idea of Lemma 1 in [14] we have the following more general result.
Lemma 2.
The constants in (6) satisfy
C t,r ≤ C t,s C t−s,r−s , for all 0 ≤ s ≤ t, r. Proof. Observe that the operator (I − Z t )K s Z t−s K r−s = 0 since K s Z t−s K r−s f ∈ Z t for any f ∈ L 2 (a, b). Thus, (I − Z t )K r = (I − Z t )K s (I − Z t−s )K r−s ≤ (I − Z t )K s (I − Z t−s )K r−s ,
and the result follows from the definition of C t,r .
Similar to Theorem 2.1 in [18] we obtain the following estimate. (6) and (7) satisfy
Lemma 3. The constants in
C t,r ≤ C t,1 C t−1,1 · · · C t−r+1,1 ≤ C r ,
for all t ≥ r − 1.
Proof. The case r = 1 is contained in Lemma 1. For the first inequality, the cases r ≥ 2 follow from Lemma 2 (with s = 1) and induction on r. The second inequality then follows from Lemma 1.
In the next subsection we consider a particularly relevant integral operator: the Volterra operator.
Error estimates for the Ritz projection
Let K be the integral operator defined by integrating from the left,
(Kf )(x) := x a f (y)dy.(8)
One can check that K * is integration from the right,
(K * f )(x) = b x f (y)dy;
see, e.g., Section 7 of [14]. Note that in this case we have (I − Z 0 )K = (I − Z 0 )K * , and so C = C 0,1 . Moreover, the space H r (a, b) can be described as H r (a, b) = P 0 + K(H r−1 (a, b)) = P 0 + K * (H r−1 (a, b)) = P r−1 + K r (H 0 (a, b)), (9) with H 0 (a, b) = L 2 (a, b) and P −1 = {0}. Thus, any u ∈ H r (a, b) is of the form u = g + K r f for g ∈ P r−1 and f ∈ L 2 (a, b). This leads to the following error estimate for the L 2projection.
Theorem 1. Let Z t be the L 2 -projector onto Z t and assume P r−1 ⊆ Z t . Then, for any
u ∈ H r (a, b) we have u − Z t u ≤ C t,r ∂ r u .
Proof. Since P r−1 ⊆ Z t and using (9), we obtain
u − Z t u = g + K r f − Z t (g + K r f ) = (I − Z t )K r f ≤ C t,r f ,
and the result follows from the identity ∂ r u = f .
From the definition of Z t in (5), with K as in (8), it follows that P r−1 is a subspace of Z t for any t ≥ r − 1. Hence, Theorem 1 and Lemma 3 imply the following result.
Corollary 1. Let Z t be the L 2 -projector onto Z t . Then, for any u ∈ H r (a, b) we have u − Z t u ≤ C t,r ∂ r u ≤ C r ∂ r u , for all t ≥ r − 1.
We now focus on a different projector which is very natural in the context of Galerkin methods. For any q = 0, . . . , t we define the projector
R q t : H q (a, b) → Z t by (∂ q R q t u, ∂ q v) = (∂ q u, ∂ q v), ∀v ∈ Z t , (R q t u, g) = (u, g), ∀g ∈ P q−1 .(10)
We remark that R q t is the Ritz projector for the q-harmonic problem. Observe that this projector satisfies ∂ q R q t = Z t−q ∂ q , where Z t−q denotes the L 2 -projector onto Z t−q . With the aid of the Aubin-Nitsche duality argument we arrive at the following estimate.
Theorem 2. Let u ∈ H q (a, b) be given, and let R q t be the projector onto Z t defined in (10). Then, for any = 0, . . . , q we have
∂ (u − R q t u) ≤ C t−q,q− ∂ q u − Z t−q ∂ q u ,for all t ≥ q such that P q− −1 ⊆ Z t−q .
Proof. Let u ∈ H q (a, b) be given and define w as the solution to the Neumann problem
(−1) q− ∂ 2(q− ) w = u − R q t u, w (q− ) (a) = w (q− ) (b) = · · · = w (2(q− )−1) (a) = w (2(q− )−1) (b) = 0.
Using integration by parts, q − times, we have
∂ (u − R q t u) 2 = (∂ (u − R q t u), ∂ (u − R q t u)) = (∂ (u − R q t u), (−1) q− ∂ ∂ 2(q− ) w) = (∂ q (u − R q t u), ∂ q w) = (∂ q (u − R q t u), ∂ q (w − v)), for any v ∈ Z t , since (∂ q (u − R q t u), ∂ q v) = 0. Using ∂ (u − R q t u) = ∂ 2q− w and the Cauchy-Schwarz inequality, we obtain ∂ (u − R q t u) ∂ 2q− w ≤ ∂ q (u − R q t u) ∂ q (w − v) . If we let v = R q t w, then Theorem 1 implies that ∂ q (w − R q t w) = ∂ q w − Z t−q ∂ q w ≤ C t−q,q− ∂ 2q− w , since P q− −1 ⊆ Z t−q . Thus, ∂ (u − R q t u) ≤ C t−q,q− ∂ q (u − R q t u) = C t−q,q− ∂ q u − Z t−q ∂ q u , which completes the proof.
Theorem 1 in combination with Theorem 2 results in a more classical error estimate for the Ritz projection.
Spline spaces of arbitrary smoothness
In this section we show error estimates, with explicit constants, for spline spaces of arbitrary smoothness defined on arbitrary knot sequences. To do this we make use of a theorem in [20] for polynomial approximation.
Lemma 4. Let u ∈ H r (a, b) be given. For any p ≥ r − 1, let P p be the L 2 -projector onto P p . Then, u − P p u ≤ b − a 2 r (p + 1 − r)! (p + 1 + r)! ∂ r u .
Proof. This follows from Theorem 3.11 in [20] since the L ∞ -norm of the weight-function is bounded by 1.
Lemma 5. Let u ∈ H r (a, b) be given. For any p ≥ r − 1 and knot sequence Ξ, let S −1 p be the L 2 -projector onto S −1 p,Ξ . Then, u − S −1 p u ≤ h 2 r (p + 1 − r)! (p + 1 + r)! ∂ r u .
Proof. This follows from Lemma 4 together with a scaling argument.
Example 2. For r = 1 we have
u − S −1 p u ≤ h 2 (p + 1)(p + 2) ∂u .
We are now ready to derive an error estimate for the L 2 -projection onto an arbitrarily smooth spline space S k p,Ξ . We start by observing that if Z 0 = S −1 p−k−1,Ξ we have Z k+1 = S k p,Ξ , for the sequence of spaces in (5). Specifically,
S k p,Ξ = P 0 + K(S k−1 p−1,Ξ ) = P 0 + K * (S k−1 p−1,Ξ )
, k ≥ 0, and from Lemma 5 (and Example 2) we deduce that
C 0,r ≤ h 2 r (p − k − r)! (p − k + r)! , C ≤ h 2 (p − k)(p − k + 1) ,(13)
for any r such that P r−1 ⊆ Z 0 = S −1 p−k−1,Ξ ; the argument is similar to the one in the proof of Theorem 1. We then define the constant c p,k,r for p ≥ r − 1 as follows. If k ≤ p − 2, we let
c p,k,r := 1 2 r 1 (p − k)(p − k + 1) r , k ≥ r − 2, 1 (p − k)(p − k + 1) k+1 (p + 1 − r)! (p − 1 + r − 2k)! , k < r − 2, and if k = p − 1, we let c p,p−1,r := 1 π r .
By combining Theorem 1.1 of [18] with Theorem 1 and Corollary 1 we obtain the following error estimate.
Theorem 3. Let u ∈ H r (a, b) be given. For any knot sequence Ξ, let S k p be the L 2 -projector onto S k p,Ξ for −1 ≤ k ≤ p − 1. Then, u − S k p u ≤ c p,k,r h r ∂ r u , for all p ≥ r − 1.
Proof. For k = p − 1, this result has been shown in Theorem 1.1 of [18]; see inequality (2). Now, let k ≤ p − 2. For r ≤ k + 2, the result follows from Corollary 1 (with t = k + 1) and the bound for C in (13). On the other hand, for r > k + 2, we use Theorem 1 (with t = k + 1), since P r−1 is a subspace of Z k+1 = S k p,Ξ for all p ≥ r − 1. Then, applying Lemma 2 (with t = k + 1) and Lemma 3 (with t = k + 1) we get
C k+1,r ≤ C k+1,k+1 C 0,r−k−1 ≤ C k+1 C 0,r−k−1 ,
and the bounds in (13) complete the proof.
Remark 1. We can bound c p,k,r for k ≤ p − 2 as follows. For r ≤ k + 2, we have
c p,k,r ≤ 1 2(p − k) r ,
while for r > k + 2, using the Stirling formula (see, e.g., the proof of Corollary 3.12 in [20]), we get
c p,k,r ≤ 1 2(p − k) r e 2 (r−k−1) 2 p−k ≤ e 4(p − k) r .
As a consequence, the estimate in Theorem 3 can be simplified to
u − S k p u ≤ e h 4(p − k) r ∂ r u ,(14)
for all p ≥ r − 1. This is in agreement with the estimate in Theorem 2 of [2].
Remark 2. Numerical experiments reveal that smoother spline spaces exhibit a better approximation behavior per degree of freedom; see, e.g., [11]. It was observed in [2], however, that a simple error estimate like (14) does not capture this behavior properly. The sharper estimate in Theorem 3 seems to provide a more accurate description of this behavior. Now, let (a, b) = (0, 1). Assuming a uniform knot sequence Ξ and h 1, the spline dimension can be measured by
n := dim(S k p,Ξ ) = p − k h + k + 1 p − k h .(15)
Hence, the estimate in Theorem 3 can be rephrased as
u − S k p u c p,k,r p − k n r ∂ r u ,(16)
for all p ≥ r − 1. As illustrated in Example 3 ( Figure 1) and Example 4 ( Figure 2), numerical evaluation indicates that
c p,k 1 ,r (p − k 1 ) r < c p,k 2 ,r (p − k 2 ) r , k 1 > k 2 .
This confirms that, for fixed spline degree, smoother spline spaces have better approximation properties per degree of freedom, even for low smoothness of the functions to be approximated. We refer the reader to [5] for a more exhaustive theoretical comparison of the approximation power of spline spaces per degree of freedom in the extreme cases k = −1, 0, p − 1.
Example 3. Let r = 3. Figure 1 depicts the numerical values of c p,k,3 (p − k) 3 for different choices of p and k. We clearly see that the smallest values are attained for maximal spline
smoothness k = p − 1, namely c p,p−1,3 = (1/π) 3 ≈ 0.0323.
Example 4. Consider now the maximal Sobolev smoothness r = p + 1. Figure 2 depicts the numerical values of c p,k,p+1 (p − k) p+1 for different choices of p and k. For any fixed p, one notices that the values are decreasing for increasing k, and hence the smallest values are attained for maximal spline smoothness k = p − 1.
By utilizing Lemma 4 once again we can further sharpen the error estimate in Theorem 3. Let us now define C h,p,k,r by
C h,p,k,r := min c p,k,r h r , b − a 2 r (p + 1 − r)! (p + 1 + r)! ,(17)
for p ≥ max{r − 1, k + 1}. The following result shows that C t,r ≤ C h,p,k,r for Z t = S k p,Ξ .
Corollary 5. Let u ∈ H r (a, b) be given. For any knot sequence Ξ, let S k p be the L 2 -projector onto S k p,Ξ for −1 ≤ k ≤ p − 1. Then, u − S k p u ≤ C h,p,k,r ∂ r u , for all p ≥ r − 1.
Proof. Since P p ⊆ S k p,Ξ , the result immediately follows from Lemma 4 and Theorem 3. Note that for k = −1, the constant C h,p,k,r equals c p,k,r h r for any p, h and r. However, for large k and p (compared to 1/h) the second argument in (17) can become smaller than the first. The error estimate in Corollary 5 will in this case then coincide with the error estimate for global polynomial approximation. We will look closer at this case in the next section; see in particular
c p,k,r (p − k) r p = 2, r = 3 p = 3, r = 3 p = 4, r = 3 p = 5, r = 3 p = 6, r = 3 p = 7, r = 3 p = 8, r = 3 p = 9, r = 3 p = 10, r = 3 Figure 1: Numerical values of c p,k,r (p − k) r for r = 3 and different choices of p ≥ r − 1 and −1 ≤ k ≤ p − 1.
For any fixed p, one notices that the values are decreasing for increasing k. This means that the smoother spline spaces perform better in the error estimate (16) for fixed spline dimension. In many applications one would be interested in finding a single spline function that can provide a good approximation of all derivatives of u up to a given number q. Derivative estimates for the L 2 -projection could be obtained under some quasi-uniformity assumptions on the knot sequence that ensure stability in H 1 (a, b); see, e.g., [9]. However, these assumptions can be avoided by using a Ritz projection. As a special case of (10) we define, for any q = 0, . . . , k + 1, the Ritz projector R q,k p :
H q (a, b) → S k p,Ξ by (∂ q R q,k p u, ∂ q v) = (∂ q u, ∂ q v), ∀v ∈ S k p,Ξ , (R q,k p u, g) = (u, g), ∀g ∈ P q−1 .(18)
As a consequence of Theorem 2 we have the following estimate.
Theorem 4. Let u ∈ H q (a, b) be given. For any degree p, knot sequence Ξ and smoothness q − 1 ≤ k ≤ p − 1, let R q,k p be the projector onto S k p,Ξ defined in (18). Then, for any = 0, . . . , q, we have ∂ (u − R q,k p u) ≤ C h,p−q,k−q,q− ∂ q u − S k−q p−q ∂ q u , for all p ≥ 2q − − 1. Proof. Since P q− −1 ⊆ Z k+1−q = S k−q p−q,Ξ for p ≥ 2q− −1, the result follows from Theorem 2 (with t = k + 1).
By applying Corollary 2 in a similar way, we arrive at an error estimate in the desired form.
Corollary 6. Let u ∈ H r (a, b) be given. For any degree p, knot sequence Ξ and smoothness −1 ≤ k ≤ p − 1, let R q,k p be the projector onto S k p,Ξ defined in (18) for q = 0, . . . , min{k + 1, r}. Then, for any = 0, . . . , q, we have
∂ (u − R q,k p u) ≤ C h,p−q,k−q,q− C h,p−q,k−q,r−q ∂ r u , for all p ≥ max{q, r − 1, 2q − − 1}.
Remark 3. Using the definition of C h,p,k,r together with the argument in Remark 1 we can simplify the result in Corollary 6 as follows. For any q = 0, . . . , min{k + 1, r} and = 0, . . . , q, we have
∂ (u − R q,k p u) ≤ c p−q,k−q,q− c p−q,k−q,r−q h r− ∂ r u , ≤ e h 4(p − k) r− ∂ r u ,
for all p ≥ max{q, r − 1, 2q − − 1}. Since this estimate is explicit in h and p, it is very useful for h-p refinement.
Example 5. Similar to Example 1 we let q = 1. Then, for any u ∈ H r (a, b) and k ≥ 0 we have the stability estimates
∂R 1,k p u = S k−1 p−1 ∂u ≤ ∂u , R 1,k p u ≤ u + C h,p−1,k−1,1 ∂u ≤ u + e h 4(p − k) ∂u ,
and, as in Remark 3, the error estimates
u − R 1,k p u ≤ C h,p−1,k−1,1 C h,p−1,k−1,r−1 ∂ r u ≤ e h 4(p − k) r ∂ r u , ∂(u − R 1,k p u) ≤ C h,p−1,k−1,r−1 ∂ r u ≤ e h 4(p − k) r−1 ∂ r u ,
for all p ≥ r − 1. Thus, R 1,k p u provides a good approximation of both the function u itself, and its first derivative. Example 6. Let q = 2 and r = 3. For R 2,k p u to approximate u ∈ H 3 (a, b) in the L 2 -norm, Corollary 6 requires the degree to be at least 2q − 1 = 3, and not r − 1 = 2 as one might expect. In view of (18), this is consistent with the common assumption to solve the biharmonic equation with piecewise polynomials of at least cubic degree for obtaining an optimal rate of convergence in L 2 ; see, e.g., p. 118 in [23]. Remark 4. In the spirit of Remark 2, the above error estimates for the Ritz projection can also be used to investigate the approximation behavior per degree of freedom. Let (a, b) = (0, 1), and assume a uniform knot sequence Ξ and h 1. Then, keeping the dimension formula (15) in mind, the first inequality in Remark 3 can be rephrased as: for any q = , . . . , min{k + 1, r}, we have
∂ (u − R q,k p u) c p−q,k−q,q− c p−q,k−q,r−q p − k n r− ∂ r u ,(19)
for all p ≥ max{q, r − 1, 2q − − 1}. As illustrated in Example 7 (Figure 3), numerical evaluation of the constant in (19) indicates that smoother spline spaces have better approximation properties per degree of freedom, not only in the L 2 norm but also in more general H q (semi-)norms.
Example 7. Let q = 1 and consider the maximal Sobolev smoothness r = p + 1. Figure 3 depicts the numerical values of c p−q,k−q,q− c p−q,k−q,r−q (p − k) r− for = 0, 1 and different choices of p and k. For any fixed p and , one notices that the values are decreasing for increasing k, and hence the smallest values are attained for maximal spline smoothness k = p − 1. Since this trend is happening for both = 0, 1, it means that higher smoothness performs better per degree of freedom, in both the L 2 and H 1 norms, for any fixed p.
Note that the graphs look like the ones in Figure 2 using the L 2 -projection. This is not a coincidence because one can check that
c p−1,k−1,1− c p−1,k−1,p = c p− ,k− ,p+1− ,
for 0 ≤ k ≤ p − 1 and = 0, 1. Figure 3: Numerical values of c p−q,k−q,q− c p−q,k−q,r−q (p − k) r− for r = p + 1, q = 1, = 0, 1, and different choices of p ≥ 1 and q − 1 ≤ k ≤ p − 1. For any fixed p, one notices that the values are decreasing for increasing k. This means that the smoother spline spaces perform better in the error estimate (19) for fixed spline dimension.
Remark 5. The last observation in Example 7 can be generalized as follows. In case of maximal Sobolev regularity r = p + 1 and max{q − 1, 2q − − 2} ≤ k ≤ p − 1, we have
c p−q,k−q,q− c p−q,k−q,p+1−q = c p− ,k− ,p+1− .
Spline spaces of maximal smoothness
In this section we delve deeper into the behavior of the error estimates for the space of maximally smooth splines, i.e., k = p − 1. In particular, we investigate more carefully the p-dependence. Let us define the constant C h,p,r by C h,p,r := C h,p,p−1,r with C h,p,k,r in (17), or more explicitly by
C h,p,r := min h π r , b − a 2 r (p + 1 − r)! (p + 1 + r)! ,(20)
for p ≥ r − 1. As a generalization of Corollary 6.3 in [25] we obtain the following result.
Corollary 7. Let u ∈ H r (a, b) be given. For any knot sequence Ξ, let S p be the L 2 -projector onto S p,Ξ . Then,
u − S p u ≤ C h,p,r ∂ r u ,(21)u − S p u ≤ C h,p,1 C h,p−1,1 · · · C h,p−r+1,1 ∂ r u ,(22)
for all p ≥ r − 1. (21) is the case k = p − 1 of Corollary 5. For (22), we first observe that if Z 0 = S 0,Ξ we have Z p = S p,Ξ for the sequence of spaces in (5). From Lemma 3 (with t = p) we then obtain (22) for u ∈ H r (a, b).
Proof. The estimate
The first argument in the definition of C h,p,r only depends on h and r, while the second argument only depends on p and r. Hence, it is clear that the second argument is smaller than the first for large enough p with respect to h. This is illustrated in the next examples.
p(p + 1)(p + 2)(p + 3) < 1 p 2 < h π 2 .
In this case the error estimate (21) is better than (2). It is easy to see that for fixed p and small enough h, both estimates in Corollary 7 coincide. Moreover, for fixed h and large enough p, (21) is a sharper estimate than (22). On the other hand, as we illustrate in the next example, there are certain choices of h and p such that (22) is sharper than (21).
Example 10. Let r = 2 and b − a = 2. Then, assuming π (p + 1)(p + 2) < h < π
p(p + 3) ,(23)
we have
C h,p,1 C h,p−1,1 = h π 1 (p + 1)(p + 2) < min h π 2 , 1 p(p + 1)(p + 2)(p + 3) = C h,p,2 .
As a consequence, when h satisfies (23), the error estimate (22) is sharper than (21).
The estimates in Corollary 7 hint towards a complex interplay between h and p in the sense that for a strongly refined grid (very small h), increasing the degree p might give little or no benefit, and vice versa.
Remark 6. Using the Stirling formula (in the same way as in Remark 1), we have
(p + 1 − r)! (p + 1 + r)! ≤ e 2(p + 1) r . Thus, C h,p,r ≤ min h π r , e(b − a) 4(p + 1) r = min h π , e(b − a) 4(p + 1) r ,
and by taking the harmonic mean of the two quantities in the above bound, we get
u − S p u ≤ 2eh(b − a) eπ(b − a) + 4h(p + 1) r ∂ r u ,(24)
for all p ≥ r − 1. Even though this estimate is less sharp than the result in Corollary 7, it has the benefit of always decreasing as the grid is refined and/or as the degree is increased.
Remark 7. For small values of r (compared to p) we can improve upon the estimate in Remark 6 as follows. Since C h,p−i+1,1 ≤ C h,p−r+1,1 for i = 1, . . . , r, Corollary 7 implies that
u − S p u ≤ (C h,p−r+1,1 ) r ∂ r u ,
for all p ≥ r − 1. By taking the harmonic mean of the two quantities in the bound
C h,p−r+1,1 ≤ min h π , b − a 2(p − r + 2) , we obtain u − S p u ≤ 2h(b − a) π(b − a) + 2h(p − r + 2) r ∂ r u ,
for all p ≥ r − 1. This estimate is sharper than (24) if p > e e−2 (r + 2 e − 2). Note that this is always the case if p ≥ 4(r − 1).
We now look at some error estimates for the Ritz projection. Corollary 3 and Lemma 3 lead to the following result. Corollary 8. Let u ∈ H q (a, b) be given. For any knot sequence Ξ, let R q p be the projector onto S p,Ξ = Z p defined in (10). Then, for any = 0, . . . , q we have
∂ (u − R q p u) ≤ C h,p−q,q− ∂ q u − S p−q ∂ q u , ∂ (u − R q p u) ≤ C h,p−q,1 C h,p−q−1,1 · · · C h,p−2q+ +1,1 ∂ q u − S p−q ∂ q u , for all p ≥ max{q, 2q − − 1}.
Similarly, using Corollary 4 and Lemma 3 we obtain the following estimate.
Corollary 9. Let u ∈ H r (a, b) be given. For any q = 0, . . . , r and knot sequence Ξ, let R q p be the projector onto S p,Ξ = Z p defined in (10). Then, for any = 0, . . . , q we have
∂ (u − R q p u) ≤ C h,p−q,q− C h,p−q,r−q ∂ r u , ∂ (u − R q p u) ≤ (C h,p−q,1 · · · C h,p−2q+ +1,1 ) (C h,p−q,1 · · · C h,p−r+1,1 ) ∂ r u , for all p ≥ max{q, r − 1, 2q − − 1}.
Remark 8. As a generalization of Theorem 3.1 in [18], it follows from Corollary 9 that for any q = 0, . . . , r and = 0, . . . , q,
∂ (u − R q p u) ≤ h π r− ∂ r u ,(25)
for all p ≥ max{q, r − 1, 2q − − 1}. Not only is this a very simple and explicit estimate, but it is also very useful for h refinement. Note that the error estimate for periodic splines in Theorem 4.1 of [18] is of the same form as (25) for the corresponding Ritz projection in the case of periodic boundary conditions. Remark 9. Following a similar argument as in Remark 6, we get for any q = 0, . . . , r and = 0, . . . , q,
∂ (u − R q p u) ≤ 2eh(b − a) eπ(b − a) + 4h(p − q + 1) r− ∂ r u ,
for all p ≥ max{q, r − 1, 2q − − 1}. In addition, following a similar argument as in Remark 7, we get for any q = 0, . . . , r and = 0, . . . , q,
∂ (u − R q p u) ≤ 2h(b − a) π(b − a) + 2h(p + 2 − max{2q − , r}) r− ∂ r u ,
for all p ≥ max{q, r − 1, 2q − − 1}. The latter estimate is sharper than the former one if p > e e−2 (max{2q − , r} + 2(1−q) e − 2). Even though these two estimates are less sharp than the result in Corollary 9, they have the benefit of always decreasing as the grid is refined and/or as the degree is increased. They are therefore useful estimates for h-p-k refinement.
Reduced spline spaces
The goal of this section is to prove error estimates for the Ritz projection onto certain reduced spline spaces of maximal smoothness studied in [12,14,15,18,22,25]. To do that we first prove a general result for any integral operator K using ideas from [12,14].
General error estimates
Let K be any integral operator as in (4), and let X 0 and Y 0 be any finite dimensional subspaces of L 2 (a, b). We then define the subspaces X p and Y p in an analogous way to (5), by
X p := K(Y p−1 ), Y p := K * (X p−1 ),(26)
for p ≥ 1. Finally, for any p ≥ 0, let X p be the L 2 -projector onto X p and Y p be the L 2 -projector onto Y p .
Lemma 6. For any
p ≥ 1 we have K − KY p ≤ K * − K * X p−1 ≤ K − X 0 K , p odd, K * − Y 0 K * , p even. Proof. First, note that K − KY p = K * − Y p K * = sup f ≤1 K * f − Y p K * f .
Next, observe that K * X p−1 maps into Y p and since Y p K * f is the best approximation of K * f in Y p we must have
sup f ≤1 K * f − Y p K * f ≤ sup f ≤1 K * f − K * X p−1 f = K * − K * X p−1 .
This shows that K − KY p ≤ K * − K * X p−1 . Similarly, by swapping the roles of K and K * we have K * − K * X p ≤ K − KY p−1 . The result then follows from induction on p.
Error estimates for reduced spline spaces
In [12,14,18,25] error estimates for certain reduced spline spaces were shown. Here we prove a generalization of these results for the Ritz projections. Specifically, in [14] and [18] the spaces S p,Ξ,0 and S p,Ξ,1 , defined by
were studied. We further define the related spaces S p,Ξ,0 and S p,Ξ,1 by S p,Ξ,0 := {s ∈ S p,Ξ : ∂ α s(a) = ∂ α s(b) = 0, 0 ≤ α < p, α even},
S p,Ξ,1 := {s ∈ S p,Ξ : ∂ α s(a) = ∂ α s(b) = 0, 0 ≤ α < p, α odd}.(28)
For uniform knot sequences, the spaces S p,Ξ,1 are exactly the reduced spline spaces investigated in [25] (see Definition 5.1 of [25]). Observe that S p,Ξ,0 ⊆ S p,Ξ,0 where equality holds for p odd and S p,Ξ,1 ⊆ S p,Ξ,1 where equality holds for p even. Observe further that in the case p = 0 all the spaces in (27) and (28) equal the standard spline space S 0,Ξ except for S 0,Ξ,0 . For a specific (degree-dependent) knot sequence Ξ it was shown in [14] that the spline spaces in (27) are optimal for certain n-width problems. Later it was shown in [18] that if n is the dimension of these optimal spaces, then they converge to the space spanned by the n first eigenfunctions of the Laplacian (with either Dirichlet or Neumann boundary conditions) as their degree p increases. Convergence in the case of periodic boundary conditions was also studied in [18].
Staying consistent with the notation in [12,14,17] we define the integral operator K 1 by
K 1 := (I − P 0 )K,
where P 0 denotes the L 2 -projector onto P 0 , and K is the integral operator in (8). One can verify that if u = K 1 f then ∂u = f and u ⊥ 1. Moreover, since K * 1 = K * (I − P 0 ) it follows that if u = K * 1 f then ∂u = (P 0 − I)f and u(a) = u(b) = 0. Using these properties it was shown in [14] that S p,Ξ,0 = K * 1 (S p−1,Ξ,1 ),
S p,Ξ,1 = P 0 ⊕ K 1 (S p−1,Ξ,0 ),(29)
for all p ≥ 1, since the derivative of a spline is a spline of one degree lower on the same knot sequence. For the spline spaces in (28) we deduce by the same argument that
S p,Ξ,0 = K * 1 (S p−1,Ξ,1 ), S p,Ξ,1 = P 0 ⊕ K 1 (S p−1,Ξ,0 ),(30)
for all p ≥ 1. Let S p,i : L 2 (a, b) → S p,Ξ,i , i = 0, 1, denote the L 2 -projector. Analogously to (10) we define, for p ≥ 1, the Ritz projector R p,0 : H 1 0 (a, b) → S p,Ξ,0 by (∂R p,0 u, ∂v) = (∂u, ∂v), ∀v ∈ S p,Ξ,0 ,
and the Ritz projector R p,1 : H 1 (a, b) → S p,Ξ,1 by (∂R p,1 u, ∂v) = (∂u, ∂v), ∀v ∈ S p,Ξ,1 , (R p,1 u, 1) = (u, 1).
Using the above definitions, together with (29), we find that R p,0 = K * 1 S p−1,1 and R p,1 = P 0 + K 1 S p−1,0 .
Lastly, we define the quantity h by
h := max{2h 0 , h 1 , h 2 , . . . , h N −1 , 2h N }.
To prove the error estimates for our Ritz projections onto the sequences of spaces in (27) we make use of the next lemma.
Lemma 7. For any u ∈ H 1 (a, b) we have u − S 0,1 u ≤ h π ∂u ,
and for
any v ∈ H 1 0 (a, b) we have v − S 0,0 v ≤ h π ∂v .
Proof. These results follow from the Poincaré inequality. See Theorem 1.1 and Lemma 8.1 in [18] for the details.
Using the above lemma together with Lemma 6 we obtain the desired error estimates.
Theorem 5. Let p ≥ 0 be given. Then, for any u ∈ H 1 (a, b) we have Moreover, using the definition of K 1 we observe that H 1 (a, b) = P 0 ⊕ K 1 (L 2 (a, b)). Thus, any function u ∈ H 1 (a, b) can be decomposed as u = c + K 1 f for c ∈ P 0 and f ∈ L 2 (a, b). Using Lemma 7 we then find that
u − R p,1 u ≤ h π ∂u , p odd, u − R p,1 u ≤ h π ∂u , p even, and for any v ∈ H 1 0 (a, b) we have v − R p,0 v ≤ h π ∂v , p odd, v − R p,0 v ≤ h π ∂v ,Kf − S 0,1 Kf = u − S 0,1 u ≤ h π ∂u = h π f ,
since P 0 ⊂ S p,Ξ,1 , and so K − S 0,1 K ≤ h/π. Furthermore, it was shown in [14] that H 1 0 (a, b) = K * 1 (L 2 (a, b)) and so any function v ∈ H 1 0 (a, b) can be written as v = K * 1 g for g ∈ L 2 (a, b). Again, using Lemma 7, we find that
K * g − S 0,0 K * g = v − S 0,0 v ≤ h π ∂v = h π g ,
and K * − S 0,0 K * ≤ h/π. The result then follows from Lemma 6 since R p,0 = K * 1 S p−1,1 and R p,1 = P 0 + K 1 S p−1,0 .
Let S p,i : L 2 (a, b) → S p,Ξ,i , i = 0, 1, denote the L 2 -projector. We then define the Ritz projector R p,0 : H 1 (a, b) → S p,Ξ,0 in a completely analogous way to (31) and R p,1 : H 1 (a, b) → S p,Ξ,1 in a completely analogous way to (32). As before, using (30) we find that R p,0 = K * 1 S p−1,1 and R p,1 = P 0 + K 1 S p−1,0 .
Theorem 6. Let p ≥ 0 be given. Then, for any u ∈ H 1 (a, b) we have
u − R p,1 u ≤ h π ∂u , and for any v ∈ H 1 0 (a, b) we have v − R p,0 v ≤ h π ∂v ,
Proof. This result follows from a similar argument as in the proof of Theorem 5. The main change being that in the case p = 0 we have S 0,Ξ,0 = S 0,Ξ , and so K * −S 0,0 K * ≤ h/π.
Remark 10. The reduced spline spaces defined in (27) and (28) all satisfy the boundary conditions stated in Theorem 9.1 of [18]. Hence, any element s in such spaces satisfies the following inverse inequality:
s ≤ 2 √ 3 h min s ,
where h min is the minimum knot distance.
Remark 11. As the error estimates in Theorems 5 and 6 are complemented with the inverse inequality in Remark 10, the reduced spline spaces defined in (27) and (28) can be used to design fast iterative (multigrid) solvers for linear systems arising from spline discretization methods [15,22].
Tensor-product spline spaces
In this section we describe how to extend our error estimates to the case of tensor-product spline spaces. We start by introducing some notation. Consider the d-dimensional domain Ω := (a 1 , b 1 ) × (a 2 , b 2 ) × · · · × (a d , b d ), and let · Ω denote the L 2 (Ω)-norm. Moreover, we deal with the standard Sobolev spaces on Ω defined by
H r (Ω) := {u ∈ L 2 (Ω) : ∂ α 1 1 · · · ∂ α d d u ∈ L 2 (Ω), 1 ≤ α 1 + · · · + α d ≤ r}.
For i = 1, . . . , d, let Z t i be a finite dimensional subspace of L 2 (a i , b i ) as in (5) with K as in (8), and define the tensor-product space Z t := Z t 1 ⊗ Z t 2 ⊗ · · · ⊗ Z t d . We only investigate projectors onto Z t of the form Π := Π 1 ⊗ Π 2 ⊗ · · · ⊗ Π d . To simplify notation, we use the following convention: when applying the univariate operator Π i to a d-variate function u, we mean that Π i acts on the i-th variable of u while the others are considered as parameters. In this perspective, we have Π = Π 1 • Π 2 • · · · • Π d .
We first study error estimates for the L 2 (Ω)-projection onto Z t , denoted by Z t :
= Z t 1 ⊗ Z t 2 ⊗ · · · ⊗ Z t d .
The following result can be concluded from the univariate error estimates using a standard argument (see, e.g., [2,5,18,20,25]), but for the sake of completeness we include the argument here.
Lemma 8. For any u ∈ L 2 (Ω) we have u − Z t u Ω ≤ d i=1 u − Z t i u Ω .
Proof. We only consider the case d = 2. The generalization to arbitrary d is straightforward. From the triangle inequality we obtain
u − Z t 1 ⊗ Z t 2 u Ω ≤ u − Z t 1 u Ω + Z t 1 u − Z t 1 ⊗ Z t 2 u Ω ≤ u − Z t 1 u Ω + Z t 1 u − Z t 2 u Ω ≤ u − Z t 1 u Ω + u − Z t 2 u Ω ,
since the L 2 (Ω)-operator norm of Z t 1 is equal to 1.
Combining Lemma 8 with Theorem 1 leads to the following error estimate for tensorproduct spaces.
Theorem 7. Assume P r−1 ⊆ Z t i for all i = 1, . . . , d. Then, for any u ∈ H r (Ω) we have u − Z t u Ω ≤ d i=1 C t i ,r ∂ r i u Ω .
Remark 12. For simplicity let Ω = (0, 1) d . Note that Theorem 7 actually holds for all functions u in the larger Sobolev space
d i=1 L 2 (0, 1) i−1 ⊗ H r (0, 1) ⊗ L 2 (0, 1) d−i ⊇ H r (Ω).
We make use of a similar Sobolev space in Section 7.2.
For tensor-product spline spaces of arbitrary smoothness, let S k p := S k 1 p 1 ⊗· · ·⊗S k d p d denote the L 2 (Ω)-projector onto S k p,Ξ := S k 1 p 1 ,Ξ 1 ⊗ · · · ⊗ S k d p d ,Ξ d . For maximally smooth spline spaces, let S p := S p 1 ⊗ · · · ⊗ S p d denote the L 2 (Ω)-projector onto S p,Ξ := S p 1 ,Ξ 1 ⊗ · · · ⊗ S p d ,Ξ d . Error estimates for these spaces can be immediately obtained by replacing C t i ,r in Theorem 7 with the constants derived in Corollaries 5 and 7. Let h i denote the maximal knot distance in Ξ i for i = 1, . . . , d.
Corollary 10. For any u ∈ H r (Ω) we have u − S k p u Ω ≤ d i=1 C h i ,p i ,k i ,r ∂ r i u Ω , and u − S p u Ω ≤ d i=1 C h i ,p i ,r ∂ r i u Ω , u − S p u Ω ≤ d i=1 C h i ,p i ,1 C h i ,p i −1,1 · · · C h i ,p i −r+1,1 ∂ r i u Ω ,
for all p i ≥ r − 1.
Example 11. Let h := max{h 1 , h 2 , . . . , h d }. Then, for any u ∈ H r (Ω) we have
u − S p u Ω ≤ d i=1 h i π r ∂ r i u Ω ≤ h π r d i=1 ∂ r i u Ω ,
for all p i ≥ r − 1.
Let us now focus on error estimates for tensor products of the Ritz projection in (10). For simplicity of notation, we only consider the case q = 1 and d = 2. Define the tensorproduct Ritz projector R t : H 1 (a 1 , b 1
) ⊗ H 1 (a 2 , b 2 ) → Z t 1 ⊗ Z t 2 by
Example 12. Let R k p := R 1,k 1 p 1 ⊗ R 1,k 2 p 2 be the tensor-product Ritz projector onto S k p,Ξ , and let h := max{h 1 , h 2 } and p − k := min{p 1 − k 1 , p 2 − k 2 }. Then, for any u ∈ H r (Ω), r ≥ 2, we have
u − R k p u Ω ≤ e h 4(p − k) r ( ∂ r 1 u Ω + ∂ r 2 u Ω + ∂ r 12 u Ω ) ,
where we slightly abuse notation by letting
∂ r 12 u Ω := min ∂ 1 ∂ r−1 2 u Ω , ∂ r−1 1 ∂ 2 u Ω , for all p 1 , p 2 ≥ r − 1.
Example 13. Let R p := R 1 p 1 ⊗ R 1 p 2 be the tensor-product Ritz projector onto S p,Ξ , and let h := max{h 1 , h 2 }. Then, for any u ∈ H 2 (Ω) and for 0 ≤ 1 , 2 ≤ 1 we have
∂ 1 1 ∂ 2 2 (u − R p u) Ω ≤ h π 2− 1 − 2 ∂ 2 1 u Ω + ∂ 2 2 u Ω + ∂ 1 ∂ 2 u Ω ,
for all p 1 , p 2 ≥ 1. In general, for any u ∈ H r (Ω), r ≥ 2, and for 0 ≤ 1 , 2 ≤ 1 we have
∂ 1 1 ∂ 2 2 (u − R p u) Ω ≤ h π r− 1 − 2 ∂ r 1 u Ω + ∂ r 2 u Ω + ∂ 1 ∂ r−1 2 u Ω + ∂ r−1 1 ∂ 2 u Ω , for all p 1 , p 2 ≥ r − 1.
Similar results hold for the tensor products of the reduced spline spaces in Section 5.2. The results of this section can also be generalized to higher order Ritz projections in a straightforward way.
Mapped geometry
Motivated by IGA, in this section we consider error estimates for spline spaces defined on a mapped (single-patch) domain. Let Ω = (0, 1) d be the reference domain, Ω the physical domain, and G : Ω → Ω ⊂ R d the geometric mapping defining Ω. We assume that the mapping G is a bi-Lipschitz homeomorphism. As a general rule, we indicate quantities and operators that refer to the (mapped) physical domain by means of˜. In particular, the derivative operator with respect to physical variables is denoted by∂.
Define the space Z t as the push-forward of the tensor-product space Z t with respect to the mapping G. Specifically, let
Z t := {s • G −1 : s ∈ Z t }.(33)
Furthermore, for any projector Π : L 2 (Ω) → Z t we let Π : L 2 ( Ω) → Z t denote the projector defined by Πũ := (Π(ũ • G)) • G −1 , ∀ũ ∈ L 2 ( Ω).
Using a standard substitution argument we obtain the following result.
Lemma 11. Forũ ∈ L 2 ( Ω) and G ∈ (W 1,∞ (Ω)) d let u :=ũ • G ∈ L 2 (Ω). Then, for any projector Π we have
ũ − Πũ Ω ≤ det ∇G L ∞ (Ω) u − Πu Ω .
Smooth geometry
Similar to [15,24] we can easily extend the results from Section 6 if we take the geometry map G to be sufficiently globally smooth. Specifically, in this subsection we assume G ∈ (W r,∞ (Ω)) d , which implies that u :=ũ • G ∈ H r (Ω) wheneverũ ∈ H r ( Ω). We further assume G −1 ∈ (W 1,∞ ( Ω)) d .
We define the mapped L 2 -projector Z t : L 2 ( Ω) → Z t by taking Π = Z t in (34). Then, combining Lemma 11 and Theorem 7 gives rise to the following estimate.
Lemma 12. Let G ∈ (W r,∞ (Ω)) d . Then, for anyũ ∈ H r ( Ω) we have ũ − Z tũ Ω ≤ det ∇G L ∞ (Ω) d i=1 C t i ,r ∂ r i (ũ • G) Ω ,
for all t i ≥ r − 1.
Using a slightly simplified version of the multivariate Faà di Bruno formula in [7] and substituting back to the physical domain, we obtain an error estimate in a more classical form. To this end, we set G := (G 1 , . . . , G d ) and define
C G := det ∇G L ∞ (Ω) det ∇G −1 L ∞ ( Ω) ,
and C G,i,r,j := m(k m,1 + · · · + k m,d ) = r .
I(r,j) r! r m=1 ∂ m i G 1 k m,1 · · · ∂ m i G d k m,d k m,1 ! · · · k m,d ! m! k m,1 +···+k m,d L ∞ (Ω) ,(35)
Theorem 9. Let G ∈ (W r,∞ (Ω)) d and G −1 ∈ (W 1,∞ ( Ω)) d . Then, for anyũ ∈ H r ( Ω) we have
ũ − Z tũ Ω ≤ C G 1≤|j|≤r d i=1 C t i ,r C G,i,r,j ∂ j 1 1 · · ·∂ j d dũ Ω ,
for all t i ≥ r − 1.
Proof. By means of the multivariate Faà di Bruno formula in [7] we can express the highorder partial derivatives in Lemma 12 as
∂ r i (ũ • G) = 1≤|j|≤r (∂ j 1 1 · · ·∂ j d dũ ) • G I(r,j) r! r m=1 ∂ m i G 1 k m,1 · · · ∂ m i G d k m,d k m,1 ! · · · k m,d ! m! k m,1 +···+k m,d .
This gives
ũ − Z tũ Ω ≤ det ∇G L ∞ (Ω) d i=1 C t i ,r 1≤|j|≤r C G,i,r,j (∂ j 1 1 · · ·∂ j d dũ ) • G Ω ,
and a standard substitution argument completes the proof.
In the spirit of Corollary 10, using results from Sections 3 and 4, the above theorem can be used to obtain error estimates for mapped L 2 -projections onto spline spaces of any smoothness. Indeed, we just need to replace C t i ,r with the corresponding constants, e.g., the ones derived in Corollaries 5 and 7.
Example 14. Let d = 1. Given the geometry map G, we have
C G,1,r,j = I(r,j) r! r m=1 ∂ m G km k m ! m! km L ∞ (Ω) , where I(r, j) := (k 1 , . . . , k r ) ∈ Z r ≥0 : r m=1 k m = j, r m=1 mk m = r .
Observe that C G,1,r,j can be compactly expressed in terms of (exponential) partial Bell polynomials B r,j (x 1 , . . . , x r−j+1 ) by C G,1,r,j = B r,j (∂G, ∂ 2 G, . . . , ∂ r−j+1 G) L ∞ (Ω) ; see, e.g., Section 3.3 in [6]. These Bell polynomials can be easily computed by the following recurrence relation:
B r,j (x 1 , . . . , x r−j+1 ) = 1 j r−1 i=j−1 r i x r−i B i,j−1 (x 1 , . . . , x i−j+2 ),
where B 0,0 = 1 and B r,0 = 0 for r ≥ 1. In particular, we have
B 1,1 (x 1 ) = x 1 , B 2,1 (x 1 , x 2 ) = x 2 , B 2,2 (x 1 ) = (x 1 ) 2 , B 3,1 (x 1 , x 2 , x 3 ) = x 3 , B 3,2 (x 1 , x 2 ) = 3x 1 x 2 , B 3,3 (x 1 ) = (x 1 ) 3 .
Example 15. Let d = 2. For r = 1 and i = 1, 2 we have
C G,i,1,(1,0) = ∂ i G 1 L ∞ (Ω) , C G,i,1,(0,1) = ∂ i G 2 L ∞ (Ω) .
For r = 2 and i = 1, 2 we have
C G,i,2,(1,0) = ∂ 2 i G 1 L ∞ (Ω) , C G,i,2,(0,1) = ∂ 2 i G 2 L ∞ (Ω) , C G,i,2,(2,0) = (∂ i G 1 ) 2 L ∞ (Ω) , C G,i,2,(0,2) = (∂ i G 2 ) 2 L ∞ (Ω) , C G,i,2,(1,1) = 2(∂ i G 1 )(∂ i G 2 ) L ∞ (Ω) .
Similar results can be obtained for tensor-product Ritz projections in the presence of a mapped geometry. As before, it is a matter of applying the Ritz estimates from Section 6 in combination with the multivariate Faà di Bruno formula [7]. We omit these results to avoid repetition. We just illustrate this with an example.
Example 16. Let d = 2 and r = 2. Recall from Example 13 that for any u ∈ H 2 (Ω) and for 0 ≤ 1 , 2 ≤ 1 we have
∂ 1 1 ∂ 2 2 (u − R p u) Ω ≤ h π 2− 1 − 2 ∂ 2 1 u Ω + ∂ 2 2 u Ω + ∂ 1 ∂ 2 u Ω ,
for all p 1 , p 2 ≥ 1 and h := max{h 1 , h 2 }. We define the mapped Ritz projector R p : H 2 ( Ω) → Z t by taking Π = R p in (34). Assume G ∈ (W 2,∞ (Ω)) 2 and G −1 ∈ (W 1,∞ ( Ω)) 2 . From Theorem 9 (and Example 15) we know estimates for ∂ 2 1 (ũ • G) Ω and ∂ 2 2 (ũ • G) Ω , and we can compute similar ones for ∂ 1 ∂ 2 (ũ • G) Ω . Then, for anyũ ∈ H 2 ( Ω) and for 0 ≤ 1 , 2 ≤ 1 we obtain
∂ 1 1∂ 2 2 (ũ − R pũ ) Ω ≤ C G h π 2− 1 − 2 1≤|j|≤2 C G,1,2,j + C G,2,2,j + C G,12,2,j ∂ j 1 1∂ j 2 2ũ Ω , for all p 1 , p 2 ≥ 1, where C G,12,2,(1,0) = ∂ 1 ∂ 2 G 1 L ∞ (Ω) , C G,12,2,(0,1) = ∂ 1 ∂ 2 G 2 L ∞ (Ω) , C G,12,2,(2,0) = (∂ 1 G 1 )(∂ 2 G 1 ) L ∞ (Ω) , C G,12,2,(0,2) = (∂ 1 G 2 )(∂ 2 G 2 ) L ∞ (Ω) , C G,12,2,(1,1) = (∂ 1 G 1 )(∂ 2 G 2 ) + (∂ 2 G 1 )(∂ 1 G 2 ) L ∞ (Ω) .
Bent geometry
In IGA the geometry map G is commonly taken to be componentwise a spline function from the same space as our approximation space. However, the results in the previous subsection can require the geometry map to be in a smoother subspace. We will overcome the issue in this subsection.
For r ≥ 1 and k ≥ 0 we define the univariate bent Sobolev space
H r,k Ξ (0, 1) := {u ∈ H min{r,k+1} (0, 1) : u ∈ H r (ξ j , ξ j+1 ), j = 0, 1, . . . , N }.
Note that for k ≥ r − 1 we have H r,k Ξ (0, 1) = H r (0, 1). Then, similar to the space in Remark 12, we define the (L 2 -extended) multivariate bent Sobolev space
H r,k Ξ (Ω) := d i=1 L 2 (0, 1) i−1 ⊗ H r,k i Ξ i (0, 1) ⊗ L 2 (0, 1) d−i .
Following [3] we also introduce the mesh-dependent norm
· 2 Ω,Ξ := σ∈M Ξ · 2 σ ,
where M Ξ is the collection of the (open) elements defined by Ξ and · σ denotes the L 2 -norm on the element σ. Furthermore, for k i ≥ 0, i = 1, . . . , d, we define the bent geometry function class ∞ (0, 1). The space G r,k Ξ (Ω) contains the spline space S k p,Ξ , and it also allows for several other interesting piecewise spaces such as NURBS spaces based on S k p,Ξ . If we assume G ∈ (G r,k Ξ (Ω)) d , then u :=ũ • G ∈ H r,k Ξ (Ω) for u ∈ H r ( Ω). Having u not in H r (Ω) is a potential problem for applying the error estimates we derived in Section 6, but this can be fixed by making use of Lemma 3.1 in [4]. For completeness we provide a short proof here as well.
G r,k Ξ (Ω) := {G ∈ W k+1,∞ (Ω) : G ∈ W r,∞ (σ), σ ∈ M Ξ }, where W k+1,∞ (Ω) := W k 1 +1,∞ (0, 1) ⊗ · · · ⊗ W k d +1,
Lemma 13. For k ≤ r − 2, there exists an operator Γ : H r,k Ξ (0, 1) → S k r−1,Ξ such that u − Γu ∈ H r (0, 1) for all u ∈ H r,k Ξ (0, 1). Proof. Let u ∈ H r,k Ξ (0, 1) for some k ≤ r − 2, and let ∂ − u (∂ + u) denote the limit from the left (right) of the -th order derivative of u. From the definition of the bent Sobolev space we know that u is C k continuous at any interior knot ξ j , j = 1, . . . , N . Now, we define
ϕ j,k (x) := (∂ k+1 + u − ∂ k+1 − u)(ξ j ) (k + 1)! max{0, (x − ξ j ) k+1 }.
It is easy to check that ϕ j,k ∈ S k r−1,Ξ and that u − ϕ j,k is C k+1 continuous at the knot ξ j . Repeating this argument and taking Γu = N j=1 r−2 l=k ϕ j,l , it follows that u − Γu is C r−1 continuous at each interior knot. Since Γu ∈ S k r−1,Ξ we also know that u − Γu ∈ H r,k Ξ (0, 1), and so u − Γu ∈ H r (0, 1). Similar to Proposition 3.1 in [4] we then obtain the following error estimate.
Lemma 14.
Let u ∈ H r,k Ξ (0, 1) be given. Then, u − S k p u ≤ C h,p,k,r ∂ r u (0,1),Ξ , for all p ≥ r − 1.
Proof. For k ≥ r − 1, the result immediately follows from Corollary 5 by recalling that H r,k Ξ (0, 1) = H r (0, 1) in this case. Assume now k ≤ r − 2. Since S k r−1,Ξ ⊆ S k p,Ξ we deduce from Lemma 13 and Corollary 5 that
u − S k p u 2 = u − Γu − S k p (u − Γu) 2 ≤ (C h,p,k,r ∂ r (u − Γu) ) 2 = (C h,p,k,r ) 2 N j=0
∂ r u 2 (ξ j ,ξ j+1 ) = C h,p,k,r ∂ r u (0,1),Ξ 2 , and the result follows by taking the square root of both sides.
The univariate error estimate in Lemma 14 can be easily extended to the multivariate tensor-product spline setting.
Lemma 15. Let u ∈ H r,k Ξ (Ω) be given. Then,
u − S k p u Ω ≤ d i=1 C h i ,p i ,k i ,r ∂ r i u Ω,Ξ ,
for all p i ≥ r − 1.
Proof. Using Lemma 8 we have
u − S k p u Ω ≤ d i=1 u − S k i p i u Ω ,
and the result follows by applying Lemma 14 in each direction separately.
In the case of maximal spline smoothness, i.e., k i = p i − 1 for all i, the constants C h i ,p i ,k i ,r in the above lemma can be replaced by the constants used in Corollary 7.
Using the argument of Theorem 9 we then arrive at the desired error estimates for a bent geometry. To this end, we need to redefine the constants C G,i,r,j in (35) using the mesh-dependent norm · L ∞ (Ω),Ξ := max
σ∈M Ξ · L ∞ (σ) .(36)
Theorem 10. Let G ∈ (G r,k Ξ (Ω)) d and G −1 ∈ (W 1,∞ ( Ω)) d . Then, for anyũ ∈ H r ( Ω) we have
ũ − S k pũ Ω ≤ C G 1≤|j|≤r d i=1 C h i ,p i ,k i ,r C G,i,r,j ∂ j 1 1 · · ·∂ j d dũ Ω ,
for all p i ≥ r − 1.
Similar results can be obtained for tensor-product Ritz projections in the presence of a bent geometry. As before, it is a matter of applying the Ritz estimates from Section 6 in combination with the operator in Lemma 13 and proper (Ritz extended) multivariate bent Sobolev spaces. We omit these results to avoid repetition. We just illustrate this with an example similar to Example 16.
Example 17. Let d = 2 and r = 2. Assuming G ∈ (S p,Ξ ) 2 and G −1 ∈ (W 1,∞ ( Ω)) 2 , for anỹ u ∈ H 2 ( Ω) and for 0 ≤ 1 , 2 ≤ 1 we have
∂ 1 1∂ 2 2 (ũ − R pũ ) Ω ≤ C G h π 2− 1 − 2 1≤|j|≤2
C G,1,2,j + C G,2,2,j + C G,12,2,j ∂ j 1 1∂ j 2
2ũ Ω , for all p 1 , p 2 ≥ 1 and h := max{h 1 , h 2 }. The constants in the above sum are the same as the ones in Examples 15 and 16 but in the mesh-dependent norm (36).
Multi-patch geometry
In this section we generalize our error estimates to the case of multi-patch domains with C 0 continuity across the patches. The arguments here are based on those found in [3,24].
We start by explaining the general framework in the univariate case. Let Z t be a finite dimensional subspace of L 2 (a, b) as in (5) with K as in (8). For t ≥ 1 we define the projector Q t : H 1 (a, b)
→ Z t by Q t u := u(a) + KZ t−1 ∂u,(37)
where K is the integral operator in (8) and Z t the L 2 -projector onto Z t . As we shall see momentarily, the projection (37) is closely related to the Ritz projection for q = 1 in (10) and satisfies essentially the same properties. Additionally, we observe that Q t u(a) = u(a) and
Q t u(b) = u(a) + b a Z t−1 ∂u(x)dx = u(a) + b a ∂u(x)dx = u(b).(38)
Thus, Q t can be equivalently expressed as
Q t u = u(b) − K * Z t−1 ∂u.(39)
The interpolation at the boundary will be used to enforce C 0 continuity across the patches. Similar to the case q = 1 of Corollary 2 we have the following error estimate.
Lemma 16. Let u ∈ H r (a, b) for r ≥ 1 be given. Then, u − Q t u ≤ C t−1,1 C t−1,r−1 ∂ r u , ∂(u − Q t u) ≤ C t−1,r−1 ∂ r u , for all t ≥ 1 such that P r−2 ⊆ Z t−1 .
Proof. By the fundamental theorem of calculus we have u = u(b) − K * v for v ∈ H r−1 (a, b). Thus, using (39),
u − Q t u = K * v − K * Z t−1 v = K * (I − Z t−1 )v .
Moreover, v ∈ H r−1 (a, b) can be written as v = g + K r−1 f for g ∈ P r−2 and f ∈ L 2 (a, b). Using P r−2 ⊆ Z t−1 and (I − Z t−1 ) 2 = (I − Z t−1 ) we obtain
K * (I − Z t−1 )v = K * (I − Z t−1 )K r−1 f ≤ K * (I − Z t−1 ) (I − Z t−1 )K r−1 f = (I − Z t−1 )K (I − Z t−1 )K r−1 f = C t−1,1 C t−1,r−1 ∂ r u ,
which proves the first inequality. The second inequality follows from Theorem 1 since ∂Q t = Z t−1 ∂.
Error estimates for spline spaces can be immediately obtained by replacing the constants in Lemma 16 with the constants derived for q = 1 in Corollaries 6 and 9.
We now move on to the bivariate case (d = 2). As before, we let t = (t 1 , t 2 ) and define the tensor-product projector Q t : H 1 (a 1 , b 1
) ⊗ H 1 (a 2 , b 2 ) → Z t 1 ⊗ Z t 2 by Q t := Q t 1 ⊗ Q t 2 .
Remark 13. As in Theorem 3.4 of [24], we conclude from (37) and (38) that for all u ∈ H 1 (a 1 , b 1 ) ⊗ H 1 (a 2 , b 2 ),
• u and Q t u coincide at the four corners of [a 1 , b 1 ] × [a 2 , b 2 ], and
• Q t u restricted to any boundary edge of Ω = (a 1 , b 1 ) × (a 2 , b 2 ) coincide with the univariate projection onto that edge, e.g., Q t u(a 1 , ·) = Q t 2 u(a 1 , ·).
Using the same argument as for Theorem 8 we obtain the following error estimates for Q t .
Theorem 11. Let u ∈ H r (Ω) for r ≥ 2 be given. If P r−2 ⊆ Z t 1 −1 ∩ Z t 2 −1 for t 1 , t 2 ≥ 1, then u − Q t u Ω ≤ C t 1 −1,1 C t 1 −1,r−1 ∂ r 1 u Ω + C t 2 −1,1 C t 2 −1,r−1 ∂ r 2 u Ω + C t 1 −1,1 C t 2 −1,1 min C t 2 −1,r−2 ∂ 1 ∂ r−1 2 u Ω , C t 1 −1,r−2 ∂ r−1 1 ∂ 2 u Ω , and ∂ 1 (u − Q t u) Ω ≤ C t 1 −1,r−1 ∂ r 1 u Ω + C t 2 −1,1 C t 2 −1,r−2 ∂ 1 ∂ r−1 2 u Ω , ∂ 2 (u − Q t u) Ω ≤ C t 1 −1,1 C t 1 −1,r−2 ∂ r−1 1 ∂ 2 u Ω + C t 2 −1,r−1 ∂ r 2 u Ω , ∂ 1 ∂ 2 (u − Q t u) Ω ≤ C t 1 −1,r−2 ∂ r−1 1 ∂ 2 u Ω + C t 2 −1,r−2 ∂ 1 ∂ r−1 2 u Ω .
In the spirit of Corollary 10, using results from Sections 3 and 4, the above theorem can be used to obtain similar error estimates for spline spaces of any smoothness.
Finally, we are ready to consider the multi-patch setting in IGA. We assume that the physical domain Ω ⊂ R 2 is divided into M non-overlapping patches Ω i , i = 1, . . . , M . The patches are conforming, i.e., the intersection of the closures of Ω i and Ω j for i = j is either (a) empty, (b) one common corner, or (c) the union of one common edge and two common vertices. Following [24], we define the bent Sobolev space in the physical domain We assume that for each i = 1, . . . , M there is a geometry map G i : Ω = (0, 1) 2 → Ω i , which can be continuously extended to the closure of Ω, such that • G i ∈ (G r,k Ξ (Ω)) 2 and G −1 i ∈ (W 1,∞ ( Ω i )) 2 (see Section 7.2), and
• for any interface Γ ij shared by Ω i and Ω j , the parameterizations G i and G j are identical along that interface, i.e., G −1 i | Γ ij = R ij • G −1 j | Γ ij where R ij is a rigid motion of the unit square to itself.
Similar to (33) we define Z t,i := {s • G −1 i : s ∈ Z t,i }, and, following [3,24], we require that these function spaces are fully matching on the interfaces, i.e., for eachs i ∈ Z t,i there existss j ∈ Z t,j such that along any interface Γ ij shared by Ω i and Ω j we haves i | Γ ij =s j | Γ ij . Remark 14. Under the assumptions on the geometry maps, the fully matching requirement at the interface Γ ij is simply satisfied whenever for l = i, j the univariate spaces Z tm l ,l , m l ∈ {1, 2}, associated with G −1 l ( Γ ij ) coincide. For instance, if G −1 i ( Γ ij ) is a horizontal edge while G −1 j ( Γ ij ) is a vertical one, then Z t 1 ,i = Z t 2 ,j . With the patch spaces Z t,i in place, we define the continuous isogeometric multi-patch space Z t : Ω → R as the continuously glued collection of those patch spaces, i.e., Z t := {s ∈ C 0 ( Ω) :s| Ω i ∈ Z t,i , i = 1, . . . , M }.
We let Q t,i : H 2 ( Ω i ) → Z t,i denote the projector defined by Q t,iũ := (Q t,i (ũ • G i )) • G −1 i , ∀ũ ∈ H 2 ( Ω i ), and for anyũ ∈ H 2,1 ( Ω) we define Q t (ũ) by ( Q tũ )| Ω i := Q t,iũ .
With the same line of arguments as in Proposition 3.8 of [3] (see also Lemma 3.4 of [24]), by using Remark 13 together with the requirement that the patch spaces are fully matching, it follows that Q tũ can be extended to a continuous function across the patch-interfaces and hence this is a projector onto Z t . Similar to the mapped Ritz projection in the previous section we can now obtain error estimates for the projector Q t . As a continuation of Example 17 we can for instance obtain the following result.
Example 18. Let d = 2 and r = 2. Assume G i ∈ (S p,Ξ ) 2 and G −1 i ∈ (W 1,∞ ( Ω i )) 2 for i = 1, . . . , M . Then, for anyũ ∈ H 2 ( Ω i ) and for 0 ≤ 1 , 2 ≤ 1 we have
∂ 1 1∂ 2 2 (ũ − Q pũ ) Ω i ≤ C G i h π 2− 1 − 2 1≤|j|≤2
C G i ,1,2,j + C G i ,2,2,j + C G i ,12,2,j ∂ j 1 1∂ j 2
2ũ Ω i ,
Figure 4 .
4
Figure 2 :
2Numerical values of c p,k,r (p − k) r for r = p + 1 and different choices of p ≥ 1 and −1 ≤ k ≤ p − 1. For any fixed p, one notices that the values are decreasing for increasing k. This means that the smoother spline spaces perform better in the error estimate(16) for fixed spline dimension.
Example 8 .= 2 .
82Let r = 2 and b − a Then
Figure 4 :
4The two arguments of C h,p,r in(20) are equal for h = h * p,r , depicted in normalized form (divided by the interval length b − a) for different choices of r ≥ 1 and p ≥ r − 1. Lower values of h will activate the first argument of C h,p,r , while higher values of h the second argument of C h,p,r .Example 9.Figure 4depicts the values h * p,r /(b − a) ∈ [choices of r and p. It follows that the two arguments of C h,p,r in(20)are equal for h = h * p,r . For smaller values of h we have C h,p,r = (h/π) r , and then (21) coincides with (2). Otherwise, for larger values of h, (21) coincides with the estimate for global polynomial approximation in Lemma 4. Assuming a uniform knot sequence, we observe that the latter only holds for a rather small number of knot intervals N = (b − a)/h with respect to p. For instance, if p = 10 and r = 11, then h * p,r /(b − a) ≈ 0.17 and so N must be less than or equal to 5 for the estimate in(21) to coincide with the estimate for global polynomial approximation. Similarly, one can check that if p = 10 and r = 1, then N must be less than or equal to7.
S p,Ξ,0 := {s ∈ S p,Ξ : ∂ α s(a) = ∂ α s(b) = 0, 0 ≤ α ≤ p, α even}, S p,Ξ,1 := {s ∈ S p,Ξ : ∂ α s(a) = ∂ α s(b) = 0, 0 ≤ α ≤ p, α odd},
p even. Proof. Define the spaces S p by S p := {s ∈ S p,Ξ,1 : s ⊥ 1}. Using (27) we find that if K 1 plays the role of the generic integral operator K in Section 5.1, then the spaces S p are examples of the X p in (26) and the spaces S p,Ξ,0 are examples of the Y p in (26) .
where j := (j 1 , . . . , j d ) andI(r, j) := (k 1,1 , . . . , k 1,d , k 2,1 , . . . , k 2,d , . . . , k r,1 , . . . , k r,d ) ∈ Z r×d ≥0 :
H 2,1 ( Ω) by H 2,1 ( Ω) := {ũ ∈ H 1 ( Ω) :ũ| Ω i ∈ H 2 ( Ω i ), i = 1, . . . , M }.
R t := R 1 t 1 ⊗ R 1 t 2 .Note that H 1 (a 1 , b 1 ) ⊗ H 1 (a 2 , b 2 ) consists of functions u ∈ L 2 (Ω) such that ∂ 1 u ∈ L 2 (Ω),∂ 2 u ∈ L 2 (Ω) and ∂ 1 ∂ 2 u ∈ L 2 (Ω). We thus have H 2 (Ω) ⊂ H 1 (a 1 , b 1 ) ⊗ H 1 (a 2 , b 2 ) ⊂ H 1 (Ω). Lemma 9. Let u ∈ H 1 (a 1 , b 1 ) ⊗ H 1 (a 2 , b 2 ) be given. Then, for all t 1 , t 2 ≥ 1 we have u − R t u Ω ≤ u − R 1 t 1 u Ω + u − R 1 t 2 u Ω + C t 1 −1,1 ∂ 1 u − R 1 t 2 ∂ 1 u Ω , ∂ 1 (u − R t u) Ω ≤ ∂ 1 (u − R 1 t 1 u) Ω + ∂ 1 u − R 1 t 2 ∂ 1 u Ω , ∂ 1 ∂ 2 (u − R t u) Ω ≤ ∂ 1 ∂ 2 u − Z t 1 −1 ∂ 1 ∂ 2 u Ω + ∂ 1 ∂ 2 u − Z t 2 −1 ∂ 1 ∂ 2 u Ω .
AcknowledgementsThis work was supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C18000100006). C. Manni and H. Speleers are members of Gruppo Nazionale per il Calcolo Scientifico, Istituto Nazionale di Alta Matematica.Proof. Using (12) we obtainand since ∂ 1 commutes with R 1 t 2 , the first result follows. Similarly, using(11)we obtainand the second result follows. For the third result we use the commuting relationand we apply Lemma 8.By using Corollary 2 we can achieve error estimates in the desired form. If the function u is only assumed to be in H 1 (a 1 , b 1 ) ⊗ H 1 (a 2 , b 2 ) then one obtains the "unbalanced" estimate:for all t 1 , t 2 ≥ 1. This can be resolved by requiring higher Sobolev smoothness.Lemma 10. Let u ∈ H 2 (Ω) be given. Then, for all t 1 , t 2 ≥ 1 we haveMoreover, for all 1 + 2 = 1 we haveThe lemma can be generalized as follows.In the spirit of Corollary 10, using results from Sections 3 and 4, the above theorem can be used to obtain error estimates for tensor-product Ritz projections onto spline spaces of any smoothness. We end this section with two examples.Remark 15. Ifũ ∈ H 2,1 ( Ω) is zero at the boundary then it follows from Remark 13 and the definition of Q t that Q tũ is also zero at the boundary. Thus, we can obtain the same error estimates in the case of Dirichlet boundary conditions.ConclusionsIn this paper we have provided a priori error estimates with explicit constants for approximation in spline spaces of arbitrary smoothness defined on arbitrary knot sequences and their isogeometric extensions. More precisely, we have considered error estimates in Sobolev (semi-)norms for L 2 and Ritz projections of any function in H r onto univariate and multivariate spline spaces, addressing single-patch and C 0 multi-patch configurations.Our results improve upon existing error estimates in the literature as they fill the gap of the smoothness[2]and allow for more flexible hp refinement for spline spaces of maximal smoothness[18,25]. Moreover, they are consistent with the numerical evidence that smoother spline spaces exhibit a better approximation behavior per degree of freedom, which has been observed when solving practical problems by the IGA paradigm. Our error estimates also pave the way for extending to arbitrary smoothness and to arbitrary knot sequences the theoretical comparison, recently performed in[5], of the approximation power of different piecewise polynomial spaces commonly employed in Galerkin methods for solving partial differential equations. In case of a mapped domain, the error estimates explicitly highlight the influence of the (derivatives of the) geometry map on the approximation properties of the considered isogeometric spaces.Besides their direct theoretical interest, the presented results may have an impact on several practical aspects of the IGA paradigm, including the convergence analysis under different kinds of refinements, the definition of good mesh quality metrics, and the design of fast iterative (multigrid) solvers for the resulting linear systems. We finally note that the range of possible applications of the presented results is not confined to the IGA context, since standard C 0 tensor-product finite elements are also covered as special cases.
Here h := max{h 1 , h 2 }. The constants in the above estimate are the same as the ones in Example 17. By squaring both sides of (40) and summing over all the patches one can obtain a global estimate forũ ∈ H. . . , M , 21all p 1 , p 2 ≥ 1 and i = 1, . . . , M . Here h := max{h 1 , h 2 }. The constants in the above estimate are the same as the ones in Example 17. By squaring both sides of (40) and summing over all the patches one can obtain a global estimate forũ ∈ H 2,1 ( Ω).
Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes. Y Bazilevs, L Beirão Da Veiga, J A Cottrell, T J R Hughes, G Sangalli, Math. Models Methods Appl. Sci. 16Y. Bazilevs, L. Beirão da Veiga, J. A. Cottrell, T. J. R. Hughes, and G. Sangalli, Iso- geometric analysis: Approximation, stability and error estimates for h-refined meshes, Math. Models Methods Appl. Sci. 16 (2006), 1031-1090.
Some estimates for h-p-krefinement in isogeometric analysis. L Beirão Da Veiga, A Buffa, J Rivas, G Sangalli, Numer. Math. 118L. Beirão da Veiga, A. Buffa, J. Rivas, and G. Sangalli, Some estimates for h-p-k- refinement in isogeometric analysis, Numer. Math. 118 (2011), 271-305.
Mathematical analysis of variational isogeometric methods. L Beirão Da Veiga, A Buffa, G Sangalli, R Vázquez, Acta Numer. 23L. Beirão da Veiga, A. Buffa, G. Sangalli, and R. Vázquez, Mathematical analysis of variational isogeometric methods, Acta Numer. 23 (2014), 157-287.
Anisotropic NURBS approximation in isogeometric analysis. L Beirão Da Veiga, D Cho, G Sangalli, Comput. Methods Appl. Mech. Engrg. L. Beirão da Veiga, D. Cho, and G. Sangalli, Anisotropic NURBS approximation in isogeometric analysis, Comput. Methods Appl. Mech. Engrg. 209-212 (2012), 1-11.
Approximation in FEM, DG and IGA: a theoretical comparison. A Bressan, E Sande, 10.1007/s00211-019-01063-5Numer. Math. A. Bressan and E. Sande, Approximation in FEM, DG and IGA: a theoretical com- parison, Numer. Math. (2019), https://doi.org/10.1007/s00211-019-01063-5.
Advanced combinatorics: The art of finite and infinite expansions, D. L Comtet, Reidel Publishing CompanyL. Comtet, Advanced combinatorics: The art of finite and infinite expansions, D. Reidel Publishing Company, 1974.
A multivariate Faà di Bruno formula with applications. G M Constantine, T H Savits, Trans. Amer. Math. Soc. 348G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with ap- plications, Trans. Amer. Math. Soc. 348 (1996), 503-520.
Isogeometric analysis: Toward integration of CAD and FEA. J A Cottrell, T J R Hughes, Y Bazilevs, John Wiley & SonsJ. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs, Isogeometric analysis: Toward inte- gration of CAD and FEA, John Wiley & Sons, 2009.
The stability in L p and W 1 p of the L 2 -projection onto finite element function spaces. M Crouzeix, V Thomée, Math. Comp. 48M. Crouzeix and V. Thomée, The stability in L p and W 1 p of the L 2 -projection onto finite element function spaces, Math. Comp. 48 (1987), 521-532.
L Engvall, J A Evans, arXiv:1810.06975Mesh quality metrics for isogeometric Bernstein-Bézier discretizations, preprint. L. Engvall and J. A. Evans, Mesh quality metrics for isogeometric Bernstein-Bézier discretizations, preprint, arXiv:1810.06975.
n-Widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method. J A Evans, Y Bazilevs, I Babuska, T J R Hughes, Comput. Methods Appl. Mech. Engrg. 198J. A. Evans, Y. Bazilevs, I. Babuska, and T. J. R. Hughes, n-Widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method, Comput. Methods Appl. Mech. Engrg. 198 (2009), 1726-1741.
Optimal spline spaces of higher degree for L 2 n-widths. M S Floater, E Sande, J. Approx. Theory. 216M. S. Floater and E. Sande, Optimal spline spaces of higher degree for L 2 n-widths, J. Approx. Theory 216 (2017), 1-15.
On periodic L 2 n-widths. J. Comput. Appl. Math. 349, On periodic L 2 n-widths, J. Comput. Appl. Math. 349 (2019), 403-409.
Optimal spline spaces for L 2 n-width problems with boundary conditions, Constr. Approx. 50, Optimal spline spaces for L 2 n-width problems with boundary conditions, Con- str. Approx. 50 (2019), 1-18.
Robust multigrid for isogeometric analysis based on stable splittings of spline spaces. C Hofreither, S Takacs, SIAM J. Numer. Anal. 55C. Hofreither and S. Takacs, Robust multigrid for isogeometric analysis based on stable splittings of spline spaces, SIAM J. Numer. Anal. 55 (2017), 2004-2024.
Isogeometric analysis: Mathematical and implementational aspects, with applications, Splines and PDEs: From Approximation Theory to Numerical Linear Algebra. T J R Hughes, G Sangalli, M Tani, Lecture Notes in Mathematics. T. Lyche, C. Manni, and H. Speleers2219Springer International Publishing AGT. J. R. Hughes, G. Sangalli, and M. Tani, Isogeometric analysis: Mathematical and implementational aspects, with applications, Splines and PDEs: From Approximation Theory to Numerical Linear Algebra (T. Lyche, C. Manni, and H. Speleers, eds.), Lecture Notes in Mathematics, vol. 2219, Springer International Publishing AG, 2018, pp. 237-315.
n-Widths in approximation theory. A Pinkus, Springer-VerlagA. Pinkus, n-Widths in approximation theory, Springer-Verlag, 1985.
Sharp error estimates for spline approximation: Explicit constants, n-widths, and eigenfunction convergence. E Sande, C Manni, H Speleers, Math. Models Methods Appl. Sci. 29E. Sande, C. Manni, and H. Speleers, Sharp error estimates for spline approximation: Explicit constants, n-widths, and eigenfunction convergence, Math. Models Methods Appl. Sci. 29 (2019), 1175-1205.
L L Schumaker, Spline functions: Basic theory. Cambridge University Pressthird ed.L. L. Schumaker, Spline functions: Basic theory, third ed., Cambridge University Press, 2007.
C Schwab, p-and hp-finite element methods: Theory and applications in solid and fluid mechanics. Clarendon PressC. Schwab, p-and hp-finite element methods: Theory and applications in solid and fluid mechanics, Clarendon Press, 1999.
Shadrin, Inequalities of Kolmogorov type and estimates of spline interpolation on periodic classes W m 2. A Yu, Math. Notes. 48A. Yu. Shadrin, Inequalities of Kolmogorov type and estimates of spline interpolation on periodic classes W m 2 , Math. Notes 48 (1990), 1058-1063.
Robust multigrid solvers for the biharmonic problem in isogeometric analysis. J Sogn, S Takacs, Comput. Math. Appl. 77J. Sogn and S. Takacs, Robust multigrid solvers for the biharmonic problem in isoge- ometric analysis, Comput. Math. Appl. 77 (2019), 105-124.
An analysis of the finite element method. G Strang, G Fix, Prentice-HallG. Strang and G. Fix, An analysis of the finite element method, Prentice-Hall, 1973.
Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations. S Takacs, Math. Models Methods Appl. Sci. 28S. Takacs, Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations, Math. Models Methods Appl. Sci. 28 (2018), 1899-1928.
Approximation error estimates and inverse inequalities for B-splines of maximum smoothness. S Takacs, T Takacs, Math. Models Methods Appl. Sci. 26S. Takacs and T. Takacs, Approximation error estimates and inverse inequalities for B-splines of maximum smoothness, Math. Models Methods Appl. Sci. 26 (2016), 1411- 1445.
| [] |
[
"Semantic Robustness of Models of Source Code",
"Semantic Robustness of Models of Source Code"
] | [
"Goutham Ramakrishnan [email protected] \nUniversity of Wisconsin-Madison\n\n",
"Jordan Henkel [email protected] \nUniversity of Wisconsin-Madison\n\n",
"Zi Wang \nUniversity of Wisconsin-Madison\n\n",
"Aws Albarghouthi \nUniversity of Wisconsin-Madison\n\n",
"Somesh Jha [email protected] \nUniversity of Wisconsin-Madison\n\n",
"Thomas Reps [email protected] \nUniversity of Wisconsin-Madison\n\n"
] | [
"University of Wisconsin-Madison\n",
"University of Wisconsin-Madison\n",
"University of Wisconsin-Madison\n",
"University of Wisconsin-Madison\n",
"University of Wisconsin-Madison\n",
"University of Wisconsin-Madison\n"
] | [] | Deep neural networks are vulnerable to adversarial examples-small input perturbations that result in incorrect predictions. We study this problem for models of source code, where we want the network to be robust to source-code modifications that preserve code functionality. (1) We define a powerful adversary that can employ sequences of parametric, semantics-preserving program transformations;(2) we show how to perform adversarial training to learn models robust to such adversaries; (3) we conduct an evaluation on different languages and architectures, demonstrating significant quantitative gains in robustness. * Equal Contribution Preprint. Under review. | 10.1109/saner53432.2022.00070 | [
"https://export.arxiv.org/pdf/2002.03043v2.pdf"
] | 211,068,816 | 2002.03043 | 63dc08b8efd67cf788cd17301bdfaf74d9450d93 |
Semantic Robustness of Models of Source Code
Goutham Ramakrishnan [email protected]
University of Wisconsin-Madison
Jordan Henkel [email protected]
University of Wisconsin-Madison
Zi Wang
University of Wisconsin-Madison
Aws Albarghouthi
University of Wisconsin-Madison
Somesh Jha [email protected]
University of Wisconsin-Madison
Thomas Reps [email protected]
University of Wisconsin-Madison
Semantic Robustness of Models of Source Code
Deep neural networks are vulnerable to adversarial examples-small input perturbations that result in incorrect predictions. We study this problem for models of source code, where we want the network to be robust to source-code modifications that preserve code functionality. (1) We define a powerful adversary that can employ sequences of parametric, semantics-preserving program transformations;(2) we show how to perform adversarial training to learn models robust to such adversaries; (3) we conduct an evaluation on different languages and architectures, demonstrating significant quantitative gains in robustness. * Equal Contribution Preprint. Under review.
Introduction
While deep neural networks have been widely adopted in many areas of computing, it has been repeatedly shown that they are vulnerable to adversarial examples [33,10,17,27]: small, seemingly innocuous perturbations to the input that lead to incorrect predictions. Adversarial examples raise safety and security concerns, for example, in computer-vision models used in autonomous vehicles [14,8] or for user authentication [31]. Significant progress has recently been made in identifying adversarial examples and training models that are robust to such examples. However, the majority of the research has targeted computer-vision tasks [11,25,33], a continuous domain. In this paper, we study the problem of robustness to adversarial examples in the discrete domain of deep neural networks for source code. With the growing adoption of neural models for programming tasks, robustness is becoming an important property. Why do we want robust models of code? There are many answers, ranging from usability to security. Consider, for instance, a model that explains in English what a piece of code is doing-the code-captioning task. A developer using such a model to navigate a new code base should not receive completely different explanations for similar pieces of code. For a concrete example, consider the behavior of the state-ofthe-art code2seq model [3] on the Java code in Fig. 1, where the prediction changes after logging print statements are added. Alternatively, imagine the security-critical setting of malware classification. We do not want a small modification to the malware's binary to cause the model to deem it safe.
With images, the threat model involves small changes that are imperceptible to a human. With code, there is no analogous notion of a change imperceptible to a human. Consequently, we consider attacks based on semantics-preserving transformations. Because the original program's semantics is preserved, the program that results from the attack must have the same behavior as the original.
Most-related work. We are not the first to study this problem. Wang and Christodorescu [35] demonstrated the drop in accuracy of deep models over source code when applying standard transformations and refactorings; however, they did not propose defenses in their work. Recently, Yefet et al. [37] developed a gradient-based attack that is specific to variable-name substitution or dead-code insertion, similar in spirit to attacks on natural-language models [13], which can efficiently estimate gradients for token substitution and insertion. Yefet et al. [37] propose a defense based on outlier detection, but do not consider arbitrary program transformations or adversarial training. Zhang et al. [39] use the Metropolis-Hastings algorithm to perform adversarial identifier renaming. However, they do not consider other program transformations, and their defense is akin to dataset augmentation with adversarial examples, instead of a robust optimization [25]. Parallel work [9] is discussed in Sec. 2.
Our aim in this paper is to consider the general setting of an adversary that can apply a sequence of source-code transformations. Specifically, our goal is to answer the following question:
Can we train models that are robust to sequences of semantics-preserving transformations?
k-adversaries & k-robustness. We begin by defining the notion of a k-adversary, one that can select a sequence of k transformations from a prespecified set of semantics-preserving transformations T , and apply them to an input program. The adversary succeeds if it manages to change the prediction of the neural network. For example, a transformation may add dead code to a program, replace for loops with while loops, change variable names, replace one API call with an equivalent one, etc.
The primary challenge in implementing a k-adversary is the combinatorial search space of possible sequences of transformations. Further, some transformations are parametric, e.g., insert a print statement with string s, which blows up the search space even further. To implement a k-adversary in practice, we exploit the insight that we can break up the search into two pieces: (1) enumerative search through transformation sequences, and (2) gradient-based optimization to discover transformation parameters. Specifically, given a sequence of transformations, we partially apply them to a given program, without supplying parameters, resulting in a program sketch [32]: a program with unknown holes (tokens or AST leaves). Then, gradient-based optimization, like that of Yefet et al. [37] or Ebrahimi et al. [13], can be used to discover a worst-case instantiation of the holes.
Adversarially training k-robust models. To train a k-robust model-one that is robust to kadversaries-we adapt the robust-optimization objective of Madry et al. [25] to our setting: Instead of computing the loss for a program x, we compute the worst-case loss resulting from a k-adversary. While this approach is theoretically sound, it is hopelessly inefficient for two reasons.
First, a k-adversary is an expensive operation within the training loop, because it involves enumerating transformation sequences. Thus, even for small values of k, the search space can make training impractically slow. We demonstrate experimentally that training on a small value, k = 1, results in models that are robust to larger values of k. This phenomenon allows us to efficiently train adversarially without incurring a combinatorial explosion in the number of transformation sequences.
Second, program transformations are typically defined as tree transformations over abstract syntax trees (ASTs). However, the neural network usually receives as input some other program format, some of which lose program structure or entire portions of the program-for example, tokens or subtokens of program text, or randomly sampled paths through the AST [4,3]. Therefore, modeling an in-training adversary is very time consuming, because it requires invoking external, non-differentiable program-analysis tools, and converting back and forth between ASTs and training formats.
To work around training complexity, we generate sequences of transformations offline and partially apply them to programs to generate program sketches. During training, we need only consider the generated program sketches, performing gradient-based attacks to search for optimal parameters of the transformations. This approach allows us to avoid applying AST transformations during training.
Evaluation. We have developed an extensible framework for writing transformations and performing adversarial training [15]. We evaluate our approach on radically different programming languages-Java and Python-and architectures-using tokenized and AST program representations. Our results show that (1) adversarial training improves robustness and (2) training with a 1-adversary results in a model that is robust on larger values of k, enabling efficient training.
Contributions. We summarize our contributions below:
• We define k-transformation robustness for source-code tasks: robustness to an adversary that is allowed k transformations to an input program. We show how to implement a k-adversary by combining enumeration and gradient-based optimization over program sketches.
• We propose the first adversarial-training method for neural models of code, adapting the robust-optimization objective of Madry et al. [25]. To efficiently model the adversary during training, we pre-generate program sketches by partially applying transformations.
• We build an extensible framework for adversarially training models of code. We thoroughly evaluate our approach on a variety of datasets and architectures. Our results demonstrate improvements in robustness and the power of training with weak adversaries.
Related Work
In concurrent work, 2 Bielik and Vechev [9] combine adversarial training with abstention and AST pruning to train robust models of code. There are a number of key differences with our work: (1) We consider a richer space of transformations for the adversary, including inserting parameterized dead-code. (2) We use a strong gradient-based adversary and program sketches for completing transformations, while they use a greedy search through the space of transformations with a small number of candidates. (3) Our adversarial training approach is more efficient, as it does not solve an expensive ILP problem to prune ASTs or train multiple models, but it is possible that we can incorporate their AST pruning in our framework.
Adversarial examples. In test-time attacks, an adversary perturbs an example so that it is misclassified by a model (untargeted attack) or the perturbed example is classified as an attacker-specified label (targeted) [5,10,21,12,11]. Initially, test-time attacks were explored in the context of images. Our discrete domain is closer to test-time attacks in natural language processing (NLP). There are several test-time attacks in NLP that consider discrete transformations, such as substituting words or introducing typos [24,26,13,40,16]. A key difference between our domain and NLP is that in the case of programs one has to worry about semantics-the program has to work even after transformations.
Deep learning for source code. Recent years have seen huge progress in deep learning for sourcecode tasks-see Allamanis et al. [2]. Here, we considered (sub-)tokenizing the program, analogous to NLP, and using a variant of recurrent neural networks. This idea has appeared in numerous papers, e.g., the pioneering work of Raychev et al. [29] for code completion. We have also considered the AST paths encoding pioneered by Alon et al. [4,3]. Researchers have considered more structured networks, like graph neural networks [1] and tree-LSTMs [41]. These would be interesting to consider for future experimentation in the context of adversarial training. The task we evaluated on, code summarization, was first introduced by Allamanis et al. [1].
Transformation Robustness
We now formally define k-adversaries robustness and our robust-optimization objective.
Learning problem. We assume a data distribution D over X × Y, where X is the space of samples and Y is the space of labels. As is standard, we wish to solve the following optimization problem:
argmin w∈H E (x,y)∼D L(w, x, y)
where H is the hypothesis space and L is the loss function. Once we have solved the optimization problem given above, we obtain a w * , which yields a function F w : X → Y.
Abstract syntax trees. We are interested in tasks where the sample space X is that of programs in some programming language. We do not constrain the space of outputs Y-it could be a finite set of labels for classification, a natural-language description, etc.
We view a program x as an abstract syntax tree (AST)-a standard data structure for representing programs. The internal nodes of an AST typically represent program constructs, such as while loops Transformations. A transformation t of a program x transforms it into another program x . Typically, t is defined over ASTs, making it a tree-to-tree transformation. Formally, we think of a transformation as a function t : R × X → X , where R is a space of parameters to the transformation. For example, if t changes the name of a variable, it may need to receive a new variable name as a parameter.
For our goal of semantic robustness, we will focus on semantics-preserving transformations, i.e., ones that do not change the behavior of the program. Consider, e.g., the transformation t shown in Fig. 2 (left), where we replace x > 0 with 0 < x (the transformed subtree is highlighted). Note, however, that our approach is not tied to semantics-preserving transformations, and one could define transformations that, for example, introduce common typos and bugs that programmers make.
Program sketches. It will be helpful for us to think of how to partially apply a transformation t to a program x, without supplying parameters. Intuitively, this operation should result in a set of programs, one for each possible parameter to t. We assume that parameters only affect leaves of the transformed AST. Therefore, when we partially apply a transformation t, it results in a tree with unknown leaves. Equivalently, we can think of such tree as a program sketch: a program with holes. For example, if t changes names of program variables to new given names, applying t partially to our running example if (x > 0) y = false results in the program sketch
if ( 1 > 0) 2 = false
where i are distinct unknown variable names (holes) to be filled in the sketch. The AST of this program sketch is shown in Fig. 2 (right).
k-adversary. Given a set of transformations T , a k-adversary is an oracle that finds a sequence of transformations of size k that maximizes the loss function.
We use q to denote a sequence of transformations t 1 , . . . , t n , and their corresponding parameters r 1 , . . . , r n , where t i ∈ T and r i ∈ R. We use q(x) to denote the program t n (. . . t 2 (r 2 , t 1 (r 1 , x))), i.e., the result of applying all transformations in q to x. Let Q k denote the set of all sequences of transformations and parameters of length k. Given a program x with label y, the goal of the adversary is to transform x to maximize the loss; formally, the adversary solves the following objective function:
max q∈Q k L(w, q(x), y)(1)
Robust-optimization objective. Now that we have formally defined a k-adversary, we can solve a robust-optimization problem [6] to learn a model that is robust to such attacks-we call this k-robustness. Specifically, we solve the following problem:
argmin w∈H E (x,y)∼D max q∈Q k L(w, q(x), y) objective of k-adversary(2)
Informally, instead of computing the loss of a pair (x, y), we consider the worst-case loss resulting from a k-adversary-generated transformation to x. Such robust-optimization objectives have been used for training robust deep neural networks [25]. Our setting of source code, however, results in unique challenges, which we address next.
Adversaries and Efficient Training
We now show how to practically implement a k-adversary and efficiently train k-robust models.
Deploying k-adversaries
It is typically quite expensive to solve the optimization objective of a k-adversary (Eq. (1)). Even if we have k = 1 and a single transformation, the search space can be very large. For example, say we have a transformation that changes names of function arguments. For a function with n arguments, the space of possible parameters to the transformation is roughly |size of vocabulary| n , and vocabulary size is easily in the thousands for recent datasets. Indeed, it can be easily shown that the Eq. (1) is PSPACE-hard, via a reduction from the PSPACE-complete problem of checking that the intersection of a set of automata is empty [23]. (See proof in the Appendix.) This is in contrast to the vision domain, where finding adversarial examples is NP-complete [22].
Algorithm 1 k-adversary for program x with label y
1: Let x = x 2: for all sequences t1, . . . , t k do 3: Let sketch z[·] be t k (. . . t2(·, t1(·, x))) 4: Let r = argmax r∈R L(w, z[r], y) 5: if L(w, z[r], y) > L(w, x , y) then x = z[r] 6: return x
Clearly, a naïve enumeration approach is impractical for implementing a k-adversary. And, unlike with robustness in the vision domain, the space of AST transformations is not differentiable. Nonetheless, we observe that we can break up the search space into two pieces: (1) an enumerative search over sequences of transformations t 1 , . . . , t k , and (2) a gradient-based search over transformation parameters, r 1 , . . . , r k , which allows us to efficiently traverse parameter space.
The full algorithm is shown in Algorithm 1. Given a sequence of transformations t 1 , . . . , t k , the algorithm partially applies it to the input program x. This results in a program sketch z[·]. Recall that a sketch is a program with holes in the leaves of the AST, and so we use [·] to denote a parameter that z[·] can take to fill its holes; as with transformations, we use R to denote the set of possible parameters to z[·]. At this point, we can use existing algorithms to discover a complete program z[r] that maximizes the loss function (approximately).
For example, if the sketch is represented as a sequence of tokens for an LSTM, the holes are simply missing tokens to be inserted in certain locations. As such, attacks from natural language processing, like HotFlip [13], can be used. In our implementation, we use a version of a recent algorithm by Yefet et al. [37] that performs a gradient-based search for picking a token replacement. We first fill in the holes of the program sketch z[·] with temporary special tokens s. We replace the input embedding lookup layer with a differentiable tensor multiplication of one-hot inputs and an embedding matrix. This ensures differentiability up to the input layer v, allowing us to (1) take a gradient-ascent step in the direction that maximizes loss, and (2) approximate the worst token replacement; formally,
v ← v + η · ∇ v L(w, z[s], y) ; r = argmax v
In other words, we pick the replacement r as the token with the maximum value in the vector v , obtained after a gradient-ascent step on the one-hot input v corresponding to each special token in z[s]. We impose additional semantic constraints, e.g., in a sketch like if ( 1 > 0) 2 = false, we enforce that 1 and 2 receive different replacements. See appendix for full details.
Adversarial training
To train a model that is robust to the k-adversary we defined in Algorithm 1, we can solve the robust-optimization problem in Eq. (2), where the inner maximization objective-the k-adversary's objective-is approximated using Algorithm 1.
argmin w∈H E (x,y)∼D max q∈Q k L(w, q(x), y) approximate using Algorithm 1 (3)
Practically, for every program x in a mini-batch, we need to run Algorithm 1 to compute a transformed program x exhibiting worst-case loss. This approach is wildly inefficient during training for two reasons: (1) the combinatorial search space of the k-adversary in Algorithm 1, and (2) the mismatch between program formats for transformation and for training. We discuss both below.
Size of the search space. Given a set of transformations T , Algorithm 1 runs for |T | k iterations, a space that grows exponentially with k. In practice, we observe that it suffices to train with a weak k = 1 adversary, and still be quite robust to stronger adversaries. Therefore, the number of iterations [38,25,36] and NLP [18].
∆F1 (Q 1 R / Q 1 G ) ∆F1 (Q 1 R / Q 1 G ) ∆F1 (Q 1 R / Q 1 G ) ∆F1 (Q 1 R / Q 1 G )AddDeadCode
Representation mismatch. There is usually a mismatch between the program format needed for applying transformations and the program format needed for training a neural network. A program x is an AST and the adversary's transformations are defined over ASTs, but the neural network expects as input a different representation-for example, sequences of (sub)tokens, or as in a recent popular line of work [4,3], a sampled set of paths from one leaf of the AST to another.
Therefore, in Algorithm 1, we have to translate back and forth between ASTs and their neural representation. To be specific, line 3 of Algorithm 1 computes a sequence of transformations over ASTs, resulting in a program sketch z[·]. Then, line 4 requires a neural representation of z[·] to solve the maximization problem. This approach is expensive to employ during training: in every training step, we have to apply transformations using an external program-analysis tool and convert the transformed AST to its neural representation.
We address this challenge as follows: To avoid calling program-analysis tools within the training loop, we pre-generate all possible program sketches considered by the adversary in Algorithm 1. That is, for every program x in the training set, before training, we generate the set of program sketches )))} to avoid performing AST transformations during training. As such, the robust-optimization objective in Eq. (3) reduces to the following:
S x = {z[·] | z[·] = t k (. . . t 2 (·, t 1 (·, xargmin w∈H E (x,y)∼D max r∈R,z[·]∈Sx L(w, z[r], y)(4)
Experimental Evaluation
Research questions. We designed our evaluation to answer the following research questions: (Q1) Does adversarial training improve robustness to semantics-preserving transformations? (Q2) Does training with a k-adversary improve robustness against stronger adversaries (larger k)?
Code-summarization task. We consider the task of automatic code summarization [1]-the prediction of a method's name given its body-which is a non-trivial task for deep-learning models. The performance of models in this task is measured by the F1 score metric.
We experimented with two model architectures: (i) a sequence-to-sequence BiLSTM model (seq2seq) and (ii) Alon et al.'s [3] state-of-the-art code2seq model. The seq2seq model takes sub-tokenized programs as inputs. We built upon the implementation of IBM [20], and trained models for 10 epochs. code2seq is a code-specific architecture that samples paths from the program AST in its encoder, and uses a decoder with attention for predicting output tokens. We used the code2seq TensorFlow code [3], along with their Java and Python 3 path extractors. We train code2seq models for 20 epochs.
Datasets. We conducted experiments on four datasets in two languages: Java, statically typed with types explicitly stated in the code, and Python, a dynamically typed scripting language. The datasets originated from three different sources: (i) code2seq's java-small dataset (c2s/java-small) [3], (ii) GitHub's CodeSearchNet Java and Python datasets (csn/java, csn/python) [19], and (iii) SRI Lab's Py150k dataset (sri/py150) [30]. For computational tractability of adversarial training, we randomly subsample each dataset to train/validation/test sets of 150k/10k/20k each.
Program Transformations. We used two separate code-transformation pipelines: for Java, based on Spoon [28], and for Python, based on Astor [7]. These frameworks apply program transformations to generate program sketches, as described in Sec. 4.2. We implemented a suite of 8 semantics-preserving transformations, listed in Table 1. All transformations except UnrollWhiles are parameterized; therefore, applying the transformations creates program sketches with holes. The transformations are configured to produce only a single hole (e.g., the WrapTryCatch transformation encloses the target method body in a single try/catch statement and replaces the name of the variable holding the caught exception with a single hole). Refer to appendix for exact details.
Adversaries. We consider two variants of the adversary defined in Algorithm 1, a weak and a strong adversary that search through sequences of length k:
Random (weak) adversary (Q k R ): uses a random choice for the parameter r in line 4, i.e., it randomly fills the hole in a sketch;
Gradient-based (strong) adversary (Q k G ): uses the gradient-based approach described in Sec. 4.1 to fill sketch holes with parameters that maximize loss in line 4. Table 1 presents the drops in test performance from attacking normally trained seq2seq and code2seq models with each of the 8 transformations independently, i.e., using a 1-adversary that only has a single transformation in its set T . We observe that the gradient-based attacks are significantly stronger than the random attacks (most cause 2x-15x greater drops in F1). Among the most effective transformations are AddDeadCode, InsertPrintStatements, and WrapTryCatch, which introduce new tokens in the program that confuse the predictor. Note that while the UnrollWhiles transformation has no effect in seq2seq's test performance by itself, it increases the strength of the other attacks when used in tandem with them.
Robustness evaluation
Our approach & baselines. We will use Tr-Q 1 G to denote models trained using our approach (Eq. (4)) with the gradient-based 1-adversary. We consider a series of progressively stronger baselines:
Normal training (Tr-Nor): Models are trained using a standard (non-robust) objective. Data augmentation (Tr-Aug): Minimize the composite loss i L(w, x i , y i ) + L(w, x i , y i ), where, for each x i , a program x i is constructed using a random transformation from Q 1 .
Random adversarial training (Tr-Q 1 R ): Similar to our adversarial-training method, except it uses a random choice for parameters r (line 4 of Algorithm 1), instead of one that maximizes loss.
Robustness against 1-adversary. Table 2 compares our approach and the three baselines on two metrics: (1) the test F1 score (Nor), and (2) the adversarial F1 score (denoted Q 1 G ), where every test point is attacked with the gradient-based k = 1 adversary with all the transformations in Table 1.
We make the following key observations: Tr-Q 1 G models are significantly more robust to attack than all the baselines, especially for seq2seq. Consider the seq2seq models trained on the c2s/java-small dataset. While the normally trained model suffers a drop in F1 of 14 points, from 37.8 to 23.3, Tr-Q 1 G only sees a drop of 8.7 points. Its final F1 of 32.0 is markedly higher than the other models. On the csn/python dataset, the F1 of seq2seq Tr-Nor drops 44% from 28.9 to 16.2. Tr-Q 1 G starts significantly higher at 36.2, and experiences a drop of only 4 points to achieve an F1 score of 32.2 under a Q 1 G attack. On average across datasets, Tr-Q 1 G achieves gains in F1 over Tr-Nor of around 6 (27%) and 14 (74%) points for code2seq and seq2seq, respectively. c2s/java-small csn/java csn/python sri/py150 Figure 3: A comparison of normally trained (Tr-Nor, ), trained with dataset augmentation (Tr-Aug, ), adversarially trained with random parameters (Tr-Q 1 R , ), and adversarially trained (Tr-Q 1 G , ) models on four adversaries, four datasets, and two model architectures (seq2seq and code2seq).
Nor Q 1 R Q 5 R Q 1 G Q 5 G 10 20 30 40 F1 (code2seq) Nor Q 1 R Q 5 R Q 1 G Q 5 G Nor Q 1 R Q 5 R Q 1 G Q 5 G Nor Q 1 R Q 5 R Q 1 G Q 5 G
Our results answer Q1 in the affirmative: adversarial training increases robustness to attack via semantics-preserving program transformations.
Robustness against a 5-adversary. We also studied the robustness of the models against a much stronger adversary. In particular, we considered random and gradient-based 5-adversaries, denoted by Q 5 R and Q 5 G , respectively. Because there are an intractable number (8 5 ≈ 32k) number of transformation sequences of length 5, we use a randomly sampled set of sequences of length 5 (see Appendix for details). In addition, we also evaluated the models against a random 1-adversary, denoted by Q 1 R . The graphs shown in Fig. 3 summarize our findings. In general, the strength of the adversaries increases along the x-axis. Interestingly we observe that the Q 1 G attack is often stronger than the Q 5 R attack, reaffirming the strength of the gradient-based approach to completing sketches and maximizing loss. While the Tr-Q 1 R and Tr-Q 1 G models achieve comparable robustness to the random adversaries, Tr-Q 1 G consistently outperforms Tr-Q 1 R against the gradient-based adversaries. The performance of all models drops significantly against the Q 5 G attack, however the drops suffered by the Tr-Q 1 G models are much less compared to the others, thus displaying remarkable robustness to the very strong attack. For example, for the Q 5 G attack on seq2seq models for csn/java, the F1 score of Tr-Nor drops from 32.3 to 9.8, while the F1 of Tr-Q 1 G drops just 8.1 points from 37.8 to 29.7.
Our results answer Q2 in the affirmative: adversarial training with a small k can improve robustness against a stronger adversary-at least for the case of training with k = 1 and attacking with k = 5.
seq2seq vs. code2seq. It is interesting to compare the robustness of the seq2seq and code2seq models. Although the baseline test F1 scores of code2seq models are much higher than the respective seq2seq models, the models are just as vulnerable to attack. Whereas adversarial training makes the seq2seq model significantly more robust, the robustness gains seen in code2seq are much less. This phenomenon is especially evident in the Python datasets, where there is a gap of ≈16 F1 points between the Tr-Nor and Tr-Q 1 G models under Q 1 G attack for seq2seq, but only 2.8-5.8 F1 points for code2seq (see Table 2). The phenomenon can also be seen in the code2seq graphs in Fig. 3.
We conjecture that this difference arises because the code2seq model uses sampled program AST paths to make its prediction. A single program transformation may affect several AST paths at once, thus making the code2seq model especially vulnerable to attack, even after adversarial training.
Conclusion
To the best of our knowledge, this work is the first to address adversarial training for source-code tasks via a robust-optimization objective. Our approach is general, in that it considers adversaries that perform arbitrary and parameterized sequences of transformations. For future work, it would be interesting to study the effects of adversarial training on other models, e.g., graph neural networks. It would also be interesting to explore other source-code tasks and transformations, such as automated refactoring.
A PSPACE-hardness of a k-adversary
Let D 1 , . . . , D m be m deterministic finite automata (DFAs) over a finite alphabet Σ. Let n be the size of the largest automaton, measured as the number of states. Let L(D i ) be the language defined by D i . The problem of deciding whether ∩ i L(D i ) is empty is PSPACE-complete [23].
Construct a program x with m variables v 1 , . . . , v m with assignment statements that assign each v i the initial state of the DFA D i . Our reduction uses just one transformation t parameterized by the alphabet Σ. t(α, x) (where α ∈ Σ) transforms the program x by updating the rhs of the assignment statements to the next states of all automata, i.e., t simulates a single step of all the automata D i . We also include another transformation that is the identity function, i.e., t(·, x) = x. We assume a special stuck state. Now we construct a 0 − 1 loss function as follows: L(x) = 1 iff the assignment statements in x correspond to accepting states in all DFAs D i .
Note that the intersection is empty iff there is no string of length at most n m that is accepted by all automata. (E.g., for m = 1, if D 1 is non-empty, there has to be a string of length n, the number of states.) Therefore, it is easy to see that
max q∈Q k L(x) is 1 iff ∩ n i=1 L(D i ) is not empty, where k = n m .
Since PSPACE is closed under complement, we obtain the following theorem. Figure 4 shows an overview of the framework we built to support our experiments and bootstrap future efforts in the space of semantically robust models of source code. In general, the framework supports arbitrary input datasets. For each input dataset, a normalization procedure must be supplied to translate the input data into a consistent encoding. Once data is normalized, the rest of the framework can be utilized to (i) apply program transformations to arbitrary depth, (ii) train models (normally), (iii) apply either gradient or random targeting to the holes in transformed programs (for seq2seq and code2seq), (iv) adversarially train models, and (v) adversarially attack and evaluate the trained models. The source code is available anonymously [15].
B.2 Details of Program Transformations
We employ a total of eight semantics preserving transforms. Each program transform produces, as output, a program sketch (that is, a program with one or more holes). To turn a program sketch into an complete program, each hole must be replaced with a valid token. For each of the eight transforms we implemented, valid hole replacements take the form of either variable names or string literals. Detailed descriptions of our eight semantics preserving transforms are given below: Figure 4: An overview of our framework, which enables our experimental evaluation and bootstraps efforts towards training more robust models of source code. With its plug-and-play architecture it can, in the future, be extended with more models, data sources, and program transforms.
InsertPrintStatements:
A single System.out.println("<HOLE>"), in Java, and print('<HOLE>') in Python, is appended to the beginning or end of the target program. The insertion location (either beginning, or end) is chosen at random.
C Experimental Details
C.1 Datasets
We conduct our experiments on four datasets in two different languages (java, python): c2s/javasmall, csn/java, csn/python and sri/py150. Each dataset contains over 0.5M data points (method body-method name pairs). The original sizes of the datasets are shown in Table 3.
Adversarial training in our domain of source code is an expensive process: we have to pre-generate several transformed versions of each data point as program sketches, and then repeatedly run the gradient-directed attack to fill in the holes in the sketches. Running extensive experiments across the 4 datasets and 2 models (seq2seq, code2seq) was computationally intractable, both in terms of time and space. Thus, we randomly subsample the four datasets to have train/validation/test sets of 150k/10k/20k each. The datasets remain sizeable, and thus we find that this only has a minimal effect on model performance.
C.2 Models seq2seq
The seq2seq models were given sub-tokenized programs as input, which were obtained by splitting up camel and snake-case, and other minor preprocessing. We trained 2-layer BiLSTM models, with 512 units in the encoder and decoder, with embedding sizes of 512.
code2seq code2seq is the current state-of-the-art model for the task of code summarization. We built upon the implementation from the authors of code2seq 4 , and use the original model parameters. For both seq2seq and code2seq, we use input and output vocabularies of size 15k and 5k respectively. All models were trained and evaluated using NVIDIA GPUs (GeForce RTX 2080 Ti and Tesla V100).
2
C.3 Adversarial Training
Adversarial training is known to improve robustness to attack at the cost of degraded performance on clean data [34]. To maintain performance on clean data, we train with the following composite loss:
i λ · L(x i , y i ) + (1 − λ) · L adv (x i , y i )
where L is the normal training loss and L adv is the loss from the robust optimization objective described in Sec. 4. The λ hyperparameter controls the trade-off between performance and robustness, we picked λ = 0.4 in our experiments after a grid search.
The robust optimization objective for adversarial training requires choosing the worst transformation for each data point in each mini-batch of training. While this can be efficiently implemented in computer vision using PGD, it is very expensive to do so in our setting. For tractability of training, we do the following:
• We apply the gradient attack periodically during training to generate new token replacements for the program sketches. For seq2seq we do this after every epoch, and for code2seq we do this after every two epochs.
• Instead of picking the worst transformation for each individual point in a mini-batch, we pick the transformation that does the worst for all the points on the whole (i.e. highest loss over the batch).
C.4 Gradient Attack
We adapt the attack for code models from Yefet et al. [37], to find replacements for the holes in the program sketches. We represent holes in the program sketches by special tokens. A sketch obtained after applying more than one transform would contain multiple variants of the special tokens, as the holes must be filled in independently. The summary of the attack on seq2seq is described below:
• During the training of the model, the special tokens are added to the input vocabulary. The embedding layer in the model encoder is replaced by an embedding matrix. The input tensor of shape (batch_size, max_seq_len) is converted to its one-hot representation v, a tensor of shape (batch_size, max_seq_len,|source_vocab|). The embedding look-up is replaced by a multiplication between v and the embedding matrix, the rest of the encoder remains the same. This step is necessary for differentiability upto the token-level input representation.
• For simplicity, we implement the attack to depth=1 and width=1. After one forward, we calculate the loss and backpropagate its gradient to the v layer. We take a gradient-ascent step on v, to yield v . We approximate the worst token replacement for each special token by averaging gradients over each occurrence in the input and choose the token corresponding to the maximum. We impose an additional semantic constraint that the token replacements made for different special tokens are distinct.
In code2seq, the subtoken embedding layer is replaced by the tensor multiplication.
C.5 Implementing a tractable 5-Adversary
In Sec. 5.1, we evaluate the robustness of the trained models on the 5-adversary Q 5 G , i.e. an adversary that performs 5 semantics preserving transformations on the input. With our suite of 8 transforms, it results in a total of 8 5 = 32768 possible sequences against which we need to evaluate each test data point. This is computationally intractable for two reasons:
• For each of the 20k test points in each dataset, we choose the sequence of transforms which results in the greatest loss as the attack. To evaluate against each of the 8 5 transforms, it would require millions of forward passes through the model. • Each sequence of transforms is generated in two steps: (i) application of transformations to generate program sketches, and (ii) gradient attack to fill in the holes of the sketches. Both these steps are expensive and time consuming. In particular, generating 10 transforms for 20k data points requires 10 minutes for step (i) and 120 minutes for step (ii) for the code2seq model (seq2seq is somewhat faster, but still very slow). In addition, after generating program sketches and using a gradient attack to fill holes, we still must perform adversarial evaluation which, for code2seq on 200k data points (20k points × 10 transformed variants per point), takes an additional 60 minutes.
Due to the above reasons, we choose to implement the 5-adversary by randomly sampling 10 sequences of transformations of length 5. We investigate the effect of increasing the number of transforms sampled, and we find that the strength of the attack saturates. This is seen in Table 4 for the Tr-Q 1 G model on the csn/java dataset. Because of this prominent saturation, we chose to use only 10 samples to keep total evaluation time to less than 24 hours (using 4 V100 GPUs). Table 5 contain numbers corresponding to the graphs in Fig. 2 of the main text. Each table presents results for the different models under attacks of increasing strength (Q 1 R , Q 5 R , Q 1 G , Q 5 G ). We note that, in all but one case, the adversarially trained model (Tr-Q 1 G ), performs the best under attack.
D Full Robustness Evaluation Results
for (Object elem: this.elements) if (elem.equals(target)) System.out.println("Found"); return i; i++; return -1;
Figure 1 :
1code2seq[3] correctly predicts function name: indexOfTarget. After the highlighted logging statements are added, it predicts search.
Figure 2 :
2Left: Example AST transformation (no parameters). Right: AST of program sketch with two holes and conditionals, and the leaves of the tree denote variable names, constants, fields, etc.Fig. 2 (left)shows the AST representation of the code snippet if (x > 0) y = false.
6 .
6UnrollWhiles: a single, randomly selected, while loop in the target program has its loop body unrolled exactly one step. No holes are created by this transform. 7. WrapTryCatch: the target program is wrapped by a single try { ... } catch (...) { ... } statement. The catch statement passes along the caught exception. A hole is used in the place of the name of the caught exception variable (e.g., catch (Exception <HOLE>)).
Table 1 :
1Our suite of semantics-preserving transformations, along with the drops in F1 obtained by random (Q 1 R ) and gradient-based (Q 1 G ) adversaries per transformation, for normally trained models on c2s/java-small.Transform
seq2seq (F1: 37.8)
code2seq (F1: 41.4)
Transform (Cont.)
seq2seq (F1: 37.8)
code2seq (F1: 41.4)
Table 2 :
2Evaluation of our approach and baselines across four datasets. Numbers in brackets are the difference in adversarial F1 compared to the normally trained model (higher is better).Model
Training
c2s/java-small
csn/java
csn/python
sri/py150
Nor
Q 1
G
Nor
Q 1
G
Nor
Q 1
G
Nor
Q 1
G
seq2seq
Tr-Nor
37.8
23.3
32.3
17.2
28.9
16.2
34.3
22.0
Tr-Aug 38.8 27.8 [+4.4] 32.6
21.4 [+4.3]
29.8
20.9 [+4.7]
33.7
24.0 [+2.0]
Tr-Q 1
R
40.0 27.5 [+4.2] 38.1 30.1 [+12.9] 36.6 29.8 [+13.5] 41.8 33.4 [+11.4]
Tr-Q 1
G
40.7 32.0 [+8.7] 37.8 32.8 [+15.6] 36.2 32.2 [+16.0] 41.8 37.1 [+15.1]
code2seq
Tr-Nor
41.4
24.6
39.2
19.6
37.4
21.0
39.7
23.7
Tr-Aug 42.3 23.4 [-1.2] 39.4
19.7 [+0.1]
36.8
20.8 [-0.2]
39.7
24.4 [+0.7]
Tr-Q 1
R
42.3 28.1 [+3.5] 40.0
23.5 [+3.9]
37.6
22.5 [+1.6]
40.0
26.6 [+2.9]
Tr-Q 1
G
43.6 31.6 [+7.0] 40.6
27.5 [+7.9]
37.3
23.8 [+2.8]
40.2
29.5 [+5.8]
Table 3 :
3The original sizes of the datasets
Number of Samples Adversarial F1 ∆F1 (Baseline: 37.8)10
29.7
-8.1
20
28.5
-9.3
30
27.9
-9.9
40
27.6
-10.2
50
27.4
-10.4
Table 4 :
4An exploration of increasing sample counts for Q 5 G attacks, on the Tr-Q 1 G model on the csn/java test set. Note that the attack strength saturates very quickly.
Table 5 :
5Evaluation of our approach and baselines across four datasets. Numbers in brackets are the difference in adversarial F1 compared to the normally trained model (higher is better). (a) Results for the random 1-adversary (Q 1 R ). 40.0 33.6 [+4.1] 38.1 34.0 [+10.6] 36.6 33.3 [+11.6] 41.8 37.8 [+8.5] Tr-Q 1 G 40.7 34.8 [+5.2] 37.8 34.3 [+10.8] 36.2 33.5 [+11.8] 41.8 38.1 [+8.8] Tr-Aug 38.8 31.7 [+12.9] 32.6 26.3 [+10.4] 29.8 24.2 [+9.0] 33.7 27.9 [+2.7] Tr-Q 1 R 40.0 30.4 [+11.6] 38.1 30.6 [+14.7] 36.6 30.8 [+15.5] 41.8 34.7 [+9.5] Tr-Q 1 G 40.7 31.3 [+12.5] 37.8 32.1 [+16.1] 36.2 31.6 [+16.4] 41.8 35.4 [+10.2] Tr-Aug 38.8 27.8 [+4.4] 32.6 40.0 27.5 [+4.2] 38.1 30.1 [+12.9] 36.6 29.8 [+13.5] 41.8 33.4 [+11.4] Tr-Q 1 G 40.7 32.0 [+8.7] 37.8 32.8 [+15.6] 36.2 32.2 [+16.0] 41.8 37.1 [+15.1] Tr-Aug 42.3 23.4 [-1.2] 39.4 40.0 21.7 [+10.2] 38.1 25.1 [+15.3] 36.6 25.3 [+14.9] 41.8 28.1 [+11.6] Tr-Q 1 G 40.7 26.3 [+14.8] 37.8 29.7 [+19.9] 36.2 29.4 [+19.0] 41.8 33.4 [+16.8] 43.6 27.3 [+11.9] 40.6 22.0 [+10.0] 37.3Model
Training
c2s/java-small
csn/java
csn/python
sri/py150
Nor
Q 1
R
Nor
Q 1
R
Nor
Q 1
R
Nor
Q 1
R
seq2seq
Tr-Nor
37.8
29.5
32.3
23.5
28.9
21.7
34.3
29.3
Tr-Aug 38.8 33.8 [+4.3] 32.6
29.1 [+5.7]
29.8
27.2 [+5.5]
33.7 30.5 [+1.2]
Tr-Q 1
R
code2seq
Tr-Nor
41.4
29.5
39.2
27.3
37.4
25.0
39.7
28.7
Tr-Aug 42.3 29.4 [-0.1] 39.4
26.9 [-0.4]
36.8
25.5 [+0.5]
39.7 29.1 [+0.4]
Tr-Q 1
R
42.3 32.1 [+2.6] 40.0
29.6 [+2.3]
37.6
26.0 [+1.0]
40.0 30.8 [+2.1]
Tr-Q 1
G
43.6 33.5 [+4.0] 40.6
30.4 [+3.1]
37.3
26.7 [+1.7]
40.2 31.2 [+2.4]
(b) Results for the random 5-adversary (Q 5
R ).
Model
Training
c2s/java-small
csn/java
csn/python
sri/py150
Nor
Q 5
R
Nor
Q 5
R
Nor
Q 5
R
Nor
Q 5
R
seq2seq
Tr-Nor
37.8
18.8
32.3
15.9
28.9
15.2
34.3
25.2
code2seq
Tr-Nor
41.4
24.4
39.2
22.0
37.4
22.8
39.7
24.4
Tr-Aug 42.3
23.4 [-1.0]
39.4
20.5 [-1.5]
36.8
22.8 [+0.0]
39.7
24.8 [+0.4]
Tr-Q 1
R
42.3
29.9 [+5.5]
40.0
26.2 [+4.2]
37.6
23.9 [+1.1]
40.0
28.2 [+3.8]
Tr-Q 1
G
43.6
31.0 [+6.6]
40.6
27.2 [+5.2]
37.3
24.6 [+1.8]
40.2
28.4 [+4.0]
(c) Results for the targeted 1-adversary (Q 1
G ).
Model
Training
c2s/java-small
csn/java
csn/python
sri/py150
Nor
Q 1
G
Nor
Q 1
G
Nor
Q 1
G
Nor
Q 1
G
seq2seq
Tr-Nor
37.8
23.3
32.3
17.2
28.9
16.2
34.3
22.0
21.4 [+4.3]
29.8
20.9 [+4.7]
33.7
24.0 [+2.0]
Tr-Q 1
R
code2seq
Tr-Nor
41.4
24.6
39.2
19.6
37.4
21.0
39.7
23.7
19.7 [+0.1]
36.8
20.8 [-0.2]
39.7
24.4 [+0.7]
Tr-Q 1
R
42.3 28.1 [+3.5] 40.0
23.5 [+3.9]
37.6
22.5 [+1.6]
40.0
26.6 [+2.9]
Tr-Q 1
G
43.6 31.6 [+7.0] 40.6
27.5 [+7.9]
37.3
23.8 [+2.8]
40.2
29.5 [+5.8]
(d) Results for the targeted 5-adversary (Q 5
G ).
Model
Training
c2s/java-small
csn/java
csn/python
sri/py150
Nor
Q 5
G
Nor
Q 5
G
Nor
Q 5
G
Nor
Q 5
G
seq2seq
Tr-Nor
37.8
11.5
32.3
9.8
28.9
10.4
34.3
16.5
Tr-Aug 38.8 23.5 [+12.0] 32.6
15.6 [+5.8]
29.8
14.4 [+4.0]
33.7
19.1 [+2.6]
Tr-Q 1
R
code2seq
Tr-Nor
41.4
15.5
39.2
12.1
37.4
15.6
39.7
18.2
Tr-Aug 42.3
14.5 [-0.9]
39.4
11.8 [-0.3]
36.8
15.8 [+0.2]
39.7
17.8 [-0.4]
Tr-Q 1
R
42.3
22.4 [+7.0]
40.0
17.6 [+5.5]
37.6
17.6 [+2.0]
40.0
22.1 [+3.9]
Tr-Q 1
G
20.6 [+5.0]
40.2
25.1 [+6.9]
A preprint of our work appeared earlier on arXiv than[9].
We modified the Python extractor (making it similar to the Java one), resulting in improved performance.
https://github.com/tech-srl/code2seq
A convolutional attention network for extreme summarization of source code. CoRR, abs/1602.03001. Miltiadis Allamanis, Hao Peng, Charles A Sutton, Miltiadis Allamanis, Hao Peng, and Charles A. Sutton. A convolutional attention network for extreme summarization of source code. CoRR, abs/1602.03001, 2016. URL http://arxiv. org/abs/1602.03001.
A survey of machine learning for big code and naturalness. Miltiadis Allamanis, T Earl, Premkumar Barr, Charles Devanbu, Sutton, ACM Computing Surveys (CSUR). 514Miltiadis Allamanis, Earl T Barr, Premkumar Devanbu, and Charles Sutton. A survey of machine learning for big code and naturalness. ACM Computing Surveys (CSUR), 51(4):1-37, 2018.
Uri Alon, Shaked Brody, Omer Levy, Eran Yahav, arXiv:1808.01400Generating sequences from structured representations of code. 2arXiv preprintUri Alon, Shaked Brody, Omer Levy, and Eran Yahav. code2seq: Generating sequences from structured representations of code. arXiv preprint arXiv:1808.01400, 2018.
Learning distributed representations of code. Uri Alon, Meital Zilberstein, Omer Levy, Eran Yahav, Proceedings of the ACM on Programming Languages. 2Uri Alon, Meital Zilberstein, Omer Levy, and Eran Yahav. code2vec: Learning distributed representations of code. Proceedings of the ACM on Programming Languages, 3(POPL):1-29, 2019.
Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. Anish Athalye, Nicholas Carlini, David Wagner, arXiv:1802.00420arXiv preprintAnish Athalye, Nicholas Carlini, and David Wagner. Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. arXiv preprint arXiv:1802.00420, 2018.
Robust optimization. Aharon Ben-Tal, Laurent El Ghaoui, Arkadi Nemirovski, Princeton University Press28Aharon Ben-Tal, Laurent El Ghaoui, and Arkadi Nemirovski. Robust optimization, volume 28. Princeton University Press, 2009.
. Berkerpeksag, Berkerpeksag. berkerpeksag/astor, Jan 2020. URL https://github.com/berkerpeksag/ astor.
Practical black-box attacks on deep neural networks using efficient query mechanisms. Arjun Nitin, Warren Bhagoji, Bo He, Dawn Li, Song, European Conference on Computer Vision. SpringerArjun Nitin Bhagoji, Warren He, Bo Li, and Dawn Song. Practical black-box attacks on deep neural networks using efficient query mechanisms. In European Conference on Computer Vision, pages 158-174. Springer, 2018.
Adversarial robustness for code. Pavol Bielik, Martin Vechev, Pavol Bielik and Martin Vechev. Adversarial robustness for code, 2020.
Evasion attacks against machine learning at test time. Battista Biggio, Igino Corona, Davide Maiorca, Blaine Nelson, Nedim Šrndić, Pavel Laskov, Giorgio Giacinto, Fabio Roli, Joint European conference on machine learning and knowledge discovery in databases. SpringerBattista Biggio, Igino Corona, Davide Maiorca, Blaine Nelson, Nedim Šrndić, Pavel Laskov, Giorgio Giacinto, and Fabio Roli. Evasion attacks against machine learning at test time. In Joint European conference on machine learning and knowledge discovery in databases, pages 387-402. Springer, 2013.
Towards evaluating the robustness of neural networks. Nicholas Carlini, David Wagner, Security and Privacy (SP), 2017 IEEE Symposium on. IEEENicholas Carlini and David Wagner. Towards evaluating the robustness of neural networks. In Security and Privacy (SP), 2017 IEEE Symposium on, pages 39-57. IEEE, 2017.
Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models. Pin-Yu Chen, Huan Zhang, Yash Sharma, Jinfeng Yi, Cho-Jui Hsieh, Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security. the 10th ACM Workshop on Artificial Intelligence and SecurityACMPin-Yu Chen, Huan Zhang, Yash Sharma, Jinfeng Yi, and Cho-Jui Hsieh. Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models. In Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security, pages 15-26. ACM, 2017.
Hotflip: White-box adversarial examples for text classification. Javid Ebrahimi, Anyi Rao, Daniel Lowd, Dejing Dou, Javid Ebrahimi, Anyi Rao, Daniel Lowd, and Dejing Dou. Hotflip: White-box adversarial examples for text classification, 2017.
Robust physical-world attacks on deep learning visual classification. Kevin Eykholt, Ivan Evtimov, Earlence Fernandes, Bo Li, Amir Rahmati, Chaowei Xiao, Atul Prakash, Tadayoshi Kohno, Dawn Song, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionKevin Eykholt, Ivan Evtimov, Earlence Fernandes, Bo Li, Amir Rahmati, Chaowei Xiao, Atul Prakash, Tadayoshi Kohno, and Dawn Song. Robust physical-world attacks on deep learning visual classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1625-1634, 2018.
Our data-generation framework for adversarial training. Frameworkurl, FrameworkURL. Our data-generation framework for adversarial training, Feb 2020. URL https://anonymous.4open.science/r/4ac75222-3673-418e-8b86-9f36ba29fc59.
Bae: Bert-based adversarial examples for text classification. Siddhant Garg, Goutham Ramakrishnan, Siddhant Garg and Goutham Ramakrishnan. Bae: Bert-based adversarial examples for text classification, 2020.
J Ian, Goodfellow, arXiv:1412.6572Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprintIan J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversar- ial examples. arXiv preprint arXiv:1412.6572, 2014.
Achieving verified robustness to symbol substitutions via interval bound propagation. Po-Sen Huang, Robert Stanforth, Johannes Welbl, Chris Dyer, Dani Yogatama, Sven Gowal, Krishnamurthy Dvijotham, Pushmeet Kohli, 10.18653/v1/D19-1419Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing. Kentaro Inui, Jing Jiang, Vincent Ng, and Xiaojun Wanthe 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language ProcessingHong Kong, ChinaAssociation for Computational LinguisticsPo-Sen Huang, Robert Stanforth, Johannes Welbl, Chris Dyer, Dani Yogatama, Sven Gowal, Krishnamurthy Dvijotham, and Pushmeet Kohli. Achieving verified robustness to symbol substitutions via interval bound propagation. In Kentaro Inui, Jing Jiang, Vincent Ng, and Xiaojun Wan, editors, Proceedings of the 2019 Conference on Empirical Methods in Natu- ral Language Processing and the 9th International Joint Conference on Natural Language Processing, EMNLP-IJCNLP 2019, Hong Kong, China, November 3-7, 2019, pages 4081- 4091. Association for Computational Linguistics, 2019. doi: 10.18653/v1/D19-1419. URL https://doi.org/10.18653/v1/D19-1419.
Codesearchnet challenge: Evaluating the state of semantic code search. Hamel Husain, Ho-Hsiang Wu, Tiferet Gazit, Miltiadis Allamanis, Marc Brockschmidt, Hamel Husain, Ho-Hsiang Wu, Tiferet Gazit, Miltiadis Allamanis, and Marc Brockschmidt. Codesearchnet challenge: Evaluating the state of semantic code search, 2019.
IBM. Ibm/pytorch-seq2seq. IBM. Ibm/pytorch-seq2seq, Jan 2020. URL https://github.com/IBM/pytorch-seq2seq.
Black-box adversarial attacks with limited queries and information. Andrew Ilyas, Logan Engstrom, Anish Athalye, Jessy Lin, arXiv:1804.08598arXiv preprintAndrew Ilyas, Logan Engstrom, Anish Athalye, and Jessy Lin. Black-box adversarial attacks with limited queries and information. arXiv preprint arXiv:1804.08598, 2018.
Reluplex: An efficient smt solver for verifying deep neural networks. Guy Katz, Clark Barrett, L David, Kyle Dill, Julian, Kochenderfer, International Conference on Computer Aided Verification. SpringerGuy Katz, Clark Barrett, David L Dill, Kyle Julian, and Mykel J Kochenderfer. Reluplex: An efficient smt solver for verifying deep neural networks. In International Conference on Computer Aided Verification, pages 97-117. Springer, 2017.
Lower bounds for natural proof systems. Dexter Kozen, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977). IEEEDexter Kozen. Lower bounds for natural proof systems. In 18th Annual Symposium on Foundations of Computer Science (sfcs 1977), pages 254-266. IEEE, 1977.
Discrete adversarial attacks and submodular optimization with applications to text classification. Qi Lei, Lingfei Wu, Pin-Yu Chen, Alexandros G Dimakis, Inderjit S Dhillon, Michael Witbrock, SysML. Qi Lei, Lingfei Wu, Pin-Yu Chen, Alexandros G. Dimakis, Inderjit S. Dhillon, and Michael Witbrock. Discrete adversarial attacks and submodular optimization with applications to text classification. In SysML, 2019.
Towards deep learning models resistant to adversarial attacks. Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, Adrian Vladu, International Conference on Learning Representations. Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id=rJzIBfZAb.
Did the model understand the question?. Ankur Pramod Kaushik Mudrakarta, Mukund Taly, Kedar Sundararajan, Dhamdhere, In ACL. Pramod Kaushik Mudrakarta, Ankur Taly, Mukund Sundararajan, and Kedar Dhamdhere. Did the model understand the question? In ACL, 2018.
Practical black-box attacks against machine learning. Nicolas Papernot, Patrick Mcdaniel, Ian Goodfellow, Somesh Jha, Ananthram Berkay Celik, Swami, Proceedings of the 2017 ACM on Asia Conference on Computer and Communications Security. the 2017 ACM on Asia Conference on Computer and Communications SecurityACMNicolas Papernot, Patrick McDaniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Anan- thram Swami. Practical black-box attacks against machine learning. In Proceedings of the 2017 ACM on Asia Conference on Computer and Communications Security, pages 506-519. ACM, 2017.
Spoon: A Library for Implementing Analyses and Transformations of Java Source Code. Software: Practice and Experience. Renaud Pawlak, Martin Monperrus, Nicolas Petitprez, Carlos Noguera, Lionel Seinturier, 10.1002/spe.234646Renaud Pawlak, Martin Monperrus, Nicolas Petitprez, Carlos Noguera, and Lionel Seinturier. Spoon: A Library for Implementing Analyses and Transformations of Java Source Code. Software: Practice and Experience, 46:1155-1179, 2015. doi: 10.1002/spe.2346. URL https://hal.archives-ouvertes.fr/hal-01078532/document.
Code completion with statistical language models. Veselin Raychev, Martin Vechev, Eran Yahav, Proceedings of the 35th ACM SIGPLAN Conference on Programming Language Design and Implementation. the 35th ACM SIGPLAN Conference on Programming Language Design and ImplementationVeselin Raychev, Martin Vechev, and Eran Yahav. Code completion with statistical language models. In Proceedings of the 35th ACM SIGPLAN Conference on Programming Language Design and Implementation, pages 419-428, 2014.
Probabilistic model for code with decision trees. Veselin Raychev, Pavol Bielik, Martin Vechev, 10.1145/2983990.2984041Proceedings of the 2016 ACM SIGPLAN International Conference on Object-Oriented Programming, Systems, Languages, and Applications, OOPSLA 2016. the 2016 ACM SIGPLAN International Conference on Object-Oriented Programming, Systems, Languages, and Applications, OOPSLA 2016New York, NY, USAAssociation for Computing MachineryVeselin Raychev, Pavol Bielik, and Martin Vechev. Probabilistic model for code with decision trees. In Proceedings of the 2016 ACM SIGPLAN International Conference on Object-Oriented Programming, Systems, Languages, and Applications, OOPSLA 2016, page 731-747, New York, NY, USA, 2016. Association for Computing Machinery. ISBN 9781450344449. doi: 10.1145/2983990.2984041. URL https://doi.org/10.1145/2983990.2984041.
Accessorize to a crime: Real and stealthy attacks on state-of-the-art face recognition. Mahmood Sharif, Sruti Bhagavatula, Lujo Bauer, Michael K Reiter, Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security. the 2016 ACM SIGSAC Conference on Computer and Communications SecurityACMMahmood Sharif, Sruti Bhagavatula, Lujo Bauer, and Michael K Reiter. Accessorize to a crime: Real and stealthy attacks on state-of-the-art face recognition. In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pages 1528-1540. ACM, 2016.
Combinatorial sketching for finite programs. Armando Solar-Lezama, Liviu Tancau, Rastislav Bodik, Sanjit Seshia, Vijay Saraswat, Proceedings of the 12th international conference on Architectural support for programming languages and operating systems. the 12th international conference on Architectural support for programming languages and operating systemsArmando Solar-Lezama, Liviu Tancau, Rastislav Bodik, Sanjit Seshia, and Vijay Saraswat. Combinatorial sketching for finite programs. In Proceedings of the 12th international conference on Architectural support for programming languages and operating systems, pages 404-415, 2006.
Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, arXiv:1312.6199Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. arXiv preprintChristian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfel- low, and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013.
Robustness may be at odds with accuracy. Dimitris Tsipras, Shibani Santurkar, Logan Engstrom, Alexander Turner, Aleksander Madry, International Conference on Learning Representations. Dimitris Tsipras, Shibani Santurkar, Logan Engstrom, Alexander Turner, and Aleksander Madry. Robustness may be at odds with accuracy. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id=SyxAb30cY7.
Coset: A benchmark for evaluating neural program embeddings. Ke Wang, Mihai Christodorescu, Ke Wang and Mihai Christodorescu. Coset: A benchmark for evaluating neural program embeddings, 2019.
Eric Wong, Leslie Rice, J Zico Kolter, arXiv:2001.03994Fast is better than free: Revisiting adversarial training. arXiv preprintEric Wong, Leslie Rice, and J Zico Kolter. Fast is better than free: Revisiting adversarial training. arXiv preprint arXiv:2001.03994, 2020.
Adversarial examples for models of code. Noam Yefet, Uri Alon, Eran Yahav, arXiv:1910.07517arXiv preprintNoam Yefet, Uri Alon, and Eran Yahav. Adversarial examples for models of code. arXiv preprint arXiv:1910.07517, 2019.
You only propagate once: Accelerating adversarial training via maximal principle. Dinghuai Zhang, Tianyuan Zhang, Yiping Lu, Zhanxing Zhu, Bin Dong, Advances in Neural Information Processing Systems. Dinghuai Zhang, Tianyuan Zhang, Yiping Lu, Zhanxing Zhu, and Bin Dong. You only prop- agate once: Accelerating adversarial training via maximal principle. In Advances in Neural Information Processing Systems, pages 227-238, 2019.
Generating adversarial examples for holding robustness of source code processing models. Huangzhao Zhang, Zhuo Li, Ge Li, Lei Ma, Yang Liu, Zhi Jin, Huangzhao Zhang, Zhuo Li, Ge Li, Lei Ma, Yang Liu, and Zhi Jin. Generating adversarial examples for holding robustness of source code processing models. 2020. URL https: //www.aaai.org/Papers/AAAI/2020GB/AAAI-ZhangH.6730.pdf.
Adversarial attacks on deep learning models in natural language processing: A survey. Wei Emma Zhang, Z Quan, Sheng, Chenliang Alhazmi, Li, arXiv:1901.06796arXiv preprintWei Emma Zhang, Quan Z Sheng, AHOUD Alhazmi, and CHENLIANG LI. Adversarial attacks on deep learning models in natural language processing: A survey. arXiv preprint arXiv:1901.06796, 2019.
Neuralaugmented static analysis of android communication. Jinman Zhao, Aws Albarghouthi, Vaibhav Rastogi, Somesh Jha, Damien Octeau, Proceedings of the 2018 26th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering. the 2018 26th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software EngineeringJinman Zhao, Aws Albarghouthi, Vaibhav Rastogi, Somesh Jha, and Damien Octeau. Neural- augmented static analysis of android communication. In Proceedings of the 2018 26th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering, pages 342-353, 2018.
AddDeadCode: a single statement of the form if (false) { int <HOLE> = 0; } is appended to the beginning or end of the target program. The insertion location (either beginning. or end) is chosen at randomAddDeadCode: a single statement of the form if (false) { int <HOLE> = 0; } is ap- pended to the beginning or end of the target program. The insertion location (either begin- ning, or end) is chosen at random.
RenameLocalVariables: a single, randomly selected, local variable declared in the target program has its name replaced by a hole. RenameLocalVariables: a single, randomly selected, local variable declared in the target program has its name replaced by a hole.
RenameParmeters: a single, randomly selected, parameter in the target program has its name replaced by a hole. RenameParmeters: a single, randomly selected, parameter in the target program has its name replaced by a hole.
RenameFields: a single, randomly selected, referenced field (this.field in Java, or self.field in Python) has its name replaced by a hole. RenameFields: a single, randomly selected, referenced field (this.field in Java, or self.field in Python) has its name replaced by a hole.
ReplaceTrueFalse: a single, randomly selected, boolean literal is replaced by an equivalent expression containing a single hole. e.g., ("<HOLE>" == "<HOLE>") to replace trueReplaceTrueFalse: a single, randomly selected, boolean literal is replaced by an equivalent expression containing a single hole (e.g., ("<HOLE>" == "<HOLE>") to replace true).
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[
"A Short Guide to Orbifold Deconstruction",
"A Short Guide to Orbifold Deconstruction"
] | [
"Peter Bantay [email protected] \nInstitute for Theoretical Physics\nEötvös Lóránd University\nPázmány Péter s. 1/AH-1117BudapestHungary\n"
] | [
"Institute for Theoretical Physics\nEötvös Lóránd University\nPázmány Péter s. 1/AH-1117BudapestHungary"
] | [] | We study the problem of orbifold deconstruction, i.e., the process of recognizing, using only readily available information, whether a given conformal model can be realized as an orbifold, and the identification of the twist group and the original conformal model. | 10.3842/sigma.2019.027 | [
"https://arxiv.org/pdf/1810.00145v4.pdf"
] | 104,292,145 | 1810.00145 | 6fd6f246b55deaf668ce26ead12f3111c42df1c9 |
A Short Guide to Orbifold Deconstruction
Published online April 09, 2019
Peter Bantay [email protected]
Institute for Theoretical Physics
Eötvös Lóránd University
Pázmány Péter s. 1/AH-1117BudapestHungary
A Short Guide to Orbifold Deconstruction
Published online April 09, 201910.3842/SIGMA.2019.027Received September 28, 2018, in final form March 27, 2019;Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 027, 10 pagesconformal symmetryorbifold models 2010 Mathematics Subject Classification: 11F9913C05
We study the problem of orbifold deconstruction, i.e., the process of recognizing, using only readily available information, whether a given conformal model can be realized as an orbifold, and the identification of the twist group and the original conformal model.
Introduction
Orbifolding [14,15,16], i.e., the gauging of (discrete) symmetries of a conformal model, is one of the most important procedures for obtaining new models from known ones, especially since it respects such important properties as the central charge, unitarity and rationality 1 .
Unfortunately, the precise determination of the properties of the orbifold could be quite involved in general, and effective techniques are only known for special types, like holomorphic [1,2,12,13] or permutation orbifolds [3,5,8,9,20,25].
The present note addresses the inverse problem: instead of trying to determine the structure of an orbifold from the knowledge of the original model and the action of the twist group, we ask for effective procedures to recognize whether a conformal model could be obtained as an orbifold, and if so, to identify the original model together with the relevant twist group. That such procedures could exist is actually not that surprising, as simple current extensions [4,21] are nothing but an example of this for Abelian twist groups. But the question is how the corresponding results could be extended to more general orbifolds, and what kind of new structures emerge in the process. We should stress that our approach in the present note is computational: we aim to identify in a down-to-earth manner the relevant conformal models with the help of some simple, mostly numerical data (like conformal weights, quantum dimensions and fusion rules of the primary fields) [10], providing explicit expressions for the characteristic quantities of the deconstructed model. We have no doubt that the whole process could be described elegantly in more abstract terms [19,24], but the relevant techniques seem (at least to us) less amenable to practical computations.
In the next section we'll summarize those general features of orbifold constructions that will form the basis of the deconstruction procedure. Then we'll turn to the description of the procedure itself, while the last section provides an outlook on questions not addressed in this note. We stress that our exposition is informal, presenting only the most important ideas and their mutual interrelations, but omitting formal arguments. Most of these arguments can be supplied readily, while some of them seem to require more effort, but with hindsight one could This paper is a contribution to the Special Issue on Moonshine and String Theory. The full collection is available at https://www.emis.de/journals/SIGMA/moonshine.html 1 It has even been argued that all rational conformal models may be obtained either as GKO cosets or as orbifolds thereof.
argue that a strong supporting evidence of the ideas to be presented is that they lead to a coherent algorithmic procedure giving meaningful answers that agree with the correct ones in all known cases which could be checked by alternative means. Should some of our assertions prove to be wrong, one would expect that the whole procedure would lead to completely meaningless results in most cases. And it goes without saying that in case of Abelian orbifolds we just get back a suitable simple current extension [21].
Generalities on orbifolds
Consider a (unitary) conformal model whose chiral algebra V is a well behaved (rational, C 2cofinite, etc.) vertex operator algebra [18,23,26,29]. Given a subgroup G < Aut(V) of automorphisms of V, the G-orbifold [13,14] is obtained by identifying the states that can be transformed into each other by an element of G. The Hilbert space of the orbifold is a direct sum of twisted modules of the chiral algebra V (these consist of those operators which are local with respect to V only up to an element of the twist group G), while its chiral algebra is the
fixed-point subalgebra V G = {v ∈ V | gv = v for all g ∈ G}.
There is a natural action of G on the set of all twisted modules under which an element h ∈ G sends a g-twisted module M to a hgh -1 -twisted module h(M ). This shows that the set of all g-twisted modules, as g runs over a given conjugacy class of the twist group, is G-stable under this action, leading to a partition of the orbifold's Hilbert space into sectors labeled by the conjugacy classes of G, each of which usually splits into several G-orbits. Since the twisted modules inside a given G-orbit are all related by the action of an automorphism of V, it follows that they have pretty similar properties (while still not being isomorphic), e.g., their conformal weights h M , quantum dimensions d M and trace functions [17,29] Z M (τ ) = Tr M e 2πiτ (L 0 − c /24) all coincide. This means that it is enough to know these data for just one representative module M from the orbit.
The stabilizer G M = {h ∈ G | h(M ) ∼ = M } of a g-twisted module M is the subgroup of those elements h ∈ G for which h(M ) is isomorphic to M ; clearly, it is a subgroup of the centralizer C G (g) = {h ∈ G | gh = hg}, and the stabilizers of different modules belonging to the same Gorbit are all conjugate to each other. Note that the length of the orbit of M , i.e., the number of different modules contained in it, is just the index [G : G M ] in G of its stabilizer. By the previous reasoning, we have on each twisted module M an action of its stabilizer G M , which is usually not a linear representation, but only a projective one, with associated 2-cocycle ϑ M ∈ Z 2 (G M , C). This projective representation decomposes into homogeneous components corresponding to the irreducible projective representations ξ p ∈ Irr (G M |ϑ M ) of the stabilizer, and each such homogeneous component corresponds to a primary field of the orbifold.
To recapitulate, twisted modules are organized into sectors labeled by conjugacy classes of the twist group G, and each such sector splits into orbits of G, with each orbit characterized by the stabilizer G M of one representative module M from the orbit and by the associated 2cocycle ϑ M . In turn, each irreducible projective representation ξ p ∈ Irr (G M |ϑ M ) corresponds to a primary field of the orbifold. We'll call the set of primaries originating from a particular G-orbit the block corresponding to that orbit. By the above, the primaries of the orbifold are partitioned into blocks, each corresponding to a G-orbit of twisted modules, and the primaries inside a given block correspond to the (projective) irreducible representations of the stabilizer of the orbit. We shall use the notation b o to denote the G-orbit of twisted modules corresponding to a block b. Note that |b o | = [G : G M ] and |b| = |Irr (G M |ϑ M ) | by the above, for any
M ∈ b o .
If M ∈ b o is a g-twisted module of V with g ∈ G of order n, then the eigenvalues of the operator nL 0 are integrally spaced on M , i.e., the difference of two L 0 eigenvalues is an integer multiple of 1 /n. We define the order of a block b as the smallest positive integer n such that the eigenvalues of the operator nL 0 are integrally spaced on any M ∈ b o . In particular, untwisted modules, are characterized by the property that their L 0 eigenvalues are integrally spaced, providing a simple criterion singling out untwisted modules from twisted ones. This also means that a block corresponding to a G-orbit of untwisted modules is characterized by the fact that the conformal weights of its members differ by integers, i.e., its order is 1.
In particular, the vacuum module belongs to the untwisted sector and is left fixed by all elements of G, hence it forms in itself a G-orbit of length 1 whose stabilizer is the whole of G, with trivial associated cocycle (because the twist group acts linearly on V). It follows that the block corresponding to this vacuum module (the vacuum block b 0 ) consists of primaries in oneto-one correspondence with the irreducible representations of G, whose quantum dimensions are integers (since they equal the dimension of the corresponding irreducible representations), and whose conformal weights are integers too, because the conformal weight of the vacuum module is zero (recall that we consider unitary models).
Since the vacuum acts as the identity of the fusion product, the fusion of an element of any given block b with an element of the vacuum block b 0 will contain only primaries from block b. This means that two primaries p and q belong to the same block precisely when N q αp > 0 for at least one element α ∈ b 0 of the vacuum block. In particular, the vacuum block generates a closed subring of the fusion ring of the orbifold, which is identical to the representation ring (Grothendieck ring) of the twist group, since the fusion product of any two of its members is the same as the tensor product of the corresponding irreducible representations.
Orbifold deconstruction
Armed with the above, we can now attack the problem of orbifold deconstruction, i.e., the identification of the original model and the twist group from the sole knowledge of data related to the orbifold. We start from a (unitary) conformal model for which we know the fusion coefficients N r pq , conformal weights h p , quantum dimensions d p and chiral characters χ p (τ ) of the primary fields, and we wish to identify it as a non-trivial orbifold of some other conformal model. Notice that one and the same model might have several deconstructions, reflecting the fact that the same model can be realized as an orbifold in many distinct ways; for example, if the twist group G has a (non-trivial) normal subgroup N G, then a G-orbifold can be obtained as a G/N -orbifold of an N -orbifold. This indicates that there exists a whole hierarchy of deconstructions, whose most interesting members are the maximal ones leading to primitive models, i.e., models that cannot be obtained as a non-trivial orbifold of some other model, hence cannot be deconstructed any further.
The starting point of the deconstruction procedure is the observation, made at the end of the previous section, that the vacuum block of an orbifold has very special properties: it is a set of primaries closed under the fusion product, and all its members have integer conformal weight and quantum dimension. For an arbitrary conformal model, we'll call a set of primaries with these properties a twister. By the above, the vacuum block of an orbifold is a twister, and we assume in the sequel that all twisters arise as the vacuum block of a suitable orbifold realization of the model under study. 2 For each twister there is a different deconstruction of the model, and maximal deconstructions correspond to maximal twisters not contained in any other twister.
The trivial twister (that contains the vacuum primary only) gives rise to a trivial deconstruction resulting in the original model.
Let's now consider the deconstruction with respect to a particular twister g, assumed to be non-trivial. The sole knowledge of the twister allows the determination of a host of important information about the twist group. For example, because the quantum dimensions of the elements of g equal the dimensions of the corresponding irreducible representations of the twist group, and since the order of a group equals, by Burnside's famous theorem [22,28], the sum of the squared dimensions of its irreducible representations, it follows that the order of the twist group should equal the spread g = α∈g d 2 α of the twister. Along the same lines, the number of conjugacy classes of the twist group equals the size |g| of the twister, i.e., the number of its elements. Since the twister generates a subring of the fusion ring isomorphic to the representation ring of the twist group, and the knowledge of the representation ring determines the character table of the underlying group [27], even the character table of the twist group and the size of its conjugacy classes may be computed from this information, and this is usually sufficient to identify the twist group up to isomorphism. 3 Once the twist group has been identified, the next problem is the determination of the primary fields of the original model and their most important attributes, like conformal weights, quantum dimensions and chiral characters. We know that these primaries originate in the untwisted sector of the orbifold, corresponding to untwisted modules organized into G-orbits, and to each such orbit corresponds a block of the orbifold, whose elements are in one-to-one correspondence with the projective irreducible representations of the stabilizer subgroup of the orbit (more precisely, of one module from it). These blocks, viewed as set of primaries of the orbifold, have the characteristic property that the primaries p and q belong to the same block precisely when N q αp > 0 for at least one element α of the vacuum block. But the twister g chosen for deconstruction is precisely the vacuum block of the orbifold realization of our model, hence the blocks my be recovered from the knowledge of g alone: these are the maximal sets b of primaries such that p, q ∈ b implies N q αp > 0 for at least one element α ∈ g. Actually, there is no need to compute all the blocks for deconstruction, since only those corresponding to untwisted modules are needed for identifying the deconstructed model. But we know that blocks corresponding to a G-orbit of untwisted modules have order 1, i.e., the conformal weights of their members can differ only by integers, providing a simple criterion to single them out.
Having identified the blocks in the untwisted sector, i.e., the G-orbits of (untwisted) modules of the deconstructed model, what remains is to determine the properties of the primary fields corresponding to these modules. Since they lie on the same G-orbit and are thus related by an automorphism of the relevant chiral algebra, they have the same conformal weights, quantum dimensions and chiral characters, so it is enough to determine these data for just one of them, but one should keep in mind that they correspond to different primaries of the deconstructed theory, hence their multiplicity (the length |b o | of the corresponding G-orbit) should be taken into account. Finally, to finish the identification of the deconstructed model, one needs to find out its fusion rules. This can be accomplished by assigning to each block b of the untwisted sector a block-fusion matrix N(b) with matrix elements
N(b) r q = 1 g p∈b d p d M N r pq .
Such matrices form a ring, i.e., the product of any two of them may be expressed as a sum
N(a)N(b) = c N c ab N(c)
over the untwisted blocks, where the block-fusion coefficients N c ab ∈ Z + are given by the sums
N c ab = A∈a o , B∈b o N C AB (3.1)
for any representative module C ∈ c o . Note that these block-fusion coefficients come near to provide the fusion rules of the deconstructed model, but for one thing: they do not describe the fusion of the individual modules, but only that of the direct sum of the modules contained in the orbits corresponding to the individual blocks. This shouldn't come as a surprise since, after all, orbifolding amounts to identifying the modules related by a symmetry, so that their individual properties are lost in the process, except for those (like conformal weights and quantum dimensions) which are the same for all modules on the same orbit. Nevertheless, this aggregated version of the fusion rules, supplemented by the knowledge of the conformal weights, quantum dimensions and chiral characters is enough in many examples to identify uniquely the deconstructed model, at least in those amenable to direct computations. 4 4 Example: the 3-fold symmetric product of the moonshine module
Consider the self-dual c = 72 conformal model made up of three identical copies of the moonshine module V . Clearly, any permutation of these identical copies is a symmetry, hence one may orbifold this model with respect to the symmetric group S 3 of degree 3, resulting in the 3-fold symmetric product V S 3 of the moonshine module [6,11]. The properties of this symmetric product are well understood, since it is by construction a permutation orbifold [5], and at the same time a holomorphic orbifold [13], because the moonshine module is self-dual. V S 3 is known to have 8 different primaries [5], whose most important properties are summarized in the following table (with ζ = e 2πi 3 denoting a primitive third root of unity).
label conformal weight dimension character It follows that the only primaries that can belong to a twister are among 1 + , 1 − , 2, ρ 1 and σ − , since only these have simultaneously integer quantum dimension and conformal weight.
1 + 0 1 1 6 J(τ ) 3 + 3J(τ )J (2τ ) + 2J (3τ ) 1 − 4 1 1 6 J(τ ) 3 − 3J(τ )J (2τ ) + 2J (3τ ) 2 2 2 1 3 J(τ ) 3 − J (3τ ) ρ 0 8 3 2 1 3 J τ 3 + ζJ τ +1 3 + ζJ τ +2 3 ρ 1 4 2 1 3 J τ 3 + J τ +1 3 + J τ +2 3 ρ 2 10 3 2 1 3 J τ 3 + ζJ τ +1 3 + ζJ τ +2 3 σ + 3 2 3 1 2 J(τ ) J τ 2 + J τ +1 2 σ − 3 3 1 2 J(τ ) J τ 2 − J τ +1 2
With the notations Ξ = 1 + ⊕ 1 − and Σ = 2 ⊕ ρ 0 ⊕ ρ 1 ⊕ ρ 2 , the fusion rules read It follows that there are precisely three non-trivial twisters: the maximal twisters {1 + , 1 − , 2} and {1 + , 1 − , ρ 1 }, both of spread 6 and size 3, and their intersection {1 + , 1 − }. We note that the existence of two maximal twisters is related to the automorphism of the fusion rules that exchanges the primaries 2 and ρ 1 while leaving all other primaries invariant. Let's take a closer look at the different deconstructions corresponding to these cases. 5 The twister {1 + , 1 − } consists of two simple currents corresponding to the one-dimensional representations of S 3 , hence the deconstructed model will be a simple current extension of V S 3 that could be determined alternatively using the techniques of [21]. Obviously, the twist group is isomorphic to Z 2 . There are 6 different blocks, whose properties can be read off the following table (the multiplicity of a block b is the length |b o | of the associated orbit, cf. Section 3). block order multiplicity dimension trace function
1 + 1 − 2 ρ 0 ρ 1 ρ 2 σ + σ − 1 + 1 + 1 − 2 ρ 0 ρ 1 ρ 2 σ + σ − 1 − 1 − 1 + 2 ρ 0 ρ 1 ρ 2 σ − σ + 2 2 2 2 ⊕ Ξ ρ 1 ⊕ ρ 2 ρ 0 ⊕ ρ 2 ρ 0 ⊕ ρ 1 σ + ⊕ σ − σ + ⊕ σ − ρ 0 ρ 0 ρ 0 ρ 1 ⊕ ρ 2 ρ 0 ⊕ Ξ 2 ⊕ ρ 2 2 ⊕ ρ 1 σ + ⊕ σ − σ + ⊕ σ − ρ 1 ρ 1 ρ 1 ρ 0 ⊕ ρ 2 2 ⊕ ρ 2 ρ 1 ⊕ Ξ 2 ⊕ ρ 0 σ + ⊕ σ − σ + ⊕ σ − ρ 2 ρ 2 ρ 2 ρ 0 ⊕ ρ 1 2 ⊕ ρ 1 2 ⊕ ρ 0 ρ 2 ⊕ Ξ σ + ⊕ σ − σ + ⊕ σ − σ + σ + σ − σ + ⊕ σ − σ + ⊕ σ − σ + ⊕ σ − σ + ⊕ σ − 1 + ⊕ Σ 1 − ⊕ Σ σ − σ − σ + σ + ⊕ σ − σ + ⊕ σ − σ + ⊕ σ − σ + ⊕ σ − 1 − ⊕ Σ 1 + ⊕ Σ{1 + , 1 − } 1 1 1 1 3 J(τ ) 3 + 2J (3τ ) {2} 1 2 1 1 3 J(τ ) 3 − J (3τ ) {ρ 0 } 1 2 1 1 3 J τ 3 + ζJ τ +1 3 + ζJ τ +2 3 {ρ 1 } 1 2 1 1 3 J τ 3 + J τ +1 3 + J τ +2 3 {ρ 2 } 1 2 1 1 3 J τ 3 + ζJ τ +1 3 + ζJ τ +2 3 {σ + , σ − } 2 1 3 J(τ )J τ 2
The untwisted sector consists of 9 modules arranged into 4 orbits of length 2 and one fixedpoint (the vacuum block). Inspecting their conformal weights and trace functions, one recognizes that the deconstructed model is nothing but the permutation orbifold V A 3 of the moonshine module by the alternating group A 3 of degree 3 (the commutator subgroup of S 3 ); this is further corroborated by the fusion rules computed using equation (3.1). We note that there is a unique Z 2 -twisted module in this case, corresponding to the block of order 2.
For the twister {1 + , 1 − , 2}, the fusion rules are
1 + 1 − 2 1 + 1 + 1 − 2 1 − 1 − 1 + 2 2 2 2 1 + ⊕ 1 − ⊕ 2
leading to the following character table for the twist group
1 2 3 1 + 1 1 1 1 − 1 −1 1 2 2 0 −1
from which can one infer that the twist group is isomorphic to S 3 in this case. Actually, this already follows from the fact that it is a group of order g = 6 with |g| = 3 different conjugacy classes, and all such groups are isomorphic to S 3 . There are 2 blocks besides the vacuum block, and the following table summarizes their most important properties. block order multiplicity dimension trace function
{1 + , 1 − , 2} 1 1 1 J(τ ) 3 {σ + , σ − } 2 3 1 J(τ )J τ 2 {ρ 0 , ρ 1 , ρ 2 } 3 2 1 J τ 3
Since there is just one block in the untwisted sector (i.e., of order 1), it follows that the deconstructed model is self-dual, with trace function equal to J(τ ) 3 . From this we can conclude that the deconstructed model is made up from 3 identical copies of the moonshine module V .
For the twister {1 + , 1 − , ρ 1 }, the situation is pretty similar to the previous one after one performs the exchange 2 ↔ ρ 1 . The fusion rules read
1 + 1 − ρ 1 1 + 1 + 1 − ρ 1 1 − 1 − 1 + ρ 1 ρ 1 ρ 1 ρ 1 1 + ⊕ 1 − ⊕ ρ 1
leading once again to the character table
1 2 3 1 + 1 1 1 1 − 1 −1 1 ρ 1 2 0 −1
showing that the twist group is isomorphic to S 3 in this case too. Once again, there are 3 blocks: the vacuum block {1 + , 1 − , ρ 1 }, and the blocks {σ + , σ − } and {ρ 0 , 2, ρ 2 } of respective orders 2 and 3. These share the properties of their counterparts for the twister {1 + , 1 − , 2}, except for their trace functions, which read in this case block trace function
{1 + , 1 − , ρ 1 } J(τ ) 3 − 2c 1 J(τ ) − 2c 2 {σ + , σ − } J(τ )J τ 2 {ρ 0 , 2, ρ 2 } J τ 3 + c 1 J(τ ) + c 2
upon taking into account the replication identity 6
J τ 3 + J τ + 1 3 + J τ + 2 3 + J (3τ ) = J(τ ) 3 − 3c 1 J(τ ) − 3c 2
for the modular invariant function J(τ ), where c 1 = 196884 and c 2 = 21493760. Note that this second deconstruction differs markedly from the previous one, although their twist groups are isomorphic. In particular, since the polynomial P (x) = x 3 − 2c 1 x − 2c 2 has 3 different roots, it follows that the action of the twist group is not a permutation action in this case. This exemplifies that one and the same conformal model (in our case V S 3 ) may be obtained as an orbifold of quite different models, with different (possibly non-isomorphic) twist groups.
Summary and outlook
As described above, there is an effective algorithmic procedure for orbifold deconstruction, i.e., the realization of a given conformal model as an orbifold of some other model. Different deconstructions correspond to different twisters of the model under study, where non-maximal deconstructions lead to models that can be further deconstructed themselves, corresponding to orbifolding in stages. Deconstructions corresponding to maximal twisters (i.e., not contained in any other twister) are the ones that may lead to primitive models that cannot be realized as non-trivial orbifolds. 7 The deconstruction algorithm provides us with a more-or-less unique 6 This illustrates the fact that the existence of different maximal deconstructions of a permutation orbifold is related to the existence of non-trivial replication identities [7]. 7 It may happen that orbifolding by stages breaks down, i.e., the orbifold of an orbifold is not an orbifold of the original model, in which case the corresponding maximal deconstruction is itself a non-trivial orbifold. This phenomenon shows up, as put forward by one of the referees, in the 16-fold tensor power of the Ising model, which has a maximal deconstruction to the SO(16) Wess-Zumino model at level 1, itself a Z2-orbifold of the E8 Wess-Zumino model of the same level.
identification of the twist group and of the original model, up to some ambiguities related to the projective realization of the twist group and to the individual fusion rules of the untwisted modules. It should be emphasized that this ambiguity is a finite one, which means that, as a last resort, one can in principle go through a tedious case-by-case analysis of the allowed possibilities to find out the correct one.
Besides its intrinsic interest, what could be the benefits of the deconstruction procedure in the study of 2D CFT? There is one such benefit that is more or less obvious: if we can identify both the deconstructed model and the twist group unambiguously, then we get an example of an orbifold construction where the result, being the input of the deconstruction procedure, is known right from the start, making possible a thorough investigation of the orbifolding process. Actually, the procedure can be extended so as to give precise results not only about the modules in the untwisted sector (the ones that are of interest for identifying the deconstructed model), but also about the structure of all of the twisted modules. This is particularly important, since the structure of g-twisted modules does only depend on the conjugacy class of g in Aut(V), hence the deconstruction of a G-orbifold gives valuable information about the structure of all orbifolds (of the deconstructed model) whose twist group contains elements that are conjugate in Aut(V) to some element of G.
Finally, let us note that the above ideas might be used in attempts to classify rational conformal models. For one thing, the classification problem can be reduced to that of primitive models (the ones that don't have nontrivial twisters), since all others are orbifolds of these, and the latter can be classified by group theoretic means. On the other hand, even primitive models can be grouped together if they arise from maximal deconstructions of one and the same conformal model, i.e., if some of their orbifolds are identical. An interesting problem is to develop simple criteria testing whether two primitive models are related in this way, and to understand what kind of common structures are responsible for such behavior.
Since they are related by an element of Aut(V), the modules M belonging to the G-orbit b o corresponding to a block b all have the same conformal weight h M = min {h p | p ∈ b}, o | denotes the length of the G-orbit b o . Note that this last result would provide an explicit form of the chiral characters should we know the quantum dimensions d M and the orbit lengths |b o |. Unfortunately, their determination could be tricky in general because the 2-cocycle ϑ M ∈ Z 2 (G M , C) is usually non-trivial, and the irreducible representations ξ p ∈ Irr (G M |ϑ M ) corresponding to the primaries p ∈ b are not ordinary representations, but only projective ones. Nevertheless, exploiting general properties of projective representations it is possible to set up a set of rules that allow their unambiguous computation in most instances. In particular, d M is an algebraic integer ≥ 1 such that the quantum dimensions d p are all integer multiples of |b o |e M d M for p ∈ b, where e M denotes the multiplicative order of the cohomology class of ϑ M (which is obviously the same for all modules M ∈ b o ). These observations are usually sufficient to pin down the precise values of d M and |b o |.
Table 1 .
1Primaries of V S 3 .
Table 2 .
2Fusion rules of V S 3 .
While we have no formal argument supporting this assumption, all available computational results corroborate it.
We note that, while the representation ring does not determine the group uniquely up to isomorphism (a famous example being that of the dihedral group D4 of order 8 and the group Q of unit quaternions, with identical representation rings), the braided structure of the module category of V allows one to determine the λ-ring structure of the representation ring, and in particular the powers of the conjugacy classes, leading to the possibility of identifying the twist group unambiguously.
We should also point out that, since the block-fusion matrices depend explicitly on the quantum dimensions dM but not on the orbit lengths |b o |, the integrality of the block-fusion coefficients N c ab provides an extra condition for determining the precise value of the dimensions in case they are not fixed by the previous considerations.
An important feature of this example that simplifies tremendously the analysis is that, because the relevant groups are pretty small, all the cocycles are trivial, so one does only encounter ordinary representations during the procedure.
Orbifolds and Hopf algebras. P Bantay, 10.1016/0370-2693(90)90676-WPhys. Lett. B. 245Bantay P., Orbifolds and Hopf algebras, Phys. Lett. B 245 (1990), 477-479.
Hopf algebras, and the Moonshine. P Bantay, Orbifolds, 10.1007/BF00403544Lett. Math. Phys. 22Bantay P., Orbifolds, Hopf algebras, and the Moonshine, Lett. Math. Phys. 22 (1991), 187-194.
Characters and modular properties of permutation orbifolds. P Bantay, 10.1016/S0370-2693(97)01464-0arXiv:hep-th/9708120Phys. Lett. B. 419Bantay P., Characters and modular properties of permutation orbifolds, Phys. Lett. B 419 (1998), 175-178, arXiv:hep-th/9708120.
Simple current extensions and mapping class group representations. P Bantay, 10.1142/S0217751X9800007XarXiv:hep-th/9611124Internat. J. Modern Phys. A. 13Bantay P., Simple current extensions and mapping class group representations, Internat. J. Modern Phys. A 13 (1998), 199-207, arXiv:hep-th/9611124.
Permutation orbifolds. P Bantay, 10.1016/S0550-3213(02)00198-0arXiv:hep-th/9910079Nuclear Phys. B. 633Bantay P., Permutation orbifolds, Nuclear Phys. B 633 (2002), 365-378, arXiv:hep-th/9910079.
Symmetric products, permutation orbifolds and discrete torsion. P Bantay, 10.1023/A:1024453119772arXiv:hep-th/0004025Lett. Math. Phys. 63Bantay P., Symmetric products, permutation orbifolds and discrete torsion, Lett. Math. Phys. 63 (2003), 209-218, arXiv:hep-th/0004025.
Character relations and replication identities in 2d conformal field theory. P Bantay, 10.1007/JHEP10(2016)020arXiv:1603.05511020, 13 pages. 2016Bantay P., Character relations and replication identities in 2d conformal field theory, J. High Energy Phys. 2016 (2016), no. 10, 020, 13 pages, arXiv:1603.05511.
Twisted sectors for tensor product vertex operator algebras associated to permutation groups. K Barron, C Dong, G Mason, 10.1007/s002200200633arXiv:math.QA/9803118Comm. Math. Phys. 227Barron K., Dong C., Mason G., Twisted sectors for tensor product vertex operator algebras associated to permutation groups, Comm. Math. Phys. 227 (2002), 349-384, arXiv:math.QA/9803118.
Systematic approach to cyclic orbifolds. L Borisov, M B Halpern, C Schweigert, 10.1142/S0217751X98000044arXiv:hep-th/9701061Internat. J. Modern Phys. A. 13Borisov L., Halpern M.B., Schweigert C., Systematic approach to cyclic orbifolds, Internat. J. Modern Phys. A 13 (1998), 125-168, arXiv:hep-th/9701061.
Conformal field theory, Graduate Texts in Contemporary Physics. Di Francesco, P Mathieu, P Sénéchal, D , 10.1007/978-1-4612-2256-9Springer-VerlagNew YorkDi Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997.
Elliptic genera of symmetric products and second quantized strings. R Dijkgraaf, G Moore, E Verlinde, H Verlinde, 10.1007/s002200050087arXiv:hep-th/9608096Comm. Math. Phys. 185Dijkgraaf R., Moore G., Verlinde E., Verlinde H., Elliptic genera of symmetric products and second quantized strings, Comm. Math. Phys. 185 (1997), 197-209, arXiv:hep-th/9608096.
Quasi Hopf algebras, group cohomology and orbifold models. R Dijkgraaf, V Pasquier, P Roche, 10.1016/0920-5632(91)90123-VNuclear Phys. B Proc. Suppl. 18Dijkgraaf R., Pasquier V., Roche P., Quasi Hopf algebras, group cohomology and orbifold models, Nuclear Phys. B Proc. Suppl. 18B (1990), 60-72.
The operator algebra of orbifold models. R Dijkgraaf, C Vafa, E Verlinde, H Verlinde, 10.1007/BF01238812Comm. Math. Phys. 123Dijkgraaf R., Vafa C., Verlinde E., Verlinde H., The operator algebra of orbifold models, Comm. Math. Phys. 123 (1989), 485-526.
The conformal field theory of orbifolds. L Dixon, D Friedan, E Martinec, S Shenker, 10.1016/0550-3213(87)90676-6Nuclear Phys. B. 282Dixon L., Friedan D., Martinec E., Shenker S., The conformal field theory of orbifolds, Nuclear Phys. B 282 (1987), 13-73.
Strings on orbifolds. L Dixon, J Harvey, C Vafa, E Witten, 10.1016/0550-3213(85)90593-0Nuclear Phys. B. 261Dixon L., Harvey J., Vafa C., Witten E., Strings on orbifolds, Nuclear Phys. B 261 (1985), 678-686.
L Dixon, J Harvey, C Vafa, E Witten, 10.1016/0550-3213(86)90287-7Strings on orbifolds. IIDixon L., Harvey J., Vafa C., Witten E., Strings on orbifolds. II, Nuclear Phys. B 274 (1986), 285-314.
Modular-invariance of trace functions in orbifold theory and generalized moonshine. C Dong, H Li, G Mason, 10.1007/s002200000242arXiv:q-alg/9703016Comm. Math. Phys. 214Dong C., Li H., Mason G., Modular-invariance of trace functions in orbifold theory and generalized moon- shine, Comm. Math. Phys. 214 (2000), 1-56, arXiv:q-alg/9703016.
Vertex operator algebras and the Monster. I Frenkel, J Lepowsky, A Meurman, Pure and Applied Mathematics. 134Academic Press, IncFrenkel I., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathe- matics, Vol. 134, Academic Press, Inc., Boston, MA, 1988.
Correspondences of ribbon categories. J Fröhlich, J Fuchs, I Runkel, C Schweigert, 10.1016/j.aim.2005.04.007arXiv:math.CT/0309465Adv. Math. 199Fröhlich J., Fuchs J., Runkel I., Schweigert C., Correspondences of ribbon categories, Adv. Math. 199 (2006), 192-329, arXiv:math.CT/0309465.
Orbifolds by cyclic permutations in Gepner type superstrings and in the corresponding Calabi-Yau manifolds. J Fuchs, A Klemm, M G Schmidt, 10.1016/S0003-4916(05)80001-6Physics. 214Fuchs J., Klemm A., Schmidt M.G., Orbifolds by cyclic permutations in Gepner type superstrings and in the corresponding Calabi-Yau manifolds, Ann. Physics 214 (1992), 221-257.
A matrix S for all simple current extensions. J Fuchs, A N Schellekens, C Schweigert, 10.1016/0550-3213(96)00247-7arXiv:hep-th/9601078Nuclear Phys. B. 473Fuchs J., Schellekens A.N., Schweigert C., A matrix S for all simple current extensions, Nuclear Phys. B 473 (1996), 323-366, arXiv:hep-th/9601078.
Character theory of finite groups. I M Isaacs, Pure and Applied Mathematics. 69Academic PressIsaacs I.M., Character theory of finite groups, Pure and Applied Mathematics, Vol. 69, Academic Press, New York -London, 1976.
Vertex algebras for beginners. V Kac, University Lecture Series. 10Amer. Math. SocKac V., Vertex algebras for beginners, University Lecture Series, Vol. 10, Amer. Math. Soc., Providence, RI, 1997.
. A KirillovJr, arXiv:math.QA/0401119On G-equivariant modular categories. Kirillov Jr. A., On G-equivariant modular categories, arXiv:math.QA/0401119.
Orbifolds by cyclic permutations of tensor product conformal field theories. A Klemm, M G Schmidt, 10.1016/0370-2693(90)90164-2Phys. Lett. B. 245Klemm A., Schmidt M.G., Orbifolds by cyclic permutations of tensor product conformal field theories, Phys. Lett. B 245 (1990), 53-58.
Introduction to vertex operator algebras and their representations. J Lepowsky, H Li, 10.1007/978-0-8176-8186-9Progress in Mathematics. Boston, MABirkhäuser Boston, Inc227Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathe- matics, Vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004.
K Lux, H Pahlings, 10.1017/CBO9780511750915Representations of groups. A computational approach. CambridgeCambridge University Press124Lux K., Pahlings H., Representations of groups. A computational approach, Cambridge Studies in Advanced Mathematics, Vol. 124, Cambridge University Press, Cambridge, 2010.
J.-P Serre, 10.1007/978-1-4684-9458-7Linear representations of finite groups. New York -HeidelbergSpringer-Verlag42Serre J.-P., Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York -Heidelberg, 1977.
Modular invariance of characters of vertex operator algebras. Y Zhu, 10.1090/S0894-0347-96-00182-8J. Amer. Math. Soc. 9Zhu Y., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237-302.
| [] |
[
"String Embeddings of the Pentagon",
"String Embeddings of the Pentagon"
] | [
"S Echols [email protected] \nDepartment of Physics\nCal Poly State University\n93407San Luis ObispoCA\n"
] | [
"Department of Physics\nCal Poly State University\n93407San Luis ObispoCA"
] | [] | The Pentagon Model is an explicit supersymmetric extension of the Standard Model, which involves a new strongly-interacting SU (5) gauge theory at TeV-scale energies. We discuss embeddings of the Pentagon Model into string theory, specifically N = 1 supersymmetric type IIa intersecting D-brane models, M-theory compactifications of G 2 holonomy, and heterotic orbifold constructions.1 The analysis is only under analytic control if m ISS << Λ Nc | null | [
"https://arxiv.org/pdf/1005.0164v1.pdf"
] | 117,056,161 | 1005.0164 | 0eb469a00468d20cfc0e1326bd9f9dc42180d2d9 |
String Embeddings of the Pentagon
2 May 2010
S Echols [email protected]
Department of Physics
Cal Poly State University
93407San Luis ObispoCA
String Embeddings of the Pentagon
2 May 2010Preprint typeset in JHEP style -PAPER VERSION
The Pentagon Model is an explicit supersymmetric extension of the Standard Model, which involves a new strongly-interacting SU (5) gauge theory at TeV-scale energies. We discuss embeddings of the Pentagon Model into string theory, specifically N = 1 supersymmetric type IIa intersecting D-brane models, M-theory compactifications of G 2 holonomy, and heterotic orbifold constructions.1 The analysis is only under analytic control if m ISS << Λ Nc
Introduction
The string model-building program has a number of goals. First, if completely realistic models are found, this would provide a proof that string theory may be a unified theory of all particles and interactions. Further, the study of the surviving low-energy spectra of various string models might lead to the identification of general patterns (such as symmetries or exotic particle content) present in a large class of realistic vacua. Additionally, it might lead to new ideas for addressing problems such as CP violation, fermion mass mixings, or even dark matter and dark energy. Perhaps most significant is the hope that the discipline will lead to experimentally testable predictions. The last of these is especially provocative at a time when we find ourselves on the verge of a plethora of new data from the LHC.
The Pentagon Model of TeV physics successfully addresses a number of low energy phenomenological issues, and we would therefore like to find it as an effective field theory of a string construction. Such a search is the subject of this paper.
Though previous search [1] has produced promising results towards embedding the Pentagon into a grand unified theory (GUT), the method nevertheless relies on arguments involving operators at the Planck scale where in reality the theory breaks down and becomes unreliable. Thus, the purpose of this paper is to explore the possibility of embedding the Pentagon Model into a string theory directly. In practice this translates into choosing one specific string theory with a given geometry and turning the crank to find the resulting particle spectrum of that theory, and comparing it with the Pentagon. There are currently five different types of string theories for which it is known how to calculate the low-energy chiral spectrum: orbifold constructions in heterotic string theory, G 2 compactifications of M-theory, intersecting D-brane models in either type IIA or IIB string theory, and F-theory models. In this chapter we will consider the first three of these approaches. Type IIB and their dual F-theory models might certainly be of interest, but are left to future work.
Though a search for the Pentagon has never before been performed, it should be noted that each of these approaches has yielded only mediocre results in previous searches for the standard model and various GUTs. While many models have been discovered which may contain the desired particle content and gauge symmetries, one must also contend with other issues such as problems with the existence of chiral exotics, symmetry breaking and Higgs fields, and finding proper U (1) charges and Yukawa couplings. While extensive research has been devoted to these questions, only perhaps a handful of models have satisfactorily addressed all of these issues. As the purpose of this search is merely to establish the viability of the existence of the Pentagon, our strategy has been to search for models that are 'at least as good as state of the art'. In other words, we must begin by searching the various string theories for our desired particle content. If we were to find a massless particle spectrum corresponding to that of the Pentagon, we would then turn our attention to the phenomenological aspects of these models.
Unfortunately, we have found that the existence of the Pentagon as a low energy spectrum of these theories is impossible at worst and inconclusive at best. The difficulty seems to arise due to the requirement that we obtain both chiral and vector-like particles, as will be discussed. The paper is constructed as follows: We first briefly review the contents of the Pentagon Model in section 2. In section 3 we will consider models of N = 1 globally supersymmetric type IIA intersecting D6-brane construction. In [10], Cvetič, Papadimitriou, and Shiu (CPS) have performed an extensive search for N = 1 supersymmetric three-family SU (5) Grand Unified Models in the type IIA con-text, and we therefore use their work as the starting point for our search. In Section 4 we consider the G 2 lift of these models onto eleven dimensional M-theory compactifications. We find a possible candidate for the Pentagon in this context, but cannot provide a proof for its existence. Section 5 is devoted to heterotic orbifold models. It appears difficult to find a consistent model supporting an SU (5) × SU (5) gauge group with charged matter in the bifundamental representation, so we instead focus our search for the chiral spectrum of the Pyramid Model. The results of some of the more promising models are listed but we were unable to find an exact replication of the low energy model, though we were unable to rule out the possibility. In Section 6 we write some concluding remarks.
The Pentagon
The Original Model
The Pentagon is a supersymmetric model of TeV scale physics [2,3], whose foundation is the Minimally Supersymmetric Standard Model (MSSM). The Pentagon model was origianally constructed to address standard issues with the MSSM, such as SUSY breaking, the µ problem, the flavor problem, CP violation, and baryon violation . In addition to an SU (5) grand unified version of the MSSM, a new strongly interacting 'Pentagon' SU (5) super-QCD with five flavors of pentaquarks is introduced as a hidden sector which mediates SUSY breaking through Standard Model gauge couplings. An hypothetical meta-stable N F = N C = 5 vacuum of the theory is used to employ the SUSY breaking mechanism of Intriligator, Seiberg, and Shih (ISS) to construct an effective theory for Cosmological SUSY breaking (CSB). It naturally introduces a µ term of the right order of magnitude, contains a discrete R-symmetry which eliminates all unwanted dimension 4 and 5 Baryon and Lepton violating operators, and resolves the SUSY flavor problem. Strong CP violating phases remain in the model (in addition to standard neutrino see-saw and CKM matrix phases), but these are potentially addressed with the addition of an axion.
The Lagrangian of the Pentagon model contains several pieces. The standard MSSM Lagrangian is implemented as usual: the kinetic energy terms for the matter and Higgs fields arise in the Kahler potential, L 1 = d 4 θ[P * e V P + Q * e V Q + L * e V L + (Ū ) * e VŪ + (D) * e VD + (Ē) * e VĒ and the gauge superpotential produces the kinetic terms for the gauge fields and gauginos,
L 2 = d 2 θ( τ i W i α ) 2 .
Yukawa couplings for the Standard Model fermions and a mass term for the Higgsino are contained in the superpotential,
L 3 = d 2 θλ u H u QŪ + λ d H d QD + λ L H d LĒ + λ mn M U L m L n H 2 u + h.c.
In addition to the MSSM Lagrangian, the Pentagon model includes an additional superpotential for the pentaquarks (transforming as P ∼ [5,5] andP ∼ [5,5] under the SU (5) P × SU (5) GU T gauge group) and an additional singlet field S with discrete R-charge 2:
L 4 = d 2 θP A iP j A (m ISS δ i j + g S SY i j ) + g µ SH u H d + g T S 3 .
The scale M U is taken to lie in the range M U ∼ 10 14 − 10 15 GeV to successfully implement the neutrino seesaw effect. m ISS is assumed to be induced by CSB in the UV sector of the theory (we will discuss CSB further in the next section),
m ISS = γ λ 1/4 M P Λ 5 .
λ is the cosmological constant, M P the Planck mass, and Λ 5 the confinement scale of the Pentagon gauge group. To be consistent with CSB, Λ 5 ∼ 1.5 TeV. γ is an unknown constant of order one. ISS proved that for a theory of SUSY QCD with N C + 1 ≤ N F ≤ 3N C 2 , the mass term m ISS TrPP induces a meta-stable SUSY violating ground state with SUSY order parameter F ∼ m ISS Λ N C 1 [4]. They further argued that a similar meta-stable state might exist for a theory with N F = N C , though its properties could not be calculated analytically. The Pentagon therefore has a stationary point of its effective potential with a non-zero vacuum energy of order m 2 ISS Λ 2 5 . SUSY breaking is communicated to the Standard Model via two mechanisms. The dominant contribution to gaugino masses as well as the masses of the squarks and sleptons is through conventional gauge mediation. The Higgs superfields also contribute tree level masses to squarks and sleptons due to non-zero F terms.
The singlet field S is thought to be the remnant of an SU (5) adjoint, transforming like the hypercharge generator of SU (3) × SU (2) × U (1). Its coupling to the Standard Model therefore implies that the GUT SU (5) is broken to SU (3)×SU (2)×U (1). It also ties the properties of the meta-stable SUSY violating vacuum to electroweak symmetry breaking through its F-term, predicting SU (2)×U (1) → U (1) EM with |h u | ∼ |h d | ∼ Λ 5 , tan β ∼ 1. Furthermore, the VEV of S can give rise to a natural µ term.
The SUSic m ISS → 0 limit of the theory admits an anomaly free R-symmetry which is identified with the discrete Z N R-symmetry required by the rules of CSB. The SUSY degrees of freedom transform non-trivially under an R-symmetry, it follows that N = 4 to accommodate all terms in the superpotential. In models of CSB, the discrete R-symmetry guarantees Poincaré invariance; it also has the effect of preventing all unwanted dimension 4 and 5 baryon and lepton violating operators leading to proton decay 2 . The Z 4 also forbids various dimension 5 flavor combinations, so quark and lepton flavor changing processes arise from dimension 6 operators. Thus, similar to generic gauge mediated models, flavor changing neutral currents are suppressed below experimental limits.
R-parity preservation implies that the LSP is the gravitino. Estimates of the scale of SUSY breaking give a gravitino mass of order 5 × 10 −3 eV, consistent with Big Bang Nucleosynthesis. It is far too small, however, to be a viable dark matter candidate, and it is strongly coupled enough that the NLSP will decay too rapidly to be of cosmological importance. Thus there is no conventional MSSM dark matter candidate. On the other hand, ISS show that the Pentabaryons, dimension one fields made of five pentaquarks, have a non-vanishing expectation value in the meta-stable vacuum. Pentabaryon number is therefore spontaneously broken, B = Λ 5 e ib/Λ 5 , and the associated Goldstone boson, the penton, is cosmologically long-lived. If the penta-baryon asymmetry produced in the early universe is sufficiently large, the penton can be the dark matter. Furthermore, the pentabaryon and baryon numbers are coupled by QCD interactions, providing a possible connection between the dark matter and the observed baryon asymmetry of the universe. We will discuss the issues of dark matter and baryogenesis further in the next section.
The Pyramid
After its invention, it was noticed that the Pentagon model may suffer from a number of troubling issues. Most importantly, Λ 5 ∼ 1.5 leads to a Landau pole before gauge coupling unification. In fact, a calculation of the two loop β functions for the running of Standard Model couplings requires both Λ 5 , m ISS > 10 3 TeV [5]. This is inconsistent with the conditions of CSB. Another problem has to do with stellar phenomenology. The penton gains mass through a dimension 7 operator; if the scale associated with this operator is too large, stars will produce an overabundance of pentons leading to unobserved stellar cooling [6].
The successor of the Pentagon, the Pyramid model, was constructed to address these issues [7]. The Pyramid model employs an SU (3) 4 gauge symmetry, each factor being represented by the vertices of a pyramid quiver diagram. Standard Model particles exist as broken multiplets running around the base of the pyramid-singlets of a new Pyramid SU (3) P gauge group, but fitting into complete multiplets of a conventional trinification GUT. In such models, a single generation of fermions comes in the representation
(3, 1,3) ⊕ (3, 3, 1) ⊕ (1,3, 3) under the trinification SU 1 (3) × SU 2 (3) × SU 3 (3)
. This respects a Z 3 permutation symmetry, and can be embedded precisely into the 27 of E 6 . SU 3 (3) is identified with the color symmetry of the Standard Model, electroweak symmetry comes from an SU (2) subgroup of SU 2 (3), and hypercharge is a linear combination of generators from both SU 2 (3) and SU 1 (3). Gauge coupling unification is guaranteed if all matter comes in complete representations of SU (3) 3 × Z 3 and this symmetry is preserved by Yukawa couplings.
Analogous to pentaquarks, trianons are introduced to implement the ISS mechanism of meta-stable SUSY breaking and to mediate SUSY breaking to the MSSM. Trianons transform under both the Pyramid SU (3) P and the trinification symmetry: Because they respect the Z 3 symmetry, one loop perturbative coupling unification is preserved.
T 1 +T 1 = (3,
The remainder of the construction of the theory is in complete parallel with the Pentagon model. The singlet field S can give rise to a µ term, and its F-terms gives a VEV to the meson fields that are responsible for electroweak symmetry breaking. A discrete R-symmetry exists as a consequence of CSB which forbids all dangerous dimension 4 and 5 operators. Gaugino and squark masses are estimated to lie in an acceptable range for phenomenology. The pyrmabaryons themselves are expected to be the prime dark matter candidate, although spontaneous breaking of pyrmabaryon number does occur in the model. The Goldstones of this broken symmetry are called the pyrmions which, in contrast to the pentons, avoid constraints from stellar cooling.
Although the majority of this paper is devoted toward developing the Pentagon model, we do address how the Pyramid model can be extended to accommodate these developments.
Intersecting type IIA D-branes
Brief review
Intersecting D-brane models provide a very nice geometric picture for some of the fundamental ingredients of any low energy effective field threory 3 . In particular, they provide a mechanism for generating not only gauge symmetries but also chiral fermions, where family replication is achieved by multiple topological intersection numbers of various D-branes. To be more specific, the spectrum of open strings stretched between the intersecting D-branes contains the chiral particles which are localized at the intersections. In this section we will consider specifically the construction of four dimensional N = 1 supersymmetric type IIA orientifolds with D6-branes intersecting at angles.
Type IIA superstring theory exists in 10 space-time dimensions, six of which must be compactified to make contact with the observed world. The theory contains both closed and open strings as well as extended charged objects of higher dimension-the D-branes. Fluctuations of these objects can be described as open strings attached to the D-branes. The endpoints of the strings give Chan-Paton factors, which can be viewed as a U (1) gauge field with momentum only along (and therefore confined to) the brane. By placing N D-branes on top of each other the gauge fields on the branes will transform in the adjoint representation of the gauge group U (N ). If these fields are carried by D6-branes, three dimensions must remain uncompactified for these fields to be free to move in four dimensional Minkowski space-time. This means that in the six dimensional transverse compact space the branes are three dimensional and wrap a three dimensional cycle. In general, two such branes will intersect at a point in the compactified space.
An open string extended between the two branes can be shown to have only one fermionic degree of freedom. Taking into account an open string with the opposite orientation between the two D6 branes, one is left with two fermionic degrees of freedom corresponding to one chiral Weyl fermion from the four dimensional point of view. In the same way, strings extended at the intersection of two stacks of branes, with N and M D6-branes per stack respectively, will give rise to a chiral fermion transforming in the bifundamental representation of U (N ) × U (M ). While the gauge fields are confined to the branes, gravity still propagates throughout the bulk. Thus, the Dbranes interact gravitationally, which means that they will contribute positively to the vacuum energy. To cancel this contribution, we must introduce negative tension objects known as orientifold planes. Both the D-branes and the orientifold planes carry R-R charge, which must vanish for consistency. This gives rise to tadpole cancellation conditions, which must be satisfied along with certain supersymmetry conditions for the theory to be consistent.
The simplest compactification scheme is six dimensional toroidal compactification factorized as the product of three rectangular two-tori, T 6 = T 2 × T 2 × T 2 , and to assume that the D6-branes are the products of one-cycles in each of the three twotori. This allows us to specify the branes by wrapping numbers (n i , m i ) along the fundamental cycles [a i ] and [b i ] on the ith T 2 . Next we introduce the orientifold O6plane, and allow it to wrap along each of the [a i ] cycles (as well as the transverse uncompactified space). The introduction of the orientifold plane mods the theory by world-sheet parity as well as an anti-holomorphic involution, so that the 06-plane is localized at the fixed plane of the local reflection (n i , m i ) → (n i , −m i ). However, in this scenario, if the D6-branes do not lie entirely parallel to the 06-plane everywhere, the tension of these branes in the perpendicular directions cannot be canceled. Thus, no non-trivial globally supersymmetric consistent models can be constructed on these manifolds.
This problem can be alleviated by extending the orientifold planes into all perpendicular directions via orbifolding [9]. The simplest examples of such models are orientifolds of toroidal type IIA orbifolds T 2 × T 2 × T 2 /(Z 2 × Z 2 ). Using the notation of [9,10], the orbifold twists are v = (1/2, −1/2, 0) and w = (0, 1/2, −1/2), acting on the complex coordinates of the three two-tori as
Θ : (z 1 , z 2 , z 3 ) → (−z 1 , −z 2 , z 3 ) ω : (z 1 , z 2 , z 3 ) → (z 1 , −z 2 , −z 3 ).
Orientifolding mods the theory by the orientifold action ΩR, where Ω is world-sheet parity and R acts as
R : (z 1 , z 2 , z 3 ) → (z 1 ,z 2 ,z 3 ).
As with the case of toroidal compactification, the action of the orientifold requires the O6-plane to lie along the three [a i ] cycles. However, orbifolding creates three new classes of O6-planes, each associated with the combined action of the orientifold and the orbifold:
ΩR : [Π 1 ] = [a 1 ][a 2 ][a 3 ], ΩRΘ : [Π 2 ] = [b 1 ][b 2 ][a 3 ], ΩRω : [Π 3 ] = [a 1 ][b 2 ][b 3 ], ΩRΘω : [Π 4 ] = [b 1 ][a 2 ][b 3 ].
The complex structure of the tori is arbitrary but must be consistent with the orientifold projection. This admits only two choices, each torus may be rectangular (with the lattice vectors e 1 ⊥ e 2 ) or tilted such that e 1 = e 1 + e 2 /2, e 2 = e 2 . To describe both choices in a common notation, a generic one cycle can be written as n i a [a i ] + l i a [b i ], with l i a = m i a for a rectangular torus and l i a = 2m i a +n i a for a tilted torus. The homology class of a three cycle is just the product of three one cycles,
[Π a ] = 3 i=1 (n i a [a i ] + 2 −β i l i a [b i ])
where the factor 2 −β i is included to account for tilted tori (β i = 1 if the ith torus is tilted, zero otherwise). The orientifold action maps a one cycle (n i a , l i a ) to its image (n i a , −l i a ), thus for any stack of D-branes we must also include its image
[Π a ] = 3 i=1 (n i a [a i ] − 2 −β i l i a [b i ]).
Finally, we define
[Π O6 ] = 8[Π 1 ] − 2 3−β 1 −β 2 [Π 2 ] − 2 3−β 2 −β 3 [Π 3 ] − 2 3−β 1 −β 3 [Π 4 ].
The coefficients reflect the number of images of each O6 plane that must be included. With these definitions we are equipped to consider the open-string spectrum of the theory. Chiral sectors are defined by the objects between which the strings in the sector are extended. Adjoint fields are given by strings with endpoints on a single brane, thus the gauge group is found in the aa sector. As mentioned, in toroidal theory a stack of N a D6-branes gives rise to a U (N a ) gauge group. In the orbifold theory, the Θ action breaks this to U (N a /2) × U (N a /2), and the ω action identifies these factors, leaving the gauge group U (N a /2). However, in the special case of branes coincidental with some of the O6-planes, the symmetry is enhanced to a U Sp(N a ) gauge group. Massless strings extended between these branes will necessarily be vector-like, and so they have gained the name 'filler branes' because they can contribute an RR tadpole charge without adding to the particle spectrum.
The ab+ba sector gives chiral supermultiplets in the bi-fundamental representation (N a /2, N b /2). The multiplicity of these states is given by the topological intersection number
I ab = [Π a ][Π b ] = 2 −k 3 i=1 (n i a l i b − n i b l i a )
with k = β 1 + β 2 + β 3 . Similarly, the ab + b a sector (the prime indicates the ΩR image) gives I ab chiral fields in the representation (N a /2, N b /2), with
I ab = [Π a ][Π b ] = −2 −k 3 i=1 (n i a l i b + n i b l i a ).
The sign of I signifies the chirality of the particle, with a negative intersection number corresponding to a left-handed fermion. D6-branes can also intersect with their images. Naively one might assume that strings extended from a stack a to stack a would give particles transforming as (N a /2, N a /2), but the orientifold projection leads to two index symmetric and antisymmetric tensor representations of U (N a /2). The intersection number between a stack and its image is given by
I aa = [Π a ][Π a ] = −2 3−k 3 i=1 (n i a l i a ).
However, massless strings will also stretch between a stack of branes and image at the orientifold planes, and so we must take into account the intersection
I aO6 = [Π a ][Π O6 ] = 2 3−k (−l 1 a l 2 a l 3
a + l 1 a n 2 a n 3 a + n 1 a l 2 a n 3 a + n 1 a n 2 a l 3 a ).
The final result for the net number of symmetric and anti-symmetric representations is found by anomaly cancellation:
1 2 (I aa − 1 2 I aO6 ) symmetric 1 2 (I aa + 1 2 I aO6 ) antisymmetric.
The resulting chiral spectrum is listed in table 1. A consistent supersymmetric theory must satisfy both tadpole and supersymmetry constraints. Cancelation of RR tadpoles follows from the cancellation of D6-brane and O6-plane charge, which implies
a N a [Π a ] + a N a [Π a ] − 4[Π O6 ] = 0.
To preserve supersymmetry, each D6-brane must be related to the orientifold plane by an SU (3) rotation. Because the D6-branes are taken to be products of one-cycles, each cycle will lie at some angle θ i with respect to the horizontal direction in the ith torus. The condition ensures that the total angle of rotation is an element of SU (3). The angles θ i can be expressed in terms of the wrapping numbers as
θ 1 + θ 2 + θ 3 = 0 mod 2π Sector Representation Multiplicity ab + ba (N a /2, N b /2) I ab = 2 −k 3 i=1 (n i a l i b − n i b l i a ) ab + b a (N a /2, N b /2) I ab = −2 −k 3 i=1 (n i a l i b + n i b l i a ). aa + aO6 (N a ⊗ N a ) s 1 2 (I aa − 1 2 I aO6 ) (N a ⊗ N a ) a 1 2 (I aa + 1 2 I aO6 )sin θ i = 2 −β i l i R i 2 L i (n i , l i ) , cos θ i = n i R i 1 L i (n i , l i ) ,
where R i 1 , R i 2 are the radii of the horizontal and vertical directions of the ith torus, and
L i (n i , l i ) = (2 −β i l i R i 2 ) 2 + (n i R i 1 ) 2
is the total length of the one-cycle on the ith torus.
Search Strategy and Results
In [10], Cvetič, Papadimitriou, and Shiu (CPS) have performed an extensive search for N = 1 supersymmetric three-family SU (5) Grand Unified Models using the above construction. Therefore, we have used their work as the starting point for our search for the Pentagon model. In this section we will discuss what adjustments must be made to the CPS models in order to accommodate the inclusion of the Pentagon SU (5) gauge group and our required matter multiplets. Based on simple assumptions that these adjustments lead us to, the existence of the Pentagon Model is ruled out. In particular, the number of stacks required to obtain the Pentagon spectrum introduces a problem with the complex moduli, and the simplest solution to this problem is not consistent with both tadpole and supersymmetry constraints. Relaxing these assumptions leads to models that are far more complicated and which must be evaluated on a case-by-case basis. Thus, while the construction of consistent models in the context of the Pentagon is not a forbidden possibility, it is left to future research. We are looking to build a low energy phenomenological model that is 'at least as good' as the various CPS models, but with a few additional requirements. The CPS models are all four-dimensional chiral models with N=1 SUSY constructed from IIA orientifolds on T 2 ×T 2 ×T 2 /(Z 2 ×Z 2 ). They satisfy consistency conditions (tadpole cancellation), preserve supersymmetry, and contain three generations of SU GU T (5) matter (or to be more precise, they all have 3 generations of the 10 a representation of the SU GU T (5), but with varying number 5 fundamental representations). These models also have various phenomenological challenges, including the existence of substantial numbers of chiral exotics as well as issues with the Higgs fields and Yukawa couplings.
We are willing to accept these shortcomings for the present purpose, but there are other requirements that must be satisfied to reproduce the low energy spectrum of the Pentagon. Primarily, we require the existence of an additional stack of D-branes to give the SU P (5) of the Pentagon model, and total topological intersection number zero between this stack and the stack of the GUT SU (5) (this is because of the vector-like nature of the Pentaquarks, which transform as either (5,5) or (5,5) plus c.c. under SU P (5) × SU GU T (5)). This last requirement is satisfied by having the two stacks parallel on the first T 2 (by choice), but we wish to impose the additional constraint that the intersection number equal one on the remaining two Torii, so that there is only a single point at which the vector-like Pentaquarks may arise, thereby prohibiting additional unwanted generations of the Pentaquarks which could be disastrous for coupling unification and possibly introduce Landau poles. We also assume that the two parallel stacks on the first T 2 are actually lying right on top of each other, and further that they lie parallel to the orientifold plane. The first of these ensures that the pentaquarks remain massless, and the latter that they have no intersections with their orientifold images 4 (which would lead to exotic pentaquark-like fields charged under both SU (5)s). See figure 4.1. Finally, we would like to have two U (1) stacks of D6-branes, the intersections of which would provide the singlets of the Pentagon. The desired particle content is summarized in table 2.
Let us begin by briefly listing the constraints relevant to our criteria. Following CPS, we define the parameters A a = −n 1 a n 2 a n 3 a , B a = n 1 a l 2 a l 3 a , C a = l 1 a n 2 a l 3 a , D a = l 1 a l 2 a n 3 ã A a = −l 1 a l 2 a l 3 a ,B a = l 1 a n 2 a n 3 a ,C a = n 1 a l 2 a n 3 a ,D a = n 1 a n 2 a l 3 a . Then the tadpole cancellation conditions can be rewritten as
−16 = −2 k N (1) + a N a A a = −2 k N (2) + a N a B a 4
Of course, the SU (5) stacks and their images will still be parallel, so their positions must be fixed at positions on the first torus such that string states stretching between a brane and its image are massive, i.e. non-zero distance.
Stack
Gauge Group D6-brane Stack a:
Pentagon
SU (5) N a = 10 b: SM GUT SU (5) N b = 10 c: U (1) N c = 2 d: U (1) N d = 2 SU P (5) × SU GU T (5)
Topological Intersection Particle Representation (Multiplicity) (5,5) + (5,5) I ab = 0 (1, 10 a ) 1 2 (I bb + 1 2 I bO6 ) = 3 (1,5) I bc + I bc + I bd + I bd = −3 Singlets I cd + I cd = 0
= −2 k N (3) + a N a C a = −2 k N (4) + a N a D a .
The N (i) correspond to the number of 'filler branes' wrapping the ith orientifold plane, which is for our purposes arbitrary and can be used to reduce the total tadpole charge to the desired -16. They always contribute negatively, so our biggest obstacle will be to ensure that the sum of the charges from our stacks be large enough. The number of branes in each stack determines the gauge group, so in our case we want N a = 10,
N b = 10, N c = 2, N d = 2.
The extra factor of two is due to the orientifold projection. This allows us to write out the tadpole constraints more explicitly: (1) or to simplify things
10A a + 10A b + 2A c + 2A d = −16 + 2 k N10A a + 10A b + 2A c + 2A d > −16
and similarly for B, C, D.
The supersymmetry constraints must be satisfied for each stack individually. The condition θ 1 + θ 2 + θ 3 = 0 mod 2π is equivalent to sin(θ 1 + θ 2 + θ 3 ) = 0 and cos(θ 1 + θ 2 + θ 3 ) > 0, which can be rewritten in terms of our new variables as
x AÃa + x BBa + x CCa + x DDa = 0 A a /x A + B a /x B + C a /x C + D a /x D < 0 with similar expressions for stacks b,c,d. The x A , x B , x C , x D are related to the complex moduli of the tori χ i = (R 2 /R 1 ) i .
Only three are independent (for simplicity we can set x A = 1), and each must be positive. While we are free to adjust the moduli, each stack of branes introduces a new constraint. Thus generically three stacks of branes completely fix the three moduli of the tori, and so there is no freedom in adding a fourth stack of branes. For this reason, CPS only consider configurations with up to three non-trivial stacks, and this is a significant problem which must be addressed in our model. CPS classify the possible brane wrapping configurations into four types, based on the number of tori in which the stack of branes is parallel to one of the orientifold planes (i.e. the number or ns or ls equal to zero). Type I has 3 zeros and so is completely parallel to one of the orientifold planes, these are the so-called 'filler branes'. Type II has two zeros, there are no SUSic configurations with two zeros. Type III has one zero, so a type III stack is parallel to the orientifold plane in one of the three tori. In this case, exactly two of A, B, C, D and two ofÃ,B,C,D are zero. Without loss of generality we can choose n 1 = 0, so that A = B =C =D = 0, CD = −ÃB, and imposing the SUSY conditions we find
C < 0, D < 0,ÃB < 0, x B = −Ã/B.
Finally, type IV has no zeros, and so AÃ = BB = CC = DD = constant = 0. Also, SUSY requires that only one of A, B, C, D is positive, and
x A /A a + x B /B a + x C /C a + x D /D a = 0.
Assuming a maximum of three non-trivial stacks of branes, CPS show that the SU GU T (5) brane stack must be type III, and taking n 1 a = 0 they find C a = −1, D a < −4 with k = 1 or k = 2. Tadpole conditions then require a second stack with D b > 0 and must therefore be type IV, with A b , B b , C b negative. We still have the freedom to add a third stack of either type III or type IV, with the requirement C c ≤ 0.
In our case, since we are interested in four stacks, we would like two of the stacks to obey the same equation for the moduli, i.e. to have equal angles with respect to the orientifold plane. The natural choice then seems to be the GUT and Pentagon stacks since they must be parallel anyway. Our strategy then will be to consider the two SU (5) stacks to be parallel in the first torus, but with the orientation of one stack in the second torus parallel to that of the other stack in the third and vice versa (see figure 4.1). So for example, two stacks with the winding numbers
SU P (5) : (0, 3) × (1, 1) × (2, 1) SU GU T (5) : (0, 3) × (2, 1) × (1, 1)
would obey the same moduli equations; that is, the supersymmetry constraints on these stacks fix only one of the three moduli because the total angle of the stacks are the same. You will notice that this configuration has the two stacks parallel in the first torus (and therefore total topological intersection number of zero), while the number of intersection points in the last two tori is one as we would like, and if you calculate the number of chiral fields in the antisymmetric representation of the SU GU T (5) you will find the desired three families. In this example we would have C = −3, D = −6,Ã = 3,B = −6, x B = 1/2 for the Pentagon stack (stack a), and C = −6, D = −3,Ã = 3,B = −6, x B = 1/2 for the GUT stack (stack b). Unfortunately, this model is just one example of an entire class of similar models which suffer an incompatibility between the tadpole constraints and the SUSY conditions, as follows.
We have mentioned that stacks a and b must be parallel to the orientifold plane in order to avoid pentaquark-like exotics. This follows from the fact that we have demanded the number of intersection points in the second and third tori to be exactly one. If the two stacks were not parallel to the O6-plane in T 2 1 , stack a would certainly intersect with the image of stack b in that torus (with the reverse being true as well). We might have hoped that the topological intersection could still be zero if there is a cancellation (n i a l i b + n i b l i a ) = 0, i = 2 or 3, but this cannot be true if (n i a l i b − n i b l i a ) = 1, i = 2, 3. Since n, l are integers, there is no way to add two numbers to get one and subtract them to get zero. Thus, both the Pentagon and GUT stacks must be type III.
However, if this is true, we cannot simultaneously satisfy the tadpole and supersymmetry conditions. Let us enumerate some of the requirements on a and b if they are both to be type III. First, since the number of antisymmetric tensor representations for stack a is given by I aa + 1/2I aO6 = 3, and for type III I aa = 0, I aO6 = 2 3−k (à +B), we find that either k = 1 withà +B = 3 or k = 2 withà +B = 6, and in each casẽ AB < 0 as before. Second, if we are to have the two stacks parallel in the first torus (by choice), we must have either n 1 a = n 1 b = 0 or l 1 a = l 1 b = 0 in order for them to satisfy the same moduli equations. We will choose the former for convenience. Finally, the requirement that we have only one intersection between stack a and b in the last two tori implies n 2 a l 2 b − l 2 a n 2 b = 1, n 3 a l 3 b − l 3 a n 3 b = 1. These conditions are solvable yet very confining, the simplest solution of which was given in the example above.
The problem then is this. We know that the values of C and D are both negative integers for both stacks a and b, and in fact C a = D b , C b = D a for the type of configurations where stack a and b obey the same moduli equations as suggested above. This alone already implies that C a + C b = D a + D b < −2, but if the stacks are to satisfy the requirements of the previous paragraph the statement is more severe with
C a + C b = D a + D b < −9.
Recall that the tadpole condition instructs us to multiply these factors by the number of membranes in each stack, which again is N a = N b = 10, so that at best we have −90 + 2C c + 2C d > −16 and similarly for D. In other words, either C c or C d as well as D c or D d must be large and positive. This immediately rules out the possibility that stacks c and d are type III, because we know that for type III C and D are less than or equal to zero. For type IV stacks, only one of A, B, C, D can be positive and still satisfy SUSY conditions, so our only hope is that say C c > 0 and D d > 0 and that both values are large. Unfortunately, even this doesn't work. If C c > 0 then D c will contribute negatively to the D tadpole conditions, so of course we must require |D d | > |D c |, similarly |C c | > |C d |. This then leads to a problem with the moduli. For stacks c and d we have
A c + x B /B c + x C /C c + x D /D c = 0 A d + x B /B d + x C /C d + x D /D d = 0.
We have already solved for x B previously (it is positive), so the the first two terms in each of these equations sum to a negative number. Multiply the equations by C c D c and C d D d respectively, and we can rewrite these as
|D c |x C − |C c |x D = P −|D d |x C + |C d |x D = Q,
where P, Q are positive numbers. Summing the two equations we find
(|D c | − |D d |)x C + (|C d | − |C c |)x D = P + Q
implying that at least one of x C or x D is negative. This argument is analogous to the arguments in CPS given to forbid case (iv), k = 2 and case (i), k = 3 for type IV branes and case (i) for type III branes. Therefore, stacks a and b cannot be required to solve the same moduli equation. In fact, the argument is even stronger: stacks a and b cannot both be type III. This implies the existence of pentaquark-like exotics.
What if we relax this last requirement, i.e. not demanding stacks a and b to be type III? In order for two stacks to solve the same moduli equations, they must both be of the same type, so the question leads us to consider the compatibility of two stacks of type IV branes. The answer in this case is simple, and in fact applies regardless of the gauge groups supported on the stacks or their intersection number. As we have seen, the moduli equations for type IV can be written
A + x B /B + x C /C + x D /D = 0,
and if any two stacks are to both obey the same equation we must have
A 1 = A 2 , B 1 = B 2 , C 1 = C 2 , D 1 = D 2 .
In this case, as far as the tadpole conditions and supersymmetry constraints are concerned, we can then just consider these two stacks as a single stack with N = N 1 + N 2 and A = 2A 1 , etc. But we already know the requirements for a consistent model with three stacks. In particular, the GUT stack must be type III by the requirement of correct family multiplicity. Therefore, the possibility of the SU GU T (5) and SU P (5) stacks solving the same moduli equation is completely excluded. If we wish any other combination of type IV branes to solve the same moduli equation, the problem again reduces to the discussion of three stacks 5 .
We are left with two possible approaches. The first would be to return to the possibility of constructing a consistent model with three stacks of D6-branes. We would then have to either assume that two of the stacks exactly obey the same moduli, supersymmetry, and tadpole equations (as suggested above), or to abandon one of the U (1) stacks and argue that the Pentagon singlets arise from another mechanism. However, the question of SU (5) GUT theories with three stacks of branes was exactly the subject of the CPS search. The GUT stack must be type III, and a second stack must be type IV. In their paper, they have listed all 149 possible solutions for models with a third stack of type III, none of which contain a second SU (5) gauge group. Thus, the SU P (5) and any U (1) factors must arise from stacks of type IV. According to CPS, this would require a very extensive search that would have to be conducted on a case by case basis, a search that CPS didn't endeavor to attempt. In any case, we believe it likely that a proof could be constructed to show that this possibility is inconsistent due to the severity of the constraints imposed by adding a second SU (5) gauge group. This will be the subject of a future investigation.
The second possible approach is to allow a different combination of stacks to obey the same moduli equation. As we have argued, both these stacks would have to be type III or else the problem again reduces to a question of three stacks. We know that at minimum one of the stacks has to be type IV to satisfy the tadpole conditions, and this stack would likely have to sustain the SU P (5). If we make this assumption, we would have to find a non-trivial combination of two stacks with N a = 10 and N b = 2 which satisfy the same moduli equation. The argument would then parallel that of two type III stacks given above, but with some of the assumptions made there relaxed. If such a search were to fail, we would have to completely abandon the hope that two of our four stacks exactly solve the same moduli equation, and would be forced to find a system of equations in which the fourth stack obeys an equation which is a non-trivial linear combination of the other three. We have not yet found a strategy for systematically attacking this problem.
In any case, we know that these models will contain many undesired chiral exotics. CPS have shown that the existence of 15 sym representations are unavoidable in models with 10 a s. We have further demonstrated that our models will contain chiral pentaquark-like exotics, charged as (5,5) under the Pentagon and GUT gauge groups. These particles are surely phenomenologically untenable.
Furthermore, though our search for the particle content of the Pentagon has proven somewhat inconclusive to this point, we find it likely that no self-consistent solution exists. The major culprit for this difficulty seems to be the tadpole constraints imposed by requiring two SU (5) gauge groups. Generally speaking, the larger the gauge group, the more negatively a stack will contribute to the RR-charge. This fact, combined with the requirement that we find one (and only one) vector-like pair of pentaquarks, seems to be too great an obstacle to overcome. This leads us to believe that these constraints might be softened if we were to consider searching for the particle spectrum of the Pyramid model, for which we would need an SU (3) 4 gauge group arising from four stacks of N = 6 D6-branes. At the current time we are in the preliminary stages of such a search, and the approach seems promising.
M-theory on G 2 Manifolds
Because they carry no fluxes or additional charge sources, D6-branes and O6-planes are seen to be pure geometrical artifacts in the strong coupling limit. This suggests that one might consider an M-theory description of chiral particles arising at points in the manifold. In particular, we are led to believe that N = 1 globally supersymmetric type IIA intersecting D6-brane models lift up to eleven dimensional M-theory compactifications on singular G 2 manifolds [11]. D6-branes and O6-planes wrap smooth supersymmetric three cycles in the IIA compactifications, and one fibers each of these by a suitable noncompact hyperkähler four-manifold to obtain the G 2 holonomy space. In the Mtheory language, these are codimension four ADE-orbifold singularities spanning three cycles in the G 2 compactification manifold, and must be ALE (asymptotically locally Euclidean) spaces. N overlapping D6-branes correspond to an A-type ALE singularity, D-type singularities arise for D6-branes overlapping O6-planes. Chiral fermions exist at isolated co-dimension seven singularities, which would correspond to the G 2 lift of the intersection points of D6-branes and O6-planes in the IIA picture. Just as in the IIA constructions, family replication is given by the number of these singular points in the manifold. When a point on the manifold shrinks to a conical singularity, the symmetry supported along that fiber will be enhanced at the singularity. To determine the chiral representations arising there, we decompose the adjoint of the group associated with higher symmetry with respect to that of the lower [12].
Specifically [13], we will obtain chiral fields in the representation R of group G if at certain points on the manifold the G singularity is enhanced to a groupĜ = G ⊗ U (1). Away from these points, the Lie algebra ofĜ will decompose aŝ g → g ⊕ o ⊕ r ⊕r where g and o are the Lie algebras of G and U (1), r transforms as R (and of charge 1 under U (1)), andr the complex conjugate. However, Acharya and Witten have shown that the net number of chiral zero modes is one, meaning that only either r orr will appear in the low energy theory (depending on how the chirality is fixed) 6 . The group G need not be simple, it may be any semi-simple product of the groups obtained by deleting one node from the Dynkin diagram ofĜ. The representations found at a particular singularity will not always be free from anomalies; however one can show that when this is the case there must exist another point (or set of points) elsewhere on the manifold supporting particles which render the theory anomaly free.
For A and D type singularities we are led to the representations listed in Table 3. Note that these are in agreement with the picture we have from IIA intersecting D6-brane models. For two three-cycles intersecting at a singular point in the G 2 manifold, supporting gauge groups SU (N ) and SU (M ) respectively, we are left with chiral Resolution Chiral Representation fermions in the bifundamental representation (N,M ). This is just as we would expect from the intersection of two stacks with N and M D6-branes. Similarly, the resolution SO(2N ) → SU (N ) leads to the antisymmetric representation of SU (N ), corresponding to particles which would be found at the intersection of a stack of N D6-branes with an O6-plane. However, the parallel should not be taken too literally. Unlike the IIA picture, three cycles in a seven-manifold do not generically intersect, so the existence of multiply charged particles will only be found in specially constructed geometries.
SU (N + 1) → SU (N ) [N ] SU (N + M ) → SU (N ) × SU (M ) [N,M ] SO(2N ) → SU (N ) [N ⊗ N ] a SO(2(N + 1)) → SO(2N ) [2N ]
Can the Pentagon model be embedded in a G 2 compactification of M-theory? To answer this, we must find a 4-d theory with an SU P (5) × SU SM (5) gauge group and chiral fermions in the representations (5,5) and (5, 5) (or possibly (5, 5) and (5,5)), 3 × (1, 10), 3 × (1,5), a pair of Higgs (1, 5), (1,5), and a singlet field S (which we will ignore for the moment-we might assume it arises as some modulus of the geometry); all of which arise at singularities in the G 2 manifold. There is no single point that can sustain a symmetry which unfolds to the SU (5) × SU (5) gauge group plus all of the desired matter, so our model will necessarily have to be a patchwork of fields lying at different points in the manifold. This is not necessarily a problem; such a geometry would surely be less generic, but it could help explain the large family hierarchies of the standard model. The proximity of matter multiplets to Higgs fields would vary from point to point, creating a natural hierarchy in the Yukawa couplings.
It is clear that the desired components of our model can be derived in this construction, and our search for such a model in the IIA context points to the answer. Consider a three-cycle in the G 2 manifold sustaining an SU (5) ADE-orbifold singularity. If at certain points along this cycle we find a conical singularity, the symmetry will be enhanced. If the enhanced gauge group is SU (6), the symmetry will unfold as SU (6) → SU (5) ⊕ o ⊕ 5 ⊕5. Let us suppose the zero modes are the5s, and that there are four of such points. Anomaly cancellation ensures that elsewhere on the manifold we can find representations of opposite chirality, but let us assume we find only one such point (leaving us with one 5 representation). Let the additional anomalies be canceled by 10 a representations, arising at singularities where the symmetry has been enhanced to SO(10), i.e. SO(10) → SU (5) ⊕ o ⊕ 10 ⊕10. Now assume that there exists an additional three-cycle on the manifold supporting a new SU (5) symmetry, and that these two cycles somewhere intersect at a point. If this special point happens to lie at a conical singularity, the symmetry will be enhanced to SU (10), which will resolve as SU (10) → SU (5) × SU (5) ⊕ o ⊕ (5,5) ⊕ (5, 5). Here we will find the pentaquarks, and elsewhere on the manifold we must find the anti-pentaquarks to cancel anomalies. Thus, this G 2 manifold will support two SU (5) symmetries as well as the entire matter content of the Pentagon model, with no exotics (see Table 4).
Location
Supported Gauge Group Enhanced Singularity Matter Content Three-Cycle 1 (10) [5,5] Points [5,5] Clearly, such a model will be highly non-generic. We may desire a model which sustains (at minimum) a pervasive symmetry G = SU (5)×SU (5) throughout the entire manifold, allowing the gauge group to be defined throughout the bulk. However, such a requirement complicates the model significantly. As we have seen, the pentaquarks can be found at points where the symmetry is enhanced to SU (10), but this leaves no freedom to derive the chiral fields of the model 7 . On the other hand, the SU (5)×SU (5) gauge group can be obtained from Higgsing an SU (10) with Wilson lines, so let us have the G 2 manifold support a pervasive G = SU (10).
SU (5) 5 × SU (6) 4 ×5 1 × 5 3 × SO(10) 3 × 10 a Three-Cycle 2 SU (5) Intersection SU (5) × SU (5) 2 × SU
One possibility is that G uplifts toĜ = SU (11), in which caseĝ → su(10) ⊕ o ⊕ 10⊕10. With Wilson lines, the 10 will further decompose as (1, 5)+(5, 1). Five of these points (with the proper chirality) will provide us with the standard model 3 × (1,5) as well as the Higgs fields. However, we are left with some potentially undesirable particles, namely the four (5, 1)s and the (5, 1). We can imagine that the (5, 1) will mass up with one of the (5)s, but we are still left with 3 × (5, 1). These particles are not necessarily problematic, as they have no standard model quantum numbers or interactions. In fact, they may have the potential of providing us with a dark matter candidate. For this to be possible, we would need to find λ ijk s (where λ ijk 10 i5j5k ) such that the U (3) × U (3) flavor symmetry is broken to a conserved U (1) with T r[T 3 ] = 0. For now, let us just assume that these fields pose no phenomenological problems for the model.
If at another point we findĜ = SO (20), we would be left with SU (10) + o + 45 +45 where the 45 a is the antisymmetric tensor representation of SU (10). Higgsing the SU (10), the 45 decomposes as (5, 5) + (1, 10) + (10, 1) under SU P (5) × SU G (5), leaving us with candidates for the pentaquarks and the standard model 10 a . However, in order to have three standard model generations there must be three separate points on the manifold with aĜ = SO(20) singularity. Thus we obtain (one half of) the desired pentaquarks (charged as (5,5) as opposed to (5,5)), as well as the 3 × (1, 10) of the standard model; we also are left with an undesirable two additional copies of (5,5) and with three (10, 1). The latter of these is possibly interesting (as discussed above), but the former spells disaster. Fortunately, with proper Z 4 R-charge assignments (two of the three (5,5) with R-charge 0 and one with charge 2), we might imagine one pair massing up and leaving us with just the single generation of pentaquarks. Furthermore, this Z 4 could conceivably ensure that the 10s do not gain mass.
Of course, so far this model is not anomaly free. This is perhaps fortunate, because we are guaranteed to find the vector-like pair for the pentaquarks elsewhere on the manifold. We might be tempted to think the anomalies of the numerous 5 and 10 representations exactly cancel (there are equal numbers of5s with 5 + 10 a s), but there would then be no way to construct the anti-pentaquarks (5,5). Thus we are forced to consider an additional three points elsewhere in the manifold supportingĜ = SO (20) singularities giving rise to the conjugate pairs. But this would lead to vector-like pairs 10 +10, which is clearly unacceptable. This would also force us to include additional 5 representations to cancel the anomalies of the5s, and whether or not these pairs gain mass or remain light, this is certainly phenomenologically untenable.
One potential solution to this problem would be to soften our requirement for a pervasive SU (10) gauge symmetry to a pervasive SU (5) × SU (5), but containing an entire fiber with enhanced symmetry SU (10). The pentaquarks of this model would arise at points away from this fiber, but also enhanced to an SU (10) symmetry, and unfolding as SU (10) → SU (5) × SU (5) ⊕ o ⊕ (5,5) ⊕ (5, 5) plus its complex conjugate. Along the fiber, certain points would have 'worsened' singularities with the enhanced symmetriesĜ = SO (20), SU (11). This would give rise to all of the desired components listed above, with the anomalies of the5s and 10 a s exactly canceling. The (5, 5)s arising from the SO(20) conical singularities would be exotics in this model. However, if for every (5,5) representation there were an additional five singularities with SU (11) symmetry, each producing a (5, 1) + (1,5) representation, the anomalies of these undesired particles would cancel. Furthermore, with correct R-charges, we might hope that all of these exotics gain mass. Of course, this model seems no more aesthetically viable than those without a pervasive symmetry listed above.
We therefore believe that the best candidate for our model is a G 2 manifold supporting two three-cycles with SU (5) gauge symmetry intersecting at exactly two points. One of these cycles will support the standard model GUT SU (5), and there will be points along this cycle at which the symmetry is enhanced to either SU (6) or SO (10) giving rise to the standard GUT matter. The intersection points must lie at special points on the manifold where the K3 structure has an enhanced symmetry, such that we find SU (10) → SU (5)×SU (5)⊕o⊕(5,5)⊕(5, 5) at one intersection and the vector-like partners at the other. Though this type of geometry is certainly highly non-generic, it arises naturally from a lift of type IIA intersecting D6-branes. In such a model, we expect to find chiral particles in the bifundamental representation exactly at such points where two three-cycles intersect. Indeed, the fact that such a construction is possible in the M-theory context suggests that we might hope to find a consistent model of intersecting D6-branes. Conversely, the difficulty we have found in constructing such models might suggest that the existence of such an M-theory geometry is dubious. Unfortunately, the tools necessary to perform an explicit calculation are unknown, and the existence of such a model remains in question.
Heterotic Orbifold Constructions
Some of the most realistic phenomenological string models have been produced in the framework of heterotic orbifolds [15]. This presents us with a promising approach toward discovering a phenomenologically viable low energy model. In this section we will briefly review the heterotic orbifold construction 8 , and then proceed to discuss our search strategy and preliminary results.
Brief Review
Heterotic string theory is a theory of closed strings, combining the supersymmetric right moving string with the left moving bosonic string. As the (uncompactified) theory must exist in 10 space-time dimensions, the extra 16 dimensions of the bosonic string are interpreted as internal degrees of freedom. To satisfy modular invariance (and anomaly cancellation) these 16 left movers must live on a 16 dimensional Euclidean even selfdual lattice, which we choose to be the root lattice of E 8 × E 8 (although the SO(32) root lattice has been shown to be of interest as well [17]).
The low energy effective field theory will consist of those states which survive to low energies. In particular, any state with mass at the string scale will not be observed at low energy, so we are only concerned with string states of zero mass. Further, since the physical heterotic string states are the direct product of the right movers with the left movers, both the right-moving and left-moving string states must be massless. Working in the light cone gauge, we find that there is a total of 16 massless right movers, 8 in the NS sector which transform as an SO(8) vector and 8 in the R sector transforming as the SO(8) spinor. When tensored with the left movers, the vector representation will produce the boson of the 10 dimensional supersymmetric chiral fields, while the spinor gives its fermionic superpartner. The 8 bosonic left moving oscillators corresponding to the space-time degrees of freedom create massless states when acting on the left moving ground state; when tensored with the right movers, these form the N = 1 supergravity multiplet. Similarly, the 16 internal degrees of freedom bosonic oscillators acting on the left moving ground state form the 16 uncharged gauge bosons of E 8 × E 8 (and their superpartners) when tensored with the right movers. Finally, there are the massless 240 + 240 charged gauge bosons (plus superpartners) of E 8 × E 8 , which come from the tensor product of the right movers with those left moving states having internal momenta satisfying (p I ) 2 = 2. This is exactly the condition for the root vectors of E 8 , ensuring that these states lie on the E 8 × E 8 root lattice. Altogether, the massless heterotic string states form a ten-dimensional N = 1 supergravity theory with E 8 × E 8 gauge group.
As the Pentagon model is a four dimensional low energy effective field theory, six of the space-time dimensions must be compactified. The simplest way to achieve this is to wrap each of these extra coordinates on a circle, which is topologically equivalent to compactifying on the 6-torus T 6 . However, torus compactification schemes in general do not lead to realistic models in four dimensions. In particular, the SO(8) spinor of the 10 dimensional heterotic theory compactified on T 6 gives a total of 4 gravitinos in 4 dimensions, thus leading to N = 4 supersymmetry. To obtain a chiral theory with N = 1 supersymmetry, one may compactify on an orbifold
O = T 6 /P ⊗ T E 8 ×E 8 /G,
where the space-time and internal degrees of freedom are differentiated to admit a clear space-time interpretation. Formally, an orbifold is defined to be the quotient of a torus over a discrete set of isometries of the torus, called the point group P. The simplest of these is the symmetric abelian orbifold, where the point group is chosen to be the cyclic group Z N with N = 3, 4, 6, 7, 8, 12. The lattice on which P acts as an isometry will be the root lattices of semi-simple Lie algebras of rank 6. The space group S is defined to be the point group P plus the translations given by these lattice vectors, such that T 6 /P = R 6 /S. The action of the space group on the (complex) space-time degrees of freedom can be written as Z a → e (2πiv a ) Z a + n α e α , a = 1, 2, 3
where v, the generator of the discrete group Z N , is called the twist vector, and the e α are the lattice vectors of the root lattice spanning T 6 . Thus, two points on R 6 are identified if they differ by the action of the space group. Points that are invariant under the action of the space group are known as fixed points of the orbifold. To ensure that exactly one 4 dimensional space-time supersymmetry survives, ±v 1 ± v 2 ± v 3 = 0 mod 2 with none of the v a vanishing.
G is called the gauge twisting group. Modular invariance requires the action of the space group to be embedded into the gauge degrees of freedom. This means that in general the internal gauge group of the orbifold will be a subgroup of the E 8 × E 8 gauge group of the uncompactified heterotic theory. To realize this embedding, the orbifold twist vector is associated with a shift vector V in the E 8 × E 8 root lattice, while the torus shifts e α are embedded as shifts W α . Since the W α correspond to gauge transformations associated with non-contractible loops, they are interpreted as Wilson lines. The action of the gauge twisting group G on the gauge degrees of freedom is
X I → X I + 2π(kV I + n α W I α ).
The combined action of S ⊗ G is known as the orbifold group. Not all gauge twists and discrete Wilson lines are physically allowed. Modular invariance automatically guarantees the anomaly freedom of orbifold models. For the partition function to be modular invariant, it must satisfy the following conditions:
(V 2 − v 2 ) = 0 mod 2 V · W α = 0 mod 1 W α · W β = 0 mod 1, α = β W 2 α = 0 mod 2.
These conditions are known as 'strong modular invariance'. In reality one need only satisfy 'weak modular invariance', where these conditions are slightly relaxed:
N (V 2 − v 2 ) = 0 mod 2 N V · W α = 0 mod 1 N W α · W β = 0 mod 1, α = β N W 2 α = 0 mod 2.
However, if weak modular invariance is satisfied, we can in general bring V and W α to a form which obeys strong modular invariance by adding E 8 × E 8 lattice vectors. This has the advantage of simplifying the projection conditions on physical states. 9 N is the order of the orbifold (and of the cyclic group Z N ). Cyclic group multiplication rules require that N successive rotations of the orbifold act as the identity N v = 0 mod 1, and that N V belongs to the E 8 × E 8 lattice.
The gauge transformations are required to be a symmetry of the system. To calculate which states survive orbifolding, we must consider the action these transformations have on the states with right-and left-moving momentum. Neither the shifts nor the twists act on the oscillators. The generator of translation is
e ip·X |0 = |P ,
so a shift in the coordinate degrees of freedom acts as a phase rotation on the states. For the right movers, |q → e 2πiq·(kv) |q and for the left movers, |P → e 2πip·(kV +nαWα) |P .
States that are invariant with respect to the orbifold group transform trivially (with a phase of 1) under every element of the group, i.e. for all k, n α = 0, ..., N − 1. Only invariant states are consistent with the geometry of the underlying orbifold space; all other states must be projected out.
The massless spectrum consists of all massless closed string states consistent with the geometry of the orbifold. This includes the massless strings of the original heterotic theory which survive the projection conditions, as well as additional new states which arise due to the non-trivial geometry of the orbifold. The former form the untwisted sector and are free to move throughout the orbifold, while the latter are known as twisted sector states and are confined to the fixed points.
First consider the untwisted sector. As mentioned earlier, orbifolding projects out three of the supersymmetries, and we are left with an N = 1 supergravity multiplet (as well as certain modulus fields which are not relevant to the current discussion). The 16 uncharged gauge bosons correspond to the Cartan generators of the E 8 × E 8 algebra. By construction, the gauge twists and Wilson lines must commute with the Cartan subalgebra, thus all uncharged gauge bosons (and gaugino partners) survive the orbifold projection. Furthermore, the rank of the algebra can never be reduced by 9 There are exceptions to this rule, such as when V = 0. the shift embedding. The charged gauge bosons of the heterotic string give rise to both the unbroken gauge group as well as charged matter states. As these are states with both right-and left-moving momenta, they transform under the orbifold group as |q ⊗ |P → e 2πi(q·(kv)+p·(kV +nαWα)) |q ⊗ |P .
The momenta of the right movers are given by their SO (8) The underline denotes that all permutations are included. Only states with an even number of minus signs are included for the fermions. Gauge bosons in four dimensions have two transverse polarizations, and so require oscillators in the uncompactified directions, i.e. q = (±1, 0, 0, 0) in common notation. 10 Similarly, the gaugino states must have the right movers q = ±(1/2, 1/2, 1/2, 1/2). Thus, the right movers of the four dimensional gauge bosons (and gauginos) are invariant under the orbifold action, q · v = 0. The left movers, then, must satisfy p · (kV ) = 0 mod 1 p · (n α W α ) = 0 mod 1 for all k, n α . Not all of the charged gauge bosons of the heterotic string will satisfy these conditions, so the gauge group is broken; those that do survive (along with the 16 Cartan generators) form the generators of the unbroken gauge group on the orbifold. However, there are additional states which satisfy q · (kv) + p · (kV + n α W α ) = 0 mod 1 without fulfilling q · v = 0. These states are interpreted as charged matter, and the root vectors p are their weights with respect to the unbroken gauge group.
A twisted string is one that closes only by imposing the space group symmetry,
Z a (τ, σ + 2π) = e (2πikv a ) Z a (τ, σ) + n α e α ,
i.e. by performing both twists and lattice shifts. Thus, they must be localized at the fixed points. Each of these states is dependent on the required number of twists, thus k 10 In complex coordinates, the first component refers to the uncompactified directions and the last three components to the coordinates on the six-torus T 6 . The lightcone coordinates are gauge fixed and are omitted. Typically, the first component is omitted when writing the twist vector, v, as it must be zero. labels the N − 1 twisted sectors (k = 0 corresponds to the untwisted sector). Similarly, the presence of Wilson lines is determined by the corresponding lattice shifts required at each fixed point. Wilson lines affect the mass equation for the left movers (as we will see below), so this has the effect of changing the representations found at different fixed points. Modifying the boundary conditions for the twisted sector changes the mode expansions for the right and left movers, which in turn shifts the weights of the states, q → q + kv and p → p + (kV + n α W α ). As a result, the level matching condition for the massless states now reads
1 2 (q + kv) 2 − 1 2 + δc = 1 4 m 2 R = 1 4 m 2 L = 1 2 (p + kV + n α W α ) 2 + N L − 1 + δc = 0.
N L is the number operator for the left movers, and is allowed to be fractional as a consequence of a non-trivial twist. To be more specific, N L = a (η a N La +η a N * La ), where η a = kv a mod 1 with 0 ≤ η a < 1,η a = −kv a mod 1 with 0 ≤η a < 1, and N La , N * La are oscillator numbers of the left movers in the z a andz a directions. δc is a shift in the zero point energy, and is given by
δc = 1 2 3 a=0 η a (1 − η a ).
Once the massless spectrum of the twisted sectors is calculated, projection conditions must be applied. Among the massless representations, physical states are selected by the generalized GSO projection operator. In a theory with non-trivial Wilson lines, the momentum shift is dependent on the fixed point under consideration as discussed earlier. Therefore, the GSO projection should be applied to each state individually. This can be written:
P (k, γ, n α ) = 1 N N −1 l=0
[∆(k, γ, n α )] l with ∆(k, γ, n α ) = φγe 2πi[(P +kV +nαWα)·(V +nαWα)−(r+kv)·v]
Here, k labels the twisted sector, and the n α label the order of the Wilson lines relevant for the given state (corresponding to the number of lattice shifts required for the point to be invariant under the space group action). γ is the eigenvalue of the state under the action of k orbifold twists. For prime orbifolds (e.g. Z 3 , Z 7 ) this factor is trivial, γ = 1. For non-prime orbifolds, physical states are defined by linear combinations of massless states living at fixed points which transform into each other under the space group action. These physical states can be shown to have definite eigenvalue γ = e (2πqγ ) , q γ = 0, 1/n, 2/n...1 under the rotation. The oscillator phase is
φ = e 2πi a va(N La −N * La ) .
For any non-trivial phase ∆, the contributions of ∆ l in the sum for P will all add up to zero. Thus, only states satisfying ∆(k, γ, n α ) = 1 will survive the projection. Equivalently, the projection condition can be written:
(P + kV + n α W α ) · (V + n α W α ) − (r + kv) · v + a v a (N La − N * La ) + q γ = 0 mod 1.
For states with q γ = 0 (i.e. for prime orbifolds), one can use the modular invariance equations to show that this condition is in fact automatically satisfied for all states satisfying the mass equation. Thus, all massless representations of the prime orbifolds are in fact physical states, and the GSO projector need not be calculated.
The above construction provides the rules for calculating the entire low-energy spectrum of the heterotic orbifold theory. For calculational convenience, we have automated the process using Mathematica, and included it as an Appendix 11 . The required input is simply the orbifold twist vector, the gauge shift, and the Wilson lines; the program will then check modular invariance, calculate the gauge group and output the surviving simple roots and Cartan matrix, and calculate the surviving states in both the untwisted and twisted sectors, displaying the highest weight representations in Dynkin label notation. The surviving gauge groups and representations may be interpreted by comparison with, for example, the extensive tables of [18].
Search Strategy and Results
Heterotic models based on Z N orbifolds are well known and have been discussed extensively in the literature. There are a finite number of gauge groups obtainable from E 8 × E 8 for a particular orbifold, and these have all been systematically classified and their matter contents calculated. In [19], the authors have tabulated the results for every inequivalent modular invariant gauge shift (with no Wilson lines) for each discrete orbifold. Wilson lines complicate the theory significantly, as they provide a mechanism to further break down the gauge symmetries of the models as well as to change the representations found at different fixed points, thereby greatly increasing the number of inequivalent models. Still, the rules are well understood and a large number of these models have been calculated. The prime orbifold Z 3 is particularly well known as it has the simplest transformation properties under the orbifold group. Therefore, we have chosen the Z 3 orbifold as the starting point for our search. Calculations of twisted sector states in Z 3 models are greatly simplified due to the fact that GSO projectors need not be calculated if strong modular invariance is satisfied. Conversely, the simplicity of the projectors in this case allow a straightforward calculation of physical states, sug-gesting we may employ weak modular invariance to ease the constraints on our models. We have elected to follow the latter approach.
There are only five possible breakings of E 8 by N = 3 modular invariant gauge shifts (without Wilson lines): E 6 × SU (3), SU (9), E 7 × U (1), SO(14) × U (1), and E 8 (unbroken). Clearly, the SU (5) factors of the Pentagon model would have to arise from different E 8 s, and there is only a limited number of Wilson lines that would provide the desired symmetry. However, there is a very large number of ways to fit SU (3) 4 into E 8 × E 8 . Furthermore, it would seem natural for the Z 3 symmetry of the trinification model to arise as the result of the geometry of the orbifold. Thus, we have elected to confine our search to the particle spectrum of the Pyramid model. That is, we wish to find the low energy gauge group SU on a Z 3 orbifold with twist vector (1/3, 1/3, −2/3). The matter content of standard trinification fits naturally into a 27 representation of E 6 . Thus, we will further assume that three of the SU (3) factors fit into an E 6 subgroup of a single E 8 . There is only one gauge shift that will break E 8 to E 6 × SU (3) on a Z 3 orbifold, V = (2/3, 1/3, 1/3, 0, 0, 0, 0, 0) (there are other modular invariant gauge shifts that have the same effect, but they are all equivalent to the one listed by shifts in the lattice). Of course, in our models V has 16 components, 8 degrees of freedom corresponding to each E 8 . The full vector must satisfy (strong) modular invariance, and the condition (V 2 − v 2 ) = 0 mod 2 provides little freedom for the last eight components. Thus we will choose V = (2/3, 1/3, 1/3, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0), leaving the second E 8 unbroken for the moment. This will change with the addition of Wilson lines.
In fact, realistic trinification models have been discovered under these assumptions [20]. These models do not, unfortunately, exactly reproduce the spectrum of the Pyramid model. In particular, while some of these do include a fourth SU (3) gauge group, none contain a full set of vector-like trianons. However, their models do provide useful guidelines for directing our search. There are only two models listed in the tables of [19] which give an SU (3) 3 , but these do not have the correct matter content. Thus, one is forced to consider a model with Wilson lines. In [21] It would be convenient if the entire Pyramid model fit into a single E 8 , with the second E 8 remaining hidden. For that to be true, the gauge shift would break E 8 to the Pyramid SU (3) times the Standard Model GUT E 6 , and Wilson lines (specifically W 2 above) would further break E 6 to the desired trinification SU (3). Unfortunately, these choices do not produce the desired spectrum, and it appears that the Pyramid SU (3) will have to arise from the second E 8 . Phenomenologically this poses no problem; it does however make obtaining this model quite difficult, due to the fact that there are no fields charged under both E 8 s in the non-orbifolded heterotic string theory. Since all chiral representations obtained in the untwisted sector are merely a subgroup of the entire E 8 ×E 8 adjoint which survive the projection conditions, it is impossible to obtain representations charged under both E 8 s from the untwisted sector. However, because the momenta of the states existing at the fixed points of the lattice are shifted by the presence of Wilson lines, it is possible to obtain states charged under both E 8 s in the twisted sector.
Thus, our search strategy has been as follows. We begin with the Z 3 orbifold obtained from the gauge twist (1/3, 1/3, −2/3). We wish to break the first E 8 to E 6 × SU (3) via the gauge twist V = (2/3, 1/3, 1/3, 0, 0, 0, 0, 0)( − → 0 ). To obtain standard trinification with no chiral exotics, we must further break E 6 → SU (3) 3 and SU (3) → U (1) 2 . This can be acheived by W 1 alone, or by a combination of W 2 and additional Wilson lines, such as W 3 = (1/3, 1/3, 0, 0, 0, 0, 0, 0). Whichever we choose, we must also assign values for the eight additional components corresponding to the second E 8 such that the entire 16 component vector remains modular invariant. The second E 8 gauge group must be broken to SU (3) × G, where G is some unspecified cofactor. Phenomenologically the group G is arbitrary as long as there are no fields charged simultaneously under both it and the trinification group.
Because the lattice vector e i 1 is equivalent to e i 2 on each two-torus T 2 i by the action of the twist, the Z 3 orbifold can sustain a maximum of three Wilson lines, one for each two-torus. As we have discussed, the presence of a Wilson line differentiates states at different fixed points of the corresponding torus. Because there are 27 fixed points on the Z 3 orbifold, the multiplicity of twisted sector states in a model without Wilson lines would be 27. Models with one Wilson line will have twisted sector multiplicity 9, two Wilson lines give multiplicity 3, and three Wilson lines differentiates each fixed point individually. Models with two Wilson lines seem to suggest a geometric explanation for the family multiplicity of the Standard Model. However, we are constrained to find only a single generation of the trianons, and are therefore led to consider models with three Wilson lines. This of course complicates the task of finding three trinification generations.
Thus far we have fixed the gauge twist and have narrowed the possibilities for one of the Wilson lines. There still remains the freedom to choose two additional Wilson Lines-each of which is a 16 dimensional vector. Constraints are imposed due to the fact that the Wilson lines must obey modular invariance and by the requirement that we do not break the gauge symmetry of the first E 8 beyond SU (3) 3 . Nevertheless, this still permits a vast number of models to be calculated if we are to scroll through each possible vector in succession (perhaps on the order of > 10 10 ), making a comprehensive search rather difficult. At the present time we do not have the computing power necessary to perform such a search, though we would like to do so in the future. Actually, it is conceivable that the number of distinct Wilson lines is in fact much smaller due to the fact that many will be equivalent up to lattice shifts, but we have not found a way to use this fact to our advantage at the current time. Thus, to this point we have only endeavored to follow the more modest approach of trial and error.
Unfortunately, we have not found anything resembling the complete spectrum of the Pyramid model. While a large number of models contain the standard trinification spectrum (as we should expect considering we have specifically chosen our gauge twist and Wilson lines to enforce this), it is very difficult to obtain the trianons. We believe this is due to the difficulty of finding particles charged under both E 8 s. It is interesting to note that the models we have found closest resembling the spectrum of the Pyramid contain a chiral set of the trianon-like particles, but finding their vector-like partners has proved elusive. This is not entirely surprising, considering that the heterotic orbifold models were originally constructed to produce a chiral spectrum. This could even be a general symptom of these models (and a failure for our purposes), but a comprehensive search would have to be conducted to know this for certain.
We present two interesting results in the tables 5, 6. The first model is perhaps the most promising. It contains three complete trinification generations, as well as a number of Higgs-like fields. It also contains a single generation of (an incomplete set of) chiral trianon-like particles, but it does not contain their vector-like partners. The model also contains a few chiral exotics. The second model is interesting in that it contains a single complete set of chiral-like trianons (i.e. one half of the 6 total). It also complains a completely vector-like trinification spectrum. It should be noted that the GSO projectors have not been implemented on the spectra of these models (the projectors will only project out states, and the spectra are incomplete to begin with), and the spectra listed are therefore anomalous.
The next step in our research will be to clearly establish the number of inequivalent If such a search fails to produce the desired spectrum, we will be forced to perform a similar search in the other Z N orbifolds, probably forcing us to abandon our desire for the Z 3 trinification symmetry to be an artifact of the geometry of the manifold 12 . Regardless of the outcome, we are still interested in a future search for the SU (5) × SU (5) gauge group and particle spectrum of the Pentagon model.
Concluding remarks
Though we have not been able to rule out the existence of the Pentagon model as a 12 It might still arise as a result of a non-prime orbifold, Z 6 = Z 2 × Z 3 or Z 12 = Z 3 × Z 4 . low energy effective field theory embedded in a string theory, we have thus far had no success in constructing such a model. In each of the embedding structures we have explored, the constraints imposed by our criterion have proven to be quite strict. In part this is due to the size of the desired gauge groups, but we believe that an even more restricting constraint is the requirement that we find both chiral and vector-like particles in the spectrum.
This requirement posed a strict constraint on the geometry of the two stacks of D6-branes supporting SU (5) gauge groups in the type IIA construction. In fact, we found that no such structure was able to satisfy the same equation for the complex structure moduli while remaining consistent with RR-tadpole charge cancelation. This problem translates into a difficulty with maintaining supersymmetry. While there may exist a more complicated geometry satisfying all of our criteria, finding such a model proved to be beyond the scope of our current search. However, another approach we are currently investigating is to embed the Pyramid model into an intersecting D6-brane construction, though the results of this search are still unclear.
We did discover a potential candidate for the existence of the Pentagon model in the case of M-theory manifolds of G 2 holonomy, though the proof of its existence is beyond our capabilities. However, such a model does not support a pervasive SU (5) × SU (5) symmetry throughout the G 2 manifold. If we include this criteria as a requirement for the model, we have shown that it becomes quite difficult to obtain both the vector-like pentaquarks and the chiral antisymmetric 10 representations of the GUT SU (5). This follows from the fact that both representations are found at a singularity which resolves as SO(20) → SU (10) + o + 45 +45. We may break the SU(10) via Wilson lines, leaving us with chiral particles in the representations (5, 5) + (1, 10) + (10, 1). Therefore, in this construction, it is impossible to find vector-like partners for the pentaquarks without simultaneously producing vector-like partners for the 10s.
We have also shown that it is difficult to obtain the vector-like trianons of the Pyramid model in a heterotic orbifold construction. While we were able to find models with a standard trinification spectrum, we found that the trianons must arise in the twisted sector of a Z 3 orbifold due to the fact that they must come from fields charged under both E 8 s of the uncompactified heterotic theory. We did not perform a systematic search through all modular invariant gauge shifts so we cannot make any conclusions about the existence of the Pyramid spectrum in these models, but we were unable to find the complete spectrum in our search and believe it likely that no gauge shift in the Z 3 will give rise to a vector-like set of trianons. If this is indeed the case, we are forced to abandon our hope that the Z 3 symmetry of the Pyramid model arises as an artifact of the geometry.
Despite our limited success, there still remain many avenues in the vast landscape of string models to explore. We are especially interested in continuing our search for the Pyramid model of TeV physics in the contexts of each of the three string theories we have investigated. We also believe that models of intersecting branes in type IIB theory and F-theory models might afford us the techniques required to build our desired low energy effective theory.
Acknowledgments
I would like to thank my dissertation advisor T. Banks for extensive discussions about this work. I would also like to thank H. Haber for discussions about group theory and representations, and Ben Dundee for his help in understanding heterotic orbifolds. This research was supported in part by DOE grant number DE-FG03-92ER40689.
Figure 1 :
1Geometeric Requirement for the Pentaquarks. Total intersection number is zero, with only a single intersection point (at the origin) in the second and third T 2 .
the authors have classified all possible Wilson line breakings of E 6 × SU (3) on a Z 3 orbifold with one Wilson line, and have tabulated the resulting gauge groups. There are only two possibilities (up to lattice shifts) for obtaining SU (3)
Table 1 :
1Chiral Spectrum from Intersecting D6-branes.
Table 2 :
2Summary of Pentagon model D6-brane content and corresponding topological intersection numbers.
Table 3 :
3Resolutions at A and D type singularities on manifolds of G 2 holonomy.
Table 4 :
4M-theory model containing the Pentagon spectrum.
Table 5 :
5Z 3 heterotic orbifold model 1. Contains Standard Model trinification and chiral trianon-like particles modular invariant Wilson lines for the Z 3 orbifold, and to perform a comprehensive search for the Pyramid model spectrum.
Table 6 :
6Z 3 heterotic orbifold model 2. Contains a chiral set of trianons and vector-like trinification.
This symmetry is explicitly broken by the ISS mass term, but the arguments of CSB lead us to believe that these operators will still be supressed when the cosmological constant is non-zero, i.e. in the SUSY broken theory.
For a review, see[8].
This also provides an alternative argument proving the existence of pentaquark-like exotics in these models. If stacks a and b are both type III, they cannot be required to solve the same moduli equations. Because two stacks cannot solve the same moduli equation if they are of different types, this responsibility falls on the type IV stacks c and d. As argued, we can then consider these as a single stack. But we know that the Pentagon does not exist in a model with three stacks, two of which are type III. At least one of stacks a,b, then, must be type IV.
The discussion is complicated in the case of semi-simple G, see[14]
There may be an exception to this statement. We are currently investigating the possibility of singularities enhanced to SU (6)×SU (5) or SO(10)×SU (5). However, it is unclear to us at the present moment whether these fields will be charged under the second SU (5) or have other undesirable U (1) charges.
In addition to the references given in[15], see[16] for further details
The program does not implement GSO projectors, and these must be checked by hand.
. × So, SU2/3,1/3,1/3,0,0,0,0,0) (0,0,0,0,0,0,0,0) W 1 (0,2/3,1/3,1/3,1/3,1/3,0,0) (2/3,0,0,0,0,0,0,0) W 2 (0,2/3,1/3,1/3,1/3,1/3,0,0) (1/3,1/3,1/3,1/3,0,0,0,0) W 3 (1/3,0,1/3,0,0,0,0,0SU (3) × SO(8) V (2/3,1/3,1/3,0,0,0,0,0) (0,0,0,0,0,0,0,0) W 1 (0,2/3,1/3,1/3,1/3,1/3,0,0) (2/3,0,0,0,0,0,0,0) W 2 (0,2/3,1/3,1/3,1/3,1/3,0,0) (1/3,1/3,1/3,1/3,0,0,0,0) W 3 (1/3,0,1/3,0,0,0,0,0)
. T Banks, S Echols, J L Jones, arXiv:0708.0022JHEP. 0710105hep-thT. Banks, S. Echols and J. L. Jones, JHEP 0710, 105 (2007) [arXiv:0708.0022 [hep-th]].
Cosmological supersymmetry breaking and the power of the pentagon: A model of low energy particle physics. T Banks, arXiv:hep-ph/0510159T. Banks, "Cosmological supersymmetry breaking and the power of the pentagon: A model of low energy particle physics," arXiv:hep-ph/0510159.
Remodeling the pentagon after the events of 2/23/06. T Banks, arXiv:hep-ph/0606313T. Banks, "Remodeling the pentagon after the events of 2/23/06," arXiv:hep-ph/0606313.
Dynamical SUSY breaking in meta-stable vacua. K A Intriligator, N Seiberg, D Shih, arXiv:hep-th/0602239JHEP. 060421K. A. Intriligator, N. Seiberg and D. Shih, "Dynamical SUSY breaking in meta-stable vacua," JHEP 0604, 021 (2006) [arXiv:hep-th/0602239].
J L Jones, arXiv:0812.2106Gauge Coupling Unification in MSSM + 5 Flavors. hep-phJ. L. Jones, "Gauge Coupling Unification in MSSM + 5 Flavors," arXiv:0812.2106 [hep-ph].
Note on the pseudo-NG-boson of meta-stable SUSY breaking. T Banks, H Haber, in preparationT. Banks and H. haber, "Note on the pseudo-NG-boson of meta-stable SUSY breaking," in preparation.
A Pyramid Scheme for Particle Physics. T Banks, J F Fortin, arXiv:0901.3578hep-phT. Banks and J. F. Fortin, "A Pyramid Scheme for Particle Physics," arXiv:0901.3578 [hep-ph].
Toward realistic intersecting D-brane models. R Blumenhagen, M Cvetic, P Langacker, G Shiu, arXiv:hep-th/0502005Ann. Rev. Nucl. Part. Sci. 55R. Blumenhagen, M. Cvetic, P. Langacker and G. Shiu, "Toward realistic intersecting D-brane models," Ann. Rev. Nucl. Part. Sci. 55, 71 (2005) [arXiv:hep-th/0502005].
Chiral four-dimensional N = 1 supersymmetric type IIA orientifolds from intersecting D6-branes. M Cvetic, G Shiu, A M Uranga, arXiv:hep-th/0107166Nucl. Phys. B. 6153M. Cvetic, G. Shiu and A. M. Uranga, "Chiral four-dimensional N = 1 supersymmetric type IIA orientifolds from intersecting D6-branes," Nucl. Phys. B 615, 3 (2001) [arXiv:hep-th/0107166].
Supersymmetric three family SU(5) grand unified models from type IIA orientifolds with intersecting D6-branes. M Cvetic, I Papadimitriou, G Shiu, arXiv:hep-th/0212177Nucl. Phys. B. 659298Erratum-ibid. BM. Cvetic, I. Papadimitriou and G. Shiu, "Supersymmetric three family SU(5) grand unified models from type IIA orientifolds with intersecting D6-branes," Nucl. Phys. B 659, 193 (2003) [Erratum-ibid. B 696, 298 (2004)] [arXiv:hep-th/0212177].
M Cvetic, G Shiu, A M Uranga, arXiv:hep-th/0111179Chiral type II orientifold constructions as M theory on G(2) holonomy spaces. M. Cvetic, G. Shiu and A. M. Uranga, "Chiral type II orientifold constructions as M theory on G(2) holonomy spaces," arXiv:hep-th/0111179.
M-theory dynamics on a manifold of G(2) holonomy. M Atiyah, E Witten, arXiv:hep-th/0107177Adv. Theor. Math. Phys. 61M. Atiyah and E. Witten, "M-theory dynamics on a manifold of G(2) holonomy," Adv. Theor. Math. Phys. 6, 1 (2003) [arXiv:hep-th/0107177].
Chiral fermions from manifolds of G(2) holonomy. B S Acharya, E Witten, arXiv:hep-th/0109152B. S. Acharya and E. Witten, "Chiral fermions from manifolds of G(2) holonomy," arXiv:hep-th/0109152.
J L Bourjaily, S Espahbodi, arXiv:0804.1132Geometrically Engineerable Chiral Matter in M-Theory. hep-thJ. L. Bourjaily and S. Espahbodi, "Geometrically Engineerable Chiral Matter in M-Theory," arXiv:0804.1132 [hep-th].
Supersymmetric standard model from the heterotic string. II. W Buchmuller, K Hamaguchi, O Lebedev, M Ratz, arXiv:hep-th/0606187Nucl. Phys. B. 785149W. Buchmuller, K. Hamaguchi, O. Lebedev and M. Ratz, "Supersymmetric standard model from the heterotic string. II," Nucl. Phys. B 785, 149 (2007) [arXiv:hep-th/0606187].
A mini-landscape of exact MSSM spectra in heterotic orbifolds. O Lebedev, H P Nilles, S Raby, S Ramos-Sanchez, M Ratz, P K S Vaudrevange, A Wingerter, arXiv:hep-th/0611095Phys. Lett. B. 64588O. Lebedev, H. P. Nilles, S. Raby, S. Ramos-Sanchez, M. Ratz, P. K. S. Vaudrevange and A. Wingerter, "A mini-landscape of exact MSSM spectra in heterotic orbifolds," Phys. Lett. B 645, 88 (2007) [arXiv:hep-th/0611095].
The Heterotic Road to the MSSM with R parity. O Lebedev, H P Nilles, S Raby, S Ramos-Sanchez, M Ratz, P K S Vaudrevange, A Wingerter, arXiv:0708.2691Phys. Rev. D. 7746013hep-thO. Lebedev, H. P. Nilles, S. Raby, S. Ramos-Sanchez, M. Ratz, P. K. S. Vaudrevange and A. Wingerter, "The Heterotic Road to the MSSM with R parity," Phys. Rev. D 77, 046013 (2008) [arXiv:0708.2691 [hep-th]].
Searching for realistic 4d string models with a Pati-Salam symmetry: Orbifold grand unified theories from heterotic string compactification on a Z(6) orbifold. T Kobayashi, S Raby, R J Zhang, arXiv:hep-ph/0409098Nucl. Phys. B. 7043T. Kobayashi, S. Raby and R. J. Zhang, "Searching for realistic 4d string models with a Pati-Salam symmetry: Orbifold grand unified theories from heterotic string compactification on a Z(6) orbifold," Nucl. Phys. B 704, 3 (2005) [arXiv:hep-ph/0409098].
SUSY GUT model building. S Raby, AIP Conf. Proc. 1078128S. Raby, "SUSY GUT model building," AIP Conf. Proc. 1078, 128 (2009).
Aspects of Grand Unification in Higher Dimensions. Akin Wingerter, ; Patrick, K S Vaudrevange, BONNIR-2005-05 ISSN-0172-8741PhD thesisGeometrical Aspects of Heterotic Orbifolds. Diploma thesis BONN-IB-2005-08Akin Wingerter. Aspects of Grand Unification in Higher Dimensions. 2005. PhD thesis BONNIR-2005-05 ISSN-0172-8741. Patrick K. S. Vaudrevange. Geometrical Aspects of Heterotic Orbifolds. 2005. Diploma thesis BONN-IB-2005-08.
Exploring the SO(32) heterotic string. H P Nilles, S Ramos-Sanchez, P K S Vaudrevange, A Wingerter, arXiv:hep-th/0603086JHEP. 060450H. P. Nilles, S. Ramos-Sanchez, P. K. S. Vaudrevange and A. Wingerter, "Exploring the SO(32) heterotic string," JHEP 0604, 050 (2006) [arXiv:hep-th/0603086].
Group Theory For Unified Model Building. R Slansky, Phys. Rept. 791R. Slansky, "Group Theory For Unified Model Building," Phys. Rept. 79, 1 (1981).
. Y Katsuki, Y Kawamura, T Kobayashi, N Ohtsubo, Y Ono, K Tanioka, TABLES OF Z(N) ORBIFOLD MODELS. Y. Katsuki, Y. Kawamura, T. Kobayashi, N. Ohtsubo, Y. Ono and K. Tanioka, "TABLES OF Z(N) ORBIFOLD MODELS,"
Three family Z(3) orbifold trinification, MSSM and doublet-triplet splitting problem. J E S Kim ; K, J E Choi, Kim, arXiv:hep-ph/0310158arXiv:hep-ph/0305002Trinification from superstring toward MSSM. 56787J. E. Kim, "Trinification from superstring toward MSSM," arXiv:hep-ph/0310158. K. S. Choi and J. E. Kim, "Three family Z(3) orbifold trinification, MSSM and doublet-triplet splitting problem," Phys. Lett. B 567, 87 (2003) [arXiv:hep-ph/0305002].
Z(3) orbifold construction of SU(3)**3 GUT with sin**2(Theta(0)(W)) = 3/8. J E Kim, arXiv:hep-th/0301177Phys. Lett. B. 56435J. E. Kim, "Z(3) orbifold construction of SU(3)**3 GUT with sin**2(Theta(0)(W)) = 3/8," Phys. Lett. B 564, 35 (2003) [arXiv:hep-th/0301177].
Dynkin diagram strategy for orbifolding with Wilson lines. K S Choi, K Hwang, J E Kim, arXiv:hep-th/0304243Nucl. Phys. B. 662476K. S. Choi, K. Hwang and J. E. Kim, "Dynkin diagram strategy for orbifolding with Wilson lines," Nucl. Phys. B 662, 476 (2003) [arXiv:hep-th/0304243].
| [] |
[
"Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks",
"Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks"
] | [
"Ryo Ohkawa ",
"Hokuto Uehara "
] | [] | [] | For a toric Deligne-Mumford (DM) stack X , we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism F : X → X on a 2-dimensional toric DM stack X , we show that the push-forward F * O X of the structure sheaf generates the bounded derived category of coherent sheaves on X .We also choose a full strong exceptional collection from the set of direct summands of F * O X in several examples of two dimensional toric DM orbifolds X . | 10.1016/j.aim.2013.04.023 | [
"https://arxiv.org/pdf/1205.6861v3.pdf"
] | 119,627,974 | 1205.6861 | 2dcb03fa52a7eb4b5ffa476150aebd477eef627c |
Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks
15 Jun 2013
Ryo Ohkawa
Hokuto Uehara
Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks
15 Jun 2013Dedicated to Professor Yujiro Kawamata on the occasion of his 60th birthday
For a toric Deligne-Mumford (DM) stack X , we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism F : X → X on a 2-dimensional toric DM stack X , we show that the push-forward F * O X of the structure sheaf generates the bounded derived category of coherent sheaves on X .We also choose a full strong exceptional collection from the set of direct summands of F * O X in several examples of two dimensional toric DM orbifolds X .
Introduction
There are many smooth projective varieties X (or more generally algebraic stacks) whose derived categories D b (X) have elements generating D b (X) (see Definition 4.1). The direct sums of full exceptional collections give examples of such. Furthermore for a given vector bundle E, if E generates D b (X) and it satisfies that Hom i D b (X) (E, E) = 0 for i = 0, the derived category D b (X) is equivalent to the derived category of the module category of its endomorphism algebra End D b (X) (E) (cf. [TU,Lemma 3.3]), and then we can apply the representation theory of finite dimensional algebras to study D b (X). It is important to find a "good" generator of a given triangulated category.
In [Bo], Bondal announces that there exists a derived equivalence between abelian categories of coherent sheaves on a toric variety X and constructible sheaves on a real torus with a stratification associated to X. He uses the assertion (without proof) that the Frobenius push-forward F * O X of the structure sheaf generates the bounded derived category D b (X). The purpose of this note is to give a rigorous proof of it for the 2-dimensional stacky case:
Main Theorem (=Theorem 7.1). Let X be a two dimensional toric Deligne-Mumford (DM) stack. Then the vector bundle F * O X generates D b (X ) for a Frobenius morphism F on X with a sufficiently divisible degree.
The generators we find in Theorem 7.1 get along well with the birational geometry in some sense (cf. Lemma 4.4). Actually we make full use of the birational geometry to obtain Theorem 7.1. The proof of Theorem 7.1 is divided into 4 steps:
(i) First we reduce the proof to the case X is a toric DM orbifold. This step works for arbitrary dimensional case.
(ii) We introduce the notion of the associated weighted blow up on a toric DM orbifold X with a weighted blow up on a toric variety X which is the coarse moduli space of X . In the two dimensional case we take a toric resolution of X, and consider the associated birational morphism on X . We use Lemma 4.4 for this morphism to reduce the proof to the case X is smooth.
(iii) Use the strong factorization theorem to connect X and P 2 by birational morphisms, and consider the associated birational morphisms. We use Lemma 4.4 and a technical Lemma 6.4 to reduce the proof to the case X is a root stack of P 2 .
(iv) Finally we show the statement for the case X is a root stack of P 2 .
The construction of this note is as follows. In §2 we recall the construction and some fundamental properties of toric DM stacks, following [BCS]. In §3 we recall the notion of a root stack and rigidification of toric DM stacks. In §4 we explain how to compute the direct summands of the Frobenius push-forward F * O X for toric DM stacks X , and put the step (i) into practice. In §5 we put the step (iv) into practice. In §6 we define the associated birational morphisms as in step (ii), and put the step (ii) into practice. In §7 we put the step (iii) into practice and complete the proof of Theorem 7.1. In §8 we use Theorem 7.1 to show existences of full strong exceptional collections in several examples.
We freely use terminology defined in §2.1 and §2.2 after these subsections. We always work over the complex number field C, and note that (certain generalized) Frobenius morphisms can be defined on toric varieties over C (or actually any fields, see §4.2).
The authors began this research during their stay in the Max Planck Institute for Mathematics. They appreciate the Max Planck Institute for Mathematics for their hospitality and stimulating environment. R.O. thanks Zheng Hua, Akira Ishii, Hiroshi Iritani, Yoshiyuki Kimura, Hiraku Nakajima, So Okada and Kazushi Ueda for useful discussions. He also thanks Research Institute for Mathematical Sciences, Kyoto University, and Institut des HautesÉtudse Scientifiques. He thanks Isamu Iwanari for answering his question about Remark 2.4. He is supported by GCOE 'Fostering top leaders in mathematics', Kyoto University. H.U. is supported by the Grants-in-Aid for Scientific Research (No.23340011). The authors are very grateful to the referees for the careful reading of the paper.
Notation For any vector space C s and any m ∈ Z >0 , we denote by ∧m : C s → C s a map defined by ∧m(z 1 , . . . , z s ) = (z m 1 , . . . , z m s ). We denote by Z m the cyclic group Z/mZ. For any real vector v = t (v 1 , . . . , v s ) ∈ R s , we put ⌊v⌋ = t (⌊v 1 ⌋, . . . , ⌊v s ⌋) ∈ Z s , where ⌊v i ⌋ is the integer satisfying
⌊v i ⌋ ≤ v i < ⌊v i ⌋ + 1.
For a commutative ring R and l, m ∈ Z >0 , we denote by M (l, m, R) the set of l × m matrices in R. We often identify a matrix A = (a i,j ) ∈ M (l, m, R) with a linear map
R m → R l t (x 1 , . . . , x m ) → A t (x 1 , . . . , x m ).
The symbol δ i,j stands for the Kronecker delta. For a 1 , . . . , a l ∈ R, we denote by diag(a 1 , . . . , a l )
the diagonal matrix (a i δ i,j ) ∈ M (l, l, R).
For a group homomorphism π : N → N ′ between finitely generated abelian groups, we denote by π R : N R → N ′ R the associated R-linear map between vector spaces N R = N ⊗R and N ′ R = N ′ ⊗R. For a fan ∆ in N ⊗ Q, ∆(1) denotes the set of 1-dimensional cones in ∆. X ∆ denote the toric variety associated with the fan ∆.
For a DM stack X , we denote by D b (X ) the bounded derived category of coherent sheaves on X .
Toric DM stacks
Toric DM stacks are introduced in [BCS]. They are motivated by the Cox's construction of toric varieties ( [Co]). In this section we recall some definitions and facts. See [FMN] and [Iw1] for other definitions.
Definitions
Let N be a finitely generated abelian group of rank n. We have an exact sequence of abelian groups
0 → N tor → N →N → 0,
where N tor is the subgroup of torsion elements in N . A stacky fan Σ = (∆, β) in N consists of a simplicial fan ∆ in N ⊗ Q and a group homomorphism
β : Z s → N such that for the canonical basis f 1 , . . . , f s of Z s , each β(f i ) generates the cone ρ i in N R , where ∆(1) = {ρ 1 , . . . , ρ s }.
In this note we always assume that ∆ is complete 1 .
We define a toric DM stack associated to Σ as follows. We take the mapping cone Cone(β) of β in the derived category of Z-modules and its derived dual Cone(β) ⋆ = R Hom Z (Cone(β), Z). We put DG(β) := H 1 (Cone(β) ⋆ ) and G(= G Σ ) := Hom Z (DG(β), C * ).
Remark 2.1. Note that in the Cox's construction [Co], N is torsion free and every β(f i ) is primitive. In this case DG(β) is just the Chow group A n−1 (X ∆ ) of the toric variety X ∆ .
There exists an exact triangle
Cone(β) ⋆ → R Hom Z (N, Z) → R Hom Z (Z s , Z)( ∼ = Z s ), which induces cl : Z s → DG(β).(1)
Applying Hom Z (−, C * ) to it, we obtain a map Hom Z (cl, C * ) : G → (C * ) s . Hence by the natural action of (C * ) s on C s , we have a G-action on C s .
Let S := C[z 1 , . . . , z s ] be the coordinate ring of C s . For each cone σ in ∆, we put z σ = β R (f i ) / ∈σ z i and U σ := C s \ {z σ = 0}. We have a G-invariant subspace U (= U Σ ) := σ∈∆ U σ .
The toric Deligne-Mumford (DM) stack X = X Σ associated to the stacky fan Σ is defined by the quotient stack
X := [U/G].
This is actually a DM stack by [BCS,Proposition 3.2]. When N is torsion free, the stack X has the trivial generic stabilizer group, and we call X a toric DM orbifold. In this case, let us denote by v i the primitive vectors in ρ i ∈ ∆(1), and also denote by D i (resp. D i ) the toric divisors on X = X ∆ (resp. X = X Σ ) corresponding to the cone ρ i (see their precise definitions in §2.2 and §2.3). Take the positive integers b i satisfying β(f i ) = b i v i . We often denote the toric DM orbifold X by
X X, b i D i .
Remark 2.2. If N is torsion free, then the map Hom Z (cl, C * ) becomes an inclusion. Assume furthermore that every β(f i ) is primitive and the fan ∆ is non-singular, i.e. every cone is generated by a subset of a basis of N . Then we can see that G acts on U freely, and actually the converse is also true. In this case, X is an algebraic space. Since the toric variety X = X ∆ is a coarse moduli space of X ( [BCS,Proposition 3.7]), X coincides with X. Thus in this situation, the construction of toric DM stacks is same as the original construction of toric varieties given in [Co].
Note that the orbifold X X, D i coincides with its coarse moduli space X if and only if X is smooth. In general the orbifold X X, D i is the canonical stack with the coarse moduli space X. We put X can = X X, D i .
We denote by Coh G U the category of G-equivariant coherent sheaves on U . Then we have an equivalence of categories
Coh X ∼ = Coh G U F → F U(2)
by [Vi,(7.21)]. By this equivalence we identify Coh X and Coh G U , and define
H 0 (U, F) := H 0 (U, F U )
for any F ∈ Coh X . Define G ∨ to be the group of characters on G, that is, G ∨ := Hom Z (G, C * ), and then note that the map cl in (1) can be regarded as the map from Z s to G ∨ , since G ∨ is naturally isomorphic to DG(β). The G-action on C s = Spec S induces an eigenspace decomposition
S = χ∈G ∨ S χ , where we put S χ := k∈Z s ≥0 cl(k)=χ Cz k(3)
and z k := z k 1 1 · · · z ks s for k = t (k 1 , · · · , k s ) ∈ Z s . We call the G ∨ -graded ring S the homogeneous coordinate ring of X . The point in X corresponding to the G-orbit of (z 1 , . . . , z s ) ∈ U is denoted by [z 1 : · · · : z s ]. We call it the homogeneous coordinate of X .
We denote by grS the category of G ∨ -graded finitely generated S-modules and by tor S the full subcategory of grS consisting of S-modules annihilated by some powers of the ideal (z σ | σ ∈ ∆).
We associate a coherent sheaf F on X with a G ∨ -graded S-module
M := χ∈G ∨ H 0 (U, F) χ ,
where H 0 (U, F) χ is the set of sections having an eigenvalue χ ∈ G ∨ for the action of G. The module M is finitely generated by [BH1,Lemma 4.7] and a unique G ∨ -graded S-module, up to tor S, satisfyingM | U ∼ = F U in Coh G U . Hence we have an equivalence of categories
Coh X ∼ = grS/ tor S F → M = χ∈G ∨ H 0 (U, F) χ .(4)
Here grS/ tor S is the quotient category (cf. [HMS,Appendix A.2]).
Picard groups and toric divisors of toric DM stacks
For χ ∈ G ∨ = Hom Z (G, C * ), we define a G-action on the trivial bundle U × C by
G × U × C → U × C (g, z, t) → (gz, χ(g)t).(5)
We obtain a G-equivariant trivial line bundle on U , which defines a line bundle L χ on X via the equivalence (2). This gives an isomorphism G ∨ ∼ = Pic X as [BHu,Proposition 3.3]. Note that Borisov and Hua show this fact under the assumption N tor = 0, but a similar proof works in the case N tor = 0. Henceforth we often identify Pic X , G ∨ and DG(β), and their elements corresponding to each other are denoted as
DG(β) ∼ = G ∨ ∼ = Pic Xw → χ w = χ → L χ .(6)
We describe Pic X more explicitly. Take a projective resolution of N ;
0 → Z r A → Z n+r → N → 0
for a matrix A = (a i,j ) ∈ M (n + r, r, Z). Then the map β : Z s → N is determined by some matrix B = (b i,j ) ∈ M (n + r, s, Z), and the mapping cone of β is described as
Cone(β) = {· · · → 0 → Z s ⊕ Z r (B|A) → Z n+r → 0 → · · · },
where the term Z n+r fits into the degree 0 part. Let us denote by f 1 , . . . , f s and g 1 , . . . , g r the canonical bases of Z s and Z r respectively. Since
DG(β) = H 1 (Cone(β) ⋆ ) = coker t (B|A),(7)
we have
Pic X ∼ = DG(β) = Zf * 1 ⊕ · · · ⊕ Zf * s ⊕ Zg * 1 ⊕ · · · ⊕ Zg * r s j=1 b i,j f * j + r j=1 a i,j g * j | i = 1, . . . , n + r .(8)
We denote byf * i andḡ * j the classes of dual bases f * i and g * j in DG(β) respectively. By (7), the group G = Hom Z (DG(β), C * ) is described as
G = t = t 1 . . . t s+r ∈ (C * ) s+r s j=1 t b i,j j × r j=1 t a i,j s+j = 1 for i = 1, . . . , n + r .(9)
The G-action on U ⊂ C s is defined by
t · (z 1 , . . . , z s ) = (t 1 z 1 , . . . , t s z s )(10)
for t ∈ G. Hence we can see that the group
H(= H Σ ) := {t ∈ G | t 1 = · · · = t s = 1} ∼ = t s+1 . . . t s+r ∈ (C * ) r r j=1 t a i,j s+j = 1 for i = 1, . . . , n + r
acts on U trivially, and it is the generic stabilizer group of X = [U/G]. Note that H is a finite group, since rk A = r.
For each i = 1, . . . , s, we have a Cartier divisor D i (= D X i ) := [{z i = 0}/G] (in the notation of §2.3, D i = [z i = 0]) on X = [U/G] corresponding to the ray ρ i , which satisfies that O X (D i ) ∼ = L χf * i in the notation in (6). We call D i the toric divisors corresponding to the cone ρ i . When X is a variety, we often denote D i by D i . Put (3)) corresponding to O X (kD) via the isomorphism in (4), where the G ∨ -grading of Sz −k is given by cl : Z s → G ∨ as in (3).
D(= D X ) := D 1 . . . D s and kD = s i=1 k i D i for k = s i=1 k i f * i ∈ Z s . We consider a G ∨ -graded S-module Sz −k (recall that z −k = z −k 1 1 · · · z −ks s as in
For w ∈ Z s ⊕ Z r and a G ∨ -graded S-module M , we define a G ∨ -graded S-module
M (w) = χ∈G ∨ M (w) χ by M (w) χ := M χ+χw ,
wherew is the image of w in DG(β), and χw ∈ G ∨ is defined in (6). Then we have an isomorphism of G ∨ -graded S-modules Sz −k ∼ = S (k) . For l ∈ Z r denote by O X (kD) l the line bundle which corresponds to the graded S-module S (k + l) by (4). Here we regard k + l as an element of Z s ⊕ Z r . Note that O X (kD) l is the line bundle L χ k+l in (6). When k = 0, we
put O X ,l = O X (0D) l .
Henceforth we freely use terminology defined in §2.1 and 2.2. We often use the superscript ′ for objects associated with a toric DM stack X ′ . For instance, Σ ′ = (∆ ′ , β ′ ) stands for the stacky fan in finitely generated abelian group N ′ defining X ′ .
Closed substacks of toric DM stacks
Let Σ = (∆, β) be a stacky fan and X = X Σ the associated toric DM stack. For any non-zero cone τ ∈ ∆, we consider the abelian group
N (τ ) := N β(f i ) | ρ i ⊂ τ
and denote by π τ : N → N (τ ) the quotient map. We consider the fan
∆ τ := π τ,R (σ) | σ + τ ∈ ∆, σ ∈ ∆ = π τ,R (σ) τ ⊂ σ ∈ ∆ in N (τ ) R .
Then the map π τ,R gives a one to one correspondence between the set
ρ i ρ i + τ ∈ ∆, ρ i ∩ τ = 0 and the set ∆ τ (1) = π τ,R (ρ i ) ρ i + τ ∈ ∆, ρ i ∩ τ = 0 .
By this identification, we regard the set ∆ τ (1) as a subset of ∆(1), and hence we have (1) ) .
Z ∆τ (1) ⊂ Z ∆(1) = Z s under the identifications Z ∆τ (1) ∼ = Hom Z (Z ∆τ (1) , Z) and Z ∆(1) ∼ = Hom Z (Z ∆(1) , Z). Then we define a stacky fan Σ τ = (∆ τ , β τ ) in N (τ ), where β τ := π τ • (β| Z ∆τ
The toric DM stack X Στ associated to the stacky fan Σ τ defines a closed substack of X as follows.
Reorder the set ∆(1) so that ∆ τ (1) = {π τ,R (ρ 1 ), . . . , π τ,R (ρ sτ )} and τ = ρ sτ +1 + · · · + ρ l for some s τ , l with l > s τ ≥ 0. For the homogeneous coordinate [z 1 : · · · : z s ] of X , we define a subset V τ of U and a subgroup G τ of G by
V τ := { t (z 1 , . . . , z s ) ∈ U | z sτ +1 = · · · = z l = 0, z l+1 = · · · = z s = 1} G τ := { t (t 1 , . . . , t s+r ) ∈ G ⊂ (C * ) s+r | t l+1 = · · · = t s = 1}.
For a cone σ ∈ ∆ satisfying τ ⊂ σ, we have U σ ∩ V τ = ∅, which implies that
V τ = ( τ ⊂σ∈∆ U σ ) ∩ {z sτ +1 = · · · = z l = 0, z l+1 = · · · = z s = 1}.
Hence we have a natural isomorphism U Στ ∼ = V τ induced by the inclusion
C ∆τ (1) = C sτ ֒→ C ∆(1) = C s (z 1 , . . . , z sτ ) → (z 1 , . . . , z sτ , l−sτ 0, . . . , 0, s−l 1 . . . , 1). Recall the definitions in §2.2 of A = (a 1 | . . . |a r ) ∈ M (n + r, r, Z) and B = (b 1 | . . . |b s ) ∈ M (n + r, s, Z), where a i , b i are column vectors in Z n+r . Define A τ = (b sτ +1 | . . . |b l |a 1 | . . . |a r ) ∈ M (n + r, r + l − s τ , Z) and B = (b 1 | . . . |b sτ ) ∈ M (n + r, s τ , Z).
Then the short exact sequence
0 → Z r+l−sτ Aτ → Z n+r → N (τ ) → 0
gives a projective resolution of N (τ ), and B τ defines the map β τ as B defines β. By the explicit description (9), we have an isomorphism G Στ ∼ = G τ . Then the isomorphism between U Στ and V τ becomes G Στ ∼ = G τ -equivariant, and hence we get an isomorphism
X Στ (= [U Στ /G Στ ]) ∼ = [V τ /G τ ] of stacks. Now put V (τ ) := {z sτ +1 = · · · = z l = 0 in U } (⊂ τ ⊂σ∈∆ U σ ).
Then we have the following lemma.
Lemma 2.3. We have an isomorphism [V τ /G τ ] ∼ = [V (τ )/G] of stacks. In particular, there is a closed embedding ι : X Στ ∼ = [V (τ )/G] ֒→ X = [U/G]. Proof. Embeddings V τ ⊂ V (τ ), G τ ⊂ G gives a morphism ϕ : [V τ /G τ ] → [V (τ )/G] of stacks. We show that ϕ is an isomorphism. To show that ϕ is essentially surjective, since [V (τ )/G] is a sheaf, it is enough to show that for any object P of [V (τ )/G](W ) over a scheme W there exists anétale covering {W i → W } i of W such that P | W i is in the essential image of ϕ W i : [V τ /G τ ](W i ) → [V (τ )/G](W i ) for any i. First
take anétale covering such that P | W i is given by a trivial principal G-bundle W i × G → W i and a G-equivariant morphism ψ:
V (τ ) ψ ←− W i × G −→ W i .
Furthermore for each i, we may assume that there exists a cone σ ∈ ∆ satisfying τ ⊂ σ, such that im ψ ⊂ U σ ∩ V (τ ).
Then by the explicit description (9) and (10), we can take anétale cover
W ′ i → W i such that the morphism W ′ i −→ W i ψ| W i ×id G −→ U σ ∩ V (τ )
is also decomposed as
W ′ i −→ U σ ∩ V τ ֒−→U σ ∩ V (τ ) up to the G-action on U σ . Hence P | W ′ i belongs to the essential image of ϕ W ′ i . Note that for the G-action on U the subgroup of elements of G keeping V τ is equal to G τ .
Hence we can see that ϕ is fully faithful. This completes the proof.
We call X Στ the toric substack of X corresponding to the cone τ . We often denote it by
[z sτ +1 = · · · = z l = 0].
For k = t (k 1 , . . . , k s ) ∈ Z s and l = t (l 1 , . . . , l r ) ∈ Z r , we put
k τ = t (k 1 , . . . , k sτ ) ∈ Z sτ , l τ = t (k sτ +1 , . . . , k l , l 1 , . . . , l r ) ∈ Z l−sτ +r and D τ = t (ι * D 1 , . . . , ι * D sτ ). By Lemma 2.3, we have ι * O X Στ (k τ D τ ) l τ = O X (kD) l ⊗ ι * O X Στ ,(11)
Morphisms between toric DM stacks
In this subsection we consider a morphism between toric DM stacks X and X ′ . First let us consider the following morphism of triangles of the derived category of Z-modules:
· · · / / Z s β / / γ 1 =C N γ 2 / / Cone(β) γ 3 / / · · · · · · / / Z s ′ β ′ / / N ′ / / Cone(β ′ ) / / · · · ,(12)
where γ 1 is a matrix C = (c i,j ) ∈ M (s ′ , s, Z). We assume that (T1) γ 2 induces a mapγ 2 : (N , ∆) → (N ′ , ∆ ′ ) of fans, that is, for every cone σ ∈ ∆, there exists a
cone σ ′ ∈ ∆ ′ satisfying γ 2,R (σ) ⊂ σ ′ . Take j satisfying (γ 2 • β) R (f j ) ∈ σ ′ for some cone σ ′ ∈ ∆ ′ . We furthermore assume that (T2) we have γ 1 (f j ) = s ′ i=1 c i,j f ′ i ∈ β ′ R (f ′ i )∈σ ′ Z ≥0 f ′ i .
As its consequence, we know that every c i,j is non-negative. Under these assumptions we construct
a morphism ϕ : X = [U/G] → X ′ = [U ′ /G ′ ] of DM stacks as follows.
First consider the morphism
γ 1,C * := γ 1 ⊗ Z id C * : (C * ) s → (C * ) s ′ (z j ) → z ′ i = j z c i,j j .
We can naturally extend it to a morphism C s → C s ′ , which is also denoted by γ 1,C * . Then γ 1,C * induces the morphism U → U ′ as follows: take a point p = (p j ) ∈ U σ = C s \{z σ = 0} for some
cone σ ∈ ∆. If p j = 0, then j satisfies that β R (f j ) ∈ σ. Take a cone σ ′ such that γ 2,R (σ) ⊂ σ ′ .
Then the second assumption above implies that
c i,j = 0 for any i with β ′ R (f ′ i ) / ∈ σ ′ . This means that γ 1,C * (p) ∈ U ′ σ ′ .
We again denote this morphism by
γ 1,C * : U → U ′ .
On the other hand, the map γ 3 defines a group homomorphism ρ = Hom(
H 1 (γ ⋆ 3 ), C * ) ρ : G = Hom Z (H 1 (Cone(β) ⋆ ), C * ) → G ′ = Hom Z (H 1 (Cone(β ′ ) ⋆ ), C * ).
Then we have the following commutative diagram
G × U / / ρ×γ 1,C * U γ 1,C * G ′ × U ′ / / U ′ ,(13)
where the horizontal arrows are defined by the actions of G on U and G ′ on U ′ . This diagram determines a morphism
ϕ : X = [U/G] → X ′ = [U ′ /G ′ ]
of DM stacks.
Remark 2.4. By [Iw2, Theorem 1.2], we see that giving torus equivariant morphisms between toric DM orbifolds is equivalent to giving homomorphism γ 1 , γ 2 as in (12) satisfying assumptions (T1) and (T2).
Let us take projective resolutions of N and N ′ ;
0 → Z r A → Z n+r → N → 0 and 0 → Z r ′ A ′ → Z n ′ +r ′ → N ′ → 0 for A ∈ M (n + r, r) and A ′ ∈ M (n ′ + r ′ , r ′ ). The map γ 2 is determined by matrices D = (d i,j ) ∈ M (n ′ + r ′ , n + r, Z), E = (e i,j ) ∈ M (r ′ , r, Z),
which make the diagram in the right square commutative (but not necessarily in the left):
Z s C B / / Z n+r D Z r E A o o Z s ′ B ′ / / Z n ′ +r ′ Z r ′ . A ′ o o
Here B ∈ M (n + r, s) and B ′ ∈ M (n ′ + r ′ , s ′ ) give lifts of the maps β : Z s → N and β ′ : Z s ′ → N ′ respectively. By the commutativity of (12), we have a homotopy homomorphism
F = (f i,j ) : Z s → Z r ′ such that DB − B ′ C = A ′ F .
The maps γ 1,C * and ρ are described as follows:
γ 1,C * : U → U ′ (z j ) → z ′ i = j z c i,j j ρ : G → G ′ (t j ) → t ′ i ,(14)
where we put
t ′ i := s j=1 t c i,j j for i = 1, . . . , s ′ and t ′ s ′ +k := s j=i t f k,j j r l=1 t e k,l s+l for k = 1, . . . , r ′ , and z c i,j j = 1 when z j = c i,j = 0. Consequently we have ϕ * O X ′ (D ′ i ) = O X ( j c i,j D j ) and ϕ * O X ′ ,g ′ * k = O X ( j f k,j D j ) l e k,l g * l .(15)
For the homogeneous coordinate ring
S = C[z 1 , . . . , z s ] (resp. S ′ = C[z ′ 1 , . . . , z ′ s ′ ]) of X (resp. X ′ ), we define the map ϕ ♯ : S ′ → S (z ′ i ) → j z c i,j j so that ϕ ♯ (z ′k ′ ) = z γ * 1 (k ′ ) . In particular, we have ϕ ♯ (S ′ χ ′ ) ⊂ S ρ ∨ (χ ′ ) , where ρ ∨ : G ′ ∨ → G ∨ is the C * -dual map Hom Z (ρ, C * ) of ρ.
We have the following commutative diagram: (4), then a G ∨ -graded S-module M ⊗ S ′ S corresponds to the pull-back ϕ * F, where the grading is given by
· · · / / Hom Z (N, Z) / / Z s cl / / DG(β) = G ∨ / / Ext 1 Z (N, Z) / / · · · · · · / / Hom Z (N ′ , Z) γ * 2 O O / / Z s ′ γ * 1 = t C O O cl / / DG(β ′ ) = G ′∨ ρ ∨ O O / / Ext 1 Z (N ′ , Z) γ * 2 [1] O O / / · · · .(16)If a G ′ ∨ -graded S ′ -module M = χ ′ ∈G ′∨ M χ ′ gives a coherent sheaf F on X ′ by(M ⊗ S ′ S) χ = η ′ ∈G ′∨ M η ′ ⊗ S ′ S χ−ρ ∨ (η ′ )
for each χ ∈ G ∨ . On the other hand, a G ∨ -graded S-module L = χ∈G ∨ L χ gives a coherent sheaf G on X , and the sheaf ϕ * G corresponds a G ′ ∨ -graded S ′ -module S ′ N whose grading is given by
( S ′ L) χ ′ = L ρ ∨ (χ ′ ) for χ ′ ∈ G ′ ∨ . Here S ′ -module structure of S ′ N is given by ϕ ♯ . This S ′ -module structure is compatible with the G ′ ∨ -grading of S ′ L, since ϕ ♯ (S ′ χ ′ ) ⊂ S ρ ∨ (χ ′ )
. The functor S ′ (−) : grS/ tor S → grS ′ / tor S ′ is the right adjoint functor of (−) ⊗ S S ′ . Since ϕ * : Coh X → Coh X ′ is the right adjoint functor of ϕ * : Coh X ′ → Coh X , both functors S ′ (−) and ϕ * must coincide by the correspondence in (4). Hence for any G ∈ Coh X , we have a natural isomorphism
H 0 (U ′ , ϕ * G) ∼ = χ ′ ∈G ′∨ H 0 (U, G) ρ ∨ (χ ′ )(17)
in grS ′ / tor S ′ .
Lemma 2.5. Take a morphism ϕ : X → X ′ determined by matrices C = (c i,j ) ∈ M (s ′ , s, Z), D ∈ M (n ′ + r ′ , n + r, Z) and E ∈ M (r ′ , r, Z) as above. Then we have the following.
(i) Suppose that for each i there exists j such that c i,j > 0, and entries of the j-th column vector of C are all zero except c i,j . Furthermore, in the diagram (16), we assume that γ * 2 is surjective, and γ * 2 [1] is injective. Then we have ϕ * O X = O X ′ .
(ii) Assume that s = s ′ , C = (c i δ i,j ) for c i ∈ Z >0 , and that the map of fansγ 2 : (N ,
∆) → (N ′ , ∆ ′ )
is an isomorphism. Then we have R i ϕ * F = 0 for any coherent sheaf F on X and i > 0.
Proof. (i) For any χ ′ ∈ G ′ ∨ , we consider sets
S ′ = {k ′ ∈ Z s ′ ≥0 | cl(k ′ ) = χ ′ }, S = {k ∈ Z s ≥0 | cl(k) = ρ ∨ (χ ′ )}.
By the assumption on C, we see that t C : (16), combining the surjectivity of γ * 2 and the injectivity of t C, we conclude that the map t C induces a bijection between S ′ and S.
Z s ′ → Z s is injective, and ( t C) −1 (Z s ≥0 ) = Z s ′ ≥0 . When S ′ = ∅, in the diagram
On the other hand, the injectivity of the map γ * 2 [1] is equivalent to the condition ρ ∨ −1 (cl(Z s )) = cl(Z s ′ ). So if S ′ = ∅, then S = ∅. Hence we obtain an isomorphism S ′ ∼ = S ′ S by the map ϕ ♯ . This completes the proof.
(ii) For any σ ∈ ∆ ′ = ∆, we have ϕ ♯ (z ′ σ ) = z k z σ for some k ∈ Z s ≥0 , where k = (k i ) satisfies that k i = 0 if β R (f i ) ∈ σ.
Hence if we take a G ∨ -graded S-module L ∈ tor S, then S ′ L also belongs to tor S ′ . We see that an exact sequence in the abelian category grS/ tor S is given by an exact sequence of graded S-modules
0 → L 1 f → L 2 g → L 3
such that coker g belongs to tor S. Hence ϕ * : Coh X → Coh X ′ is an exact functor, which implies the assertion.
Toric DM stacks vs. toric DM orbifolds
Let us introduce two kinds of root stacks, important notions in this note. In this section, we consider the toric DM stack X = X Σ associated to a stacky fan Σ = (β, ∆) in
N := Z n ⊕ r i=1 Z a i for some a i ∈ Z >0 . As in §2.2 we take matrices A = O n×r diag(a 1 , . . . , a r )
∈ M (n + r, r, Z) and
B = (b i,j ) ∈ M (n + r, s, Z) such that N ∼ = coker A and β is defined by β : Z s B → Z n+r ։ coker A.
Root stacks of line bundles on toric DM stacks
For r ′ > r, e = t (e 1 , . . . , e r ′ −r ) ∈ Z r ′ −r >0 and a collection L = t (L 1 , . . . , L r ′ −r ) of line bundles on X , we consider the (e-th) root stack e L/X of (X , L) ( [FMN,1.3.a]), that is, the fiber product:
e L/X / / (BC * ) r ′ −r ∧e X L / / (BC * ) r ′ −r .
Here BC * is the quotient stack of a point by the trivial C * -action. By abuse of notation, we use symbols L : X → (BC * ) r ′ −r and ∧e : (BC * ) r ′ −r → (BC * ) r ′ −r to denote the morphisms induced by L and ∧e : (C * ) r ′ −r → (C * ) r ′ −r respectively.
The root stack e L/X is constructed as a toric stack in the following way. We take (
k i , l i ) ∈ Z s ⊕ Z r such that L i = O X (k i D) l i in Pic X for i = 1, . . . , r ′ − r and matrices A ′ = A 0 . . . 0 − t l 1 e 1 . . . . . . − t l r ′ −r e r ′ −r ∈ M (n + r ′ , r ′ , Z), B ′ = B − t k 1 . . . − t k r ′ −r ∈ M (n + r ′ , s, Z).
We define an abelian group N ′ = coker A ′ and a stacky fan Σ ′ = (β ′ , ∆) in N ′ by the map
β ′ : Z s B ′ → Z n+r ′ ։ coker(A ′ ). We have a commutative diagram Z s B ′ / / C=id Z s Z n+r ′ D Z r ′ A ′ o o E Z s B / / Z n+r Z r , A o o
where D and E is the projections. Then we have a morphism Ψ : X ′ = X Σ ′ → X by the result in §2.4, and Ψ satisfies that
Ψ * O X (kD) l = O X ′ (kD ′ ) l
for any (k, l) ∈ Z s ⊕ Z r by (15), where we identify l with the element in Z r ′ by the injection
Z r → Z r ⊕ Z r ′ −r ∼ = Z r ′ .
In particular, by the choice of A ′ and B ′ , we have isomorphisms
Ψ * O X (k i D) l i ∼ = (O X ′ ,g ′ * r+i ) ⊗e i for i = 1, . . . , r ′ − r. Hence Ψ and L ′ = t (O X ′ ,g ′ * r+1 , . . . , O X ′ ,g ′ * r ′ ) : X ′ → (BC * ) r ′ −r
gives a morphism X ′ → e L/X . By [Pe, Theorem 2.6], we see that this is an isomorphism.
We have the following theorem.
Theorem 3.1. For X ′ = e L/X , we have an equivalence
Coh X ′ ∼ = (l i )∈Z r ′ −r 0≤l i <e i (Ψ * Coh X ) ⊗ O X ′ , i l i g ′ * r+i ,
where in the right hand side the symbol means that if (l i ) = (m i ) ∈ Z r ′ −r , then we have
Hom(E, F ) = 0 for any E ∈ (Ψ * Coh X ) ⊗ O X ′ , i l i g ′ * r+i and F ∈ (Ψ * Coh X ) ⊗ O X ′ , i m i g ′ * r+i .
Proof. Applying [IU,Lemma 4.1] repeatedly, we obtain the assertion.
The rigidification of toric DM stacks
We define the rigidification X rig of X according to [FMN] as follows. Let us take a stacky fan Σ rig = (β rig , ∆) inN = Z n , where we define β rig : Z s →N to be a composition of β and a natural surjection N →N , and consider the toric DM orbifold X rig := X Σ rig .
We write B = (b i,j ) for b i,j ∈ Z and put L i = O X rig (− j b n+i,j D j ) for i = 1, . . . , r. Then the stack X is isomorphic to the t (a 1 , . . . , a r )-th root stack of (X rig , (L 1 , . . . , L r )). In particular, we have a morphism Ψ : X → X rig as in §3.1. We call this Ψ the rigidification morphism. By applying Theorem 3.1 to Ψ, we obtain the following corollaries:
Corollary 3.2. Let X be a toric DM stack and X rig its rigidification. Then we have an equivalence
Coh X ∼ = (l i )∈Z r 0≤l i <a i (Ψ * Coh X rig ) ⊗ O X , i l i g * i .
The following must be well-known to specialists. It is used in the proof of Lemma 4.4.
Corollary 3.3. Let X be a toric DM stack. Then Coh X has a finite homological dimension, namely any coherent sheaf on X has a locally free resolution of finite length.
Proof. Coh X rig has a finite homological dimension by the proof of [BH1,Theorem 4.6]. Hence the assertion follows from Corollary 3.2.
The following are stacky generalizations of the results for toric DM orbifolds in [Ka] and [BHu,Theorem 7.3].
Corollary 3.4. (i) Let X be a toric DM stack. Then D b (X ) has a full exceptional collection consisting of coherent sheaves.
(ii) Let X be a two dimensional toric DM stack whose rigidification X rig has an ample anticanonical divisor. Then D b (X ) has a full strong exceptional collection consisting of line bundles.
Proof. The assertion (i) (respectively (ii)) directly follows from Corollary 3.2 and the result in [Ka] (resp. [BHu,Theorem 7.3]).
Root stacks of effective Cartier divisors on toric DM stacks
For positive integers c 1 , . . . , c s , we define another stacky fan Σ ′′ = (β ′′ , ∆) in N by replacing β in Σ with
β ′′ : Z s BC → Z n+r ։ coker A,
where we put C = diag(c 1 , . . . , c s ) ∈ M (s, s, Z). We call X ′′ = X Σ ′′ the (c-th) root stack of (X , D)
([FMN, 1.3.b]) and denote it by
c D/X ,
where we define c = t (c 1 , . . . , c s ) and D = t (D 1 , . . . , D s ) is the collection of toric divisors on X as in §2.2. We have a commutative diagram
Z s BC / / C Z n+r D=id Z n+r Z r A o o E=id Z r Z s B / / Z n+r Z r . A o o
Then we have a morphism ϕ : X ′′ = c D/X → X by the result in §2.4. We call ϕ the root construction morphism. We obtain the following by Lemma 2.5 (i), (ii).
Lemma 3.5. Rϕ * O X ′′ = O X .
Next assume furthermore that X = X Σ is an orbifold. Then in the notation in §2.1, X is denoted by X X, s i=1 b i D i for some b i ∈ Z >0 and the coarse moduli space X = X ∆ of X . Then X is realized as a root stack b D/X can over the canonical stack X can = X X, D i for b = t (b 1 , . . . , b s ). Additionally if X is smooth, then X is a root stack b D/X over X by Remark 2.2.
Frobenius push-forward for toric DM stacks 4.1 Generators
Let us begin this section with important definitions. Suppose that X is a smooth complete DM stack.
Definition 4.1. For a subset S ⊂ D b (X ), S denotes the smallest full triangulated subcategory containing S of D b (X ) such that S is stable under taking direct summands and direct sums.
We say that S is a generator of D b (X ), or S generates D b (X ) if S = D b (X). If S consists
of a single element α, we just say that α is a generator of D b (X ).
Definition 4.2. (i) An object E ∈ D b (X ) is called exceptional if it satisfies Hom i D b (X ) (E, E) = C i = 0 0 otherwise.
(ii) An ordered set (E 1 , . . . , E n ) of exceptional objects is called an exceptional collection if the following condition holds;
Hom i D b (X ) (E k , E j ) = 0
for all k > j and all i. When we say that a finite set S of objects is an exceptional collection, it means that S forms an exceptional collection in an appropriate order.
(iii) An exceptional collection (E 1 , . . . , E n ) is called strong if
Hom i D b (X ) (E k , E j ) = 0
for all k, j and i = 0.
(iv) An exceptional collection (E 1 , . . . , E n ) is called full if the set {E 1 , . . . , E n } generates D b (X ).
Remark 4.3. Let X = X Σ be a toric DM orbifold associated with a stacky fan Σ = (∆, β). If the toric DM stack X = X Σ has a full exceptional collection consisting of n exceptional objects, the rank of its Grothendieck group K(X ) is n. Furthermore suppose that −K X is nef, which is equivalent to the condition that all β(f i )'s lie on the boundary of the convex hull of all β(f i )'s (see also §8 for the notion nef). Then it is proved in [BH2, Corollary 2.2 and Proposition 3.1 (iii)] that
rk K(X ) = rk N ! vol ∆,
where vol ∆ is the volume of the convex hull of all β(f i )'s.
Frobenius morphism
Below we use the terminology in §2.1 and 2.2. For a positive integer m, we consider the Frobenius morphism F (= F m ) : X → X induced by ∧m : U → U and ∧ m : G → G.
Take both of γ 1 : Z s → Z s and γ 2 : N → N in (12) as the multiplication maps by m. Then we obtain the Frobenius morphism F , which is actually a generalization of the usual Frobenius morphism.
Lemma 4.4. Let X and Y be toric DM stacks. Consider a proper morphism ϕ :
X → Y which satisfies Rϕ * O X = O Y . Then D b (X ) = F * O X implies D b (Y) = F * O Y .
Proof. We see that Coh Y has a finite homological dimension by Corollary 3.3. For any object
F ∈ D b (Y), by the assumption we have Lϕ * F ∈ F * O X . Since we have Rϕ * F * O X = F * Rϕ * O X = F * O Y , we see that F ∼ = Rϕ * Lϕ * F belongs to F * O Y .
Direct summands of Frobenius push-forward
We consider the Frobenius morphism
F : X → X ′ .
Although the target space X ′ is X itself, we use the notation X ′ so that we distinguish the domain X and the target X ′ . Theorem 4.5 for toric DM stacks generalizes the result for toric varieties in [Ac]. A similar proof in [Ac] works for the stacky case, but we give another proof using (17).
Theorem 4.5. For χ ∈ G ∨ and χ ′ ∈ G ′ ∨ , put
m(χ, χ ′ ) := ♯{j ∈ [0, m − 1] s | mχ ′ = χ − cl(j)}.
Then we have
F * L χ = χ ′ ∈G ′∨ L ⊕m(χ,χ ′ ) χ ′ .
Proof. We identify Coh X and Coh G U by (2). We consider the graded S-module
H 0 (U, L χ ) = η∈G ∨ H 0 (U, L χ ) η and recall that H 0 (U, L χ ) η ∼ = i∈Z s ≥0 cl(i)=χ+η Cz i ,
since the G-actions in (5) and (10) induce the action on H 0 (U, L χ ) given by z i → χīχ −1 z i . We
define a G ′∨ -graded S ′ -module M = η ′ ∈G ′∨ M η ′ by M η ′ := H 0 (U, L χ ) mη ′ ∼ = k∈Z s ≥0 cl(k)=χ+mη ′ Cz k .
By (17), the S ′ -module M corresponds to F * L χ by the equivalence (4).
We have a one-to-one correspondence between sets {k ∈ Z s ≥0 | cl(k) = χ + mη ′ } and
{(i, j) ∈ Z s ≥0 × [0, m − 1] s | m(cl(i) − η ′ ) = χ − cl(j)}
given by k = mi + j such that i = ⌊ k m ⌋ and j ∈ [0, m − 1] s . Thus for each η ′ ∈ G ′∨ , we have isomorphisms between components of G ′∨ -graded S ′ -modules
M η ′ ∼ = j∈[0,m−1] s i∈Z s ≥0 m(cl(i)−η ′ )=χ−cl(j) Cz mi+j ∼ = χ ′ ∈G ′∨ j∈[0,m−1] s mχ ′ =χ−cl(j) i∈Z s ≥0 cl(i)=χ ′ +η ′ Cz ′ i z j .
To obtain the second isomorphism, we put χ ′ = cl(i) − η ′ . The last component
i∈Z s ≥0 cl(i)=χ ′ +η ′ Cz ′ i z j is isomorphic to H 0 (U ′ , L χ ′ ) η ′ . Hence, summing up all η ′ ∈ G ′∨ , we have an isomorphism M ∼ = χ ′ ∈G ′∨ j∈[0,m−1] s mχ ′ =χ−cl(j) H 0 (U ′ , L χ ′ ) of S ′ -modules. Now the desired isomorphism F * L χ ∼ = χ ′ ∈G ′∨ j∈[0,m−1] s mχ ′ =χ−cl(j) L χ ′ = χ ′ ∈G ′∨ L ⊕m(χ,χ ′ ) χ ′
follows from (4).
We denote by D X (L χ ) the set of isomorphism classes of direct summands of F * L χ . For χ ∈ G ∨ and χ ′ ∈ G ′∨ , there exist some k, k ′ ∈ Z s and l, l ′ ∈ Z r such that
L χ = O X (kD) l and L χ ′ = O X ′ (k ′ D) l ′ . Then we have χ = χ k+l and χ ′ = χ ′ k ′ +l ′ as in §2.2. If m(χ, χ ′ ) > 0 then we have mχ ′ k ′ +l ′ = χ k+l − cl(j)
for some j ∈ Z s ∩ [0, m − 1] s , and hence there exists an element u ∈ Z n+r such that we have
(k + t Bu) ⊕ (l + t Au) = (mk ′ + j) ⊕ ml ′ in Z s ⊕ Z r . Thus we have k ′ = ⌊ k+ t Bu m ⌋ in Z s , l ′ = l+ t Au m in Z r and D X (L χ ) = O X ⌊ k + t Bu m ⌋D l+ t Au m u ∈ Z n+r ∩ [0, m − 1] n+r , l + t Au m ∈ Z r .(18)
When N is torsion free, we have r = 0 and N = Z n . For each i = 1, . . . , s, we take a primitive generator v i ∈ N of ρ i and positive integer b i such that β(
f i ) = b i v i . We have t Bu = t ((u, b 1 v 1 ), . . . , (u, b s v s )) , where (u, b i v i ) ∈ R is just the dot product on R n .
Consequently, (18) is a generalization of
Thomsen's result [Th] for smooth toric varieties.
The set D X (O X ) is stabilized for sufficiently divisible integers m in F = F m , and denote this set by D X . By the above result, we have
D X = O X ⌊ t Bu⌋D t Au u ∈ [0, 1) n+r , t Au ∈ Z r .(19)
Reduction to toric orbifolds
We reduce the proof of Main theorem to the orbifold's case. First take an arbitrary toric DM stack X and we use the notation in §3. Put a := a 1 · · · a r and a := (a, . . . , a) ∈ Z s . For simplicity, denote the root stack a D/X by X ′ = X Σ ′ and consider the rigidification X ′rig of X ′ . Then β ′ is defined by the map
Z s aB 1 0 → Z n+r ։ coker A, where B 1 = (b i,j ) ∈ M (n, s, Z).
Here we identify toric divisors on X ′ and X ′rig , and denote by the same symbol D ′ = t (D ′ 1 , . . . , D ′ s ). For the rigidification morphism Ψ ′ : X ′ → X ′rig , by (15)
, we have Ψ ′ * O X ′rig (kD ′ ) ∼ = O X ′ (kD ′ ) for any k ∈ Z s . Furthermore by (19), we have D X ′ = O X ′ ⌊a t B 1 u 1 ⌋D ′ diag(a 1 ,...,ar)u 2 u 1 ∈ [0, 1) n , u 2 ∈ [0, 1) r , diag(a 1 , . . . , a r )u 2 ∈ Z r = O X ′ ⌊a t B 1 u 1 ⌋D ′ l u 1 ∈ [0, 1) n , l ∈ Z r ∩ r i=1 [0, a i ) , D X ′rig = O X ′rig ⌊a t B 1 u 1 ⌋D ′ u 1 ∈ [0, 1) n .
Hence we have D X ′ = l (Ψ ′ * D X ′rig ) ⊗ O X ′ ,l , where l runs over the set Z r ∩ r i=1 [0, a i ), and we define Ψ ′ * D X ′rig in an obvious way.
Lemma 4.6. Let X be a toric DM stack and X ′rig the rigidification of the root stack X ′ = a D/X as above. If D X ′rig generates D b (X ′rig ), then D X generates D b (X ).
Proof. Note that by Lemmas 3.5 and 4.4, it is enough to show that D X ′ generates D b (X ′ ). The assumption and Theorem 3.1 implies that the set D X ′ = l (Ψ ′ * D X ′rig ) ⊗ O X ′ ,l generates D b (X ′ ).
Hence we obtain the assertion.
5 Root stacks of the projective plane
Root stacks of weighted projective spaces
First let us recall the definition of weighted projective space P(a) for a = t (a 1 , . . . , a n+1 ) ∈ Z n+1 >0 . Take a finitely generated abelian group N := Z n+1 /Za and put β to be the quotient map Z n+1 → N . Let us take the canonical basis e i 's of Z n+1 and consider the fan ∆ in N R , consisting of cones i =j R ≥0 β R (e i ) for j = 1, . . . , n + 1 and their faces. We often denote by P(a) the toric DM stack associated to the stacky fan Σ = (∆, β). This is called a weighted projective space.
For b = t (b 1 , . . . , b n+1 ) ∈ Z >0 , we consider a root stack X := b D/P(a). Take a projective
resolution of N 0 → Z a → Z n+1 → N = Z n+1 /Za → 0.
Then in terms in §2.2, X is associated with matrices A = a ∈ M (n + 1, 1), B = diag b 1 , . . . , b n+1 ∈ M (n + 1, n + 1).
Assume furthermore that a = b = c = t (c, . . . , c) ∈ Z n+1 for an integer c ∈ Z. According to (8), we have
Pic c D/P(c) ∼ = Zf * 1 ⊕ · · · ⊕ Zf * n+1 ⊕ Zg * cf * 1 + cg * , . . . , cf * n+1 + cg * O X ( l i D i ) kḡ * → l if * i + kḡ * ,
and hence we have an isomorphism
h : Pic c D/P(c) → Z ⊕ Z ⊕n+1 c l if * i + kḡ * → l i − k l 1 . . . l n+1 .(20)
For any L ∈ Pic c D/P(c), we call the first component l i − k of h(L) the degree of L and denote it by deg L.
Root stacks of the projective plane
Consider the root stack
X = b D/P 2 = X (P 2 , b 1 D 1 + b 2 D 2 + b 3 D 3 ), where b = t (b 1 .b 2 , b 3 ) and D i = [z i = 0] in P 2 . The map β : Z 3 →N = Z 2
is given by the matrix
B = b 1 0 −b 3 0 b 2 −b 3 .
Moreover we put an additional assumption that b 1 = b 2 = b 3 = c. In this case by (19), we have
D X = {O X (i(D 1 − D 3 ) + j(D 2 − D 3 ) + kD 3 ) | i, j ∈ [0, c), k = 0, −1, −2} .
By [BHu,Proposition 5.1], this set forms a full strong exceptional collection on X . In particular we know that D X generates D b (X ).
Remark 5.1. In general for root stacks
X = X (P d , d+1 i=1 b i D i ) of the projective space of arbitrary dimension d we see that D X generate D b (X ) as follows, where b i ≥ 0 and D i = [z i = 0]. By
Lemma 4.4 we may assume that b 1 = · · · = b d+1 . Then using [BHu,Proposition 5.1] the above argument holds for arbitrary dimension.
6 Weighted blow ups and Frobenius morphisms 6.1 Weighted blow ups of toric DM orbifolds Let us consider a toric DM orbifold X = X ∆ associated with a stacky fan Σ = (∆, β). Define primitive vectors v i in N for i = 1, . . . , s so that
ρ i = R ≥0 v i , namely, there exists b i ∈ Z >0 such that b i v i = β(f i ).
Then as in §2.1, X is also described as X X, s i=1 b i D i . Take a cone σ ∈ ∆ spanned by v 1 , . . . , v l , and another primitive vector v s+1 which is in the relative interior of σ; there exist positive integers h i , m ∈ Z >0 with coprime h i 's satisfying
mv s+1 = l i=1 h i v i .
Then we have a new simplicial fan ∆ ′ which is the subdivision of ∆ obtained from the the star shaped decomposition of the fan ∆, by adding the ray R ≥0 v s+1 . Then we obtain a weighted blow up of a toric variety X = X ∆ :
Ψ : X ′ = X ∆ ′ → X.
Suppose that positive integers b s+1 and c i satisfy
b s+1 v s+1 = l i=1 c i b i v i .(21)
Then define maps γ 1 , γ 2 as
γ 1 : s+1 i=1 Zf ′ i → s i=1 Zf i f ′ i → f i if i ≤ s l i=1 c i f i if i = s + 1
and γ 2 = id N , and take a toric DM orbifold X ′ = X Σ ′ associated with the stacky fan Σ ′ = (∆ ′ , β•γ 1 ).
Note that b s+1 v s+1 = β • γ 1 (f ′ s+1 ).
Then we obtain a morphism ψ : X ′ → X , called weighted blow-up of a toric DM orbifold X .
Lemma 6.1. We have Rψ * O X ′ = O X .
Proof. The assertion is shown in the proof of [BH1,Theorem 9.1].
Take a minimum integer h > 0 such that each h h i b i is an integer. We define b s+1 := hm and c i := h h i b i . Then the equation (21) is achieved, and hence we obtain a morphism ψ : X ′ → X as above. We call it the weighted blow-up of X associated with Ψ.
Remark 6.2. In the two dimensional case, recall that a torus equivariant resolution Θ : Y → X of singularities of X is a t-times composition of weighted blow-ups (cf. [Fu,§2.6]). Hence for such a resolution Θ, we can construct an associated morphism of toric DM orbifolds:
θ : Y = X (Y, s+t b i D i ) → X = X (X, s b i D i ).
Weighted blow ups of two dimensional toric DM stacks and Frobenius morphisms
We consider a two dimensional toric DM orbifold X = X Σ = X (X, s i=1 b i D i ) associated with a stacky fan Σ = (∆, β) in a free abelian group N = Ze 1 ⊕ Ze 2 .
Take a non-singular cone σ ∈ ∆ generated by vectors v 1 and v 2 , and consider the new simplicial fan ∆ ′ which is the subdivision of ∆ obtained from the the star shaped decomposition of σ, by adding the ray R ≥0 v s+1 in which we put v s+1 := v 1 + v 2 . Consider the (weighted) blow-up Ψ : X ′ → X associated with the subdivision. We may assume that v 1 = e 1 and v 2 = e 2 , and assume furthermore that b 1 = b 2 , which we denote by c. Let us define the blow-up ψ : X ′ → X associated with Ψ. Then we have
ψ * D i = D ′ i + D ′ s+1 for i = 1, 2, and ψ * D i = D ′ i for i = 1, 2.(22)
Moreover we have b s+1 = c. By v s+1 = e 1 + e 2 , we have an isomorphism
D ′ s+1 ∼ = c D/P(c, c)
for c = t (c, c) by Lemma 2.3. Let us define Q = [z 1 = z 2 = 0]. Then we have the following diagram:
X ′ ψ / / X D ′ s+1 ? j ′ O O p / / Q. ? j O O Lemma 6.3.
There exists a semi-orthogonal decomposition
D b (X ′ ) = j ′ * {L ∈ Pic D ′ s+1 | deg L = −1}, Lψ * D b (X ) .
Proof. Let us denote the category in the r.h.s. by T . We show below that any objects in ⊥ T are isomorphic to 0. This completes the proof, since T is an admissible subcategory of D b (X ′ ).
By Lemma 2.3, we have an isomorphism
Q ∼ = Q/ (Z c × Z c ) ,
where Q is a point. By a similar argument in [Or,Theorem 4.3], we see that for an object A ∈ ⊥ T , there exists an object B of D b (Q) satisfying
Lj ′ * A = p * B,(23)
and that B ∼ = 0 implies A ∼ = 0. We have
O Q,i = j * O X (i 1 D 1 + i 2 D 2 ) for i = t (i 1 , i 2 ) ∈ Z 2 . Since every object of D b (Q) is a direct sum of O Q,i [l] for some l ∈ Z,F := O D ′ s+1 (i 1 D ′ 1 + i 2 D ′ 2 ) i 1 +i 2
on D ′ s+1 , and then we have
Rp * F ∼ = O Q,i , j ′ * F ∼ = Lψ * j * O Q,i .(24)
The first isomorphism can be checked by similar computations in [BHu,Proposition 4.1], and the second is directly proved by the use of (11) and the Koszul resolution of j * O Q,i :
0 → O X ((i 1 − 1)D 1 + (i 2 − 1)D 2 ) → O X ((i 1 − 1)D 1 + i 2 D 2 ) ⊕ O X (i 1 D 1 + (i 2 − 1)D 2 ) → O X (i 1 D 1 + i 2 D 2 ) → j * O Q,i → 0.
By p * ⊣ Rp * , Lj ′ * ⊣ j ′ * , (23) and (24), we get equalities
Hom Q (B, O Q,i [l]) = Hom D s+1 (Lj ′ * A, F[l]) = Hom X ′ (A, Lψ * j * O Q,i [l]) = 0, since we have Lψ * j * O Q,i ∈ T .
This implies B ∼ = 0, which completes the proof.
Lemma 6.4. If D X generates D b (X ), then D X ′ generates D b (X ′ ).
Proof. For u ∈ R 2 , we define (19).
D u := s i=1 ⌊(u, b i v i )⌋D i , D ′ u := s+1 i=1 ⌊(u, b i v i )⌋D ′ i . Recall that D X = {O X (D u ) | u ∈ [0, 1) 2 } and D X ′ = {O X ′ (D ′ u ) | u ∈ [0, 1) 2 } byLet T ′ denote j ′ * {L ∈ Pic D ′ s+1 | deg L = −1}, and first we show that T ′ ⊂ D X ′ . Note that h(O D ′ s+1 l 1 D ′ 1 + l 2 D ′ 2 + kD ′ s+1 ) = l 1 + l 2 − k l 1 l 2 ∈ Z ⊕ Z c ⊕ Z c for the isomorphism h in (20), hence O D ′ s+1 D ′ u ∼ = O D ′ s+1 ⌊cu 1 ⌋D ′ 1 + ⌊cu 2 ⌋D ′ 2 + ⌊cu 1 + cu 2 ⌋D ′ s+1 ∈ T ′ if ⌊cu 1 ⌋ + ⌊cu 2 ⌋ − ⌊cu 1 + cu 2 ⌋ = −1.(25)
Fix l 1 , l 2 ∈ Z, and take a generic element u in the set u = t (u 1 , u 2 ) ∈ R 2 l 1 < cu 1 < l 1 + 1, l 2 < cu 2 < l 2 + 1, cu 1 + cu 2 = l 1 + l 2 + 1 .
Then for a sufficiently small real number ε > 0, we have
O X ′ D ′ u−ǫ t (1,1) = O X ′ D ′ u − D ′ s+1 .
It follows from the exact sequence (25), we obtain T ′ ⊂ D X ′ . Next we show Lψ * D b (X ) ⊂ D X ′ . By the assumption D X = D b (X ), it suffices to show that ψ * O X (D u ) ∈ D X ′ for any u ∈ R 2 . By (22), we have
0 → O X ′ D ′ u − D ′ s+1 → O X ′ D ′ u → O D ′ s+1 D ′ u → 0 (26) that O D ′ s+1 (D ′ u ) ∈ D X ′ . Since u satisfiesψ * O X (D u ) = O X ′ (D ′ u ) if ⌊(u, cv s+1 )⌋ = ⌊cu 1 ⌋ + ⌊cu 2 ⌋, O X ′ (D ′ u − D ′ s+1 ) if ⌊(u, cv s+1 )⌋ = ⌊cu 1 ⌋ + ⌊cu 2 ⌋ + 1.
Hence it is enough to consider the second case. Then we see by (25) (26). Now Lemma 6.3 completes the proof.
that O D ′ s+1 (D ′ u ) ∈ T ′ ⊂ D X ′ and hence ψ * O X (D u ) ∈ D X ′ by
Proof of Main theorem
The purpose of this note is to show the following.
Theorem 7.1. For every two dimensional toric DM stack X , the set D X generates D b (X ).
Full strong exceptional collections
In this section, we choose a full strong exceptional collection from the set D X , in several examples of one or two dimensional toric stacks X .
Examples
First let us show Lemma 8.1. Let X be a toric DM orbifold and assume that a coarse moduli space X of X is smooth. Recall that we have a root construction morphism π : X → X as in §3.3. Then for any divisor D on X , there exists a divisor D on X such that π * O X (D) = O X (D) ⊗m for some m > 0. Then we know that D is nef on X if and only if so is D on X. Under this notation, we have the following:
Lemma 8.1. If D is nef, then we have H i (X , O X (D)) = 0 for i > 0. Proof. Since O X is a direct summand of F m * O X , we have H i (X , O X (D)) ⊂H i (X , O X (D) ⊗ F m * O X ) ∼ = H i (X , F * m O X (D)) ∼ =H i (X , O X (D) ⊗m ) ∼ = H i (X , π * O X (D)) = 0,
which completes the proof. Note that the last equality is a consequence of the nef vanishing on toric varieties.
Put D nef X := {O X (D) ∈ D X | −D is nef} . If we have D X ⊂ D nef X ,
then the set D nef X forms a full strong exceptional collection by [Ue,Lemma 3.8(i)], Lemma 8.1 and Theorem 7.1.
Below we give calculation only in some typical cases and omit it in the rest. We leave it to readers.
(1) For X = P(a 1 , a 2 , a 3 ) with (a 1 , a 2 , a 3 ) = 1 or X = X (
P 1 × P 1 , b i D i ) with arbitrary b 1 , b 2 , b 3 , b 4 ∈ Z >0 , we have D nef X = D X .
Hence D X forms a full strong exceptional collection.
(2) Blow-up at a point on P 2 , and then we obtain the Hirzebruch surface F 1 . For the toric DM orbifold X = X (F 1 , D 1 + D 2 + 2D 3 + D 4 ), we have D nef X = D X , where D 1 , D 3 are fiber of the ruling and D 2 , D 4 are negative and positive sections respectively. In fact we take vectors In this case, since #D nef X = 6 < rk K(X ) = 7, the set D nef X does not form a full exceptional collection (see Remark 4.3), and thus we know D X ⊂ D nef X . However we can see the subset
0 → O X (−2D 3 − D 4 − iD 5 ) → O X (−D 3 − D 4 − iD 5 ) → O D 3 (−D 3 − D 4 ) → 0(27)
for i = 0, 1, we have O X (−2D 3 − D 4 − iD 5 ) ∈ S . Hence we obtain D X ⊂ S and hence S generates D b (X ). According to [BHu,Proposition 4.1], we show below that for any L, L ′ ∈ S, Ext i X (L, L ′ ) = 0 for i = 0 to conclude that S is a full strong exceptional collection. For every r = (r i ) ∈ Z 5 , we denote by Supp(r) the simplicial complex on five vertices {1, . . . , 5}
which consists of all subsets J ⊂ {1, . . . , 5} such that r i ≥ 0 for all i ∈ J and there exists a cone in the fan ∆ determining X that contains all v i , i ∈ J. By [BHu,Proposition 4.1] 2 we see that for any line bundle L on X , we have dim H 1 (X , L) = r (#{ connected components of Supp(r)} − 1) ,
where the summation is taken over the set of all r = (r i ) ∈ Z 5 such that O X ( 5 i=1 r i D i ) ∼ = L. For example, we know from (28) that Ext 1 X (L 1 , O X (−2D 3 − D 4 )) = H 1 (X , O X (−3D 3 − D 4 + D 5 )) = 0, since −3D 3 − D 4 + D 5 ∼ −D 1 − D 3 + D 5 and Supp( t (−1, 0, −1, 0, 1)) has 2 connected components.
By [BHu,Proposition 4.1] and easy but tedious computations as above, we can see that S forms a full strong exceptional collection. Moreover again by (28) for all i = 1, 2, 3. Hence we conclude that S is a unique subset of D X which forms a full exceptional collection. We remark that any two dimensional toric Fano orbifold has a full strong exceptional collection of line bundles by [BHu,Theorem 7.3].
8.2 Toric DM orbifolds with rank Pic X = 1.
As an application of Main Theorem, we also have the following:
Theorem 8.2. Let X be an one or two dimensional toric DM stack. Suppose furthermore that its rigidification X rig has the Picard group of rank 1. (Notice that this condition is automatically satisfied when X is 1-dimensional.) Then the set D X forms a full strong exceptional collection.
Proof. By Corollary 3.2, it suffices to show the statement for toric DM orbifolds. Take the linear function deg : Pic X → Z which takes value 1 on the positive generator of Pic X .
We first show that D X is contained in the set L of line bundles L satisfying deg K X < deg L ≤ 0.
Take an arbitrary line bundle L in O X . Then by (19)
, L is of the form O X i ⌊ (u,b i v i ) m ⌋D i for some m ∈ Z >0 and u ∈ Z n ∩ [0, m − 1] n . Since (u, b i v i ) = m⌊ (u, b i v i ) m ⌋ + r i
for some integers r i with 0 ≤ r i < m, the divisor m i ⌊ (u,b i v i ) m ⌋D i is linearly equivalent to the divisor − i r i D i . In particular, we have L ⊗m ∼ = O X (− i r i D i ), and hence we conclude that deg K X < deg L ≤ 0. By [BHu,Proposition 5.1] the set L forms a full strong exceptional collection. Since D X generates D b (X ), the set D X coincides with the collection they chose. Therefore the set D X forms a full strong exceptional collection.
it is enough to show Hom Q (B, O Q,i [l]) = 0 for any i and l. Take an invertible sheaf
divisors D 1 , D 2 , D 3 and D 4 on X . Then we obtainD X = {O X , O X (−D 3 ), O X (−2D 3 ), O X (D 3 − D 4 ), O X (−D 4 ), O X (−D 3 − D 4 ), O X (−2D 3 − D 4 )} and D nef X = D X \ {O X (D 3 − D 4 )}. In this case we can see that O X (D 3 − D 4 ) ∈ D nef X , and hence D X ⊂ D nef X .(3) Finally we take the toric DM orbifold X defined by vectors N corresponding to divisors D 1 , D 2 , D 3 , D 4 and D 5 on X . Then we obtainD X ={O X , O X (−D 3 − D 4 ), O X (−2D 3 − D 4 ), O X (D 3 − D 5 ), O X (−D 5 ), O X (−2D 3 − D 4 − D 5 ), O X (−D 3 − D 4 − D 5 ), O X (−D 4 − D 5 ), O X (D 3 − D 4 − D 5 )} and D nef X = D X \ {L 1 , L 2 , L 3 },where we define L 1 := O X (D 3 − D 5 ), L 2 := O X (−D 3 − D 4 ) and L 3 := O X (D 3 − D 4 − D 5 ).
S
:= D X \ {O X (−2D 3 − D 4 ), O X (−2D 3 − D 4 − D 5 )}of D X is a full strong exceptional collection as follows.By the exact sequence0 → O X → O X (D 3 − D 5 ) → O D 3 (D 3 ) → 0, we have O D 3 (−D 3 − D 4 ) = O D 3 (D 3 ) ∈ S .Moreover by the exact sequence
X (L i , O X (−2D 3 − D 4 )) = 0 and Ext 1 X (L i , O X (−2D 3 − D 4 − D 5 )) = 0
since we can compute push-forward by the embedding [V (τ )/G] ֒→ [U/G] as in the proof of [BH1, Theorem 9.1].
Completeness is not essential in many arguments below, but for simplicity, we assume it.
There is a typo in the statement[BHu, Proposition 4.1]. Precisely, the cohomology H p (X , L) is obtained by (rk N − p − 1)-th reduced homology of Supp(r).
Proof. By Lemma 4.6, we can reduce the proof of the statement to the case X is an orbifold. ThenConsider a torus equivariant resolution of X and the associated birational morphism between toric DM orbifolds as in Remark 6.2. Then by Lemmas 4.4 and 6.1 we can reduce to the case X is smooth.We have a root construction morphismBy lemmas 4.4 and 3.5, it suffices to check the statement for X ′ . The strong factorization theorem ([Od, Theorem 1.28 (2)]) implies that there exists a smooth complete toric surface Z and morphisms Φ, Ψ;where Φ and Ψ are compositions of torus equivariant blow-ups. Associated with Φ and Ψ, we obtain a compositions of blow-ups on toric DM orbifoldsBy Lemmas 4.4, 6.1 and 6.4, it suffices to show the statement in the case where X is the root stack X P 2 , cD P 2 1 + cD P 2 2 + cD P 2 3 of the projective plane P 2 , which is already shown in §5.2.Remark 7.2. Suppose that X is a toric DM stack and consider the following Cartesian diagram:by the flat base change theorem. Hence by Lemma 4.4, if D X ×P n generates D b (X × P n ), then D X generates D b (X ). In particular, D X generates D b (X ) for 0 or 1-dimensional toric DM stacks X by Theorem 7.1.Remark 7.3. An obstruction to extend Main Theorem to higher dimension is the following. For higher dimensional cases, we can get a statement similar to Lemma 6.4 again, when the center of a blow-up is zero-dimensional. However, when the dimension of the center is greater than zero, we do not know whether a similar statement holds.
P Achinger, arXiv:1012.2021A note on the Frobenius morphism on toric varieties. P. Achinger, A note on the Frobenius morphism on toric varieties, arXiv:1012.2021.
The orbifold Chow ring of toric Deligne-Mumford stacks. L Borisov, L Chen, G Smith, J. Amer. Math. Soc. 181L. Borisov, L. Chen, G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks. J. Amer. Math. Soc. 18 (2005), no. 1, 193-215
Derived categories of toric varieties. A I Bondal, Oberwolfach Reports. 31A. I. Bondal, Derived categories of toric varieties, Oberwolfach Reports, 3 (1), 284-286, 2006.
On the K-theory of smooth toric DM stacks, Snowbird lectures on string geometry. L Borisov, R Horja, Contemp. Math. 401Amer. Math. SocL. Borisov, R. Horja, On the K-theory of smooth toric DM stacks, Snowbird lectures on string geometry, Contemp. Math., 401, 21-42, Amer. Math. Soc., Providence, RI, 2006.
Mellin-Barnes integrals as Fourier-Mukai transforms. L Borisov, R Horja, Adv. Math. 2072L. Borisov, R. Horja, Mellin-Barnes integrals as Fourier-Mukai transforms, Adv. Math. 207 (2), 876-927, 2006.
On the conjecture of King for smooth toric Deligne-Mumford stacks. L Borisov, Z Hua, Adv. Math. 2211L. Borisov, Z. Hua, On the conjecture of King for smooth toric Deligne-Mumford stacks, Adv. Math. 221 (1), 277-301, 2009.
The homogeneous coordinate ring of a toric variety. D A Cox, J. Algebraic Geom. 4D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4, 17-50, 1995.
Smooth toric Deligne-Mumford stacks. B Fantechi, E Mann, F Nironi, J. Reine Angew. Math. 648B. Fantechi, E. Mann, F. Nironi, Smooth toric Deligne-Mumford stacks, J. Reine Angew. Math. 648, 201-244, 2010.
Introduction to toric varieties. W Fulton, Annals of Mathematics Studies. 131Princeton University PressW. Fulton, Introduction to toric varieties. Annals of Mathematics Studies, 131. Princeton University Press, Princeton, NJ, 1993. xii+157 pp.
Formal deformations and their categorical general fibre. D Huybrechts, E Macri, P Stellari, Comment. Math. Helv. 861D. Huybrechts, E. Macri, P. Stellari, Formal deformations and their categorical general fibre., Comment. Math. Helv. 86 (1), 41-71, 2011.
Logarithmic geometry, minimal free resolutions and toric algebraic stacks. I Iwanari, Publ. Res. Inst. Math. Sci. 454I. Iwanari, Logarithmic geometry, minimal free resolutions and toric algebraic stacks. Publ. Res. Inst. Math. Sci. 45 (4), 1095-1140, 2009.
The category of toric stacks. I Iwanari, Compos. Math. 1453I. Iwanari, The category of toric stacks. Compos. Math. 145 (3), 718-746, 2009.
A Ishii, K Ueda, arXiv:1104.2381The special McKay correspondence and exceptional collection. A. Ishii, K. Ueda, The special McKay correspondence and exceptional collection, arXiv:1104.2381.
Derived categories of toric varieties. Y Kawamata, Michigan Math. J. 543Y. Kawamata, Derived categories of toric varieties. Michigan Math. J. 54 (2006), no. 3, 517-535.
Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. T Oda, Ergebnisse der Mathematik und ihrer Grenzgebiete. 153Springer-Verlagviii+212 ppT. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric vari- eties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 15. Springer-Verlag, Berlin, 1988. viii+212 pp.
Projective bundles, monoidal transformations, and derived categories of coherent sheaves., (Russian) Izv. D Orlov, Russian Acad. Sci. Izv. Math. 564Ross. Akad. Nauk Ser. Mat.D. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves., (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852-862; translation in Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 133-141
A note on toric Deligne-Mumford stacks. F Perroni, Tohoku Math. J. 2F. Perroni, A note on toric Deligne-Mumford stacks. Tohoku Math. J. (2) 60 (3), 441-458, 2008.
Frobenius direct images of line bundles on toric varieties. F Thomsen, J. Algebra. 226F. Thomsen, Frobenius direct images of line bundles on toric varieties. J. Algebra. 226 (2000), 865-874.
Tilting generators via ample line bundles. Y Toda, H Uehara, Adv. Math. 233Y. Toda, H. Uehara, Tilting generators via ample line bundles, Adv. Math. 233 (2010), 1-29.
H Uehara, arXiv:1012.4086Exceptional collections on toric Fano threefolds and birational geometry. H. Uehara, Exceptional collections on toric Fano threefolds and birational geometry, arXiv:1012.4086.
Intersection theory on algebraic stacks and on their moduli spaces. A Vistoli, Invent. Math. 973A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97 (1989), no. 3, 613-670.
. Hokuto Uehara Department of Mathematics and Information Sciences. Tokyo Metropolitan UniversityHokuto Uehara Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1
. Hachioji-Shi Minamiohsawa, Tokyo , Japan e-mail address : [email protected], Hachioji-shi, Tokyo, 192-0397, Japan e-mail address : [email protected]
| [] |
[
"Adaptive Forecasting of Non-Stationary Nonlinear Time Series based on the Evolving Weighted Neuro-Neo-Fuzzy- ANARX-Model",
"Adaptive Forecasting of Non-Stationary Nonlinear Time Series based on the Evolving Weighted Neuro-Neo-Fuzzy- ANARX-Model"
] | [
"Zhengbing Hu \nSchool of Educational Information Technology\nCentral China Normal University\nWuhanChina\n",
"Yevgeniy V Bodyanskiy [email protected] \nKharkiv National University of Radio Electronics\nKharkivUkraine\n",
"Oleksii K Tyshchenko \nKharkiv National University of Radio Electronics\nKharkivUkraine\n",
"Olena O Boiko [email protected] \nKharkiv National University of Radio Electronics\nKharkivUkraine\n"
] | [
"School of Educational Information Technology\nCentral China Normal University\nWuhanChina",
"Kharkiv National University of Radio Electronics\nKharkivUkraine",
"Kharkiv National University of Radio Electronics\nKharkivUkraine",
"Kharkiv National University of Radio Electronics\nKharkivUkraine"
] | [] | An evolving weighted neuro-neo-fuzzy-ANARX model and its learning procedures are introduced in the article. This system is basically used for time series forecasting. It's based on neo-fuzzy elements. This system may be considered as a pool of elements that process data in a parallel manner. The proposed evolving system may provide online processing data streams. | 10.5815/ijitcs.2016.10.01 | [
"https://arxiv.org/pdf/1610.06486v1.pdf"
] | 11,052,666 | 1610.06486 | 62012311950e9d383a924d042df6c3bb6dc02802 |
Adaptive Forecasting of Non-Stationary Nonlinear Time Series based on the Evolving Weighted Neuro-Neo-Fuzzy- ANARX-Model
Zhengbing Hu
School of Educational Information Technology
Central China Normal University
WuhanChina
Yevgeniy V Bodyanskiy [email protected]
Kharkiv National University of Radio Electronics
KharkivUkraine
Oleksii K Tyshchenko
Kharkiv National University of Radio Electronics
KharkivUkraine
Olena O Boiko [email protected]
Kharkiv National University of Radio Electronics
KharkivUkraine
Adaptive Forecasting of Non-Stationary Nonlinear Time Series based on the Evolving Weighted Neuro-Neo-Fuzzy- ANARX-Model
Index Terms -Computational Intelligencetime series predictionneuro-neo-fuzzy SystemMachine LearningANARXData Stream
An evolving weighted neuro-neo-fuzzy-ANARX model and its learning procedures are introduced in the article. This system is basically used for time series forecasting. It's based on neo-fuzzy elements. This system may be considered as a pool of elements that process data in a parallel manner. The proposed evolving system may provide online processing data streams.
I. INTRODUCTION
Mathematical forecasting of data sequences (time series) is nowadays well studied and there is a large number of publications on this topic. There are many methods for solving this task: regression, correlation, spectral analysis, exponential smoothing, etc., and more advanced intellectual systems that sometimes require rather complicated mathematical methods and high user's qualification. The problem becomes more complicated when analyzed time series are non-stationary and nonlinear and contain unknown behavior trends, quasiperiodic, stochastic and chaotic components. The best results are shown by nonlinear forecasting models based on mathematical methods of computational intelligence [1][2][3], and, first of all, neuro-fuzzy systems [4][5] due to their approximating and extrapolating properties, learning abilities, transparency and results' interpretability. The models to be especially noted are the so-called NARX-models [6]
where y k is an estimate of forecasted time series at discrete time 1, 2, ... k
; f stands for a certain nonlinear transformation implemented by a neuro-fuzzy system, x k is an observed exogenous factor that defines a behavior of y k . It can be noticed that popular Box-Jenkins AR-, ARX-, ARMAX-models as well as nonlinear NARMAmodels can be described by the expression (1). These models have been widely studied; there are many architectures and learning algorithms that implement these models, but it is assumed that models' orders y n , x n are given a priori. These orders are previously unknown in a case of structural non-stationarity for analyzed time series, and they also have to be adjusted during a learning procedure. In this case, it makes sense to use evolving connectionist systems [7][8][9][10] that adjust not only their synaptic weights and activation-membership functions, but also their architectures. There are many algorithms that implement these learning methods both in a batch mode and in a sequential mode. The problem becomes more complicated if data are fed to the system with high frequency in the form of a data stream [11]. Here, the most popular evolving systems turn out to be too cumbersome for learning and information processing in an online mode.
As an alternative, a rather simple and effective architecture can be considered. It's the so-called ANARX-model (Additive NARX) that has the form [12,13]
(2) (here max , y x n n n )
, an original task of the forecasting system's synthesis is decomposed into many local tasks of parametric identification for node models with two input variables
y k l , x k l , 1, 2, ..., , ... l n .
Authors [12,13] used elementary Rosenblatt perceptrons with sigmoidal activation functions as such nodes. The ANARX-model provided high forecasting quality, but generally speaking it requires a large number of nodes in its architecture [14].
Some synthesis problems of forecasting neuro-fuzzy [15] and neo-fuzzy [15][16][17] systems based on the ANARXmodels are considered in this work. These systems avoid the above mentioned drawbacks.
Since we consider a case of stochastic nonlinear dynamic signals in this article, the basic novelty has to do with defining a model's delay order in an online mode.
II. A NEURO-FUZZY-ANARX-MODEL
An architecture of the ANARX-model is shown in Fig.1 It's recommended to use a neuron with two inputs (its architecture is shown in Fig.2) as a node of this system instead of the elementary Rosenblatt perceptron. As one can see, this node is the Wang-Mendel neuro-fuzzy system [18] with two inputs. It possesses universal approximation capabilities and is actually the zero-order Takagi-Sugeno-Kang system [19,20]
, l h h i l l l l l i i i h l i i i i w y k l x k l y k y k l x k l z k w w k w k z k where T 1 2 , , ..., l l l l h k k k k , 1 1 h l l l i i i i k z k z k , T 1 2 , ,..., l l l l h w w w w .
Considering that the output signal l y k of each node depends linearly on the adjusted synaptic weights l i w , one can use conventional algorithms of adaptive linear identification [21] for their tuning which are based on the quadratic learning criterion. If a training data set is non-stationary [22], one can use either the exponentially weighted recurrent least squares method
1 1 1 , 1 1 1 1 , 0 1, 1 l l l l T l l lT l l l l l l l T l lT l l w k w k k y k w k k k k k k k k k k k k k k k (3)
or the Kaczmarz-Widrow-Hoff optimal gradient algorithm (in a case of the "rapid" non-stationarity)
1 1 lT l l l l lT l y k w k k w k w k k k k .(4)
In fact, it is possible to tune 4h membership functions' parameters iy c , ix c , iy , ix of each node, but taking into consideration the fact that the signal l y k depends nonlinearly on these parameters, a learning speed can't be sufficient for non-stationary conditions in this case.
III. A NEO-FUZZY-ANARX-MODEL
If a large data set to be processed is given (within the "Big Data" conception [23] when data processing speed and computational simplicity come to the forefront, it seems reasonable to use neo-fuzzy neurons that were proposed by T. Yamakawa and his co-authors [15][16][17] instead of the neuro-fuzzy nodes in the ANARX-model. An architecture of the neo-fuzzy neuron as a node of the ANARX-model is shown in Fig.3. The neo-fuzzy neuron's advantages are a high learning speed, computational simplicity, good approximating properties and abilities to find a global minimum of a learning criterion in an online mode. Takagi-Sugeno fuzzy inference, but it's easy to notice that the neo-fuzzy neuron is much simpler constructively than the neuro-fuzzy node shown in Fig.2.
An output value of this node is formed with the help of input signals
y k l , x k l 1 1 h l l l l y x i y i y i h l ix ix i y k k k w y k l w x k l (5)
and an output signal of the ANARX-model can be written down in the form
1 1 1 n h h l l iy iy ix ix l i i y k w y k l w x k l ,
which means that since the neo-fuzzy neuron is also an additive model [24], the ANARX-model based on the neofuzzy neurons is twice additive. Triangular membership functions are usually used in the neo-fuzzy neuron. They meet the unity partitioning conditions
1 1 h iy i y k l , 1 1 h ix i x k l ,
which make it possible to simplify the node's architecture by excluding the normalization layer from it.
It was proposed to use B-splines as membership functions for the neo-fuzzy neuron [25]. They provide a higher approximation quality and also meet the unity partitioning conditions. A B-spline of the q th order can be written down: Either the algorithms (3), (4) or the procedure [26]
y k l c y k l c c y k l c y k l y k l q c c i h q 1, 1 1, , 1 1, , 1 , 1, if , for 1 0 otherwise , for 1 1, ..., . ix i x ix q ix i q x ix q ix i q x q i x i q x i x c x k l c q x k l c x k l c c x k l c x k l x k l q c c i h q 1 T T 1 1 1 , 0 1 , l l l l l l l l l l w k w k r k y k w k k k r k r k k k (6)
can be used for the neo-fuzzy neuron's learning. This procedure possesses both filtering and tracking properties. It should also be noticed that when 1 the equation (6) coincides completely with the Kaczmarz-Widrow-Hoff optimal algorithm (4). Evolving systems based on the neo-fuzzy neurons demonstrated its effectiveness for solving different tasks (especially forecasting tasks [25][26][27][28][29][30]). The twice additive system is the most appropriate choice for data processing in Data Stream Mining tasks [11] from a point of view of implementation simplicity and a processing speed.
IV. A WEIGHTED NEURO-NEO-FUZZY-ANARX-MODEL
Considering that every node [ ] l N of the ANARX-model is tuned independently from each other and represents an individual neuro(neo)-fuzzy system, it is possible to use a combination of neural networks' ensembles [31] in order to improve a forecasting quality. This approach results in an architecture of a weighted neuro-neo-fuzzy-ANARX-model (Fig.4).
where a value of the Lagrange function (7) at the saddle point is
1 T 1 * , n n L c I R I .
An implementation procedure of the algorithm (9) can meet some problems in a case of data processing in an online mode and high level of correlation between the signals l y k that leads to the ill-conditioned matrix R which has to be inverted at every time step k .
The Lagrange function (7) can be written down in the form
2 T T , 1 n k L c y k c y k c I
and a gradient algorithm for finding its saddle point based on the Arrow-Hurwicz procedure [27,32] can be written in the form
1 , , , 1 c c c k c k k L c L c k k k or T T 1ˆ2 1 1 1 2 1 , 1 1 c n c n n c k c k k y k c k y k y k k I c k k v k y k k I k k k c k I (10) where c k , k
are learning rate parameters.
The Arrow-Hurwicz procedure converges to the saddle point with rather general assumptions on the values
c k , k
, but these parameters can be optimized to speed up the learning process as that is particularly important in Data Stream Mining tasks.
That's why the first ratio in the equation (10) should be multiplied by
T y k T T 2 Tˆ12 1 c n y k c k y c k k v k y k k y I
and an additional function that characterizes criterial convergence is introduced
2 T 2 2 T 2 2 2 Tˆ2 2 12 1 c n c n y k y k c k v k k v k v k y k k y I k v k y k k y I .
A solution for the differential equation
2 T 2 T
gives the optimal learning rate values
c k in the form 2 T2 1 c n v k k v k y k k y I .
Applying this value to the expression (10), final equations can be written down
2 T T2 1 1 ,2 1 1 1 . n n n v k v k y k k I c k c k v k y k k y I k k k c k I (11)
It's easy to notice that when 1 0 k the procedure (11) coincides with the Kaczmarz-Widrow-Hoff algorithm (4).
V. EXPERIMENTS
In order to prove the effectiveness of the proposed system, a number of experiments should be carried out.
A. Electricity Demand
This data set describes 15 minutes averaged values of power demand in the full year 1997. Generally speaking, this data set contains 15000 points but we took only 5000 points for the experiment. 3000 points were selected for a training stage and 2000 points were used for testing. Prediction results of the ANARX and weighted ANARX systems are in Fig.5 and Fig.6 correspondingly (signal values are marked with a blue color; prediction values are marked with a magenta color; and prediction errors are marked with a grey line).
We used for comparison multilayer perceptrons (MLPs), radial-basis function neural networks (RBFNs), ANFIS and two proposed systems ANARX and weighted ANARX (both based on neo-fuzzy nodes). Since MLP can't work in an online mode, it processed data in two different modes. It had just one epoch in the first case (something similar to an online case), and it had 5 epochs in the second case (the MLP architecture had almost the same number of adjustable parameters when compared to the proposed systems). A number of MLP inputs was equal to 4 and a number of hidden nodes was equal to 7 in both cases. A total number of parameters to be tuned was 43 in both modes. It took about two times more time to compute the result in the second case but prediction quality was like almost two times higher. MLP (case 2) demonstrated the best result in this experiment.
Speaking of RBFNs, we also had two cases. The first-case RBFN was taken really close in the sense of parameters' number to our systems and the second-case RBFN's architecture was chosen to show the best performance. In the first case, RBFN had 3 inputs and 7 kernel functions. In the second case, it had 3 inputs as well but 12 kernel functions which generally led to higher prediction quality (+30% precision compared to RBFN in the first case) but took longer to compute the result. A number of parameters to be tuned was 36 in the first case and 61 in the second case.
ANFIS showed one of the best prediction qualities in this experiment. It had 4 inputs, 55 nodes and it was processing data during 5 epochs. It contained 80 parameters to be tuned. The proposed ANARX system based on neo-fuzzy nodes had 2 inputs, 2 nodes, 9 membership functions, and its parameter was equal to 0.62. This system had 37 parameters. Its prediction quality was rather high and it was definitely one of the fastest system on this data set. We should also notice that we used B-splines (q=2, which means that we used triangular membership functions) as membership functions for the proposed systems.
The proposed weighted ANARX system based on neo-fuzzy nodes had 2 inputs, 2 nodes, 8 membership functions; its parameter was equal to 0.9. It had 37 adjustable parameters. This system demonstrated better performance when compared to ANARX and the fastest results.
B. Monthly sunspot number
This data set was taken from datamarket.com. The data set describes monthly sunspot number in Zurich. It was collected between 1749 and 1983. It contains 2820 points. 2256 points were selected for a training stage and 564 points were used for testing. Prediction results of the ANARX and weighted ANARX systems are in Fig.7 and Fig.8 correspondingly (signal values are marked with a blue color; prediction values are marked with a magenta color; and prediction errors are marked with a grey line).
We used a set of systems which is similar to the previous experiment to compare results. MLP_1 (MLP for the first case) had 3 inputs, 6 hidden nodes and it was processing data during just one epoch. MLP_2 basically had the same parameter settings except the fact that it was processing data during 5 epochs. Both MLP systems had 31 parameters to be tuned. MLP_1 was 2 times faster than MLP_2 but its prediction quality was also almost 2 times worse.
Let's denote our RBFN architectures as RBFN_1 (RBFN for the first case) and RBFN_2 (RBFN for the second case). Both of them had 3 inputs. RBFN_1 had 4 kernel functions unlike RBFN_2 which had 19 hidden nodes. A number of parameters to be tuned was 21 in the first case and 96 in the second case. RBFN_2 showed a better prediction quality but it was much slower than RBFN_1.
ANFIS had 4 inputs and 55 nodes. It was processing data during 3 epochs. It had 80 adjustable parameters. The proposed ANARX system based on neo-fuzzy nodes had 2 inputs, 2 nodes, 4 membership functions, and its parameter was equal to 0.9. This system contained 17 parameters. Its prediction quality was rather high and it was definitely one of the fastest systems on this data set.
The proposed weighted ANARX system based on neo-fuzzy nodes had 2 inputs, 2 nodes, 4 membership functions; its parameter was equal to 0.9. This system contained 20 parameters to be tuned. This system demonstrated better performance when compared to ANARX and one of the fastest results. VI. CONCLUSION The evolving forecasting weighted neuro-neo-fuzzy-twice additive model and its learning procedures are proposed in the paper. This system can be used for non-stationary nonlinear stochastic and chaotic time series' forecasting where time series are processed in an online mode under the parametric and structural uncertainty. The proposed weighted ANARX-model is rather simple from a computational point of view and provides fast data stream processing in an online mode.
So, the proposed evolving forecasting model has demonstrated its efficiency for solving real-world tasks. A number of experiments has been performed to show high efficiency of the proposed neuro-neo-fuzzy system.
Figure 1 .
1The ANARX-model
Figure 2 .
2A neuro-fuzzy node of the ANARX-model
Figure 3 .
3A neo-fuzzy node for the ANARX-model Structural elements of the neo-fuzzy neuron are nonlinear synapses y NS , x NS that implement the zero order
It should be mentioned that B-splines are a sort of generalized membership functions: when 2 q one gets traditional triangular membership functions; when 4 q one gets cubic splines, etc.
Figure 4 .
4The weighted ANARX-model An output signal of the system can be written in the form -vector of unities.The method of undetermined Lagrange multipliers can be used for finding the vector с in a batch mode.
Figure 5 .Figure 6 .
56Identification of a nonlinear system. Prediction results of the ANARX system Identification of a nonlinear system. Prediction results of the weighted ANARX system
Figure 7 .Figure 8 .
78Identification of a nonlinear system. Prediction results of the ANARX system Identification of a nonlinear system. Prediction results of the weighted ANARX system
which have the form
1 , ...,
,
1 , ...,
y
x
y k
f y k
y k n x k
x k n
. It is formed by two lines of time delay elements 1z
1
1
z y k
y k
and n nodes [ ]
l
N which are simultaneously learned. These nodes are tuned independently
from each other. And adding new nodes or removing unnecessary ones doesn't have any influence on other neurons,
i.e. the evolving process for this system is implemented by changing a number of the nodes.
TABLE I .
ICOMPARISON OF THE SYSTEMS' RESULTSSystems
Parameters
to be tuned
RMSE
(training)
RMSE
(test)
Time,
s
MLP
(case 1)
43
0.0600
0.0700 0.4063
MLP
(case 2)
43
0.0245
0.0381 0.9219
RBFN
(case 1)
36
0.0681
0.0832 0.6562
RBFN
(case 2)
61
0.0463
0.0604 1.0250
ANFIS
80
0.0237
0.0396 0.7031
ANARX
(neo-
fuzzy
nodes)
37
0.0903
0.0922 0.4300
Weighted
ANARX
(neo-
fuzzy
nodes)
36
0.0427
0.0573 0.3750
TABLE II .
IICOMPARISON OF THE SYSTEMS' RESULTSSystems
Parameters RMSE
RMSE Time, s
to be tuned (training) (test)
MLP
(case 1)
31
0.1058
0.1407 0.2813
MLP
(case 2)
31
0.0600
0.0808 0.5938
RBFN
(case 1)
21
0.1066
0.2155 0.2219
RBFN
(case 2)
96
0.0702
0.1638 0.9156
ANFIS
80
0.0559
0.1965 0.8906
ANARX
(neo-
fuzzy
nodes)
17
0.1297
0.1350 0.2252
Weighted
ANARX
(neo-
fuzzy
nodes)
20
0.0784
0.1081 0.2981
ACKNOWLEDGMENTThe authors would like to thank anonymous reviewers for their careful reading of this paper and for their helpful comments.This scientific work was supported by RAMECS and CCNU16A02015.
L Rutkowski, Computational Intelligence. Methods and Techniques. Berlin-HeidelbergSpringer-VerlagRutkowski L. Computational Intelligence. Methods and Techniques. Springer-Verlag, Berlin-Heidelberg, 2008.
Computational Intelligence. R Kruse, C Borgelt, F Klawonn, C Moewes, M Steinbrecher, P Held, SpringerBerlinKruse R, Borgelt C, Klawonn F, Moewes C, Steinbrecher M, Held P. Computational Intelligence. Springer, Berlin, 2013.
K-L Du, M N S Swamy, Neural Networks and Statistical Learning. LondonSpringer-VerlagDu K-L, Swamy M N S. Neural Networks and Statistical Learning. Springer-Verlag, London, 2014.
Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence. J-S Jang, C-T Sun, E Mizutani, Prentice HallUpper Saddle RiverJang J-S, Sun C-T, Mizutani E. Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence. Prentice Hall, Upper Saddle River, 1997.
The Use of ANFIS and RBF to Model and Predict the Inhibitory Concentration Values Determined by MTT Assay on Cancer Cell Lines. A Rezaei, L Noori, M Taghipour, International Journal of Information Technology and Computer Science(IJITCS). 84Rezaei A, Noori L, Taghipour M. The Use of ANFIS and RBF to Model and Predict the Inhibitory Concentration Values Determined by MTT Assay on Cancer Cell Lines. International Journal of Information Technology and Computer Science(IJITCS), 2016, 8(4): 28-34.
Nonlinear System Identification. O Nelles, SpringerBerlinNelles O. Nonlinear System Identification. Springer, Berlin, 2001.
Evolving fuzzy neural networks -algorithms, applications and biological motivation. Proc. N Kasabov, SingaporeMethodologies for the Conception, Design and Application of Soft ComputingKasabov N. Evolving fuzzy neural networks -algorithms, applications and biological motivation. Proc. "Methodologies for the Conception, Design and Application of Soft Computing", Singapore, 1998:271-274.
Evolving fuzzy neural networks: theory and applications for on-line adaptive prediction, decision making and control. N Kasabov, Australian J. of Intelligent Information Processing Systems. 53Kasabov N. Evolving fuzzy neural networks: theory and applications for on-line adaptive prediction, decision making and control. Australian J. of Intelligent Information Processing Systems, 1998, 5(3):154-160.
Evolving Connectionist Systems. N Kasabov, Springer-VerlagLondonKasabov N. Evolving Connectionist Systems. Springer-Verlag, London, 2003.
Evolving Fuzzy Systems -Methodologies, Advanced Concepts and Applications. E Lughofer, SpringerBerlinLughofer E. Evolving Fuzzy Systems -Methodologies, Advanced Concepts and Applications. Springer, Berlin, 2011.
Adaptive Stream Mining: Pattern Learning and Mining from Evolving Data Streams. A Bifet, IOS PressAmsterdamBifet A. Adaptive Stream Mining: Pattern Learning and Mining from Evolving Data Streams. IOS Press, 2010, Amsterdam.
A novel Taylor series based approach for control computation in NN-ANARX structure based control of nonlinear systems. J Belikov, K Vassiljeva, E Petlenkov, S Nõmm, Proc. 27th Chinese Control Conference. 27th Chinese Control ConferenceKunming, ChinaBelikov J, Vassiljeva K, Petlenkov E, Nõmm S. A novel Taylor series based approach for control computation in NN- ANARX structure based control of nonlinear systems. Proc. 27th Chinese Control Conference, Kunming, China, 2008:474-478.
State-space control of nonlinear systems identified by ANARX and neural network based SANARX models. K Vassiljeva, E Petlenkov, J Belikov, Proc. WCCI 2010 IEEE World Congress on Computational Intelligence. WCCI 2010 IEEE World Congress on Computational IntelligenceBarcelona, SpainVassiljeva K, Petlenkov E, Belikov J. State-space control of nonlinear systems identified by ANARX and neural network based SANARX models. Proc. WCCI 2010 IEEE World Congress on Computational Intelligence, Barcelona, Spain, 2010:3816-3823.
Approximation by superpositions of a sigmoidal function. G Cybenko, Math. Control Signals Systems. 2Cybenko G. Approximation by superpositions of a sigmoidal function. Math. Control Signals Systems, 1989, 2:303-314.
A neo fuzzy neuron and its applications to system identification and prediction of the system behavior. T Yamakawa, E Uchino, T Miki, H Kusanagi, Proc. 2nd Int. Conf. on Fuzzy Logic and Neural Networks "IIZUKA-92. 2nd Int. Conf. on Fuzzy Logic and Neural Networks "IIZUKA-92Iizuka, JapanYamakawa T, Uchino E, Miki T, Kusanagi H. A neo fuzzy neuron and its applications to system identification and prediction of the system behavior. Proc. 2nd Int. Conf. on Fuzzy Logic and Neural Networks "IIZUKA-92", Iizuka, Japan, 1992:477-483.
Soft computing based signal prediction, restoration, and filtering. E Uchino, T Yamakawa, Intelligent Hybrid Systems: Fuzzy Logic. Uchino E, Yamakawa T. Soft computing based signal prediction, restoration, and filtering. In: Intelligent Hybrid Systems: Fuzzy Logic, Neural Networks, and Genetic Algorithms, 1997:331-349.
Analog implementation of neo-fuzzy neuron and its on-board learning. T Miki, T Yamakawa, Computational Intelligence and Applications. Miki T, Yamakawa T. Analog implementation of neo-fuzzy neuron and its on-board learning. In: Computational Intelligence and Applications, 1999:144-149.
Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. L-X Wang, J Mendel, IEEE Trans. on Neural Networks. 35Wang L-X, Mendel J M. Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans. on Neural Networks, 1992, 3(5):807-814.
Fuzzy identification of systems and its applications to modeling and control. T Takagi, M Sugeno, IEEE Trans. on Systems, Man, and Cybernetics. 15Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. on Systems, Man, and Cybernetics, 1985, 15:116-132.
Structure identification of fuzzy model. Fuzzy Sets and Systems. M Sugeno, G Kang, 28Sugeno M, Kang G T. Structure identification of fuzzy model. Fuzzy Sets and Systems, 1998, 28:15-33.
System Identification: Theory for the User. L Ljung, Prentice Hall, IncUpper Saddle RiverLjung L. System Identification: Theory for the User. Prentice Hall, Inc., Upper Saddle River, 1987.
Learning in nonstationary and evolving environments. R Polikar, C Alippi, IEEE Trans. on Neural Networks and Learning Systems. 251Polikar R, Alippi C. Learning in nonstationary and evolving environments. IEEE Trans. on Neural Networks and Learning Systems, 2014, 25(1):9-11.
Computational Intelligence in Big Data. Y Jin, B Hammer, IEEE Computational Intelligence Magazine. 93Jin Y, Hammer B. Computational Intelligence in Big Data. IEEE Computational Intelligence Magazine, 2014, 9(3):12-13.
The Elements of Statistical Learning. Data Mining, Inference and Prediction. J Friedman, T Hastie, R Tibshirani, SpringerBerlinFriedman J, Hastie T, Tibshirani R. The Elements of Statistical Learning. Data Mining, Inference and Prediction. Springer, Berlin, 2003.
Cascaded multiresolution spline-based fuzzy neural network. Bodyanskiy Ye, V Kolodyazhniy, Proc. Int. Symp. on Evolving Intelligent Systems. Int. Symp. on Evolving Intelligent SystemsLeicester, UKBodyanskiy Ye, Kolodyazhniy V. Cascaded multiresolution spline-based fuzzy neural network. Proc. Int. Symp. on Evolving Intelligent Systems, Leicester, UK, 2010:26-29.
An adaptive learning algorithm for a neo-fuzzy neuron. Bodyanskiy Ye, I Kokshenev, V Kolodyazhniy, Proc. 3rd Int. Conf. of European Union Soc. for Fuzzy Logic and Technology (EUSFLAT'03). 3rd Int. Conf. of European Union Soc. for Fuzzy Logic and Technology (EUSFLAT'03)Zittau, GermanyBodyanskiy Ye, Kokshenev I, Kolodyazhniy V. An adaptive learning algorithm for a neo-fuzzy neuron. Proc. 3rd Int. Conf. of European Union Soc. for Fuzzy Logic and Technology (EUSFLAT'03), Zittau, Germany, 2003:375-379.
An optimal algorithm for combining multivariate forecasts in hybrid systems. Bodyanskiy Ye, P Otto, I Pliss, S Popov, Lecture Notes in Artificial Intelligence. 2774Bodyanskiy Ye, Otto P, Pliss I, Popov S. An optimal algorithm for combining multivariate forecasts in hybrid systems. Lecture Notes in Artificial Intelligence, 2003, 2774:967-973.
The cascade neo-fuzzy architecture using cubic-spline activation functions. Viktorov Bodyanskiy Ye, Ye, Int. J. Information Theories and Applications. 163Bodyanskiy Ye, Viktorov Ye. The cascade neo-fuzzy architecture using cubic-spline activation functions. Int. J. Information Theories and Applications, 2009, 16(3):245-259.
Hybrid evolving neural network using kernel activation functions. Bodyanskiy Ye, N Teslenko, P Grimm, Proc. 17th Zittau East-West Fuzzy Colloquium. 17th Zittau East-West Fuzzy ColloquiumZittau/Goerlitz, GermanyBodyanskiy Ye, Teslenko N, Grimm P. Hybrid evolving neural network using kernel activation functions. Proc. 17th Zittau East-West Fuzzy Colloquium, Zittau/Goerlitz, Germany, 2010:39-46.
A multidimensional cascade neuro-fuzzy system with neuron pool optimization in each cascade. V Bodyanskiy Ye, O K Tyshchenko, D S Kopaliani, Int. J. Information Technology and Computer Science. 68Bodyanskiy Ye V, Tyshchenko O K, Kopaliani D S. A multidimensional cascade neuro-fuzzy system with neuron pool optimization in each cascade. Int. J. Information Technology and Computer Science, 2014, 6(8):11-17.
On combining artificial neural nets. A Sharkey, Connection Science. 8Sharkey A J C. On combining artificial neural nets. Connection Science, 1996, 8:299-313.
Adaptive generalization of neuro-fuzzy systems ensemble. Bodyanskiy Ye, A Deineko, M Stolnikova, Proc. of the Int. Conf. "Computer Science and Information Technologies. of the Int. Conf. "Computer Science and Information TechnologiesLviv, UkraineBodyanskiy Ye, Deineko A, Stolnikova M. Adaptive generalization of neuro-fuzzy systems ensemble. Proc. of the Int. Conf. "Computer Science and Information Technologies", Lviv, Ukraine, November 16-19, 2011:13-14.
| [] |
[
"Deep learning the slow modes for rare events sampling A preprint",
"Deep learning the slow modes for rare events sampling A preprint"
] | [
"Luigi Bonati ",
"Michele Parrinello ",
"\nDepartment of Physics\nETH Zurich\n8092ZurichSwitzerland\n",
"\nAtomistic Simulations\nItalian Institute of Technology\n16163GenovaItaly\n",
"\nPhysical and Computational Sciences Directorate\nGiovanniMaria Piccini Basic & Applied Molecular Foundations\nPacific Northwest National Laboratory\n99352RichlandWAUSA\n",
"\nIstituto Eulero\nUniversità della Svizzera italiana\n6900LuganoSwitzerland\n",
"\nItalian Institute of Technology\n16163GenovaItaly\n"
] | [
"Department of Physics\nETH Zurich\n8092ZurichSwitzerland",
"Atomistic Simulations\nItalian Institute of Technology\n16163GenovaItaly",
"Physical and Computational Sciences Directorate\nGiovanniMaria Piccini Basic & Applied Molecular Foundations\nPacific Northwest National Laboratory\n99352RichlandWAUSA",
"Istituto Eulero\nUniversità della Svizzera italiana\n6900LuganoSwitzerland",
"Italian Institute of Technology\n16163GenovaItaly"
] | [] | The development of enhanced sampling methods has greatly extended the scope of atomistic simulations, allowing long time phenomena to be studied with accessible computational resources. Many such methods rely on the identification of an appropriate set of collective variables. These are meant to describe the system's modes that most slowly approach equilibrium. Once identified, the equilibration of these modes is accelerated by the enhanced sampling method of choice. An attractive way of determining the collective variables is to relate them to the eigenfunctions and eigenvalues of the transfer operator. Unfortunately, this requires knowing the long-term dynamics of the system beforehand, which is generally not available. However, we have recently shown that it is indeed possible to determine efficient collective variables starting from biased simulations. In this paper, we bring the power of machine learning and the efficiency of the recently developed on-the-fly probability enhanced sampling method to bear on this approach. The result is a powerful and robust algorithm that, given an initial enhanced sampling simulation performed with trial collective variables or generalized ensembles, extracts transfer operator eigenfunctions using a neural network ansatz and then accelerates them to promote sampling of rare events. To illustrate the generality of this approach we apply it to several systems, ranging from the conformational transition of a small molecule to the folding of a mini-protein and the study of materials crystallization. | 10.1073/pnas.2113533118 | [
"https://arxiv.org/pdf/2107.03943v2.pdf"
] | 235,765,771 | 2107.03943 | 06b0a8c66ba5a4da7363116e7f4d65f1703d7e31 |
Deep learning the slow modes for rare events sampling A preprint
Luigi Bonati
Michele Parrinello
Department of Physics
ETH Zurich
8092ZurichSwitzerland
Atomistic Simulations
Italian Institute of Technology
16163GenovaItaly
Physical and Computational Sciences Directorate
GiovanniMaria Piccini Basic & Applied Molecular Foundations
Pacific Northwest National Laboratory
99352RichlandWAUSA
Istituto Eulero
Università della Svizzera italiana
6900LuganoSwitzerland
Italian Institute of Technology
16163GenovaItaly
Deep learning the slow modes for rare events sampling A preprint
The development of enhanced sampling methods has greatly extended the scope of atomistic simulations, allowing long time phenomena to be studied with accessible computational resources. Many such methods rely on the identification of an appropriate set of collective variables. These are meant to describe the system's modes that most slowly approach equilibrium. Once identified, the equilibration of these modes is accelerated by the enhanced sampling method of choice. An attractive way of determining the collective variables is to relate them to the eigenfunctions and eigenvalues of the transfer operator. Unfortunately, this requires knowing the long-term dynamics of the system beforehand, which is generally not available. However, we have recently shown that it is indeed possible to determine efficient collective variables starting from biased simulations. In this paper, we bring the power of machine learning and the efficiency of the recently developed on-the-fly probability enhanced sampling method to bear on this approach. The result is a powerful and robust algorithm that, given an initial enhanced sampling simulation performed with trial collective variables or generalized ensembles, extracts transfer operator eigenfunctions using a neural network ansatz and then accelerates them to promote sampling of rare events. To illustrate the generality of this approach we apply it to several systems, ranging from the conformational transition of a small molecule to the folding of a mini-protein and the study of materials crystallization.
K eywords Enhanced sampling | Collective variables | Machine learning Atomistic simulations and in particular molecular dynamics (MD), play an important role in several fields of science, serving as a virtual microscope that is of great help in the study of physical, chemical and biological processes. However, any time the free energy barrier between metastable states is large relative to the thermal energy k B T transitions between states become rare events, taking place on time scales too long to be simulated by standard methods [1]. This severely hampers the study of many important phenomena, such as phase transitions, chemical reactions, protein folding and ligand binding.
To alleviate this problem, different advanced sampling techniques have been developed. A large family of these methods relies on the identification of a small set of collective variables (CVs) s = s(R), that are functions of the system atomic coordinates R. In all these approaches an external bias potential V (s(R)) is added to the system in order to enhance the s(R) fluctuations [2]. If the CVs are able to activate the slowest degrees of freedom involved in the stateto-state transitions this procedure results in an enhanced sampling of the transition state. This in turn leads to an increase in the frequency with which rare events are sampled. Different ways of constructing appropriate bias potentials have been suggested. Examples are umbrella sampling [3,4], hyperdynamics [5], metadynamics [6], variational enhanced sampling [7,8], Gaussian mixture-based enhanced sampling [9] and on-the-fly probability enhanced sampling (OPES) [10].
Regardless of the method used, identifying appropriate collective variables is a crucial requisite for a successful enhanced sampling simulation [11,12]. Ideally one would choose the CVs solely on a physical and chemical basis. However, especially for complex systems, this can be rather cumbersome. For this reason, a number of data-driven approaches and signal analysis methods have been proposed for CV construction [13][14][15]. Some of these methods can be applied when the metastable states involved in the rare event are known beforehand [16][17][18], such as the folded and the unfolded states of a peptide or the reactants and products of a reaction. A line of attack in these cases has been to collect a number of configurations from short unbiased MD runs in the different metastable states and use these data to train a supervised classification algorithm. The classifier is then used as a CV. Our group has also contributed to this literature and developed an approach named harmonic linear discriminant analysis [16,19] that derives from Fisher's linear discriminant analysis (LDA). Later we have further improved this method by applying a non-linear version of LDA (Deep-LDA) [20]. The greater flexibility provided by the neural network architectures is of great help in dealing with complex problems [21,22]. These methods have proven to be successful in spite of the fact that they do not necessitate prior knowledge of reaction paths or transition states.
Clearly, if we had access to the transition dynamics we could further improve the CV effectiveness by making use of this dynamical information. To this purpose, several methods have been suggested to extract CVs from reactive simulations in which the system translocates spontaneously from one metastable state to another. Among all these methods those based on the variational approach to conformational dynamics (VAC) [23][24][25][26][27][28][29][30][31][32][33], both in its linear and non linear versions, are of particular relevance here. In Ref. [34] it has been argued that the resulting variables are natural reaction coordinates since they a) perform a dimensionality reduction, b) are determined by the sampling dynamics and c) are maximally predictive of the system evolution. Furthermore, an interesting feature of these CVs is that they measure the progress along any pathway connecting the metastable states, rather than focusing on a single path [34]. Hence these variables can be of great help both to understand and to enhance MD simulations. However, a difficulty in using VAC-generated CVs is that it becomes superfluous to perform enhanced sampling if unbiased reactive trajectories are already available. Thus one is in a chicken-and-egg situation: to find good CVs one needs to collect unbiased state-to-state transitions, but to promote transitions good CVs are needed [35].
A solution to this conundrum may come from an iterative approach, in which the CVs are computed using data generated in a previous enhanced sampling simulation [36][37][38][39][40][41][42][43][44], even if this initial run is far from optimal. In our group we have followed this strategy and modified the VAC protocol to identify the slow modes from biased trajectories [29,45]. In this way we can identify the modes that hinder convergence and enhance their sampling.
Here we generalize the approach of Ref. [29] in two ways. First, we employ a non-linear variant of VAC, which greatly increases its variational flexibility. Second, we propose new strategies for the collection of the initial trajectories, such as sampling generalized ensembles rather than using trial CVs, and for making full use of the information gathered during the initial trajectory. We also employ OPES to construct the bias, which has several advantages over metadynamics and other methods. These improvements lead to a general procedure that is proven to be effective in the study of a variety of rare events.
We organize the structure of this paper as follows. First, we give a brief account of the VAC theory, highlighting the points that are most relevant to our work. Then, we discuss how we can use neural networks as trial functions for the variational principle and how to adapt it when starting from enhanced sampling simulations. We initially test our method on the didactically informative example of the alanine dipeptide and then move on to more substantial applications such as folding a small protein and studying a crystallization process.
Collective variables as eigenfunctions of the transfer operator
A molecular dynamics simulation can be seen as a dynamical process that takes a density distribution p t (R) at time t and evolves it towards the equilibrium Boltzmann one
µ (R) = e −βU (R) dR e −βU (R) .
Here β is the inverse temperature and U (R) the interaction potential. An analysis of sampling dynamics can be done by studying the properties of the transfer operator T τ . We assume the dynamics to be reversible, thus satisfying the detailed balance condition. In the following, we quote some of the properties of T τ and refer the interested reader to the literature for a more formal discussion [24,46].
The transfer operator is defined by its action on the deviation of the probability distribution p t (R) from its Boltzmann value µ(R) as measured by u t (R) = pt(R) µ(R) :
u t+τ (R) = T τ • u t (R) (1) = 1 µ (R) dR P (R t+τ = R|R t = R ) u t (R ) µ (R )(2)
The action of the transfer operator depends both on the equilibrium distribution µ and the transition probability P which is a property of the sampling dynamics. Thus, we will have different operators even when sampling the same equilibrium distribution, e.g. with standard or enhanced sampling molecular dynamics.
The transfer operator T τ is self-adjoint with respect to the Boltzmann measure. This implies that its eigenvalues {λ i } are real and that its eigenfunctions {Ψ i (R)} form an orthonormal basis:
T τ • Ψ i (R) = λ i Ψ i (R)(3)
where the orthonormality condition reads:
Ψ i , Ψ j µ = dR Ψ i (R) Ψ j (R) µ (R) = δ ij(4)
Furthermore, its eigenvalues are positive and bounded from above : λ 0 = 1 > λ 1 ≥ ... ≥ λ i ≥ ... In particular, the eigenfunction corresponding to the highest eigenvalue λ 0 = 1 is the function Ψ 0 = 1. This trivial solution correspond to the fact that the Boltzmann distribution is the fixed point of T τ . In fact, if we apply k times T τ to a generic density ψ t we find
ψ t+kτ (R) = T k τ • ψ t (R) = i Ψ i , ψ t µ λ k i Ψ i (R)(5)
from which we see that if we let k → ∞ only the contribution coming from the λ 0 = 1 eigenfunction survives. The eigenvalues can be reparametrized as λ i = e −τ /ti where t i is an implied timescale measuring the decay time of the i-th eigenfunction. Thus, the leading eigenvalues are to be associated with the longest implied timescales, meaning that their corresponding eigenfunctions have a slow relaxation towards the equilibrium. For this reason, the first eigenfunctions are good CV candidates since they describe the slow dynamical process that we need to accelerate.
Of course in a multidimensional system there is no chance of exactly diagonalizing the transfer operator. Nonetheless, we can use a variational approach akin to the Rayleigh-Ritz principle in quantum mechanics. In the present context the variational principle reads
λ i ≥ ψ i , T τ •ψ i µ ψ i ,ψ i µ = ψ i (R t )ψ i (R t+τ ) ψ i (R t )ψ i (R t ) =λ i(6)
whereλ i are lower bounds for the true eigenvalues andψ i are variational eigenfuctions satisfying the orthogonality condition:
ψ i ,ψ j µ = ψ i (R t )ψ i (R t ) = δ ij ∀j = 0, ..., i − 1.
The equality holds only whenψ i coincides with the exact eigenfunctions. In addition, we have used the property that the matrix elements of T τ can be written in terms of timecorrelation function [25]. Thus they can be straightforwardly computed from the sampling trajectories.
Time lagged independent component analysis
In order to solve the variational problem we have to choose a set of trial functions. As a first step we select a set of N d descriptors {d j (R)} and build the variational eigenfunctions ψ i (R) as a linear combination of them:
ψ i (R) = N d j=1 α ij d j (R)(7)
where the expansion coefficients are the variational parameters. This amounts to applying to the problem the timelagged indipendent component analysis (TICA) method [23,25,26], a signal analysis technique that, given a set of variables, aims at finding the linear combination for which their autocorrelation is maximal. By imposing the variational functions to have zero mean we ensure that they are orthogonal to the trivial ψ 0 (R) = 1 solution. As in quantum mechanics, finding the variational solution to a problem in which a trial wave function is expressed as a linear expansion leads to solving an eigenvalue problem [46]. Recalling that the matrix elements of the transfer operator can be expressed as time correlation functions, the generalized eigenvalue problem can be written as
C (τ ) α i =λ i C (0) α i(8)
where
C ij (τ ) = d i (R t ) d j (R t+τ ) C ij (0) = d i (R t ) d j (R t ) .(9)
A neural network ansatz for the basis functions
Instead of using a predetermined set of descriptors as basis functions, as done in TICA, we employ a neural network (NN) to learn the basis functions through a non-linear transformation of the descriptors in a lower dimensional space. In this way, we can exploit the flexibility of neural networks to drastically improve the variational power of ansatz functions, and at the same time extend the VAC method to the case of a large number of descriptors.
The architecture of the NN follows here the implementation of Ref. [33], where descriptors d(t) = d(R t ) and d(t + τ ) = d(R t+τ ) are fed one after the other into a fully-connected neural network parametrized by a set of parameters θ, obtaining as outputs the corresponding latent variables h θ (d(t)) and h θ (d(t + τ )), respectively (see Fig. 1). The average of the latent variables h θ is subtracted to obtain mean free descriptors. Then, these values are used to compute the time-lagged covariance matrices, from which the eigenvaluesλ i (θ) and the corresponding eigenfunctions are obtained as solution of Eq. 8. This information is used to optimize the parameters of the NN as to maximise the first D eigenvalues, by minimizing the following loss function with gradient descent methods:
L = − D i=1λ 2 i (θ)(10)
which corresponds also to the so-called VAMP-2 score [32].
This architecture is an end-to-end framework that takes as input a set of descriptors and returns as output the few TICA eigenfunctions of interest. Since these CVs are given by the combination of NN basis functions and the TICA method, we will refer to them in the following as Deep-TICA CVs.
Extending TICA to enhanced sampling simulations
Most of the developments and applications of VAC have been focused on the analysis of long unbiased MD runs. Our purpose is different since we want to identify the slowest dynamical processes from enhanced sampling simulations. The application of an external potential can be seen as an importance sampling technique that samples a modified probability distribution in which the transition rate is accelerated. To recover the equilibrium properties over the Boltzmann distribution one needs to perform the reweighting procedure [2,27]. When the bias is in a quasi-static regime, the expectation value of any operator O can be written as:
O (R) = O (R) e βV (s(R)) V e βV (s(R)) V(11)
where · V represents a time average in the biased simulation. Another way of looking at the reweighting procedure is to rewrite Eq. 11 as an ordinary time average in a time t scaled by the value of the bias potential:
O (R) = T 0 dt O (R t ) e βV (s(Rt))
T 0 dt e βV (s(Rt))
= 1 T T 0 dt O (R t )(12)
where we have performed a change of variable dt = e βV (s(Rt)) dt, and T = T 0 dt e βV (s(Rt)) represents the total scaled time. This means that we can interpret the enhanced sampling simulation as a dynamics which samples the Boltzmann distribution on the t time scale. Thus the VAC procedure can be straightforwardly applied provided that the correlation functions of Eq. 8 are calculated in t time [29,45]. Additional details are reported in the materials and methods section.
It is important to note that, even though the enhanced sampling simulation in t time asymptotically samples the Boltzmann distribution, its sampling speed does differ from the unbiased one. As a result, the spectrum of the transfer operator will be different, since the degrees of freedom that have already been accelerated in the initial simulation will have a smaller contribution. In fact, our group has previously shown that a successful enhanced sampling simulation leads to small leading eigenvalues of the transfer operator as the slow modes are accelerated [29].
Among the many enhanced sampling methods that allow the reweighting procedure of eqs. 11-12 we choose here to use OPES for a variety of reasons that will be discussed later, but first, we sketch the main features of this method. OPES [10,47] first builds an on-the-fly estimate of the equilibrium probability distribution P (s) and the bias is then chosen to drive the system towards a desired target distribution: p tg (s).
V (s) = − 1 β log p tg (s) P (s) .(13)
At convergence, the free energy surface (FES) as a function of s is computed from F (s) = −k B T log P (s). An appropriate choice of the target distribution allows one to sample a variety of ensembles including the well-tempered P (s) 1/γ [48] with γ > 1, uniform distribution, up to generalized ensembles [47]. The latter possibility is used here to sample the multithermal ensemble, where configurations relevant to a preassigned range of temperatures are sampled. This is similar to replica-exchange methods, but does not involve abrupt exchange of configurations. In the OPES version, the multithermal ensemble is sampled by using the potential energy U (R) as collective variable.
One important factor that made us choose OPES is that it reaches the quasi-static regime more rapidly than metadynamics [10]. Thus, the bias varies more smoothly and the noise in the calculation of scaled time t correlation functions is reduced.
A recommended strategy
We outline here the key steps of our recommended procedure, see also Fig This procedure can be iterated but usually at this stage the FES is well converged.
In the examples presented in this work, we enhance the fluctuations of the eigenfunction associated with the slowest mode (Deep-TICA 1) while using the others for analysis purposes.
Unlike the approaches developed earlier [29,45], in step (3) we also add the bias V * (s 0 ) to the Hamiltonian. In this way we take into account the fact that the slow modes computed in step (2) reflect the rate of convergence to the Boltzmann distribution sampled by reweighting (Eq. 11) from the trajectories generated using the Hamiltonian H + V * (s 0 ).
Results and discussion
Alanine dipeptide from multicanonical simulations and CV-biased dynamics
A simple yet informative test of enhanced sampling methods is offered by the study of the conformational equilibrium of alanine dipeptide in vacuum. At room temperature, this small peptide exhibits two metastable states, namely the more stable C 7eq composed of two substates and the less populated C 7ax . The conformational transition between the two states is well described by the torsional angles φ and ψ, with the former being close to an ideal CV. However, since our scope is mostly didactical we shall on purpose ignore this information and build efficient CVs using as descriptors all the heavy atom interatomic distances. To illustrate the flexibility and power of the method, we consider here two scenarios that differ in the way the initial reactive trajectories are generated.
We start with illustrating the first strategy which consists of using OPES to sample the multithermal ensemble. As can be seen from Fig. 2a this procedure is not very efficient and promotes only a small number of transitions, and thus the free energy estimate is noisy (Fig. S4). Rather remarkably, this limited information is enough to extract the slow modes of the system using Deep TICA and obtain CVs that are efficient in promoting sampling. We find the training to be robust concerning the choice of lag-time (Fig. S1), and also to the number of configurations used (Fig. S2). The leading Deep-TICA 1 variable is associated with the transition between C 7eq and C 7ax , while the second describes the transition between the C 7eq substates (Fig. S6).
Subsequently, a new OPES multithermal simulation is performed biasing in addition also Deep-TICA 1. The first remarkable result is a two hundred-fold increase in the number of transitions per unit time as compared to the initial simulation (Fig. 2b). The system immediately reaches a quasi-static regime in which the average interval between inter-state transitions is about 25ps, to be compared to the pure multithermal simulation in which the same rate was as large as 3 ns. As a consequence of this speed-up, the free energy difference between metastable states converges in just 1 ns to the reference value within 0.1 k B T . A quantitative analysis of the convergence of the simulations can be found in the supporting information (SI, Fig. S4), together with a comparison of the simulations with and without the static bias potential V * (s 0 ) and the free energy surface as a function of the two Deep-TICA CVs (Fig. S3).
The time to convergence is comparable to what one finds when using as CVs the physically informed dihedral angles φ and ψ [10], but here it is the result of a procedure that is generally applicable and does not require any previous understanding of the system. There is also a clear improvement with respect to discriminant-based CVs that use the same set of descriptors [20].
Notably, the Deep-TICA CV promotes sampling along the two different pathways connecting C 7eq and C 7ax (Fig. 2c), as it measures the sampling progress along all transition pathways [34]. Combined with the OPES ability to set an upper bound to the value of the added bias, this results in focusing the sampling on the most interesting parts of the FES which are the minima and the transition region.
The above-described procedure was based on the ability of the initial multithermal simulation to induce transitions between the local minima. However, in many cases multithermal simulations are not able to induce even a single transition and the use of a CV to generate a reactive trajectory is called for. Thus we found instructive to exemplify the performance of Deep-TICA when the initial biased simulation is driven by a CV. Again we want to challenge the method and we choose the angle ψ as starting CV. A cursory look at the alanine dipetide FES of Fig. 2c makes one realize that ψ is a very poor CV being almost perpendicular to the direction of the most likely transition paths. For this reason, it is an exemplary case of a CV that should not be used [49]. The low quality of the CV is reflected in the fact that we need to simulate the system for 5 microseconds to observe a handful of transitions (Fig. 3a). As before, we feed these scant data to the Deep-TICA machinery and compute the highest eigenfunctions of the transfer operator. When we perform a new OPES calculation biasing Deep-TICA 1, the simulation immediately reaches a diffusive regime similarly to the previous example (Fig. 3b), which allows converging the free energy in a very short timescale of 1 ns (Fig. S5). This is possibly an extreme example but shows that remarkable speedups can be attained when the slow modes are correctly identified and their sampling accelerated.
In real life, one tries not to use CVs as bad as ψ but the use of suboptimal CVs is far from rare. In this respect, the Deep-TICA method holds the promise of remedying a poor initial CV choice.
As discussed in the previous sections, the eigenfunctions that we obtain describe the slowly converging modes of the sampling dynamics performed at step (1). In the SI we investigated this point by comparing the CVs extracted from the multicanonical simulation and those obtained from the ψ-biased dynamics, highlighting the effect of initial simulation (Fig. S6). Furthermore, it should be noted that for these cases in which the quality of the initial CV s 0 is poor the advantage of using the bias V * (s 0 ) from the previous simulation is less significant but still non-negligible (see Figs. S4-S5).
A blind approach to chignolin folding
Chignolin is one of the smallest proteins that can be folded into a stable structure. Here we focus on its variant CLN025, which has been extensively studied using molecular simulations by performing long simulations on the Anton supercomputer [50] and using enhanced sampling techniques [51][52][53][54].
Once again we pretend that we are unaware of the progress made in the understanding of Chignolin behavior and follow the same blind approach pursued in the first alanine dipeptide simulation reported above. That is, in the exploratory phase we perform an OPES multithermal sampling, this time boosted by the use of multiple replicas (Fig. 4a). This leads to observing a few folding-unfolding events. Using these trajectories we construct a Deep TICA CV using as descriptors all the 4278 interatomic distances between heavy atoms. Since enhancing the sampling of a CV that depends on thousands of descriptors would have been computationally inefficient, we decided to reduce their number by selecting the most relevant one for the leading CV via a sensitivity analysis [20]. In this way, we selected 210 descriptors (which are reported also in Fig. 1). We retrain the NN using this reduced set and find out that the leading eigenvalue is only decreased by just 0.5 %, thanks to the variational flexibility of the NN. Interestingly, the selected distances involve both backbone and side chain atoms, suggesting that also the latter have a significant role in the folding process.
As expected, the first CV (Deep-TICA 1) describes the folded to unfolded transition that is the slowest mode of the system. Instead, the second one (Deep-TICA 2) characterizes the fine structure of the folded state, as we will discuss later. This can be seen in Fig. 4b, where we colored the points sampled in the initial trajectory with the value of the backbone C α root-mean-square deviation (RMSD). It should be noted that we find no evidence of stable misfolded states along the dominant folding collective variables, in agreement with simulations using the same force-field [50,53,56].
Performing a new simulation with a bias potential along Deep-TICA 1 results in an enhanced sampling of the transition region (Fig. S8), with a 20-fold increase in the rate of folding events compared to the multithermal simulation (Fig. 4c). Due to the use of the multithermal approach all the free energy profiles in the chosen range of temperatures can be calculated with great accuracy (Fig. S10-S12). Indeed, the average statistical error calculated with a weighted blockaverage technique [47] is about 0.5 kJ/mol, improving on classifier-based approaches applied to the same system [54].
In particular, we find an excellent agreement with an unbiased 106 µs reference trajectory [50] at T = 340K (Fig. S9).
Note that a study of the protein behavior at lower temperatures using standard MD would have been significantly more difficult to perform.
In Fig. 5 we report the FES relative to T = 340K plotted as a function of the two leading eigenfunctions, together with their projections along with Deep-TICA CVs. We find two major states divided by a significant free energy barrier, which correspond to the unfolded and folded basins. However, the latter exhibits a fine substructure that can be traced back to the Threonine sidechains (THR6 and THR8) occupying different dihedral conformations (Fig. S13). At 340K these states can interconvert on the timescale of nanoseconds, but, of course, this time is significantly slower at lower temperatures (see Fig. S11). Notably, the most likely conformation is stabilized by the presence of a hydrogen bond between the two THR sidechains (Table S3), as previously observed in a structural analysis study for wild-type chignolin [57]. This is remarkable because it was discovered without any prior knowledge of neither structural conformation of the system nor the dynamics of folding, and it suggests that going beyond backbone-only structural descriptors is necessary to obtain an accurate representation of the folding dynamics.
Improved data-driven description of silicon crystallization
Silicon crystallization is a first-order phase transition hindered by a large free energy barrier. This implies that in step (1) of the Deep-TICA procedure one has to resort to CV-based simulations to harness reactive trajectories. Our group has previously investigated the application of TICA to simulate Na and Al crystallization [58]. The study of Si
solid line), confronted with the reference value obtained from a long unbiased MD trajectory at 340K [50] (dotted line). Note that the projection of Deep-TICA 2 is obtained by integrating only the region of space with Deep-TICA 1 > 0.65 (marked by a dotted line in the central panel), to highlight the barriers between the folded metastable states.
crystallization is however more difficult, due to the directional nature of the bonds and the ease with which defective and glassy structures can arise.
To address this problem, we make use of a recently developed set of descriptors that have proven to be useful in a machine-learning context [22]. These are the peaks of the three-dimensional structure factor of a crystal that is commensurate with the MD simulation box. Compared with spherically averaged structure factors used in Ref. [58,59], these descriptors have the advantage that they facilitate the formation of crystal structures aligned with the axes of the box. Since they measure the presence of long-range order in the system they are a natural choice in the study of crystallization.
In Ref. [22] these peaks have been combined into a CV using the Deep-LDA classification method. The question that we address here is whether we can improve upon the Deep-LDA description and obtain a CV that incorporates dynamical information. In order to make a fair comparison between Deep-LDA and Deep-TICA we use the same set of descriptors chosen with a well-defined universal concept. That is, we use in both cases the first 95 S(k) Bragg peaks with modulus k ≤ 7 A −1 (Fig. S15), which amounts as fixing the CV spatial resolution.
The Deep-LDA CV is trained using short MD simulations in the liquid and the cubic diamond states. Afterward, an
Figure 6: Comparison between a Deep-LDA driven simulation (left) and the one based on the present Deep-TICA approach (right). In the top row we report the time evolution of (a) Deep-LDA CV in the initial simulation and of (b) Deep-TICA CV in the improved one. In both panels the points are colored according to the fraction of diamond-like atoms in the system, computed as in Ref. [61]. Grey shaded lines indicate the values of the two CVs in unbiased simulations of the liquid (bottom lines) and solid (top lines). Panels (c) and (d) report the correlation between the two data-driven CVs and the fraction of diamond-like atoms. White circles denote the mean values of the two CVs in the liquid and solid states, while the dotted grey line interpolates between them. In panel (d) we report also a few snapshots of the crystallization process made with OVITO [62].
OPES simulation is performed, which promotes a few crystallization and melting events, though the system struggles to find its way to the liquid state (Fig. 6a). From this trajectory a Deep-TICA CV is extracted and subsequently used to enhance sampling together with the final static bias V * (s 0 ) (Fig. 6b). Similar to what happened in the previous examples, this procedure leads to an increase in the number of transitions between the solid and liquid states, which allows converging the free energy estimate already after 20 ns. The statistical uncertainty on the free energy difference is reduced compared to the Deep-LDA simulation (Fig. S16). Furthermore, the free energy difference between the two states is close to zero at T=1700K, in excellent agreement with the melting point of the interatomic potential [59,60].
As a final comment, we argue that the different degree of sampling efficiency between the two data-driven CVs has to be rooted in their different training objectives. Being trained as a classifier, Deep-LDA very accurately discriminates between the solid and the liquid phase, but it has no information about the transition region which connects them. Consequently, in almost the entire range of values spanned by the variable during the simulation, the system is either completely in the crystalline or the liquid phase (Fig. 6c). The Deep-TICA CV, besides classifying the states, reflects also the transition dynamics. In fact, in Fig. 6d we see that it describes more smoothly the transition between the two phases, as Deep-TICA is linearly correlated with the number of crystalline atoms, which is a relevant quantity in the classical nucleation theory framework [63].
Conclusions
The extension of the variational principle of conformation dynamics to enhanced sampling data [29] represents a promising way to address the chicken-and-egg dilemma intrinsic to the determination of collective variables. Here, we leverage the flexibility of neural networks and recent developments in advanced sampling techniques to construct a general and robust protocol. The Deep-TICA method allows us to analyze a biased simulation trajectory, extract the slow modes which hinder its convergence, and subsequently accelerate them. This can be used to extract CVs from generalized ensemble simulations and to complement approximate CVs constructed based on physical considerations or in a datadriven manner. Besides improving sampling, this method provides us with atomistic details on the rare events dynamics. Remarkably, our work underlines the fact that even a partial information about the transition pathways does go a long way to solve the rare event problem. In fact, the test on the alanine dipeptide benchmark shows that the procedure is applicable even when starting from a very poor initial enhanced sampling simulation. This promises to be of great help in the study of realistic systems, where the identification of appropriate CVs is challenging. Application of the method to the more complex examples of chignolin folding and material crystallization illustrates how this acceleration allows the FES to be reconstructed with high accuracy, with no need of physical or chemical insight into the transition dynamics. We are confident that our approach can be applied to even more complex systems and that it can be of great help to the broad molecular simulation community.
Materials and methods
Time-lagged covariance matrices Given an enhanced sampling simulation we first rescale the time according to Eq. 12. We then search for pairs of configurations distant a lag-time τ in time t . Due to time reweighting, the value of τ cannot be interpreted as a physical time. However, we found consistent results for a range of lag-time values such that all desired eigenvalues did not decay to zero (Fig. S1). Note that, when Eq. 12 is discretized, time intervals become unevenly spaced in t and the calculation of the time-lagged covariance matrices requires some care. To deal with this numerical issue we resort to the procedure proposed in [45]. These pairs of configurations are saved in a dataset and later used for the NN training. Furthermore, it should be noted that while in principle the two correlation matrices are symmetric, this condition might not be satisfied when estimating them from MD simulations due to limited sampling. Here we symmetrize the matrices to enforce detailed balance as C sym ij = (Cij + Cji)/2. This choice is the simplest, although it introduces a bias [27].
Deep-TICA CVs training Deep-TICA CVs are trained using the machine learning library PyTorch [64]. As previously done for Deep-LDA and other non-linear VAC methods [33], we apply Cholesky decomposition to C(0) to convert Eq. 8 into a standard eigenvalue problem. This allows to back-propagate the gradients through the eigenvalue problem by using the automatic differentiation feature of the ML libraries. We use a feed-forward neural network composed by 2 layers and the hyperbolic tangent as activation function. The NN parameters is optimized using ADAM with a learning rate of 1e-3. To avoid overfitting, we split the dataset in training/validation and apply early stopping with a patience of 10 epochs. Furthermore, the inputs are scaled to have zero mean and variance equal to one. Also the Deep-TICA CVs are scaled in order for their range of values to be between -1 and 1. The normalization factors are calculated over the training set and saved into the model for inference. Once the training is performed, the model is serialized so that it can be used on-the-fly in a molecular dynamics simulation.
PLUMED-Pytorch interface
To use the transfer operator eigenfunctions as CVs for enhanced sampling simulations, we use a modified version of the open-source PLUMED2 [65] plug-in which we interfaced with the LibTorch C++ library, as in Ref. [20]. The model trained in Python is loaded by PLUMED to evaluate CV values and derivatives with respect to descriptors for new configurations explored during the simulation and apply a bias potential along them. Input files to run the simulations will be deposited in the PLUMED-NEST [66] repository.
Alanine dipeptide simulations Alanine dipeptide (ACE-
ALA-NME) simulations are carried out using GROMACS [67] patched with PLUMED. We use the Amber99-SB [68] force field with a time step of 2 fs. The NVT ensemble is sampled using the velocity rescaling thermostat [69] with a temperature of 300K. For the OPES multithermal simulation we sample a range of temperatures from 300K to 600K, updating the bias every PACE=500 steps. We run a 50 ns simulation, and use the last 35 ns where the bias is in a quasi static regime. We then look for configurations separated by a lag time of 0.1. The input descriptors of the Deep-TICA CVs are the 45 distances between the heavy atoms, and the following NN architecture is used: 45-30(tanh)-30(tanh)-3. We optimize the first 2 eigenvalues in the loss function. After training the CVs, a new OPES simulation is performed in the multi-thermal ensemble with the same parameters as before, combined with the multi-umbrellas OPES ensemble along Deep-TICA 1 CV with the parameters SIGMA=0.1 and BARRIER=40. Since in the simulation driven by Deep-TICA 1 the time needed to converge the multithermal bias is very short, we did not use the static static bias from the previous simulation but we optimized it together with the bias along the TICA CV.
The second example involves the OPES simulation in which the dihedral angle ψ is used as the CV. The parameters of OPES are PACE=500, SIGMA=0.15 and BARRIER=40. The first 500 ns out of a total simulation length of 5 µs are discarded while the remaining are used to compute time correlation functions with a lag-time equal to 5. The NN details are the same as in the multithermal example. Next, an OPES simulation is performed using Deep-TICA 1 as the CV, with parameters PACE=500, SIGMA=0.025 and BARRIER=30, along with the static bias V * (s) from the previous simulation.
Chignolin simulations
Simulations of the CLN025 peptide (sequence TYR-TYR-ASP-PRO-GLU-THR-GLY-THR-TRP-TYR) are performed using GROMACS patched with PLUMED. Computational setup is chosen to make a direct comparison with Ref. [50]. CHARMM22* force field [70] and TIP3P water model [71] are used, the integration timestep is 2 fs, and the target temperature of the thermostat is set to 340K. ASP,GLU residues as well as the N-and C-terminal amino acids are simulated in their charged states. Simulation box contains 1906 water molecules, together with two sodium ions that neutralize the system. The linear constraint solver (LINCS) algorithm is applied to every bond involving H atoms and electrostatic interactions are computed via the particle mesh Ewald scheme, with a cutoff of 1 nm for all non-bonded interactions.
The initial simulation is performed with OPES to simulate the multithermal ensemble in a range of temperatures from 270K to 700K. We simulate 8 replicas sharing the same bias potential to harvest more transitions. The simulation time is 250 ns, of which the first 50 ns are not used for the Deep-TICA training. A lag-time equal to 5 is used. In order to compute the scaled-time correlation functions we reweight at the simulation temperature of 340K. Note that when starting from a multithermal simulation, one could reweight also at different temperatures and extract the associated eigenfunctions. Initially, a larger NN is trained using as input all the heavy atoms distances, with an architecture 4278-256(tanh)-256(tanh)-5. After performing an analysis of the features' relevance, based on the derivatives of the leading eigenfunction with respect to the inputs [20], a smaller NN is trained using a reduced set of 210 distances and architecture 210-50(tanh)-50(tanh)-5. The list of the distances used is available in the PLUMED-NEST repository. We observe very similar results in terms of the extracted eigenvalue when using between 100 and 300 inputs. The number of optimized eigenvalues in the loss function is equal to 2. Finally, we enhance the fluctuations of the Deep-TICA 1 CV via an OPES simulation with parameters PACE=500, SIGMA=0.1 and BARRIER=30, together with the static multithermal potential from the initial simulation.
Silicon simulations
Silicon simulations are carried out using LAMMPS [72] patched with PLUMED, using the Stillinger-Weber interatomic potential [60]. A 3x3x3 supercell (216 atoms) is simulated in the NPT ensemble with a timestep of 2 fs. A thermostat with a target temperature of 1700K is used with a relaxation time of 100 fs, while the values for the barostat are 1 atm and 1 ps.
First, two 5 ns long simulations of standard MD in the solid and liquid states are performed. The values of the 95 threedimensional structure factor peaks in these configurations are computed and this information is used to construct a Deep-LDA CV, using a two-layer NN with 30 nodes per layer. A 50 ns OPES simulation biasing this variable, with PACE=500, adaptive sigma, and BARRIER=1000 is performed. The first 25 ns are not used for NN training. A lag-time of 0.5 is used. The input descriptors and architecture of the NN are the same as those used for Deep-LDA. Only the principal eigenvalue is optimized in the loss function. We then run a new OPES simulation biasing the Deep-TICA CV using the same parameters as in the initial simulation, along with the static bias potential V * (s0). The fraction of diamond-like atoms is computed in PLUMED with the Environment Similarity CV, with parameters SIGMA=0.4 LAT-TICE_CONSTANTS=5.43 MORE_THAN={R_0=0.5 NN=12 MM=24}. Multithermal simulation -Convergence and 1D free energy profiles Figure S4: Convergence analysis for the exploratory multithermal simulation (blue, first row), for the one biasing the Deep-TICA 1 CV in the multithermal ensemble (red, second row) and for an additional simulation in which only a bias potential along Deep-TICA 1 is applied with OPES (red, third row). In the first column we show the free energy difference ∆F 3 between C 7eq and C 7ax as a function of time. In the second and third column we report the free energy profile along the φ dihedral angle and the Deep-TICA 1 CV, respectively. Blue and red shaded areas indicate the statistical uncertainties from a weighted block average technique [47]. Dotted grey lines represents the reference values, obtained from two different simulations: Ref1 is obtained by performing a 100 ns OPES simulation biasing φ − ψ, while Ref2 correspond to the OPES-MultiT+Deep-TICA 1 CV extended for 50ns. In both simulations biasing Deep-TICA 1 the FES is reconstructed with very high accuracy already after a few nanoseconds.
SUPPORTING INFORMATION
A. ALANINE DIPEPTIDE SIMULATIONS
Simulation
Average Table S1: Average transition rates between C 7eq and C 7ax , obtained by dividing the simulation time by the number of transitions. A transition is recorded whenever the running average (on a 2 ps window) of the Deep-TICA 1 CV goes below 0.5 or above 0.5. Only the part of the simulations where the bias potential is quasi-static is used, to avoid counting transitions promoted by a rapidly changing external potential. It should be noted that each simulation samples a different target distribution, and this could affect the transition rate. ψ-biased simulation -Convergence and 1D free energy profiles Figure S5: Convergence analysis for the exploratory OPES simulation biasing ψ (blue, first row), for the one biasing the Deep-TICA 1 CV together with the static bias V * (ψ) (red, second row) and for an additional simulation in which only a bias potential along Deep-TICA 1 is applied with OPES (red, third row). In the first column we show the free energy difference between C 7eq and C 7ax as a function of time. In the second and third column we report the free energy profile along the φ dihedral angle and the Table S2: Average transition rates between C 7eq and C 7ax for the enhanced sampling simulations. The values are obtained by dividing the simulation time by the number of transitions. A transition is recorded every time the running average (on a 2 ps window) of the Deep-TICA 1 CV goes below 0.5 or above 0.5. Only the part of the simulations where the bias potential is quasi-static is used, to avoid counting transitions promoted by a rapidly changing external potential.
Deep-TICA CVs comparison Figure S6: Deep-TICA CVs isolines in the Ramachandran plane. The isolines have been computed from a 2D weighted histogram where the weights are the Deep-TICA CVs of the multithermal simulation (first column) and of the ψ-based OPES simulation (second column). In order to have a uniform sampling of this space, the configurations for the histogram are taken from a OPES simulation biasing φ-ψ with a flat target distribution. The two rows report the isolines of Deep-TICA 1 and 2, respectively. In addition we overlaid the isolines of the FES as a function of the Ramachandran angles, spaced every 2 k B T (white and black lines). In particular, solid black lines highlight the minima of the FES, while dashed white lines describe the higher energy regions. Only the regions where the FES ≤ 20 k B T are shown. It is noteworthy that in both the multicanonical and the ψ-biased dynamics the leading CV is associated to the transition between the C 7eq and C 7ax states, while the second one describes the transition between the two local minima separated by a much smaller free energy barrier within C 7eq . However, the isolines of the CVs extracted from the ψ-biased dynamics has no or little dependence on the ψ angle, as the latter has made a fast degree of freedom in the original simulation. , and the energy is shifted such that the zero value corresponds to the average potential energy at 340K. The points are colored with the value of the C a lpha-RMSD. In the exploratory simulation, transitions between the folded and unfolded states are scarce especially at low values of the potential energy, which correspond to low temperature configurations. We observe that biasing also Deep-TICA 1 leads to increased sampling of the transition region in a uniform manner. This allows for more efficient recovery of the expectation values of observables at thermodynamic conditions further away from the simulated one. with all the heavy atoms distances as input descriptors rather than on the limited subset. On one side, the agreement between the Deep-TICA simulation and the unbiased reference one is striking, providing evidence for the existence of these distinct folded metastable states. On the other side, the comparison with the CVs trained using all the distances confirms that the structure of the metastable states is not an artifact of using a reduced set of descriptors.
B.CHIGNOLIN SIMULATIONS
Replicas trajectories
Free energy surface -temperature dependence Figure S10: Temperature dependence of the FES as a function of the number of H-bonds between backbone atoms 4 and the end-to-end distance 5 . All the free energies are extracted from the single OPES Multi-T* + Deep-TICA 1 simulation. 5 The number of hydrogen bonds is calculated in PLUMED in a continuous way by summing the switching functions applied to the pairwise distances with parameters R0 = 4Å, N=6, M=8. The pair of atoms considered are: ASP3 N -THR8 O, GLY7 N -ASP3 O, TYR10 N -TYR1 O. 5 The end-to-end distance is computed by measuring the distance between the CA atoms of the two TYR terminal residues.
Characterization of the folded states
To extract the simulations corresponding to each folded basin we performed a weighted k-means clustering as in Fig. 4 in the manuscript, with weights w = e βV (s) . Furthermore, a cutoff on the value of Deep-TICA 1>0.65 is used. Since none of the typical backbone descriptors (such as the C α -RMSD, the gyration radius or the end-to-end distance) is able to discriminate between the three folded states, we performed an hydrogen bonds analysis including also interactions with the side chains (Table S3). Table S3: Hydrogen bond analysis, performed using the hydrogen bonds command of VMD [55] with the same criterion used in Ref. [57], namely a cutoff distance = 3.3 Å, an angle cutoff = 35°and a presence in at least 30% of the configurations for each state.
States
All these states have a common set of H-bonds, while differing in the presence or absence of specific bonds between the side chains. In particular, the most populated state (1) is characterized by the presence of an H-bond between the alcohol oxygens of the two threonine (THR) amminoacids. This implies that the interaction between the two sidechains have a stabilizing effect on the folded state, as it was observed in a structural analysis study for the wild-type chignolin protein [57]. We can further characterize these states by looking at their correlations with the dihedral angles of the THR sidechains, from which we learn that state 2 can be further decomposed in two distinct states, characterized by different equilibrium values of the torsional angles. The emerging picture implies that there is not a single folded state, but rather an ensemble of folded structures.
C. SILICON SIMULATIONS
Structure factor-based descriptors Figure S15: Structure factor peaks for cubic diamond crystal structure. Bars correspond to the average value of three dimensional structure factor peaks as a function of |k| (right y-scale). We also report the isotropic structure factor calculated with the Debye formula (left y-scale) for comparison. At variance with Debye S(k), the 3D S(k) peaks are not rotationally invariant, but measure the presence of a crystal structure commensurate with the box size and aligned with the box axis. Selected peaks for the Deep-TICA training are highlighted in red, and the associated Miller indices are reported above the figure.
Free energy profiles and free energy difference versus time Figure S16: Free energy profiles for the Deep-LDA simulation (top row) and the Deep-TICA simulation (bottom row). In the left column we report the FES as a function of Deep-LDA and Deep-TICA CV, respectively. Shaded areas correspond to the statistical uncertainties estimated with a weighted block average. In the right column we report the free energy difference versus time, estimated as in Fig. S4 .
Figure 1 :
1(top) Deep-TICA protocol used in this paper. On the left the mini-protein chignolin is shown, with lines denoting pairwise distances used as descriptors. (bottom) Neural network architecture and optimization details of Deep-TICA CVs.
Exploration. Harness a number of reactive events using a CV-based OPES simulation with a trial CV s 0 , multithermal sampling (in such a case s 0 = U (R)), or even a combination of the two. Store the final bias potential V * (s 0 ) of this initial simulation.2. CV construction. Select the descriptors to be used as inputs of the NN. Train the Deep-TICA CVs using the trajectories generated in step (1) by calculating the correlation functions in t time. 3. Sampling. Perform an OPES simulation using the leading Deep-TICA eigenfunction as CV on the Hamiltonian modified by the addition of the bias potential V * (s 0 ).
Figure 2 :
2Deep-TICA procedure applied to a multithermal simulation of alanine dipeptide. (a) Time evolution of the φ angle in the exploratory OPES multithermal simulation, colored according to the potential energy. (b) Time evolution of the same angle for the simulation in which also the bias on Deep-TICA 1 is added, colored with the value of the latter variable. It can be seen that the system immediately reaches a diffusive behaviour. (c) Ramachandran plot of the configurations explored in the Deep-TICA simulation, colored with the average value of Deep-TICA 1. Grey lines denotes the isolines of the free energy surface, spaced every 2 kBT . Note that the sampling is focused on the minima and the transition regions that connect them.
Figure 3 :
3(a) Time evolution of the ψ angle in the exploratory simulation driven by ψ. The points are colored with the values of the φ angle. (b) Time evolution of the φ angle in the final Deep-TICA simulation, colored with the value of Deep-TICA 1. This results in a diffusive simulation similar to the previous example, which is even more impressive here given the poor quality of the exploratory sampling.
Figure 4 :
4Deep-TICA procedure applied to chignolin folding. (a) Time evolution of the Cα RMSD for one replica during the initial multithermal run. The points are colored according to their potential energy value. Low energy values reflect the fact that configurations relevant at lower temperatures are sampled. (b) Scatter plot of the two leading Deep-TICA CVs in the exploratory simulation. Points are colored according to the average Cα RMSD values. A weighted k-means clustering identifies four clusters whose centers are denoted by a white X. The pale background colors reflect how space is partitioned by the clustering algorithm. Snapshot of chignolin in the folded (high values of Deep-TICA 1) and unfolded (low values) states are also shown, realized with VMD [55]. (c) Time evolution of Cα RMSD for a replica in the multithermal simulation biasing also Deep-TICA 1, colored with the value of the latter variable. Time evolution for the other replicas is reported in Fig. S7.
Figure 5 :
5Free energy surface of chignolin at T=340K as a function of the two leading Deep-TICA CVs. Above and to the right are shown the projections of the FES along the corresponding axis (
Figure S1 :
S1Deep-TICA CVs training versus lag-time for the alanine dipeptide multithermal example. Columns correspond to different values of the lag-time used. Top row: average value of the two leading eigenvalues (bars) and standard deviation over 5 repeated trainings (black lines). Mid and bottom rows: training points projected on the φ − ψ plane, colored according to the related eigenfunctions Deep-TICA 1 and 2. The first eigenfunction is always consistent across the range of lag times studied, while the second eigenfunctions starts to lose signal when the associated eigenvalue becomes too small. This suggests to choose the value of the lag-time such that all the desired eigenvalues have not decayed to zero.
Figure S2 :
S2Deep-TICA CVs training versus number of configurations used for the training for the alanine dipeptide multithermal example. The configurations are extracted every 1 ps after 15 ns. Columns correspond to different values of the lag-time used. Top row: average value of the two leading eigenvalues (bars) and standard deviation over 5 repeated trainings (black lines). Mid and bottom rows: training points projected on the φ − ψ plane, colored according to the related eigenfunctions Deep-TICA 1 and 2. Remarkably, already after a couple of transitions the Deep-TICA procedure is able to extract a very good approximation of the eigenfunctions. Multithermal simulation -2D free energy Figure S3: Free energy profile of alanine dipeptide as a function of Deep-TICA CVs from the multithermal simulation (central panel) and 1D projections on the CVs (top and left panels). The leading Deep-TICA CV describes the conformational transition between C 7eq and C 7ax , while the second CV highlights the presence of the two substates within C 7eq .
3
The free energy difference between the two states is defined as follows:∆F = 1 β log A e −βF (s) ds B e −βF (s) dswhere s = φ is the dihedral angle and F (s) is the free energy profile. The two integrals are computed over the regions corresponding to A = C7eq and B = C7ax, in this case φ < 0 and φ > 0.
Figure S7 :Figure S8 :
S7S8C α -RMSD time evolution for the OPES Multi-T simulation (top) and for the OPES Multi-T* + Deep-TICA 1 run (bottom). Each color represent a replica of the system sharing the same bias potential. The simulation time is shifted by 350 ns times the replica id, and each replica is divided by the others by a vertical dashed line. In the case of the exploratory multithermal simulation the first 50 ns of each replica are made transparent to underline they are not used for the Deep-TICA training. A horizontal dotted line at RMSD=0.15 is added to identify folding-unfolding events. Improving sampling of the transition region Scatter plot of the potential energy versus Deep-TICA 1 for the OPES Multi-T simulation (left) and for the OPES Multi-T* + Deep-TICA 1 run (right). The potential energy range corresponds to the temperature range [280K, 500K]
Figure S9 :
S9Comparison of the 2D free energies as a function of the Deep-TICA CVs at T=340K. (left) Reference FES computed from the long unbiased run performed by Ref. [50] (center) FES from OPES Multi-T* + Deep-TICA 1 simulation. (right) FES estimated from the same Deep-TICA simulation but projected on the Deep-TICA CVs trained
Figure S11 :
S11Temperature dependence of the FES as a function of the Deep-TICA CVs, estimated from the OPES Multi-T* + Deep-TICA 1 simulation.
Figure S12 :
S12Temperature dependence of the FES as a function of Deep-TICA 1, which describes the foldingunfolding transition, from the OPES Multi-T* + Deep-TICA 1 simulation. Shaded areas indicate the statistical uncertainty obtained with a weighted block average For the 340K temperature we report also the FES obtained from the reference unbiased simulation (grey dashed line).
Figure S13 :
S13Scatter plot of the sidechain diedhral angles χ 1 (THR 6) and χ 1 (THR 8) for the three folded states. The isolines of the FES at T=340K are also reported (solid lines, the color denotes the FES value).
Figure S14 :
S14Relative population of the folded states as a function of temperature, estimated by integrating the probability ditribution P (s) = e −βF (s) in each basin where s=Deep-TICA 2 and F(s) is obtained as in Fig. 5 by integration of the 2D FES with the condition Deep-TICA 1>0.65.
Deep-TICA 1 CV, respectively. Blue and red shaded areas indicate the statistical uncertainties from a weighted block average technique. Dotted grey lines represents the reference values, obtained from two different simulations: Ref1 is obtained by performing a 100 ns OPES simulation biasing φ − ψ, while Ref2 correspond to the OPES φ*+Deep-TICA 1 performed for 50ns.Simulation
Average transition rate [ps]
OPES φ
455000
OPES φ* + Deep-TICA 1
20
OPES Deep-TICA 1
30
AcknowledgmentsThe authors are grateful to Dr. Michele Invernizzi for several valuable discussions and to him and Dr. Andrea Rizzi for carefully reading the paper. The calculations were carried out on the Euler cluster of ETH Zurich.Contact details: * [email protected]; † [email protected].
Reaction Rate Theory and Rare Events. B Peters, ElsevierB. Peters. Reaction Rate Theory and Rare Events. Elsevier, 2017.
Enhancing Important Fluctuations: Rare Events and Metadynamics from a Conceptual Viewpoint. O Valsson, P Tiwary, M Parrinello, Annual Review of Physical Chemistry. 67O. Valsson, P. Tiwary, and M. Parrinello. "Enhancing Important Fluctuations: Rare Events and Metadynamics from a Conceptual Viewpoint". Annual Review of Physical Chemistry 67 (2016).
Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling. G M Torrie, J P Valleau, Journal of Computational Physics. 23G. M. Torrie and J. P. Valleau. "Nonphysical sampling distributions in Monte Carlo free-energy estimation: Um- brella sampling". Journal of Computational Physics 23 (1977).
Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias. M Mezei, Journal of Computational Physics. 68M. Mezei. "Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias". Journal of Computational Physics 68 (1987).
Accelerated molecular dynamics of infrequent events. A F Voter, Physical Review Letters. 78A. F. Voter. "Accelerated molecular dynamics of infrequent events". Physical Review Letters 78 (1997).
Escaping free-energy minima. A Laio, M Parrinello, Proceedings of the National Academy of Sciences. 99A. Laio and M. Parrinello. "Escaping free-energy minima". Proceedings of the National Academy of Sciences 99 (2002).
Variational Approach to Enhanced Sampling and Free Energy Calculations. O Valsson, M Parrinello, Physical Review Letters. 113O. Valsson and M. Parrinello. "Variational Approach to Enhanced Sampling and Free Energy Calculations". Phys- ical Review Letters 113 (2014).
Neural networks-based variationally enhanced sampling. L Bonati, Y Y Zhang, M Parrinello, Proceedings of the National Academy of Sciences of the United States of America. the National Academy of Sciences of the United States of America116L. Bonati, Y. Y. Zhang, and M. Parrinello. "Neural networks-based variationally enhanced sampling". Proceed- ings of the National Academy of Sciences of the United States of America 116 (2019).
Gaussian Mixture-Based Enhanced Sampling for Statics and Dynamics. J Debnath, M Parrinello, Journal of Physical Chemistry Letters. 11J. Debnath and M. Parrinello. "Gaussian Mixture-Based Enhanced Sampling for Statics and Dynamics". Journal of Physical Chemistry Letters 11 (2020).
Rethinking Metadynamics: From Bias Potentials to Probability Distributions. M Invernizzi, M Parrinello, Journal of Physical Chemistry Letters. 11M. Invernizzi and M. Parrinello. "Rethinking Metadynam- ics: From Bias Potentials to Probability Distributions". Journal of Physical Chemistry Letters 11 (2020).
Metadynamics. A Barducci, M Bonomi, M Parrinello, Wiley Interdisciplinary Reviews: Computational Molecular Science. 1A. Barducci, M. Bonomi, and M. Parrinello. "Metady- namics". Wiley Interdisciplinary Reviews: Computational Molecular Science 1 (2011).
Using metadynamics to explore complex free-energy landscapes. G Bussi, A Laio, Nature Reviews Physics. 2G. Bussi and A. Laio. "Using metadynamics to explore complex free-energy landscapes". Nature Reviews Physics 2 (2020).
Machine learning for collective variable discovery and enhanced sampling in biomolecular simulation. H Sidky, W Chen, A L Ferguson, Molecular Physics. 118H. Sidky, W. Chen, and A. L. Ferguson. "Machine learning for collective variable discovery and enhanced sampling in biomolecular simulation". Molecular Physics 118 (2020).
Machine learning approaches for analyzing and enhancing molecular dynamics simulations. Y Wang, J M Lamim Ribeiro, P Tiwary, Current Opinion in Structural Biology. 61Y. Wang, J. M. Lamim Ribeiro, and P. Tiwary. "Machine learning approaches for analyzing and enhancing molecular dynamics simulations". Current Opinion in Structural Biology 61 (2020).
Collective variables for the study of long-time kinetics from molecular trajectories: theory and methods. F Noé, C Clementi, Current Opinion in Structural Biology. 43F. Noé and C. Clementi. "Collective variables for the study of long-time kinetics from molecular trajectories: theory and methods". Current Opinion in Structural Biology 43 (2017).
Collective Variables from Local Fluctuations. D Mendels, G Piccini, M Parrinello, The Journal of Physical Chemistry Letters. 9D. Mendels, G. Piccini, and M. Parrinello. "Collective Vari- ables from Local Fluctuations". The Journal of Physical Chemistry Letters 9 (2018).
Automated design of collective variables using supervised machine learning. M M Sultan, V S Pande, The Journal of Chemical Physics. 149M. M. Sultan and V. S. Pande. "Automated design of collective variables using supervised machine learning". The Journal of Chemical Physics 149 (2018).
Machine Learning of Biomolecular Reaction Coordinates. S Brandt, F Sittel, M Ernst, G Stock, Journal of Physical Chemistry Letters. 9S. Brandt, F. Sittel, M. Ernst, and G. Stock. "Machine Learning of Biomolecular Reaction Coordinates". Journal of Physical Chemistry Letters 9 (2018).
Metadynamics with Discriminants: A Tool for Understanding Chemistry. G Piccini, D Mendels, M Parrinello, Journal of Chemical Theory and Computation. 14G. Piccini, D. Mendels, and M. Parrinello. "Metadynamics with Discriminants: A Tool for Understanding Chemistry". Journal of Chemical Theory and Computation 14 (2018).
Data-Driven Collective Variables for Enhanced Sampling. L Bonati, V Rizzi, M Parrinello, Journal of Physical Chemistry Letters. 11L. Bonati, V. Rizzi, and M. Parrinello. "Data-Driven Col- lective Variables for Enhanced Sampling". Journal of Phys- ical Chemistry Letters 11 (2020).
The role of water in host-guest interaction. V Rizzi, L Bonati, N Ansari, M Parrinello, Nature Communications. 12V. Rizzi, L. Bonati, N. Ansari, and M. Parrinello. "The role of water in host-guest interaction". Nature Communi- cations 12 (2021).
Collective variables for the study of crystallisation. T Karmakar, M Invernizzi, V Rizzi, M Parrinello, Molecular Physics. T. Karmakar, M. Invernizzi, V. Rizzi, and M. Parrinello. "Collective variables for the study of crystallisation". Molec- ular Physics (2021).
Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: The case of domain motions. Y Naritomi, S Fuchigami, Journal of Chemical Physics. 134Y. Naritomi and S. Fuchigami. "Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: The case of domain motions". Journal of Chemical Physics 134 (2011).
Markov models of molecular kinetics: Generation and validation. J H Prinz, Journal of Chemical Physics. 134J. H. Prinz et al. "Markov models of molecular kinetics: Generation and validation". Journal of Chemical Physics 134 (2011).
Identification of slow molecular order parameters for Markov model construction. G Pérez-Hernández, F Paul, T Giorgino, G De Fabritiis, F Noé, Journal of Chemical Physics. 139G. Pérez-Hernández, F. Paul, T. Giorgino, G. De Fabri- tiis, and F. Noé. "Identification of slow molecular order parameters for Markov model construction". Journal of Chemical Physics 139 (2013).
Improvements in Markov State Model construction reveal many non-native interactions in the folding of NTL9. C R Schwantes, V S Pande, Journal of Chemical Theory and Computation. 9C. R. Schwantes and V. S. Pande. "Improvements in Markov State Model construction reveal many non-native interactions in the folding of NTL9". Journal of Chemical Theory and Computation 9 (2013).
Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations. H Wu, Journal of Chemical Physics. 146H. Wu et al. "Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations". Journal of Chemical Physics 146 (2017).
TICA-Metadynamics: Accelerating Metadynamics by Using Kinetically Selected Collective Variables. M M Sultan, V S Pande, Journal of Chemical Theory and Computation. 13M. M. Sultan and V. S. Pande. "TICA-Metadynamics: Accelerating Metadynamics by Using Kinetically Selected Collective Variables". Journal of Chemical Theory and Computation 13 (2017).
A variational conformational dynamics approach to the selection of collective variables in metadynamics. J Mccarty, M Parrinello, The Journal of Chemical Physics. 147J. McCarty and M. Parrinello. "A variational conforma- tional dynamics approach to the selection of collective variables in metadynamics". The Journal of Chemical Physics 147 (2017).
Variational encoding of complex dynamics. C X Hernández, H K Wayment-Steele, M M Sultan, B E Husic, V S Pande, Physical Review E 97. C. X. Hernández, H. K. Wayment-Steele, M. M. Sultan, B. E. Husic, and V. S. Pande. "Variational encoding of complex dynamics". Physical Review E 97 (2018).
Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics. C Wehmeyer, F Noé, Journal of Chemical Physics. 148C. Wehmeyer and F. Noé. "Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics". Journal of Chemical Physics 148 (2018).
VAMPnets for deep learning of molecular kinetics. A Mardt, L Pasquali, H Wu, F Noé, Nature Communications. 9A. Mardt, L. Pasquali, H. Wu, and F. Noé. "VAMPnets for deep learning of molecular kinetics". Nature Commu- nications 9 (2018).
Nonlinear discovery of slow molecular modes using state-free reversible VAMPnets. W Chen, H Sidky, A L Ferguson, Journal of Chemical Physics. 150W. Chen, H. Sidky, and A. L. Ferguson. "Nonlinear dis- covery of slow molecular modes using state-free reversible VAMPnets". Journal of Chemical Physics 150 (2019).
Identification of simple reaction coordinates from complex dynamics. R T Mcgibbon, B E Husic, V S Pande, Journal of Chemical Physics. 146R. T. McGibbon, B. E. Husic, and V. S. Pande. "Iden- tification of simple reaction coordinates from complex dynamics". Journal of Chemical Physics 146 (2017).
Discovering mountain passes via torchlight: Methods for the definition of reaction coordinates and pathways in complex macromolecular reactions. M A Rohrdanz, W Zheng, C Clementi, Annual Review of Physical Chemistry. 64M. A. Rohrdanz, W. Zheng, and C. Clementi. "Discovering mountain passes via torchlight: Methods for the defini- tion of reaction coordinates and pathways in complex macromolecular reactions". Annual Review of Physical Chemistry 64 (2013).
Spectral gap optimization of order parameters for sampling complex molecular systems. P Tiwary, B J Berne, Proceedings of the National Academy of Sciences. 113P. Tiwary and B. J. Berne. "Spectral gap optimization of order parameters for sampling complex molecular sys- tems". Proceedings of the National Academy of Sciences 113 (2016).
Molecular enhanced sampling with autoencoders: On-the-fly collective variable discovery and accelerated free energy landscape exploration. W Chen, A L Ferguson, Journal of Computational Chemistry. 39W. Chen and A. L. Ferguson. "Molecular enhanced sam- pling with autoencoders: On-the-fly collective variable dis- covery and accelerated free energy landscape exploration". Journal of Computational Chemistry 39 (2018).
Protocol for Identifying Accurate Collective Variables in Enhanced Molecular Dynamics Simulations for the Description of Structural Transformations in Flexible Metal-Organic Frameworks. R Demuynck, Journal of Chemical Theory and Computation. 14R. Demuynck et al. "Protocol for Identifying Accurate Collective Variables in Enhanced Molecular Dynamics Simulations for the Description of Structural Transforma- tions in Flexible Metal-Organic Frameworks". Journal of Chemical Theory and Computation 14 (2018).
Reweighted autoencoded variational Bayes for enhanced sampling (RAVE). J M L Ribeiro, P Bravo, Y Wang, P Tiwary, The Journal of Chemical Physics. 149J. M. L. Ribeiro, P. Bravo, Y. Wang, and P. Tiwary. "Reweighted autoencoded variational Bayes for enhanced sampling (RAVE)". The Journal of Chemical Physics 149 (2018).
Unfolding Hidden Barriers by Active Enhanced Sampling. J Zhang, M Chen, Physical Review Letters. 121J. Zhang and M. Chen. "Unfolding Hidden Barriers by Active Enhanced Sampling". Physical Review Letters 121 (2018).
Past-future information bottleneck for sampling molecular reaction coordinate simultaneously with thermodynamics and kinetics". Y Wang, J M L Ribeiro, P Tiwary, Nature Communications. 10Y. Wang, J. M. L. Ribeiro, and P. Tiwary. "Past-future information bottleneck for sampling molecular reaction coordinate simultaneously with thermodynamics and ki- netics". Nature Communications 10 (2019).
Multiscale Reweighted Stochastic Embedding: Deep Learning of Collective Variables for Enhanced Sampling. J Rydzewski, O Valsson, The Journal of Physical Chemistry A. J. Rydzewski and O. Valsson. "Multiscale Reweighted Stochastic Embedding: Deep Learning of Collective Vari- ables for Enhanced Sampling". The Journal of Physical Chemistry A (2021).
SGOOP-d: Estimating kinetic distances and reaction coordinate dimensionality for rare event systems from biased/unbiased simulations. S.-T Tsai, Z Smith, P Tiwary, arXiv:2104.13560S.-T. Tsai, Z. Smith, and P. Tiwary. "SGOOP-d: Estimat- ing kinetic distances and reaction coordinate dimension- ality for rare event systems from biased/unbiased simula- tions". arXiv:2104.13560 (2021).
Chasing Collective Variables using Autoencoders and biased trajectories. Z Belkacemi, P Gkeka, T Lelièvre, G Stoltz, arXiv:2104.11061Z. Belkacemi, P. Gkeka, T. Lelièvre, and G. Stoltz. "Chas- ing Collective Variables using Autoencoders and biased trajectories". arXiv:2104.11061 (2021).
Refining Collective Coordinates and Improving Free Energy Representation in Variational Enhanced Sampling. Y I Yang, M Parrinello, Journal of Chemical Theory and Computation. 14Y. I. Yang and M. Parrinello. "Refining Collective Co- ordinates and Improving Free Energy Representation in Variational Enhanced Sampling". Journal of Chemical Theory and Computation 14 (2018).
A variational approach to modeling slow processes in stochastic dynamical systems. F Noé, F Nüske, Multiscale Modeling and Simulation. 11F. Noé and F. Nüske. "A variational approach to modeling slow processes in stochastic dynamical systems". Multiscale Modeling and Simulation 11 (2013).
Unified Approach to Enhanced Sampling. M Invernizzi, P M Piaggi, M Parrinello, Physical Review X. 10M. Invernizzi, P. M. Piaggi, and M. Parrinello. "Unified Approach to Enhanced Sampling". Physical Review X 10 (2020).
Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method. A Barducci, G Bussi, M Parrinello, Physical Review Letters. 100A. Barducci, G. Bussi, and M. Parrinello. "Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free- Energy Method". Physical Review Letters 100 (2008).
Making the best of a bad situation: a multiscale approach to free energy calculation. M Invernizzi, M Parrinello, Journal of Chemical Theory and Computation. M. Invernizzi and M. Parrinello. "Making the best of a bad situation: a multiscale approach to free energy calculation". Journal of Chemical Theory and Computation (2019).
How Fast-Folding Proteins Fold. K Lindorff-Larsen, S Piana, R O Dror, D E Shaw, Science. 334K. Lindorff-Larsen, S. Piana, R. O. Dror, and D. E. Shaw. "How Fast-Folding Proteins Fold". Science 334 (2011).
Temperature and pressure denaturation of chignol Folding and unfolding simulation by multibaricmultithermal molecular dynamics method. H Okumura, Proteins: Structure, Function and Bioinformatics. 80H. Okumura. "Temperature and pressure denaturation of chignol Folding and unfolding simulation by multibaric- multithermal molecular dynamics method". Proteins: Structure, Function and Bioinformatics 80 (2012).
Enhanced, targeted sampling of high-dimensional free-energy landscapes using variationally enhanced sampling, with an application to chignolin. P Shaffer, O Valsson, M Parrinello, Proceedings of the National Academy of Sciences. 113P. Shaffer, O. Valsson, and M. Parrinello. "Enhanced, tar- geted sampling of high-dimensional free-energy landscapes using variationally enhanced sampling, with an applica- tion to chignolin". Proceedings of the National Academy of Sciences 113 (2016).
Modeling the mechanism of CLN025 beta-hairpin formation. K A Mckiernan, B E Husic, V S Pande, Journal of Chemical Physics. 147K. A. McKiernan, B. E. Husic, and V. S. Pande. "Mod- eling the mechanism of CLN025 beta-hairpin formation". Journal of Chemical Physics 147 (2017).
Folding a small protein using harmonic linear discriminant analysis. D Mendels, The Journal of Chemical Physics. 149D. Mendels et al. "Folding a small protein using harmonic linear discriminant analysis". The Journal of Chemical Physics 149 (2018).
VMD: Visual Molecular Dynamics. W Humphrey, A Dalke, K Schulten, Journal of molecular graphics. 14W. Humphrey, A. Dalke, and K. Schulten. "VMD: Visual Molecular Dynamics". Journal of molecular graphics 14 (1996).
Force-field dependence of chignolin folding and misfolding: Comparison with experiment and redesign. P Kührová, A Simone, M Otyepka, R B Best, Biophysical Journal. 102P. Kührová, A. De Simone, M. Otyepka, and R. B. Best. "Force-field dependence of chignolin folding and misfolding: Comparison with experiment and redesign". Biophysical Journal 102 (2012).
Analysis of Structural Stability of Chignolin. Y Maruyama, A Mitsutake, Journal of Physical Chemistry B. 122Y. Maruyama and A. Mitsutake. "Analysis of Structural Stability of Chignolin". Journal of Physical Chemistry B 122 (2018).
Improving collective variables: The case of crystallization. Y Y Zhang, H Niu, G Piccini, D Mendels, M Parrinello, Journal of Chemical Physics. 150Y. Y. Zhang, H. Niu, G. Piccini, D. Mendels, and M. Parrinello. "Improving collective variables: The case of crystallization". Journal of Chemical Physics 150 (2019).
Silicon liquid structure and crystal nucleation from ab-initio deep Metadynamics. L Bonati, M Parrinello, Physical Review Letters. 121L. Bonati and M. Parrinello. "Silicon liquid structure and crystal nucleation from ab-initio deep Metadynamics". Physical Review Letters 121 (2018).
Computer simulation of local order in condensed phases of silicon. F H Stillinger, T A Weber, Physical Review B. 31F. H. Stillinger and T. A. Weber. "Computer simulation of local order in condensed phases of silicon". Physical Review B 31 (1985).
Calculation of phase diagrams in the multithermal-multibaric ensemble. P M Piaggi, M Parrinello, Journal of Chemical Physics. 150P. M. Piaggi and M. Parrinello. "Calculation of phase dia- grams in the multithermal-multibaric ensemble". Journal of Chemical Physics 150 (2019).
Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool. A Stukowski, Modelling and Simulation in Materials Science and Engineering. 18A. Stukowski. "Visualization and analysis of atomistic sim- ulation data with OVITO-the Open Visualization Tool". Modelling and Simulation in Materials Science and Engi- neering 18 (2010).
K F Kelton, A Greer, Nucleation in condensed matter: applications in materials and biology. 15K. F. Kelton and A. Greer. Nucleation in condensed mat- ter: applications in materials and biology. Vol. 15. 2010.
Automatic differentiation in PyTorch. A Paszke, Advances in Neural Information Processing Systems 32. A. Paszke et al. "Automatic differentiation in PyTorch". Advances in Neural Information Processing Systems 32. 2019.
PLUMED 2: New feathers for an old bird. G A Tribello, M Bonomi, D Branduardi, C Camilloni, G Bussi, Computer Physics Communications. 185G. A. Tribello, M. Bonomi, D. Branduardi, C. Camilloni, and G. Bussi. "PLUMED 2: New feathers for an old bird". Computer Physics Communications 185 (2014).
Promoting transparency and reproducibility in enhanced molecular simulations. M Bonomi, Nature Methods. 16M. Bonomi et al. "Promoting transparency and repro- ducibility in enhanced molecular simulations". Nature Methods 16 (2019).
GROMACS: Fast, flexible, and free. D Van Der, Spoel, Journal of Computational Chemistry. 26D. Van Der Spoel et al. "GROMACS: Fast, flexible, and free". Journal of Computational Chemistry 26 (2005).
Comparison of multiple Amber force fields and development of improved protein backbone parameters. V Hornak, Proteins: Structure, Function and Genetics. 65V. Hornak et al. "Comparison of multiple Amber force fields and development of improved protein backbone pa- rameters". Proteins: Structure, Function and Genetics 65 (2006).
Canonical sampling through velocity rescaling. G Bussi, D Donadio, M Parrinello, The Journal of Chemical Physics. 126G. Bussi, D. Donadio, and M. Parrinello. "Canonical sam- pling through velocity rescaling". The Journal of Chemical Physics 126 (2007).
How robust are protein folding simulations with respect to force field parameterization?. S Piana, K Lindorff-Larsen, D E Shaw, Biophysical Journal. 100S. Piana, K. Lindorff-Larsen, and D. E. Shaw. "How robust are protein folding simulations with respect to force field parameterization?" Biophysical Journal 100 (2011).
Comparison of simple potential functions for simulating liquid water. W L Jorgensen, J Chandrasekhar, J D Madura, R W Impey, M L Klein, The Journal of Chemical Physics. 79W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein. "Comparison of simple potential functions for simulating liquid water". The Journal of Chemical Physics 79 (1983).
Fast parallel algorithms for short-range molecular dynamics. S Plimpton, Journal of computational physics. 117S. Plimpton. "Fast parallel algorithms for short-range molecular dynamics". Journal of computational physics 117 (1995).
| [] |
[] | [
"\nNational Research University Higher School of Economics\n20 Myasnitskaya Ulitsa101000MoscowRussia\n"
] | [
"National Research University Higher School of Economics\n20 Myasnitskaya Ulitsa101000MoscowRussia"
] | [] | We give a short combinatorial proof that the nonnegative matrix factorization is an NP-hard problem. Moreover, we prove that NMF remains NP-hard when restricted to 01-matrices, answering a recent question of Moitra. | null | [
"https://arxiv.org/pdf/1605.04000v1.pdf"
] | 11,335,115 | 1605.04000 | 2e597ebc1fe2d90a69d936f16fd595a8033d0f6a |
12 May 2016
National Research University Higher School of Economics
20 Myasnitskaya Ulitsa101000MoscowRussia
12 May 2016A SHORT PROOF THAT NMF IS NP-HARD YAROSLAV SHITOV
We give a short combinatorial proof that the nonnegative matrix factorization is an NP-hard problem. Moreover, we prove that NMF remains NP-hard when restricted to 01-matrices, answering a recent question of Moitra.
The (exact) nonnegative matrix factorization is the following problem. Given an integer k and a matrix A with nonnegative entries, do there exist k nonnegative rank-one matrices that sum to A? The smallest k for which this is possible is called the nonnegative rank of A and denoted by rank + (A). We give a short combinatorial proof of a seminal result of Vavasis [6] stating that NMF is NP-hard. Moreover, we prove that NMF remains hard when restricted to Boolean matrices, answering a recent question of Moitra [4]. Theorem 1. It is NP-hard to decide whether rank + (A) k, given an integer k and a matrix A with entries in {0, 1}.
Recall that a (directed) graph G is a finite set of vertices V and edges E ⊂ V ×V . We assume that G has no loops, that is, (v, v) / ∈ E for all v ∈ V . An independent set in G is a subset U ⊂ V such that (u 1 , u 2 ) / ∈ E for all u 1 , u 2 ∈ U . The chromatic number of G, denoted by c(G), is the smallest c such that V is a union of c independent sets. The following is a classical NP-complete problem [3].
Problem 2. Given a graph G and an integer C. Is c(G) C?
To construct a reduction from Problem 2 to NMF, we define the matrix N = N (G) with 5|V | rows and columns indexed by the set
V ∪ V 1 ∪ V 2 ∪ V 3 ∪ V 4 ,
which is the union of five copies of V . For any v ∈ V , we define the entry N (v|v) as 1 and enumerate the vertices in V \ {v} as u 1 , . . . , u m ; we set the submatrix
N (v 1 , v 2 , v 3 , v 4 , v|v 1 , v 2 , v 3 , v 4 , u 1 , . . . , u m ) equal to (0.1) 1 1 0 0 1 . . . 1 0 1 1 0 1 . . . 1 0 0 1 1 1 . . . 1 1 0 0 1 1 . . . 1 1 0 0 0 x 1 . . . x m , where x i = 0 if (v, u i ) ∈ E and x i = 1 otherwise.
The entries of N that are not yet specified are equal to 0.
We denote by N the upper left 4 × 4 submatrix of (0.1); one has rank + (N ) = 4. Since every column of (0.1) is a linear combination of the first four columns taken with nonnegative coefficients, the nonnegative rank of (0.1) equals four. Proof. Let U 1 , . . . , U c be a partition of V into disjoint independent sets of G. Let H i be the matrix such that H i (α|β) = 1 if α, β ∈ U i and H i (α|β) = 0 otherwise. We see that N − H 1 − . . . − H c is a nonnegative matrix whose nonzero entries are contained in |V | disjoint submatrices of the form (0.1). Since rank + (H i ) = 1, we get rank + (N ) 4|V | + c.
Now let M 1 , . . . , M r be nonnegative rank-one matrices that sum to N . Since the set C j = {v ∈ V : M j (v|v) = 0} is independent for every j, this set is non-empty for at least c(G) values of j. Observation 3 shows that, for these j, the submatrices
M j (V 1 ∪ V 2 ∪ V 3 ∪ V 4 |V 1 ∪ V 2 ∪ V 3 ∪ V 4 ) are zero. It remains to note that N (V 1 ∪ V 2 ∪ V 3 ∪ V 4 |V 1 ∪ V 2 ∪ V 3 ∪ V 4 ) has nonnegative rank 4|V | because it is the block-diagonal matrix with |V | blocks equal to N .
Now we see that (G, C) → (N (G), 4|V (G)| + C) is a polynomial reduction from Problem 2 to NMF. Since N (G) is Boolean, the proof of Theorem 1 is complete.
Many interesting problems regarding the complexity of NMF remain open. Let us recall a remarkable result [1] providing a polynomial time algorithm for NMF with fixed nonnegative rank. However, it is not known whether such algortihm exists if we fix the conventional rank instead of nonnegative rank [2].
Despite having proved that NMF is NP-hard, we do not know anything about completeness of this problem. We note that the entries of rank-one matrices in the optimal factorization may not be rational functions of entries of the initial matrix, see [5]. Is NMF (restricted to rational matrices) NP-complete or ∃R-complete?
.
Let M be a nonnegative rank-one matrix such that M N (G). If M (v|v) = 0 for some v ∈ V , then M (u i |u j ) = 0 for all u ∈ V and i, j ∈ {1, 2, 3, 4}.Proof. By the construction, the entry N (v|u j ) can be nonzero only if u = v, but in this case we have N (u i |v) = 0. Since M N , the entries M (v|u j ) and M (u i |v) cannot be positive simultaneously, and the same holds for M (v|v) and M (u i |u j ) because M is rank-one.
Proposition 4 .
4We have rank + (N (G)) = 4|V (G)| + c(G).
Computing a nonnegative matrix factorizationprovably. S Arora, R Ge, R Kannan, A Moitra, Proceedings of the Annual Symposium on Theory of Computing. the Annual Symposium on Theory of ComputingS. Arora, R. Ge, R. Kannan, A. Moitra, Computing a nonnegative matrix factorization - provably, Proceedings of the Annual Symposium on Theory of Computing (2012) 145-162.
On the geometric interpretation of the nonnegative rank. N Gillis, F Glineur, Linear Algebra Appl. 437N. Gillis, F. Glineur, On the geometric interpretation of the nonnegative rank, Linear Algebra Appl. 437 (2012) 2685-2712.
R Karp, Reducibility Among Combinatorial Problems, Proceedings of the Symposium on the Complexity of Computer Computations. R. Karp, Reducibility Among Combinatorial Problems, Proceedings of the Symposium on the Complexity of Computer Computations (1972) 85-103.
A Moitra, Nonnegative Matrix Factorization: Algorithms, Complexity and Applications. A presentation at the International Symposium on Symbolic and Algebraic Computation. Available viaA. Moitra, Nonnegative Matrix Factorization: Algorithms, Complexity and Applications. A presentation at the International Symposium on Symbolic and Algebraic Computation (2015). Available via http://www.issac-conference.org/2015/Slides/Moitra.pdf.
Y Shitov, arXiv:1505.01893Nonnegative rank depends on the field. preprintY. Shitov, Nonnegative rank depends on the field, preprint (2015) arXiv:1505.01893.
On the complexity of nonnegative matrix factorization. S Vavasis, SIAM J. Optimization. 20S. Vavasis, On the complexity of nonnegative matrix factorization, SIAM J. Optimization 20 (2009) 1364-1377.
| [] |
[
"A four-component Camassa-Holm type hierarchy",
"A four-component Camassa-Holm type hierarchy"
] | [
"Nianhua Li \nDepartment of Mathematics\nChina University of Mining and Technology\n100083BeijingP R China\n",
"Q P Liu \nDepartment of Mathematics\nChina University of Mining and Technology\n100083BeijingP R China\n",
"Z Popowicz \nInstitute of Theoretical Physics\nUniversity of Wroc law pl. M\nBorna 9, 50-205 Wroc lawPoland\n"
] | [
"Department of Mathematics\nChina University of Mining and Technology\n100083BeijingP R China",
"Department of Mathematics\nChina University of Mining and Technology\n100083BeijingP R China",
"Institute of Theoretical Physics\nUniversity of Wroc law pl. M\nBorna 9, 50-205 Wroc lawPoland"
] | [] | We consider a 3×3 spectral problem which generates four-component CH type systems. The bi-Hamiltonian structure and infinitely many conserved quantities are constructed for the associated hierarchy. Some possible reductions are also studied.Mathematical Subject Classification: 37K10, 37K05 | 10.1016/j.geomphys.2014.05.026 | [
"https://arxiv.org/pdf/1310.1781v1.pdf"
] | 118,518,860 | 1310.1781 | c2700d6d7606cbc705497f1aa6091acce1262947 |
A four-component Camassa-Holm type hierarchy
7 Oct 2013
Nianhua Li
Department of Mathematics
China University of Mining and Technology
100083BeijingP R China
Q P Liu
Department of Mathematics
China University of Mining and Technology
100083BeijingP R China
Z Popowicz
Institute of Theoretical Physics
University of Wroc law pl. M
Borna 9, 50-205 Wroc lawPoland
A four-component Camassa-Holm type hierarchy
7 Oct 2013Bi-Hamiltonian structureConserved quantityLax pairIntegrability
We consider a 3×3 spectral problem which generates four-component CH type systems. The bi-Hamiltonian structure and infinitely many conserved quantities are constructed for the associated hierarchy. Some possible reductions are also studied.Mathematical Subject Classification: 37K10, 37K05
Introduction
The Camassa-Holm (CH) equation
u t − u xxt + 3uu x = 2u x u xx + uu xxx ,(1)
has been a subject of steadily growing literature since it was derived from the incompressible Euler equation to model long waves in shallow water by Camassa and Holm in 1993 [1]. It is a completely integrable system, which possesses Lax representation and is a bi-Hamiltonian system and especially admits peakon solutions [1] [2]. CH equation is a system with quadratic nonlinearity, and another such system was proposed in 1999 by Degasperis and Procesi [3], which reads as
m t + um x + 3u x m = 0, m = u − u xx ,(2)
now known as called DP equation whose peakon solutions were studied in [4] [14]. The following third order spectral problem for the DP equation (2) was found by Degasperis, Holm and Hone [4] ψ xxx = ψ x − λψ (3) or equivalently it may be rewritten in the matrix form as
ϕ x = 0 0 1 −λm 0 0 1 1 0 ϕ.(4)
Olver and Rosenau suggested so-called tri-Hamiltoninan duality approach to construct the CH type equations and many examples were worked out [17] (see also [5][6] [7]). In particular, they obtained a CH type equation with cubic nonlinearity
m t + [m(u 2 − u 2 x )] x = 0, m = u − u xx(5)
which, sometimes is referred as Qiao equation, was studied by Qiao [19].
Recently, Song, Qu and Qiao [21] proposed a two-component generalization of (5)
m t − [m(u x v x − uv + uv x − u x v)] x = 0, m = u − u xx , n t − [n(u x v x − uv + uv x − u x v)] x = 0, n = v − v xx(6)
based on the following spectral problem
ϕ x = 1 2 λm λn − 1 2 ϕ.(7)
This system (7) is shown to be bi-Hamiltonian and is associated with WKI equation [22]. Moreover Xia,Qiao and Zhou [23] generalized previous system to the integrable two-component CH type equation
m t = F + F x − 1 2 m(uv − u x v x + uv x u x v), n t = −G + G x + 1 2 n(uv − u x v x + uv x − u x v),(8)m = u − u xx , n = v − v xx
where F and G are two arbitrary functions. Working on symmetry classification of nonlocal partial differential equations, Novikov [15] found another CH type equation with cubic nonlinearity. It reads as
m t + u 2 m x + 3uu x m = 0, m = u − u xx .(9)
Subsequently, Hone and Wang proposed a Lax representation for (9) and showed that it is associated to a negative flow in Sawada-Kotera hierarchy [10]. Furthermore, infinitely many conserved quantities and bi-Hamiltonian structure are also presented [11]. A two-component generalization of the Novikov equation was constructed by Geng and Xue [8].
m t + 3u x vm + uvm x = 0, m = u − u xx n t + 3v x un + uvn x = 0, n = v − v xx ,(10)
with a spectral problem
ϕ x = 0 λm 1 0 0 λn 1 0 0 ϕ,(11)
which reduces to the spectral Novikov's system as m = n. They also calculated the N-peakons and conserved quantities and found a Hamiltonian structure. In this paper we will discuss the properties of the equations which follow from the following generalized spectral problem
Φ x = UΦ, U = 0 λm 1 1 λn 1 0 λm 2 1 λn 2 0 .(12)
This spectral problem was proposed by one of us (ZP) recently [18] and interestingly the resulted nonlinear systems can involve an arbitrary function. This freedom allows us to recover many known CH type equations from reductions. As we will show this spectral problem takes the spectral problem considered by Geng and Xue for three-component system [9], one and two-component Novikov's equation [11][8], and one or two component Song-Qu-Qiao equation [21] as special cases. In this sense, almost all known 3 × 3 spectral problems for the CH type equations are contained in this case, so it is interesting to study this spectral problem. The first negative flow corresponding to the spectral problem (12) is
m 1t + n 2 g 1 g 2 + m 1 (f 2 g 2 + 2f 1 g 1 ) = 0, m 2t − n 1 g 1 g 2 − m 2 (f 1 g 1 + 2f 2 g 2 ) = 0, n 1t − m 2 f 1 f 2 − n 1 (f 2 g 2 + 2f 1 g 1 ) = 0, n 2t + m 1 f 1 f 2 + n 2 (f 1 g 1 + 2f 2 g 2 ) = 0, m i = u i − u ixx , n i = v i − v ixx , i = 1, 2,(13)
where
f 1 = u 2 − v 1x , f 2 = u 1 + v 2x , g 1 = v 2 + u 1x , g 2 = v 1 − u 2x .
which will be shown to be a bi-Hamiltonian system. The paper is organized as follows. In the section 2, we will construct bi-Hamiltonian operators related to the spectral problem (12) and present a bi-Hamilotonian representation for the system (13). In the section 3, we construct infinitely many conserved quantities for the integrable hierarchy of (12). In the section 4, we consider the special reductions of our spectral problem. The last section contains concluding remarks.
Bi-Hamiltonian structures in general
Let us consider the following Lax pair
Φ x = UΦ, Φ t = V Φ,(14)
where U is defined by (12) and V = (V ij ) 3×3 at the moment is an arbitrary matrix. The compatibility condition of (14) or the zero-curvature representation
U t − V x + [U, V ] = 0,(15)
is equivalent to
λm 1t = V 12x − V 32 + λ(m 1 V 11 + n 2 V 13 − m 1 V 22 ), λm 2t = V 23x + V 21 + λ(m 2 V 22 − m 2 V 33 − n 1 V 13 ), λn 1t = V 21x + V 23 + λ(n 1 V 22 − m 2 V 31 − n 1 V 11 ), λn 2t = V 32x − V 12 + λ(n 2 V 33 + m 1 V 31 − n 2 V 22 ),(16)
with
V 11 = V 31x + V 33 − λn 2 V 21 + λn 1 V 32 , V 13 = V 33x + V 31 + λm 2 V 32 − λn 2 V 23 , V 22x = λ(n 1 V 12 + m 2 V 32 − m 1 V 21 − n 2 V 23 ),(17)2V 31x + V 33xx = λ((∂n 2 + m 1 )V 23 − (∂m 2 + n 1 )V 32 − m 2 V 12 + n 2 V 21 ), 2V 33x + V 31xx = λ((∂n 2 + m 1 )V 21 − (∂n 1 + m 2 )V 32 − n 1 V 12 + n 2 V 23 ).
Taking account of (17) and through a tedious calculation, the system (16) yields
m 1 m 2 n 1 n 2 t = (λ −1 K + λL) V 21 V 32 V 12 V 23 ,(18)with K = 0 −1 ∂ 0 1 0 0 ∂ ∂ 0 0 1 0 ∂ −1 0 , L = J + F ,(19)
and
J = 2m 1 ∂ −1 m 1 −m 1 ∂ −1 m 2 J 13 J 14 −m 2 ∂ −1 m 1 2m 2 ∂ −1 m 2 J 23 J 24 −J * 13 −J * 23 2n 1 ∂ −1 n 1 −n 1 ∂ −1 n 2 −J * 14 −J * 24 −n 2 ∂ −1 n 1 2n 2 ∂ −1 n 2 , F = (2P + S∂)(∂ 3 − 4∂) −1 P T − (2S + P ∂)(∂ 3 − 4∂) −1 S T , where J 13 = −2m 1 ∂ −1 n 1 − n 2 ∂ −1 m 2 , J 14 = m 1 ∂ −1 n 2 + n 2 ∂ −1 m 1 , J 23 = m 2 ∂ −1 n 1 + n 1 ∂ −1 m 2 , J 24 = −2m 2 ∂ −1 n 2 − n 1 ∂ −1 m 1 , P = (m 1 , m 2 , −n 1 , n 2 ) T , S = (−n 2 , n 1 , −m 2 , m 1 ) T .
By specifying the matrix V properly, we can obtain the hierarchy of equations associated with the Lax pair (14). For example, if we assume
V = −f 1 g 1 g 1 λ −g 1 g 2 f 1 λ − 1 λ 2 + f 1 g 1 + f 2 g 2 g 2 λ −f 1 f 2 f 2 λ −f 2 g 2 ,
then the resulted system, the first negative flow of the hierarchy, is exactly the equation (13) .
On the other hand taking the V matrix in time part of the Lax pair as
V = −λ 0 m 1 Γ 0 n 1 Γ 0 m 2 Γ 0 n 2 Γ 0 .(20)
we obtain
m 1t + (Γm 1 ) x − n 2 Γ = 0, m 2t + (Γm 2 ) x + n 1 Γ = 0, n 1t + (Γn 1 ) x + m 2 Γ = 0, n 2t + (Γn 2 ) x − m 1 Γ = 0,(21)
where Γ is an arbitrary function. Now the combinations of V andṼ leads us to the following system of equations
m 1t + (Γm 1 ) x + n 2 (g 1 g 2 − Γ) + m 1 (f 2 g 2 + 2f 1 g 1 ) = 0, m 2t + (Γm 2 ) x − n 1 (g 1 g 2 − Γ) − m 2 (f 1 g 1 + 2f 2 g 2 ) = 0, n 1t + (Γn 1 ) x − m 2 (f 1 f 2 − Γ) − n 1 (f 2 g 2 + 2f 1 g 1 ) = 0, n 2t + (Γn 2 ) x + m 1 (f 1 f 2 − Γ) + n 2 (f 1 g 1 + 2f 2 g 2 ) = 0, m i = u i − u ixx , n i = v i − v ixx , i = 1, 2,(22)
which has the Lax pair
Φ x = UΦ, Φ t = (V +Ṽ )Φ.
It is obvious that the operator K given by (19) is a Hamiltonian operator. Indeed we have the following theorem Theorem 1. The operators K and L defined by (19) constitute a pair of compatible Hamiltonian operators. In particular, the four-component system (13) is a bi-Hamiltonian system, namely it can be written as
m 1 m 2 n 1 n 2 t = K δH 0 δm 1 δH 0 δm 2 δH 0 δn 1 δH 0 δn 2 = L δH 1 δm 1 δH 1 δm 2 δH 1 δn 1 δH 1 δn 2 (23)
where
H 0 = (f 1 g 1 + f 2 g 2 )(m 2 f 2 + n 1 g 1 )dx, H 1 = (m 2 f 2 + n 1 g 1 )dx.(24)
Proof: It is easy to check that L is a skew-symmetric operator. Thus, what we need to do is to verify the Jacobi identity for L and the compatibility of two operators K and L. To this end, we follow Olver and use his multivector approach [16]. Let us introduce
Θ K = 1 2 θ ∧ Kθdx, Θ L = 1 2 θ ∧ Lθdx(25)
with θ = (θ 1 , θ 2 , θ 3 , θ 4 ). A tedious but direct calculation shows that
pr v Lθ (Θ L ) = 0, pr v Kθ (Θ L ) + pr v Lθ (Θ K ) = 0 (26)
hold. The details of this calculation are postponed to the Appendix.
Since we have a bi-Hamiltonian pair K and L, we may formulate an integrable hierarchy of the corresponding nonlinear evolutions equations through recursion operator. While the system (21) possesses a Lax pair, we could not expect that it is a bi-Hamiltonian system for any Γ and the constraint of Γ will appear if it is requested to commute with the negative flow .
To understand the appearance of the arbitrary function occurred in the present case, we now calculate the Casimir functions of the Hamiltonian operator L. Let
L(A, B, C, D) T = 0,(27)
and define
K 1 = m 1 A − n 1 C, K 2 = m 2 B − n 2 D,(28)K 3 = m 2 C − m 1 D, K 4 = n 2 A − n 1 B.(29)
The system (27) consists of four equations of similar type. For example the one of them is
m 1 ∂ −1 (2K 1 − K 2 ) + (∂ 3 − 4∂) −1 (2(K 1 + K 2 ) + (K 3 + K 4 ) x = n 2 (∂ −1 K 3 + ∂ 3 − 4∂) −1 (2(K 3 + K 4 ) + (K 1 + K 2 ) x )).
Solving these equations we found
K 1 = (m 2 n 2 Λ) x + (n 1 n 2 − m 1 m 2 )Λ, K 2 = −(m 1 n 1 Λ) x + (n 1 n 2 − m 1 m 2 )Λ, K 3 = (m 1 m 2 Λ) x + (m 1 n 1 − m 2 n 2 )Λ, K 4 = −(n 1 n 2 Λ) x + (m 1 n 1 − m 2 n 2 )Λ,
where Λ = k m 1 n 1 +m 2 n 2 and k is an arbitrary number. Substituting above expressions for K i into (28)-(29) and solving the resulted linear equations leads to
A = −n 1 Γ + n 1 m 1 m 2 K 3 + 1 m 1 K 1 , B = −n 2 Γ + 1 m 2 K 2 , C = −m 1 Γ + 1 m 2 K 3 , D = −m 2 Γ,
where Γ is an arbitrary function. This implies that L is a degenerate Hamiltonian operator. For the special case k = 0 and Γ = m 1 n 1 + m 2 n 2 , the Casimir is H c = − 1 2 Γ 2 dx and hence the system (21) in this case is a Hamiltonian system
m 1 m 2 n 1 n 2 t = K δHc δm 1 δHc δm 2 δHc δn 1 δHc δn 2 .
Conserved quantities
An integrable system normally possesses infinity number of concerned quantities and such property has been taken as one of the defining properties for integrability. In this section, we show that infinitely many conserved quantities can be constructed for the nonlinear evolution equations related with the spectral problem (12). Indeed, we may derive two sequences of conserved quantities utilizing the projective coordinates in the spectral problem. We can introduce these coordinates in three different manners as
I.) a = ϕ 1 ϕ 2 , b = ϕ 3 ϕ 2 , II.) σ = ϕ 2 ϕ 1 , τ = ϕ 3 ϕ 1 , III.) α = ϕ 1 ϕ 3 , β = ϕ 2 ϕ 3
Case 1: In these coordinates we obtain that
ρ = (ln ϕ 2 ) x = λn 1 a + λm 2 b,(30)
is conserved quantity with a, b satisfy
a x = λm 1 + b − aρ, b x = a + λn 2 − bρ.(31)
Substituting the Laurent series expansions in λ of a and b into (31)
a = i≥0 a i λ i , b = j≥0 b j λ j ,
then we find
a 0 = 0, a 1 = −v 2 − u 1x = −g 1 , a 2 = 0, b 0 = 0, b 1 = −u 1 − v 2x = −f 2 , b 2 = 0, and a k,x = b k − i+j=k−1 (n 1 a i a j + m 2 a i b j ), b k,x = a k − i+j=k−1 (n 1 a i b j + m 2 b i b j ), (k ≥ 3).
With the aid of a 1 , b 1 , we obtain a simple conserved quantity
ρ 1 = − (n 1 g 1 + m 2 f 2 )dx.(32)
Also, due to a 3 − a 3xx = n 1 f 2 g 1 + m 2 f 2 2 + (n 1 g 2
1 + m 2 f 2 g 1 ) x , b 3 − b 3xx = n 1 g 2 1 + m 2 f 2 g 1 + (n 1 f 2 g 1 + m 2 f 2 2 ) x ,
we obtain the next conserved quantity by
ρ 3 = n 1 a 3 + m 2 b 3 dx = v 1 (a 3 − a 3xx ) + u 2 (b 3 − b 3xx ) dx = (n 1 g 1 + m 2 f 2 )(f 1 g 1 + f 2 g 2 )dx.(33)
In addition, we may consider alternative expansions of a, b in negative powers of λ, namely
a = Σ ∞ i≥0ã i λ −i , b = Σ ∞ j≥0b j λ −j .
As above, inserting these expansions into (31) we may find recursive relations forã i ,b j . The first two conserved quantities are
ρ 0 = √ m 1 n 1 + m 2 n 2 dx,(34)ρ −1 = 2m 1 m 2 + 2n 1 n 2 + m 1 n 1x − m 1x n 1 + m 2x n 2 − m 2 n 2x 4(m 1 n 1 + m 2 n 2 ) dx.
Case 2: The quantityρ defined as
ρ = (ln ϕ 1 ) x = λm 1 σ + τ,(35)
with σ, τ satisfy σ x = λn 1 + λm 2 τ − σρ, τ x = 1 + λn 2 σ − τρ.
is conserved quantity. Expanding σ and τ in Laurent series of λ then once again we may find the corresponding conserved quantities. For instance, in the case k ≥ 0, we get
ρ 2 = 1 2 (m 1 + n 2 )(f 1 + g 2 )dx,
while in the case k ≤ 0, we obtain ρ −1 = 2m 2 n 2 2 + 2m 1 n 1 n 2 − m 2x m 1 n 2 + 4m 1x m 2 n 2 − 3n 2x m 1 m 2 + m 1x m 1 n 1 − n 1x m 2 1 4m 1 (m 1 n 1 + m 2 n 2 ) dx.
Case 3: For this case the conserved quantity is defined aŝ
ρ = (ln ϕ 3 ) x = α + λn 2 β,(37)
with α, β satisfy
α x = λm 1 β + 1 − αρ, β x = λn 1 α + λm 2 − βρ.(38)
Expanding α and β in Laurent series of λ and substituting them into (38), we may obtain the conserved quantities and apart from those found in last two cases, we havê ρ −1 = 2n 1 n 2 2 + 2m 1 m 2 n 2 − m 2x n 2 2 + 4n 2x m 1 n 1 − 3m 1x n 1 n 2 + n 2x m 2 n 2 − n 1x m 1 n 2 4n 2 (m 1 n 1 + m 2 n 2 ) dx.
Let us remark that these conserved quantities have been obtained from the x-part of the Lax pair representation only hence they are valid for the whole hierarchy. As we checked they are conserved for the system (13) as well as for the (22).
Reductions
We now consider the possible reductions of our four component spectral problem (14) and relate them to the spectral problems known in literatures.
a three component reduction
Assuming
m 1 = u 1 = 0, we have φ 1 φ 2 φ 3 x = 0 0 1 λn 1 0 λm 2 1 λn 2 0 φ 1 φ 2 φ 3 ,(39)
which, by identifying the variables as follows
n 2 = u, m 2 = v u , n 1 = w + v u x ,(40)
may be reformulated as
ϕ 1 ϕ 2 ϕ 3 x = 0 1 0 1 + λ 2 v 0 u λ 2 w 0 0 ϕ 1 ϕ 2 ϕ 3 .
This spectral problem is nothing but the one proposed by Geng and Xue [9] and the associated integrable flows are also bi-Hamiltonian [13]. It is easy to see that (40) is an invertible transformation, so to find the bi-Hamiltonian structures of the flows resulted from the spectral problem (39) we may convert those for the Geng-Xue spectral problem into the present case. Direct calculations yield
L 1 = − m 2 n 2 ∂ − ∂ m 2 n 2 ( m 2 n 2 ∂ + ∂ m 2 n 2 )∂ + S 0 −∂( m 2 n 2 ∂ + ∂ m 2 n 2 ) − S * ∂S + S * ∂ + ∂( m 2 n 2 ∂ + ∂ m 2 n 2 )∂ 1 − ∂ 2 0 ∂ 2 − 1 0 , L 2 = − 1 2 m 2 ∂ + 2m 2x m 2 ∂ 2 + 3n 1 ∂ + 2n 1x 3n 2 ∂ + 2n 2x (∂ 3 − 4∂) −1 m 2 ∂ + 2m 2x m 2 ∂ 2 + 3n 1 ∂ + 2n 1x 3n 2 ∂ + 2n 2x * + 1 2 3m 2 ∂ −1 m 2 −m 2 2 + 3m 2 ∂ −1 n 1 −3m 2 ∂ −1 n 2 m 2 2 + 3n 1 ∂ −1 m 2 m 2 ∂m 2 + 3n 1 ∂ −1 n 1 −m 2 n 2 − 3n 1 ∂ −1 n 2 −3n 2 ∂ −1 m 2 m 2 n 2 − 3n 2 ∂ −1 n 1 3n 2 ∂ −1 n 2 ,
where S = m 2 n 2 (1 − ∂ 2 ). We remark that it is not clear how to obtain above Hamilitonian pair from (19).
a two-component reduction
In this case, we assume
n 1 = m 2 , n 2 = m 1 or φ 1 φ 2 φ 3 x = 0 λm 1 1 λm 2 0 λm 2 1 λm 1 0 φ 1 φ 2 φ 3 , which yields φ 1 + φ 3 φ 2 x = 1 2λm 1 λm 2 0 φ 1 + φ 3 φ 2 .(41)
By the following change of variables
φ 1 + φ 3 = e 1 2 x ψ 1 , φ 2 = e 1 2 x ψ 2 , 2m 1 = m, m 2 = n,(41)gives ψ 1 ψ 2 x = 1 2 λm λn − 1 2 φ 1 ψ 2 ,
a spectral problem considered by Song, Qu and Qiao [21]. Therefore the bi-hamiltonian structure of Song-Qu-Qiao system (see [22]) is hidden in our bi-hamiltonian structure.
Concluding remarks
In this paper, started from a general 3×3 problem, we considered the related four-component CH type systems. We obtained the bi-Hamiltonian structure and suggested the way to construction of infinitely many conserved quantities for the integrable equations. Different reductions were also considered.
As noticed above, the positive flows allow for an arbitrary function Γ involved and such systems are interesting since different specifications of Γ lead to different CH type equations. Although the flow equations with arbitrary Γ do possess infinitely many conserved quantities, we do not expect they are (bi-) Hamiltonian systems in general case. Also, we explained the appearance of this arbitrary function by studying the kernel of one of the Hamiltonian operators and it seems that further study of such systems is needed.
A remarkable property of CH type equations is that it possess peakon solutions. One may find that the first negative flow, which does not depend on Γ, only possess stationary peakons. The systems such as (22) may admit non-stationary peakon solutions. This and other related issues may be considered in further publication.
Appendix A.
We first prove that L is also a Hamiltonian operator. For this purpose, we first define
Θ J = 1 2 θ ∧ J θdx, Θ F = 1 2 θ ∧ F θdx, A = (∂ 3 − 4∂) −1 .
By direct calculation, we have
Θ J = (n 2 θ 1 − n 1 θ 2 ) ∧ ∂ −1 (m 1 θ 4 − m 2 θ 3 ) + (m 1 θ 1 − n 1 θ 3 ) ∧ ∂ −1 (n 2 θ 4 − m 2 θ 2 ) + (n 2 θ 4 − m 2 θ 2 ) ∧ ∂ −1 (n 2 θ 4 − m 2 θ 2 ) + (m 1 θ 1 − n 1 θ 3 ) ∧ ∂ −1 (m 1 θ 1 − n 1 θ 3 ) dx, and Θ F = 1 2 (2Q − R∂) ∧ AQ + (Q∂ − 2R) ∧ AR dx = Q ∧ AQ − R ∧ AR + Q ∧ ∂AR dx, where Q = m 1 θ 1 + m 2 θ 2 − n 1 θ 3 − n 2 θ 4 , R = n 2 θ 1 − n 1 θ 2 + m 2 θ 3 − m 1 θ 4 . Since pr v Lθ (Θ L ) = pr v Lθ (Θ J ) + pr v Lθ (Θ F ),(A1)
we calculate pr v Lθ (Θ J ) and pr v Lθ (Θ F ). Indeed, we have
−pr v Lθ (Θ J ) = (∂ −1 (n 2 θ 1 − n 1 θ 2 ) ∧ ∂ −1 (m 1 θ 4 − m 2 θ 3 ) ∧ ∂ −1 Q) x +(2m 1 θ 1 + 2n 1 θ 3 − n 2 θ 4 − m 2 θ 2 ) ∧ (AR x + 2AQ) ∧ ∂ −1 (m 1 θ 1 − n 1 θ 3 ) +(2m 2 θ 3 − 2n 2 θ 1 + m 1 θ 4 − n 1 θ 2 ) ∧ (AQ x + 2AR) ∧ ∂ −1 (m 1 θ 1 − n 1 θ 3 ) +(2m 1 θ 4 − 2n 1 θ 2 + m 2 θ 3 − n 2 θ 1 ) ∧ (AQ x + 2AR) ∧ ∂ −1 (n 2 θ 4 − m 2 θ 2 ) +(m 1 θ 1 + n 1 θ 3 − 2m 2 θ 2 − 2n 2 θ 4 ) ∧ (AR x + 2AQ) ∧ ∂ −1 (n 2 θ 4 − m 2 θ 2 ) +(m 1 θ 4 − m 2 θ 3 ) ∧ (AR x + 2AQ) ∧ ∂ −1 (n 2 θ 1 − n 1 θ 2 ) −(n 1 θ 3 + n 2 θ 4 ) ∧ (AQ x + 2AR) ∧ ∂ −1 (n 2 θ 1 − n 1 θ 2 ) +(m 1 θ 1 + m 2 θ 2 ) ∧ (AQ x + 2AR) ∧ ∂ −1 (m 1 θ 4 − m 2 θ 3 ) − (n 2 θ 1 − n 1 θ 2 ) ∧ (AR x + 2AQ) ∧ ∂ −1 (m 1 θ 4 − m 2 θ 3 ) dx.
Next we consider pr v Lθ (Θ F ). For simplicity, we denote
Υ = 2m 1 θ 1 − m 2 θ 2 − 2n 1 θ 3 + n 2 θ 4 , Ω = m 1 θ 1 − 2m 2 θ 2 − n 1 θ 3 + 2n 2 θ 4 ,
then a direct calculation shows that pr v Lθ (Θ F ) can be expressed as
−pr v Lθ (Θ F ) = (m 1 θ 1 + n 1 θ 3 ) ∧ ∂ −1 Υ ∧ (2AQ + AR x ) +2(n 2 θ 1 − n 1 θ 2 − m 2 θ 3 + m 1 θ 4 ) ∧ (AR x + 2AQ) ∧ (AQ x + 2AR) −(m 2 θ 2 + n 2 θ 4 ) ∧ ∂ −1 Ω ∧ (2AQ + AR x ) +(n 2 θ 1 − n 1 θ 2 ) ∧ ∂ −1 (m 1 θ 4 − m 2 θ 3 ) ∧ (2AQ + AR x ) −(m 1 θ 4 − m 2 θ 3 ) ∧ ∂ −1 (n 2 θ 1 − n 1 θ 2 ) ∧ (2AQ + AR x ) +(m 1 θ 4 − n 1 θ 2 ) ∧ ∂ −1 Υ ∧ (2AR + AQ x ) −(n 2 θ 1 − m 2 θ 3 ) ∧ ∂ −1 Ω ∧ (2AR + AQ x ) +(n 2 θ 4 + n 1 θ 3 ) ∧ ∂ −1 (m 1 θ 4 − m 2 θ 3 ) ∧ (2AR + AQ x ) −(m 2 θ 2 + m 1 θ 1 ) ∧ ∂ −1 (n 2 θ 1 − n 1 θ 2 ) ∧ (2AR + AQ x ) dx.
Letting f = n 2 θ 1 − n 1 θ 2 , g = m 1 θ 4 − m 2 θ 3 and substituting above expansions into (??) lead to
−pr v Lθ (Θ L ) = 2(n 2 θ 1 − n 1 θ 2 − m 2 θ 3 + m 1 θ 4 ) ∧ (AR x + 2AQ) ∧ (AQ x + 2AR) +2((n 2 θ 1 − n 1 θ 2 ) ∧ ∂ −1 (m 1 θ 4 − m 2 θ 3 ) +(m 2 θ 3 − m 1 θ 4 ) ∧ ∂ −1 (n 2 θ 1 − n 1 θ 2 )) ∧ (2AQ + AR x ) +((n 2 θ 1 − n 1 θ 2 − m 2 θ 3 + m 1 θ 4 ) ∧ ∂ −1 Q +∂ −1 (n 2 θ 1 − n 1 θ 2 − m 2 θ 3 + m 1 θ 4 ) ∧ Q) ∧ (2AR + AQ x )dx = (f ∧ ∂ −1 2g + ∂ −1 f ∧ 2g) ∧ (2AQ + AR x ) +2(f + g) ∧ (AR x + 2AQ) ∧ (AQ x + 2AR) +((f + g) ∧ ∂ −1 Q + ∂ −1 (f + g) ∧ Q) ∧ (2AR + AQ x )dx = ∂ −1 (2g + R) ∧ (Q ∧ (AQ x + 2AR) − R ∧ (AR x + 2AQ)) +(2g + R) ∧ (−∂ −1 R ∧ (AR x + 2AQ) + ∂ −1 Q ∧ (AQ x + 2AR) +2(AR x + 2AQ)(AQ x + 2AR))dx = ∂ −1 (2g + R) ∧ (∂ −1 R ∧ (AR xx + 2AQ x ) − ∂ −1 Q ∧ (AQ xx + 2AR x ) −2(AR xx + 2AQ x )(AQ x + 2AR) − 2(AR x + 2AQ)(AQ xx + 2AR x ))dx = ∂ −1 (2g + R)(∂ −1 R ∧ (2AQ x + 4AR) − ∂ −1 Q ∧ (2AR x + 4AQ) −2∂ −1 R ∧ (AQ x + 2AR) − 2(AR x + 2AQ) ∧ ∂ −1 Q)dx. = 0,
where we use f − g = R for short. Thus, L given by (??) is a Hamiltonian operator.
Finally we prove the compatibility of K and L, which is equivalent to pr v Lθ (Θ K ) + pr v Kθ (Θ L ) = pr v Kθ (Θ L ) = pr v Kθ (Θ J ) + pr v Kθ (Θ F ) = 0.
To this end, we notice
−pr v Kθ (Θ J ) = (2(θ 4 ∧ θ 2 ) x + (θ 1 ∧ θ 3 ) x + θ 1 ∧ θ 2 + θ 3 ∧ θ 4 ) ∧ ∂ −1 (n 2 θ 4 − m 2 θ 2 ) +((θ 4 ∧ θ 2 ) x + 2(θ 1 ∧ θ 3 ) x − θ 1 ∧ θ 2 − θ 3 ∧ θ 4 ) ∧ ∂ −1 (m 1 θ 1 − n 1 θ 3 ) +((θ 1 ∧ θ 2 ) x − θ 1 ∧ θ 3 − θ 2 ∧ θ 4 ) ∧ ∂ −1 (m 1 θ 4 − m 2 θ 3 ) +((θ 4 ∧ θ 3 ) x + θ 1 ∧ θ 3 + θ 2 ∧ θ 4 ) ∧ ∂ −1 (n 2 θ 1 − n 1 θ 2 )dx = (−(θ 1 ∧ θ 2 + θ 3 ∧ θ 4 ) ∧ ∂ −1 Q + (θ 1 ∧ θ 3 + θ 2 ∧ θ 4 ) ∧ ∂ −1 R (2θ 4 ∧ θ 2 + θ 1 ∧ θ 3 ) ∧ (n 2 θ 4 − m 2 θ 2 ) − (θ 1 ∧ θ 2 ) ∧ (m 1 θ 4 − m 2 θ 3 ) −(θ 4 ∧ θ 2 + 2θ 1 ∧ θ 3 ) ∧ (m 1 θ 1 − n 1 θ 3 ) + (θ 3 ∧ θ 4 ) ∧ (n 2 θ 1 − n 1 θ 2 )dx = −(θ 1 ∧ θ 2 + θ 3 ∧ θ 4 ) ∧ ∂ −1 Q + (θ 1 ∧ θ 3 + θ 2 ∧ θ 4 ) ∧ ∂ −1 Rdx. and −pr v Kθ (Θ F ) = (θ 1 ∧ (θ 3x − θ 2 ) + θ 2 ∧ (θ 1 + θ 4x ) − θ 3 ∧ (θ 1x + θ 4 ) − θ 4 ∧ (θ 2x − θ 3 )) ∧(2AQ + ∂AR) + (AQ x + 2AR) ∧ ((θ 2x − θ 3 ) ∧ θ 1 − (θ 1x + θ 4 ) ∧ θ 2 +(θ 1 + θ 4x ) ∧ θ 3 − (θ 3x − θ 2 ) ∧ θ 4 )dx.
= ((∂ 2 − 4)(θ 1 ∧ θ 2 + θ 3 ∧ θ 4 )) ∧ AQ − ((∂ 2 − 4)(θ 1 ∧ θ 3 + θ 2 ∧ θ 4 )) ∧ ARdx
= (θ 1 ∧ θ 2 + θ 3 ∧ θ 4 ) ∧ ∂ −1 Q − (θ 1 ∧ θ 3 + θ 2 ∧ θ 4 ) ∧ ∂ −1 Rdx.
Therefore we arrive at pr v Kθ (Θ L ) = pr v Kθ (Θ J + Θ F ) = 0, so K and L are two compatible Hamiltonian operators.
AcknowledgmentsWe would like to thank Decio Levi for interesting discussions. NHL and QPL are supported by the National Natural Science Foundation of China (grant numbers: 10971222 and 11271366) and the Fundamental Research Funds for Central Universities. ZP thanks the China University of Mining and Technology for its support during his visit to Beijing where this work has been carried out.
An integrable shallow water equation with peaked solitons. R Camassa, D D Holm, Phys. Rev. Lett. 71R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661-1664.
A new integrable shallow water equation. R Camassa, D D Holm, J M Hyman, Adv. Appl. Mech. 31R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994) 1-33.
A Degasperis, M Procesi, Asymptotic integrability. Symmetry and Perturbation Theory A. Degasperis and G. GaetaSingaporeWorld ScientificA. Degasperis and M. Procesi, Asymptotic integrability, in: Symmetry and Perturbation Theory A. Degasperis and G. Gaeta (eds.), Singapore: World Scientific (1999), pp 23-37.
A new integrable equation with peakon solutions. A Degasperis, D D Holm, A N W Hone, Theor. Math. Phys. 133A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys. 133 (2002) 1463-74.
A S Fokas, P J Olver, P Rosenau, Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman. A. S. Fokas and I. M. GelfandBostonBirkhauserA plethora of integrable bi-Hamiltonian equationsA. S. Fokas, P. J. Olver and P. Rosenau, A plethora of integrable bi-Hamiltonian equations, in: Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman, A. S. Fokas and I. M. Gelfand (eds.), Birkhauser: Boston (1996), pp93-101.
On a class of physically important integrable equations. A S Fokas, Physica D. 87A. S. Fokas, On a class of physically important integrable equations, Physica D 87 (1995) 145-150.
Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation. B Fuchssteiner, Phys. D. 95B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: general- izations of the Camassa-Holm equation, Phys. D 95 (1996) 229-243.
An extension of integrable peakon equations with cubic nonlinearity. X G Geng, B Xue, Nonlinearity. 22X. G. Geng and B. Xue, An extension of integrable peakon equations with cubic nonlin- earity, Nonlinearity 22 (2009) 1847-1856.
A three-component generalization of Camassa-Holm equation with N-peakon solutions. X G Geng, B Xue, Adv. Math. 226X. G. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions, Adv. Math. 226 (2011) 827-839.
Prolongation algebras and Hamiltonian operators for peakon equations. A N W Hone, J P Wang, Inverse Problems. 19A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems 19 (2003) 129-145.
Integrable peakon equations with cubic nonlinearity. A N W Hone, J P Wang, J. Phys. A: Math. Theor. 41372002A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A: Math. Theor. 41 (2008) 372002.
On Bi-Hamiltonian structure of two-component Novikov equation. N H Li, Q P Liu, Phys. Lett. A. 377N. H. Li and Q. P. Liu, On Bi-Hamiltonian structure of two-component Novikov equation, Phys. Lett. A 377 (2012) 257-261.
Bi-Hamiltonian structure of a three-component Camassa-Holm type equation. N H Li, Q P Liu, J. Nonlinear Math. Phys. 20N. H. Li and Q. P. Liu, Bi-Hamiltonian structure of a three-component Camassa-Holm type equation, J. Nonlinear Math. Phys. 20 (2013) 126-134.
Multi-peakon solutions of the Degasperis-Procesi equation. H Lundmark, J Szmigielski, Inverse problems. 19H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equa- tion, Inverse problems 19 (1994) 1241-5.
Generalisations of the Camassa-Holm equation. V S Novikov, J. Phys. A: Math. Theor. 4234200214 pagesV. S. Novikov, Generalisations of the Camassa-Holm equation, J. Phys. A: Math. Theor. 42 (2009) 342002 (14 pages).
P J Olver, Applications of Lie Groups to Differential Equations. BerlinSpringer2nd ednP. J. Olver, Applications of Lie Groups to Differential Equations, 2nd edn, Berlin: Springer (1993).
Tri-Hamiltonian duality between solitons and solitary-waves solutions having compact support. P J Olver, P Rosenau, Phys. Rev. E. 53P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-waves solutions having compact support, Phys. Rev. E 53 (1996) 1900-1906.
The generalizations of the peakon's systems, talk given at the third international conference on Nonlinear Waves: theory and applications. Z Popowicz, BeijingZ. Popowicz, The generalizations of the peakon's systems, talk given at the third interna- tional conference on Nonlinear Waves: theory and applications, June 12-15, 2013, Beijing.
A new integrable equation with cuspons and W/M-shape-peaks solitons. Z J Qiao, J. Math. Phys. 479112701Z. J. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys. 47 (2006) 112701 (9 pages).
Multi-component integrable systems and invariant curve flows in certian geometries. C Z Qu, J F Song, R X Yao, SIGMA. 9119 pagesC. Z. Qu, J. F. Song and R. X. Yao, Multi-component integrable systems and invariant curve flows in certian geometries, SIGMA 9 (2013) 001 (19 pages).
A new integrable two-component system with cubic nonlinearity. J F Song, C Z Qu, Z J Qiao, J. Math. Phys. 52913503J. F. Song, C. Z. Qu and Z. J. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys. 52 (2011) 013503 (9 pages).
Tri-Hamiltonian duality between the Wadati-Konno-Ichikawa hierarchy and Song-Qu-Qiao hierarchy. K Tian, Q P Liu, J. Math. Phys. 544351310 pagesK. Tian and Q. P. Liu, Tri-Hamiltonian duality between the Wadati-Konno-Ichikawa hi- erarchy and Song-Qu-Qiao hierarchy, J. Math. Phys. 54 (2013) 043513 (10 pages).
Integrable multi-component Camassa-Holm system. B Q Xia, Z J Qiao, arXiv:1310.0268v1B. Q. Xia and Z. J. Qiao, Integrable multi-component Camassa-Holm system, arXiv: 1310.0268v1.
| [] |
[
"Notes on p-adic numbers",
"Notes on p-adic numbers"
] | [
"Stephen William Semmes \nRice University Houston\nTexas\n"
] | [
"Rice University Houston\nTexas"
] | [] | As usual the real numbers are denoted R. For each x ∈ R, the absolute value of x is denoted |x| and defined to be x when x ≥ 0 and to be −x when x ≤ 0. Thus |0| = 0 and |x| > 0 when x = 0. One can check that |x + y| ≤ |x| + |y| and |x y| = |x| |y| for every x, y ∈ R.Let N (x) be a nonnegative real-valued function defined on the rational numbers Q such that N (0) = 0, N (x) > 0 when x = 0, | null | [
"https://export.arxiv.org/pdf/math/0502560v1.pdf"
] | 118,966,615 | math/0502560 | e43e743a62bb886c94f102cc54341487678113fb |
Notes on p-adic numbers
26 Feb 2005
Stephen William Semmes
Rice University Houston
Texas
Notes on p-adic numbers
26 Feb 2005
As usual the real numbers are denoted R. For each x ∈ R, the absolute value of x is denoted |x| and defined to be x when x ≥ 0 and to be −x when x ≤ 0. Thus |0| = 0 and |x| > 0 when x = 0. One can check that |x + y| ≤ |x| + |y| and |x y| = |x| |y| for every x, y ∈ R.Let N (x) be a nonnegative real-valued function defined on the rational numbers Q such that N (0) = 0, N (x) > 0 when x = 0,
N (x y) = N (x) N (y) (1) for all x, y ∈ Q, and N (x + y) ≤ C (N (x) + N (y)) (2) for some C ≥ 1 and all x, y ∈ Q. For the usual triangle inequality one aks that this condition holds with C = 1, i.e.,
N (x + y) ≤ N (x) + N (y) (3) for all x, y ∈ Q. The ultrametric version of the triangle inequality is stronger still and asks that N (x + y) ≤ max(N (x), N (y)) (4) for all x, y ∈ Q. If N satisfies (2), l is a positive integer, and x 1 , . . . , x 2 l ∈ Q, then
N 2 l j=1 x j ≤ C l 2 l j=1 N (x j ),(5)
as one can check using induction on l.
The usual absolute value function |x| satisfies these conditions with the ordinary triangle inequality (4). If N (x) = 0 when x = 0 and N (x) = 1 when x = 0, then N (x) satisfies these conditions with the ultrametric version of the triangle inequality. For each prime number p, the p-adic absolute value of a rational number x is denoted |x| p and defined by |x| p = 0 when x = 0, and |x| p = p −j when x is equal to p j times a ratio of nonzero integers, neither of which is divisible by p. One can check that |x| p satisfies these conditions with the ultrametric version of the triangle inequality. Roughly speaking, this says that x + y has at least j factors of p when x and y have at least j factors of p.
Let a be a positive real number. Clearly max(r, t) ≤ (r a + t a ) 1/a (6) for all r, t ≥ 0, and hence r + t ≤ max(r 1−a , t 1−a ) (r a + t a ) ≤ (r a + t a ) 1/a (7) when 0 < a ≤ 1, which implies that (r + t) a ≤ r a + t a .
When a ≥ 1, (r + t) a ≤ 2 a−1 (r a + t a ), (9) because x a is a convex function on the nonnegative real numbers.
For all a > 0, N (x) a is a nonnegative real-valued function on Q which vanishes at 0, is positive at all nonzero x ∈ Q, and sends products to products. If N (x) satisfies (2), then N (x + y) a ≤ C a (N (x) a + N (y) a ) (10) when 0 < a ≤ 1 and N (x + y) a ≤ 2 a−1 C a (N (x) a + N (y) a ) when a ≥ 1. In particular, if N (x) satisfies the ordinary triangle inequality (3) and 0 < a ≤ 1, then N (x) a also satisfies the ordinary triangle inequality. If N (x) satisfies the ultrametric version (4) of the triangle inequality, then N (x) a satisfies the ultrametric version of the triangle inequality for all a > 0.
Observe that N (1) = 1,
because N (1) 2 = N (1) and N (1) > 0, and similarly N (−1) = 1.
Suppose that N (x) > 1 for some integer x. In this event N is unbounded on the integers Z, because N (x n ) = N (x) n → ∞ (13) as n → ∞, and we say that N is Archimedian. Otherwise,
N (x) ≤ 1 for all x ∈ Z,(14)
and N is non-Archimedian.
Suppose that N is Archimedian, and let A be the set of positive real numbers a such that N (x) is bounded by a constant times |x| a for every x ∈ Z. It follows from (5) that A = ∅, and (13) implies that A has a positive lower bound. For every a ∈ A and x ∈ Z, N (x) ≤ |x| a , because N (x) > |x| a implies that N (x j ) is not bounded by a constant times |x| j a . Let α be the infimum of A. By the previous remarks, α > 0 and
N (x) ≤ |x| α(15)
for every x ∈ Z.
Let n be an integer, n > 1. Every nonnegative integer can be expressed as a finite sum of the form l j=0 r j n j , where each r j is an integer and 0 ≤ r j < n. If N (n) ≤ 1, then N (x) is bounded by a constant times a power of the logarithm of 1 + |x| for x ∈ Z, contradicting the fact that A has a positive lower bound. Similarly, if N (n) < n α , which means that N (n) = n b for some b < α, then N (x) is bounded by a constant times a power of the logarithm of 1 + |x| times |x| b for x ∈ Z, contradicting the fact that α is the infimum of A. We conclude that
N (x) = |x| α (16)
for every x ∈ Z, and therefore for every x ∈ Q. Now suppose that N is non-Archimedian, and let x, y ∈ Q be given. The binomial theorem asserts that (x + y) n = n j=0 n j
x j y n−j (17) for each positive integer n, where the binomial coefficients
n j = n! j! (n − j)! (18)
are integers, and k! is "k factorial", the product of the positive integers from 1 to k, which is interpreted as being equal to 1 when k = 0. If l is a positive integer and n + 1 ≤ 2 l , then (5) implies that
N (x + y) n = N (x + y) n (19) ≤ C l n j=0 N (x) j N (y) n−j ≤ (n + 1) C l max(N (x) n , N (y) n ).
Using this one can check that N satisfies the ultrametric version of the triangle inequality (4).
If N (x) = 1 for each nonzero integer x, then N (x) = 1 for every x ∈ Q, x = 0. Otherwise, there is an integer p > 1 such that N (p) < 1, and we may choose p to be as small as possible. If p = p 1 p 2 , where p 1 , p 2 ∈ Z and p 1 , p 2 > 1, then N (p 1 ) N (p 2 ) < 1, and hence N (p 1 ) < 1 or N (p 2 ) < 1, which contradicts the minimality of p. Therefore p is prime. Of course N (x) ≤ 1 for every x ∈ Z, and the minimality of p implies that N (x) = 1 when x ∈ Z and 1 ≤ x < p. If x = p y, y ∈ Z, then
N (x) = N (p) N (y) ≤ N (p) < 1.(20)
Using the ultrametric version of the triangle inequality for N , one can check that N (x) = 1 when x ∈ Z and x is not an integer multiple of p. Moreover,
N (x) = |x| a p(21)
for every x ∈ Q, where a > 0 is determined by N (p) = p −a .
Fix a prime number p. The p-adic metric on Q is defined by
d p (x, y) = |x − y| p (22) for x, y ∈ Q. A sequence {x j } ∞
j=1 of rational numbers converges to x ∈ Q in the p-adic metric if for every ǫ > 0 there is an L ≥ 1 such that
|x j − x| p < ǫ (23) for every j ≥ L. If {x j } ∞ j=1 , {y j } ∞ j=1
are sequences of rational numbers which converge in the p-adic metric to x, y ∈ Q, then the sequences of sums x j + y j and products x j y j converge to the sum x + y and product x y of the limits of the initial sequences, by standard arguments.
A sequence {x j } ∞ j=1 of rational numbers is a Cauchy sequence with respect to the p-adic metric if for each ǫ > 0 there is an L ≥ 1 such that
|x j − x l | p < ǫ (24)
for every j, l ≥ L. Every convergent sequence in Q is a Cauchy sequence. If {x j } ∞ j=1 is a Cauchy sequence in Q with respect to the p-adic metric, then the sequence of differences x j − x j+1 converges to 0 in the p-adic metric. Of course the analogous statement also works for the standard metric |x − y|. For the padic metric, the converse holds because of the ultrametric version of the triangle inequality.
The p-adic numbers Q p are obtained by completing the rational numbers with respect to the p-adic metric, just as the real numbers are obtained by completing the rational numbers with respect to the standard metric. To be more precise, Q ⊆ Q p , and Q p is equipped with operations of addition and multiplication which agree with the usual arithmetic operations on Q and which satisfy the standard field axioms. There is an extension of the p-adic absolute value |x| p to x ∈ Q p which satisfies the same conditions as before, i.e., |x| p is a nonnegative real number for all x ∈ Q p which is equal to 0 exactly when x = 0, and |x y| p = |x| p |y| p (25) and
|x + y| p ≤ max(|x| p , |y| p )
for every x, y ∈ Q p . We can extend the p-adic metric to Q p using the same formula
d p (x, y) = |x − y| p (27)
for x, y ∈ Q p . The rational numbers are dense in Q p , in the sense that for each x ∈ Q p and ǫ > 0 there is a y ∈ Q such that
|x − y| p < ǫ. (28) A sequence {x j } ∞ j=1 of p-adic numbers converges to x ∈ Q p if for each ǫ > 0 there is an L ≥ 1 such that |x j − x| p < ǫ(29)
for every j ≥ L. One can show that sums and products of convergent sequences in Q p converge to the sums and products of the limits of the individual se-
quences. A sequence {x j } ∞ j=1 in Q p is a Cauchy sequence if for every ǫ > 0 there is an L ≥ 1 such that |x j − x l | p < ǫ (30)
for every j, l ≥ L. Convergent sequences are Cauchy sequences, and a sequence
{x j } ∞ j=1 in Q p is a Cauchy sequence if and only if lim j→∞ (x j − x j+1 ) = 0 in Q p ,(31)
because of the ultrametric version of the triangle inequality. The p-adic numbers are complete in the sense that every Cauchy sequence in Q p converges to an element of Q p .
Suppose that
x ∈ Q p , x = 0. Because Q is dense in Q p , there is a y ∈ Q such that |x − y| p < |x| p .(32)
Using the ultrametric version of the triangle inequality, one can check that
|x| p = |y| p .(33)
It follows that |x| p is an integer power of p.
An infinite series ∞ j=0 a j with terms in Q p converges if the corresponding sequence of partial sums n j=0 a j converges in Q p . It is easy to see that the partial sums for ∞ j=0 a j form a Cauchy sequence in Q p if and only if the a j 's converge to 0 in Q p , in which event the series converges. If ∞ j=0 a j , ∞ j=0 b j are convergent series in Q p and α, β ∈ Q p , then ∞ j=0 (α a j + β b j ) converges, and
∞ j=0 (α a j + β b j ) = α ∞ j=0 a j + β ∞ j=0 b j .(34)
This follows from the analogous statements for convergent sequences.
Suppose that x ∈ Q p . For each nonnegative integer n,
(1 − x) n j=0 x j = 1 − x n+1 .(35)
Here x j is interpreted as being 1 when j = 0 for every x ∈ Q p . If x = 1, then we can rewrite this identity as
n j=0 x j = 1 − x n+1 1 − x . (36) If |x| p < 1, then ∞ j=0 x j converges, and ∞ j=0 x j = 1 1 − x .(37)
Let us write p Z for the set of integer multiples of p, which is an ideal in the ring of integers Z. There is a natural mapping from Z onto Z/p Z, the integers modulo p. Addition and multiplication are well-defined modulo p, and this mapping is a homomorphism, which is to say that it preserves these operations. The fact that p is prime implies that Z/p Z is a field, which means that nonzero elements have inverses. Equivalently, for each integer x which is not a multiple of p, there is an integer y such that x y − 1 ∈ p Z.
For every w ∈ Z, ∞ j=0 p j w j converges in Q p to 1/(1 − p w), as in (37). In particular, 1/(1 − p w) is the limit of a sequence of integers in the p-adic metric. Hence if a, b ∈ Z and b − 1 ∈ p Z, then a/b is the limit of a sequence of integers in the p-adic metric. If b ∈ Z and b ∈ p Z, then there is a c ∈ Z such that b c − 1 ∈ p Z, and hence a/b + (a c)/(b c) is a limit of integers in the p-adic metric.
Of course every integer has p-adic absolute value less than or equal to 1, and the limit of any sequence of integers in the p-adic metric has p-adic absolute value less than or equal to 1. The discussion in the previous paragraph shows that every x ∈ Q with |x| p ≤ 1 is the limit of a sequence of integers in the p-adic metric. In other words,
{x ∈ Q : |x| p ≤ 1} (38)
is the same as the closure of Z in Q with respect to the p-adic metric.
Put
Z p = {x ∈ Q p : |x| p ≤ 1}.(39)
Every element of Q p is the limit of a sequence of rational numbers in the p-adic metric, because Q is dense in Q p . If x ∈ Z p , then x is the limit of a sequence of rational numbers {x j } ∞ j=1 in the p-adic metric, and |x j | p ≤ 1 for sufficiently large j by the ultrametric version of the triangle inequality. Because rational numbers with p-adic absolute value less than or equal to 1 can be approximated by integers in the p-adic metric, x is the limit of a sequence of integers in the p-adic metric. In short, Z p is equal to the closure of Z in Q p .
The elements of Z p are said to be p-adic integers. Let us write p Z p for the set of p-adic integers of the form p x, x ∈ Z p , which is the same as
p Z p = {y ∈ Q p : |y| p ≤ 1/p}. (40) Of course Z ⊆ Z p and p Z ⊆ p Z p . If x ∈ Z p , then x can be expressed as y + w, where y ∈ Z and w ∈ p Z p .
Note that sums and products of p-adic integers are p-adic integers, which is to say that Z p is a ring with respect to addition and multiplication, a subring of the field Q p . Furthermore, p Z p is an ideal in Z p , which means that sums of elements of p Z p lie in p Z p , and the product of an element of p Z p and an element of Z p lies in p Z p . Consequently, there is a quotient ring Z p /p Z p and a natural mapping from Z p onto the quotient Z p /p Z p which preserves addition and multiplication.
The inclusions of p Z, Z in p Z p , Z p , respectively, lead to a natural mapping
Z/p Z → Z p /p Z p .(41)
If one maps an element of Z into Z/p Z, and then into Z p /p Z p , then that is the same as mapping the element of Z into Z p , and then into the quotient Z p /p Z p . One can check that the mapping (41) is a ring isomorphism. To describe the inverse more explicitly, if x ∈ Z p , x = y + w, y ∈ Z, w ∈ p Z p , then the image of y in the quotient Z/p Z corresponds exactly to the image of x in the quotient Z p /p Z p .
More generally, for each positive integer j, p j Z denotes the set of integer multiples of p j , which is an ideal in Z, and p j Z p denotes the ideal in Z p consisting of the products p j x, x ∈ Z p , which is the same as
p j Z p = {y ∈ Q p : |y| p ≤ p −j }.(42)
The inclusion p j Z ⊆ p j Z p leads to a homomorphism
Z/p j Z → Z p /p j Z p(43)
between the corresponding quotients, which is an isomorphism. This uses the fact that each x ∈ Z p can be expressed as y + w, with y ∈ Z and w ∈ p j Z p , because Z is dense in Z p .
We say that C ⊆ Q p is a cell if it is of the form
C = {y ∈ Q p : |y − x| p ≤ p l }(44)
for some x ∈ Q p and l ∈ Z. The diameter of C, denoted diam C, is equal to p −l in this case, which is to say that the maximum of the p-adic distances between elements of C is equal to p −l . Note that cells are both open and closed as subsets of Q p , and hence Q p is totally disconnected. More precisely, the distances between points in a cell and points in the complement are greater than or equal to the diameter of the cell. Also, if C 1 , C 2 are cells in Q p , then C 1 ⊆ C 2 , or C 2 ⊆ C 1 , or C 1 ∩ C 2 = ∅.
If C is a cell in Q p and n is a positive integer, then C contains p n disjoint cells with diameter equal to p −n C, and C is equal to the union of these smaller cells. For instance, Z p is equal to the disjoint union of translates of p n Z p , one for each element of Z/p n Z. Every cell in Q p can be obtained from Z p by a translation and dilation, and so the decompositions for Z p yield the analogous decompositions for arbitrary cells in Q p . One can show that cells in Q p are compact, in practically the same way as for closed and bounded intervals in the real line. Consequently, closed and bounded subsets of Q p are compact.
Suppose that C is a cell in Q p and that f (x) is a continuous real-valued function on C. By standard results in analysis, f is uniformly continuous. For each positive integer n, let C 1,n , . . . , C p n ,n be the p n disjoint cells of diameter p −n diam C contained in C. This leads to the Riemann sums
p n j=1 f (x j ) p −n diam C,(45)
where x j ∈ C j,n for each j. Because of uniform continuity, these Riemann sums converge as n → ∞ to the integral of f over C, and the limit does not depend on the choices of the x j 's.
Of course a Q p -valued function on C is uniformly continuous too, since C is compact. Riemann sums for Q p -valued functions can be defined in the same way as in the previous paragraph, since p −n diam C ∈ Q. However,
|p −n | p = p n → ∞ as n → ∞,(46)
and uniform continuity is not sufficient to imply convergence of the Riemann sums as n → ∞. As in Section 12.4 in [1], one can look at Q p -valued measures on C which are bounded on the cells contained in C. Because of the ultrametric version of the triangle inequality, boundedness of the measures of small cells is adequate to get convergence of Riemann sums of continuous Q p -valued functions on C.
One can also consider continuous Q ℓ -valued functions on C, where ℓ is prime and ℓ = p. As in the previous situations, such a function is uniformly continuous because of the compactness of C. Integer powers of p have absolute value equal to 1 in Q ℓ , which implies that the absolute value of a Riemann sum of a Q ℓ -valued function on C is bounded by the maximum of the absolute value of the function on C, because of the ultrametric version of the triangle inequality. Uniform continuity of the function implies convergence of the Riemann sums, because differences of Riemann sums can be bounded by maximal local oscillations of the function.
Let f (x) be a polynomial on Q p , which means that f (x) = a n x n + · · · + a 0 (47) for some nonnegative integer n and a 0 , . . . , a n ∈ Q p . For the discussion that follows it is convenient to ask that the coefficients a j of f (x) be p-adic integers.
For each x, y ∈ Z p ,
|f (x) − f (y)| p ≤ |x − y| p .(48)
To see this one can first consider the case where y = 0, and then use the observation that the translate of a polynomial with p-adic integer coefficients by an element of Z p also has p-adic integer coefficients. Alternatively, one can use the identity
x n − y n = (x − y) n−1 j=0 x j y n−1−j (49) for n ≥ 1.
The derivative of f (x) is the polynomial f ′ (x) = n a n x n−1 + · · · + a 1 .
Thus f ′ (x) has p-adic integer coefficients, and therefore
|f ′ (x) − f ′ (y)| p ≤ |x − y| p (51)
for every x, y ∈ Z p .
Observe that
|f (x) − f (y) − f ′ (y) (x − y)| p ≤ |x − y| 2 p (52)
for every x, y ∈ Z p . One can check this first when y = 0, and then use translations to get the general case. One can also show this when f (x) = x n and then sum over n.
Suppose that z ∈ Q p , f (z) ∈ p Z p , and that |f ′ (z)| p = 1. Hensel's lemma asserts that there is a w ∈ Z p such that w − z ∈ p Z p and f (w) = 0.
To prove this we use Newton's method. Put x 0 = z, and for each j ≥ 1 choose x j according to the rule
f (x j−1 ) + f ′ (x j−1 ) (x j − x j−1 ) = 0. (53)
Equivalently,
x j = x j−1 + f (x j−1 ) f ′ (x j−1 ) .(54)
More precisely,
x j−1 ∈ Z p , f (x j−1 ) ∈ p Z p , |f ′ (x j−1 )| p = 1 (55)
by induction, which implies that
x j − x j−1 ∈ p Z p .(56)
This ensures that the analogous conditions hold for x j , by (48) and (51).
Using (52), we get
|f (x j )| p ≤ |x j − x j−1 | 2 p . (57) Since |x j+1 − x j | p ≤ |f (x j )| p , |x j+1 − x j | p ≤ |x j − x j−1 | 2 p .(58)
It follows that lim j→∞ (x j+1 − x j ) = 0 in Q p , and hence that {x j } ∞ j=1 converges to an element w of Q p . Because x j − x j−1 ∈ p Z p for every j, w − z ∈ p Z p and w ∈ Z p . Moreover, lim j→∞ f (x j ) = 0 in Q p , and therefore f (w) = 0, as desired.
Suppose now that
z ∈ Z p , |f (z)| p < |f ′ (z)| 2 p .(59)
If f (z) ∈ p Z p and |f ′ (z)| p = 1, then |f (z)| p < |f ′ (z)| 2 p , and thus these conditions are more general than the previous ones. A refined version of Hensel's lemma asserts that there is a w ∈ Z p such that f (w) = 0.
Again we put x 0 = z and choose x j according to the same rule as in the previous situation. This uses the induction hypotheses
x j−1 ∈ Z p , |f (x j−1 )| p < |f ′ (x j−1 )| 2 p = |f ′ (z)| 2 p .(60)
Under these conditions,
|x j − x j−1 | p ≤ |f (x j−1 )| p |f ′ (x j−1 )| p < |f ′ (x j−1 )| p ,(61)which implies that |f ′ (x j )−f ′ (x j−1 )| p < |f ′ (x j−1 )| p and |f ′ (x j )| p = |f ′ (x j−1 )| p . Furthermore, |f (x j )| p ≤ |x j − x j−1 | 2 p < |f (x j−1 )| p ,(62)
and hence the induction hypotheses continue to be satisfied for x j . One can check that lim j→∞ f (x j ) = 0 (63) in Q p , and that {x j } ∞ j=1 converges to w ∈ Z p such that f (w) = 0. If x, y ∈ Q p , n is a positive integer, and x = y n , then |x| p = |y| n p , which implies that x = 0 or |x| p is of the form p l n for some l ∈ Z. Of course x = y n with y = 0 when x = 0. If |x| p = p l n for some integer l, and x 1 = p −l x, then |x 1 | p = 1 and x = y n for some y ∈ Q p if and only if x 1 = (y 1 ) n for some y 1 ∈ Q p with |y 1 | p = 1.
The mapping h(y) = y n (64) satisfies h(Z p ) ⊆ Z p and |h(z) − h(w)| p ≤ |z − w| p . Therefore h induces a mapping on Z p /p Z p ∼ = Z/p Z, which also takes nth powers.
Fix a positive integer q which is prime and a ∈ Q p with |a| p = 1. Consider the polynomial
f (x) = x q − a.(65)
Thus the coefficients of f are p-adic integers, and the zeros of f are the qth roots of a. The derivative of f is
f ′ (x) = q x q−1 .(66)
In particular, for every x ∈ Q p with |x| p = 1,
|f ′ (x)| p = 1 (67) when q = p and |f ′ (x)| p = 1 p (68) when q = p.
Let (Z/p Z) * be the group of nonzero elements of Z/p Z under multiplication, a finite abelian group with p− 1 elements. It is a well-known theorem that every finite abelian group is isomorphic to a Cartesian product of finitely many cyclic groups. Another well-known theorem asserts that Z/p Z is cyclic.
If p − 1 is not an integer multiple of q, then every element of Z/p Z is a qth power. If p − 1 is not an integer multiple of q and q = p, then we can apply Hensel's lemma to get that f (x) = x q − a has a root.
Suppose that p − 1 is an integer multiple of q, which implies that q = p. A necessary condition for f (x) = x q − a to have a root in Z p is that the image of a in Z p /p Z p ∼ = Z/p Z be a qth power there. Hensel's lemma implies that this condition is sufficient too.
When q = p = 2, one can check that x 2 − 1 ∈ 8 Z 2 for every x ∈ Z 2 such that |x| 2 = 1. Conversely, the refined version of Hensel's lemma implies that every y ∈ Z 2 such that y − 1 ∈ 8 Z 2 is equal to x 2 for some x ∈ Q 2 such that |x| 2 = 1.
Let us consider real and complex-valued functions again briefly. Suppose that f (x) is a continuous function from Z p into R or C such that
f (x + y) = f (x) + f (y) (69)
for every x, y ∈ Z p . The image of f is a compact subgroup of R or C, as appropriate, under addition, and it follows that f (x) = 0 for every x ∈ Z p . Now suppose that φ(x) is a continuous function from Z p into nonzero complex numbers such that
φ(x + y) = φ(x) φ(y)(70)
for every x, y ∈ Z p . Since log |φ(x)| is a homomorphism from Z p into R with respect to addition, the remarks of the previous paragraph yield
|φ(x)| = 1 (71)
for every x ∈ Z p . If U is a small neighborhood of 1 in C, then there is a positive integer j such that
φ(p j Z p ) ⊆ U,(72)
because of the continuity of φ. Since p j Z p is a subgroup of Z p under addition, φ(p j Z p ) is a subgroup of the unit circle in C under multiplication, and it follows that φ(x) = 1 (73)
for every x ∈ p j Z p . Thus φ corresponds to a homomorphism from Z p /p j Z p ∼ = Z/p j Z as a group under addition into the unit circle in C as a group under multiplication.
Next let us consider some connections with matrices, following [1]. Of course 1 × 1 matrices are scalars, and we start with them.
Let p be a prime and let x be an element of Z p such that x = 1 and x−1 p Z p . We can express x as 1 + p j a, where j is a positive integer and |a| p = 1. Let q be a prime and consider x q .
Using the binomial theorem we can express x q as 1 + q p j a + b p 2 j , where b ∈ Z p . If q = p or j ≥ 2, then x q − 1 reduces to q p j a = 0 in Z p /p j+1 Z p , and therefore x q = 1.
Suppose that q = p, j = 1, and p = 2. Using the binomial theorem we can express x p as 1 + p 2 a + p 3 b, where b ∈ Z p . Again we get x p = 1.
When q = p = 2 and j = 1, x 2 = 1 is possible, since x may be −1. As oberseved previously, |x| 2 = 1 implies that x 2 − 1 ∈ 8 Z 2 .
To summarize, when p = 2, x q = 1, and it follows that x t = 1 for all positive integers t, by repeating the argument. When p = 2, it may be that x 2 = 1.
Otherwise, x t = 1 for all positive integers t.
Fix a positive integer n, and let M n (Q p ) be the space of n × n matrices with entries in Q p . We can add and multiply matrices in M n (Q p ) in the usual way, or multiply matrices and elements of Q p .
Let M n (Z p ) be the space of n × n matrices with entries in Z p . Sums and products of matrices in M n (Z p ) also lie in M n (Z p ), as do products of matrices in M n (Z p ) and elements of Z p . Of course I ∈ M n (Z p ), where the entries of I on the diagonal are equal to 1 and the entries off of the diagonal are equal to 0.
For each AM n (Q p ) the determinant det A is defined in the usual way and is an element of Q p . If A ∈ M n (Z p ), then det A ∈ Z p .
We say that
A ∈ M n (Q p ) is invertible if there is a B ∈ M n (Q p ) such that A B = B A = I, in which event the inverse B of A is denoted A −1 . Standard results in linear algebra imply that A ∈ M n (Q p ) is invertible if and only if det A = 0. If A ∈ M n (Z p ), then A is invertible and A −1 ∈ M n (Z p ) if and only if | det A| p = 1.
Suppose that A ∈ M n (Z p ), A = I, and A − I has entries in p Z p . Hence A = I + p j B for some positive integer j and B ∈ M n (Z p ), where at least one entry of B has p-adic absolute value equal to 1.
Let q be prime and consider A q . The binomial theorem implies that
A q − I − q p j B(74)
has entries in p 2 j Z p , and it follows that A q = I when q = p or j ≥ 2. If q = p = 2 and j = 1, then
A p − I − p 2 B(75)
has entries in p 3 Z p , and A p = I.
If q = p = 2 and j = 1, then
A 2 = I + 4 B + 4 B 2 .(76)
It may be that A 2 = I, and in any event A 2 − I has entries in 4 Z 2 , and the previous discussion applies to A 2 if A 2 = I.
Let GL n (Q p ) be the group of n × n invertible matrices with entries in Q p , and let GL n (Z p ) be the subgroup of GL(Q p ) consisting of matrices with entries in Z p whose inverse have the same property. There is a natural homomorphism from GL n (Z p ) onto GL n (Z/p Z), the group of n × n invertible matrices with entries in the field Z/p Z, induced by the ring homomorphism from Z p into Z p /p Z p ∼ = Z/p Z. Of course the determinant defines a homomorphism from GL n (Q p ) into the group of nonzero elements of Q p under multiplication, and from GL n (Z p ) into the group of p-adic numbers with p-adic absolute value equal to 1 under multiplication.
If p = 2, A ∈ GL n (Z p ), A − I has entries in p Z p , and A = I, then A q = I for every prime q, and therefore A l = I for every positive integer l. If G is a subgroup of GL n (Z p ) with only finitely many elements, then for each A ∈ G there is a positive integer l such that A l = I, and it follows that the natural homomorphism from GL n (Z p ) into GL n (Z/p Z) is one-to-one on G, because I is the only element of G which can be sent to the identity in GL n (Z/p Z).
If p = 2, A ∈ GL n (Z 2 ), A − I has entries in 2 Z 2 , and A = I, then A 2 = I or A l = I for every positive integer l. If H is a subgroup of GL n (Z 2 ) such that H has only finitely many elements and every entry of A − I is an element of p Z 2 for every A ∈ H, then A 2 = I for every A ∈ H, and hence H is abelian, since (A B) 2 = A B A B = I implies that A B = B A when A 2 = B 2 = I. The earlier remarks also imply that A = I when A ∈ H and A − I has entries in 4 Z 2 .
If G is a subgroup of GL n (Z 2 ) with only finitely many elements, then the subgroup H of G consisting of the matrices A such that A − I has entries in 2 Z 2 satisfies the conditions described in the previous paragraph. Of course H is the same as the subgroup of A ∈ G which go to the identity under the natural homomorphism from GL n (Z 2 ) to GL n (Z/2 Z).
Let GL n (Q) be the group of n × n invertible matrices with entries in Q. We can think of GL n (Q) as a subgroup of GL n (Q p ) for every prime p. If G is a subgroup of GL n (Q) with only finitely many elements, then G is actually a subgroup of GL n (Z p ) for all but finitely many p.
If H ⊆ GL n (Q 2 ) is the collection of diagonal matrices with diagonal entries ±1, then H is a subgroup of GL n (Z 2 ) and every A ∈ H satisfies A 2 = I and A − I has entries in 2 Z 2 . The conjugate of H by any element of GL 2 (Z 2 ) has the same features.
If A is a linear transformation on a vector space V over a field k with characteristic = 2 such that A 2 = I, then V is spanned by the eigenspaces of V corresponding to the eigenvalues ±1. Specifically,
P 1 = I − A 2 , P 2 = I + A 2 (77)
are the projections of V onto these eigenspaces, and any linear transformation B on V which commutes with A maps these eigenspaces onto themselves. These facts provide additional information about subgroups H of GL n (Z 2 ) such that A 2 = I for every A ∈ H.
J Cassels, Local Fields. Cambridge University PressJ. Cassels, Local Fields, Cambridge University Press, 1995.
F Gouvêa, p-Adic Numbers: An Introduction. Springer-Verlagsecond editionF. Gouvêa, p-Adic Numbers: An Introduction, second edition, Springer- Verlag, 1997.
. W Rudin, Fourier Analysis on Groups. WileyW. Rudin, Fourier Analysis on Groups, Wiley, 1990.
M Taibleson, Fourier Analysis on Local Fields. Princeton University PressM. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, 1975.
| [] |
[
"DETC2018-85449 FOLLOWER FORCES IN PRE-STRESSED FIXED-FIXED RODS TO MIMIC OSCILLATORY BEATING OF ACTIVE FILAMENTS",
"DETC2018-85449 FOLLOWER FORCES IN PRE-STRESSED FIXED-FIXED RODS TO MIMIC OSCILLATORY BEATING OF ACTIVE FILAMENTS"
] | [
"Soheil Fatehiboroujeni [email protected] ",
"Arvind Gopinath [email protected] ",
"Sachin Goyal [email protected] ",
"\nDepartment of Mechanical Engineering\nDepartment of Bioengineering\nUniversity of California Merced\n95343Quebec CityCaliforniaCanada\n",
"\nDepartment of Mechanical Engineering Health Science Research Institute\nUniversity of California Merced\nUniversity of California Merced\n95343, 95343California, California\n"
] | [
"Department of Mechanical Engineering\nDepartment of Bioengineering\nUniversity of California Merced\n95343Quebec CityCaliforniaCanada",
"Department of Mechanical Engineering Health Science Research Institute\nUniversity of California Merced\nUniversity of California Merced\n95343, 95343California, California"
] | [] | Flagella and cilia are examples of actively oscillating, whiplike biological filaments that are crucial to processes as diverse as locomotion, mucus clearance, embryogenesis and cell motility. Elastic driven rod-like filaments subjected to compressive follower forces provide a way to mimic oscillatory beating in synthetic settings. In the continuum limit, this spatiotemporal response is an emergent phenomenon resulting from the interplay between the structural elastic instability of the slender rods subjected to the non-conservative follower forces, geometric constraints that control the onset of this instability, and viscous dissipation due to fluid drag by ambient media. In this paper, we use an elastic rod model to characterize beating frequencies, the critical follower forces and the non-linear rod shapes, for prestressed, clamped rods subject to two types of fluid drag forces, namely, linear Stokes drag and non-linear Morrison drag. We find that the critical follower force depends strongly on the initial slack and weakly on the nature of the drag force. The emergent frequencies however, depend strongly on both the extent of pre-stress as well as the nature of the fluid drag. | 10.1115/detc2018-85449 | [
"https://arxiv.org/pdf/1805.08922v1.pdf"
] | 51,686,996 | 1805.08922 | 59b25771135e4a760c7900eeb7303f9b19cc2c04 |
DETC2018-85449 FOLLOWER FORCES IN PRE-STRESSED FIXED-FIXED RODS TO MIMIC OSCILLATORY BEATING OF ACTIVE FILAMENTS
August 26-29, 2018,
Soheil Fatehiboroujeni [email protected]
Arvind Gopinath [email protected]
Sachin Goyal [email protected]
Department of Mechanical Engineering
Department of Bioengineering
University of California Merced
95343Quebec CityCaliforniaCanada
Department of Mechanical Engineering Health Science Research Institute
University of California Merced
University of California Merced
95343, 95343California, California
DETC2018-85449 FOLLOWER FORCES IN PRE-STRESSED FIXED-FIXED RODS TO MIMIC OSCILLATORY BEATING OF ACTIVE FILAMENTS
August 26-29, 2018,Proceedings of the ASME 2018 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2018
Flagella and cilia are examples of actively oscillating, whiplike biological filaments that are crucial to processes as diverse as locomotion, mucus clearance, embryogenesis and cell motility. Elastic driven rod-like filaments subjected to compressive follower forces provide a way to mimic oscillatory beating in synthetic settings. In the continuum limit, this spatiotemporal response is an emergent phenomenon resulting from the interplay between the structural elastic instability of the slender rods subjected to the non-conservative follower forces, geometric constraints that control the onset of this instability, and viscous dissipation due to fluid drag by ambient media. In this paper, we use an elastic rod model to characterize beating frequencies, the critical follower forces and the non-linear rod shapes, for prestressed, clamped rods subject to two types of fluid drag forces, namely, linear Stokes drag and non-linear Morrison drag. We find that the critical follower force depends strongly on the initial slack and weakly on the nature of the drag force. The emergent frequencies however, depend strongly on both the extent of pre-stress as well as the nature of the fluid drag.
INTRODUCTION
Stability analysis of slender structures subjected to follower loads are important instantiations of nonconservative problems in the theory of elastic stability. A number of thorough surveys of the developments and achievements on the structural stability of nonconservative systems can be found in the literature [1][2][3]. Conservative loads such as gravitational or electrostatic forces can be written as gradient of a time independent potential function [4].
The elastic buckling of a straight, slender column subject to compressive forces is a classical problem involving conservative forces that has received much attention. Consider an Euler column of length L, modulus of elasticity E and moment of inertia I, clamped at one end and subject to a concentrated axial load P at the other. Let the column deform transversely with a (dimensional) lateral displacement h as a result of this forcing. Since the force is conservative, we can write a potential function V = 1 2 EI 0 h 2 ds − 1 2 P L 0 h 2 ds using which variational techniques deliver the shape of the buckled column h + (P/EI)h = 0. The critical load P cr for instability and subsequent buckling is obtained by seeking solutions to this equation subject to boundary conditions h(0) = h (0) = h (L) = 0. The critical load P cr = EIπ 2 /4L 2 can be then determined as the minimum loading that makes the potential energy V cease being positive definite. However, such static linear stability analyses are applicable only to systems where the loading forces are not time dependent.
Nonconservative loads, however, do not fit this criteria; their magnitude and direction depend on the configuration of a structure (e.g., deflection and slope), its velocity, and time. An ubiquitous example of a nonconservative force is fluid drag and viscous damping forces that depend on the velocity of a structure. Fol- Figure 1. (a) Schematic representation of a rod of unstressed length L with fixed-fixed boundary condition (clamped at both ends). The end to end distance when buckled is L ee < L. (b) The motion of material points comprising the cross-section of the rod at arc-length position, s and at time t is determined by tracking the transformations of the body-fixed frameâ i (s,t) with respect to the inertial frame of referenceê i . (c) The shape (top) and prestress (bottom) in the buckled state with L ee /L = 0.9. Pre-stress is determined by the component of the Internal force in the direction of cross-sectional normal vectorâ 3 (s,t) i.e., f 3 . lower forces are a second type of nonconservative force; these, acting either as a point force at one end or a distributed load along the structure, always act tangential to the deflection curve of a structure. Reut [5], Pfluger [6], and Beck [7] were among the first researchers to analyze the buckling of cantilevers subjected to follower forces. For instance, Beck [7] reports that the critical buckling load for a cantilever subjected to a compressive point load that always remains tangential to the free-end of the cantilever is approximately 10 times larger than for a force with constant direction. Stability analysis of slender structures subjected to follower loads is critical in several applications including pipes conveying fluid [8,9], structures which are self propelled by their own thrust [10], and rockets [11]. It is shown that motion equations of disc-brake systems [12] is identical to that of Leipholz's column which is a cantilever subjected to a compressive and uniformly distributed follower load.
More recently, follower forces have been studied in the context of connected actively driven beads. Synthetic filaments comprised of connected beads when actuated provide a mechanism to mimic the oscillatory beating of flagella and cilia [13][14][15]. While the length scales are much smaller in these applications (ranging from around 1-500 µm), dynamical principles underlying their structural stability remain similar; indeed connections between mechanics at multiple length scales have been illustrated in other biological settings [16][17][18]. For example, tunable colloidal chains that are assemblies of Janus particles with controllable polarities can be tuned to generate oscillatory beating [19][20][21] by subjecting them to tangential compressive (follower) forces. Continuum models are used in numerous studies and have shown to be effective tools for analyzing the postbuckling behavior of rods subjected to follower force. Dissipation of energy due to viscous drag plays an important role in determining the steady-state beating frequency as well as the critical value of the follower load [21][22][23][24].
Previous studies of filaments subject to follower forces-both for viscous drag dominated as well as inertial filaments-have focused on mobility and dynamics of free-free, fixed-free, and pinned-free filaments with the base state being a straight nonstressed filament or rod. The role of pre-stress in emergent oscillations driven by active distributed follower forces is yet to be elucidated. Here, we focus on this complementary scenario of a fixed-fixed rod. Specifically, a rod clamped at both ends is first pre-stressed by decreasing the end-to-end distance, thereby generating a buckled shape and then subjected to follower force. In the fixed-free scenario, the lack of constraint at the free-end allow for either lateral oscillations or steady rotations to develop in favorable conditions [21]. In the fixed-fixed scenario, the slack generated upon initial compression (which may alternately be interpreted as a pre-stress) offers the necessary degree of freedom to allow for oscillations. Our simulations are three-dimensional, but, by introducing strictly planar perturbations and loads, the oscillations remain planar.
Note that in general, stability of systems subject to nonconservative forces involve linear operators that is not selfadjoint. Critical buckling loads in such non self-adjoint systems cannot be determined using Euler's static method but rather by estimating eigenvalues of the associated dynamic problem. Alternatively, critical points and post-buckling solutions may be obtained by solving the fully non-linear, time-dependent problem as done in this work.
MODEL
Quantity
Variable Value Units Surrounding fluid density ρ f 1000 kg/m 3 Table 1. Numerical values for the geometric and elastic properties of the rod and drag coefficients used in the computations.
We consider a rod that is in its stress-free state when maintaining a straight shape. By moving one of the clamped ends of the rod toward the other end and forcing the rod to bend due to bucking as shown in Fig 1(a), we generate pre-stress in the rod. Thus, pre-stress rate is controlled by the end-to-end length of the rod, L ee < L.
Governing equations
The continuum rod model that we use [25] follows the classical approach of the Kirchhoff [26] which assumes each crosssection of the rod to be rigid. The rigid-body motion of individual cross-sections is examined by discretizing an elastic rod into infinitesimal elements along its arc-length. The position and orientation of each cross-section is determined in space s (i.e., the arc-length variable) and time t by tracking the transformation of a body-fixed frameâ i (s,t) with respect to an inertial frame of referenceê i (s,t) as shown in Fig 2(
b), where subscript i = 1, 2, 3.
Vector R(s,t) defines the position of the cross-sections relative to the inertial frame of reference. The spatial derivative of R(s,t) is denoted by vector r(s,t). Deviation of r(s,t) from the unit normal of the cross-section determines shear while the change in its magnitude quantifies stretch (extension or compression) along the arc-length s. Both shear and stretch deformations are negligible for filaments with large slenderness (length/thickness) ratio under compression. So, we assume r(s,t) =â 3 (s,t) =t(s,t), wheret(s,t) is the unit tangent vector along the arc-length. Vector κ(s,t) captures two-axes bending and torsion of the rod and vectors v(s,t) and ω(s,t) represent the translational velocity and the angular velocity of cross-sections, respectively. The stress distribution over the cross-section of the rod results in a net internal force and a net internal moment shown respectively with f (s,t) and q(s,t).
The equations of equilibrium (1) and (2) are derived by applying Newton's second law to an infinitesimal element of the rod. The compatibility equations (3) and (4) follow from the space-time continuity of the cross-section position R(s,t), and the space-time continuity of the transformation fromâ i (s,t) tô e i (s,t). In Eqns. (1)-(4) all derivatives are relative to the bodyfixed reference frame, m is the mass of the rod per unit length and I m is a 3-by-3 tensor of the moments of inertia per unit length. External force per unit length F as well as the external moment per unit length Q capture interactions of the rod with itself [27,28] or the environment such as drag force.
m( ∂ v ∂t + ω × v) − ( ∂ f ∂s + κ × f ) − F = 0 (1) I m ∂ ω ∂t + ω × I m ω − ( ∂ q ∂s + κ × q) + f × r − Q = 0 (2) ∂ r ∂t + ω × r − ( ∂ v ∂s + κ × v) = 0 (3) ∂ κ ∂t − ( ∂ ω ∂s + κ × ω) = 0(4)
The distributed follower forces and moments in this model are captured by F and Q. In the scenario of fixed-fixed rod, we consider the effect of distributed follower forces in tangential direction (alongâ 3 (s,t)) in this paper. Hereforward for simplicity of notation, we refer to this tangential follower force density by scalar F. In the scenario of fixed-free rods, there may also be point follower loads at one of the boundaries.
The differential equations of equilibrium and compatibility (1)-(4) have to be solved together with a constitutive law relating the deformations to the restoring forces. The constitutive law, for an elastic rod may, in general, take the form of an implicit algebraic constraint [29], but for an isotropic and linearly elastic rod that is not curved in the stress-free state [30] it takes an explicit form [31]:
q(s,t) = B(s) κ(s,t).(5)
The matrix B in equation (5) encodes the bending and torsional stiffness moduli of the rod. By choosing the body-fixed frames of reference to coincide with principal axes of rod cross-sections, B can be written as
B = EI 1 0 0 0 EI 2 0 0 0 GI 3 ,(6)
where E is the Young's modulus, G is the shear modulus, and I 1 , I 2 , and I 3 represent the second moment of area of the rod crosssection about its principal axes. Our choice, as implicit in Fig 1(b), implies that subscripts i = 1, 2 inâ i (s,t) represent the rod's axes of bending and i = 3 represents torsional axis.
Solution scheme
The Generalized-α method [32] is adopted to compute the numerical solution of this system, subjected to necessary and sufficient initial and boundary conditions. A detailed description of this numerical scheme applied to this formulation is given in the extant literature [33]. The important feature of this method is that it is an unconditionally stable second order accurate method for numerically stiff problems, which allows for controllable numerical dissipation. In the context of rod mechanics, it brings an improvement over box method [34] by controlling the Crank-Nicolson noise, in which numerical solution oscillates about the true solution at every (temporal or spatial step) and corrupts the subsequent computation. While in the analysis presented here, the constitutive equations are linear and local, as embodied in equations (5) and (6), the method can be adapted to analyze problems where the constitutive relationships are non-linear and nonlocal.
RESULTS
We next present results for the critical value of the follower force density F cr versus end-to-end distance L ee /L and explore how the beating frequency, ω(|F|, L ee /L) both at the critical point and for values of the follower force |F| > F cr depends on the prestress. A cylindrical rod with slenderness ratio of 800 is simulated with properties given in Table 1. We compare the findings for two types of drag forces, namely Stokes (S) drag and Morrison (M) drag forms given in Eqs. (7) and (8), respectively as explained in [25].
F S = − 1 2 ρ f d C nt × ( v ×t) + πC t ( v ·t)t (7) F M = − 1 2 ρ f d C n | v ×t|t × ( v ×t) + πC t ( v ·t)| v ×t|t(8)
In both equations ρ f and d represent the environment fluid density and diameter of the rod, respectively. Drag coefficients (per unit length) C n and C t are given in Table 1. We note that the Stokes [S] form for the drag is linear in the velocity while the Morrison form [M] is quadratic, and hence non-linear in the velocity. Thus for the same change in configuration and frequency, the Morrison form will result in a larger viscous dissipation per unit length than the Stokes form. Conversely, if we require that the same amount of energy be dissipated, the Stokes limit will be characterized by either higher frequency or by larger amplitude deformations or both.
Benchmark: Critical Force for Beck's Column
In order to benchmark the model presented in this paper, we calculate the critical buckling force for the Beck's column and compare it with the values reported. Beck's column is a cantilever subjected to a compressive point load that is always tangential to the free-end of the column. Beck's analysis, published in German [7] and reviewed in English [2] yields the following expression for the critical buckling force, P cr of a cantilever with bending stiffness, EI and length, L in absence of damping dissipation. P cr ≈ 20.51 (EI/L 2 ).
Using the formulation presented in Section 2, we investigate the value of the critical buckling force for Beck's column and compare it to the value reported in the literature. To approach the conditions of a quasi-static simulation and reduce the dynamic effects we apply a compressive follower force, which gradually increases in time, to the free-end of the cantilever. The critical force found by our computational model in absence of viscous drag is approximately P cr ≈ 20.10 (EI/L 2 ), which is within two percent error margin of the classical estimated value.
Oscillatory Beating of Fixed-Fixed Rods
In this section we present the results for the post-buckling analysis of pre-stressed rods with fixed-fixed boundary conditions for various values of the the slack (and thus, various values of the pre-stress as well as base curvature). Identifying and characterizing critical points as well as the force-frequency relationship can be potentially useful in designing accurately controllable oscillations.
We study cylindrical rods that are in stress-free state when straight. When both ends of a rod are fixed and clamped, moving one of the clamped ends of the rod toward the other end forces the rod to bend or buckle (c.f., Figure 1(a)). This process generates pre-stress in a rod and thus, pre-stress rates can be controlled by the end-to-end length of the rod, L ee -an example of this is shown in Fig. 1(c). Starting from a base state completely determined by the ratio L ee /L, we then apply uniformly distributed follower load, Fâ 3 along the rod. When the magnitude of the follower load, |F| > F cr , buckled shapes become unstable and beating oscillations emerge. Figure 2 shows the magnitudes of the critical follower load against end-to-end distance for the same rod sub- increases to values much larger than the critical values, the effect of the pre-stress diminishes, and (ii) the frequencies in the limit |F| F cr vary as |F| α where the exponent 1 < α < 4/3. jected to two types of drag forces. Surprisingly we observe that critical follower load increases as the amount of pre-stress in the rod increases even though decreasing L ee /L implies more slack. The magnitude of critical follower load found to be nearly the same for both Stokes and Morrison drags (discrepancies being ±0.1%).
Despite the low sensitivity of F cr to the nature of drag law, beating configurations as well as the steady frequency of oscillations are found to be significantly different for Stokes and Morrison drags. This can be explained by the fact that Morrison drag dissipates energy at a higher rate compared to the Stokes drag for the same frequency and mode shapes. Figure 3 illustrates how the shape of the rod evolves during one complete oscillation for both Stokes and Morrison drags. We visually observe that configurations of the rod subjected to Stokes drag consist of various shape modes. This suggests that higher order harmonics are stronger in this case. Whereas, for cases subjected to Morrison drag, higher order shapes are not recognizable visually. By looking at the Fourier transformation of any quantity we can also confirm the significance of higher order harmonics in Stokes regime compared to Morrison as is shown in Figure 4.
Finally, with the computational model proposed here we systematically investigate the effect of pre-stress and the follower force on the frequency of beating oscillations and emer-gent shapes. It is useful, at this point, to recall previous results on unstressed fixed-free cantilever type rods subject to follower forces and Stokes drag [21] with equal values of axial and normal drag. As in our simulations, fixed-free unstressed rods were found to undergo an instability to oscillatory motion beyond a critical value of the follower force; in this case, the lack of prestress implies that the instability occurs at a single critical point F cr ≈ 75.5(EI/L 3 ) (where we recall that the follower force is a force density with units of N/m) . Interestingly relaxing the boundary condition to a pinned-free rod was found to alter the dynamical picture completely. In the latter case the post-bucked state was not an oscillating filament but instead a rotating spiral. In either case, a single dimensionless parameter β ≡ FL 3 /EI encodes the role of activity in the post-buckling shapes and frequencies post criticality. For β > 75.5, the frequency Ω was found to monotonically increase with β (and hence with |F|) and eventually follow a power law relationship Ω ∼ |F| Moving now to our results for pre-stressed filaments, we plot the frequency of beating oscillations for rods under various endto-end distances in figures 5(a,b). We observe that frequency of oscillations under Stokes drag undergoes a sudden increase once the magnitude of the distributed follower load reaches a second critical limit. Such a behavior is absent under Morrison drag. Also for a region where magnitude of the distributed follower load is below 18 N/m we observe that larger pre-stress (or smaller end-to-end distance) results in smaller beating frequency. This pattern is also evident under the Stokes drag but only in the region in which the magnitude of the distributed follower load is above 18 N/m. Examining the force dependence of the beating filaments subject to Morrison type drag forces more closely in Figure 5(b), we find the frequencies in the limit |F| F cr vary as |F| α where the exponent is closer to unity than to 4/3. More interestingly, we note also the weakening dependence of the frequency on the pre-stress (slack) in the limit of large follower force densities -that is when |F|/F cr 1. This behavior is particularly evident in Figure 5(b).
DISCUSSION
In this paper we discussed the application of a computational rod model to analyze the buckling stability as well as the postbuckling oscillations of slender structures subjected to compressive follower loads. Simulations were first benchmarked with previous findings on magnitude of the critical buckling force for Beck's column. We focused on slender rods that maintain a straight shape corresponding to their stress-free state (i.e., having no intrinsic curvature and twist) with both ends clamped. By moving one end of the rod toward the other end, the structure undergoes buckling and the end-to-end distance represents a measure of the amount of pre-stress in the rod. We found that beyond a critical value of distributed and compressive follower loads the buckled shapes become unstable and oscillatory beating emerges. The magnitude of the critical follower load increases as the magnitude of the pre-stress in the structure increases. We also observed that frequency of the oscillations as well as the configuration of the rod are significantly influenced by the type of drag law used in modeling. Morrison drag induces higher dissipation rate than Stokes drag, therefore, under identical circumstances many more harmonics are discernible in the oscillations of a rod subjected to Stokes drag. Moreover, for the rods subjected to Stokes drag we observed that frequency of oscillations as a function of follower load undergoes a sudden increase once the magnitude of the distributed follower load reaches a second critical limit. Our results provide a starting point to systematically investigate the interplay between geometry, elasticity, dissipation and activity towards designing bio-inspired multifunctional, synthetic structures to move and manipulate fluid at various length scales.
Figure 2 .
2Critical load for onset of oscillations F cr versus scaled endto-end distance L ee /L for both Stokes [S] drag and Morrison [M] drag.We note that the critical loads are roughly the same over the range of pre-stress values investigated.
Figure 3 .
3Configurations of the rod in sequence (1-10) over one period of oscillation when |F| = 15 N/m and for different values of the dimensionless slack L − L ee /L. The configurations (rod shapes) for Stokes drag are shown on the top, with shapes for Morrison drag shown in the bottom row.
Figure 4 .
4Fourier transform (in the time domain) of the shear force at the mid-span length of the rod shows that higher harmonics (insets show the raw data) are damped in the case of the non-linear Morrison drag (top row) more effectively than for the linear Stokes drag (bottom row). The right column of the picture corresponds to L ee /L = 0.9 while the left column corresponds to L ee /L = 0.7. The ratio of drag coefficients for both cases is 10. Intuitively, we expect this ratio to affect the extent of dampening.
Figure 5 .
5(a) Frequency of beating oscillations for rods having various end-to-end distances L ee . We show results for both types of fluid drag -the linear Stokes [S] drag as well as the quadratic Morrison [M] drag. The frequency is plotted as a function of the distributed follower load. We note that the results for the Stokes drag features possible transitions that may be related to activation of higher order mode shapes as seen from figure 3. (b) Results for the case where the viscous drag of of the Morrison [M] form are re-plotted in logarithmic scales to illustrate two salient features -(i) as the follower force
properties. For extremely high values of |F|, self-avoidance resulted in a change in this scaling to a different power law with exponent of 2.
Dynamic stability of columns subjected to follower loads: A survey. M Langthjem, Y Sugiyama, Journal of Sound and Vibration. 2385Langthjem, M., and Sugiyama, Y., 2000. "Dynamic sta- bility of columns subjected to follower loads: A survey". Journal of Sound and Vibration, 238(5), pp. 809 -851.
Controversy Associated With the So-Called "Follower Forces. I Elishakoff, Critical Overview". Applied Mechanics Reviews. 582117Elishakoff, I., 2005. "Controversy Associated With the So- Called "Follower Forces": Critical Overview". Applied Mechanics Reviews, 58(2), p. 117.
Dynamic instabilities in mechanics of structures. V V Bolotin, Appl. Mech. Rev. 551Bolotin, V. V., 1999. "Dynamic instabilities in mechanics of structures". Appl. Mech. Rev., 55(1), pp. R1-R9.
Stability of elastic systems / Horst Leipholz. Sijthoff and Noordhoff. H H E Leipholz, Alphen aan den Rijn, the NetherlandsLeipholz, H. H. E., 1980. Stability of elastic systems / Horst Leipholz. Sijthoff and Noordhoff Alphen aan den Rijn, the Netherlands.
About the theory of elastic stability. V I Reut, Proceedings of the Odessa Institute of Civil and Communal Engineering. the Odessa Institute of Civil and Communal EngineeringReut, V. I., 1939. "About the theory of elastic stability". In Proceedings of the Odessa Institute of Civil and Communal Engineering, No. 1.
Stabilitatsprobleme der Elastostatik. A Pfluger, Springer-VerlagBerlinPfluger, A., 1950. Stabilitatsprobleme der Elastostatik. Springer-Verlag, Berlin.
Die Knicklast des einseitig eingespannten, tangential gedruckten Stabes. M Beck, ZAMP Zeitschrift fur angewandte Mathematik und Physik. 33Beck, M., 1952. "Die Knicklast des einseitig eingespan- nten, tangential gedruckten Stabes". ZAMP Zeitschrift fur angewandte Mathematik und Physik, 3(3), pp. 225-228.
M P Païdoussis, Fluid-Structure Interactions: Slender Structures and Axial Flow. Second EditionPaïdoussis, M. P., 2016. Fluid-Structure Interactions: Slender Structures and Axial Flow: Second Edition.
Pipes conveying fluid: A model dynamical problem. M P Païdoussis, G X Li, Journal of Fluids and Structures. 72Païdoussis, M. P., and Li, G. X., 1993. "Pipes conveying fluid: A model dynamical problem". Journal of Fluids and Structures, 7(2), pp. 137-204.
The Kinetic Stability of a Tangentially Loaded Strut. W G Wood, S S Saw, P M Saunders, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 313Wood, W. G., Saw, S. S., and Saunders, P. M., 1969. "The Kinetic Stability of a Tangentially Loaded Strut". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 313(1513), pp. 239- 248.
Dynamics of cylindrical structures subjected to axial flow. M Païdoussis, Journal of Sound and Vibration. 293Païdoussis, M., 1973. "Dynamics of cylindrical structures subjected to axial flow". Journal of Sound and Vibration, 29(3), pp. 365-385.
Parametric instability of a Leipholz column under periodic excitation. B Kang, C A Tan, Journal of Sound and Vibration. 2295Kang, B., and Tan, C. A., 2000. "Parametric instability of a Leipholz column under periodic excitation". Journal of Sound and Vibration, 229(5), pp. 1097-1113.
Microscopic artificial swimmers. R Dreyfus, J Baudry, M L Roper, M Fermigier, H A Stone, J Bibette, Nature. 4377060Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A., and Bibette, J., 2005. "Microscopic artificial swim- mers". Nature, 437(7060), pp. 862-865.
Tethered fleximags as artificial cilia. A Babataheri, M Roper, M Fermigier, O Du Roure, Journal of Fluid Mechanics. 678Babataheri, A., Roper, M., Fermigier, M., and Du Roure, O., 2011. "Tethered fleximags as artificial cilia". Journal of Fluid Mechanics, 678, pp. 5-13.
Flagellar kinematics and swimming of algal cells in viscoelastic fluids. B Qin, A Gopinath, J Yang, J P Gollub, P E Arratia, Scientific Reports, 5Qin, B., Gopinath, A., Yang, J., Gollub, J. P., and Arratia, P. E., 2015. "Flagellar kinematics and swimming of algal cells in viscoelastic fluids". Scientific Reports, 5.
Cell and biomolecular mechanics in silico. A Vaziri, A Gopinath, Nature Materials. 71Vaziri, A., and Gopinath, A., 2008. "Cell and biomolecular mechanics in silico". Nature Materials, 7(1), pp. 15-23.
Elastohydrodynamics of wet bristles, carpets and brushes. A Gopinath, L Mahadevan, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 467Gopinath, A., and Mahadevan, L., 2011. "Elastohydrody- namics of wet bristles, carpets and brushes". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467(2130), pp. 1665-1685.
Continuum-based computational models for cell and nuclear mechanics. A Vaziri, A Gopinath, V S Deshpande, Journal of Mechanics of Materials and Structures. 26Vaziri, A., Gopinath, A., and Deshpande, V. S., 2007. "Continuum-based computational models for cell and nu- clear mechanics". Journal of Mechanics of Materials and Structures, 2(6), pp. 1169-1191.
Colloidal caterpillars for cargo transportation. Y Sasaki, Y Takikawa, V S R Jampani, H Hoshikawa, T Seto, C Bahr, S Herminghaus, Y Hidaka, H Orihara, Soft matter. 1044Sasaki, Y., Takikawa, Y., Jampani, V. S. R., Hoshikawa, H., Seto, T., Bahr, C., Herminghaus, S., Hidaka, Y., and Orihara, H., 2014. "Colloidal caterpillars for cargo trans- portation.". Soft matter, 10(44), pp. 8813-20.
Active colloids in complex fluids. A E Patteson, A Gopinath, P E Arratia, Current Opinion in Colloid & Interface Science. 21Patteson, A. E., Gopinath, A., and Arratia, P. E., 2016. "Ac- tive colloids in complex fluids". Current Opinion in Colloid & Interface Science, 21, pp. 86 -96.
Flagellar dynamics of a connected chain of active, polar, brownian particles. R Chelakkot, A Gopinath, L Mahadevan, M F Hagan, Journal of the Royal Society, Interface. 119220130884Chelakkot, R., Gopinath, A., Mahadevan, L., and Hagan, M. F., 2014. "Flagellar dynamics of a connected chain of active, polar, brownian particles". Journal of the Royal Society, Interface, 11(92), March, p. 20130884.
Spontaneous oscillations of elastic filaments induced by molecular motors. G De Canio, E Lauga, R E Goldstein, Journal of The Royal Society Interface. 13614De Canio, G., Lauga, E., and Goldstein, R. E., 2017. "Spontaneous oscillations of elastic filaments induced by molecular motors". Journal of The Royal Society Interface, 14(136).
On the stability of elastic systems subjected to nonconservative forces. G Herrmann, R W Bungay, Journal of Applied Mechanics. 313Transactions ASMEHerrmann, G., and Bungay, R. W., 1964. "On the stabil- ity of elastic systems subjected to nonconservative forces". Journal of Applied Mechanics, Transactions ASME, 31(3), pp. 435-440.
Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella. P V Bayly, S K Dutcher, Journal of The Royal Society Interface. 12313Bayly, P. V., and Dutcher, S. K., 2016. "Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella". Journal of The Royal Society Interface, 13(123).
Nonlinear Dynamics and Loop Formation in Kirchhoff Rods with Implications to the Mechanics of DNA and Cables. S Goyal, N C Perkins, C L Lee, Journal of Computational Physics. 2091Goyal, S., Perkins, N. C., and Lee, C. L., 2005. "Nonlinear Dynamics and Loop Formation in Kirchhoff Rods with Im- plications to the Mechanics of DNA and Cables". Journal of Computational Physics, 209(1), pp. 371-389.
Uber das gleichgewicht und die bewegung eines unendlich dunnen elastischen stabes. G Kirchhoff, J. Reine Angew. Math. (Crelle). 56Kirchhoff, G., 1859. "Uber das gleichgewicht und die bewegung eines unendlich dunnen elastischen stabes". J. Reine Angew. Math. (Crelle), 56, pp. 285-343.
Nonlinear dynamic intertwining of rods with self-contact. S Goyal, N Perkins, C L Lee, International Journal of Non-Linear Mechanics. 431Goyal, S., Perkins, N., and Lee, C. L., 2008. "Non- linear dynamic intertwining of rods with self-contact". International Journal of Non-Linear Mechanics, 43(1), pp. 65-73.
Electrostatics and self-contact in an elastic rod approximation for DNA. T D Lillian, N Perkins, Journal of Computational and Nonlinear Dynamics. 6111008Lillian, T. D., and Perkins, N., 2011. "Electrostatics and self-contact in an elastic rod approximation for DNA". Journal of Computational and Nonlinear Dynamics, 6(1), p. 011008.
Constitutive-Law Modeling of Microfilaments From Their Discrete-Structure Simulations -A Method Based on an Inverse Approach Applied to a Static Rod Model. A R Hinkle, S Goyal, H J Palanthandalam-Madapusi, ASME Journal of Applied Mechanics. 7951005Hinkle, A. R., Goyal, S., and Palanthandalam-Madapusi, H. J., 2012. "Constitutive-Law Modeling of Microfilaments From Their Discrete-Structure Simulations -A Method Based on an Inverse Approach Applied to a Static Rod Model". ASME Journal of Applied Mechanics, 79, Sep, p. 051005.
Intrinsic Curvature of DNA Influences LacR-Mediated Looping. S Goyal, T Lillian, S Blumberg, J C Meiners, E Meyhofer, N C Perkins, Biophysical Journal. 9312Goyal, S., Lillian, T., Blumberg, S., Meiners, J. C., Mey- hofer, E., and Perkins, N. C., 2007. "Intrinsic Curvature of DNA Influences LacR-Mediated Looping". Biophysical Journal, 93(12), pp. 4342-4359.
Looping mechanics of rods and DNA with non-homogeneous and discontinuous stiffness. S Goyal, N Perkins, 10.1016/j.ijnonlinmec.2008.06.013International Journal of Non-Linear Mechanics. 44Goyal, S., and Perkins, N., 2008. "Looping mechanics of rods and DNA with non-homogeneous and discontinuous stiffness". International Journal of Non-Linear Mechanics, 44(doi:10.1016/j.ijnonlinmec.2008.06.013).
A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation -the Generalized-Alpha Method. J Chung, G M Hulbert, Journal of Applied Mechanics-Transactions of The ASME. 602Chung, J., and Hulbert, G. M., 1993. "A Time Integration Algorithm for Structural Dynamics with Improved Numer- ical Dissipation -the Generalized-Alpha Method". Journal of Applied Mechanics-Transactions of The ASME, 60(2), pp. 371-375.
A dynamic rod model to simulate mechanics of cables and DNA". Ph.d. dissertation in mechanical engineering. S Goyal, University of MichiganGoyal, S., 2006. "A dynamic rod model to simulate me- chanics of cables and DNA". Ph.d. dissertation in mechan- ical engineering, University of Michigan.
Application of the Generalized-Alpha Method to the Time Integration of the Cable Dynamics Equations. J I Gobat, M A Grosenbaugh, Computer Methods in Applied Mechanics and Engineering. 190Gobat, J. I., and Grosenbaugh, M. A., 2001. "Appli- cation of the Generalized-Alpha Method to the Time In- tegration of the Cable Dynamics Equations". Computer Methods in Applied Mechanics and Engineering, 190(37- 38), pp. 4817-4829.
| [] |
[
"AN OPTIMAL POLYNOMIAL APPROXIMATION OF BROWNIAN MOTION *",
"AN OPTIMAL POLYNOMIAL APPROXIMATION OF BROWNIAN MOTION *"
] | [
"James Foster ",
"Terry Lyons ",
"ANDHarald Oberhauser "
] | [] | [] | In this paper, we will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. Remarkably the coefficients obtained from the expansion of Brownian motion in this polynomial basis are independent Gaussian random variables. Therefore it is practical (requires N independent Gaussian coefficients) to generate an approximate sample path of Brownian motion that respects integration of polynomials with degree less than N . Moreover, since these orthogonal polynomials appear naturally as eigenfunctions of the Brownian bridge covariance function, the proposed approximation is optimal in a certain weighted L 2 (P) sense. In addition, discretizing Brownian paths as piecewise parabolas gives a locally higher order numerical method for stochastic differential equations (SDEs) when compared to the piecewise linear approach. We shall demonstrate these ideas by simulating Inhomogeneous Geometric Brownian Motion (IGBM). This numerical example will also illustrate the deficiencies of the piecewise parabola approximation when compared to a new version of the asymptotically efficient log-ODE (or Castell-Gaines) method. | 10.1137/19m1261912 | [
"https://arxiv.org/pdf/1904.06998v1.pdf"
] | 119,301,554 | 1904.06998 | fb4e1e2fbb8e956d775e7cf5612f2ed2dd0134c5 |
AN OPTIMAL POLYNOMIAL APPROXIMATION OF BROWNIAN MOTION *
James Foster
Terry Lyons
ANDHarald Oberhauser
AN OPTIMAL POLYNOMIAL APPROXIMATION OF BROWNIAN MOTION *
Brownian motionpolynomial approximationnumerical methods for SDEs AMS subject classifications 41A1060H3560J65
In this paper, we will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. Remarkably the coefficients obtained from the expansion of Brownian motion in this polynomial basis are independent Gaussian random variables. Therefore it is practical (requires N independent Gaussian coefficients) to generate an approximate sample path of Brownian motion that respects integration of polynomials with degree less than N . Moreover, since these orthogonal polynomials appear naturally as eigenfunctions of the Brownian bridge covariance function, the proposed approximation is optimal in a certain weighted L 2 (P) sense. In addition, discretizing Brownian paths as piecewise parabolas gives a locally higher order numerical method for stochastic differential equations (SDEs) when compared to the piecewise linear approach. We shall demonstrate these ideas by simulating Inhomogeneous Geometric Brownian Motion (IGBM). This numerical example will also illustrate the deficiencies of the piecewise parabola approximation when compared to a new version of the asymptotically efficient log-ODE (or Castell-Gaines) method.
1. Introduction. Brownian motion is a central object for modelling real-world systems that evolve under the influence of random perturbations [1]. In applications where methods discretize Brownian motion, usually only increments of the path are generated [2]. In this setting, the best L 2 (P) approximation of Brownian motion that is measurable with respect to these increments is given by the piecewise linear path that agrees on discretization points [3]. This motivates the following natural question:
Are there better discrete approximations of Brownian motion than piecewise linear?
The next simplest approximant would be a piecewise polynomial, though it is not clear whether this would be advantageous for tackling problems such as SDE simulation. This paper can be viewed as a logical continuation of [4], where a polynomial wavelet representation of Brownian motion was proposed. These wavelets were constructed to capture certain "geometrical features" of the path, namely the integrals of the Brownian motion against monomials. The goal of this paper is to convince the reader of the practical implications of these polynomials in the numerical analysis of SDEs. The paper is organised as follows. In Section 2, we shall state and prove the main result of the paper. This will be a Karhunen-Loève theorem for the Brownian bridge, where the orthogonal functions used in the approximation turn out to be polynomials. In Section 3, we will investigate some significant consequences of the main theorem. In particular, the following theorem can be proved immediately from the main result. u k dW u , for k = 0, 1, · · · , n − 1.
Then W = W n + Z n , where Z n is a centred Gaussian process independent of W n .
The above theorem has a simple yet striking conclusion, namely that polynomials can be unbiased approximants of Brownian motion. Moreover, the first non-trivial case (n = 2) already has interesting applications within the numerical analysis of SDEs. The reason is that parabolas can capture the "space-time area" of Brownian motion. Brownian motion can be expressed as a (random) parabola plus independent noise. Moreover, the approximating parabola has the same increment and time integral as the original path.
Therefore discretizing Brownian motion using a piecewise parabola gives a locally high order methodology for numerically solving one-dimensional SDEs. However, since certain triple iterated integrals of Brownian motion and time are partially matched by these parabolas, we expect this method to have only an O(h) rate of convergence (where h denotes the step size used). This gives motivation for the following theorem: Theorem 1.2. Let ı W be the unique quadratic polynomial with a root at 0 and
ı W 1 = W 1 , 1 0 ı W u du = 1 0 W u du.
Then the following third order iterated integral of Brownian motion can be estimated:
E ñ 1 0 W 2 u du W 1 , 1 0 W u du ô = 1 0 ı W 2 u du + 1 15 .
The above theorem can be directly incorporated into the stochastic Taylor method as well as the log-ODE or Castell-Gaines method (see [5], [6]). The hope is that by estimating this non-trivial iterated integral with its conditional expectation, we can design numerical methods that enjoy high orders of both strong and weak convergence. Specifically, for a general one-dimensional SDE with sufficiently regular vector fields, numerical methods that correctly utilize the above conditional expectation will have a strong convergence rate of O(h 3 In Section 4, we shall demonstrate the above ideas through various discretizations of Inhomogeneous Geometric Brownian Motion (IGBM)
dy t = a(b − y t ) dt + σy t dW t ,
where a ≥ 0 and b ∈ R are the mean reversion parameters and σ ≥ 0 is the volatility.
In mathematical finance, IGBM is an example of a short rate model that can be both mean-reverting and non-negative. As a result it is suitable for modelling interest rates, stochastic volatilities and default intensities [7]. From a mathematical viewpoint, IGBM is one of the simplest SDEs that has no known method of exact simulation [8].
By incorporating the ideas provided by the main theorem into the log-ODE method, we will produce a state-of-the-art numerical approximation of IGBM. Although the vector fields for IGBM are not bounded, our numerical evidence indicates that the method has a strong convergence rate of O(h 3 2 ) and a weak convergence rate of O(h 2 ). 1.1. Notation. Below is some of the notation that is used throughout the paper.
• W denotes a standard real-valued Brownian motion. It is often convenient to write W with an additional coordinate corresponding to time, i.e. W (0) t := t.
• Any Stratonovich SDE on the interval [0, T ] in this paper will be of the form
dy t = f 0 (y t ) dt + f 1 (y t ) • dW t , y 0 = ξ,
where y, ξ ∈ R e , and f i : R e → R e denote the vector fields.
(Similarly, Itô SDEs will be defined on fixed intervals and have the same form)
• B denotes a standard real-valued Brownian bridge on [0, 1]. • [s, t] is a general closed subinterval of [0, T ].
• h is the step size at which a numerical solution is propagated, usually h = t−s. • {e k } k≥1 are a family of Jacobi-like polynomials with deg (e k ) = k + 1 that are orthogonal with respect to the weight function w(x) := 1 x(1−x) for x ∈ (0, 1). • P (α,β) is the k-th order (α, β)-Jacobi polynomial on [−1, 1] where α, β > −1. • I k denotes the time integral of B over [0, 1] against the polynomial e k (t)w(t),
I k := 1 0 B t · e k (t) t(1 − t)
dt.
• H s,t denotes the rescaled space-time Lévy area of Brownian motion over [s, t],
H s,t := 1 h W s,u − u − s h W s,t du.
• L s,t denotes the space-space-time Lévy area of Brownian motion over [s, t],
L s,t := 1 6 Ç t s u s v s • dW r • dW v du − 2 t s u s v s • dW r dv • dW u + t s u s v s dr • dW v • dW u å .
• ı W denotes the Brownian parabola corresponding to W over a fixed interval.
ı W u = W s + u − s h W s,t + 6(u − s)(t − u) h 2 H s,t , ∀u ∈ [s, t].
• Z is the Brownian arch corresponding to W over an interval (Z := W − ı W ).
Main result.
It was shown in [4] that Brownian motion can be generated using Alpert-Rokhlin multiwavelets (see [9]). The mother functions that generate this wavelet basis are supported on [0, 1] and are defined using polynomials as follows:
Definition 2.1 (Alpert-Rokhlin wavelets). For q ≥ 1, define the q functions φ q,1 , · · · , φ q,q : [0, 1] → R as piecewise polynomials of degree q − 1 with pieces on [0, 1 2 ], [ 1 2 , 1] and which satisfies the following conditions for 1 ≤ p ≤ q and t ∈ [0, 1 2 ) :
φ q,p (t) = (−1) q+p−1 φ q,p (1 − t), (2.1) 1 0 φ q,p (t)φ q,r (t) dt = δ qr , for 1 ≤ r ≤ q, (2.2) 1 0 t k φ q,p (t) dt = 0, for 0 ≤ k ≤ q − 1. (2.
3)
The Alpert-Rokhlin multiwavelets of order q can now be generated by translating and scaling the mother functions φ q,p .
φ q,p nk (t) := 1 √ 2 n φ q,p (2 n t − k),
for n ≥ 0 and k ∈ {0, · · · , 2 n − 1}.
Whilst our results will not be presented in terms of the above wavelets, we shall see that the polynomials of interest are directly related to the conditions (2.1), (2.2) and (2.3).
The main result of this paper gives an effective method for approximately Brownian sample paths using a class of Jacobi-like polynomials. The proof is based on the new discovery that these polynomials can be viewed as eigenfunctions of an integral operator defined by the Brownian bridge covariance function. These polynomials, which lie at the heart of this paper, will also help us interpret the geometrical features that certain normally distributed iterated integrals encode about the Brownian path. e i e j dµ = δ ij , with δ ij denoting the Kronecker delta, such that B admits the following representation
B = ∞ k=1 I k e k ,(2.
4)
where {I k } is the collection of independent centered Gaussian random variables with
I k := 1 0 B t · e k (t) t(1 − t) dt,(2.
5)
and
Var(I k ) = 1 k(k + 1) .
Moreover, {e k } is an optimal orthonormal basis of L 2 ([0, 1], µ) for approximating B by truncated series expansions with respect to the following weighted L 2 (P) norm
X L 2 µ (P) := E ñ 1 0 (X s ) 2 dµ(s) ô ,
where X is a square µ-integrable process.
Proof. Our argument is that of the Karhunen-Loève theorem in general L 2 spaces. Note that B is a square µ-integrable process as
E ñ 1 0 (B s ) 2 dµ(s) ô = 1 0 E (B s ) 2 dµ(s) = 1 0 s(1 − s) · 1 1 − s du = 1 < ∞.
Let K B denote the covariance function for the standard Brownian bridge on [0, 1]. One can show by direct calculation that
K B 2 L 2 ([0,1] 2 , µ) = 1 0 1 0 (min(s, t) − st) 2 dµ(s)dµ(t) = 1 3 π 2 − 3 < ∞.
It now follows that the linear operator T K :
L 2 ([0, 1] 2 , µ) → L 2 ([0, 1] 2 , µ) given by (T K f )(t) := 1 0 K B (s, t)f (s) dµ(s),
is well-defined and continuous. Furthermore the function k B on [0, 1] defined as
k B (x) := K B (x, x) satisfies 1 0 |k B (x)| dµ(x) = 1 0 x(1 − x) · 1 x(1 − x) dx = 1 < ∞.
It is now possible to apply Mercer's theorem for kernels on general L 2 spaces (see [10]). It then follows from Mercer's theorem that there exists an orthonormal set {e k } k≥1 of L 2 ([0, 1], µ) consisting of eigenfunctions of T K such that the corresponding sequence of eigenvalues {λ n } n≥1 is non-negative. Moreover, the eigenfunctions corresponding to non-zero eigenvalues are continuous on [0, 1] and the kernel K B has the representation
K B (s, t) = ∞ k=1 λ k e k (s)e k (t), (2.6)
where the series convergences absolutely and uniformly on compact subsets of [0, 1].
In the next part of the proof, we'll show that each e k is a polynomial of degree k + 1.
As each e k is an eigenfunction of T K , we have
1 0 min(s, t) − st s(1 − s) e k (s) ds = λ k e k (t). (2.7)
Since e k ∈ L 2 ([0, 1], µ), it follows that e k (0) = 0 and e k (1) = 0 for each k ≥ 1. Therefore by using the Leibniz integral rule to twice differentiate both sides of (2.7), we see that e k must satisfy the following differential equation
λ n e k (t) = − 1 t(1 − t) e k (t). (2.8)
Since e k = 0, we have that λ k = 0. Differentiating both sides of the ODE (2.8) gives
t(1 − t) d 2 dt 2 (e k ) + (1 − 2t) d dt (e k ) + 1 λ k e k (t) = 0. For x ∈ [0, 1], we define the function y k (x) := e k Å 1 2 (1 + x) ã .
Thus y k satisfies the following differential equation
(1 − x 2 )y k (x) − 2xy k (x) + 1 λ k y k (x) = 0. (2.9)
Remarkably, this is the Legendre differential equation [11]. It follows from classical theory that 1 λ k = k(k + 1), and y k is proportional to the k-th Legendre polynomial. Therefore e k is a constant multiple of the k-th shifted Legendre polynomial and hence each e k is a polynomial of degree k + 1.
We can now define the following integrals for k ≥ 1,
I k := 1 0 B t · e k (t) t(1 − t) dt.
It follows from Fubini's theorem that
E[I k ] = 0, E[I i I j ] = E ñ 1 0 1 0 B s B t e i (s) e j (t) dµ(s) dµ(t) ô = 1 0 1 0 E[B s B t ] e i (s) e j (t) dµ(s) dµ(t) = 1 0 e j (t) Ç 1 0 K B (s, t) e i (s) dµ(s) å dµ(t) = λ i δ ij .
Since each I k is defined by a linear functional on the same Gaussian process B, we see from the above that {I k } is a collection of uncorrelated (and therefore independent) Gaussian random variables with
E[I k ] = 0, Var(I k ) = 1 k(k + 1) .
Finally, the L 2 (P) convergence we require follows as
E B t − N k=1 I k e k (t) 2 = k B (t) + E N i,j=1 I i I j e i (t)e j (t) − 2E B t N k=1 I k e k (t) = k B (t) + N i=1 λ i e 2 i (t) − 2E N i=1 1 0 B s B t e i (s)e i (t) dµ(s) = k B (t) − N i=1 λ i e 2 i (t),
which converges to 0 by Mercer's theorem (2.6).
All that is remains is to prove optimality for the truncated series expansions of (2.4).
Let {f k } k≥1 denote an orthonormal basis of L 2 ([0, 1], µ) such that B = ∞ k=1 J k f k , where J k := 1 0 B t f k (t) dµ(t), ∀k ≥ 1.
For n ≥ 1, define the error process r n :
= ∞ k=n+1 J k f k .
Then the square L 2 (P) norm of the error process admits the following expansion,
r n (t) 2 L 2 (P) = E ∞ i=n+1 ∞ j=n+1 J i J j f i (t)f j (t) = ∞ i=n+1 ∞ j=n+1 E ñ 1 0 1 0 B s B t f i (s)f j (t) dµ(s) dµ(t) ô f i (t)f j (t) = ∞ i=n+1 ∞ j=n+1 Ç 1 0 1 0 K B (s, t) f i (s)f j (t) dµ(s) dµ(t) å f i (t)f j (t).
Integrating the above with respect to µ and using the orthogonality of {f k } gives
r n 2 L 2 µ (P) = 1 0 r(t) 2 L 2 (P) dµ(t) = ∞ k=n+1 1 0 1 0 K B (s, t)f k (s)f k (t) dµ(s) dµ(t).
Note that an optimal orthonormal basis of L 2 ([0, 1], µ) solves the following problem
min f k r n 2 L 2 µ (P) subject to f k L 2 ([0,1],µ) = 1.
By introducing Lagrange multipliers ν k , we wish to find functions {f k } that minimize
E n [{f k }] := ∞ k=n+1 1 0 1 0 K B (s, t)f k (s)f k (t)dµ(s)dµ(t) − ν k Ç 1 0 f k (s) 2 dµ(s) − 1 å .
We will now consider the following square integrable functions, defined for s, t ∈ (0, 1):
f k (t) := f k (t) · 1 t(1 − t) ,K B (s, t) := K B (s, t) · 1 s(1 − s) · 1 t(1 − t) .
Therefore it is enough to find a family of functions
{f k } in L 2 ([0, 1]) that minimizes E n [{f k }] := ∞ k=n+1 1 0 1 0K B (s, t)f k (s)f k (t) ds dt − ν k Ç 1 0 f k (s) 2 ds − 1 å .
To find a minimizer, we set the functional derivative ofẼ n with respect tof k to zero.
∂Ẽ n ∂f k (t) = 2 1 0K B (s, t)f k (s) ds − 2ν kfk (t) = 0.
By using the definitions off k andK B , it is trivial to show the above is equivalent to
1 0 K B (s, t)f k (s) dµ(s) = ν k f k (t),
which is satisfied if and only if f k are eigenfunctions of T K .
Corollary 2.3. The above result can naturally be extended to Brownian motion. If W is a standard Brownian motion on [0, 1] and B is the associated bridge process then by Theorem 2.2, we have the following representation of W :
W = W 1 e 0 + ∞ k=1 I k e k ,(2.10)
where e 0 (t) := t for t ∈ [0, 1], and the random variables {I k } are independent of W 1 . In the rest of this section, we shall study the key objects introduced in Theorem 2.2. Since each orthogonal polynomial lies in L 2 ([0, 1], µ), it must have roots at 0 and 1. Therefore e k · 1 t(1−t) is itself a polynomial but with degree k−1, and one can repeatedly apply the integration by parts formula to the stochastic integrals {I k } defined by (2.5). This enables us to express each I k in terms of iterated integrals of Brownian motion. Moreover, as e k · 1 t(1−t) has precisely k − 2 non-zero derivatives, the highest order iterated integral that is required to fully describe I k is 0<s1<···<s k <1 B s1 ds 1 · · · ds k . So by applying the integration by parts formula as above, we can construct a lower triangular n × n matrix M n with non-zero diagonal entries that characterizes the relationship between {I k } 1≤k≤n and a set of n iterated integrals of Brownian motion.
Ö I 1 . . . I n è = M n Ö 0<s1<1 B s1 ds 1 . . . 0<s1<···<sn<1 B s1 ds 1 · · · ds n è . (2.11)
Since M n is an invertible matrix, it follows that the column vectors appearing in (2.11) both encode the same information about the Brownian bridge. This will now enable us to establish a fundamental connection between Brownian motion and polynomials.
Theorem 2.4. Consider the following unbiased estimator of a one-dimensional Brownian motion over the interval [0, 1],
W n t := E ñ W t W 1 , 0<s1<1 W s1 ds 1 , · · · , 0<s1<···<sn−1<1 W s1 ds 1 · · · ds n−1 ô . (2.12)
Then W n is the unique polynomial of degree n with a root at 0 that matches the increment and n − 1 iterated time integrals of the Brownian path appearing in (2.12).
Proof. It is an immediate consequence of (2.11) that W n t = E[W t | W 1 , I 1 , · · · , I n−1 ]. Hence by (2.10) and independence of the random variables {W 1 , I 1 , · · · }, we have that
W n t = W 1 e 0 + n−1 k=1 I k e k .
Thus W n is indeed a polynomial of degree n with a root at 0 and that matches the increment of the Brownian path. Without loss of generality we can now assume n ≥ 2.
All that remains is to argue W n matches the n − 1 iterated integrals given in (2.12).
Using the orthgonality of {e k }, it follows that for 1 ≤ k ≤ n − 1:
I k = 1 0 (W t − W 1 e 0 ) · e k (t) t(1 − t) dt = 1 0 (W n t + ∞ m=n I m e m − W 1 e 0 ) · e k (t) t(1 − t) dt = 1 0 (W n t − W 1 e 0 ) · e k (t) t(1 − t) dt + ∞ m=n I m 1 0 e k (t) e m (t) t(1 − t) dt = 1 0 (W n t − W 1 e 0 ) · e k (t) t(1 − t) dt.
Thus W n matches the integrals of Brownian motion against polynomials with degree at most n − 1. By the same argument used to derive (2.11), the result then follows. On the other hand, it was shown that the defining eigenfunction property of each e k implies that its derivative e k is proportional to the k-th shifted Legendre polynomial. Hence the family {e k } are the (normalized) shifted (α, β)-Jacobi polynomials but with α = β = −1. Since Jacobi polynomials are typically studied with α, β > −1, it is necessary to check that there are no complications when the parameters approach −1. (2.13)
Naturally, for the above definition to make sense we require the following lemma.
Lemma 2.6. Let P (α,β) k denote the k-th order (α, β)-Jacobi polynomial on [−1, 1]. Then for k ≥ 2, there exists a real-valued polynomial P k such that P k −P
(α,β) k ∞ → 0 as α, β → −1 + .
Proof. Consider the following integral relationship for (α, β)-Jacobi polynomials with α, β > −1 (see [12]).
P (α,β) k (x) = k + α + β + 1 2 x −1 P (α+1,β+1) k−1 (u) du, for all k ≥ 2.
(2.14)
Define a real-valued polynomial P k on [−1, 1] by
P k (x) := k − 1 2 x −1 P (0,0) k−1 (u) du, for all k ≥ 2. (2.15)
Since the terms of (α, β)-Jacobi polynomials depend continuously on (α, β), we have P (0,0) n = lim α,β→0 P (α,β) n for n ≥ 1. The result then follows from (2.14) and (2.15).
Using the above definition for (-1, -1)-Jacobi polynomials, we can give an explicit formula for the polynomials {e k } k≥1 which appear in Theorem 2.2 and Corollary 2.3.
Theorem 2.7. Suppose that each e k has a positive leading coefficient. Then
e k (t) = 1 k » k(k + 1)(2k + 1) P (-1,-1) k+1 (2t − 1), ∀t ∈ [0, 1], ∀k ≥ 1.
Proof. The following result is stated in [12]:
1 −1 (1 − x) α (1 + x) β P (α,β) n (x) 2 dx = 2 α+β+1 2n + α + β + 1 Γ(k + α + 1) Γ(n + β + 1) Γ(n + α + β + 1) n! ,
for n ≥ 1 and α, β > −1. Applying the change of variables, t := 1 2 (x + 1), we have
1 0 t β (1 − t) α P (α,β)
n (2t − 1) 2 dt = 1 2n + α + β + 1 Γ(n + α + 1) Γ(n + β + 1) Γ(n + α + β + 1) n! , for n ≥ 1 and α, β > −1. Taking the limit α, β → −1 + gives
1 0 1 t(1 − t) Ä P (-1,-1) n (2t − 1) ä 2 dt = 1 2n − 1 (n − 1)! (n − 1)! n! (n − 2)! = 1 2n − 1 n − 1 n , for all n ≥ 2.
Therefore by setting k := n − 1, we have
1 0 1 t(1 − t) Å 1 k » k(k + 1)(2k + 1) P (-1,-1) n (2t − 1) ã 2 dt = 1, for all k ≥ 1.
Note that the construction of P k in the proof of Lemma 2.6 implies that each e k (t) is proportional to P (-1,-1) k+1 (2t − 1). The result now follows from the above calculations. To conclude this section, we will address the relationship between the orthogonal Jacobi-like polynomials {e k } and the Alpert-Rokhlin wavelets given in Definition 2.1.
Since each e k is proportional to the k-th shifted Legendre polynomial, the family of polynomials {e k } is orthogonal with respect to the standard L 2 ([0, 1]) inner product. This orthogonality is exactly what is needed to satisfy the conditions (2.2) and (2.3). So for any q ≥ 1 there exists an Alpert-Rokhlin mother function of order q that is a piecewise polynomial where both pieces can be rescaled and translated to give e q−1 .
Applications to SDEs. Consider the Stratonovich SDE on the interval [0, T ]
dy t = f 0 (y t ) dt + f 1 (y t ) • dW t ,
(3.1)
y 0 = ξ,
where ξ ∈ R e and f i denote bounded C ∞ vector fields on R e with bounded derivatives. It then follows from the standard Picard iteration argument that there exists a unique strong solution y to (3.1). An important tool in the numerical analysis of this solution is the stochastic Taylor expansion (see chapter 5 of [13] for a comprehensive review).
For the purposes of this paper, we only require the following specific Taylor expansion.
Theorem 3.1 (High order Stratonovich-Taylor expansion). Let y denote the unique strong solution to (3.1) and let 0 ≤ s ≤ t. Then y t can be expanded as follows:
y t = y s + f 0 (y s ) h + f 1 (y s ) W s,t + 1 2 f 1 (y s )f 1 (y s )W 2 s,t + 1 2 f 0 (y s )f 0 (y s )h 2 (3.2) + f 0 (y s )f 1 (y s ) t s u s • dW v du + f 1 (y s )f 0 (y s ) t s u s dv • dW u + 1 6 (f 1 (y s )f 1 (y s )f 1 (y s ) + f 1 (y s ) (f 1 (y s ), f 1 (y s ))) W 3 s,t + (f 0 (y s )f 1 (y s )f 1 (y s ) + f 0 (y s )(f 1 (y s ), f 1 (y s ))) t s u s v s • dW r • dW v du + (f 1 (y s )f 0 (y s )f 1 (y s ) + f 1 (y s )(f 0 (y s ), f 1 (y s ))) t s u s v s • dW r dv • dW u + (f 1 (y s )f 1 (y s )f 0 (y s ) + f 1 (y s )(f 1 (y s ), f 0 (y s ))) t s u s v s dr • dW v • dW u + 1 24 (f 1 (y s )f 1 (y s )f 1 (y s )f 1 (y s ) + f 1 (y s )f 1 (y s ) (f 1 (y s ), f 1 (y s )) + 3f 1 (y s ) (f 1 (y s )f 1 (y s ), f 1 (y s )) + f 1 (y s ) (f 1 (y s ), f 1 (y s ), f 1 (y s ))) W 4 s,t + R 4 (h, y s ),
where h := t−s and the remainder term has the following uniform estimate for h < 1,
sup ys∈R e R 4 (h, y s ) L 2 (P) ≤ C h 5 2 ,(3.
3)
where the constant C depends only on the vector fields of the differential equation.
From a numerical perspective, the most challenging terms presented in (3.2) are those that involve non-trivial third order iterated integrals of Brownian motion and time. Moreover, the most significant source of discretization error that high order numerical methods will experience is generally due to approximating these stochastic integrals. By representing Brownian motion as a (random) polynomial plus independent noise, we shall derive a new optimal and unbiased estimator for these third order integrals. Then W = W n + Z n , where Z n is a centred Gaussian process independent of W n .
Furthermore, Z n has the following covariance function:
cov(Z n s , Z n t ) = K B (s, t) − n−1 k=1 λ k e k (s)e k (t), for s, t ∈ [0, 1] .
where K B denotes the standard Brownian bridge covariance function and {λ k }, {e k } are the same eigenvalues and eigenfunctions defined by Theorem 2.2.
Proof. It follows from the integration by parts formula that W n matches the increment and n − 1 iterated time integrals of Brownian motion that appear in (2.12). Hence W n is also the polynomial defined in Theorem 2.4 and W = W n + Z n where
W n = W 1 e 0 + n−1 k=1 I k e k , Z n = ∞ k=n I k e k .
Then by Theorem 2.2, Z n is a centered Gaussian process that is independent of W n . In addition, the covariance function defining Z n can be directly computed as follows:
cov(Z n s , Z n t ) = cov ∞ i=n I i e i (s), ∞ j=n I j e j (t) = ∞ k=n λ k e k (s)e k (t) = K B (s, t) − n−1 k=1
λ k e k (s)e k (t), for s, t ∈ [0, 1] .
Note that the final line is achieved using the representation of K B given by (2.6).
The above theorem has a significant conclusion, namely that there exist unbiased polynomial approximants of Brownian motion for which the error process can be uniformly estimated in an L 2 (P) sense. In particular, this theorem already gives interesting applications in the case when n = 2 and motivates the following definitions:
Definition 3.3.
The standard Brownian parabola ı W is the unique quadratic polynomial on [0, 1] with a root at 0 and satisfying
ı W 1 = W 1 , 1 0 ı W u du = 1 0 W u du.K Z (s, t) = min (s, t) − st − 3st(1 − s)(1 − t), for s, t ∈ [0, 1] .H s,t := 1 h W s,u − u − s h W s,t du,
where h = t − s. Since e 1 (t) = √ 6 t(1 − t), one can see that H 0,1 corresponds to √ 6 6 I 1 as defined in Theorem 2.2. Therefore H s,t ∼ N 0, 1 12 h and is independent of W s,t . In particular, the Brownian arch has less variance at its midpoint compared to most points in [s, t] by which we mean that |{u ∈ [s, t] : Var(Z u ) ≤ Var(Z 1 2 (s+t) )}| < 1 2 h . This is in contrast to the Brownian bridge, which has most variance at its midpoint. In fact, the Brownian parabola gives a relatively uniform estimate of the original path. Using these new definitions, we can study the high order integrals appearing in (3.2). Proof. By the natural Brownian scaling it is enough to prove the result on [0, 1]. Note that W = ı W + Z where the parabola ı W is completely determined by (W 1 , H 1 ) and Z is independent of (W 1 , H 1 ). This gives a decomposition for the LHS of (3.4).
E ñ 1 0 W 2 u du W 1 , H 1 ô = E ñ 1 0 Ä ı W u + Z u ä 2 du W 1 , H 1 ô = E ñ 1 0 ı W 2 u du + 2 1 0 ı W u Z u du + 1 0 Z 2 u du W 1 , H 1 ô = 1 0 ı W 2 u du + 2 1 0 ı W u E [Z u ] du + 1 0 E Z 2 u du = 1 0 (uW 1 + 6u(1 − u)H 1 ) 2 du + 1 0 u − u 2 − 3u 2 (1 − u) 2 du.
The result now follows by evaluating the above integrals.
The above theorem has practical applications for SDE simulation as W s,t and H s,t are independent Gaussian random variables and can be easily generated or approximated. That said, we must first recall how the various iterated integrals in (3.2) are connected.
Definition 3.7. The space-space-time Lévy area of Brownian motion over an interval [s, t] is defined as
L s,t := 1 6 Ç t s u s v s • dW r • dW v du − 2 t s u s v s • dW r dv • dW u + t s u s v s dr • dW v • dW u å .
Theorem 3.8. Let H s,t and L s,t denote the Lévy areas of Brownian motion given by definitions 3.5 and 3.7 respectively. Then the following integral relationships hold,
t s u s • dW v du = 1 2 hW s,t + hH s,t , t s u s dv • dW u = 1 2 hW s,t − hH s,t , t s u s v s • dW r • dW v du = 1 6 hW 2 s,t + 1 2 hW s,t H s,t + L s,t , t s u s v s • dW r dv • dW u = 1 6 hW 2 s,t − 2L s,t , t s u s v s dr • dW v • dW u = 1 6 hW 2 s,t − 1 2 hW s,t H s,t + L s,t .
Proof. The result follows from numerous applications of integration by parts.
We can now present the new unbiased estimator for third order iterated integrals of Brownian motion and time. The proposed estimator is fast to compute and the best L 2 (P) approximation of these integrals that is measurable with respect to (W s,t , H s,t ). Proof. The result follows immediately from Theorem 3.6 and Theorem 3.8.
Therefore in order to propagate a numerical solution of (3.1) over an the interval [s, t], one can generate (W s,t , H s,t ) exactly and then approximate L s,t using Theorem 3.9. However, there are many numerical methods we can choose for solving a given SDE. For the purposes of this paper, we will consider the following two numerical methods:
Definition 3.10 (High order log-ODE method). For a fixed number of steps N we can construct a numerical solution {Y k } 0≤k≤N of (3.1) by setting Y 0 := ξ and for each k ∈ [0 . . N − 1], defining Y k+1 to be the solution at u = 1 of the following ODE:
dz du = f 0 (z)h + f 1 (z)W t k ,t k+1 + [f 1 , f 0 ] (z) · hH t k ,t k+1 (3.6) + [f 1 , [f 1 , f 0 ]] (z) · E L t k ,t k+1 | W t k ,t k+1 , H t k ,t k+1 , z 0 = Y k .
where h := T N , t k := kh and [ ·, · ] denotes the standard lie bracket of vector fields.
Definition 3.11 (The parabola-ODE method). For a fixed number of steps N we can construct a numerical solution {Y k } 0≤k≤N of (3.1) by setting Y 0 := ξ and for each k ∈ [0 . . N − 1], defining Y k+1 to be the solution at u = 1 of the following ODE:
dz du = f 0 (z)h + f 1 (z) W t k ,t k+1 + (6 − 12u) H t k ,t k+1 , (3.7) z 0 = Y k ,
where h := T N and t k := kh. In both numerical methods the true solution y at time t k can be approximated by Y k . Whilst there are different ways of interpolating between the successive approximations Y k and Y k+1 , for this paper we shall simply interpolate between such points linearly. To analyse the above methods, we shall first note the key differences between them. The first important distinction between the two methods is a purely practical one. Although these methods both involve computing a numerical solution of an ODE, the parabola method does not require one to explicitly resolve the vector field derivatives. The second significant difference can be seen in the Taylor expansions of the methods.
Y log 1 = y t − [f 1 , [f 1 , f 0 ]] (y s ) L s,t − E L s,t W s,t , H s,t + O(h 5 2 ). (3.8)
Similarly, let Y para denote the one-step approximation given by the parabola-ODE method on the interval [s, t] with the same initial value. Then for sufficiently small h
Y para 1 = y t − [f 1 , [f 1 , f 0 ]] (y s ) Å L s,t − 3 5 H 2 s,t ã + O(h 5 2 ). (3.9)
Note that O(h 5 2 ) denotes terms which can be estimated in an L 2 (P) sense as in (3.3). Proof. In order to derive (3.8), we must compute the Taylor expansion of (3.6). Let F denote the vector field defined in (3.6) that was constructed from f 0 and f 1 . Then F is smooth, and it follows from the classical Taylor's theorem for ODEs that One can define the degree of each term in the above Taylor expansion by counting the number of times functions from {F, F , F , · · · } appear. Therefore, we can see that the higher order terms are precisely the terms with degree strictly greater than four.
Since the largest component of F is f 1 (·)W s,t , both F and its derivatives are O(h 1 2 ). Hence the "higher order terms" in the above Taylor expansion are O(h Arguing (3.9) is fairly straightforward and does not require extensive computations. Using the substitution Z u := z 1 h (u−s) for u ∈ [s, t], the ODE (3.7) can be rewritten as By emulating the derivation of the Stratonovich-Taylor expansion (3.2), it is possible to Taylor expand (3.10) in the same fashion. The only difference is that Stratonovich integrals with respect to W are replaced by Riemann-Stieltjes integrals against ı W . Therefore each term in the expansion of (3.10) can be estimated in L 2 (P) by applying the natural Brownian scaling to the corresponding iterated integral of ı W with time. As before, the largest difference are the O(h 2 ) terms involving third order integrals. Fortunately, iterated integrals of the Brownian parabola can be computed explicitly:
dZ u = f 0 (Z u ) du + f 1 (Z u ) d ı W u ,(3.t s u s v s • dW r • dW v du − t s u s v s d ı W r d ı W v du = L s,t − 3 5 H 2 s,t .
The result (3.9) is now a direct consequence of Theorem 3.8 along with the above.
Theorem 3.12 shows that both methods give a one-step approximation error of O(h 2 ). This means that the log-ODE and parabola-ODE methods are both locally high order; however there is a significant difference in how these methods propagate local errors. The reason is that the O(h 2 ) components of the log-ODE local errors give a martingale, whilst the O(h 2 ) part for each parabola-ODE local error has non-zero expectation. Thus the log-ODE method is globally high order whilst the parabola method is not. However, since the parabola-ODE method is straightforward to implement and locally high order, one could expect it to perform well compared to other low order methods.
In the numerical example, we shall see that the parabola method has the same order of convergence as the piecewise linear approach but gives significantly smaller errors.
To conclude this section, we will present the orders of convergence for both methods.
Definition 3.13 (Strong convergence). A numerical solution Y for (3.1) is said to converge in a strong sense with order α if there exists a constant C such that
Y N − y T L 2 (P) ≤ Ch α ,
for all sufficiently small step sizes h = T N . Definition 3.14 (Weak convergence). A numerical solution Y for (3.1) is said to converge in a weak sense with order β if for any polynomial p there exists C p > 0 such that
|E [ p (Y N )] − E [ p (y T )] | ≤ C p h β ,
for all sufficiently small step sizes h = T N . Theorem 3.15 (Orders of convergence). For a general SDE (3.1), the log-ODE method convergences in a strong sense with order 1.5 and a weak sense with order 2. The parabola-ODE method convergences in both strong and weak senses with order 1.
Proof. Note that Theorem 3.12 establishes the Taylor expansions of both methods. The strong convergence can then be shown as in the proof of Theorem 11.5.1 in [13]. Moreover, the proof of Theorem 11.5.1 also provides the orders of strong convergence. Similarly weak convergence follows directly from the Taylor expansions (3.8) and (3.9), and the rate of convergence can be shown as in the proof of Theorem 14.5.2 in [13].
In this numerical example, we shall use the same parameter values as in [7], namely a = 0.1, b = 0.04, σ = 0.6 and y 0 = 0.06. We will also fix the time horizon at T = 5. Below is the definition of the error estimators used to analyse the numerical methods.
Definition 4.1 (Strong and weak error estimators). For each N ≥ 1, let Y N denote a numerical solution of (4.1) computed at time T using a fixed step size h = T N . We can define the following estimators for quantifying strong and weak convergence:
S N := … E Ä Y N − Y f ine T ä 2 ,(4.
3)
E N := E Y N − b + − E Y f ine T − b + , (4.4)
where the above expectations are approximated by standard Monte-Carlo simulation and Y f ine T is the numerical solution of (4.1) obtained at time T using the log-ODE method with a "fine" step size of min h 10 , T 1000 . The fine step size is chosen so that the L 2 (P) error between Y f ine T and the true solution y is negligible compared to S N . Note that Y N and Y f ine T are both computed with respect to the same Brownian paths.
We will now present our results for the numerical experiment that is described above. From the above graph we see that the log-ODE method gives the best performance. In addition, whilst the other numerical methods share the same order of convergence it is evident there are magnitudes of difference between their respective accuracies. For example, the parabola method is seven times more accurate than piecewise linear. The above graph demonstrates that the log-ODE method is especially well-suited for weak approximation as it achieves a second order convergence rate in this example. Surprisingly, the other three methods all perform with a comparable level of accuracy. We expect the log-ODE and parabola methods to have about twice the computational cost as the other methods because each step requires generating two random variables. Table 4.1 confirms this and thus sampling is the main bottleneck for the methods. So overall, the numerical evidence supports our claim that the high order log-ODE method is currently a state-of-the-art method for the pathwise discretization of IGBM.
Conclusion.
There are primarily three significant results given in this paper:
• An efficient strong polynomial approximation of Brownian motion The main result allows one to construct a "smoother" Brownian motion as a finite sum of (-1, -1)-Jacobi polynomials with independent Gaussian weights. Moreover, it was shown that the approximation is optimal in a weighted L 2 (P) sense and the surrounding noise is an independent centered Gaussian process.
• Unbiased approximation of third order Brownian iterated integrals Iterated integrals of Brownian motion and time are important objects in the study of SDEs as they appear naturally within stochastic Taylor expansions. We have derived the L 2 (P)-optimal estimator for a class of such integrals that is measurable with respect to the path's increment and space-time Lévy area.
• Simulation of Inhomogeneous Geometric Brownian Motion (IGBM) IGBM is a mean-reverting short rate model used in mathematical finance and also one of the simplest SDEs that has no known method of exact simulation. By incorporating the new iterated integral estimator into the log-ODE method we have developed a high order state-of-the-art numerical method for IGBM.
Moreover, the results of this paper immediately produce the following open problems:
• Is it possible to generalize the main theorem to fractional Brownian motion?
• What are the most efficient Runge-Kutta methods for general one-dimensional SDEs that correctly use the new estimator for third order iterated integrals?
Fig. 1 . 1 .
11Sample paths of Brownian motion with corresponding polynomial approximations. * Research supported by the Engineering and Physical Sciences Research Council [EP/N509711/1]. † Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG. ([email protected], [email protected], [email protected])
Theorem 1. 1 .
1Let W denote a standard real-valued Brownian motion on [0, 1]. Let W n be the unique n-th degree polynomial with a root at 0
Fig. 1 .
12.
Theorem 2. 2 (
2A polynomial Karhunen-Loève theorem for the Brownian bridge). Let B denote a Brownian bridge on [0, 1] and consider the Borel measure µ given by for all open intervals (a, b) ⊂ [0, 1]. Then there exists a family of orthogonal polynomials {e k } k≥1 with deg (e k ) = k+1 and 1 0
Fig. 2 . 1 .
21Brownian motion can be expressed as a sum of polynomials with independent weights. Moreover, these polynomials are orthogonal and capture different time integrals of the original path.
Fig. 2. 2 .
2Sample paths of Brownian motion with corresponding polynomial approximations. Although Theorem 2.2 and Corollary 2.3 are interesting results from a theoretical point of view, both lack an explicit construction of the orthogonal polynomials {e k }.
Definition 2. 5 .
5For k ≥ 2, the k-th order (-1, -1)-Jacobi polynomial, P
Theorem 3. 2 .
2Let W denote a standard real-valued Brownian motion on [0, 1]. Let W n be the unique n-th degree polynomial with a root at 0 dW u , for k = 0, 1, · · · , n − 1.
Definition 3. 4 .
4The standard Brownian arch Z is the process Z := W − ı W . By Theorem 3.2, Z is the centred Gaussian process on [0, 1] with covariance function
Definition 3 . 5 .
35The rescaled space-time Lévy area of Brownian motion over the interval [s, t] is the average excursion experienced by the associated bridge process,
Fig. 3 . 1 .
31Brownian motion can be expressed as a (random) parabola plus independent noise. The approximating parabola has the same increment and space-time Lévy area as the original path.
Fig. 3 . 2 .
32Variance profile of the standard Brownian arch.
Theorem 3 . 6 (
36Conditional expectation of non-trivial Brownian time integral).
E
Theorem 3. 9 (
9Conditional expectation of Brownian space-space-time Lévy area).E L s,t W s,t , H
Theorem 3 . 12 .
312Let Y log be the one-step approximation defined by the log-ODE method on the interval [s, t] with initial value Y log 0 = y s . Then for sufficiently small h
FF
(y s ) (F (y s ), F (y s )) + 1 6 F (y s )F (y s )F (y s )+ 1 24 F (y s )F (y s )F (y s )F (y s ) + 1 24F (y s )F (y s ) (F (y s ), F (y s (y s ) (F (y s )F (y s ), F (y s )) + 1 24 F (y s ) (F (y s ), F (y s ), F (y s )) + higher order terms.
the only terms of degree four that are not O(h 5 2 ) are those involving W 4 s,t . It is now enough to analyse just the terms appearing in the first line of the expansion. By substituting the formula for F given by (3.6) into the first line and then rearranging the resulting terms, we are left with a Taylor expansion for Y log 1 that resembles (3.2). The largest difference are the O(h 2 ) terms that correspond to third order integrals. Hence (3.8) follows from Theorem 3.8 with the definitions of [f 1 , f 0 ] and [f 1 , [f 1 , f 0 ]].
Fig. 4 . 1 .
41S N computed with 100,000 sample paths as a function of step size h = T N .
Fig. 4 . 2 .
42E N computed with 500,000 sample paths as a function of step size h = T N .
By applying the natural scaling of Brownian motion, one can define the Brownian parabola and Brownian arch processes over any interval [s, t] with finite size h = t − s. Whilst the Brownian arch can be viewed in a similar light to the Brownian bridge, there are clear qualitative and quantitative differences in their covariance functions.
Table 4 .1
4Simulation times for computing 100,000 sample paths with 100 steps per path using a single-threaded C++ program on a desktop computer.Log-ODE Parabola-ODE Piecewise Linear Milstein
Computation time (s)
2.44
2.95
1.48
1.18
) as well as a weak convergence rate of O(h 2 ). Since the piecewise parabola approach doesn't entirely match the expectation of this iterated integral, it will have a lower order of convergence than the log-ODE method.
As before, we shall be discretizing the SDE on a uniform partition with mesh size h. For IGBM, the log-ODE method has the advantage that it leads to a closed formula since the vector field lie brackets that appear in the ODE (3.6) turn out to be constant. Similarly, the parabola-ODE and piecewise linear methods are both straightforward to implement numerically due to analytical tractability of the SDE governing IGBM. We have included the (non-negative) Milstein method into the numerical experiment as a benchmark to test how the proposed methods compare to a well-known method. That said, the Euler-Maruyama method will not be displayed since it has a similar computational cost as Milstein's method but has a lower order of strong convergence.A numerical example.We shall demonstrate the ideas presented so far through various discretizations of Inhomogeneous Geometric Brownian Motion (IGBM)where a ≥ 0 and b ∈ R are the mean reversion parameters and σ ≥ 0 is the volatility.As the vector fields are smooth, the SDE (4.1) can be expressed in Stratonovich form:whereã := a + 1 2 σ 2 andb := 2ab 2a+σ 2 denote the "adjusted" mean reversion parameters. IGBM is an example of a one-factor short-rate model and has seen recent attention in the mathematical finance literature as an alternative to popular models (see[7],[8]). IGBM is also one of the simplest SDEs that has no known method of exact simulation. We will investigate the strong and weak convergence rates of the following methods:1. Log-ODE method (see definition3.10)Due to the analytically tractable vector fields of (4.2), this method becomesParabola-ODE method (see definition 3.11)Due to the analytically tractable vector fields of (4.2), this method becomesThe integral above will be computed by 3-point Gauss-Legendre quadrature.3. Piecewise linear method (see[14]for definition and proof of convergence) Due to the analytically tractable vector fields of (4.2), this method becomes
R M Mazo, Brownian Motion: Fluctuations, Dynamics and Applications. OxfordClarendron PressR. M. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications, Clarendron Press, Oxford, 2002.
An algorithmic introduction to numerical simulation of stochastic differential equations. D J Higham, SIAM Review. 43D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, Volume 43, 2001.
The maximum rate of convergence of discrete approximations for stochastic differential equations. J M C Clark, R J Cameron, Stochastic Differential Systems Filtering and Control. Springer25J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discrete approxi- mations for stochastic differential equations, Stochastic Differential Systems Filtering and Control, Volume 25 in Lecture Notes in Control and Information Sciences, Springer, 1980.
A multiscale guide to Brownian motion. D S Grebenkov, D Belyaev, P W Jones, Journal of Physics A. 49D. S. Grebenkov, D. Belyaev and P. W. Jones, A multiscale guide to Brownian motion, Journal of Physics A, Volume 49, 2015.
The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations. F Castell, J G Gaines, Annales de l'Institut Henri Poincar. 32F. Castell and J. G. Gaines, The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations, Annales de l'Institut Henri Poincar, Volume 32, 1996.
Stochastic Lie group integrators. S J A Malham, A Wiese, Siam Journal on Scientific Computing. 30S. J. A. Malham and A. Wiese, Stochastic Lie group integrators, Siam Journal on Scientific Computing, Volume 30, 2008.
Approximation Methods for Inhomogeneous Geometric Brownian Motion. L Capriotti, Y Jiang, G Shaimerdenova, International Journal of Theoretical and Applied Finance. 22L. Capriotti, Y. Jiang and G. Shaimerdenova, Approximation Methods for Inhomogeneous Geometric Brownian Motion, International Journal of Theoretical and Applied Finance, Volume 22, 2019.
Three One-Factor Processes for Option Pricing with a Mean-Reverting Underlying: The Case of VIX. B Zhao, C Yan, S Hodges, Financial Review. 54B. Zhao, C. Yan and S. Hodges, Three One-Factor Processes for Option Pricing with a Mean-Reverting Underlying: The Case of VIX, Financial Review, Volume 54, 2019.
G Beylkin, R Coifman, V Rokhlin, Wavelets in Numerical Analysis, Wavelets and Their Applications. Jones and BartlettG. Beylkin, R. Coifman, and V. Rokhlin, Wavelets in Numerical Analysis, Wavelets and Their Applications, Jones and Bartlett, 1992.
Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem. C Carmeli, A De Vito, E Toigo, Analysis and Applications. 4C. Carmeli, A. De Vito, and E. Toigo, Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem, Analysis and Applications, Volume 4, 2006.
The Classical Orthogonal Polynomials. B G S Doman, World ScientificB. G. S. Doman, The Classical Orthogonal Polynomials, World Scientific, 2016.
E W Weisstein, Jacobi Polynomial, From MathWorld -A Wolfram Web Resource. E. W. Weisstein, Jacobi Polynomial, From MathWorld -A Wolfram Web Resource, http://mathworld.wolfram.com/JacobiPolynomial.html.
. P E Kloeden, E Platen, Numerical Solution of Stochastic Differential Equations. SpringerP. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992.
On the Convergence of Ordinary Integrals to Stochastic Integrals. E Wong, M Zakai, Annals of Mathematical Statistics. 36E. Wong, M. Zakai, On the Convergence of Ordinary Integrals to Stochastic Integrals, Annals of Mathematical Statistics, Volume 36, 1965.
| [] |
[
"Convergence of Adaptive Biasing Potential methods for diffusions",
"Convergence of Adaptive Biasing Potential methods for diffusions"
] | [
"Michel Benaïm tion.*[email protected] \nInstitut de Mathématiques\nUniversité de Neuchâtel\nRue Emile Argand 11CH-2000NeuchâtelSwitzerland\n",
"Charles-Edouard Bréhier [email protected] \nUMR 5208\nUniv Lyon\nUniversité Claude Bernard Lyon 1\nCNRS\nInstitut Camille Jordan\n43 blvd. du 11 novembre 1918F-69622Villeurbanne cedexFrance\n"
] | [
"Institut de Mathématiques\nUniversité de Neuchâtel\nRue Emile Argand 11CH-2000NeuchâtelSwitzerland",
"UMR 5208\nUniv Lyon\nUniversité Claude Bernard Lyon 1\nCNRS\nInstitut Camille Jordan\n43 blvd. du 11 novembre 1918F-69622Villeurbanne cedexFrance"
] | [] | We prove the consistency of an adaptive importance sampling strategy based on biasing the potential energy function V of a diffusion process dX 0 t = −∇V (X 0 t )dt + dW t ; for the sake of simplicity, periodic boundary conditions are assumed, so that X 0 t lives on the flat d-dimensional torus. The goal is to sample its invariant distributionwhere V t is the new (random and time-dependent) potential function, acts only on some coordinates of the system, and is designed to flatten the corresponding empirical occupation measure of the diffusion X in the large time regime.The diffusion process writes dX t = −∇V t (X t )dt + dW t , where the bias V t − V is function of the key quantity µ t : a probability occupation measure which depends on the past of the process, i.e. on (X s ) s∈ [0,t] . We are thus dealing with a self-interacting diffusion.In this note, we prove that when t goes to infinity, µ t almost surely converges to µ. Moreover, the approach is justified by the convergence of the bias to a limit which has an intepretation in terms of a free energy.The main argument is a change of variables, which formally validates the consistency of the approach. The convergence is then rigorously proven adapting the ODE method from stochastic approxima- | 10.1016/j.crma.2016.05.011 | [
"https://arxiv.org/pdf/1603.08088v1.pdf"
] | 88,515,690 | 1603.08088 | 40c7c7bda6e96becc6dc0d8fcb2b67520bd87ca9 |
Convergence of Adaptive Biasing Potential methods for diffusions
26 Mar 2016 20 décembre 2017
Michel Benaïm tion.*[email protected]
Institut de Mathématiques
Université de Neuchâtel
Rue Emile Argand 11CH-2000NeuchâtelSwitzerland
Charles-Edouard Bréhier [email protected]
UMR 5208
Univ Lyon
Université Claude Bernard Lyon 1
CNRS
Institut Camille Jordan
43 blvd. du 11 novembre 1918F-69622Villeurbanne cedexFrance
Convergence of Adaptive Biasing Potential methods for diffusions
26 Mar 2016 20 décembre 2017arXiv:1603.08088v1 [math.PR] 1
We prove the consistency of an adaptive importance sampling strategy based on biasing the potential energy function V of a diffusion process dX 0 t = −∇V (X 0 t )dt + dW t ; for the sake of simplicity, periodic boundary conditions are assumed, so that X 0 t lives on the flat d-dimensional torus. The goal is to sample its invariant distributionwhere V t is the new (random and time-dependent) potential function, acts only on some coordinates of the system, and is designed to flatten the corresponding empirical occupation measure of the diffusion X in the large time regime.The diffusion process writes dX t = −∇V t (X t )dt + dW t , where the bias V t − V is function of the key quantity µ t : a probability occupation measure which depends on the past of the process, i.e. on (X s ) s∈ [0,t] . We are thus dealing with a self-interacting diffusion.In this note, we prove that when t goes to infinity, µ t almost surely converges to µ. Moreover, the approach is justified by the convergence of the bias to a limit which has an intepretation in terms of a free energy.The main argument is a change of variables, which formally validates the consistency of the approach. The convergence is then rigorously proven adapting the ODE method from stochastic approxima-
Introduction
Computing the average µ(ϕ) = D ϕ(x)µ(dx) of a function ϕ : D → R, with respect to a probability distribution µ defined on D ⊂ R d , is typically a challenging task in many applications (e.g. chemistry, statistical physics, see e.g. [5]), since usually d is large and µ is multimodal.
In the sequel, we assume that D = T d = (R/Z) d is the flat d-dimensional torus, and that µ writes
µ(dx) = µ β (dx) = exp −βV (x) Z(β) dx,(1)
where V : T d → R is a smooth potential function, β ∈ (0, +∞) is the inverse temperature, dx denotes the Lebesgue measure on T d and Z(β) is a normalizing constant. In this context, the multimodality of µ β follows, in the case of so-called energetic barriers, from the existence of several local minima of V .
A standard approach to computing µ β (ϕ) is to consider the following SDE on T d (overdamped Langevin dynamics) :
dX 0 t = −∇V (X 0 t )dt + 2β −1 dW t , X 0 0 = x.(2)
where W (t) t≥0 is standard Brownian Motion on T d . Indeed, it is wellknown that, for any continuous function ϕ : T d → R almost surely
1 t t 0 ϕ(X 0 r )dr → t→+∞ T d ϕ(x)µ β (dx),(3)
However, this convergence may be very slow, when β is large and V has several minima : the stochastic process X 0 is then metastable, and hopping from the neighborhood of one local minimum of V to another is a rare event which may have a strong influence on the estimation of averages µ β (ϕ). Many strategies based on importance sampling techniques -self-healing umbrella-sampling [7], well-tempered metadynamics [1], Wang-Landau algorithms, adaptive biasing force, etc... -have been proposed and applied to improve the convergence to equilibrium of stochastic processes in order to compute approximations of µ β . We refer for instance to [6] and references therein for a mathematical review.
In this work, we focus on an Adaptive Biasing Potential (ABP) method, given by the system (4). The method was designed in [4,7] for problems in chemistry, and up to our knowledge no rigorous general mathematical analysis has been performed so far. Precisely, in (2), V is replaced with a time-dependent and random potential function V t which is modified adaptively, using the history of the process up to time t : A t depends on the values of the associated stochastic process X r for all 0 ≤ r ≤ t. Here,
V t = V − A t • ξ,
where, for some m ∈ {1, . . . , d − 1}, A t : T m → R and ξ : T d → T m is a smooth function, referred to as the reaction coordinate mapping. In applications, usually m ∈ {1, 2, 3}. To simplify further the presentation, we assume that ξ(x 1 , . . . , x d ) = (x 1 , . . . , x m ) ; in this case, z = (x 1 , . . . , x m ) = ξ(x 1 , . . . , x d ) (resp. z ⊥ = (x m+1 , . . . , x d )) is interpreted as the slow (resp. fast) variable.
The dynamics of the ABP method is given by the following system
dX t = −∇ V − A t • ξ (X t )dt + 2β −1 dW (t) µ t = µ 0 + t 0 exp −βAr•ξ(Xr) δ Xr dr 1+ t 0 exp −βAr•ξ(Xr) dr exp −βA t (z) = T d K z, ξ(x) µ t (dx), ∀z ∈ T m ,(4)
where a smooth kernel function K : T m × T m → (0, +∞), which is such that T m K(z, ζ)dz = 1, ∀ζ ∈ T m , is introduced to define a smooth function A t from the distribution µ t . The unknows in (4) are the stochastic processes t → X t ∈ T d , t → µ t ∈ P(T d ) (the set of Borel probability distributions on T d , endowed with the usual topology of weak convergence of probability distributions), and t → A t ∈ C ∞ (T m ) (the set of infinitely differentiable functions on T m ). In addition to (4), arbitrary (and deterministic, for simplicity) initial conditions X t=0 = x, µ t=0 = µ 0 and A t=0 = A 0 are prescribed. The third equation in (4) introduces a coupling between the evolutions of the diffusion X t and of the weighted empirical distribution µ t : then X can be seen as a self-interacting diffusion process, like in [3].
Our main result is the consistency of the ABP approach.
Theorem 1.1 Almost surely, µ t → t→+∞ µ β , in P(T d ).
With standard arguments, Theorem 1.1 yields almost sure convergence of A t in C k (T m ), for all k ∈ N.
Corollary 1.2 Set exp −βA ∞ = K(·, ξ(x))µ β (dx). Then almost surely, A t → t→+∞ A ∞ , in C k (T m ), ∀ k ∈ N.
The limit A ∞ is an approximation of the function known as the free energy A ⋆ (see (5)), which depends on V , β and ξ. As explained in Section 2, the construction of the adaptive dynamics (4) is motivated by an efficient non-adaptive biasing method, (6), which depends on A ⋆ . Computing A ⋆ is the aim of many algorithms in molecular dynamics (see [6]), and adaptive methods are among the most used in practice. Our result, Theorem 1.1, answers positively the important question of the consistency of ABP method.
The remaining part of the article is organized as follows. In Section 2, we define the free energy function A ⋆ , and explain why non-adaptive and adaptive biaising methods which are related to this function are interesting in the context of metastable dynamics (2). In Section 3, we detail the strategy for the proof of Theorem 1.1 : we prove a stability estimate for A t , and then introduce a random change of variables, based on a change of time. We are then in position to adapt the strategy of proof from [3] in our setting, which is based on the ODE method from stochastic approximation. The main essential role of the change of variables is the identification of the limit flow.
The main result Theorem 1.1 holds in a more general setting, with appropriate modifications, than that of the present paper. For instance, the overdamped Langevin dynamics may be defined on the non-compact space R d instead of T d ; one can also consider the (hypoelliptic) Langevin dynamics, or infinite-dimensional dynamics (parabolic SPDEs). It is also possible to study the efficiency of the method in terms of a Central Limit Theorem. These generalizations will be studied in [2].
Free energy and construction of the ABP dynamics (4)
The aims of this section are to explain first how the ABP method (4), is constructed in a consistent way (the limit in Theorem 1.1 is µ β ) ; and second why it is expected to be efficient (a rigorous analysis of the efficiency is out of the scope of this work).
Observe that exp −βA ∞ (z) = T m K(z, ζ) exp −βA ⋆ (ζ, β) dζ, where A ⋆ (·, β) is the free energy (at temperature β −1 ), defined by : for all z ∈ T m
exp −βA ⋆ (z, β) = T d−m exp −βV (z, z ⊥ ) Z(β) dz ⊥ .(5)
Usually, K(z, ζ) = K ǫ (z, ζ) = 1 ǫK (ζ − z)/ǫ , where ǫ ∈ (0, 1) andK : R m → (0, +∞) is symmetric, smooth, with compact support in [−1/2, 1/2] ; then A ǫ ∞ converges to A ⋆ (·, β), in C ∞ . Choosing ǫ sufficiently small, A t almost surely approximates the free energy A ⋆ (·, β) when t → +∞, thanks to Corollary 1.2.
Equation (5) means that exp −βA ⋆ (z, β) dz ∈ P(T m ) is the image µ β ξ −1 (·) of µ β by ξ. The free energy gives an effective potential along ξ, which is chosen in practice such that ξ(X 0 t ) t≥0 is metastable ; this is related to µ β ξ −1 (·) being metastable, for instance when A ⋆ (·, β) has several local minima. This is why in many applications, computing free energy differences A ⋆ (z 1 , β) − A ⋆ (z 2 , β) is essential, see [6]. The free energy function also theoretically provides efficient importance sampling algorithms ; however these algorithms can only be implemented if A ⋆ is explicitly known, and adaptive strategies allow to circumvent this practical difficulty. Define biased probability distribution and dynamics
µ ⋆ β = exp −β V (x) − A ⋆ (ξ(x), β) Z(β) dx dX ⋆ t = −∇ V − A ⋆ (ξ(·), β) (X ⋆ t )dt + 2β −1 dW (t),(6)
by replacing the original potential function V with the biased potential function V − A ⋆ ξ(·), β in (1) and (2). Note that µ ⋆ β is the unique invariant distribution of X ⋆ . By construction, it is easy to check that the image by ξ of µ ⋆ β is the uniform distribution dz on T m , i.e. the associated free energy is equal to 0. Now define (unweighted) empirical distributions associated with (2) and (6) respectively :
ρ 0 t = 1 t t 0 δ X 0 r dr , ρ ⋆ t = 1 t t 0 δ X ⋆ r dr.
Then, by (3), the image by ξ of ρ 0 t , resp. ρ ⋆ t , converges almost surely in P(T m ), to exp −βA ⋆ (z, β) dz, resp. dz. Thus the dynamics in (6) reaches asymptotically a flat histogram property in the z = ξ(x) direction ; the exploration of T m is thus faster for ξ(X ⋆ ) than for ξ(X 0 ), and in turn the convergence of X ⋆ to µ ⋆ β is expected to be faster than the convergence of X 0 to µ β .
Finally, the construction of the ABP method (4), in particular the use of weighted empirical distributions µ t , is motivated by the following almost sure convergence : for any continuous ϕ :
T d → R, 1 t t 0 exp −βA ⋆ (ξ(X ⋆ r ), β) ϕ(X ⋆ r )dr 1 t t 0 exp −βA ⋆ (ξ(X ⋆ r ), β) dr → t→+∞ µ ⋆ β ϕ exp −βA ⋆ (ξ(·), β) = µ β (ϕ).
(7) Theorem 1.1 thus extends this consistency property from a non-adaptive (6) to an adaptive dynamics (4).
Proof of Theorem 1.1
In this section, we provide the main ideas of the proof of Theorem 1.1.Some technical arguments are skipped, and will be fully detailed in [2], in a more general framework. We first state an important property of A t , and then introduce a change of variables, which helps us identifying a more standard form for self-interacting diffusion processes. We then adapt in our context the arguments from [3], to establish the consistency of the ABP approach thanks to the ODE method from stochastic approximation theory.
Properties of the ABP dynamics (4)
Our first task in the study of the ABP dynamics is to study the wellposedness of the equation, i.e. the existence of a unique global solution t → (X t , µ t , A t ) ∈ T d × P(T d ) × C ∞ (T m ). In order to apply a standard fixed point/Picard iteration strategy, it is essential to control the Lipschitz constant of ∇ A t •ξ (first equation in (4)). This key stability property is ensured as follows. Let m = min z,ζ∈T m K(z, ζ), and M (n) = max z,ζ |∂ n z K(z, ζ)| for n ∈ {0, 1}, where ∂ n z denotes the differential of order n, and introduce
A = A ∈ C ∞ (T m ) | min z∈T m e −βA(z) ≥ m, max z∈T m |∂ n z e −βA(z) | ≤ M (n) , n = 0, 1 .
Then A is left invariant by the evolution t → A t , i.e. A 0 ∈ A implies A t ∈ A for all t ≥ 0, almost surely.
Change of variables
The stochastic process t → µ t , with values in P(T d ), is the unique solution of the random Ordinary Differential Equation (ODE), interpreted in a weak sense (considering continuous bounded test functions) :
dµ t dt = θ ′ (t) 1 + θ(t) δ Xt − µ t ,(8)
where θ(t) = t 0 exp −βA r (ξ(X r )) dr. The random function θ : [0, +∞) → [0, +∞) is a C 1 -diffeomorphism : indeed for all t ≥ 0, almost surely θ ′ (t) = exp −βA t (ξ(X t )) ∈ [m, M ]. This fundamental property allows us to apply the following change of variables :
s = θ(t) , t = θ −1 (s) ; Y s = X t , ν s = µ t , B s = A t .(9)
Observe that s = θ(t) → t→+∞ +∞ and that t = θ −1 (s) → s→+∞ +∞, almost surely. Instead of studying the asymptotic behavior of µ t when t → +∞, it thus equivalent to study the asymptotic behavior of ν s when s → +∞. In the new variables (9), the ABP dynamics (4) writes
dY s = −∇ V − B s • ξ (Y s )e βBs(ξ(Ys)) ds + 2β −1 e βBs(ξ(Ys)) dW (s) ν s = ν 0 + s 0 δ Yr dr 1+s exp −βB s (z) = T d K(z, ξ(x))ν s (dx),(10)
whereW is a new standard Brownian motion on T d , defined from W and θ. Notice that ν s is a nonweighted empirical distribution and that s → ν s satisfies the simpler random ODE
dν s ds = 1 1 + s δ Ys − ν s .(11)
The change of variable (9) both removes θ(t) from (8) as well as the weigths exp −βA t (ξ(X t )) = θ ′ (t) from (4). Thanks to Equation (10), an analogy with the framework of [3] can now be made. Even though we cannot directly apply the results therein, due to the specific form of the dynamics on Y , we follow the same strategy for the analysis of ν s when s → +∞ : we use the ODE method.
Application of the ODE method and sketch of proof of Theorem 1.1
The guideline of the so-called ODE approach we wish to apply is as follows : there is an asymptotic time-scale separation between the (fast) evolution of Y s and the (slow) evolution of ν s (and of B s ). The asymptotic behavior of ν s is then determined by a so-called limit ODE, where δ Ys is replaced in (11) with the unique invariant probability distribution of the following SDE on T d , dY B s = −∇ V − B • ξ (Y s )e βB(ξ(Ys)) ds + 2β −1 e βB(ξ(Ys)) dW (s), (12) i.e. the first (fast) equation of (10) where the slowly varying variable B s is frozen at an arbitrary B ∈ A. In fact, we have the following fundamental result : the invariant distribution of (12) does not depend on B.
Proposition 3.1 For any smooth B : T m → R, the unique invariant distribution of (12) is µ β .
for a given family f n n≥1 of C ∞ functions, which is dense in C 0 (T d ). i.e. almost surely s → ν s is an asymptotic pseudo-trajectory of the semiflow Γ.
We refer to [3] for a proof of a similar result in a different context, and to [2] for a detailed proof in a more general context ; the main difference between the two situations is the use of a specific Poisson equation related to the generator of (12).
To conclude, observe that d ν exp(s) , µ β ≤ ∆(s−S, S)+d Γ S (ν exp(s) ), µ β . Letting first s, then S, go to +∞, Proposition 3.2 implies the main result of this paper, Theorem 1.1.
AcknowledgementsThe authors would like to thank Tony Lelièvre and Gabriel Stoltz for helpful comments. The work is partially supported by the Swiss National Foundation, Grants : 200020 149871 and 200021 163072.Proposition 3.1 is essential and its proof is very simple. Indeed, introduce the generator L B X of X B , resp. the unique invariant distribution of1 is a consequence of the following identity : for any smooth φ, ψ :We now outline the end of the proof of Theorem 1.1, adapting the arguments from[3]in our original case ; details in a more general setting are given in[2]. The ODE method suggests us to define Γ(σ, s, ν) = Γ σ−s (ν), for any σ ≥ s and ν ∈ P(To state (without proof) our last techincal result, we recall that weak convergence in P(T d ) is associated with the following metric d µ 1 , µ 2 =
Welltempered metadynamics : A smoothly converging and tunable freeenergy method. Alessandro Barducci, Giovanni Bussi, Michele Parrinello, Physical review letters. 100220603Alessandro Barducci, Giovanni Bussi, and Michele Parrinello. Well- tempered metadynamics : A smoothly converging and tunable free- energy method. Physical review letters, 100(2) :020603, 2008.
Convergence of adaptive biasing potential methods for diffusion processes. Michel Benaïm, Charles-Edouard Bréhier, in preparationMichel Benaïm and Charles-Edouard Bréhier. Convergence of adaptive biasing potential methods for diffusion processes. in preparation.
Self-interacting diffusions. Michel Benaïm, Michel Ledoux, Olivier Raimond, 122Probab. Theory Related FieldsMichel Benaïm, Michel Ledoux, and Olivier Raimond. Self-interacting diffusions. Probab. Theory Related Fields, 122(1) :1-41, 2002.
Free energy calculations : An efficient adaptive biasing potential method. Bradley Dickson, Frédéric Legoll, Tony Lelièvre, Gabriel Stoltz, Paul Fleurat-Lessard, J. Phys. Chem. B. 114Bradley Dickson, Frédéric Legoll, Tony Lelièvre, Gabriel Stoltz, and Paul Fleurat-Lessard. Free energy calculations : An efficient adaptive biasing potential method. J. Phys. Chem. B, 114 :5823-5830, 2010.
Molecular dynamics. Ben Leimkuhler, Charles Matthews, With deterministic and stochastic numerical methods. ChamSpringer39Ben Leimkuhler and Charles Matthews. Molecular dynamics, volume 39 of Interdisciplinary Applied Mathematics. Springer, Cham, 2015. With deterministic and stochastic numerical methods.
A mathematical perspective. Tony Lelièvre, Mathias Rousset, Gabriel Stoltz, Imperial College PressLondonFree energy computationsTony Lelièvre, Mathias Rousset, and Gabriel Stoltz. Free energy com- putations. Imperial College Press, London, 2010. A mathematical pers- pective.
Self-healing umbrella sampling : a nonequilibrium approach for quantitative free energy calculations. Simone Marsili, Alessandro Barducci, Riccardo Chelli, Piero Procacci, Vincenzo Schettino, The Journal of Physical Chemistry B. 11029Simone Marsili, Alessandro Barducci, Riccardo Chelli, Piero Procacci, and Vincenzo Schettino. Self-healing umbrella sampling : a non- equilibrium approach for quantitative free energy calculations. The Journal of Physical Chemistry B, 110(29) :14011-14013, 2006.
| [] |
[
"Optimal Installation for Electric Vehicle Wireless Charging Lanes",
"Optimal Installation for Electric Vehicle Wireless Charging Lanes"
] | [
"Hayato Ushijima-Mwesigwa \nSchool of Computing\nClemson University\nClemson SCUSA\n",
"MDZadid Khan \nDepartment of Civil Engneering\nClemson University\nClemsonSCUSA\n",
"Mashrur A Chowdhury \nDepartment of Civil Engneering\nClemson University\nClemsonSCUSA\n",
"Ilya Safro [email protected] \nSchool of Computing\nClemson University\nClemson SCUSA\n"
] | [
"School of Computing\nClemson University\nClemson SCUSA",
"Department of Civil Engneering\nClemson University\nClemsonSCUSA",
"Department of Civil Engneering\nClemson University\nClemsonSCUSA",
"School of Computing\nClemson University\nClemson SCUSA"
] | [] | R ange anxiety, the persistent worry about not having enough battery power to complete a trip, remains one of the major obstacles to widespread electric-vehicle adoption. As cities look to attract more users to adopt electric vehicles, the emergence of wireless in-motion car charging technology presents itself as a solution to range anxiety. For a limited budget, cities could face the decision problem of where to install these wireless charging units. With a heavy price tag, an installation without a careful study can lead to inefficient use of limited resources. In this work, we model the installation of wireless charging units as an integer programming problem. We use our basic formulation as a building block for different realistic scenarios, carry out experiments using real geospatial data, and compare our results to different heuristics. Reproducibility: all datasets, algorithm implementations and mathematical programming formulation presented in this work are available at https://github.com/hmwesigwa/smartcities.git | null | [
"https://arxiv.org/pdf/1704.01022v3.pdf"
] | 33,917,849 | 1704.01022 | 9c4ce72ec140c0ae85511eb9b1c1424cddd90ea6 |
Optimal Installation for Electric Vehicle Wireless Charging Lanes
May 16, 2017
Hayato Ushijima-Mwesigwa
School of Computing
Clemson University
Clemson SCUSA
MDZadid Khan
Department of Civil Engneering
Clemson University
ClemsonSCUSA
Mashrur A Chowdhury
Department of Civil Engneering
Clemson University
ClemsonSCUSA
Ilya Safro [email protected]
School of Computing
Clemson University
Clemson SCUSA
Optimal Installation for Electric Vehicle Wireless Charging Lanes
May 16, 2017Resource AllocationWireless ChargingElectric VehiclesTransportation PlanningNetwork Analysis
R ange anxiety, the persistent worry about not having enough battery power to complete a trip, remains one of the major obstacles to widespread electric-vehicle adoption. As cities look to attract more users to adopt electric vehicles, the emergence of wireless in-motion car charging technology presents itself as a solution to range anxiety. For a limited budget, cities could face the decision problem of where to install these wireless charging units. With a heavy price tag, an installation without a careful study can lead to inefficient use of limited resources. In this work, we model the installation of wireless charging units as an integer programming problem. We use our basic formulation as a building block for different realistic scenarios, carry out experiments using real geospatial data, and compare our results to different heuristics. Reproducibility: all datasets, algorithm implementations and mathematical programming formulation presented in this work are available at https://github.com/hmwesigwa/smartcities.git
Introduction
The transportation sector is the largest consumer in fossil fuel worldwide. As cities move towards reducing their carbon footprint, electric vehicles (EV) offer the potential to reduce both petroleum imports and greenhouse gas emissions. The batteries of these vehicles however have a limited travel distance per charge. Moreover, the batteries require significantly more time to recharge compared to refueling a conventional gasoline vehicle. An increase in the size of the battery would proportionally increase the driving range. However, since the battery is the single most expensive unit in an EV, increasing its size would greatly increase the price discouraging widespread adaptation. Given the limitations of on-board energy storage, concepts such as battery swapping have been proposed as possible approaches to mitigate these limitations. In the case of battery swapping, the battery is exchanged at a location that stores the equivalent replacement battery. This concept, however, leads to issues such as battery ownership in addition to significant swapping infrastructure costs.
An alternative method to increase the battery range of the EV is to enable power exchange between the vehicle and the grid while the vehicle is in motion. This method is sometimes referred 1 arXiv:1704.01022v2 [math.OC] 12 May 2017 to as dynamic charging [42,26]. Dynamic charging can significantly reduce the high initial cost of EV by allowing the battery size to be downsized which would be used to complement other concepts such as battery swapping to reduce driver range anxiety.
Our contribution Given the effectiveness and advances in dynamic charging technology, cities face the challenge of budgeting and deciding on what locations to install these wireless charging lanes (WCL). We define a road segment as the one-way portion of a road between two intersections. A city would contain thousands of road segments. Each road segment will possibly have a different length and driving conditions. The problem of deciding the optimal road segments to install wireless charging units in order maximize the range of the EV's driving within the city becomes a non-trivial one. In this paper, we formulate the WCL installation problem as an integer programming model that is build upon taking into account different realistic scenarios. We compare the computational results for the proposed model to faster heuristics and demonstrate that our approach provides significantly better results for fixed budget models. Using a standard optimization solver with parallelization, we provide solutions for networks of different sizes including the Manhattan road network.
Related Work
A wireless power transfer (WPT) is the transmission of electrical energy from a power source to an electrical load without the use of physical conductors. Wireless transmission is useful to power electrical devices where wires are inconvenient, hazardous, or inaccessible, which is the case for charging EV battery. Wireless charging can be divided into two types, namely, static, and dynamic. Static charging refers to fixed charging stations where vehicles have to pull out of the road and park to access the charging. The dynamic charging occurs while an EV is in transit. Strips of charging coils are placed on the road and EVs get charged as they go over the coils. Dynamic charging is an emerging technology that offers the advantages of reducing range anxiety and charging time [42].
There have been previous studies related to optimal placement of wireless charging units in a simple circle topology road network. One prominent study focuses on optimal system design of the online electric vehicle (OLEV) that utilizes wireless charging technology [20]. In this study, a particle swarm optimization (PSO) method is used to find a minimum cost solution considering the battery size, total number of WCLs (power transmitters) and their optimal placement as decision variables. The model is calibrated to the actual OLEV system and the algorithm generates reliable solutions. However, the formulation contains a non-linear objective function making it computationally challenging for multi-route networks. Moreover, speed variation is not considered in this model, which is typical in a normal traffic environment. The OLEV and its wireless charging units were developed in Korea Advanced Institute of Science and Technology (KAIST) [16]. At Expo 2012, an OLEV bus system was demonstrated, which was able to transfer 100KW (5 × 20KW pick-up coils) through 20 cm air gap with an average efficiency of 75%. The battery package was successfully reduced to 1/5 of its size due to this implementation [15].
Recently, the authors in [4] formulate the installation of charging lanes in road network topologies with an objective to minimize the total system travel times which they define as the total social cost. In their work, the impact of charging infrastructure into the drivers travel choices is considered and a mathematical program with complementarity constraints is developed. The model is applied on two networks, namely, one with 19 and the other with 76 road segments. However, the mathematical models with complementarity constraints are often difficult to solve [28] especially for large problems. Therefore, the issue of scalability of the approach arises. While the proposed approach is certainly promising, the objective function that minimizes the total social cost could potentially benefit by taking into account the objective functions like the one we provide to consider, for example, public transportation vehicles that have fixed routes.
Optimization of electric vehicle charging stations (EVCS) placement in transportation networks have been investigated in depth for many studies. A review of the numerous optimization techniques employed in the last decade to determine the optimal EVCS placement and sizing is presented by [14]. Some of the most popular methods are genetic algorithms [29], integer (linear) programming [13], particle swarm optimization [36], ant colony optimization [34], greedy algorithms [22]. The problem of EVCS placement is formulated and four solution methods are evaluated by [22]. While the iterative and effective mixed integer programming methods yield the most accurate solution, the greedy algorithm provides faster solution at a reasonable accuracy. A location optimization algorithm based on a modified genetic algorithm is presented and evaluated for a practical scenario by [30]. Another method named OCEAN and its faster version OCEAN-C have been presented by [45] as better alternatives to baseline methods for EVCS placement. A mobile data driven genetic algorithm based solution is presented for EVCS placement by [40]. An EVCS placement problem is studied by [44] for an urban public bus system and they found backtracking algorithm and greedy algorithm schemes to be better than the others.
A number of studies have focused on the methods of the real world application of dynamic charging and its overall impact on the transportation network. A new scheme for charging of EVs based on wireless charging is presented by [31]. Integration of control strategy at traffic intersections is discussed to maximize charging while minimizing waiting delays. All strategies, in this study, show benefit over no charging scenario. The vehicle to vehicle (V2V) and vehicle to infrastructure (V2I) communications are effective ways to implement the wireless charging solution in a mixed traffic environment. An efficient WPT system using V2I communication for efficient distribution of power among EVs is presented by [37]. Simulation results indicate that the fog based Balanced State of Charge (BSoC) system has less communication latency, more BSoC of EVs, and less packet drop rate than the conventional system. Simulation of a traffic network of EVs equipped with connected vehicle technology reveals significant benefits over conventional systems (without connectivity) [17]. An ant colony optimization based multi-objective routing algorithm is presented by [27]. The V2V and V2I communications are used for finding the best route, and for intelligently recharging on the move. This study reveals that connected EVs could reduce not only the total travel time and the energy consumption, but also the recharged volume of electricity and corresponding cost.
A few studies focus on the financial aspect of the implementation of a dynamic charging system. A smart charge scheduling model is presented by [26] maximizes the net profit to each EV participant while simultaneously satisfying energy demands for individual's trips. A thorough analysis of the costs associated with the implementation of a dynamic WPT infrastructure and a business model for the development of a new EV infrastructure are presented by [7].
There have been many studies on the design, application and future prospects of wireless power transfer for electric vehicles [35,1,25]. Some energy companies are teaming up with automobile companies to incorporate wireless charging capabilities in EVs. Examples of such partnerships include Tesla-Plugless and Mercedez-Qualcomm. The universities, research laboratories and companies have invested in research work for developing efficient wireless charging systems for electric vehicles and testing them in a dynamic charging scheme. Notable institutions include Auckland University 3 [5], HaloIPT (Qualcomm) [23], Oak Ridge National laboratory (ORNL) [19], MIT (WiTricity) and Delphi [18]. However, there is still a long way to go for a full commercial implementation, since it requires to make significant changes to the current transportation infrastructure.
State of Charge of an EV
State of charge (SOC) is the equivalent of a fuel gauge for the battery pack in a battery electric vehicle (BEV) and hybrid electric vehicle (HEV). In the optimization model, the objective is to have a simple function that calculates the change of SOC for a road segment with and without wireless charging units installed. The SOC determination is a complex non-linear problem and there are various techniques to address it [3,43,21]. As discussed in the literature, the SOC of an EV battery can be determined in real time using different methods, such as terminal voltage method, impedence method, coulomb counting method, neural network, support vector machines, and Kalman fitering. The input to the models are physical battery parameters, such as terminal voltage, impedence, and discharging current. However, the SOC related input to our optimization model is the change in SOC of the EV battery to traverse a road segment rather than the absolute value of the real time SOC of the EV battery. So, we formulate a function that approximates the change in SOC of an EV to traverse a road segment using several assumptions, as mentioned in the following. The units of SOC are assumed to be percentage points (0% = empty; 100% = full). The change in SOC is assumed to be proportional to the change in battery energy. This is a valid assumption for very small road segments that form a large real road network, which is the case in this analysis (range of 0.1 to 0.5 mile).
We compute the change in SOC of an EV as a function of the time t spent traversing a road segment by
∆SOC t = E end − E start E cap ,(1)
where E start and E end is the energy of the battery (KWh) before and after traversing the road segment respectively and E cap is the battery energy capacity. Following computation of the battery energy given in [37], we, however, assume that the velocity of an EV is constant while traversing the road segment. This gives us
E end − E start = (P 2t · η)t − P 1t t,(2)
where P 1t is the power consumption (KW) while it traverses the road segment, and P 2t is the power delivered to the EV in case a WCL is installed on the road segment, otherwise P 2t is zero. In order to take into account the inefficiency of charging due to factors such as misalignment between the primary (WCL) and secondary (on EV) charging coils and air gap, an inefficiency constant η is assumed. The power consumption P 1t varies from EV to EV. In this work, we take an average power consumption calculated by taking the average mpge (miles per gallon equivalent) and battery energy capacity rating from a selected number of EV. We took the average of over 50 EVs manufactured in 2015 or later. For each EV, its fuel economy data was obtained from [39]. For P 2t and η, the power rating of the WCL, and the efficiency factor, we average the values from [1], Table 2, where the authors make a comparison of prototype dynamic wireless charging units for electric vehicles.
Optimization Model Development
The purpose of developing a mathematical model of the WCL installation problem is to construct an optimization problem that maximizes the battery range per charge within a given budget and road network. This, in turn, will minimize the driver range anxiety within the road network. In this section, we first define the road segment graph model with notation, then we discuss the modeling assumptions.
Road Segment Graph
Consider a physical network of roads within a given location. We define a road segment as the one-way portion of a road between two intersections. Let G = (V, E) be a directed graph with node set V such that v ∈ V if and only if v is a road segment. Two road segments u and v are connected with a directed edge (u, v) if and only if the end point of road segment u is adjacent to the start point of road segment v. We refer to G as the road segment graph. For a given road segment graph and budget constraint, our goal is to find a set of nodes that would minimize driver range anxiety within the network.
Modeling Assumptions
For a road segment graph G, we assume that each node has attributes such as average speed and distance that are used to compute the average traversal time of the road segment. Note that since nodes represent road segments, an edge represents part of an intersection, thus, the weight of an edge does not have a typical general purpose weighting scheme associated to it (such as a length). For a given pair of nodes s and t, where s represents the origin and t the destination of a user within the road network, respectively, we assume that the user will take the fastest route from s to t. As a result, we assign the weight of each edge (u, v) in the road segment graph with a value equal to the average traversal time of road segment u. Then we can find a shortest path with road segment graph that represents the fastest route from the start point of road segment s to the end point of t.
We assume that SOC of any EV whose journey starts at the beginning of a given road segment is fixed. For example, we may assume that if a journey starts at a residential area, then any EV at this starting location will be fully charged or follows a charge determined by a given probability distribution which would not significantly change the construction of our model. For example, in real applications, one could choose the average SOC of EV's that start at that given location. In our empirical studies, for simplicity, we first assume that all EVs start fully charged. We later give results for studies where we take the initial SOC to be chosen uniformly at random. We also assume that SOC takes on real values such that 0 ≤ SOC ≤ 1 at any instance where SOC = 1 implies that the battery is fully charged and SOC = 0 implies that the battery is empty.
For any two road segments s and t in G, we assume that there is a unique shortest path between them. Since we are using traversal time as a weighting scheme of the road segment graph, this is a reasonable assumption. If there exists two road segments such that this assumption is not realistic, then one can treat these paths using distinct routes and include them in the model since we will define a model based on distinct routes.
We call a route infeasible within a network if any EV that starts its journey at the beginning of this route (starts fully charged in our empirical studies), will end with a final SOC ≤ α, where 0 ≤ α ≤ 1. The constant α is a global parameter of our model called a global SOC threshold.
Introducing different types of EV and more than one type of α would not significantly change the construction of the model.
Given the total length of all road segments in the network, T , we define the budget, 0 ≤ β ≤ 1, as a part of T for which funds available for WCL installation exist. For example, if β = 0.5, the city planners have enough funds to install WCL's across half the road network. We use our model and its variations to answer the following problems that the city planners are interested in.
1. For a given α, determine the minimum budget, β, together with the corresponding locations, needed such that the number of infeasible routes is zero.
2. For a given α and β, determine the optimal installation locations to minimize the number of infeasible routes.
We assume that minimizing the number of infeasible routes would reduce the driver range anxiety within the network.
Single Route Model Formulation
Let Routes be the set of all possible fastest routes between each pair of nodes i, j ∈ V . In our model, we assume that the fastest route between a pair of nodes is represented by a unique route. For each route r ∈ Routes, assume hat each EV whose journey is identical to this route has a fixed initial SOC, and a variable final SOC, termed iSOC r , and f SOC r , respectively, depending on whether or not WCL's were installed on any of the road segments along the route. The goal of the optimization model is to guarantee that either f SOC r ≥ α where α is a global threshold or f SOC r is as close as possible to α for a given budget. The complete optimization model takes all routes into account. Given that realistic road segment graphs have a large number of nodes, taking all routes into account may overwhelm the computational resources, thus, the model is designed to give the best solution for any number of routes considered. We describe the model by first defining it for a single route and then generalizing it to multiple routes. For simplicity, we will assume that the initial SOC, iSOC r = 1, for each route r, i.e., all EV's start their journey fully charged. This assumption can easily be adjusted with no significant changes to the model. Since we assume that SOC takes discrete values, we define a unit of increase or decrease of SOC as the next or previous discrete SOC value respectively. For simplicity, we will also assume a simple SOC function in this section. We will assume that SOC of an EV traversing a given road segment increases by one unit if a WCL is installed, otherwise it decreases by one unit. A more realistic SOC function can be incorporated into the following model without any major adjustments, as presented in the following.
For a single route r ∈ Routes, with iSOC r = 1, consider the problem of determining the optimal road segments to install WCL's in order to maximize f SOC r within a limited budget constraint. Define a SOC-state graph, socG r , for route r, as an acyclic directed graph whose vertices describe the varying SOC an EV on a road segment would have depending on whether or not the previously visited road segment had a WCL installed. More precisely, let r = (u 1 , u 2 , . . . , u k ), for u i ∈ V with i = 1, . . . , k and k > 0. Let nLayers ∈ N represent the number of discrete values that the SOC can take. For each u i ∈ r, let u i,j be node in socG r for j = 1, . . . , nLayers representing the nLayers discrete values that the SOC can take at road segment u i . Let each node u i,j have out-degree at most 2, representing the two different scenarios of whether or not a WCL is installed at road segment u i . The edge (u i,j 1 , u i+1,j 2 ) has weight 1 represents the scenario if a WCL is installed at u i and has weight 0 if a WCL is not installed. An extra nodes are added accordingly to capture the output from the final road segment u k , we can think of this as adding an artificial road segment u k+1 . Two dummy nodes, s and t, are also added to the graph socG r to represent the initial and final SOC respectively. There is one edge of weight 0 between s and u 1,j * where u 1,j * represents the initial SOC of an EV on this route. Each node u k,j for each j is connected to node t with weight 0. Consider a path p from s to t, namely, p = (s, u 1,j 1 , u 2,j 2 , . . . , u k,j k , t), then each node in p represents the SOC of an EV along the route. We use this as the basis of our model. Any feasible s − t path will correspond to an arrival at a destination with an SOC above a given threshold. A minimum cost path in this network would represent the minimal number of WCL installations in order to arrive at the destination. Let socG r = (V , E ) with weighted edges w ij , then the minimum cost path can be formulated as follows:
minimize ij∈E w ij x ij subject to j x ij − j x ji = 1, if i = s; −1, if i = t; 0, otherwise x i ∈ {0, 1}
where j x ij − j x ji = 0 ensures that we have a path i.e., number of incoming edges is equal to 7 number of out going edges.
R i = 1, if k w ik ≥ 1 0, if k w ik = 0.
In other words, install a WCL if at least one route requires it. Under these constraints for a single route, an optimal solution to the minimum cost path from s to t would be a solution for the minimum number of WCL's that need to be installed, in order for EV to arrive at the destination with its final SOC greater than a specified threshold.
General Model Description and Notations
In this section, we describe the model when multiple routes are taken into consideration. Similar to the model for a single route, we define a different graph G r = (V, E r ) for each route, r, together with one global constraint. Note that each graph G r is defined over the same node set V however, the edge set E r is dependent on the route. Consider all fastest routes in the network. Construct a SOC-state graph for each of these routes. We can think of this network as nRoutes distinct graphs interconnected by at least one constraint.
Decision Variables:
We first describe the different notation necessary for describing the model.
w (r,u,v) · x (r,u,v)
Then,
R s = 1, if p(s) ≥ 1 0, otherwise(3)
models the installation decision.
Objective function:
For the problem of minimizing the budget, the objective function is simply given by
nRoads s=1 c s · R s .(4)
where c s is the cost of installing a WCL at road segment s. For the problem of minimizing the number of infeasible routes for any fixed budget, we modify the SOC graph such that there exists an s − t path for any budget. We accomplish this by adding an edge of weight 0 between the nodes u i,nLayers to t for all i = 1, . . . , k + 1, in each route, where k is the number of road segments in the route. We then define the boundary nodes of the SOC graph to be all the nodes that are adjacent to node t. Let B be the set of all boundary nodes and B r be the boundary nodes with respect to route r. Assign each node in u i,j ∈ B r weights according to the function:
w(u i,j ) = 1, if s(u i,j ) ≥ α 0, otherwise
where s(u i,j ) represents the discretized SOC value that node u i,j represents and |r| is the number of road segments in route r. In the weighting scheme above, we make no distinctions between two infeasible routes. However, a route in which an EV completes, say, 90% of the trip would be preferable to one in which an EV completes, say, 10% of the trip. One can easily take this preference into account and weight the boundary nodes by the function
w(u i,j ) = 1, if s(u i,j ) ≥ α i−|r| |r| , otherwise,(5)
where the term i−|r| |r| measures how close an EV comes to completing a given route, i.e, if i−|r| |r| = −1, then the EVs SOC falls below α after traversing the first road segment.
For simplicity, we relabel the nodes in the SOC graph in a canonical way indexed by the set N. For a node i in the SOC graph, let w(r, i, i) represent the weight of i. Then the objective is maximize charge of each route, which is given by nRoutes r=1 (u,t)∈Br w (r,u,u) · x (r,u,t)
Budget Constraint:
Cost of installation cannot exceed a budget B. Since this technology is not yet widely commercialized, we can discuss only the estimates of the budget for WCL installation. Currently, the price of installation per kilometer ranges between a quarter million to several millions dollars [7]. For simplicity, in our model the cost of installation at a road segment is assumed to be proportional to the length of the road segment which is likely to be a real case. Thus, a budget would represent a fraction of the total length of all road segments. nRoads s=1 c s · R s ≤ B
Route Constraints:
The constraints defining an s-t path
j x r,i,j − j x r,j,i = 1, if i = s; −1, if i = t; 0, otherwise
Model
The complete model formulation for minimizing the number of infeasible routes, for a fixed budget is given by:
maximize nRoutes r=1 (u,t)∈Br w (r,u,u) · x (r,u,t) subject to nRoads s=1 c s · R s ≤ B j x (r,i,j) − j x (r,j,i) = 1, if (r, i) = (r, s) −1, if (r, i) = (r, t) 0, otherwise r = 1, . . . ,w (r,u,v) · x (r,u,v)
and M, a large constant and 0 < < 1 are used to model the logic constraints given in equation (3).
Since we are interested in reducing the number of routes with a final SOC less than α, we can take the set of routes in the above model to be the all the routes that have a final SOC below the given threshold. We evaluate this computationally and compare it with several fast heuristics.
Heuristics
Integer programming is NP-hard in general and since the status of the above optimization model is unknown, we have little evidence to suggest that it can be solved efficiently. For large road networks, it may be desirable to use heuristics instead of forming the above integer program. In particular, since we know the structure of the network, one natural approach may be to apply concepts from network science to capture the features of the best candidates for a WCL installation. In this section, we outline different heuristics for deciding on the a set of road segments. We then compare these structural based solutions to the optimization model solution, and demonstrate the superiority of proposed model. Different centrality indexes is one of the most studied concepts in network science [32]. Among them, the most suitable to our application are betweenness and vertex closeness centralities. In [6], a node closeness centrality is defined as the sum of the distances to all other nodes where the distance from one node to another is defined as the shortest path (fastest route) from one to another. Similar to interpretations from [2], one can interpret closeness as an index of the expected time until the arrival of something "flowing" within the network. Nodes with a low closeness index will have short distances from others, and will tend to receive flows sooner. In the context of traffic flowing within a network, one can think of the nodes with low closeness scores as being well-positioned or most used, thus ideal candidates to install WCL.
The betweenness centrality [6] of a node k is defined as the fraction of times that a node i, needs a node k in order to reach a node j via the shortest path. Specifically, if g ij is the number of shortest paths from i to j, and g ikj is the number of i-j shortest paths that use k, then the betweenness centrality of node k is given by
i j g ikj g ij , i = j = k,
which essentially counts the number of shortest paths that path through a node k since we assume that g ij = 1 in our road network because edges are weighted according to time. For a given road segment in the road segment graph, the betweenness would basically be the road segments share of all shortest-paths that utilize that the given road segment. Intuitively, if we are given a road network containing two cities separated by a bridge, the bridge will likely have high betweenness centrality. It also seems like a good installation location because of the importance it plays in the network. Thus, for a small budget, we can expect the solution based on the betweenness centrality to give to be reasonable in such scenarios. There is however an obvious downfall to this heuristic, consider a road network where the betweenness centrality of all the nodes are identical. For example, take a the cycle on n nodes. Then using this heuristic would be equivalent to choosing installation locations at random. A cycle on n vertices can represent a route taken by a bus, thus, a very practical example. Figure 2 shows an optimal solution from our model to minimize the number of infeasible routes with a budget of at most four units. The eigenvector centrality [33] of a network is also considered. As an extension of the degree centrality, a centrality measure based on the degree of the node, the concept behind the eigenvector centrality is that the importance of a node is increased if it connected to other important nodes. In terms of a road segment graph, this would translate into the importance of a road segment increasing if its adjacent road segments are themselves important. For example, if a road segment is adjacent to a bridge. One drawback of using this centrality measure is that degree of nodes in road segment graphs is typically small across the graph. However, it stll helps to find regions of potentially heavy traffic.
Data collection and post-processing
The geospatial data is collected from OpenStreetMap (OSM). It is a free collaborative project to generate editable maps of any location on earth [9]. A region of interest can be selected on OpenStreetMap user interface, and all available data can be generated for the selected region. OpenStreetMap offers different formats for the user. In this case, the selected format is XML format. Also, there is a website named planet.osm that already contains captured OSM XML files for different cities in different parts of the world. The contents of an OSM XML file are described in Table 1.
A script is developed to process the raw XML data and extract meaningful information. First, the road network is filtered using the tag 'highway' that specifies the characteristics of a road. The 'highway' tag must also contain sub-tags such as 'motorway', 'trunk', 'primary', 'secondary', 'tertiary', 'road', 'residential', 'living_street' etc. This eliminates the unnecessary parts of the network, as the model only deals with the road network. Then, the filtered road network is segmented into small road segments having two end points (points with specific latitude and longitude). In the segmented graph, each node represents a road, and connectivity between nodes represent connection of roads. Two nodes in a graph are connected if they share a point with the same latitude and longitude. Then, the adjacency matrix of the segmented graph is extracted representing the network's connectivity (i.e. if two road segments have one point in common, they are connected). This is a sparse matrix with 1 as its only non-zero entry.
Finally, the length of each road segment is calculated using the haversine formula. The haversine formula gives great-circle distances between two points on a sphere from their longitudes and latitudes. It is a special case of a more general formula in spherical trigonometry, the law of haversines, relating the sides and angles of spherical triangles. For any two points on a sphere, the distance (d) between the points is calculated using the following equations [41]. a = sin 2 (∆φ/2) + cos φ 1 · cos φ 2 · sin 2 (∆λ/2) (8)
c = 2 arctan 2( √ a, √ 1 − a) (9) d = R · c(10)
where, φ 1 , φ 2 are latitudes, ∆φ is a difference of latitudes, λ 1 and λ 2 are longitudes, ∆λ is a difference of longitudes, and R is the Earth's radius (mean radius = 3959 miles).
Results and Discussion
In this section, we discuss the results of the proposed model used to solve the WCL installation problem. We use Pyomo, a collection of Python packages by [11,10] to model the integer program. As a solver, we used CPLEX 12.7 [12] with all our results attained with an optimality gap of at most 10%. Designing a fast customized solver is not the central goal of this paper. However, it is clear that introducing customized parallelization and using advanced solvers will make the proposed model solvable for the size of a large city in urban area.
The measurement of WCL installation effectiveness on a particular road segment depends on the SOC function used. However, the SOC function varies from EV to EV and is dependent on such factors as vehicle and battery type and size together with the effectiveness of the charging technology used. However, the purpose of this paper is to propose a model that is able to accommodate any SOC function.
Small networks
In order to demonstrate the effectiveness of our model, we begin with presenting the results on two small toy graphs in which all road segments are identical. We incorporate a simple toy SOC function, one in which the SOC increases and decreases by one unit if a wireless charging lane is installed or not installed, respectively. For the first toy graph (see Figure 3(a)), we are interested in determining the minimum budget such that all routes are feasible. For this we assume that a fully charged battery has four different levels of charge 0, 1,2, and 3, where a fully charged battery contains three units. This would imply that the SOC-layered graph would contain four layers. The parameter α is fixed to be 0. For the second toy graph (see Figure 3(b)), we are interested in minimizing the number of infeasible routes with a varying budget. For this example, we take the number of layers for the SOC-layered graph to be five while also taking α = 0. Figure 3: Directed toy graphs of 26 and 110 vertices used for problems 1 and 2, respectively. The bold end points on the edges of (a) represent edge directions. The graphs are subgraphs of the California road network taken from the dataset SNAP in [24] In experiments with the first graph, we take all routes into consideration and compute an optimal solution which is compared with the betweenness and eigenvector centralities. We rank the nodes based on their centrality indexes, and take the smallest number of top k central nodes that ensure that all routes are feasible. The installation locations for each method are shown in Figure 4 of which the solution to our model uses the smallest budget. We observe a significant difference in the required budget to ensure feasibility of the routes (see values B in the figure).
For the second graph, we vary the available budget β. The results are shown in Figure 5. The plots also indicate how the optimal solution affects the final SOC of all other routes. The solutions to our model were based on 100 routes, with length at least 2, that were sampled uniformly without repetitions. We observe that our solution gives a very small number of infeasible routes for all budgets. We notice that for a smaller budget, taking a solution based on betweenness centrality gives a similar but slightly better solution than that produced by our model. However, this insignificant difference is eliminated as we increase the number of routes considered in our model. Note that if our budget was limited to one WCL, then the node chosen using the betweenness centrality would likely be a good solution because this would be the node that has the highest number of shortest paths traversed through it compared to other nodes. As we increase the budget, the quality of our solution is considerably better than the other techniques. For budgets close to 50% in Figure 5 (d) and (e), our model gives a solution with approximately 90% less infeasible routes compared to that of the betweenness centrality heuristic. This is in spite of only considering about 1% of all routes as compared to betweenness centrality that takes all routes into account.
Experiments with Manhattan network
In the above example, the input to our model is a road segment graph with identical nodes, and a simple SOC function. We next test our model with real data and the SOC function defined in Section 1.1. We extract data of lower Manhattan using Openstreet maps. The data is preprocessed by dividing each road into road segments. Each road in Openstreet maps is categorized into one of eight categories presented in Table 2, together with the corresponding speed limit for a rural or urban setting. For this work, we consider roads from categories 1 to 5 as potential candidates for installing wireless charging lanes due to their massive exploitation. Thus, we remove any intersections that branch off to road categories 6 to 8. The resulting road segment network contains 5792 nodes for lower Manhattan. We also study a neighborhood of lower Manhattan that forms a graph of 914 nodes. The graphs are shown in Figure 6. Similar to experiments on the second toy example, we carry out experiments on the Manhattan network using 200 routes. We sample routes that have a final SOC less than the threshold α uniformly at random without repetitions. . Due to relatively small driving radius within the Manhattan neighborhood graph shown in Figure 6 (b), we increase the length of each road segment by a constant factor in order to have a wider range of a final SOC within each route. We take α = 0.8 and 0.85 with a corresponding budget of β = 0.1 and 0.2 respectively for the Manhattan neighborhood graph while α = 0.7 and β = 0.1 for the lower Manhattan graph. We compare our results with the heuristic of choosing installation locations based on their betweenness centrality. In our experiments, the betweenness centrality produces significantly better results than other heuristics, so it is used as our main comparison.
For a threshold α = 0.8 in the Manhattan neighborhood graph, there are 42,001 infeasible routes with no WCL installation. With a budget β = 0.1, our model was able to reduce this number to 4,957. Using the heuristic based on betweenness centrality, the solution found contained 21,562 infeasible routes. For a budget of β = 0.2 with threshold α = 0.85, there were 170,393 infeasible routes without a WCL installation, 57,564 using the betweenness centrality heuristic and only 14,993 using our model. Histograms that demonstrate the distributions of SOC are shown in Figure 7. The green bars represent a SOC distribution without any WCL installation. The red and blue bars represent SOC distributions after WCL installations based on the betweenness centrality heuristic and our proposed model, respectively.
In Figure 8, we demonstrate the results for the lower Manhattan graph, with α = 0.7 and β = 0.1. Due to the large number of routes, we sample about 16 million routes. From this sample, our model gives a solution with 10% more infeasible routes compared to the heuristic based on betweenness centrality. Note that in this graph, there are about 13 million infeasible routes. From these routes, we randomly chose less than 1000 routes for our model without any sophisticated technique for choosing these routes, while the heuristic based on betweenness centrality takes all routes into account. The plot in Figure 8, shows the number of routes whose final SOC falls below a given SOC value. Similar to Figure 5 (a), the results demonstrate that for a relatively small budget, our model gives a similar result compared to the betweenness centrality heuristic.
Experiments with Random Initial SOC
In the preceding experiments, EVs were assumed to start their journey fully charged. However, the assumption in our model was that the initial SOC be any fixed value. Thus, as an alternative scenario, one can take the initial SOC to follow a given distribution selected either by past empirical data or known geographic information about a specific area. For example, one could assume higher values in residential areas compared to non-residential areas. In this work, we carried out experiments where the initial SOC was chosen uniformly at random in the interval (a, 1). The left endpoint of the interval was chosen such that the final SOC associated to any route would be positive. In order to give preference to longer routes, we define Ω k as the set of all routes with distance greater than µ + kσ, where µ is the average distance of a route with standard deviation σ, for some real number k. We then study the average of the final SOC of all routes in Ω k which we denote asx k .
In the Manhattan neighborhood graph, Figure 6 (b), we chose 1200 routes as an input to the model. These routes were chosen uniformly at random from Ω k . In our solution analysis, we took a = 0.4 and computedx k . Without any installation, we hadx k ≈ 0.39 for k ≥ 2 while 0.5 ≤x k ≤ 0.51 based the betweenness heuristic with a installation budget of 20%. However, our model gives us 0.58 ≤x k ≤ 0.71 given the same 20% installation budget. The value ofx k in this case significantly increases for an increase in k or in the number of routes sampled from Ω k .
Conclusion and Future Work
In this work, we have presented an integer programming formulation for modeling the WCL installation problem. With a modification to the WCL installation optimization model, we present a formulation that can be used to answer two types of questions. First, determining a minimum budget to reduce the number of infeasible routes to zero, thus, assuring EV drivers of arrival at their destinations with a battery charge above a certain threshold. Second, for a fixed budget, minimizing the number of infeasible routes and thus reducing drivers' range anxiety. Our experiments have shown that our model gives a high quality solution that typically improves various centrality based heuristics. The best reasonable candidate (among many heuristics we tested) that sometimes not significantly outperforms our model is the betweenness centrality. In our model, the routes were chosen randomly based on whether their final SOC is below α or not. We notice that a smarter way of choosing the routes leads to a better solution, for example choosing the longest routes generally provided better solutions. In future research, a careful study on the choice of routes to include in the model will give more insight into the problem. For a more comprehensive study of this model and its desired modifications, we suggest to evaluate their performance using a large number of artificially generated networks using [8,38]. • Contains the version of the API (features used)
• Contains the generator that distilled this file (the editor tool)
Node
• Set of single points in space defined by unique latitude, longitude and node id according to the World Geodetic System (WGS84)
• WGS84 is the reference coordinate system (for latitude, longitude) used by GPS (Global Positioning System)
• Data contains tags of each node Way • An ordered list of nodes which normally also has at least one tag or is included within a relation
• A way can have between 2 and 2,000 nodes, unless there is some error in data
• A way can be open or closed, a closed way is one whose last node is also the first • Data contains the references to its nodes and tags of each way Relation • One or more tags and an ordered list of one or more nodes, ways and/or relations as members which is used to define logical or geographic relationships between other elements
• Data contains the references to its members for each relation and tags of each relation.
Figure 1 :
1Figures 1shows an example for a SOC graph constructed from a route with three road segments. Example of socG r with r = (u 1 , u 2 , u 3 ) and nLayers = 4. u 4 is an artificial road segment added to capture the final SOC from u 3 . The nodes in the set B = {u i,j |i = 4 or j = 4} are referred to as the boundary nodes. The out going edges of each node u i,j are determined by an SOC function. Each node represents a discretized SOC value. For example, the nodes u i,! and u i,4 represent an SOC value of 1 and 0, respectively.
nRoadSegs number of road segments nLayers number of discrete values SOC can take nRoutes number of routes taken into account nN odes number of nodes in soc-state graph For a given route, r, define a SOC graph G r = (V, E r ) where the edges E r are defined according to the SOC function. The weights for edges in E r are given byw r,i,j = 1, if WCL is installed in respective road segment for i 0, otherwiseThen the decision variables of the model are given by R s = 1, if at least one route requires a WCL installation 0, otherwise for s = 1, . . . , nRoadSegsx r,i,j = 1, if edge (r, i, j) is in an s-t path in G r 0, otherwise for r = 1, . . . , nRoutesFor the decision variable R s on the installation of a WCL at road segment s, we install a WCL if at least one route requires an installation within the different s-t paths for each route. For road segment s, and for any set of feasible s-t paths, let p(s) be the number of routes that require a WCL installation, then p(s) is given by
Figure 2 :
2Optimal solution with a four unit installation budget. The thick ends of the edges are used to indicate the direction of the edge. Taking α = 0 without any installation, there are 70 number of infeasible routes. An optimal installation of 5 WCLs would ensure zero infeasible routes. With an optimal installation of 4 WCLs, the nodes colored in red, there would have 12 infeasible routes.
Figure 4 :
4Comparison of the different methods. The minimum number of WCL installation needed to eliminate all infeasible routes is B. The nodes colored red indicate location of WCL installation. In (a), we demonstrate the result given by our model requiring a budget of 12 WCL'sin order to have zero infeasible routes. In (b) and (c), we demonstrate solutions from the betweenness and eigenvector heuristics that give budgets of 20 and 23 WCL's, respectively.
Figure 5 :
5In each figure from (a) to (f) we show plots of number of routes ending with final SOC below a given value. The model was solved by optimizing 100 routes chosen uniformly at random with α = 0 with a varying budget. The y-intercept of the different lines shows the number of infeasible routes for the different methods. Our model gives a smaller number in all cases. The plots go further and show how a specific solution affects the SOC of all routes. As the budget approaches 50%, we demonstrate that our model gives a significant reduction to the number of infeasible routes while also improving the SOC in general of the feasible routes 19
Figure 6 :Figure 7 :
67Road segment graphs from real geospatial data: a node, drawn in blue, represents a road segment. Two road segments u and v are connected by a directed edge (u, v) if and only if the end point of u is that start point of v Histograms showing the number of infeasible routes for different values of α and β for the Manhattan neighborhood graph. The vertical line indicates the value of α. In (a) with a budget of 10%, our model gives a solution with at least 50% less infeasible routes compared to the betweenness heuristic. In (b), we demonstrate how the effects of a 20% budget on the SOC distribution within the network. In (c), our model gives a solution with at least 25% less infeasible routes.
Figure 8 :
8The number of routes ending with final SOC below a given value in the lower Manhattan graph. The solution was obtained with α = 0.7 and β = 0.1. The blue curve shows the SOC distribution when no WCL are installed. Green and red curves show the SOC distribution after an installation using the proposed model and the betweenness heuristic, respectively. Plot (b) a gives closer look into (a) for the SOC values below 0.7.
Decision Variable for Installation Let R i be the decision variable for installation of a WCL at road segment i. Then, for a single route, we have R i =k
w ik ∈ {0, 1}. For multiple routes we
have
Table 1 :
1Contents of an OSM XML file.• Introduces the UTF-8 character encoding for the fileItem
Sub-item
Features
XML Suffix
OSM Elements Node,
Way
and
Relation
Table 2 :
2Road category with corresponding speed in Miles/HrCategory Road Type
Urban Speed Rural Speed
1
Motorway
60
70
2
Trunk
45
55
3
Primary
30
50
4
Secondary
20
45
5
Tertiary
15
35
6
Residential/Unclassified 8
25
7
Service
5
10
8
Living street
5
10
AcknowledgmentThis research is supported by the National Science Foundation under Award #1647361. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the of the National Science Foundation.
A review of wireless power transfer for electric vehicles: Prospects to enhance sustainable mobility. Zicheng Bi, Tianze Kan, Chris Chunting, Yiming Mi, Zhengming Zhang, Gregory A Zhao, Keoleian, Applied Energy. 179Zicheng Bi, Tianze Kan, Chunting Chris Mi, Yiming Zhang, Zhengming Zhao, and Gregory A Keoleian. A review of wireless power transfer for electric vehicles: Prospects to enhance sus- tainable mobility. Applied Energy, 179:413-425, 2016.
Centrality and aids. P Stephen, Borgatti, Connections. 181Stephen P Borgatti. Centrality and aids. Connections, 18(1):112-114, 1995.
The state of charge estimating methods for battery: A review. Wen-Yeau Chang, ISRN Applied Mathematics. Wen-Yeau Chang. The state of charge estimating methods for battery: A review. ISRN Applied Mathematics, 2013, 2013.
Optimal deployment of charging lanes for electric vehicles in transportation networks. Zhibin Chen, Fang He, Yafeng Yin, Transportation Research Part B: Methodological. 91Zhibin Chen, Fang He, and Yafeng Yin. Optimal deployment of charging lanes for electric vehicles in transportation networks. Transportation Research Part B: Methodological, 91:344- 365, 2016.
Magnetic field design for low emf and high efficiency wireless power transfer system in on-line electric vehicle. D H Cho, Kim, DH Cho and J Kim. Magnetic field design for low emf and high efficiency wireless power transfer system in on-line electric vehicle. 2011.
. C Linton, Freeman, Centrality in social networks conceptual clarification. Social networks. 13Linton C Freeman. Centrality in social networks conceptual clarification. Social networks, 1(3):215-239, 1978.
Infrastructure cost issues related to inductively coupled power transfer for electric vehicles. S Jasprit, Parth Gill, Mashrur Bhavsar, Jennifer Chowdhury, Joachim Johnson, Ryan Taiber, Fries, Procedia Computer Science. 32Jasprit S Gill, Parth Bhavsar, Mashrur Chowdhury, Jennifer Johnson, Joachim Taiber, and Ryan Fries. Infrastructure cost issues related to inductively coupled power transfer for electric vehicles. Procedia Computer Science, 32:545-552, 2014.
Multiscale network generation. Alexander Gutfraind, Ilya Safro, Lauren Ancel Meyers, Information Fusion (Fusion), 2015 18th International Conference on. IEEEAlexander Gutfraind, Ilya Safro, and Lauren Ancel Meyers. Multiscale network generation. In Information Fusion (Fusion), 2015 18th International Conference on, pages 158-165. IEEE, 2015.
Openstreetmap: User-generated street maps. Mordechai Haklay, Patrick Weber, IEEE Pervasive Computing. 74Mordechai Haklay and Patrick Weber. Openstreetmap: User-generated street maps. IEEE Pervasive Computing, 7(4):12-18, 2008.
Pyomo-optimization modeling in python. E William, Carl Hart, Jean-Paul Laird, David L Watson, Woodruff, Springer Science & Business Media67William E Hart, Carl Laird, Jean-Paul Watson, and David L Woodruff. Pyomo-optimization modeling in python, volume 67. Springer Science & Business Media, 2012.
Pyomo: modeling and solving mathematical programs in python. E William, Jean-Paul Hart, David L Watson, Woodruff, Mathematical Programming Computation. 33William E Hart, Jean-Paul Watson, and David L Woodruff. Pyomo: modeling and solving mathematical programs in python. Mathematical Programming Computation, 3(3):219-260, 2011.
Cplex optimization studio. Ibm Ilog, IBM ILOG. Cplex optimization studio. URL: http://www- 01.ibm.com/software/commerce/optimization/cplex-optimizer, 2014.
Optimization for allocating bev recharging stations in urban areas by using hierarchical clustering. Andy Ip, Simon Fong, Elaine Liu, Advanced Information Management and Service (IMS), 2010 6th International Conference on. IEEEAndy Ip, Simon Fong, and Elaine Liu. Optimization for allocating bev recharging stations in urban areas by using hierarchical clustering. In Advanced Information Management and Service (IMS), 2010 6th International Conference on, pages 460-465. IEEE, 2010.
A review of techniques for optimal placement and sizing of electric vehicle charging stations. Hussain Md Mainul Islam, Azah Shareef, Mohamed, Przegląd Elektrotechniczny. 918Md Mainul Islam, Hussain Shareef, and Azah Mohamed. A review of techniques for optimal placement and sizing of electric vehicle charging stations. Przegląd Elektrotechniczny, 91(8):122- 126, 2015.
Creating innovation with systems integration, road and vehicle integrated electric transportation system. Young Jae Jang, Young Dae Ko, Seungmin Jeong, Systems Conference (SysCon). IEEEYoung Jae Jang, Young Dae Ko, and Seungmin Jeong. Creating innovation with systems integration, road and vehicle integrated electric transportation system. In Systems Conference (SysCon), 2012 IEEE International, pages 1-4. IEEE, 2012.
Optimal design of the wireless charging electric vehicle. Young Jae Jang, Young Dae Ko, Seungmin Jeong, Electric Vehicle Conference (IEVC). IEEEYoung Jae Jang, Young Dae Ko, and Seungmin Jeong. Optimal design of the wireless charging electric vehicle. In Electric Vehicle Conference (IEVC), 2012 IEEE International, pages 1-5. IEEE, 2012.
Utilizing real-time information transferring potentials to vehicles to improve the fast-charging process in electric vehicles. Jennifer Johnson, Mashrur Chowdhury, Yiming He, Joachim Taiber, Transportation Research Part C: Emerging Technologies. 26Jennifer Johnson, Mashrur Chowdhury, Yiming He, and Joachim Taiber. Utilizing real-time information transferring potentials to vehicles to improve the fast-charging process in electric vehicles. Transportation Research Part C: Emerging Technologies, 26:352-366, 2013.
High efficient inductive power supply and pickup system for on-line electric bus. Guho Jung, Boyune Song, Seungyong Shin, Seokhwan Lee, Jaegue Shin, Yangsu Kim, Sungjeub Jeon, Electric Vehicle Conference (IEVC). IEEEGuho Jung, Boyune Song, Seungyong Shin, Seokhwan Lee, Jaegue Shin, Yangsu Kim, and Sungjeub Jeon. High efficient inductive power supply and pickup system for on-line electric bus. In Electric Vehicle Conference (IEVC), 2012 IEEE International, pages 1-5. IEEE, 2012.
Design of a regulator for multi-pick-up systems through using current offsets. Yangsu Kim, Youngdong Son, Seungyong Shin, Jaegue Shin, Boyune Song, Seokhwan Lee, Guho Jung, Seongjeub Jeon, Electric Vehicle Conference (IEVC). IEEEYangsu Kim, Youngdong Son, Seungyong Shin, Jaegue Shin, Boyune Song, Seokhwan Lee, Guho Jung, and Seongjeub Jeon. Design of a regulator for multi-pick-up systems through using current offsets. In Electric Vehicle Conference (IEVC), 2012 IEEE International, pages 1-6. IEEE, 2012.
The optimal system design of the online electric vehicle utilizing wireless power transmission technology. Dae Young, Young Jae Ko, Jang, IEEE Transactions on Intelligent Transportation Systems. 143Young Dae Ko and Young Jae Jang. The optimal system design of the online electric vehicle uti- lizing wireless power transmission technology. IEEE Transactions on Intelligent Transportation Systems, 14(3):1255-1265, 2013.
Comparison of two battery equivalent circuit models for state of charge estimation in electric vehicles. Narayani Koirala, Fengxian He, Weixiang Shen, Industrial Electronics and Applications (ICIEA), 2015 IEEE 10th Conference on. IEEENarayani Koirala, Fengxian He, and Weixiang Shen. Comparison of two battery equivalent circuit models for state of charge estimation in electric vehicles. In Industrial Electronics and Applications (ICIEA), 2015 IEEE 10th Conference on, pages 17-22. IEEE, 2015.
Electric vehicle charging station placement: Formulation, complexity, and solutions. Y S Albert, Yiu-Wing Lam, Xiaowen Leung, Chu, IEEE Transactions on Smart Grid. 56Albert YS Lam, Yiu-Wing Leung, and Xiaowen Chu. Electric vehicle charging station place- ment: Formulation, complexity, and solutions. IEEE Transactions on Smart Grid, 5(6):2846- 2856, 2014.
On-line electric vehicle using inductive power transfer system. Sungwoo Lee, Jin Huh, Changbyung Park, Nam-Sup Choi, Gyu-Hyeoung Cho, Chun-Taek Rim, IEEE Energy Conversion Congress and Exposition. IEEESungwoo Lee, Jin Huh, Changbyung Park, Nam-Sup Choi, Gyu-Hyeoung Cho, and Chun-Taek Rim. On-line electric vehicle using inductive power transfer system. In 2010 IEEE Energy Conversion Congress and Exposition, pages 1598-1601. IEEE, 2010.
Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Jure Leskovec, J Kevin, Anirban Lang, Michael W Dasgupta, Mahoney, Internet Mathematics. 61Jure Leskovec, Kevin J Lang, Anirban Dasgupta, and Michael W Mahoney. Community struc- ture in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics, 6(1):29-123, 2009.
Wireless power transfer for electric vehicle applications. IEEE journal of emerging and selected topics in power electronics. Siqi Li, Chunting Chris Mi, 3Siqi Li and Chunting Chris Mi. Wireless power transfer for electric vehicle applications. IEEE journal of emerging and selected topics in power electronics, 3(1):4-17, 2015.
Optimizing the performance of vehicle-to-grid (v2g) enabled battery electric vehicles through a smart charge scheduling model. Z Li, P Chowdhury, Y Bhavsar, He, International Journal of Automotive Technology. 165Z Li, M Chowdhury, P Bhavsar, and Y He. Optimizing the performance of vehicle-to-grid (v2g) enabled battery electric vehicles through a smart charge scheduling model. International Journal of Automotive Technology, 16(5):827-837, 2015.
Connectivity supported dynamic routing of electric vehicles in an inductively coupled power transfer environment. Zhiyun Li, Kakan Dey, Mashrur Chowdhury, Parth Bhavsar, IET Intelligent Transport Systems. Zhiyun Li, Kakan Dey, Mashrur Chowdhury, and Parth Bhavsar. Connectivity supported dynamic routing of electric vehicles in an inductively coupled power transfer environment. IET Intelligent Transport Systems, 2016.
Mathematical programs with equilibrium constraints. Jong-Shi Zhi-Quan Luo, Daniel Pang, Ralph, Cambridge University PressZhi-Quan Luo, Jong-Shi Pang, and Daniel Ralph. Mathematical programs with equilibrium constraints. Cambridge University Press, 1996.
An overview of genetic algorithms for the solution of optimisation problems. Simon Mardle, Sean Pascoe, Computers in Higher Education Economics Review. 131Simon Mardle, Sean Pascoe, et al. An overview of genetic algorithms for the solution of optimisation problems. Computers in Higher Education Economics Review, 13(1):16-20, 1999.
An optimization location scheme for electric charging stations. Sara Mehar, Sidi Mohammed Senouci, Smart Communications in Network Technologies (SaCoNeT), 2013 International Conference on. IEEE1Sara Mehar and Sidi Mohammed Senouci. An optimization location scheme for electric charging stations. In Smart Communications in Network Technologies (SaCoNeT), 2013 International Conference on, volume 1, pages 1-5. IEEE, 2013.
Toward a wireless charging for battery electric vehicles at traffic intersections. Shahram Mohrehkesh, Tamer Nadeem, 14th International IEEE Conference on Intelligent Transportation Systems (ITSC). IEEEShahram Mohrehkesh and Tamer Nadeem. Toward a wireless charging for battery electric vehicles at traffic intersections. In 2011 14th International IEEE Conference on Intelligent Transportation Systems (ITSC), pages 113-118. IEEE, 2011.
Networks: An Introduction. Mark Newman, Oxford University Press, IncNew York, NY, USAMark Newman. Networks: An Introduction. Oxford University Press, Inc., New York, NY, USA, 2010.
The mathematics of networks. E J Mark, Newman, The new palgrave encyclopedia of economics. 2Mark EJ Newman. The mathematics of networks. The new palgrave encyclopedia of economics, 2(2008):1-12, 2008.
Optimal placement of ev fast charging stations considering the impact on electrical distribution and traffic condition. Prakornchai Phonrattanasak, Nopbhorn Leeprechanon, Green Energy for Sustainable Development (ICUE), 2014 International Conference and Utility Exhibition on. IEEEPrakornchai Phonrattanasak and Nopbhorn Leeprechanon. Optimal placement of ev fast charg- ing stations considering the impact on electrical distribution and traffic condition. In Green Energy for Sustainable Development (ICUE), 2014 International Conference and Utility Exhi- bition on, pages 1-6. IEEE, 2014.
Overview of wireless power transfer for electric vehicle charging. Chun Qiu, Chunhua Chau, Liu, Chan, Electric Vehicle Symposium and Exhibition (EVS27), 2013 World. IEEEChun Qiu, KT Chau, Chunhua Liu, and CC Chan. Overview of wireless power transfer for electric vehicle charging. In Electric Vehicle Symposium and Exhibition (EVS27), 2013 World, pages 1-9. IEEE, 2013.
Particle swarm optimization: technique, system and challenges. Dian Palupi Rini, Siti Mariyam Shamsuddin, Siti Sophiyati Yuhaniz, International Journal of Computer Applications. 141Dian Palupi Rini, Siti Mariyam Shamsuddin, and Siti Sophiyati Yuhaniz. Particle swarm opti- mization: technique, system and challenges. International Journal of Computer Applications, 14(1):19-26, 2011.
An efficient wireless power transfer system to balance the state of charge of electric vehicles. Ankur Sarker, Chenxi Qiu, Haiying Shen, Andrea Gil, Joachim Taiber, Mashrur Chowdhury, Jim Martin, Mac Devine, 45th International Conference on. IEEEParallel Processing (ICPP)Ankur Sarker, Chenxi Qiu, Haiying Shen, Andrea Gil, Joachim Taiber, Mashrur Chowdhury, Jim Martin, Mac Devine, and AJ Rindos. An efficient wireless power transfer system to balance the state of charge of electric vehicles. In Parallel Processing (ICPP), 2016 45th International Conference on, pages 324-333. IEEE, 2016.
Generating scaled replicas of real-world complex networks. L Christian, Michael Staudt, Ilya Hamann, Alexander Safro, Henning Gutfraind, Meyerhenke, International Workshop on Complex Networks and their Applications. SpringerChristian L Staudt, Michael Hamann, Ilya Safro, Alexander Gutfraind, and Henning Meyer- henke. Generating scaled replicas of real-world complex networks. In International Workshop on Complex Networks and their Applications, pages 17-28. Springer, 2016.
Optimizing the deployment of electric vehicle charging stations using pervasive mobility data. M Mohammad, Hongmou Vazifeh, Paolo Zhang, Carlo Santi, Ratti, arXiv:1511.00615arXiv preprintMohammad M Vazifeh, Hongmou Zhang, Paolo Santi, and Carlo Ratti. Optimizing the de- ployment of electric vehicle charging stations using pervasive mobility data. arXiv preprint arXiv:1511.00615, 2015.
Calculate distance and bearing between two latitude/longitude points using haversine formula in javascript. Movable Type Scripts. Chris Veness, Chris Veness. Calculate distance and bearing between two latitude/longitude points using haversine formula in javascript. Movable Type Scripts, 2011.
Wireless power transfer for electric vehicles, present and future trends. D M Vilathgamuwa, Sampath, Plug in electric vehicles in smart grids. SpringerDM Vilathgamuwa and JPK Sampath. Wireless power transfer for electric vehicles, present and future trends. In Plug in electric vehicles in smart grids, pages 33-60. Springer, 2015.
Comparison of kalman filter-based state of charge estimation strategies for li-ion batteries. Weizhong Wang, Deqiang Wang, Xiao Wang, Tongrui Li, Ryan Ahmed, Saeid Habibi, Ali Emadi, Transportation Electrification Conference and Expo (ITEC). IEEEWeizhong Wang, Deqiang Wang, Xiao Wang, Tongrui Li, Ryan Ahmed, Saeid Habibi, and Ali Emadi. Comparison of kalman filter-based state of charge estimation strategies for li-ion batteries. In Transportation Electrification Conference and Expo (ITEC), 2016 IEEE, pages 1-6. IEEE, 2016.
Electric vehicle charging station placement for urban public bus systems. Xiumin Wang, Chau Yuen, Naveed Ul Hassan, Ning An, Weiwei Wu, IEEE Transactions on Intelligent Transportation Systems. 181Xiumin Wang, Chau Yuen, Naveed Ul Hassan, Ning An, and Weiwei Wu. Electric vehicle charging station placement for urban public bus systems. IEEE Transactions on Intelligent Transportation Systems, 18(1):128-139, 2017.
Optimal electric vehicle charging station placement. Yanhai Xiong, Jiarui Gan, Bo An, Chunyan Miao, Ana Lc Bazzan, IJCAI. Yanhai Xiong, Jiarui Gan, Bo An, Chunyan Miao, and Ana LC Bazzan. Optimal electric vehicle charging station placement. In IJCAI, pages 2662-2668, 2015.
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